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ENCYCLOPEDIA METROPOLITANA
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UNIVERSAL DICTIONARY OF KNowLEDGE,
OR;
arººrº
Q9m au QBriginal ºlam :
COMPRISING THE TWOFOLD ADVANTAGE OF
A PHILOSOPHICAL AND AN ALPHABETICAL ARRANGEMENT,
WITH APPROPRIATE ENGRAVINGS.
ED ITED BY
T H E R E.V. EDWA R D S ME DL EY, M.A.,
IATE FELLOW OF SIDNEY COLLEGE, CAMBRIDGE ;
THE REV. HUGH JAMES ROSE, B.D.,
PRINCIPAL OF KING's COLLEGE, LONDON ;
THE REV. H E N R Y JOHN R O SE, B. D.,
LATE FELLow of ST. JoHN's COLLEGE, CAMBRIDGE.
VOLUME I.
[PURE SciENCEs, Vol. 1.]
LONDON :
B. FELLOWES; F. AND J. RIVINGTON; DUNCAN AND MALCOLM.; SUTTABY AND CO.; E. HODGSON; J. DOWDING;
G. LAWFORD; J. M. RICHARDSON; J. BOHN; T. ALLMAN ; J. BAIN; S. HoDGSON; F. C. WESTLEY; L.A. LEWIS;
T. HODGES; AND H. WASHBOURNE; ALSO J. H. PARKER, AND T. LAYCOCK, OXFORD;
- AND J. AND J. J. DEIGHTON, CAMBRIDGE.
1845.
1.ONDON :-PRINTED BY WILLIAM CLOWES AND SONS, STAMFORD STREET.
CONTENTS TO WOL. I.
GENERAL INTRODUCTION. A. PRELIMINARY TREATISE ON METHOD. By S. T. Col.ERIDGE, Esq.
- PAGE
GRAMMAR . . . . . . . By Sir JoHN STODDART, LL.D., Judge of the Vice-Admiralty Court, Malta . l
LOGIC . . . . . . . . By Rev. RICHARD WHATELY, D.D., Principal of St. Alban's Hall, Oxford . 193
RHETORIC . . . . . . . By the same . . . . . . . . . . . . . . . . . . . 241
GEOMETRY . . . . . . By PETER BARLow, Esq., F.R.S., Professor at the Royal Military Academy,
Woolwich & e º O C & © tº e s e s a 304 .
ARITHMETIC . . . . . . By Rev. GEORGE PEACOCK, M.A., Fellow and Tutor of Trinity College,
Cambridge * * * * * * tº e º e º nº e e º a tº 369
ALGEBRA . . . . . . . By Rev. DIONYSIUS LARDNER, LL.D., F.R.S. London and Edinburgh; Professor
of Natural Philosophy in the University of London . . . . . . 524
GEOMETRICAL ANALYSIS . By the same . . . . . . . . . . . . . . . . . . . 632
THEORY OF NUMBERS . . By Professor BARLow . . . . . . . . . . . . . . . . . 641
TRIGONOMETRY . . . . By GEORGE BIDDELL AIRY, Esq., M.A., Plumian Professor of Astronomy,
Cambridge © º e te © º e º g º º e s º _, e © º e - 672
ANALYTICAL GEOMETRY . By Rev. HENRY PARR HAMILTON, M.A., Fellow of Trinity College, Cambridge 709
CoNIC SECTIONs . . . . By the same . . . . . . . . . . . . . . . . . . . 136
DIFFERENTIAL CALCULUS . By A. LEvy, Esq., M.A., F.G.S., Lecturer on Natural Philosophy and Mathe-
matics in the University of Liege . . . . . . . . . o 771
INTEGRAL CALCULUS tº º By the same & º e & te * * tº c tº e & • & © e º O s 802
P R E F A (; E
TO THE
ENCYCLOPAEDIA METROPOLITANA.
GENERAL OBSERVATIONS.
As the Encyclopædia Metropolitana is now placed before the public as a complete work, it
appears essential to offer a few remarks on the objects proposed in this great undertaking,
and the manner in which its early professions have been realized. The Prospectus,
written by the late eminent poet and philosopher, S. T. Coleridge, and Dr. Stoddart, and
the Introductory Essay on the Principles of Method, which accompanied the first part
of the work, sufficiently explain the plan on which it was intended to conduct the Ency-
clopaedia. The scheme put forth in those two remarkable productions certainly proceeded
on a more enlarged and philosophical view, both of the general relations existing between
different branches of human knowledge, and of the proper mode of exhibiting those relations
and the principles of each science in an Encyclopædia, than had ever formed the basis
of any similar work. A very brief historical notice respecting Encyclopaedias will confirm
this assertion. f
“With the Ancients,” it was remarked in the Prospectus, “the term ENCYCLOPEDIA
explained itself. It was really Instruction in a Cycle, i. e., the cycle of the seven liberal
Arts and Sciences that constituted the course of education for the higher class of
citizens; grammar being the first, and each of the others having its particular place in the
cycle determined by its dependency on the preceding.” No work of this nature, however,
has descended to us from ancient times, although the name of Encyclopaedia has sometimes
been applied to the Antiquities of Varro and the Historia Naturalis of Pliny. Speusippus,
the Academic, and Aristotle, in his last work on the Sciences (repi ériotăuov), are referred
to by Krug" as having been amongst the earliest compilers of similar works. But in the
Middle Ages they were not uncommon under the title of Summa, Specula, &c. One of
* In his Philosophical Lexicon.
Vi - P R E F A C E.
the most celebrated of these is the Speculum historiale, naturale et doctrinale, by Pincent
of Beauvais (Vincentius Bellovacensis), in the XIIIth century, to which a Speculum morale
was afterwards added. In the XVIth century several works of an Encyclopaedic character
appeared, such as Ringelberg's Cyclopaedia, Basle, 1541; Paulus de Scala Epistemon,
Basle, 1559; Reisch's Margarita Philosophica, Martini Idea Philosophica, &c. The work
of Ringelberg, a small thick volume, nearly represents the ancient notion of an Ency-
clopædia, and consists of concise treatises on Grammar, Logic, Rhetoric, &c. The nature
of the work may, in some degree, be perceived from the title, which runs thus: Joachimi
Fortii Ringelbergii Andoverpiani Lucubrationes, vel potius absolutissima dykvk\otrauðeta,
nempe Liber de ratione studii utrusque lingua, Grammatica, Dialectica, Rhetorica,
Mathematica, et sublimioris Philosophica, Multa, &c. It is possible that this work of
Ringelberg may have led the way to Alsted's more elaborate Encyclopaedia, which is
generally referred to as the most celebrated of the early Encyclopædias. Its author, John
Henry Alsted, born in Herborn of Nassau, 1588, was one of the Divines who attended the
Synod of Dort. His Encyclopaedia, after several smaller editions had appeared, was pub-
lished at Lyons in 1649, in 4 volumes, folio. Its plan is not unlike that of Ringelberg,
but the subjects it embraces are more varied, and each is more elaborately treated. It is
preceded by an analysis and compendium of the whole work. It contains thirty-five books.
The 1st book is entitled Hexilogia (or doctrina de habitu mentis); the 2d, Technologia;
3rd, Archelogia; 4th, Didactica ; 5th, Lexicons and Nomenclature of each Science ; 6th,
Grammar; 7th, Rhetoric, &c. In the early part of the work will be found Short Gram-
mars and Lexicons of Latin, Greek, and Hebrew.
It is sufficient to mention these among the earlier Encyclopaedias to show how near
they approach to the ancient idea attached to the word Encyclopaedia, and how far they
differ from more modern works which bear the same name.
In recent times, in fact, the term has almost exclusively been applied to dictionaries of
general knowledge, or works in which the arts and sciences, and most branches of human
knowledge, are treated of in alphabetical order. In France many dictionaries of this kind
appeared towards the end of the XVIIth and during the course of the XVIIIth century,
among which the Dictionnaire Universel of M. l'Abbé Furetière (Amst. 1690, with a
Preface by Bayle), afterwards published by M. de Beauval, and subsequently re-edited by
M. Brutel de la Riviere at the Hague, in 1727, bears the highest character. The celebrated
Dictionnaire de Trevoua (as we learn from the preface by the last editor of the above
Dictionary) was only a pirated edition of this work. It is, like most of the general dic.
tionaries of the same age and country, chiefly confined to the definition of scientific terms,
with a very brief account of each science, &c. But in England, about the beginning of
the last century, the Leavicon Technicum of Harris, and the Cyclopaedia of Chambers, pre-
pared the way for the more elaborate and extensive undertakings which have appeared
during the last fifty years in so great numbers. In most of these the alphabetical arrange-
ment has been adopted, although it has been adhered to with greater strictness in some
instances than in others. The chief difference has consisted in this circumstance,—that in
P R E F A C E. - vii
some of these works, indeed in most of them, treatises of more or less completeness are given
under the general name of each science, such as OPTICs, ASTRONOMY, SURGERY, &c.; and
a reference to the treatise is made under each of the technical terms which belong to it;
while in others, under these technical terms, a short account is given of the meaning of the
word, and the most useful information respecting the portion of the science to which it
belongs is inserted there. Thus in this latter case the laws of Refraction, the treatment
of Aneurism, and the doctrine of Precession would be given under those terms respectively;
while in the former plan nothing but a definition would be given, with a reference to
OPTICs, SURGERY, and ASTRONOMY. The former plan is the most common, and, as it is
easy to perceive, partakes more of a scientific and systematic character. Still some of the
disadvantages of any mere alphabetical arrangement pointed out in the original Prospectus
to this Encyclopædia must remain under whatever modifications it may be adopted, and
with whatever ability it may be executed. In some of the smaller Encyclopædias an attempt
has been made to obviate this inconvenience by a division into various branches of know-
ledge, and by giving, in separate volumes, the historical and geographical articles in one
dictionary, the arts and sciences in another, and so forth. But this arrangement has not
formed the basis of any very extensive undertaking in our own language.
The plan of treating each science separately has, however, been adopted in the latest and
most elaborate work published in France—the Encyclopédie Méthodique, the publication
of which commenced in the year 1782, but was not concluded till about ten years ago.
This great work consists of 201 volumes, including 47 volumes of plates. It is, however,
nothing more or less than a collection of classified dictionaries, with a few dissertations
interspersed. For example, the section devoted to Law, and called Jurisprudence, consists
of a Law Dictionary, in ten volumes, to which a Preliminary Discourse is prefixed: the
“ Histoire Naturelle” is also in ten volumes, of which the First Volume consists of a
Preliminary Discourse, followed by a Dictionary of Quadrupeds; a Discourse on Orni-
thology, followed by a Dictionary of Birds, which is concluded in the Second Volume.
The Second Volume contains, besides the conclusion of the Dictionary of Birds, a Dis-
course on Ophiology, with a Dictionary of Serpents. The Third Volume contains Fishes;
and Vols. 4 to 10 contain Insects, on the same plan as the preceding volumes. History
consists of an Historical Dictionary in 6 volumes; and the whole Encyclopædia consists of
Dictionaries arranged in the same manner. Of the earlier French Encyclopédie, to the
name of which so much infamy attaches, it is not necessary here to speak. It was alpha-
betical in its arrangement, and the Encyclopédie Méthodique was probably intended to
supersede its use by a more methodical system and better principles.
The last work to which we shall call attention is the celebrated Encyklopädie of Ersch
and Grüber. Germany offers great facilities for the execution of any literary work requiring
the combination of men of varied acquirements and indefatigable industry; and it would be
impossible to deny that articles of first-rate merit are to be found in this work written by
German scholars and mathematicians of the highest character. But at present it is difficult
b 2
viii -- P R E F A C E.
to form any judgment upon the work as a whole. For although the alphabet has been
drawn up into three brigades, and an attack on each commenced with the courage and per-
severance characteristic of Germans, the enemy's position is not yet stormed,—in other
words, the work, after these operations have proceeded for about a quarter of a century, is
still incomplete, much of the alphabet is unpublished, and some of the most important
sciences remain to be treated.
It is scarcely worth while here to do more than just to notice the class of works which
have latterly been common in Germany under the name of Conversations-Lexicon, some of
which have found their way by means of translation into other countries. The scientific
portions are usually very superficial, hardly advancing beyond the mere definitions and the
class of information supplied in the French Dictionnaire Universel, already described (see
p. vi.); while on subjects of historical and miscellaneous information a great deal of useful
matter, though sometimes not untinged with unsound principles, is brought forward in a
popular and attractive manner. No notice is here taken of Oriental Encyclopædias, as they
scarcely affect European Literature. There is a list of them, with much information on
the subject, by V. Hammer Purgstall, in Ersch and Grüber's Encyklopädie, Art. Ency-
klopädie [orientalische].
From this brief review of the various classes of works bearing the name of Encyclopædia,
it will be seen that no great work has ever yet taken the same ground with the present
undertaking, and attempted to make a separation between those subjects which demand an
alphabetical arrangement and those which are far more conveniently treated in a systematic
manner. A very few words will be sufficient to place this in a clear light. It is presumed
that Encyclopædias are required by different classes of readers. By some they will be looked
upon as repertories of general information; and to this class of readers the facility of reference
afforded by the alphabetical arrangement is, no doubt, a matter of convenience. And yet,
if their reference is for the purpose of acquainting themselves with some of the principles of
a science, or refreshing their memory on some point connected with its details, it is quite
obvious that it can make no difference to them whether that science is found placed in its
alphabetical order, or in a separate volume with other sciences to which it bears a close
relation. Indeed, if mere facility of reference were the only object, and the reader has
neither time nor grasp of mind to take in more than is contained under the single term to
which he refers, then the old plan, now almost abandoned in England, of giving sciences
piecemeal, must have the preference over every arrangement which gives the technical terms
and the details of any science in one comprehensive treatise, whether inserted in its alpha-
betical order or ranged with its sister sciences. But no work of real value ought to contem-
plate so limited an utility, nor attempt to meet a demand for such desultory and superficial
information. One step, therefore, is clearly gained when the several details and technical
terms belonging to one science are gathered together into one treatise, even when the place
of that treatise is determined by no regard to system, nor to any other circumstance than
the first letter in its name. But the Encyclopaedia Metropolitana makes another step in
advance, and that advance is of more importance than at first sight it seems to be.
P. R. E. F. A. C. E. ix
One of the advantages offered by this arrangement is, that it brings the work under the
class of publications really deserving the name of an Encyclopædia, i.e., instruction in a
methodical order. The sciences which are capable of mutual dependency are thus brought
into one volume; and those who really desire instruction in them may read them in their
natural sequence, and ensure by that means a progressive proficiency in them. It is not
a small advantage, particularly in the exact sciences, to find such an arrangement adopted
as would enable a student to pursue them even without the assistance of a tutor. It may
safely be affirmed, that any person of good mathematical abilities, who followed the course
of treatises in the first and second volumes of Pure and Mixed Sciences in this Ency-
clopaedia would become by that means a mathematician of a very high character, and be
enabled to master the most difficult and delicate speculations of continental mathematicians.
If, again, in Sciences, where, although there is a mutual dependency, yet each science may
be pursued separately by one acquainted with a few mathematical truths, the advantages of
this systematic treatment are so great, must it not be tenfold greater in regard to all
historical information, where nothing can be isolated, but all is intimately connected. In
the alphabetical arrangement a concise history of each country may be given under its
name, but then it is isolated from all collateral matter and all contemporary history. But
even this system is not always adopted; but the clumsy and unscientific mode of exhibiting
the history of each country under the name of its sovereigns is often followed. To obviate
the inconvenience arising from this fragmentary kind of history, the Encyclopaedia Metro-
politana has exhibited the history of the world at first in a series of biographical sketches,
and then in a continuous history of each remarkable country, combined with an ecclesiastical
history remarkably full and rich in the most interesting epochs of the Christian Church.
But of these portions of the work, the Scientific and the Historical, we shall have to speak
more in detail hereafter ; our concern at present lies only with the mode of arrangement.
These two portions of the work, however, still leave untouched a considerable portion of
that Miscellaneous information for which it is usual to refer to an Encyclopaedia; and
accordingly a very large proportion of the work is devoted to this class of subjects, and
combined with the most philosophical dictionary of the English language hitherto published.
While, therefore, we deprecate the practice of extravagantly lauding every particular article
furnished to this Encyclopaedia, we feel justified in observing, that the plan on which
it was projected, by a peculiar adjustment of the systematic and alphabetical arrange-
ments, has happily avoided the greater inconveniences of each, while it has at the same
time combined their chief advantages. That which is capable of being learned systemati-
cally is so exhibited, while any portion of it may be referred to with ease as a separate
article; and the alphabetical arrangement has been restricted to that which scarcely admits
of any other with convenience to the reader. The plan may have some slight inconve-
niences of its own, but these advantages far more than counterbalance them. Indeed one
of the greatest disadvantages entailed on the work, viz., the fragmentary manner in which
each portion was published in the separate parts, is now wholly removed by the completion
of the Encyclopædia, and its formation into volumes. .
x P. R. E. F. A. C. E.
It may be proper here to state exactly the nature of the several divisions of the work, as
set forth in the original Prospectus, and as subsequently modified in one or two slight
particulars.
FIRST DIVISION.
Universal Grammar.
Logic:—Rhetoric.
FORMAL. Mathematics.
PURE SCIENCES.
smárºs Metaphysics.
2 Wols. Morals.
REAL. Law.
Theology.
SECOND DIVISION.
|Mechanics.
Hydrostatics.
r MIXED. K. Pneumatics.
. | Optics.
Astronomy.
Magnetism:—Electro-Magnetism.
Electricity, Galvanism.
/* Heat.
' I. A Light.
y EXPERIMENTAL Chemistry.
MIXED AND APPLIED PHILOSOPHY. Sound.
SCIENCES. Meteorology.
tººl*: < Figure of the Earth.
6. Wols. Tides and Waves.
Architecture.
| Sculpture.
II | Painting.
• j Heraldry.
THE FINE ARTS, Numismatics.
| Poetry.
Music.
º U Engraving,
l APPLIED. K ſ Agriculture.
Horticulture.
Commerce.
IIH. & Political Economy.
THE USEFUL ARTS, Y Carpentry. . -
Fortification.
Naval Architecture.
& Manufactures.
IV. ſ Inanimate:—Crystallography, Geology, Mineralogy.
NATURAL HISTORY. Insentient :—Phytonomy, Botany. -
Animate :-Zoology.
V Anatomy.
APPLICATION OF J Materia Medica.
Nārū’īāisióñy. l."
^e urgery.
THIRD DIVISION.
BIOGRAPHICAL AND Biography CBRONoLogic ALLY arranged, interspersed with introductory Chapters of National
HISTORICAL. History, Political Geography, and Chronology, and accompanied with correspondent Maps and
º & Charts. The far larger portion of HISTORY being thus conveyed, not only in its most interesting,
ºm but in its most philosophical, because most natural and real form; while the remaining and con-
5 Vols. Unecting facts are interwoven in the several preliminary chapters.
FOURTH DIVISION.
Alphabetical, Miscellaneous, and Supplementary:-containing a GAzETTEER, or complete
MISCELLANEOUS AND | Vocabulary of Geography: and a Philosophical and Etymological LExicon of the English Lan-
LEXICOGRAPHICAL. guage, or the History of English Words;–the citations arranged according to the Age of the
• . ." -- * Works from which they are selected, yet with every attention to the independent beauty or value
of the sentences chosen, which is consistent with the higher ends of a clear insight into the original
12 Wors. U and acquired meaning of every word.
INDEX-Being a digested and complete Body of Reference to the whole Work.
*
We now proceed to speak of each portion separately.

P. R. E. F. A. C. E. xi
SCIENTIFIC IDIVISION.
P U R E AND MIX E D SCIENCES.
The principles on which the plan of the Encyclopædia Metropolitana was formed
having been already explained, we now proceed to consider the various divisions in detail,
beginning with the scientific portion of the work. *.
It is obvious that, besides the mere separation of scientific from historical and literary
matter, another very remarkable division is afforded by the nature of the sciences them-
selves. On this natural line of distinction the subdivisions of the scientific volumes of the
Encyclopædia Metropolitana are founded.
In the first place, those sciences are grouped together, the principles of which belong to
the pure Reason, (e.g., Algebra, Geometry, Grammar, Logic, &c.). They are combined
together as preliminary to the knowledge of those which depend partly on abstract prin-
ciples and partly on close observation of the phenomena around us, and thus belong to
the truths received by the Understanding.” And here, again, there is also a manifest
difference between those sciences in which so great a progress has been made by the human
mind that their fundamental principles may be considered permanently fixed, such as
Mechanics, Hydrostatics, Astronomy, &c.; and those which depend chiefly on observation
of the external world and a large collection of facts and a careful induction from those
facts, such as Geology, and perhaps Chemistry. This distinction has not been overlooked
in the Encyclopædia Metropolitana.
It would be idle to pretend to give treatises upon the latter which shall permanently
embody all the principles which belong to them. This would be to profess to perform
impossibilities. All that can be done is to represent their present condition ; and the
names of those who have contributed these portions of the Encyclopædia are a sufficient
guarantee that this is effectually provided for.
But with respect to the exact sciences, whether pure or mixed, more is required, and
much more has here been performed. The principles of these sciences have long been esta-
blished; but the efficiency of any treatise depends much on the mode in which they are
exhibited, and the value of the whole series in some degree on the manner in which they
are combined. In both these respects the Encyclopædia Metropolitana may challenge
competition with any existing work. The order in which these sciences are exhibited, and
the plan on which the mathematical portion of the Encyclopaedia is conceived, resemble
considerably that of the series of Elementary Treatises projected many years ago for the
University of Cambridge by Dr. Wood the late Dean of Ely, and Professor Vince; but
with this difference, that the present volumes are far more comprehensive in the subjects
they embrace, and far more elaborate and scientific in their execution. But this very simi-
* The Reason and the Understanding are here distinguished according to the views of German philosophers, and much in
the same manner as in the works of the late S. T. Coleridge.
xii P. R. E. F. A. C. E.
larity shows that the Encyclopædia Metropolitana has attained one of its professed objects,
systematic instruction and scientific information conveyed—not in a confused mass, but in
the natural sequence of the sciences.
Indeed this portion of the work has met with a degree of approbation in many
quarters, but especially in the University of Cambridge, which no other Encyclopaedia
has ever yet received. And this preference relates, we may observe, to sciences which
have obtained a stated position, and are not liable to be superseded by any new discoveries.
Geology and Chemistry indeed, and other sciences founded on observation and experiment,
are constantly enlarging their boundaries and changing even some of their elementary
principles. But no such change takes place, or indeed, we may confidently assert, ever
can take place, in pure Mathematics, or the more exact branches of the Mixed Sciences.
The utmost which may be expected in these, is some extension of their present boun-
daries. The principles already established may implicitly contain results not yet developed
from them, and some of the known elementary principles may perhaps be thrown into
a different form, but they are established on too firm a basis ever to be overturned.
Again, as physical science employs in its advancement some of the results of those refined
speculations in pure mathematics which are at present only truths belonging to the
Reason, and have no connection with the world in which we live, there may be dis-
covered another set of results which may give to the mind of man a more ample dominion
over the phenomena of the material world. Still these are results which require nothing
to be unlearned; they are a mere advance in the quantity of our knowledge, and in the
number of the results we can elicit from them. The student, who has really mastered
these sciences in the systematic form in which they are arranged here, will never in the
course of the longest life find occasion to unlearn any portion of what he has here ac,
quired, and will find no difficulty whatever in adding to his stores any new results which
the mental energy and labour of mankind may hereafter develop from principles now
known. -
We have been thus particular in stating the advantages of the arrangement adopted,
because we deem it a matter of considerable importance; but we now proceed to speak
more in detail of the execution of each portion.
The distinction between Pure and Mixed Mathematics is of primary importance. In
the manner in which mathematical inquiries are now conducted, our progress in mixed
mathematical science mainly depends on our command of the principles of pure mathe-
matics. It is indeed almost an acknowledged fact, that, in some respects, we have a super-
fluity of knowledge of these principles. Our application of Mathematics to Natural
Philosophy is so far from having exhausted all our stores of Pure Mathematics, that
although there are still many problems too intricate for solution with our present means,
yet there is also a large mass of results in pure mathematics which as yet have no specific
application, and may be considered as stores reserved for future use.
P. R. E. F. A. C. E. - xiii
The mere names of the authors of the Treatises on Pure Mathematics are sufficient to
prove that the work is worthy of the present state of science, and that its most important
Treatises are contributed by those who have themselves been foremost in the onward march
of science. The elaborate Treatise on ARITHMETIC, by the present Dean of Ely (Dr.
Peacock), Lowndian Professor of Mathematics in the University of Cambridge, is interest-
ing alike to the scholar, the mathematician, and the speculator in metaphysics. The
brief but comprehensive Treatise on TRIGONOMETRY, by Professor Airy, now Astronomer
Royal, although on so elementary a subject, is of considerable value from the general
elegance of its demonstrations.
The publications of the Rev. H. P. Hamilton on ANALYTICAL GEOMETRY and CoNIC
SECTIONs, and that of Professor Barlow on the THEORY of NUMBERs, are so well known
and so highly esteemed that any eulogium on the essays supplied by these gentlemen on
these subjects respectively would be entirely superfluous.
The Treatises of Professor Levy on the DIFFERENTIAL and INTEGRAL CALCULUs are
written with a comprehensive brevity which recommends them as an introduction to those
important branches of Analytical Mathematics, and are calculated to carry the student to a
very high point of proficiency.
The GEOMETRY, ALGEBRA, and GEOMETRICAL ANALYSIS complete the volume in a
manner worthy of the treatises with which they are associated.
These sciences are however in some degree elementary; and although by them the
student would be so far advanced as to enter upon the works of some of the ablest
analysts, it would be unworthy of such a publication as the Encyclopædia Metropolitana to
leave either untouched or imperfectly treated the more refined applications of the higher
Calculus. It will be found accordingly, that in the second volume of pure sciences the
highest branches of mathematical analysis have been treated by writers conversant with all
its intricacies, and that the mathematical student is furnished in them with results of far
greater variety and of a more subtle nature than can at present be used in the application of
analysis to Mixed Mathematics. On this subject it is unnecessary to do more than just to
enumerate the names of the treatises and their respective authors, whose eminence in ma-
thematical attainments is universally acknowledged.
The CALCULUs of VARIATIONs, and the CALCULUS of FINITE DIFFERENCEs, supplied by
the Rev. T. G. Hall, Professor of Mathematics in King's College, London, are treated
with the clearness which his long and successful course of mathematical teaching has
enabled him to give to these refined and subtle portions of analysis.
The CALCULUS of FUNCTIONs and the THEORY of PROBABILITY are the work of Professor
De Morgan. The former of these subjects may at present be considered almost in its infancy;
but there can be no doubt that this author has here brought forward much that is calcu-
lated to expedite its development. The Treatise on Probabilities (a subject which has
C
xiv. P R E F A C E.
exercised the talents of the greatest mathematicians even down to the times of La Place)
is, as might be expected, one of the most complete in any language. -
And lastly, the Treatise on DEFINITE INTEGRALs completes the series of these elaborate
essays on the higher branches of mathematical analysis. The name of Professor Moseley
is a sufficient warrant that this essay is also of the highest character.
Without wishing, therefore, to offer any undue eulogium on the treatises enumerated
above, we may confidently ask that portion of the public which is qualified to judge of
their merits, to compare the whole system of Pure Mathematics here presented to them
with that in any similar work, whether of this country or of the continent, on the grounds
of arrangement, clearness, ability, and completeness. From any ordeal of this sort, how-
ever severe, this Encyclopædia will not shrink; and it is confidently believed that no
parties connected with it would have reason to regret the comparison.
From Pure Mathematics we proceed in natural order to their application to physical
phenomena. Of these sciences, some belong to the more elementary branches of physical
knowledge, and others to a higher and more advanced stage. Now the treatises on—
HYDRODYNAMICs, MECHANICs, HYDROSTATICs, OPTICS, PLANE ASTRONOMY,
have been written by Professor Barlow with an express view to this distinction. They
are elementary enough to enable any student, with a competent knowledge of Pure Mathe-
matics, to overcome their difficulties; and yet they are so based on scientific principles, that
they will also prepare him to enter readily on the higher branches of Mixed Mathematics.
In Mechanics, more especially, a foundation is laid for the succeeding investigations of
Physical Astronomy, which is in fact only one of the higher branches of Analytical Physics.
While, however, for these portions of the work we claim only that high share of appro-
bation due to the presentation of ascertained results and knowledge already acquired, in an
elegant and useful form, there are some treatises in the volumes devoted to the Mixed
Sciences which demand a separate notice, as enlarging the boundaries of our scientific know-
ledge. Of this class are the Treatises on LIGHT and Sound, by Sir J. F. W. Herschel.
The Treatise on LIGHT, by Sir J. F. W. Herschel, from the position it has already
obtained in the scientific world, both in England and on the Continent, cannot require any
comment or recommendation here. We shall merely cite it as furnishing the best refuta-
tion to the words of its author respecting the decline of Science in England.*
The simple mention of Sir J. F. W. Herschel's name is a sufficient recommendation to
the Treatise on PHYSICAL ASTRONOMY. It proves at once that it must be an Essay of the
highest order of merit, and worthy of the present state of the Science. Indeed the name
of Sir J. F. W. HERSCHEL stands so confessedly at the head of Physical Science in Eng-
land, that the conductors of this Encyclopædia may justly be proud that he has contributed
“so largely to its pages. .
* Herschel–Essay on Sound. Mixed Sciences, vol. ii. p. 810.
P. R. E. F. A. C. E. XV
But although Plane and Physical Astronomy had been thus ably treated, it was con-
sidered that something more was required; and the late Captain Kater kindly furnished the
very useful and able Treatise on NAUTICAL ASTRONOMY, a subject with which his acquaint-
ance was at once profound and practical. -
, MAGNETISM and ELECTRo-MAGNETISM are treated by Professor Barlow with the same
ability and research which he has displayed in the other essays contributed by him. It
cannot be needful to recommend the Essay on GALVANISM, as Dr. Roget's scientific cha-
racter is too firmly established to leave any doubt as to its merit.
The author of the Treatises on ELECTRICITY, HEAT, and CHEMISTRY, the late Rev. F.
Lunn, was one whose merits as an experimental philosopher and chemist were not so exten-
sively known as they deserved to be; but in Cambridge, in a considerable circle of persons
qualified to judge in these matters, his talents were justly appreciated, and his acquirements
acknowledged to be of the highest order. The treatises themselves, it is believed, will
amply justify their favourable anticipations. - w -
º
The third volume of Mixed Sciences is chiefly devoted to the Fine Arts; but there are
two or three essays in the early part of the volume which belong to the more exact sciences,
viz., the Essay on the FIGURE OF THE EARTH, by the present Astronomer Royal, and his
Treatise on the TIDEs. With regard to the former, much novelty was hardly to be
expected; but the Editor believes he is justified in stating that this Treatise contains the
most complete combination and discussion of observations relating to the subject which
has yet appeared in England. But the treatise into which this great mathematician
has thrown all his power is the Theory of the Tides. It was remarked in 1833, by
Mr. Lubbock, on the subject of the tides, that “there is no branch of physical astronomy
in which so much remains to be accomplished.”* The Astronomer Royal, in this treatise,
has made a large step in advance in this science; he has, at all events, demonstrated the
unsoundness of the equilibrium theory and the inapplicability of the theory of Laplace.
The latter he has explained in such a manner as to bring it within the reach of good
mathematicians; whereas, in the manner it was presented by its author in the Mécanique
Céleste, none but persons of very high mathematical ability and undaunted perseverance
would venture to encounter its difficulties. Still the theory was inapplicable; and the
Astronomer Royal gave all the leisure he could command, for some years, to the con-
sideration of these questions, and to an endeavour to place this great problem on a firm
foundation. The Editor does not pretend to speak on this point from his own knowledge;
but the terms in which some of the most distinguished mathematicians of Cambridge have
spoken to him of this treatise prove that they consider it to have advanced the knowledge
of this difficult subject in no ordinary degree. Indeed, the Editor believes that he may
confidently assert that every previous treatise on the subject is entirely superseded by this
theory, and that it will prove, for many years to come, the only sound foundation of our
knowledge of the Theory of the Tides.
* “Report on the Tides,” published in the Report of the First and Second Meetings of the British Association for the
Advancement of Science. - * ,
c 2
xyi - P. R. E. F. A. C. E.
A few more treatises in these volumes require separate mention,-the METEOROLOGY
of the late Mr. Harvey, and the CRYSTALLOGRAPHY of Mr. Brooke. Although not anxious
to quote opinions on articles in this Enclyclopaedia, the Editor may be permitted to call
attention to the following incidental notice of the article in Professor Forbes's Report on
Meteorology, addressed to the British Association at its second meeting:—“We shall
occasionally avail ourselves of the information contained in this work (the Elémens de
Physique of M. Pouillet), as well as of a useful compendium of facts contained in the
article Meteorology, in the Encyclopaedia Metropolitana, now in the course of publication."
—Report, p. 206. The testimony of Professor Forbes is of first-rate authority, and above
all suspicion. -
Of the CRYSTALLOGRAPHY and MINERALOGY of Mr. Brooke it is not necessary to speak
particularly, but we may again quote the same volume of Reports for the testimony of a
competent witness to the value of Mr. Brooke's labours in these sciences:—
“Mr. Phillips and Mr. Brooke have contributed to the stock of crystallography
observations more numerous and exact, probably, than any other two names could
rival.”—Dr. Whewell's Report on Mineralogy at the Second Meeting of the British
- Association.
The names of Mr. Phillips and Dr. Daubeny will sufficiently recommend the Treatise
on GEOLOGY, as exhibiting an adequate representation of that science at the time of its
publication. And, even in this hasty enumeration, the Essays on CARPENTRY, by
P. Nicholson, Esq.; on ForTIFICATION, by Major Michell and Captain Procter; and on
NAVAL ARCHITECTURE, by the late Mr. Harvey, must not be passed over. We can
only say here, as in so many other instances, the names guarantee the value of their
contributions.
Before we leave this class of Mixed Sciences, we must call attention to the novel feature
exhibited in the sixth volume of the series, viz., a systematic account of the ARTs and
MANUFACTUREs of Great Britain. There is, probably, no writer who would be able to
do such ample justice to so extensive a range of matter, requiring both theoretical and practical
knowledge, as Mr. Barlow; but that nothing might be wanting to the completeness of this
portion of the work, Professor Babbage was engaged to give a Preliminary Discourse on the
Principles of Manufactures ; and it may confidêntly be asked, to what other source could
the conductors of the work have appealed on so difficult and general a subject where the
answer to that appeal would have afforded such entire confidence in the result?
We have now enumerated all the articles in these volumes which appertain to the more
exact sciences and to those connected with physical phenomena. The remainder are
devoted to another class of subjects, Natural History, Physiology, Medical Sciences,
the Useful Arts, Belles Lettres, and the Fine Arts.
The Treatises on BotANY and HoRTICULTURE are supplied by G. Don, Esq., whose
profound acquaintance with every department of knowledge which belongs to the vegetable
P R E F A C E. xvii
kingdom is known to all botanists and florists. The Treatise on PoliticAL EconoMY was
written by N. W. Senior, Esq.
The following enumeration of the remaining Treatises in the volumes devoted to the
Mixed and Applied Sciences will show that the range of subjects to which attention is
directed is wide and comprehensive, and the intrinsic merit of the Essays themselves will
prove that no pains have been spared to do justice to these interesting topics. They em-
brace a Series of Treatises on ARCHITECTURE, Sculpture, PAINTING, ENGRAVING, HE-
RALDRY, NUMISMATICs, PoETRY, MUSIC, AGRICULTURE, and CoMMERCE. -
In the first volume of Pure Sciences Sir J. Stoddart has given a lucid and able summary
of the General Principles of GRAMMAR, of which it is unnecessary to speak in detail.
The Logic and RHEToRIC of the present Archbishop of Dublin require no commendation
here, as they have already, for many years, been published in a separate form, and taken
their place among the standard works of our language.
The Treatise on LAw is the work of three gentlemen, Richard Jebb, Esq., Professor
Graves, and Archer Polson, Esq. It was originally intended that the whole should have
been executed by Mr. Jebb; but ill health having rendered it inconvenient to him to furnish
the conclusion, it was intrusted to Professor Graves and Mr. Polson, who were fully
acquainted by Mr. Jebb with the plan on which he projected it, and kindly undertook to
complete its execution. The portion accomplished by Mr. Jebb embraces one of the most
difficult portions of philosophy—the general foundations of law and morals; and the
Editor is happy to state that testimony from the very highest quarters has been given
to the profoundness of the views entertained by Mr. Jebb, and the ability with which they
are developed. .
In regard to two of the Treatises in the volumes devoted to Pure Sciences, viz., the
MoRAL AND METAPHYSICAL PHILosophy, and the OUTLINEs OF THEOLOGY, a few words
of explanation are required. They appear, it must be acknowledged, under a form different
from that which seems to be contemplated in the original scheme of this work.
That scheme apparently was intended to comprise formal and scientific treatises on these
important subjects; but every person at all conversant with these matters will acknowledge
that such a Treatise could have but little value, if it were confined to the limits which a
general work like the present must necessarily prescribe. A course was therefore adopted,
by which, it is hoped, the most important principles of these sciences are brought forward
in the manner most likely to conduce to the advantage of those who study them. In the
present state of metaphysical knowledge, it would be presumptuous to put forth any system
of Metaphysics; but a general History of Moral and Metaphysical Philosophy affords the
most convenient opportunity for displaying the principles on which the greatest philoso-
phers have hitherto endeavoured to form their systems, for pointing out their difficulties,
and for marking how far each has contributed to the progress of the science. Such a
sketch, however, required the hand of a master; and the Editor confidently believes that the
rvii - P R E F A C E.
Treatise on Moral and Metaphysical Philosophy which is here given is calculated fully to
sustain the deservedly high reputation of the Rev. F. D. Maurice. .
Of the Outlines of Theology, it does not become the Editor to say more than that to
acknowledge with gratitude the very able assistance of Professor Corrie, to whom two
ehapters are due. Much of the matter which usually falls under the head of Theology had
already been anticipated in the Miscellaneous and Historical Departments; and it was the
object of the Editor to devote the comparatively small space which he could command only
to the most important portions of the subject, and to render this Treatise as practically
useful as possible. He has endeavoured to avoid passing controversies, but to bring forward
the sound and genuine doctrines of the Church of England; and perhaps he may be allowed
to add that, in pursuance of this object, he has spared no pains or labour.
HISTORICAL DIVISION.
From the time that this Encyclopædia was consigned to the management of Archdeacon
Lyall, and subsequently to that of the Rev. Edward Smedley, its Historical Division
became enriched with contributions from some of the most eminent writers of the day.
It will be impossible, in the rapid sketch of the contents of the several volumes which a
Preface admits, to specify every paper; but as every contribution (except in part of the
first volume) is assigned to its proper author at the beginning of each volume, such a course
is unnecessary, either for the information of the public, or as a tribute of respect to the
distinguished authors themselves. It will be observed, on a general survey of their names,
that ample care has been taken to enlist among the contributors to this department
writers not only of splendid endowments, but also of the highest attainments in different
classes of historical knowledge. There will be found contributions from Bishop Blomfield,
Dr. Whewell, Serjeant Talfourd, Dr. Arnold, Dr. Hinds, Rev. J. A. Jeremie, Rev. G. C.
Renouard, Rev. J. H. Newman, Bishop Russell, Archdeacon Hale, Archdeacon Lyall,
Rev. J. B. S. Carwithen, Dr. Hampden, Rev. R. Garnet, Major Mountain, Rev. J. H.
B. Mountain, Dr. W. C. Taylor, Captain Procter, Rev. J. E. Riddle, Rev. T. G. Ormerod,
T. Roscoe, Esq., W. M'Pherson, Esq., Rev. R. L. Browne, Rev. H. Thompson, Rev. J. G.
Dowling, Rev. J. W. Blakesley, Rev. J. B. Ottley, W. Lowndes, Esq., Q.C., &c. &c.
A good work on general history has long been a great desideratum in our literature.
The summaries of Tytler and Russell are too brief, and the Universal History, independently
of the heavy manner in which it is written, is too long. It is presumed that the historical
volumes of the Encyclopædia Metropolitana will be found to meet this want in an efficient
manner. The histories are written by men of undoubted ability; and historical dissertations,
such as those on the Crusades and the Feudal System, are introduced into the text at the
most convenient periods, for the illustration of the subjects involved.
In the original Prospectus it was intimated that the History would be given in the
form of Biography, chronologically arranged. Such an arrangement, however conve-
P R E F A C E. 4. xix.
nient in regard to Ancient History, when the History of Greece or Rome was virtually the
history of the world, would scarcely admit of any modification by which a modern unit
versal history could be treated biographically. The interests even of Europe alone are too
complicated in modern times to be treated in any other way than by a separate history of
each country. Accordingly, it will be found that the former plan has been exchanged for
national histories from about the middle of the third volume, an exchange which every
reader will acknowledge to have been not only advantageous, but imperatively required.
The first volume, beginning from the earliest accounts of mankind, brings down the
History to about the year 200, B.C. It contains, besides the usual course of Ancient
History, an Essay on Greek Philosophy, connected with the life of Socrates, by the present
Bishop of London; and a Life of Archimedes, with a Sketch of Greek Mathematics, by
Dr. Whewell; with many other papers, which it is obviously impossible here to specify.
The second volume continues the secular history to the age of the Antonines, and lays a
foundation for the future chapters of Ecclesiastical History in an elaborate account of the
first appearance of Christianity, and of the apostolic age. The following dissertations, un-
connected with the general course of the History, but of great importance in a philoso-
phical point of view, may be particularly specified as giving great value to this volume,
PLATO, ARISTOTLE, SENECA, the STOICs, CICERo, RoMAN PHILOSOPHY, HISTORIANS OF
RoME, SEXTUs EMPIRICUs, the PyRRHoNISTs, &c. Nor would it be proper to pass over,
without a distinct reference, the elaborate History of Latin Poetry, which has been gene-
rally acknowledged as a valuable accession to our literature. , -
The third volume contains an account of the Decline of the Roman Empire, the Rise of
the Empire of Charlemagne, and of the Modern System of Europe, as well as an elaborate
History of Mohammed, and the origin of Saracenic Power. It brings down the History
to about the end of the thirteenth century, and comprises, besides the Secular History, an
ample Ecclesiastical History of the same period. The historical dissertations with which
it is enriched are Essays on MoHAMMED ; on the HERESIES OF THE SECOND AND THIRD
CENTURIES; PLOTINUS and the LATER ECLECTICs; the CRUSADES; the FEUDAL SYSTEM ;
º THOMAS AQUINAs, and the Scholastic System.
The fourth and fifth volumes continue the Modern History to the settlement of Europe
under the Treaties of 1815. The Ecclesiastical History is also continued to the same
period. z
The Editor would also desire to call attention to the copious Chronological Tables
inserted at convenient intervals in this division of the work.
The historical volumes of the Encyclopædia Metropolitana, it will be seen, have been
formed on the principle of giving an accurate and ample general history. As every
Encyclopædia is now expected to embody a large amount of history, the only question
left for consideration was, how to meet this demand in the most efficient manner. The plan
XX - P R E F A C E.
most commonly adopted in works of this nature, of giving the history of each country in
the article assigned by alphabetical order to that country, appeared liable to some objec-
tions, which might be obviated by removing the history to separate volumes, and giving
to it a certain degree of continuity. The convenience of this method is obvious; and the
names of the contributors employed upon this important portion of the work bear ample
testimony to the exertions which must have been made to obtain the co-operation of so
©
many writers of high endowments.
MISCELLANEOUS PORTION.
Although the Miscellaneous Division of this Encyclopædia occupies a larger number of
volumes than any other, it requires a less extended notice. It will be impossible to mention
separately every article, or even every contributor of merit; but all that is required in this
Preface is to explain in some degree the principle on which this portion of the work was
executed, and to indicate the authors of some of the most remarkable series of papers.
The most remarkable features in this division of the Encyclopædia are clearly—
1. The English Lexicon.
2. The Geography.
3. The Natural History.
4. The strictly Miscellaneous Articles.
It is unnecessary here to speak in any detail on the subject of the Lexicon. Its plan was
duly, described in the Prospectus and the special Preface to the Lexicon itself. To that
plan a steady adherence has been maintained; and the universal approbation with which this
Lexicon has been received, precludes the necessity of enlarging either on the plan itself or
on the gigantic labour involved in its execution. The plan of giving the quotations of
each word chronologically has the advantage of embodying in a philosophical Lexicon a
history of our own language. They are generally full of interest ; but the labour of
searching them out and arranging them is one of which those who have never engaged
in any similar occupation can form no adequate notion. Once achieved, the work is
performed for ever; and Dr. Richardson may be contented to think that he has here left
a KTºwa Śs del of infinite value to his countrymen.
Before we speak of the Geography and Natural History, and the articles on Law, we may
be allowed to insert a few words relating to the highly gifted individual to whom this
Encyclopædia owes so many of its advantages and attractions; we mean the late Rev.
Edward Smedley. Besides the advantages derived from the confidence reposed in him
during his editorship by so many men of distinguished literary merit, he not only threw
into the historical volumes of this book very elaborate chapters, containing the results of
deep historical research, but gave to the Miscellaneous Division a series of articles which
embodied a vast store of curious and recondite information, communicated in a manner at
once instructive and agreeable. The copious stores of his own mind, and his vast fund of
P R E F A C E. - - XXi
acquired knowledge, enabled him to enrich this department of the Encyclopaedia with a class
of articles which stamp a peculiar character on those volumes of the work which he super-
intended,—a character which it would have been in vain to seek to supply from any other
source. His death was a loss to literature in general; it left a void which it was difficult
to supply, and we may be thankful that it was not more severely felt in this Encyclopædia.
The arrangements he had already made were so efficient that the succeeding Editors found
little difficulty in carrying on what he had begun, and completing what he had either over-
looked or left unfinished. The editorship was placed on his decease in the hands of the late
Rev. Hugh James Rose, B.D., Principal of King's College, London. It would not become
the present Editor to speak of one so closely connected with himself, of the high purpose
which he ever set before him in all his undertakings, and the noble endowments with which
those high purposes were ever prosecuted. Of these it would be a grateful task to speak, but
this is not a fitting place. We confine ourselves here to the simple fact that he made such
engagements as materially benefited the work, and facilitated the completion of it on the
plan which had been projected and adhered to as far as was practicable.
We proceed now to add a few words on some of the most remarkable sections of the
Miscellaneous Division. And first, on the Geography.
It will be observed, that the arrangement in this department, although in the alphabetical
portion of the work, is not strictly alphabetical. It has been the practice, through the chief
portion of the Encyclopaedia, to describe whole regions at once, and give accounts of remark-
able places and smaller divisions of territory under the larger geographical division to which
they belong. Thus, for example, if the reader wished to turn to the account of SMYRNA,
he must look under NATOLIA ; for UTRECHT, he would look at NETHERLANDS; and so
forth. That this is a sacrifice in some degree of facility of reference, cannot be denied; but
at the same time it gives a more philosophical and systematic consistency to the geogra-
phical section; and, as the work is now complete, the Indea will obviate every difficulty of
this character. In any case in which it is uncertain where a town or district may be
described, a single reference to the Index will be enough.
For the whole of the articles on Geography, the proprietors feel that they may fairly
advance the claim of having obtained the co-operation of persons more than competent to
bring forward whatever is most valuable for a work like this from all usually accessible
sources of information. In this respect the Encyclopædia Metropolitana claims to take a
high station among similar works; and the names of those gentlemen who have contributed
the articles on European and American Geography are a sufficient pledge of the ability
and care with which they are executed. The gentlemen to whose labours this department
is chiefly indebted, are the following:—T. Myers, Esq., Captain Bonnycastle, R.E.,
C. Vignoles, Esq., C.E., H. Lloyd, Esq., G. H. Smith, Esq., A. Jacob, Esq., W. D.
Cooley, Esq., and Cyrus Redding, Esq.
But there is one class of geographical articles which demands an especial mention. They
are indeed sui GENERIs, and may be said to be wholly without a rival in any similar work
d
xxii .P. R. E. F. A. C. E.
in our own language. These are the articles on Ancient, Oriental, and African Geo-
graphy, which, throughout the work, were supplied by the Rev. G. C. Renouard, late
Fellow of Sidney-Sussex College, Cambridge, and formerly Chaplain at Smyrna. It is not
merely the extensive familiarity with every class of language, ancient and modern, and with
all the storehouses of information in them, which give the value to his researches, but it
is the extraordinary zeal and industry which he has invariably bestowed in conjunction
with these great advantages on his favourite pursuit of geography. No one but the Editor
of this Encyclopædia is probably aware of the amount of time and labour bestowed by Mr.
Renouard on each of these articles. This circumstance, and his extensive familiarity with
the original sources of information in all languages, render his contributions unique in the
history of similar undertakings; and the Editor believes that if these essays were collected
together, and published as a system of Oriental Geography, they would surpass in accuracy
and value anything at present existing in our own or any other European language.
We pass on now to the Section of Natural History. This is divided chiefly into Botany
and Zoology. In these two sciences the Genera will be found described in their alpha-
betical order, while their scientific arrangement and the principles of the sciences form
part of the treatises in the volumes devoted to the Mixed Sciences.
For these two departments, the services of several eminent naturalists were engaged.
In Botany, T. Edwards, Esq., and Mr. Don, &c. In Zoology, T. Bell, Esq., F.L.S., &c.,
J. E. Gray, Esq., F.L.S., &c., of the British Museum, J. F. Stephens, Esq., and Mr.
South. To Mr. South the Encyclopædia is much indebted for the very great accuracy
with which he has composed his descriptions, and for the varied and interesting informa-
tion he has interwoven with the subject of most of these articles.” It will also be observed
that a very copious Law Dictionary is incorporated with this portion of the work, fur-
nished by a variety of able contributors engaged in the study and the practice of the
Law. The articles supplied by each contributor are indicated in the volumes in which they
. OCCUll”. ,
Besides the miscellaneous articles of the late Editor, the Geographical Gazetteer, and
the Law Dictionary, included in this portion of the Encyclopædia, a large number of arti-
cles, some of them of very great importance and value, will be found scattered through the
volumes of the Miscellaneous Division, which it is obviously impossible here to particularize.
Attention may, however, be called, amongst a variety of others, to the Biblical articles, by
the Rev. T. H. Horne; to the Philological and Oriental articles, by the Rev. G. C. Renouard;
the Scientific articles (as e. g., Dialling, Surveying, Weights and Measures, &c.), by Mr.
Barlow; Meteoric Stones, by Professor Miller; Stove and Ventilation, by C. Hood, Esq.,
F.R.S., &c.; Stucco, by T. L. Donaldson, Professor of Architecture in University College,
London; the Theological articles, by Archdeacon Hale ; Writing, and other articles, by
the Rev. R. Garnet; and to a number of others, which cannot here be enumerated, but for
* These will sometimes be found to supersede other articles on similar subjects. Thus, Balaena includes an account of Whale
Fisheries, &c. . - - , , - *
P R E F A C E. Xxiii
which the able and distinguished writers will receive due credit in the volumes to which
their labours belong. - $º
MEDICAL VOLUME,
Every portion of the Encyclopædia has now been considered except the Physiological
and Medical Volume.
The ZooLogy combines GENERAL PHYSIOLOGY with CoMPARATIVE ANATOMY, and is the
work of J. F. South, Esq., Surgeon of St. Thomas's Hospital, (assisted in one portion of
Physiology, by F. Le Gros Clark, Esq., and T. Solly, Esq., both of St. Thomas's Hospital).
For this treatise one merit, and that not of any ordinary kind, may be claimed. It is
usual, in works of this kind, to give the best information derived from the best authorities.
But Mr. South, whose acquaintance with these authorities is most extensive, on comparing
the descriptions in books of the very highest character with the specimens themselves (par-
ticularly those of Osteology), preserved in the Museum of the College of Surgeons, found
that he could never entirely rely upon them, and accordingly determined to describe, in
every instance in which it was practicable, from the specimens themselves. Of the labour
thus entailed upon him, and of the value which this circumstance must give to his details,
it is unnecessary to say one single word.
Of the ANATOMY, by Mr. South and Mr. Le Gros Clark, and the MATERIA MEDICA, by
Dr. G. Johnson, it may be said that their names are sufficient pledge that these Treatises
are of first-rate character.
The Treatise on MEDICINE, by Dr. Robert Williams, of St. Thomas's Hospital, is an
attempt to give a more philosophical view of the classification of disease than has hitherto
been taken in any works of modern date. The work of Dr. Williams on Morbid Poisons,
and his essays read before the College of Physicians, have obtained him the highest
reputation among the members of his own profession. No person can read his treatise
without a deep interest; and the Editor is willing to believe that it will add to the fame of
its author, and invest him with the credit of having triumphed over obstacles hitherto
thought an insuperable bar to any philosophical arrangement of disease.
To W. Bowman, Esq., the Encyclopædia is indebted for an able outline of
SURGICAL PRACTICE. His qualifications for treating that subject are amply testified by his
long experience as Demonstrator at King's College, London, and by his publication on
Physiology in conjunction with Dr. Todd. This volume, the contents of which will, it is
hoped, prove interesting to all classes of readers, is closed by a comprehensive Treatise on
VETERINARY ART, by W. C. Spooner, Esq.
Before concluding this Preface, there are two subjects to which some allusion is re-
quired,—the Plates which accompany the work, and the general Index.
The Plates are for the most part the work of those two eminent engravers, Messrs.
Lowry. They speak for themselves, and require only a simple inspection to prove their
xxiv. P. R. E. F. A. C. E.
beauty and excellence, and the ample justice which the engraver has done to the subject
before him.
With regard to the Index, it is proper to observe that it was begun at an early period
in the publication of the Encyclopaedia, when it was intrusted to the Rev. J. Hindle, who,
after completing his references to the portion then published, added those which were
required for the succeeding Parts, as each appeared. The consequence is, that the Index,
instead of being a hasty work got up under the disadvantage of an overwhelming mass
of references to arrange in a short time, occupied the attention of a very competent
person for several years. It is hoped that if it does not fulfil the promise of giving a
reference to the English name of every scientific subject, it will be found to contain amply
sufficient to facilitate a reference to all that is most important and interesting.
The foregoing enumeration of the principal parts of the Encyclopædia embodies all the
observations which the Editor considers it necessary to make in recommending the work to
the patronage of the public. The exertions made by the Proprietors to procure the just
fulfilment of the high expectations formed of the work, and of the promises they had made,
as well as the perseverance with which they have conducted this important publication to
its completion, annidst the many obstacles which must necessarily arise in so extensive an
undertaking, entitle them to high consideration from that portion of the Public which is
interested in works of a sterling and substantial character. From the present position of
Literature, and the system now in fashion of publishing small and superficial works which
may be cheaply produced, and are really of no intrinsic value, it is probable that a long
period must elapse before any similar undertaking will be entered upon, from the enormous
outlay of capital it requires, and the uncertainty of remuneration which it offers. It is
hoped, therefore, that this great national work, for such it really is, may meet with that
patronage which the Proprietors feel confident it fairly and fully deserves. They feel
assured that, whether it be viewed as a whole or in its separate divisions, it embodies a mass
of information at once extensive, accurate, and scientifically arranged, which must place it in
a pre-eminent and triumphant position. Whatever its measure of success may be in a pe-
cuniary point of view, they may justly feel a high gratification in having been instrumental,
under Providence, in bringing to a successful termination a work which, whether its lite-
rary merit or the soundness of its moral and religious views be regarded, must ever be
eonsidered as an inestimable benefit to their country and a permanent ornament to its lite-
rature.
H. J. ROSE.
-- explanation. It is current amongst us as the title of various Dictionaries of Science, whose professed
GENERAL INTRODUCTION:
OR,
A PR E LIMIN A R Y TREATIS E O N M ETH OD.
Non simpliciter nil Sciri posse; sed nil nisi certo ordine certà vià Sciri posse. BAcon.
P 2
SECTION I.
ON THE PHILOSOPHICAL PRINCIPLES OF METHOD,
THE word ENCYCLOPEDIA is too familiar to Modern Literature to require, in this place, any detailed Section I.
\-V-2
Nature of
object is to furnish a compendium of Human Knowledge, whatever may be their plan. But to methodize the Work.
such a compendium has either never been attempted, or the attempt has failed, from the total disregard
of those general connecting principles, on which Method essentially depends. In presenting, therefore,
to the Public an entirely new Work, intended to be Methodically arranged, we are not insensible to the
difficulties of our undertaking; but we trust that we have found a clue to the labyrinth in those con-
siderations which we are now about to submit to the reader. -
As METHOD is thus avowed to be the principal aim and distinguishing feature of our publication, it
becomes us at the commencement, clearly to explain what we mean in this Introduction by that word;
to exhibit the Principles on which alone a correct Philosophical Method can be founded; to illustrate
those principles by their application to distinct studies and to the History of the Human Mind; and lastly
to apply them to the general concatenation of the several Arts and Sciences, and to the most perspicuous,
elegant, and useful manner of developing each particular study. Such are the objects of this Essay,
which we conceive must form a necessary Introduction to a Work, that is designated in its title from the
place whence it originates, the ENcyclopædia Metropolitana; but claims from its mode of execution
to be also called “a Methodical Compendium of Human Knowledge.”
The word METHOD (wé9930s) being of Grecian origin, first formed and applied by that acute, ingenious, The word
and accurate People, to the purposes of Scientific arrangement, it is in the Greek Language that we must Method.
seek for its primary and fundamental signification. Now, in Greek, it literally means a way, or path, of
transit. Hence the first idea of Method is a progressive transition from one step in any course to another;
and where the word Method is applied with reference to many such transitions in continuity, it necessarily
implies a Principle of UNITY witH PROGRESSION. But that which unites, and makes many things one in the
Mind of Man, must be an act of the Mind itself, a manifestation of intellect, and not a spontaneous and
uncertain production of circumstances. This act of the Mind, then, this leading thought, this “key
note” of the harmony, this “subtile, cementing, subterraneous” power, borrowing a phrase from the
b
2 INTRODUCTION.
intº nomenclature of legislation, we may not inaptly call the INITIATIVE of all Method. It is manifest, that Section.”
J.- the wider the sphere of transition is, the more comprehensive and commanding must be the initiative :
and if we would discover a universal Method by which every step in our progress through the whole
circle of Art and Science should be directed, it is absolutely necessary that we should seek it in the very
interior and central essence of the Human intellect.
** To this point we are led by mere reflection on the meaning of the word Method. We discover that
Method, it cannot, otherwise than by abuse, be applied to a dead and arbitrary arrangement, containing in itself
no Principle of progression. We discover, that there is Science of Method; and that that Science, like
all others, must necessarily have its Principles ; which it therefore becomes our duty to consider, in so
far at least as they may be necessary to the arrangement of a Methodical Encyclopædia.
† º All things, in us, and about us, are a Chaos, without Method: and so long as the mind is entirely
tions. passive, so long as there is a habitual submission of the Understanding to mere events and images, as
such, without any attempt to classify and arrange them, so long the Chaos must continue. There may be
transition, but there can never be progress; there may be sensation, but there cannot be thought: for
the total absence of Method renders thinking impracticable; as we find that partial defects of Method
proportionably render thinking a trouble and a fatigue. But so soon as the Mind becomes accustomed
to contemplate, not things only, but likewise relations of things, there is immediate need of some path or
way of transit from one to the other of the things related;—there must be some law of agreement or of
contrast between them ; there must be some mode of comparison; in short there must be Method. We
may, therefore, assert that the relations of things form the prime objects, or so to speak, the materials of
Method : and that the contemplation of those relations is the indispensable condition of thinking
Methodically. * -
Of these relations of things, we distinguish two principal kinds. One of them is the relation by which
we understand that a thing must be: the other, that by which we merely perceive that it is. The one, we
call the relation of Law, using that word in its highest and original sense, namely, that of laying down a
rule to which the subjects of the Law must necessarily conform. The other, we call the relation of
THEoRY.
Relation of
Law The relation of LAw is in its absolute perſection conceivable only of God, that Supreme Light, and
Living Law, “in whom we live and move, and have our being ;” who is #y ºrgyri, and ºrph rāy ºrdvray.
But yet the Human Mind is capable of viewing some relations of things as necessarily existent; that is to
say, as predetermined by a truth in the Mind itself, pregnant with the consequence of other truths in an
indefinite progression. Of such truths, some continue always to exist in and for the Mind alone,
forming the Pure Sciences, moral or intellectual; whilst others, though originating in the Mind, constitute
what are commonly called the great Laws of Nature, and form the groundwork of the Mixed Sciences, such
as those of Mechanics and Astronomy.
Relation of The second relation is that of THEoRY, in which the existing forms and qualities of objects, discovered
” by observation, suggest a given arrangement of them to the Mind, not merely for the purposes of more
easy remembrance and communication; but for those of understanding, and sometimes of controlling
them. The studies to which this class of relations is subservient, are more properly called Scientific Arts
than Sciences. Medicine, Chemistry, and Physiology are examples of a Method founded on this second
sort of relation, which, as well as the former, always supposes the necessary connection of cause and effect.
Fine Arts. The relations of Law and Theory have each their Methods. Between these two, lies the Method of
the FINE ARTs, a Method in which certain great truths, composing what are usually called the Laws of
Taste, necessarily predominate; but in which there are also other Laws, dependent on the external objects
of sight and sound, which these Arts embrace. To prove the comparative value and dignity of the
first relation, it will be sufficient to observe that what is called “tinkling” verse is disagreeable to the
accomplished Critic in Poetry, and that a fine Musical taste is soon dissatisfied with the Harmonica, or any
similar instrument of glass or steel, because the body of the sound, (as the Italians phrase it,) or that
ON THE SCIENCE OF METHOD, 3
Introdue effect which is derived from the materials, encroaches too far on the effect derived from the proportions section I.
—- of the notes, which proportions are, in fact, Laws of the Mind, analogous to the Laws of Arithmetic and ~~
Geometry. - - -
We have stated, that Method implies both an uniting and a progressive power. Now the relations º of
of things are not united in Human conception at random—humano capiti—cervicem equinam ; but there
is some rule, some mode of union, more or less strictly necessary. Where it is absolutely necessary, we
have called it a relation of Law; and as by Law we mean the laying down the rule, so the rule laid down
we call, in the ancient and proper sense of the word, an Idea ; and consequently the words Idea and Ideas.
Law are correlative terms, differing only as object and subject, as Being and Truth. It is extremely
necessary to advert to this use of the word Idea ; since, in Modern Philosophy, almost any and every
exercise of any and every mental faculty, has been abusively called by this name, to the utter confusion
and unmethodizing of the whole Science of the Human Mind, and indeed of all other Knowledge whatsoever.
The Idea may exist in a clear, distinct, definite form, as that of a circle in the Mind of an accurate Definite or
Geometrician; or it may be a mere instinct, a vague appetency towards something which the Mind inces- instinctive.
santly hunts for, but cannot find, like a name which has escaped our recollection, or the impulse which
fills the young Poet's eye with tears, he knows not why. In the infancy of the Human Mind, all our
ideas are instincts; and Language is happily contrived to lead us from the vague to the distinct, from the
imperfect to the full and finished form: the boy knows that his hoop is round, and this, in after years,
helps to teach him, that, in a circle, all the lines drawn from the centre to the circumference are equal.
It will be seen, in the sequel, that this distinction between the instinctive approach toward an Idea, and
the Idea itself, is of high importance in Methodizing Art and Science.
From the first, or initiative Idea, as from a seed, successive Ideas germinate. Thus, from the Idea of Principle of
a triangle, necessarily follows that of equality between the sum of its three angles and two right angles.”
This is the Principle of an indefinite, not to say infinite, progression ; but this progression, which is truly
Method, requires not only the proper choice of an initiative, but also the following it out through all its
ramifications. It requires, in short, a constant wakefulness of Mind ; so that if we wander but in a
single instance from our path, we cannot reach the goal, but by retracing our steps to the point of diver-
gency, and thence beginning our progress anew. Thus, a ship beating off and on an unknown coast,
often takes, in nautical phrase, “a new departure;” and thus it is necessary often to recur to that
regulating process, which the French Language so happily expresses by the word s'orienter, i. e. to find
out the East for ourselves, and so to put to rights our faulty reckoning.
The habit of Method should always be present and effective; but in order to render it so, a certain State of
training, or education of the Mind, is indispensably necessary. Events and images, the lively and spirit- º to
stirring machinery of the external world, are like light, and air, and moisture, to the seed of the Mind,
which would else rot and perish. In all processes of mental evolution the objects of the senses must
stimulate the Mind; and the Mind must in turn assimilate and digest the food which it thus receives from
without. Method, therefore, must result from the due mean, or balance, between our passive impressions
and the Mind’s reaction on them. So in the healthful state of the Human body, waking and sleeping,
rest and labour, reciprocally succeed other, and mutually contribute to liveliness, and activity, and
strength. There are certain stores proper, and, as it were, indigenous to the Mind, such as the Ideas of
number and figure, and the logical forms and combinations of conception or thought. The Mind that is
rich and exuberant in this intellectual wealth, is apt, like a miser, to dwell upon the vain contemplation
of its riches, is disposed to generalize and Methodize to excess, ever Philosophizing, and never descending
to action;–spreading its wings high in the air above some beloved spot, but never flying far and wide
over earth and sea, to seek food, or to enjoy the endless beauties of Nature; the fresh morning and the
warm noon, and the dewy eve. On the other hand, still less is to be expected, toward the Methodizing
of Science, from the man who flutters about in blindness, like the bat ; or is carried hither and thither,
like the turtle sleeping on the wave, and fancying, because he moves, that he is in progress.
b 2
4. INTRODUCTION.
intº The paths in which we may pursue a Methodical course are manifold: at the head of each stands its Section I
< …— peculiar and guiding Idea; and those Ideas are as regularly subordinate in dignity, as the paths to which TT
. º they point are various and eccentric in direction. The world has suffered much, in modern times, from
- a subversion of the natural and necessary order of Science: from elevating the terrestrial, as it has
been called, above the celestial; and from summoning Reason and Faith to the bar of that limited Physical
experience, to which, by the true laws of Method, they owe no obedience. The subordination, of which
we here speak, is not that which depends on immediate practical utility: for the utility of Human powers,
in their practical application, depends on the circumstances of the moment; and at one time strength is
essential to our very existence, at another time skill: and even Caesar in a fever could cry,
- Give me some drink Titinius,
As a sick girl.
In truth there is scarcely any one of the powers or faculties with which the Divine Goodness has endowed
his creatures, which may not in its turn be a source of paramount benefit and usefulness; for every thing
around us is full of blessings: nor is there any line of honest occupation in which we would dare to affirm,
that by a proper exercise of the talent committed to his charge, an individual might not justly advance
himself to highest praise. But we now allude to the subordination which necessarily arises among the
different branches of Knowledge, according to the difference of those Ideas by which they are initiated and
Gradation directed; for there is a gradation of Ideas, as of ranks in a well-ordered State, or of commands in a well-
** regulated army; and thus above all partial forms, there is one universal form of good and FAIR, the
xzxo~gyzſoy of the Platonic Philosophy. Hence the expressions of Lord Bacon, who in his great Work,
the Novum Organum, speaks so much and so often of the lumen siccum, the pure light, which from a
central focus, as it were, diffuses its rays all around, and forms a lucid sphere of Knowledge and of Truth.
Metaphy- We distinguish Ideas into those of essential property, and those of natural existence; in other words,
i. into Metaphysical and Physical Ideas. Metaphysical Ideas, or those which relate to the essence of things
as possible, are of the highest class. Thus, in accurate language, we say, the essence of a circle, not
its nature; because, in the conception of forms purely Geometrical, there is no expression or implication
of their actual existence: and our reasoning upon them is totally independent of the fact, whether any
such forms ever existed in Nature, or not. Physical Ideas are those which we mean to express, when
we speak of the nature of a thing actually existing and cognizable by our faculties, whether the thing
be material or immaterial, bodily or mental. Thus, the laws of memory, the laws of vision, the laws of
vegetation, the laws of crystallization, are all Physical Ideas, dependent for their accuracy, on the more
or less careful observation of things actually existing.
In speaking of the word Nature, however, we must distinguish its two principal uses, viz. first, that to
Nature.
which we have adverted, and according to which it signifies whatever is requisite to the reality of a thing
as existent, such as the nature of an animal or a tree, distinguished from the animal or tree itself: and
secondly, the sum total of things, as far as they are objects of our senses. In the first of these two
meanings, the word Nature conveys a Physical Idea, in the other only a material or sensible impression.
Mere ar- Even natural substances, it is true, may be classed and arranged for various purposes, in a certain
*g” order. Such mere arrangement, however, is not properly Methodical, but rather a preparation toward
Method; as the compilation of a Dictionary is a preparation for classical study.
The limits of our present Essay will not allow us to do more than briefly to touch the chief topics of
a general dissertation on Method; but enough we trust has here been said, to render intelligible the
principles on which our Methodical Encyclopædia must be constructed. We have shown that a Method,
which is at all comprehensive, must be founded on the relations of things : that those relations are of
two sorts, according as they present themselves to the Human Mind as necessary, or merely as the result
of observation. The former we have called relations of Law, the latter of Theory. Where the former
alone are in question, the Method is one of necessary connection throughout; where the latter alone,
though the connection be considered as one of cause and effect, yet the necessity is less obvious,
ON THE SCIENCE OF METHOD, 5
Introduc. and the connection itself less close. We have observed, that in the Fine Arts there is a sort of middle Section II.
J Method, inasmuch as the first and higher relations are necessary, the lower are only the results of obser-
vation. The great principles of all Method we have shown to be two, viz. Union and Progression. The
relations of things cannot be united by accident: they are united by an Idea either definite or instinctive.
Their union, in proportion as it is clear, is also progressive. The state of Mind adapted to such progress
holds a due mean between a passiveness under external impression, and an excessive activity of mere
reflection; and the progress itself follows the path of the Idea from which it sets out; requiring, however,
a constant wakefulness of Mind, to keep it within the due limits of its course. Hence the orbits of
Thought, so to speak, must differ among themselves as the initiative Ideas differ; and of these latter, the
great distinctions are into Physical and Metaphysical. Such, briefly, are the views by which we have
been guided, in our present attempt to Methodize the great mass of Human Knowledge.
SECTION II.
ILLUSTRATION OF THE PRECEDING PRINCIPLES.
THE Principles which have been exhibited in the preceding Section, and in respect to which we claim
no other merit, than that of having drawn them from the purest sources of Philosophy, ancient and modern,
are, we trust, sufficiently plain and intelligible in themselves; but as the most satisfactory mode of proving
their accuracy, we proceed to illustrate them by a consideration of some particular studies, pursuits, and
opinions; and by a reference to the general History of the Human Mind.
And first, as to the general importance of Method;—what need have we to dilate on this fertile topic?
For it is not solely in the formation of the Human Understanding, and in the constructions of Science and
Literature, that the employment of Method is indispensably necessary; but its importance is equally felt,
and equally acknowledged, in the whole business and economy of active and domestic life. From the Fº
cottager's hearth or the workshop of the artisan, to the Palace or the Arsenal, the first merit, that which ū
admits neither substitute nor equivalent, is, that every thing is in its place. Where this charm is want-
ing, every other merit either loses its name, or becomes an additional ground of accusation and regret.
Of one, by whom it is eminently possessed, we say proverbially, that he is like clockwork. The resem-
blance extends beyond the point of regularity, and yet falls short of the truth. Both do, indeed, at once
divide and announce the silent and otherwise indistinguishable lapse of time; but the man of Methodical
industry and honourable pursuits, does more; he realizes its ideal divisions, and gives a character
and individuality to its moments. If the idle are described as killing time, he may be justly said to
call it into life and moral being, while he makes it the distinct object not only of the consciousness,
but of the conscience. He organizes the hours, and gives them a soul: and to that, the very essence
of which is to fleet, and to have been, he communicates an imperishable and a spiritual nature. Of the
good and faithful servant, whose energies, thus directed, are thus Methodized, it is less truly affirmed,
that he lives in Time, than that Time lives in him. His days, months, and years, as the stops and
punctual marks in the records of duties performed, will survive the wreck of worlds, and remain extant
when Time itself shall be no more. :
Let us carry our views a step higher. What is it that first strikes us, and strikes us at once in a Conver-
man of education, and which, among educated men, so instantly distinguishes the man of superior sation.
Mind? Not always the weight or novelty of his remarks, nor always the interest of the facts which
he communicates; for the subject of conversation may chance to be trivial, and its duration to be
short. Still less can any just admiration arise from any peculiarity in his words and phrases; for
every man of practical good sense will follow, as far as the matters undér consideration will permit
him, that golden rule of Caesar—Insolens verbum, tanquam scopulum, evitare. The true cause of the
impression made on us is, that his mind is Methodical. We perceive this, in the unpremeditated and
evidently habitual arrangement of his words, flowing spontaneously and necessarily from the clearness
6 INTRODUCTION,
intº of the leading Idea; from which distinctness of mental vision, when men are fully accustomed to it, Section II.
J they obtain a habit of foreseeing at the beginning of every sentence how it is to end, and how all its
parts may be brought out in the best and most orderly succession. However irregular and desultory •
the conversation may happen to be, there is Method in the fragments. - -
tº. Let us once more take an example which must come “home to every man's business and bosom.”
Is there not a Method in the discharge of all our relative duties P And is not he the truly virtuous and
truly happy man, who seizing first and laying hold most firmly of the great first Truth, is guided by
that divine light through all the meandring and stormy courses of his existence? To him every
relation of life affords a prolific Idea of duty; by pursuing which into all its practical consequences,
he becomes a good servant or a good master, a good Subject or a good Sovereign, a good son or a good
father; a good friend, a good patriot, a good Christian, a good man --
Scientific It cannot be deemed foreign from the purposes of our Disquisition, if we are anxious, before we
discoveries.
leave this part of the subject, to attract the attention of our readers to the importance of speculative
meditation (which never will be fruitful unless it be Methodical) even to the worldly interests of
mankind. We can recall no incident of Human History that impresses the imagination more deeply
than the moment, when Columbus, on an unknown ocean, first perceived that startling fact, the
change of the magnetic needle ! How many such instances occur in History, where the Ideas of Nature
(presented to chosen minds by a Higher Power than Nature herself) suddenly unfold, as it were, in
prophetic succession, systematic views destined to produce the most important revolutions in the state
of Man The clear spirit of Columbus was doubtless eminently Methodical. He saw distinctly that
great leading Idea, which authorized the poor pilot to become “a promiser of kingdoms:” and he
pursued the progressive developement of the mighty truth with an unyielding firmness, which taught
him to “rejoice in lofty labours.” Our readers will perhaps excuse us for quoting, as illustrative of
what we have here observed, some lines from an Ode of Chiabrera which in strength of thought and
lofty majesty of Poetry, has but “few peers in ancient or in modern Song.”
CoLUMBUs.
Certo, dal cor, ch' alto Destin nom scelse,
Son l'imprese magnanime neglette ;
Ma le bell' alme alle bell' opre elette;
Sanno gioir nelle fatiche eccelse:
Ne biasmo popolar, frale catena,
Spirto d'onore il suo cammin raffrena.
Cosi lunga stagion per modi indegni
Europa disprezzó l'inclita speme:
Schermendo il vulgo (e seco i Regi insieme,
Nudo nocchier promettitor di Regni ;
Ma per le sconosciute onde marine
L'invitta prora ei pur sospinse al fine.
Qual wom, che torni al gentil consorte,
Tal ei da sua magion spiegó l'antenne
L'ocean corse, e i turbini sostenne
Vinse le crude imagini di morte ;
Poscia, dell' ampio mar spenta la guerra,
Scorse la diamzi favolosa Terra.
Allor dal cavo Pin scende veloce
E di grand Orma il nuovo mondo imprime;
Né men ratto per l'Aria erge sublime,
Segno del Ciel, insuperabil Croce;
E porse umile esempio, onde adorarla
Debba sua Gente. CHIABRERA, vol. i.
ON THE SCIENCE OF METHOD, 7
|
Introduc- We do not mean to rest our argument on the general utility or importance of Method. Every Science Section II.
J and every Art attests the value of the particular Principles on which we have above insisted. In Mathe- Nº
matics they will, doubtless, be readily admitted; and certainly there are many marked differences between ties and
Mathematical and Physical studies; but in both a previous act and conception of the Mind, or what we’”
have called an initiative, is indispensably necessary, even to the mere semblance of Method. In Mathe-
matics, the definition makes the object, and pre-establishes the terms, which alone can occur in the after
reasoning. If an existing circle, or what is supposed to be such, be found not to have the radii from the
centre to the circumference perfectly equal: it will in no manner affect the Mathematician's reasoning on
the properties of circles; it will only prove that the figure in question is not a circle according to the
previous definition. A Mathematical Idea, therefore, may be perfect. But the place of a perfect Idea
cannot be exactly supplied, in the Sciences of experiment and observation, by any theory built on gene-
ralization. For what shall determine the Mind to one point rather than another; within what limits, and
from what number of individuals, shall the generalization be made 2 The theory must still require a
prior theory for its own legitimate construction. The Physical definition follows and does not precede
the reasoning. It is representative, not constitutive, and is indeed little more than an abbreviature of
the preceding observation, and the deductions therefrom. But as the observation, though aided by
experiment, is necessarily limited and imperfect, the definition must be equally so. The history of
theories, and the frequency of their subversion by the discovery of a single new fact, supply the best
illustrations of this truth. - - -
But in Experimental Philosophy, it may be said, how much do we not owe to accident? Doubtless: Electricity.
but let it not be forgotten, that if the discoveries so made stop there; if they do not excite some
master IDEA ; if they do not lead to some LAw; (in whatever dress of theory or hypothesis the
fashions and prejudices of the time may disguise or disfigure it;) the discoveries may remain
for Ages limited in their uses, insecure and unproductive. How many centuries have passed
since the first accidental discovery of the attraction and repulsion of light bodies by rubbed amber, &c.
Compare the interval with the progress made within less than a century, after the discovery of the
phenomena that led immediately to a theory of ELECTRICITY. That here, as in many other instances,
the theory was supported by insecure hypotheses; that by one theorist two heterogeneous fluids were
assumed, the vitreous and the resinous; by another, a plus and minus of the same fluid; that a third
considered it a mere modification of light; while a fourth composed the electrical aura of oxygen,
hydrogen, and caloric: all this does but place the truth we have been insisting on in a stronger and
clearer light. For, abstract from all these suppositions, or rather imaginations, that which is common
to, and involved in them all; and there will remain neither notional fluid or fluids; nor chemical com-
pounds, nor elementary matter, but the Idea of two—opposite—forces, tending to rest by equilibrium.
These are the sole factors of the calculus, alike in all the theories: these give the Law and with it the
Method of arranging the phenomena. For this reason it may not be rash to anticipate the nearest
approaches to a correct system of Electricity from those Philosophers, who, since the year 1798, have
presented the Idea most distinctly as such, rejecting the hypothesis of any material substratum, and
contemplating in all Electrical phenomena the operation of a Law which reigns through all Nature, viz.
the law of polarity, or the manifestation of one power by opposite forces. -
How great the contrast between Electricity and MAGNETrsm.! From the remotest antiquity, the attrac-Magnetism.
tion of iron by the magnet was known, and noticed ; but century after century it remained the undisturbed
property of Poets and Orators. The fact of the magnet, and the fable of the Phoenix, stood on the same
scale of utility, and by the generality of mankind, the latter was as much credited as the former, and
considered far more interesting. In the XIIIth century, however, or perhaps earlier, the polarity of the
magnet, and its communicability to iron, were discovered. We remain in doubt whether this discovery
were accidental, or the result of theory; if the former, the purpose which it soon suggested was so grand
and important, that it may well be deemed the proudest trophy ever yet raised by accident in the
8 - INTRODUCTION.
Introduc. service of mankind. But still it furnished no genuine Idea ; it led to no Law, and consequently, to no Section II.
—- Method; though a variety of phenomena, as startling as they are at present mysterious, have forced on TT’
us a presentiment of its intimate connection with other great agencies of Nature. We would not be
understood to assume the power of predicting to what extent, or in what directions, that connection may
hereafter be traced; but amidst the most ingenious hypotheses that have yet been formed on the subject,
we may notice that which, combining the three primary Laws of Magnetism, Electricity, and Galvanism,”
considers them all as the results of one common power, essential to all material construction in the
works of Nature. It is, perhaps, more an operation of the Fancy than of the Reason, which has suggested
that these three material powers are analogous to the three dimensions of space. Hypothesis, be it
observed, can ºver form the groundwork of a true Scientific method, unless where the hypothesis is
either a true Idea proposed in an hypothetical form, or at least the symbol of an Idea as yet unknown, of
a Law as yet undiscovered; and in this latter case the hypothesis merely performs the function of an
unknown quantity in Algebra, and is assumed for the purpose of submitting the phenomena to a Scientific
calculus. But to recur to the contrast presented by Electricity and Magnetism, in the rapid progress of
the former, and the stationary condition of the latter: What is the cause of this diversity Fewer
theories, fewer hypotheses have not been advanced in the one than in the other; but the theories and
fictions of the Electricians contained an Idea, and all the same Idea, which has necessarily led to METHOD;
implicit indeed, and only regulative hitherto, but which requires little more than the dismission of the
imagery to become constitutive, like the Ideas of the Geometrician. On the contrary, the assumptions
of the Magnetists (as for instance, the hypothesis that the earth itself is one vast magnet, or that an
immense magnet is concealed within it; or that there is a concentric globe within the earth, revolving
on its own independent axis) are but repetitions of the same fact or phenomenon, looked at through a
magnifying glass; the reiteration of the problem, not its solution. This leads to the important consider-
ation, so often dwelt upon, so forcibly urged, so powerfully amplified and explained by our great
Countryman Bacon, that one fact is often worth a thousand. Satis scimus, says he, aa.iomata recte
inventa, tota agmina operum secum trahere. Hence his indignant reprobation of the vis experimentalis,
capca, stupida, vaga, praerupta | Hence his just and earnest exhortations to pursue the experimenta
lucifera, and those alone; discarding for their sakes, even the fructifera experimenta. The Natural
Philosopher, who cannot, or will not see, that it is the “enlightening” fact, which really causes all the
others to be facts, in any Scientific sense—he who has not the head to comprehend, and the soul to
reverence this parent experiment—he to whom the Évpx2 is not an exclamation of joy and rapture, a
rich reward for years of toil and patient suffering—to him no auspicious answer will ever be granted by
the Oracle of Nature.
zoology. ... We have said that improgressive arrangement is not Method : and in proof of this we appeal to the
notorious fact, that ZooLogY, soon after the commencement of the latter half of the last century, was
falling abroad, weighed down and crushed as it were by the inordinate number and multiplicity of facts
and phenomena apparently separate, without evincing the least promise of systematizing itself by any
inward combination of its parts. John HUNTER, who had appeared, at times, almost a stranger to the
grand conception, which yet never ceased to work in him, as his genius and governing spirit, rose at
length in the horizon of Physiology and Comparative Anatomy. In his printed Works, the finest elements
of system seem evermore to flit before him, twice or thrice only to have been seized, and after a
momentary detention, to have been again suffered to escape. At length, in the astonishing preparations
for his Museum, he constructed it, for the Scientific apprehension, out of the unspoken alphabet of
Nature. Yet notwithstanding the imperfection in the annunciation of the Idea, how exhilarating have
been the results. It may, we believe, be affirmed, with safety, that whatever is grandest, in the views
* See the experiments of Coulumb, Brugmans, and Goethe. To which may be added, should they be
confirmed, the curious observations on Crystallization, first made in Corsica, and since pursued in France.
ON THE SCIENCE OF METHOD, - 9
intº of Cuvier, is either a reflection of this light, or a continuation of its rays, well and wisely directed, Section II.
*- through fit media, to its appropriate object. - : \-N-'
Botany. From Zoology, or the laws of animal life, to BotANY, or those of vegetable life, the transition is easy
and natural. In this pursuit, how striking is the necessity of a clear Idea, as initiative of all Method |
How obvious the importance of attention to the conduct of the Mind in the exercise of Method itself!
The lowest attempt at Botanical arrangement consists in an artificial classification of plants, for the pre-
paratory purpose of a nomenclature; but even in this, some antecedent must have been contributed by
the Mind itself; some purpose must be in view; or some question at least must have been proposed to
Nature, grounded, as all questions are, upon some Idea of the answer. As for instance, the assumption,
That two great sexes animate the world.
For no man can confidently conceive a fact to be universally true who does not proportionally anticipate
its necessity, and who does not believe that necessity to be demonstrable by an insight into its nature,
whenever and wherever such insight can be obtained. We acknowledge, we reverence, the obligations of
Botany to LINNEUs, who adopting from Bartholinus and others the sexuality of plants, grounded thereon
a scheme of classific and distinctive marks, by which one man's experience may be communicated to
others, and the objects safely reasoned on while absent, and recognised so soon as and wherever they
occur. He invented an universal character for the Language of Botany, chargeable with no greater
imperfections than are to be found in the alphabets of every particular Language. The first requisites in
investigating the works of Nature, as in studying the Classics, are a proper Accidence and Dictionary; and
for both of these Botany is indebted to the illustrious Swede. But the inherent necessity, the true Idea
of Sex, was never fully contemplated by Linnaeus, much less that of vegetation itself. Wanting these
master-lights, he was not only unable to discern the collateral relations of the Vegetable to the Mineral and
Animal Worlds; but even in respect to the doctrine which gives name and character to his system, he only
avoided Scylla to fall upon Charybdis ; and such must be the case of every one, who in this uncertain state
of the initiative Idea, ventures to expatiate among the subordinate notions. If we adhere to the general
notion of sex, as abstracted from the more obvious modes in which the sexual relation manifests itself,
we soon meet with whole classes of plants to which it is found inapplicable. If, arbitrarily, we give it
indefinite extension, it is dissipated into the barren truism, that all specific products suppose specific
means of production. Thus a growth and a birth are distinguished by the mere verbal definition, that the
latter is a whole in itself, the former mot: and when we would apply even this to Nature, we are baffled
by objects (the flower polypus, &c. &c.) in which each is the other. All that can be done by the most
patient and active industry, by the widest and most continuous researches; all that the amplest survey of
the Vegetable realm, brought under immediate contemplation by the most stupendous collections of species
and varieties, can suggest; all that minutest dissection and exactest Chemical analysis can unfold; all
that varied experiment and the position of plants and their component parts in every conceivable relation
to light, heat, and whatever else we distinguish as imponderable substances; to earth, air, water; to the
supposed constituents of air and water, separate and in all proportions—in short all that Chemical agents
and reagents can disclose or adduce;—all these have been brought, as conscripts, into the field, with the
completest accoutrement, in the best discipline, under the ablest commanders. Yet after all that was
effected by Linnaeus himself, not to mention the labours of Caesalpinus, Ray, Gesner, Tournefort, and
the other heroes who preceded the general adoption of the Sexual system, as the basis of artificial
arrangement—after all the successive toils and enterprises of HE Dwig, JUSS1EU, MIRBEL, SMITH,
KNIGHT, ELLIs, &c. &c.—what is BotANY at this present hour? Little more than an enormous momen-
clature; a huge catalogue, bien arrange, yearly and monthly augmented, in various editions, each with
its own scheme of technical memory and its own conveniences of reference 1 The innocent amusement,
the healthful occupation, the ornamental accomplishment of amateurs ; it has yet to expect the devotion
and energies of the Philosopher. Whether the Idea which has glanced across some Minds, that the
C
10 .* - INTRODUCTION.
Introduc. harmony between the Vegetable and Animal World is not a harmony of resemblance, but of contrast, may Section II.
~~ not lead to a new and more accurate Method in this engaging Science, it becomes us nothere to determine:
but should its objective truth be hereafter demonstrated by induction of facts in an unbroken series of
correspondences in Nature, we shall then receive it as a Law of organic existence; and shall thence obtain
another splendid proof, that with the knowledge of Law alone dwell power and prophecy, decisive experi-
ment, and Scientific Method. { - - .
Chemistry. Such, too, is the case with the substances of the LABoratoRY, which are assumed to be incapable of
decomposition. They are mere exponents of some one Law, which the Chemical Philosopher, whatever
may be his Theory, is incessantly labouring to discover. The Law, indeed, has not yet assumed the form
of an Idea in his Mind; it is what we have called an Instinct; it is a pursuit after unity of Principle,
through a diversity of forms. Thus as “ the lunatic, the lover, and the Poet,” suggest each other to
Shakspeare's Theseus, as soon as his thoughts present him the oNE ForM, of which they are but varieties;
so water and flame, the diamond, the charcoal, and the mantling champagne, with its ebullient sparkles,
are convoked and fraternized by the theory of the Chemist. This is, in truth, the first charm of Chemistry,
and the secret of the almost universal interest excited by its discoveries. The serious complacency which is
afforded by the sense of truth, utility, permanence, and progression, blends with and ennobles the exhila-
rating surprise and the pleasurable sting of curiosity, which accompany the propounding and the solving
of an enigma. It is the sense of a Principle of connection given by the Mind, and sanctioned by the
correspondency of Nature. Hence the strong hold which in all Ages Chemistry has had on the imagina-
tion. If in the greatest Poets we find Nature idealized through the creative power of a profound yet
observant meditation, so through the meditative observation of a DAvy, a Wollaston, a HATCHETT, or
a MURRAY, - -
By some connatural force,
Powerful at greatest distance to unite
With secret amity things of like kind,
we find Poetry, as it were, substantiated and realized.
This consideration leads us from the paths of Physical Science into a region apparently very different.
Poetry. Those who tread the enchanted ground of PoETRY, oftentimes do not even suspect that there is such a
thing as Method to guide their steps. Yet even here we undertake to show that it not only has a neces-
sary existence, but the strictest Philosophical application; and that it is founded on the very Philosophy
which has furnished us with the Principles already laid down. It may surprise some of our readers,
especially those who have been brought up in Schools of foreign taste, to find that we rest our proof of
these assertions on one single evidence, and that that evidence is SHAKs PEARE, whose Mind they have
probably been taught to consider as eminently immethodical. In the first place, Shakspeare was not
only endowed with great native genius, (which indeed he is commonly allowed to have been,) but what
is less frequently conceded, he had much acquired knowledge. “His information,”
says Professor
WILDE, “ was great and extensive, and his reading as great as his knowledge of Languages could reach.
Considering the bar which his education and circumstances placed in his way, he had done as much to
acquire knowledge as even Milton. A thousand instances might be given, of the intimate knowledge that
Shakspeare had of facts. I shall mention only one. I do not say, he gives a good account of the Salic
law, though a much worse has been given by many antiquaries. But he who reads the Archbishop of
Canterbury's speech in Henry the Fifth, and who shall afterwards say, that Shakspeare was not a man of
great reading and information, and who loved the thing itself, is a person whose opinion I would not ask
or trust upon any matter of investigation.” Then, was all this reading, all this information, all this
knowledge of our great Dramatist, a mere rudis indigestaque moles? Very far from it. Method, we have
seen, demands a knowledge of the relations which things bear to each other, or to the observer, or to the
state and apprehension of the hearers. In all and each of these was Shakspeare so deeply versed, that
in the personages of a Play, he seems “to mould his mind as some incorporeal material alternately into
ON THE SCIENCE OF METHOD, 11
ſ
Introduc. all their various forms.” In every one of his various characters we still feel ourselves communing
tion.
with the same Human Nature. Every where we find individuality: no where mere portrait. The excel-
lence of his productions consists in a happy union of the universal with the particular. But the
universal is an Idea. Shakspeare, therefore, studied mankind in the Idea of the Human race; and he
followed out that Idea into all its varieties, by a Method which never failed to guide his steps aright.
Let us appeal to him, to illustrate by example, the difference between a sterile and an exuberant Mind,
in respect to what we have ventured to call the Science of Method. On the one hand observe Mrs.
Quickly's relation of the circumstances of Sir John Falstaff's debt: ,
FALSTAFF. What is the gross sum that I owe thee?
Mrs. QUICKLY. Marry, if thou wert an honest man, thyself and the money too. Thou didst swear to me
upon a parcel-gilt goblet, sitting in my dolphin chamber, at the round table, by a sea-coal fire, on Wednesday in
Whitsun week, when the prince broke thy head for likening his father to a singing man in Windsor—thou didst
swear to me then, as I was washing thy wound, to marry me and make me my lady thy wife. Canst thou deny
it 2 Did not goodwife Keech, the Butcher's wife, come in then and call me gossip Quickly 2—coming in to
borrow a mess of vinegar: telling us she had a good dish of prawns—whereby thou didst desire to eat some—
whereby I told thee they were ill for a green wound,” &c. &c. &c.
(Henry IV. P. I. Act II. Scene I.)
On the other hand consider the narration given by Hamlet to Horatio, of the occurrences during his
proposed transportation to England, and the events that interrupted his voyage. (Act V. Scene II.)
HAM. Sir, in my heart there was a kind of fighting
That would not let me sleep: methought I lay
Worse than the mutines in the bilboes. Rashly,
And prais'd be rashness for it Let us know,
Our indiscretion sometimes serves us well,
When our deep plots do fail : and that should teach us
There's a Divinity that shapes our ends,
Rough-hew them how we will.
HoR. That is most certain.
HAM. Up from my cabin,
My sea-gown scarf'd about me, in the dark
Grop'd I to find out them; had my desire ;
Finger'd their pocket; and, in fine, withdrew
To my own room again : making so bold,
My fears forgetting manners, to unseal
Their grand commission; where I found, Horatio,
A royal knavery—an exact command, -
Larded with many several sorts of reasons,
Importing Denmark's health, and England's too,
With, ho! such bugs and goblins in my life,
That on the supervize, no leisure bated,
No, not to stay the grinding of the axe,
My head should be struck off
HoR. Is’t possible?
HAM. Here's the commission.—Read it at more leisure.
I sat me down ;
Devis’d a new commission ; wrote it fair.
-à ... → ---- -º - -- *— ” - sº-º- is " -
* 6 Tāv Šavré prüxnv čaet Wav táva doºgatov uopºpa's rouci Ma’s poppugas.
- - THEMISTIUS.
Section II.
\-y-”
c 2
12 INTRODUCTION.
Introduc-
tion.
^-yº”
I once did hold it, as our statists do,
A baseness to write fair, and labour'd much
How to forget that learning ; but, sir, now
It did me yeoman's service. Wilt thou know
The effect of what I wrote 2
HoR. Aye, good my lord.
HAM. An earnest conjuration from the king,
As England was his faithful tributary;
As love between them, like the palm, might flourish ;
As peace should still her wheaten garland wear,
And many such like As's of great charge—
That on the view and knowing of these contents
He should the bearers put to sudden death,
No shriving time allowed.
HoR. How was this sealed ?
HAM. Why, even in that was heaven ordinant.
I had my father's signet in my purse,
Which was the model of that Danish seal:
Folded the writ up in the form of the other;
Subscribed it; gave’t the impression; plac'd it safely,
The changeling never known. Now, the next day
Was our sea-fight; and what to this was sequent
Thou knowest already.
If, overlooking the different value of the matter in these two narrations, we consider only the form, it
must be confessed, that both are Immethodical. We have asserted that Method results from a balance
between the passive impression received from outward things, and the internal activity of the Mind in
reflecting and generalizing; but neither Hamlet nor the Hostess hold this balance accurately. In Mrs.
Quickly, the memory alone is called into action, the objects and events recur in the narration in the
same order, and with the same accompaniments, however accidental or impertinent, as they had first
occurred to the narrator. The necessity of taking breath, the efforts of recollection, and the abrupt
rectification of its failures, produce all her pauses; and constitute most of her connections. But when
we look to the Prince of Denmark's recital the case is widely different. Here the events, with the
circumstances of time and place, are all stated with equal compression and rapidity; not one introduced
which could have been omitted without injury to the intelligibility of the whole process. If any tendency
is discoverable, as far as the mere facts are in question, it is to omission: and accordingly, the reader
will observe, that the attention of the narrator is called back to one material circumstance, which he was
hurrying by, by a direct question from the friend (How was THIS SEALED P) to whom the story is
communicated. But by a trait which is indeed peculiarly characteristic of Hamlet’s Mind, ever disposed
to generalize, and meditative to excess, all the digressions and enlargements consist of reflections, truths,
and Principles of general and permanent interest, either directly expressed or disguised in playful satire.
Instances of the want of generalization are of no rare occurrence: and the narration of Shakspeare's
Hostess differs from those of the ignorant and unthinking in ordinary life, only by its superior humour,
the Poet's own gift and infusion, not by its want of Method, which is not greater than we often meet
with in that class of Minds of which she is the dramatic representative. Nor will the excess of generali-
zation and reflection have escaped our observation in real life, though the great Poethas more conveniently
supplied the illustrations. In attending too exclusively to the relations which the past or passing events
and objects bear to general truth, and the moods of his own Mind, the most intelligent man is sometimes
in danger of overlooking that other relation, in which they are likewise to be placed to the apprehension
ON THE SCIENCE OF METHOD, 13
Introduc. and sympathies of his hearers. His discourse appears like soliloquy intermixed with dialogue. But the Section II.
—- uneducated and unreflecting talker overlooks all mental relations, and consequently precludes all ~~
- Method, that is not purely accidental. Hence,—the nearer the things and incidents in time and place,
the more distant, disjointed and impertinent to each other, and to any common purpose, will they appear
in his narration: and this from the absence of any leading thought in the narrator's own mind. On the
contrary, where the habit of Method is present and effective, things the most remote and diverse in time,
place, and outward circumstance, are brought into mental contiguity and succession, the more striking
as the less expected. But while we would impress the necessity of this habit, the illustrations adduced
give proof that in undue preponderance, and when the prerogative of the Mind is stretched into
despotism, the discourse may degenerate into the wayward, or the fantastical.
Shakspeare needed not to read Horace in order to give his characters that Methodical Unity which the
wise Roman so strongly recommends:
Si quid inexpertum scenae committis, et audes
Personam formare novam ; servetur ad imum
Qualis ab incaepto processerit, et sibi constet.
But this was not the only way in which he followed an accurate Philosophic Method: we quote the
expressions of ScHLEGEL, a foreign critic of great and deserved reputation:—“If Shakspeare deserves our
admiration for his characters, he is equally deserving of it for his exhibition of Passion, taking this word
in its widest signification, as including every mental condition, every tone, from indifference or
familiar mirth to the wildest rage and despair. He gives us the history of minds: he lays open
to us, in a single word, a whole series of preceding conditions.” This last is a profound and exquisite
remark: and it necessarily implies, that Shakspeare contemplated Ideas, in which alone are involved
conditions and consequences ad infinitum. Purblind critics, whose mental vision could not reach far
enough to comprise the whole dimensions of our Poetical Hercules, have busied themselves in measuring
and spanning him muscle by muscle, till they fancied they had discovered some disproportion. There
are two answers applicable to most of such remarks. First, that Shakspeare understood the true
language and external workings of Passion better than his critics. He had a higher, and a more Ideal,
and consequently a more Methodical sense of harmony than they. A very slight knowledge of Music will
enable any one to detect discords in the exquisite harmonies of HAYDN or MozART; and Bentley has
found more false grammar in the PARADISE Lost than ever poor boy was whipped for through all the
forms of Eton or Westminster; but to know why the minor note is introduced into the major key, or the
nominative case left to seek for its verb, requires an acquaintance with some preliminary steps of the
Methodical scale, at the top of which sits the author, and at the bottom the critic. The second answer is,
that Shakspeare was pursuing two Methods at once; and besides the Psychological” Method, he had also to
attend to the Poetical. Now the Poetical Method requires above all things a preponderance of pleasurable
feeling : and where the interest of the events and characters and passions is too strong to be continuous
without becoming painful, there Poetical Method requires that there should be, what Schlegel calls “a
musical alleviation of our sympathy.” The Lydian mode must temper the Dorian. This we call Method.
We said that Shakspeare pursued two Methods. Oh he pursued many, many more—“both oar and
sail”—and the guidance of the helm, and the heaving of the lead, and the watchful observation of the
stars, and the thunder of his grand artillery. What shall we say of his Moral conceptions P Not made
up of miserable clap-traps, and the tag-ends of mawkish Novels, and endless sermonizing;-but furnishing
lessons of profound meditation to frail and fallible Human Nature. He shows us Crime and Want of
* We beg pardon for the use of this insolens verbum; but it is one of which our Language stands in great need.
We have no single term to express the Philosophy of the Human Mind: and what is worse, the Principles of that
Philosophy are commonly called Metaphysical, a word of very different meaning.
14 INTRODUCTION.
Introduc. Principle clothed not with a spurious greatness of soul; but with a force of intellect which too often section II
S-- imposes but the more easily on the weak, misjudging multitude. He shows us the innocent mind of *N**
Othello plunged by its own unsuspecting and therefore unwatchful confidence, into guilt and misèry not to
be endured. Look at Lear, look at Richard, look in short at every Moral picture of this mighty Moralist!
Whoso does not rise from their attentive perusal “a sadder and a wiser man”—let him never dream that
he knows any thing of Philosophical Method. g . . .
Nay, even in his style, how Methodical is our “sweet Shakspeare.” Sweetness is indeed its predomi-
nant characteristic; and it has a few immethodical luxuriances of wit; and he may occasionally be
convicted of words, which convey a volume of thought, when the business of the scene did not absolutely
require such deep meditation. But pardoning him these dulcia vitia, who ever fashioned the English
Language, or any Language, ancient or modern, into such variety of appropriate apparel, from “the
gorgeous pall of scepter'd tragedy,” to the easy dress of flowing pastoral? *
More musical than lark to shepherd's ear,
When wheat is green and hawthorn buds appear.
Who, like him, could so Methodically suit the very flow and tone of discourse to characters lying so widely
apart in rank, and habits, and peculiarities, as Holofernes and Queen Catharine, Falstaff and Lear?
When we compare the pure English style of Shakspeare with that of the very best writers of his day, we
stand astonished at the Method by which he was directed in the choice of those words and idioms, which
are as fresh now as in their first bloom; nay, which are at the present moment at once more emergetic,
‘more expressive, more natural, and more elegant, than those of the happiest and most admired living
speakers or writers. . y - - 4 . . . . . - * x
But Shakspeare was “not Methodical in the structure of his Fable.” Oh gentle critic be advised.
Do not trust too much to your professional dexterity in the use of the scalping knife and tomahawk.
Weapons of diviner mould are wielded by your adversary: and you are meeting him here on his own
peculiar ground, the ground of Idea, of Thought, and of inspiration. The very point of this dispute is Ideal.
The question is one of Unity ; and Unity, as we have shown, is wholly the subject of Ideal law. There
are said to be three great Unities which Shakspeare has violated; those of Time, Place, and Action. Now
the Unities of Time and Place we will not dispute about. Be ours the Poet,
* qui pectus inaniter angit
Irritat, mulcet, falsis terroribus implet
Ut magus, et modo me Thebis, modo ponit Athenis.
The Dramatist who circumscribes himself within that Unity of Time which is regulated by a stop-watch,
may be exact, but is not Methodical; or his Method is of the least and lowest class. But
Where is he living clipt in with the sea, -
That chides the banks of England, Wales, or Scotland,
who can transpose the scenes of Macbeth, and make the seated heart knock at the ribs with the same
force as now it does, when the mysterious tale is conducted from the open heath, on which the Weird
Sisters are ushered in with thunder and lightning, to the fated fight of Dunsinane, in which their victim
expiates with life, his credulity and his ambition P. To the disgrace of the English Stage, such attempts
have indeed been made on almost all the Dramas of Shakspeare. Scarcely a season passes which does
not produce some to repoy ºpérepoy of this kind in which the mangled limbs of our great Poet are thrown
together “in most admired disorder.” There was once a noble Author, who, by a refined species of
murder, cut up the play of Julius Caesar into two good set Tragedies. Voltaire, we believe, had the
grace to make but one of it; but whether his Brutus be an improvement on the model from which it
was taken, we trust, after what we have already said, we shall hardly be expected to discuss. -
ON THE SCIENCE OF METHOD. 15
* Thus we have seen, that Shakspeare's mind, rich in stores of acquired knowledge, commanded all Section II.
~~ these stores and rendered them disposable, by means of his intimate, acquaintance with the great Laws of \-N-7
Thought, which form and regulate Method. We have seen him exemplifying the opposite faults of
Method in two different characters; we have seen that he was himself Methodical in the delineation of
character, in the display of Passion, in the conceptions of Moral Being, in the adaptations of Language,
in the connection and admirable intertexture of his ever-interesting Fable. Let it not, after this, be said,
that Poetry—and under the word Poetry we will now take leave to include all the Works of the higher
Imagination, whether operating by measured sound, or by the harmonies of form and colour, or by
words, the more immediate and universal representatives of Thought—is not strictly Methodical; nay, does
not owe its whole charm, and all its beauty, and all its power, to the Philosophical Principles of Method.
But what of Philosophy herself? Shall she be exempted from the Laws, which she has imposed on all Philosophy
the rest of the known Universe 2 Longé absit ' To Philosophy properly belongs the EDUCATION of the
Mind: and all that we have hitherto said may be regarded as an indication (we have room for no more)
of the chief Laws and regulative Principles of that education. Philosophy, the “Parent of Life,”
according to the expression of the wise Roman Orator; the “Mother of Good Deeds and of Good
Sayings,” the “Medicine of the Mind,” is herself wholly conversant with Method. g
True it is that the Ancients, as well as the Moderns, had their machinery for the extemporaneous
coinage of intellect, by means of which the scholar was enabled to make a figure on any and all subjects,
on any and all occasions. They too had their glittering vapours, which (as the Comic Poet tells us) fed
a host of Sophists— -
- peºd) at 0sal övěpáatv ćpºofs,
Atwep Yvºum v kai čudaeśw kai vo9, juſv trapéxoval,
Ka; tepareiav, kal repòefw, cat kpoãatv, cat cardxny-tv.
APIXTOq). Nep, 316.
Great goddesses are they to lazy folks,
Who pour down on us gifts of fluent speech,
Sense most sententious, wonderful fine effect,
And how to talk about it and about it, -
Thoughts brisk as bees, and pathos soft and thawing.
But the Philosophers held a course very different from that of the Sophists. We shall not trouble our
readers with a comparative view of many Systems; but we shall present to their admiration one mighty
Ancient, and one illustrious Modern, PLATO and BAcon. These two varieties will sufficiently exemplify
the species.
Of PLATO's Works, the larger and more valuable portion have all one common end, which compre-Plato.
hends and shines through the particular purpose of each several Dialogue; and this is, to establish the
sources, to evolve the Principles, and to exemplify the Art of METHOD. This is the clue, without which
it would be difficult to exculpate the noblest productions of the “Divine” Philosopher from the charge
of being tortuous and labyrinthine in their progress, and unsatisfactory in their ostensible results. The
latter indeed appear not seldom to have been drawn, for the purpose of starting a new problem, rather
than of solving the one proposed as the subject of previous discussion. But with the clear insight, that
the purpose of the writer is not so much to establish any particular truth, as to remove the obstacles, the
continuance of which is preclusive of all truth, the whole scheme assumes a different aspect, and justifies
itself in all its dimensions. We see that the EDUCATION of the Intellect, by awakening the Method of
self-developement, was his proposed object, not any specific information that can be conveyed into it
from without. He desired not to assist in storing the passive Mind, with the various sorts of knowledge
most in request, as if the Human, Soul were a mere repository, or banqueting room, but to place it in
such relations of circumstance as should gradually excite its vegetating and germinating powers to
[6 INTRODUCTION.
Introduc. produce new fruits of Thought, new conceptions, and imaginations, and Ideas. Plato was a Poetic Section II.
—- Philosopher, as Shakspeare was a Philosophic Poet. In the Poetry, as well as in the Philosophy, of
both, there was a necessary predominance of Ideas; but this did not make them regardless of the actual
existences around them. They were not visionaries, nor mystics; but dwelt in “the sober certainty” of
waking knowledge. It is strange, yet characteristic of the spirit that was at work during the latter half
of the last century, that the writings of PLATO should be accused of estranging the Mind from plain
experience and substantial matter-of-fact, and of debauching it by fictions and generalities. Plato,
whose Method is inductive throughout, who argues on all subjects not only from, but in and by, inductions
of facts who warns us indeed against the usurpation of the Senses, but far oftener, and with more
unmitigated hostility, pursues the assumptions, abstractions, generalities, and verbal legerdemain of the
Sophists. Strange but still more strange, that a notion, so groundless, should be entitled to plead in its
behalf the authority of Lord BAcon, whose scheme of Logic, as applied to the contemplation of Nature,
is Platonic throughout! It is necessary that we should explain this circumstance at some length, in order
to establish, by the concurrence of authorities, vulgarly supposed to be contradictory, the truth of a
System which we have already maintained on so many other grounds. -
What Lord Bacon was to England, Cicero was to Rome—the first and most eloquent advocate of
Philosophy. It is needless to remind the classical scholar of that almost Religious veneration with which
the accomplished Roman speaks of Plato, whom, indeed, he calls, in one instance, deus ille noster,
and in other places, “the Homer of Philosophers;” their “Prince;” the “most weighty of all who
ever spoke, or ever wrote;” “most wise, most holy, divine.” This last appellation, too, it is well known,
long remained, even among Christians, as a distinguishing epithet of the great ornament of the Socratic
School. Why Bacon should have spoken detractingly of such a man; why he should have stigmatized
him with the name of “Sophist,” and described his Philosophy, (with the tyrant Dionysius,) as verba
otiosorum senum ad imperitos juvenes, it is much easier to explain than to justify, or even to palliate.
He was, perhaps, influenced, in part, by the tone given to thinking Minds by the Reformation; the
founders and fathers of which saw in the Aristotelians, or Schoolmen, the antagonists of Protestantism,
and in the Italian Platonists (as they conceived) the secret enemies of Christianity itself. In part, too,
Bacon may have formed his notions of Plato's doctrines from the absurdities of his mis-interpreters,
rather than from an unprejudiced and diligent study of his Works.-Be it remembered, however, that
this unfairness was not less manifested to his contemporaries; that his treatment of GILBERT was cold,
invidious, and unjust; and that he seems to have disdained to learn either the existence or the name of
Shakspeare. At this conduct no one can be surprised, who has studied the life of this
wisest, brightest, meanest of mankind.
But our present business is not with his weaknesses, or his failings, but with those Philosophical Principles,
which, especially as displayed in the Novum Organum, have deservedly obtained for him the veneration
of succeeding Ages.
Those who talk superficially about Bacon's Philosophy, that is to say, nineteen-twentieths of those
who talk about it at all, know little more than his induction, and the application which he makes of his
own Method, to particular classes of Physical facts; applications, which are at least as crude, for the Age
of Gilbert, Galileo, and Kepler, as were those of Aristotle (whom he so superciliously reprehends) for the
Age of Philip and Alexander. Or they may perhaps have been struck with his recommendation of
tabular collections of particulars; and hence have placed him at the head of a Body of men, but too
numerous in modern days—the Minute Philosophers. We need scarcely say, that this is venturing his
reputation on a very tottering basis. Let any unprejudiced Naturalist turn to Bacon's questions and
proposals for the investigation of single problems; to his “Discourse on the Winds;” or to what may
almost be called a caricature of his scheme, in the “Method of improving Natural Philosophy,” by
3 5
ON THE SCIENCE OF METHOD, 17
intº Robert Hooke” (the history of whose Philosophical life is alone a sufficient answer to all such schemes) Section II
101l,
w —and then let him fairly say whether any desirable end could reasonably be hoped for, from this
process—whether by this mode of research any important discovery ever was made, or ever could be
made P Bacon, indeed, always takes care to tell us, that the sole purpose and object of collecting
together these particulars, is to concentrate them, by careful selection, into universals: but so immense
is their number, and so various and almost endless the relations in which each is to be separately
considered, that the life of an antediluvian Patriarch would be expended, and his strength and spirits
wasted, long before he could commence the process of simplification, or arrive in sight of the Law, which
was to reward the toils of the over-tasked PsychE.f .
Had Bacon done no more, than propose these impracticable projects, we should have been far from
sharing the sentiments of respect every where attached to his Philosophical character. But he has
performed a task of infinitely greater importance, by constructing that Methodical System, which is so
elegantly developed in the Movum Organum. It is this, which we propose to compare with the Principles
long before enunciated by Plato. In both cases, the inductions are frequently as crude and erroneous, as
might readily be anticipated from the infant state of Natural History, Chemistry, and Physiology, in their
several Ages. In both cases, the proposed applications are often impracticable; but setting aside these
considerations, and extracting from each writer that which constitutes his true Philosophy, we shall be
convinced that it is identical, in regard to the Science of Method, and to the grounds and conditions of
that Science. We do not see, therefore, how we can more appropriately conclude this section of our
inquiry, than by a brief statement of our renowned Countryman's own Principles of Method, conveyed, for
the greater part, in his own words: or in what more precise form we can recapitulate the substance of the
doctrines asserted and vindicated in the preceding pages. For we rest our strongest pretensions to appro-
bation on the fact, that we have only re-proclaimed the coinciding precepts of the Athenian Verulam, and
the British Plato. - -
In the first instance, Lord Bacon, equally with ourselves, demands, as the motive and guide of every Their com-
Philosophical experiment, what we have ventured to call the intellectual or mental initiative; namely, *
some well-grounded purpose, some distinct impression of the probable results, some self-consistent anti-
* We refer particularly to p. 22 to 42 of the above-mentioned Work; and we would, above all, notice the
following admirable specimen of confused and disorderly minuteness:—“The history of potters, tobacco-pipe-
makers, glaziers, glass-grinders, looking-glass-makers or foilers, spectacle-makers and optic-glass makers, makers
of counterfeit pearl and precious stones, bugle-makers, lamp-blowers, colour-makers, colour-grinders, glass-painters, .
enamellers, varnishers, colour-sellers, painters, limmers, picture-drawers, makers of baby heads, of little bowling
stones or marbles, fustian-makers, (query, whether Poets are included in this trade P) music-masters, tinsey-makers,
and taggers.-The history of schoolmasters, writing-masters, printers, book-binders, stage-players, dancing-
masters, and vaulters, apothecaries, chirurgeons, seamsters, butchers, barbers, laundresses, and cosmetics / &c. &c.
&c. &c. (the true nature of each of which being exactly determined,) will, HUGELY FACILITATE our INQUIRIES IN
PHILOSOPHY | | |''
In parallel, or rather in contrast, with the advice of Mr. Robert Hooke, may be fairly placed that of the celebrated
Dr. WATTs, which was thought by Dr. Knox to be worthy of insertion in the Elegant Extracts, vol. ii. p. 456,
under the head of - 4
DIRECTIONs conceRNING our IDEAs.
“Furnish yourselves with a rich variety of Ideas. Acquaint yourselves with things ancient and modern, things
Natural, Civil, and Religious ; things of your native land, and of foreign countries; things domestic and national ;
things present, past, and future; and above all, be well acquainted with God and yourselves; with animal nature,
and the workings of your own spirits. Such a general acquaintance with things will be of very great advantage.”
f See the beautiful allegoric tale of Cupid and Psyche in the original of Apuleius. The tasks imposed on the
hapless nymph, through the jealousy of her mother-in-law, and the agency by which they are at length self-per-
formed, are noble instances of that hidden wisdom “where more is meant than meets the ear !”
d
18 ... INTRODUCTION.
Introduc- cipation, the ground of the prudens quaestio, (the forethoughtful inquiry,) which he affirms to be the Section II.
-- prior half of the knowledge sought, dimidium scientiae. With him, therefore, as with us, an Idea is an TT'
experiment proposed, an experiment is an Idea realized. For so he himself informs us:—Weque sci-
entiam molimur tam sensu, vel instrumentis, quam experimentis; eienim experimentorum longé major
est subtilitas, quam sensus ipsius, licet instrumentis exquisitis adjuti. Nam de iis loquimur experimentis,
quae, ad intentionem ejus quod quaeritur, perité, et secundum artem excogitata et apposita sunt. Itaque
perceptioni sensus immediatae et propriae non multúm tribuimus: seded rem deducimus, ut sensus tan-
tüm de experimento, experimentum de re judicet. The meaning of this last sentence is intelligible
enough; though involved in antithesis, merely because Bacon did not possess, like Shakspeare, a good
Method in his style. What he means to say is, that we can apprehend, through the organs of sense,
only the sensible phenomena produced by the experiment; but by the mental power, in virtue of which
we shaped the experiment, we can determine the true import of the phenomena.
Now, he had before said, that he was speaking only of those experiments, which were skilfully adapted
to the intention, or purpose of him, who conducted the research. But what is it that forms the intention,
or purpose, and adapts thereto the experiment? What Bacon calls lua intellectiás ; viz. the Understand-
ing of the individual man, who makes the experiment. This light, however, as he argues at great length,
is obscured by Idols, which are false and spurious notions. His peculiar use of the word Idols, is again
a proof of faulty Method in his style; for it gives a sort of pedantic air to his reasonings; but in truth,
he means no more by it, than what Plato means by Opinion, (86%2,) which the latter calls “a medium
between knowledge and ignorance.” So, Bacon distinguishes the Idols of the Mind into various kinds,
(Idola spectis, tribits, fori, theatri,) that is, Opinions derived from the passions, prejudices, and peculiar
habits of each man's Understanding: and as these Idols, or Opinions, confessedly produce a sort of mental
obscurity, or blindness; so, the ancient and the modern master of Philosophy both agree in prescribing
and to restore
remedies and operations calculated to remove this disease; to couch the “Mind's eye;’
it to the enjoyment of a purer vision. Bacon establishes an unerring criterion between the Ideas and the
Idols of the Mind; namely, that the latter are empty notions, but the former are the very seals and im-
presses of Nature; that is to say, they always fit and cohere with those classes of things to which they
belong; as the Idea of a circle fits and coheres with all true circles. His words are these: Non leve
quiddam interest inter humanae Mentis Idola, et divinae Mentis Ideas, hoc est, inter placita quaedam
inania, et veras signaturas atque impressiones factas in creaturis, prout ratione sand et sicci luminis,
quam, docendi causa, interpretem Naturae vocare consuevimus, inveniumtur. Novum Organum, xxiii.
and xxvi. . &
Some Idols, says Bacon, are adventitious to the Mind; others innate. And here, we may observe,
that he goes somewhat further than the mere doctrine of innate Ideas, by holding that of innate Idols.
However, we say not this in disparagement of his system, which is clear and correct; nor, on the other
hand, do we mean to espouse all its parts, which must be left to speak for themselves. What he means
by innate Idols, he thus illustrates:—not only do the rays of Truth, from without, fall obliquely on the
mirror of the Mind, but that mirror itself is not pure and plain; it discolours, it magnifies, it diminishes,
it distorts. Hence, he uses the words intellectus humanus, mens hominis, &c. in a sense now peculiar,
but in his day conformable to the language of the Schools, to signify not Intellect in general, or Mind in
its perfection, but the Intellect or Mind of man, weakened and corrupted, as it is, more or less, in every
individual. A necessary consequence of this corruption, is the arrogance, which leads Man to take the
forms and mechanism of his own reflective faculty, as the measure of Nature, and of the Deity. Of all
Idols, or of all Opinions, this is the most difficult to remedy, or extirpate; and therefore, in this view, the
Intellect of Man is more prone to error, than even his Senses. Such is the sound and incontrovertible
doctrine of Bacon; but herein he does no more than repeat what both Plato and Heraclitus had long
before urged, with most impressive argument. The forms of the reflective faculty are subjective; the
truths to be embraced are objective: but according to Plato, as well as to Bacon, there can be no hope
ON THE SCIENCE OF METHOD. 19
Introduc- of any fruitful and secure Method, so long as forms, merely subjective, are arbitrarily assumed to be the Section II.
" , moulds of objective Truth, the seals and impresses of Nature. - . S---
What then I Does Bacon abandon the hope of rectifying the obliquities of the Human Intellect ; or
does he suggest, that they will be remedied by the casual operation of external impressions P. Neither.
of these. He considers, that its weaknesses and imperfections require to be strengthened and made
perfect by a higher power; and that this is possible to be done. He supposes, that the Intellect of the
individual, or homme particulier, may be refined by the Intellect of the Ideal Man, or homme général.
He assumes, that as the evidence of the Senses is corrected by the Judgment, so the evidence of the
Judgment, beset with Idols, may be corrected by the Judgment, walking in the light of Ideas. It is surely
superfluous to urge, that this corrector and purifier of all reasoning, this inextinguishable Pole star—
Which never in the ocean waves was wet :
whether it be called, as by Bacon, lumen siccum, or as by Plato, vows, or pås wbspoy, is one and the same
light of Truth, the indispensable condition of all pure Science, contemplative or experimental. Hence,
it will not surprise us, that Plato so often denominates Ideas living Laws, in and by which the Mind has
its whole true being and permanence; or that Bacon, vice versá, names the Laws of Nature, Ideas; and
represents the great leading facts of Science as signatures, impressions, and symbols of those Ideas. A
distinguishable power self-affirmed, and seen in its unity with the Eternal Essence, is, according to
Plato, an IDEA : and the discipline by which the Human Mind is purified from its Idols, and raised to
the contemplation of Ideas, and thence to the secure and progressive investigation of Truth and reality,
by Scientific Method, comprehends what the same Philosopher so highly extols, under the title of Dia-
lectic. According to Lord Bacon, as describing the same Truth, applied to Natural Philosophy, an Idea
would be defined as—Intuitio, sive inventio, quae in perceptione sensus non est (ut quae pura et sicci
luminis Intellectioni sit propria) Idearum Divinae Mentis, prout in creaturis, per signaturas suas, sese
patefaciant. “That,” saith the judicious HookER, “which doth assign to each thing the kind, that
which determineth the force and power, that which doth appoint the form and measure of working, the
same we term a LAw,”
From all that has been said, it seems clear, that the only difference between Plato and Bacon was,
that, to speak in popular language, the one more especially cultivated Natural Philosophy, the other
Metaphysics. Plato treated principally of Truth, as manifested in the world of Intellect; Bacon of the
same Truth, as manifested in the world of Sense; but far from disagreeing, as to the mode of attaining
that Truth, far from differing in their great views of the education of the Mind, they both proceeded on
the same principles of unity and progression; and consequently both cultivated alike the Science of
Method, such as we have here described it. If we are correct in these statements, then may we boast
to have solved the great problem of conciliating ancient and modern Philosophy.
That the Method, of which we have hitherto treated, is not arbitrarily assumed in any, or all of the Historical
pursuits, to which we have adverted; nor is peculiar to these in particular, but is founded in the Laws ".
and necessary conditions of Human existence, is further to be inferred from a general view of the History
of the Human race. As in the individual, so in the whole community of Mankind, our cogitations have
an infancy of aimless activity; and a youth of education and advance towards order; and an opening
manhood, of high hopes and expectations; and a settled, staid, and sober middle age, of ripened and
deliberate judgment.
“The antiquity of time was the youth of the world and of knowledge,” said Bacon. In that early age, First period
the obedience of the will was first taught to Man. He was required to look up, in submission, to that
Spirit of Truth, which, after all, we find to be at the head of wisdom. This innocent age was happily
prolonged, among those, whose first care was to cultivate the Moral sense, and to seek in Faith the evi-
dence of things not seen. To them were propounded a Spiritual Creator, and a Spiritual worship, and
the assured hope of a future and Spiritual existence; and therefore they were less curious to watch the
d 2
20 - INTRODUCTION.
Introduc. motions of the stars, or to become “artificers in brass and iron,” or to “handle the harp and the Section II
—- organ.” They were less wise in their generation, than the “mighty men of old, the men of renown ;” \-N-7
but their Ideas were plain, and distinct; they were “just and perfect men;” and they “walked with
God;” whilst, of the others, “every imagination of the thoughts of the heart was only evil continually.”
For the latter wilfully chose an opposite Method: they determined to shape their convictions and deduce
their knowledge from without, by exclusive observation of outward things, as the only realities. Hence
they became rapidly civilized. They built cities, and refined on the means of sensual gratification, and
the conveniences of courtly intercourse. They became the great masters of the agreeable, which fra-
ternized readily with cruelty and rapacity: these being, indeed, but alternate moods of the same sensual
selfishness. Thus, both before and after the flood, the vicious of Mankind receded from all true cultiva-
tion, as they hurried towards civilization. Finally, as it was not in their power to make themselves wholly
beasts, and to remain without a semblance of Religion, and yet, as they were faithful to their original
maxim, determined to receive nothing as true, but what they derived, or believed themselves to derive
from their senses, or (in modern phrase) what they could prove à posteriori, they became Idolaters of
the Heavens, and of the material elements; and finally, out of the Idols of the Mind, they formed mate-
rial Idols: and bowed down to stocks and stones, as to the unformed and incorporeal Divinity.
Second A new era next appeared, representative of the youth and approaching manhood of the Human Intel-
period. lect: and again Providence, as it were, awakened men to the pursuit of an Idealized Method, in the
developement of their faculties. Orpheus, Linus, Musæus, and the other Mythological Bards, or perhaps
Brotherhoods of Bards impersonated under individual names, whether deriving their light, imperfectly and
indirectly, from the inspired writings of the Hebrews, or graciously visited, for high and important pur-
poses, by a dawning of Truth in their own breasts, began to spiritualize Polytheism, and thereby to prevent
it from producing all its natural, barbarizing effects. Hence the Mysteries and Mythological Hymns;
which, on the one hand, gradually shaped themselves into Epic Poetry and History, and, on the other,
into Tragedy and Philosophy: whilst to the lifeless Statuary of the Egyptians was superadded a Prome-
thean animation ; and the Ideal in Sculpture soon extending itself to Painting, and to Architecture, the
Fine Arts at once shot up to perfection, by a Method founded wholly on a mental initiative, and con-
ducted throughout its progress by the developement of Ideas. This rapid advance, in all things
which owe their existence and character to the Mind's own acts, intellectual or imaginative, forms a
singular contrast with the rude and imperfect manner in which those acts were applied to the investiga-
tion of Physical Laws and phenomena. While Phidias, Apelles, Homer, Demosthenes, Thucydides, and
Plato, had, each in his individual sphere, attained almost the summit of conceivable excellence, the
Natural History and the Natural Philosophy of the whole World may be said to have lain dormant;
especially if we compare them with the efforts which the Moderns made in these directions, in the very
morning of their strength.
Romans. Of the Roman era it is scarcely necessary to speak at large, inasmuch as the Romans were con-
fessedly mere imitators of the Greeks in every thing relating to Science and Art. They sustained a very
important part in the Civil, and Military, and Ecclesiastical History of Mankind ; and their devotion to
these objects was, in their own eyes, a sufficient apology for their want of originality in what they held to
be far inferior pursuits. -
Eaccudent alii spirantia mollius aera :
Credo equidem, vivos ducent de marmore vultus:
Tu regere imperio populos, Romane, memento.
Still less will it be expected, that we should devote much space to the consideration of those Dark Ages,
which brought the countless hordes of sensual Barbarians from their Northern forests to meet, in the
Southern and middle parts of Europe, the spiritualizing influence of Christianity: but one remarkable
effect of that influence we cannot suffer to pass unnoticed. We allude to the gradual abolition of
domestic slavery, in virtue of a Principle essential to Christianity, by which a person is eternally
oN THE SCIENCE OF METHop. 21
Introduc differenced from a thing; so that the Idea of a Human Being necessarily excludes the Idea of property, in ºt" ":
-— that Being. -
We come down, then, to the great period of the REFORMATION, which, regarded as an epoch in the Reforma-
education of the Human Mind, was second to none for its striking and durable effects. The defenders"
of a simple and Spiritual worship, against one which was full of outward forms and ceremonies; the
partisans of Religious liberty, against the dominion of a Visible Head over the whole Christian Church;
and generally speaking, the advocates of the Ideal and internal, against the external, or imaginative;
maintained a zealous, and in great part of Europe, a prosperous conflict. But the revolution of Thought,
and its effects on the Science of Method, were soon visible beyond the pale of the Church or the Cloister:
and the Schoolmen were attacked as warmly in their Philosophical, as they had before been in their
Ecclesiastical character. It is needless to dwell on the various attempts towards introducing into Learning
a totally new Method. That of our illustrious Countryman, BAcon, was completely successful: and we
have already shown, that it was, in truth, the completion of the Ideal System, by applying the same
Method to external Nature, which Plato had before applied to intellectual existence.
It is only in the union of these two branches of one and the same Method, that a complete and Modern
genuine Philosophy can be said to exist. To this consideration the great mind of Bacon does not seem Philosophy
to have been fully awake; and hence, not only is the general scope of his Work directed almost exclu-
sively to the contemplation of Physical Ideas; but there are occasional expressions, which seem to have
misled many of his followers into a belief, that he considered all Wisdom and all Science both to begin
and to end with the object of the senses. In this gross error are laid the foundations of the modern
French School, which has grown up into the monstrous puerilities of CoNDILLAC and CoNDORCET; men
whose names it would be absolutely ridiculous to mention in a History of Science, if their pupils did
not unhappily compensate, in number, what their Works want in common sense and intelligibility; and
if upon such Writers, the French Nation did not mainly rest its pretensions to give the law to Europe,
in matters of Science and Philosophy.
SECTION III.
APPLICATION OF THE PRINCIPLES OF METHOD TO THE GENERAL CONCATENATION AND
DEVELOPEMENT OF STUDIES. -
WE have already dwelt so much on the general importance of Method—we have recurred to it so fre-
quently—we have placed it in so many various lights, that we ought perhaps to apologize for venturing
on one more attempt to illustrate our meaning, partly in the way of simile, and partly of example. Let
us, however, imagine an unlettered African, or rude, but musing Indian, poring over an illumined
manuscript of the inspired volume ; with the vague, yet deep impression, that his fates and fortunes are,
in some unknown manner, connected with its contents. Every tint, every group of characters, has its
several dream. Say that, after long and dissatisfying toils, he begins to sort, first, the paragraphs that
appear to resemble each other; then the lines, the words; nay, that he has at length discovered, that the
whole is formed by the recurrence and - interchange of a limited number of ciphers, letters, marks and
points, which, however, in the very height and utmost perfection of his attainment, he makes twenty-
fold more numerous than they are, by classing every different form of the same character, intentional or
accidental, as a separate element. And yet the whole is without soul or substance, a talisman of super-
stition, or a mockery of Science; or is employed perhaps, at last, to feather the arrows of death, or to
shine and flutter amid the plumes of savage vanity. The poor Indian too truly represents the state of -
learned and systematic ignorance—arrangement guided by the light of no leading Idea; mere orderliness
without METHoD ! -
But see, the friendly missionary arrives | He explains to him the nature of written words, translates
22 INTRODUCTION.
Introduc. them for him into his native sounds, and thence into the thoughts of his heart: how many of these Section III.
-- thoughts are then first unfolded into consciousness, which yet the awakening disciple receives not as S-V-
aliens! Henceforward the book is unsealed for him; the depth is opened; he communes with the spirit
of the volume, as with a living oracle. The words become transparent: he sees them, as though he saw
them not; whilst he mentally devours the meaning they contain. From that moment, his former chime-
rical and useless arrangement is discarded, and the results of Method are to him life and truth. -
If some particular studies are yet confessedly deficient in the vivifying power of Method, we much fear
that the attempts to bind together the whole Body of Science have been, in certain instances, worse than
immethodical. A slight glance at the particular department of Literature which we have chosen, especially
as it has been filled on the Continent; from the memorable combination of Deistical talent in the Dic-
tionnaire Encyclopédique, to a Work, on the same principles, said to be now publishing in France, will
demonstrate, that the best interests of Mankind have suffered serious injury from this cause; that the
fountains of education may be poisoned, where the stream appears to flow on with increasing power and
smoothness; and that the insinuation of sceptical principles into Works of Science, is fraught with the
greatest danger to posterity. *.
To oppose an effectual barrier to the rage for desultory knowledge, on the one hand, and to support
that body of independent attachment to the best principles of all knowledge, which happily distinguishes
this country, on the other, the ENCYCLOPEDIA METRopolitan A has been projected.
We do not undertake, what the most gigantic efforts of Man could not achieve, a Universal Dic-
tionary of Knowledge, in the most absolute sense of the terms. But estimating the importance of our
task rather by the principles of unity and compression, than by those of variety and extent, we have
laboured to build upon what is essential, that which is obviously useful, and upon both whatever is elegant
or agreeable in Science; and this, we conceive, cannot be well and usefully effected, but by such a
Philosophical Method, as we have already indicated.
We have shown that this METHoD consists in placing one or more particular things or notions, in
subordination, either to a preconceived universal Idea, or to some lower form of the latter; some class,
order, genus, or species, each of which derives its intellectual significancy, and Scientific worth, from
being an ascending step toward the universal; from being its representative, or temporary substitute.
Without this master-thought, therefore, there can be no true Method: and according as the general
conception more or less clearly manifests itself throughout all the particulars, as their connective and
bond of unity; according as the light of the Idea is freely diffused through, and completely illumines, the
aggregate mass, the Method is more or less perfect.
The first preconception, or master-thought, on which our plan rests, is the moral origin and tendêncy
of all true Science; in other words, our great objects are to exhibit the Arts and Sciences in their Philo-
sophical harmony; to teach Philosophy in union with Morals; and to sustain Morality by Revealed
Religion. . s
There are, as we have before noticed, two sorts of relation, on the due observation of which all Method
depends. The first is that which the Ideas or Laws of the Mind bear to each other; the second, that
which they bear to the external world: on the former are built the Pure Sciences; on the latter those
which we call Mixed and Applied. -
Pure The Pure Sciences, then, represent pure acts of the Mind, and those only ; whether employed in con-
Sciences. templating the forms under which things in their first elements are necessarily viewed and treated by the
Mind; or in contemplating the substantial reality of those things. -
Formal and Hence, in the Pure Sciences, arises the known distinction of formal and real; and of the first, some
real. teach the elementary forms, which the Mind necessarily adopts in the processes of reasoning; and others,
those under which alone all particular objects can be grasped and considered by the Mind; either as
distinguishable in quantity and number, or as occupying parts of space. The real Sciences, on the other
hand, are conversant with the true nature and existence, either of the created Universe around us; or of
ON THE SCIENCE OF METHOD. 23
Introduc- the guiding Principles within us, in their various modifications and distinguishing movements; or, lastly, Section III.
* , with the real nature and existence of the great Cause of all. - - \-º-Ve-
We begin, then, with that class of Pure Sciences which we have called formal; and of these, the first Grammar.
two that present themselves to us, are Grammar and Logic. By Grammar we are taught the rules of
that speech, which serves as the medium of Mental intercourse between man and man; by Logic, the
Mental operations are themselves regulated and bound together, in a certain Method or order. As the
communication of knowledge is the more immediate object of our present discussion, so we begin with
that Science by which it is regulated in its forms. Grammar, then, apart from the mere material consi-
deration of the sound of words, or shape of letters, and regarding speech only as a thing significant,
teaches that there are certain laws regulating that signification; laws which are immutable in their very
nature for the relation which a noun bore to a verb, or a substantive to an adjective, was the same in
the earliest days aspéray &v98&noy, in the first intelligible conversations of men, as it is now; nor can it
ever vary so long as the powers of Thought remain the same in the Human Mind. This, then, is a Pure
Science proceeding from a simple or elementary Idea of the form necessary for the conveyance of a single
thought, and thence spreading and diffusing itself over all the relations of significant Language.
Grammar brings us, naturally, to the Science of Logic, or the knowledge of those forms which the con- Logic.
ceptions of the Mind assume in the processes of reasoning. And it is manifest that this Science is no
less subject than the former, to fixed laws; for the reasoning power in Man can operate only within those
limits which Almighty Wisdom has thought fit to prescribe. It is a discursive faculty, moving in a given
path, and by allotted means. There is no possibility of subverting or altering the elementary rules of
Logic; for they are not hypothetical, or contingent, or conventional, but positive and necessary.
Under the general term Mathematics are comprised the Sciences of Geometry, which is conversant Mathema.
about the laws of figure, or limitations of space; and Arithmetic, which concerns the laws of number. tics.
Now these laws are purely Ideal. It is not externally to us that the general notion of a square, or a
triangle, of the number three, or the number five, exists; nor do we seek for external proof of the rela-
tions of those notions; but on the contrary, by contemplating them as . Ideas in the Mind, we discover
truths which are applicable to external existence.
The Sciences which we have hitherto noticed relate to the forms of our Mental conceptions; but it is Metaphy-
natural for Man to seek to comprehend the principles and conditions of real existence, both with regard to :*
the Universe in general, with regard to his own internal mover, or conscience, and, above all, with regard *ē,
to the cause, by which conscience and the whole Universe were called into being, and continue to exist,
namely, GoD. Hence, as we advance from form to reality, the Sciences of Metaphysics and Morals first
present themselves to view, and these lead us forward to the summit of Human Knowledge; for at the head
of all Pure Science stands Theology, of which the great fountain is Revelation. It is obvious that both
Metaphysics and Morals are conversant solely about those relations which we have called Relations of
Law; for it would be a contradiction to say, that a real existence could be, at the same time, a mere
theory or hypothesis. These Sciences have, therefore, all the purity and all the certainty, which belong
to that which is positive and absolute; and as far as they are distinctly apprehended by the Mind, they
approach the nearest to that clear intellectual light, which, in the peculiar phraseology of Lord Bacon,
is called lumen siccum. In the proper Philosophical Method, the reality of our speculative knowledge,
exhibited in the Science of Metaphysics, unites itself at last with the reality of our Ethical sentiments
displayed in that of Morals; and both together are at once lost and consummated in Theology, which
rises above the light of Reason to that of Faith, -
These are all the Sciences which embrace solely relations of law: and it is plain that in these, not Mixed and
only the initiative, but every subsequent step, must be an act of the Mind alone. But when we descend §.
to the second order of relations, namely, those which we bear to the external world, Theory is imme-
diately introduced; new Sciences are formed, which in contradistiction from the Pure, are called the
Mixed and Applied Sciences; and in these, new considerations relative to Method necessarily find a place.
24 INTRODUCTION.
Introduc- Every Physical Theory is in some measure imperfect, because it is of necessity progressive; and because Section III.
*— we can never be assured that we have exhausted the terms, or that some new discovery may not affect ——
the whole scheme of its relations. The discoveries of the ponderability of air, of its compound nature,
of the increased weight of the calces, of the gases in general, of Electricity, and more recently the stu-
pendous influences of Galvanism on the successive Chemical Theories; are all so many exemplifications
of this truth. The doctrines of vortices, of an universal ether, of a two-fold magnetic fluid, &c. are
Theories of Gravitation : but the Science of Astronomy is founded on the Law of Gravitation, and remains
unaffected by the rise and fall of the Theories. In the lowest condition of Method, the initiative is supplied
by a hypothesis ; of which we may distinguish two degrees. In the former, a fact of actual experience
is taken, and placed experimentally as the common support of certain other facts, as equally present in
all: thus, that oxygen is a principle of acidification and combustion, is an experienced fact; and became
a hypothesis, by the assumption that it is the sole principle of acidification and combustion. In the
latter, a fact is imagined : as, for instance, an atom or physical point, preternaturally hard, and therefore
infrangible in the Corpuscular Philosophy; or a primitive unalterable figure, in some systems of Crystalli-
zation.
In all this, we see, that Knowledge is a matter not of necessary connection, but of a connection arising
from observation, or supposition ; that is, it consists not of Law, but of Theory, or Hypothesis. True
theory is always in the first and purest sense a locum tenens of Law; when it is not, it degenerates into
hypothesis, and hypothesis melts away into conjecture. Both in Law and in Theory, there must be a
mental antecedent; but in the latter, it may be an image or conception received through the senses, and
originating from without; yet even then there is an inspiring passion, or desire, or instinctive feeling of
the truth, which is the immediate and proper offspring of the Mind. Now, we may consider the facts
which are to be reduced to Theory, as arranged over the whole surface of a plane circle. If by carrying
the power of Theory to a near identity with Law, we find the centre of the circle, then proceeding toward
the circumference, our insight into the whole may be enlarged by new discoveries; it never can be
wholly changed. A magnificent example of this has been realized in the Science of Astronomy; a recent
addition of facts has been effected by the discovery of other Planets, and our views have been rendered
more distinct by the solution of the apparent irregularities of the Moon's motion, and their subsumption
under the general Law of Gravitation. But the Newtonian was not less a System before than since the
discovery of the Georgium Sidus; not by having ascertained its circumference, but by having found its
centre; the living and salient point, from which the Method of discovery diverges, the Law in which
endless discoveries are contained implicitly, and to which, as they afterwards arise, they may be
referred in endless succession.
These reasonings, it is hoped, will sufficiently explain the nature of the transition, from the Pure
Sciences to the Mixed and Applied Sciences, and will serve to trace the inseparable connection of the
latter with the constitution of the Human Mind. And as each of these great divisions of Knowledge
has its own department in the grand Moral Science of Man, it is obvious that a scheme, which, like our
own, not only contains each separately, but combines both as indivisible, the one from the other; must
present, in the most advantageous point of view, whatever is useful and beautiful in either. In speaking
of the Mixed and Applied Sciences, we must be permitted, however, to remark that the word Science is
evidently used in a looser and more popular form, than when we denominate Mathematics, or Meta-
physics, a Science; for we known not, for instance, the truth of any general result of observation in
Nosology, as we know that two and two make four, or that a Human person cannot be identical with
another Human person. And in like manner, when the word Law is used with relation to the Mixed
and Applied Sciences; as when we speak of any supposed Law of Vegetation; we use a more popular
language than when we speak of a Law of the Conscience, which is not to be prevaricated. The strictness
of ancient Philosophy, therefore, refused the name of Science to these pursuits: and it might at least
be convenient, if in speaking generally of the Pure, the Mixed, and the Applied Sciences, we gave
ON THE SCIENCE OF METHOD, 25
Introduc. them the common name of Studies, inasmuch as we study them all alike, but we do not know them all Section III.
*— with the same sort of knowledge. ve-
Of these, then, (be they Studies or Sciences,) we call those Mired in which certain Ideas of the Mind Mixed,
are applied to the general properties of bodies, solid, fluid, and aerial; to the power of vision, and to
the arrangement of the Universe; whence we obtain the Sciences of Mechanics, Hydrostatics, Pneumatics,
Optics, and Astronomy. It is matter not of certain Science, but of observation, that such properties
do really exist in bodies, that vision is effected in such or such a manner, and that the Universe is
disposed in this or that relative position, and subjected to certain movements of its parts. Therefore,
these Sciences may vary, and notoriously have varied; and though Kepler would demonstrate that Euclid
Copernicized, or had some knowledge of the System afterwards adopted by Copernicus; yet of this there
is little proof: and certainly, for many Ages after Euclid, it was the universal opinion, that the Earth was
the fixed and immovable centre of the Universe. Nor have we here unadvisedly used the word
opinion ; since, as we before showed, it is the ancient expression, signifying a medium between Know-
ledge and Ignorance: and well did that acute Italian exclaim, Opinione, regina del mondo –for as it is
impossible that Ignorance, which cannot govern itself, should govern any thing else; so to expect that all
the world should be wise enough to submit to the government of Wisdom, would be to show that we had
followed very little Method in our study either of History of living men, or even of ourselves.
When certain Ideas, or images representative of Ideas, are applied still more particularly, not to Applied.
the investigation of the general and permanent properties of all bodies, but of certain changes in
those properties, or of properties existing in bodies partially, then we popularly call the Studies relative to
such matters by the name of Applied Sciences ; such are Magnetism, Electricity, Galvanism, Chemistry,
the Laws of Light and Heat, &c. We have already so fully shown the uncertainty of the first Principles †Ph.
in these Studies, and have so distinctly traced the cause of that uncertainty, in every case, to a want of losophy.
clearness in the first Idea or Mental initiative of the Science, that it will be unnecessary here to do more
than refer to our preceding observations.
We come now to another class of Applied Sciences, namely, those which are applied to the purposes Fine Arts.
of pleasure, through the medium of the Imagination; and which are commonly called the FINE ARTs.
These are Poetry, Painting, Music, Sculpture, Architecture. We have before said, that the Method to
be observed in these, holds a sort of middle place between the Method of Law, or Pure Science, and
the Method of Theory. In regard to the Mixed Sciences, and to the first class of Applied Sciences, the
Mental initiative may have been received from without; but it has escaped some Critics, that in the Fine
Arts the Mental initiative must necessarily proceed from within. Hence we find them giving, as it were,
recipes to form a Poet, by placing him in certain directions and positions; as if they thought that every
deer-stealer might, if he pleased, become a Shakspeare, or that Shakspeare's mind was made up of
the shreds and patches of the books of his day; which by good fortune he happened to read in such an
order, that they successively fitted into the scenes of Macbeth, Othello, the Tempest, As you like it, &c.
Certainly the Fine Arts belong to the outward world, for they all operate by the images of sight and
sound, and other sensible impressions; and without a delicate tact for these, no man ever was, or could
be either a Musician, or a Poet: nor could he attain to excellence in any one of these Arts: but as
certainly he must always be a poor and unsuccessful cultivator of the Arts, if he is not impelled first by
a mighty, inward power, a feeling, quod nequeo monstrare, et sentio tantum ; nor can he make great
advances in his Art, if in the course of his progress, the obscure impulse does not gradually become a
bright, and clear, and living Ideal
Pursuits of utility, we daily find are capable of being reduced to Method. Thus Political Economy, Useful
and Agriculture, and Commerce, and Manufactures, are now considered scientifically; or as the more S.
prevalent expression is, Philosophically. It may, perhaps, be difficult, at first, to persuade the experi-
mental Agriculturist, that he also pursues, or ought to pursue, an Ideal Method: nor do we mean by
this that he must deal only in ideal sheep and oxen, and in the groves and meads of Fairy Land. But
6.
26 INTRODUCTION.
Introduc. these Studies, soberly considered, will be found wholly dependent on the Sciences of which we have Section III.
—- already treated. It is not, surely, in the Country of ARKwRIGHT, that the Philosophy of Commerce can \-y-
be thought independent of Mechanics: and where DAvy has delivered Lectures on Agriculture, it would
be folly to say that the most Philosophic views of Chemistry were not conducive to the making our valleys
laugh with corn. -
#. We have already spoken of LINNAEus, the illustrious Swede, to whom the three kingdoms, as they are
aptly called, of Natural History, are so deeply indebted: and if, with all his great talents, he yet failed
in establishing the united empire of those three mighty monarchies, on firm laws and a fixed
constitution; we have shown, that it was only owing to a want of precision in the first Ideas of his
theory. -
flººr Natural History itself becomes a rule for dependent pursuits, such as those of Medicine (under which
are Pharmacy and the Materia Medica) and Surgery, in which is included Anatomy. That in these
and the other theoretical studies, so much still remains to be done, ought not to be a subject for regret;
but, on the contrary, for a laudable and generous ambition. Yet that ambition should be regulated and
moderated by a due consideration of the place, which the particular pursuit in question holds in the
great circle of the Sciences; and by observing the only proper Method which can be pursued for its
improvement. If, in what we have here said, we have done any thing towards the excitement, the regu-
lation, and the assistance of that ambition; if we have faintly sketched an outline of the great laws of
Method, which bind together the various branches of Human Knowledge, we may not improperly indulge
a hope that the ensuing Work, in its progress, will be found conducive to the promotion of the best
interests of Mankind.
History and Our Plan would not completely meet the views of those to whom such Works as the following are
* eminently useful and agreeable, if besides the Philosophic Method already described, we did not present
some view of the actual History of Mankind. We have therefore devoted a large portion of our labours
to the History of the Human Race, on a new, and we trust it will be found an improved System.
Biography and History tend to the same points of general instruction, in two ways: the one exhibiting
Human Principles and Passions, acting upon a large scale; the other showing them as they move in a
smaller circle, but enabling us to trace the orbit which they describe with greater precision. The one brings
Man into contact with Society, actuated by the interests which agitate and stimulate him in the various
social combinations of his existence; and Human Nature presents itself in the varied shapes impressed
upon it by the different ranks which it occupies. The other brings before us the individual, when he
stands alone, his passions asleep, his native impulses under no external excitement; in the undress of
one who has retired from the stage, on which he felt he had a part to sustain; and even the Monarch,
forgetting the pomp and circumstance of his royalty, remembers here only that he is a Man. Assuredly
the great use of History is to acquaint us with the Nature of Man. This end is best answered by the
most faithful portrait; but Biography is a collection of portraits. At the same time there must be
some mode of grouping and connecting the individuals, who are themselves the great landmarks in the
Map of Human Nature. It has therefore occurred to us, that the most effectual mode of attaining the chief
objects of Historical knowledge, will be occasionally to present History in the form of Biography, chrono-
logically arranged. This will be preceded by a general Introduction on the Uses of History, and on the line
which separates its early Facts from Fable; and it will in the course of its progress, be interspersed with
connecting Chapters on the events of large and distinguishing periods of time as well as on Political
Geography and Chronology. Thus will a large portion of History be conveyed, not only in its most
interesting, but in its most Philosophical and real form; while the remaining facts will be interwoven
in the preliminary and connecting Chapters. If in tracing thus the “eventful History” of Man, and
particularly of our own Country, we should perceive, as we must necessarily do in all that is Human,
evils and imperfections; these will not be without their uses, in leading us back to the importance of
intellectual Method as their grand and sovereign remedy. Hence shall we learn its proper national
ON THE SCIENCE OF METHOD 27
Introduce application, namely, the education of the Mind, first in the Man and citizen, and then, inclusively, in Section III.
* , the State itself.
| Such are our views in the Philosophical and Historical branches of our Work. Of the Miscellaneous Alphabetic
or Alphabetical Division we have little to add. But well aware that Works of this nature are not solely ...”
useful to those who have leisure and inclination to study Science in its comprehensiveness and unity;
but are also valuable for daily reference on particular points, suggested by the desires or business of the
individual; we could not hold ourselves dispensed from consulting the convenience of a numerous and
most respectable class of Readers; while the preceding remarks will go to prove that for many local and
supplementary illustrations of Science, no other depositary could be furnished.
As the Philosophical arrangement is, however, most conducive to the purposes of intellectual research
and information, as it will most naturally interest men of Science and Literature; will present the circle
of Knowledge in its harmony; will give that unity of design and of elucidation, the want of which we
have most deeply felt in other Works of a similar kind, where the desired information is divided into
innumerable fragments scattered over many volumes, like a mirror broken on the ground, presenting,
instead of one, a thousand images, but none entire; this Division must of necessity have that prominence
in the prosecution of our design, which our conviction of its importance to the due execution of the
plan demands; and every other part of the arrangement must be considered as subordinate to this
principal organization. With respect to the whole Work, it should be observed, that in what concerns
references we are guided by Principle, not by caprice; nor do we ever recur to them as our only means
of escape from an exigency. Throughout the ENCYCLoPEDIA METRoPolitanA, the Philosophical
arrangement predominates and regulates the Alphabetical arrangement, and the references, whether to it
or from it, are auxiliary. We never refer from the first and second Divisions to the fourth, or from the
first to the second, for the explanation of a term, the establishment of a Principle, or the demonstration
of a Proposition. The reference, whenever it occurs, unless it be retrospective, is not for the purpose of
essential information, but for that which is collateral and subordinate. The theory of the Balance, for
example, is given where it ought to be in the Treatise on Mechanics; but they who wish to acquaint
themselves with the various constructions of Balances for the purposes of Commerce or Philosophy,
knowing that these cannot be introduced into a Scientific Treatise without destroying the symmetry of
its parts by a suspension of the Logical order, will naturally turn, whether there be a reference or not, to
the Alphabetical Department of the Work. So again, the Principles of the Telescope are given in the
Treatise on Optics; the varieties of construction in the Alphabetical department: the Principles of the
Thermometer, when treating of the effects of Heat; its varieties of construction in the Alphabetical
Department. Practical detail, and niceties or peculiarities of construction, can seldom be interwoven
with propriety among the regular deductions of a Methodical Treatise: in all cases, where they cannot,
our general Principle, as it comprehends proportion, accuracy, utility, and convenience, demands a
reference, whether expressed or not, to the appropriate place for all that is subservient; that is, to the
fourth or Alphabetical Division.
This final Division of our Work will bring the whole into unison with the two great impulses of
Modern times, Trade and Literature. These, after the dismemberment of the Roman Empire, gradually
reduced the conquerors and the conquered at once into several nations and a common Christendom.
The natural Law of increase, and the instincts of Family, may produce Tribes, and under rare and
peculiar circumstances, Settlements and Neighbourhoods: and Conquests may form Empires. But without
Trade and Literature, combined, there can be no Nation; without Commerce and Science, no bond of
Nations. As the one has for its object the wants of the body, real or artificial, the desires for which
are for the greater part excited from without; so the other has for its origin, as well as for its object,
the wants of the Mind, the gratification of which is a natural and necessary condition of its growth and
sanity. In the pursuits of Commerce the Man is called into action from without in order to appropriate
the outward world, as far as he can bring it within his reach, to the purposes of his corporeal nature.
28 INTRODUCTION.
Introduc. In his Scientific and Literary character he is internally excited to various studies and pursuits, the Section III,
tion.
º , groundwork of which is in himself. \-y-º
This, again, will conduct us to the distinguishing object of the present undertaking; in endeavouring
to explain which we have dwelt long upon General Principles; but not too long, if we have established
the necessity of what we conceive to be the main characteristic of every just arrangement of Knowledge.
Our Method embraces the two-fold distinction of Human activity to which we have adverted;—the
two great directions of Man and Society, with their several objects and ends. Without advocating the
exploded doctrine of perfectibility, we cannot but regard all that is Human in Human Nature, and all that
in Nature is above herself, as together working forward that far deeper and more permanent revolution
in the Moral World, of which the recent changes in the Political World may be regarded as the
pioneering whirlwind and storm. But woe to that revolution which is not guided by the historic sense;
by the pure and unsophisticated knowledge of the past; and to convey this Methodically, so as to aid
the progress of the future, has been already announced as the distinguishing claim of the ENCYCLOPEDIA
METROPOLITANA.
January, 1818.
N. B.-The ENCYCLOPEDIA METROPolitanA on passing into new hands, after the publication of the first five
Parts, underwent certain modifications of the Plan described in the above Introduction, which may be observed in
its progress. -
y
..
•
\"
Grammar.
ENCYCLOPEDIA METROPOLITANA;
OR, THE
UNIVERSAL DICTIONARY OF KNOWLEDGE,
ON AN ORIGINAL PLAN.
º jirgt £3tbígton.
G R A M M A. R.
-º--
INTRODUCTORY SECTION.
GRAMMAR is a word used to signify both the Pure
S—— Science of Universal Grammar common to all Languages,
Significa-
tion, Uni-
versal and
Particular.
Definition
and the Applied Sciences of Particular Grammar re-
stricted each to its particular Language or Dialect.
It is only of Grammar, in the first of these accepta-
tions, that we mean, at present, to treat. In a me-
thodical view of the Pure and Applied Sciences, it is
essentially necessary to begin with the former : nor
... can any Particular Grammar be well and thoroughly
`s understood, without some previous knowledge of Uni-
º
**Versal Grammar, as its foundation.
Grammar, then, in its most comprehensive sense,
may be defined, the Science of the relations of Language
considered as significant. We say “ of the relations of
Language,” because the knowledge of Languages, in
so far as it regards the mere acquisition and remem-
brance of terms, is an affair of the attention and of the
memory; whereas to understand the relations which
those terms bear to each other, is the business of a
Science especially directed toward that end. We say
too “ of Language considered as significant;” because
Language has other properties besides that of signifi-
cation. Words, for instance, may be made up of
longer or shorter sounds, may be delivered with
Varieties of accentuation, and may be uttered in a softer
or louder voice; but these and many other circum-
stances relative to Language, do not properly fall under
WOL. I
the Science of Grammar, although some of them may
be considered as its adjuncts, or dependencies.
Of the term “Language,” which we have used in
our definition, we must speak more at large. As the
word “Grammar,” though introduced into English
from the French, is derived from the Greek verb 'ſpáq,w,
“I write;” so the word “Language,” which comes
immediately to us from the French word langage,
originates in the Latin lingua, “ the tongue;” and
therefore anciently signified only the use of the tongue
in speech. A just analogy, however, has extended its
meaning to all intentional modes of communicating the
movements of the Mind ; thus we use the expressions,
“ articulate language,” “ written language,” “ the
language of gesture,” &c.; and this analogy suggests
some considerations, which will be found important,
in developing the first principles of Grammatical
Science.
Man is formed as well internally, as externally, for
the communication of thoughts and feelings. He is
urged to it by the necessity of receiving, and by the
desire of imparting, whatever is useful or pleasant.
His wants and wishes cannot be satisfied by individual
power: his joys and sorrows cannot be limited to in-
dividual sensation. The fountains of his wisdom and
of his love spontaneously flow, not only to fertilize the
neighbouring soil, but to augment the distant ocean.
B
Introduc-
tory Sec-
tion.
*~~~
Language.
2
G R A M M A. R. *-a-
Grammar.
~~~~
Speech.
But the Mind of Man which is within him, can only
be communicated by objects which are without, by
gestures, sounds, characters more or less expressive
and permanent, instruments not merely useful for this
particular purpose, but many times pleasing in them-
selves, or rendered so, by the long-continued operation
of habit. These Reason adopts, she combines, she
arranges ; and the result is a Language.
Speech, or the language of articulate sounds, is the
most wonderful, the most delightful of the Arts, thus
taught by Nature and Reason. It is also the most
perfect. It enables us, as it were, to express things
beyond the reach of expression, the infinite range of
being, the exquisite fineness of emotion, the intricate
subtilties of thought. Of such effect are those sha-
dows of the soul, those living sounds, which we call
words ! Compared with them, how poor are all other
monuments of human power, or perseverance, or skill,
or genius ! They render the mere clown an artist;
Nations immortal; Orators, Poets, Philosophers divine !
The Dialects or systems of speech adopted by various
races of men, in different Ages and Countries, have
been, in many respects, strikingly distinguishable.
We may remark the copious Arabic, the high-sounding
Spanish, the broad Dutch, the voluble French, the soft
Italian : we may trace minute gradations from the mo-
nosyllables of the Chinese, to the long paragraph words
of the Sanscrit; or we may rise, still more gradually,
in the scale of expression, from the barbarous mutter-
Method of
study.
Lord Bacon.
ing of a poor Esquimaux in his solitary canoe, to the
thunders of Athenian eloquence, and those delightful
strains of our own Shakspeare, which are
& musical as is Apollo's lute,
And a perpetual feast of nectar'd sweets.”
Nor is this all : a thousand collateral circumstances
tend still further to diversify the numerous spoken
Languages of the World. Not only does time produce
gradual progress, or sudden change in their forms; but
their effect is endlessly modified by combination with
other arts of expression, with looks and actions, with
sights and sounds. -
In this labyrinth of interesting observations, what
objects have we to pursue? what clue to guide us?
Shall we be content to learn one or two Dialects by
rote ; to burthen the memory without exercising the
understanding 2 Or, if we would rise above this, to a
knowledge of their construction, must we draw our
general principles from the minute comparison of those
numberless particulars, which the longest life would
be too short even to contemplate, and which the united
wisdom of Ages has never attempted to arrange?
The very statement of these questions is a suffi-
cient solution of them. They indicate at once the me-
cessity of assuming some comprehensive Principles as
the rule and basis of our further inquiries. These
first elements of our reasoning must afterwards be fol-
lowed out into all their concrete forms. The History of
Language must verify the Science; but the Science must
precede ; for such, in the order of Nature, is the course
of all our knowledge. General notions, vague and
indistinct, come first; they form, as it were, the
channels into which our daily observations flow ; and
these observations again correct and strengthen our
former motions, and render them sources of clear and
abundant knowledge.
LoRD BAcon, indeed, says, that “ that would be the
most noble kind of Grammar, which would be formed,
if a man profoundly skilled in many Languages, vulgar
as well as learned, were to treat of the various proper-
ties of each, and to show their several excellencies and
defects.” But it is obvious that his Lordship here
speaks only of the last result of the Grammarian's stu-
dies; it is previously necessary, not only to learn the
words of the Languages which are to be arranged and
compared, but to acquire the arts of arrangement and
comparison. -
The first step toward a perfect arrangement is to
comprehend the whole subject-matter under a general
idea; and from what we have already said, it is mani-
fest that the idea of speech is included in the still more
general idea of Language, which comprehends the
Principles common to speech, with gesture, writing, &c.
The various arts to which these Principles are capable
of application may be considered as branches of one
great family; they are all derived from the same source,
always analogous to, sometimes associated or inter-
woven with each other ; and hence, like the sister
Graces, they will appear to the greatest advantage
together.
The general idea of Language, applicable to all these
various modes of its exercise, is, as we have said, a
communication of the thoughts and feelings of the Mind.
But how can we understand the communication, unless
we have some idea of the thing communicated 2 And
which shall we consider as the original and shaping
power of a word, the sound, or the thought 2 These
questions cannot bear a moment's reflection. If the word
were parent to the thought, a parrot or a speaking auto-
maton might be made to understand gravitation as well
as Sir Isaac Newton. And yet there are men, in the pre-
sent day, calling themselves Grammarians and Philoso-
phers, who have pushed absurdity so far as to assert,
that the faculty of Reason itself depends wholly on
speech Assuredly to know the powers and employ-
ments of the tongue conduces greatly to strengthen and
facilitate the operations of the Mind; but we cannot
understand the former until we have made considerable
progress in the knowledge of the latter.
Introduc-
tory Sec-
tion.
\-N-"
The late Mr. HoRNE TookE, in his well known Work, Horne
The Diversions of Purley, speaks thus:– “The busi-
ness of the Mind (as far as it concerns Language) is
very simple. It extends no farther than to receive im-
pressions; that is, to have sensations or feelings.
What are called its operations are merely the operations
of Language.” Let us here ask, What can possibly be
meant by “the operations of Language,” as distinct from
those of the Mind 2 Who is Language 2 How does he
operate 2 If my Mind, as far as concerns Language, do
nothing but receive impressions, how comes it to pass
that I ever open my lips?
happens it that I utter articulate sounds; that those
sounds form words ; that those words are arranged in
a certain order; and that that order is absolutely es-
sential to my being understood? How does Language
operate, so as to shape itself into nouns and verbs:
and those the very nouns and verbs which I happen
to want; and all the while, without any privity or inter-
ference of mine, or any act whatsoever of my Mind?
It is proper, however, to observe, that in respect
to the general principles here adverted to, Mr. Tooke
has neither the merit, nor the demerit, of originality.
He is so far a follower of Condillac and the writers
of that School, of whose general opinions the follow-
ing passage may afford a sufficient specimen ; and
And when I speak, how
Tooke,
Condillac.
G R A M M A. R. 3
racterise the signs. We are taught that the portrait is Introduc-
Grammar, we select it from a Work published in 1803, by a w
the original, and the man the copy, that without the º Sec-
*Y* Member of the French National Institute, and re-
10 Il.
edited and corrected in 1804 by another Member of
the same learned Body, at present a Peer of France.
“We cannot distinguish our sensations,” says the au-
thor, “but by attaching to them signs which represent
and characterise them. This is what made Condillac
say, that we cannot think at all without the help of
Language. I repeat it, without signs there exists
neither thought, nor perhaps even, to speak properly,
any true sensation. In order to distinguish a sensation,
we must compare it with a different sensation : now
their relation cannot be expressed in our Mind, unless
by an artificial sign, since it is not a direct sensation.”
CoNDILLAC, who is here quoted with so much ap-
probation, began to write in 1749. He pretended to
found his doctrine on the principles of Lock E.; and we
presume it has at last received its final perfection from
the hands of M. DESTUTT-TRACY, the noble editor of
CABANIs.
It is hardly possible to expose the absurdity of such
statements, without descending from the gravity of a
serious disquisition. We shall simply analyze the ex-
tract which we have just made, applying to its Princi-
ples (if principles they may be called) a few obvious
exemplifications: and, if the result should appear to
border too much on the ridiculous, we trust that the
imputation of folly will rest with the original authors
of a system so perfectly incoherent.
1. “We cannot distinguish our sensations but by
attaching to them signs which represent and charac-
terise them.” We might first ask what is a sign 2 Is it
a sensation, or somewhat else? If a sensation, is it
direct, or indirect? How do we distinguish one sign
from another? What part do signs perform in our
mental operations?—and many other such questions;
but passing over these difficulties, we will come to our
author's own reasoning ; and from the Principle which
he here lays down, it must follow, that if a native of
Scotland should see a brook, (which in that Country is
called a burn,) and should also feel a burn occasioned
by touching any heated substance, he would not be
able to distinguish these sensations, because he would
have attached to them the same sign ; neither could
he distinguish them if he even attached to them dif.
ferent signs, e. g. rivus and ustio, unless each sign
accurately represented the thing signified ; so that the
one sign should reproduce in him the sight of flowing
water, and the other the touch of a heated body.
2. “This is what made Condillac say, that we can-
not think at all without the help of Language.” If Con-
dillac reasoned from such premises, it is no wonder that
he came to such a conclusion.
3. “Without signs there exists neither thought, nor
perhaps even, to speak properly, true sensation.” Signs,
we have before been told, are things characterising or
representing sensations. We now learn that it is, on
the contrary, the sensations which represent or cha-
* Cn ne distingue les sensations qu’en leur attachant des signes,
qui les représentent et les caractérisent.—Poilà ce qui fait dire à
CoNDILLAC qu'on ne pense point sans le seconrs des Langues—Je Ze
ºrépête, sans signes it n'eriste ni pensée, ni peut Étre méme, à pro-
prement parler, de véritable sensation—Pour distinguer une sensa-
tion il faut la comparer avec une sensation différente ; or leur
ºrapport ne peut €tre exprémé dans notre esprit que par un signe arti-
ficiel, puisque ce n'est pas une sensation directe.”—CABANIs, Rap-
ports du Physique et du Morale de l’Homme, vol. i. p. 72.
portrait there would be no man. Some doubt is ex-
pressed, whether we might mot receive some sort of
sensation from striking our heads against a post; but
it is argued that this would not be a true sensation,
that we should not really feel the blow, unless we ac-
tually cried “post,” or read the word “post,” which
would naturally explain to us the sort of blow we had
experienced.
4. “In order to distinguish a sensation, we must
compare it with another sensation.” Here is a new
rule to know whether we are alive, and in our senses,
or not. If we chance to break our shins, we must not
be too hasty in crediting the evidence of that part of our
body; we must compare the sensation with some other,
as for instance, with that of drinking a glass of Cham-
pagne, and if we find that they differ, why then we may
be assured that they are not the same.
5. “Now, their relation cannot be expressed in our
Mind, unless by an artificial sign; since it is not a
direct sensation.” What is meant by a sensation being
expressed in the Mind, it is not very easy to discover;
but the author seems to intimate that a direct sensation
may be so expressed, and that it therein differs from
the relation between two sensations, which relation he
says is not a direct sensation. We presume that he
would rank breaking his shins, or drinking Champagne,
in the class of direct sensations; these, therefore, may
be expressed in the Mind without an artificial sign;
and consequently they are not true sensations; for (by
proposition 3.) without signs there exists no true
sensation ; neither can we think at all about them,
because (by the same proposition) without signs there
is no thought. It is probably meant to be understood
that all sensations are direct or indirect. We have
seen how the qualities of the former class are explained.
Let us next consider what happens with respect to the
latter. Some sort of relation probably exists between
drinking Champagne and breaking the shins, but that
relation, we are told, cannot be expressed in the Mind
without an artificial sign. Now, as we have never
heard of any word or even hieroglyphic to express the
particular relation that exists between drinking Cham-
pagne and breaking the shins, it follows that no such
relation can be expressed in the Mind; and conse-
quently (by proposition 4.) the separate sensations of
breaking the shins and of drinking Champagne cannot
be distinguished. -
It is obvious that if these ridiculous propositions
had been stated plainly and simply, they would never
have encountered serious discussion. They have,
however, been enveloped in the mystical jargon of the
modern ideologists ; they have assumed the imposing
name of Metaphysics; and hence the ignorant multitude
have concluded, that there is something in them of
profound wisdom.
Two chief causes may be assigned for the errors of
these modern Grammarians : first, their rejection of
that Philosophy of the Mind, on which, as we conceive,
the Philosophy of Language depends; and secondly,
their confounding Historical fact with Philosophical
principle. The almost unintelligible use of the word
sensation, in the passages above quoted, and the vague
and contradictory meanings applied by these writers
to the word idea, sufficiently demonstrate their inatten-
tion to the genuine workings of the Human Mind. In
\-2-’
Causes of
modern
€ITOIS.
B 2
4 G R A M M A. R.
Grammar, tracing the History of words, they have sometimes
\-v-' shown great ingenuity; but they have erroneously con-
Ancient
Gramma-
rians.
Reason.
cluded, that because a particular word was once a
noun or a verb, it always continues such ; forgetting
that the identity of the word depends only on its
sound, whilst the distinction of the Parts of speech
relates solely to their signification ; and, consequently,
that the one is a question of the matter of Language,
the other, of its form ; or perhaps being unable to
comprehend the ancient Philosophical distinction be-
tween matter and form, and therefore, concluding that
that distinction was frivolous and unmeaning. Thus
Mr. Tooke, conceiving that our present adverb, pre-
position, and conjunction, since, was anciently the
participle, seen, or seeing, concludes that it has still
the same signification. He happens to be mistaken
in his fact; for the word since has nothing to do with
the verb to see : * but if he had been correct in this,
as he really is in many of his etymologies, the inference
from it would have been no less illogical. There is no
reason, in the nature of Language, why one word
should not successively fill the office of every Part of
speech ; and, in particular, nothing is more common
than for the same word to be both a noun and a verb.
Mr. Tooke, therefore, to be consistent, should not have
said that “there are only two sorts of words which are
necessary for the communication of our thoughts,” viz.
“nouns and verbs,” but that there is only one sort;
which would have been saying in effect there is no such
Science as Grammar in the World.
The ancient Grammarians, who treated of the Greek
and Roman Languages, as well as those who in the
Middle Ages cultivated the Arabic and its kindred Dia-
lects, and those whose disquisitions on Indian Philo-
logy have been laid open to us by recent discoveries,
all agree in founding the Science of Grammar on that
of the mental operations. Nothing but extreme va-
nity can lead us to suppose, that all the great men,
who have ever considered this subject before ourselves,
have been involved in a more than Boeotian mist of
ignorance ; and that we alone can dispel the cloud by
a single “ electric flash.” The more modest and rational
student will confess, with the amiable author of
Hermes, that “there is one TRUTH, like one Sun,
which has enlightened human intelligence through
every Age, and saved it from the darkness both of so-
phistry and error.” It may be safely adopted as a
general observation, that the man who tells you the
whole World was ignorant of any particular subject
until he arose to set them right, is himself egre-
giously in the wrong. The study of Grammar, indeed,
like all other studies, is susceptible of gradual improve-
ment; but if we admit that the Ancients had a tole-
rable insight into the powers and operations of the
Human Mind, we must acknowledge that they could
not be entirely ignorant of the modes in which those
powers and operations were manifested by Language.
An individual writer may have taken a limited view
of the subject; but that view could not be wholly
erroneous, if he was adequately versed in the Philo-
sophy of the Human Mind.
It would seem that some ancient writers considered
Tanguage merely as representing the operations of the
* Since is derived from the Anglo-Saxon word sithe, which is the
same as the German zeit and English tide, signifying time; conse-
quently since is, literally, from that time.
reasoning faculty; and they were enabled thus to
analyze and explain a great part of its construction.
In this system, which was perhaps the most ancient,
the syllogism was considered as the basis of Grammar;
Logical writers were its chief authorities; its rules
were thought applicable only to the graver composi-
tions, such as Laws, Books of Civil institution, History,
and Treatises of the useful Arts and Sciences: the more
animated compositions of Rhetoric and Poetry, and the
common discourses of daily life, were considered as a
kind of barbarous confusion, beyond the pale of Gram-
matical law. - • ‘’
Introduc-
tory Sec-
tion.
\-v-f
But Man could not forget that he was a creature of Passion.
Passion, as well as of Reason ; and seeing that the former
was as capable of being reduced to rule as the latter,
that it was equally clear in its Principles, and equally
certain in its operation, he could not but admit its in-
fluence on the rules of speech. The syllogism had sup-
plied the two sorts of words, which, Mr. Tooke says, are
alone “necessary for the communication of our
thoughts;” but in matters of Passion the animated inter-
jection is quite as necessary as the simple name of a
thing or attribute; and in like manner the imperative is
a verbal form of no less importance, than that which
merely indicates or asserts existence.
Again, the Mind, whilst it steadily contemplates Modifica-
certain objects, passes rapidly and almost unconsciously tion anº
over those various relations which serve to modify and connection.
connect those objects with other existences. These
vague and hasty glances of the Mind, these slight and
subordinate hints, as it were, give occasion to cor-
respondent distinctions in Language. Hence arise
whole classes of words called adverbs, conjunctions,
prepositions, &c.; and thus have Grammarians settled
the Parts of Speech, which we shall hereafter consider
more at large.
Thus far the Ancients went, and for the most part
went right, in their view of Language. Recent authors
have rashly called in question the utility of these
learned labours. It is not to be denied that the many
new sources of information opened to us in modern
times, the numerous Dialects, barbarous and polished,
which we have the means of studying, the progress of
the same Language through many successive Ages,
which we are enabled historically to trace, and in short,
the extended sphere of our experimental investigations
in Language, may have served to correct some errors
and oversights even in our Scientific views of Universal
Grammar. Let no man ever presume to suppose that
his reasoning powers may not be sharpened, his judg-
ment rendered clearer, or his taste more refined by the
lessons of experience. The moment that we think
there is nothing more to be learned, we give a decisive
proof of ignorance. As the Moderns, however, fail
most in the Philosophy of Language, the Ancients failed
most in its History. They are rarely to be relied on as
Etymologists: whilst the Moderns, who have enjoyed so
much better opportunities of cultivating this branch of
the Science, have obtained in it a decided superiority.
They have discovered that most of those auxiliary
words, which are employed in aiding the construction
of nouns and verbs, were once nouns and verbs them-
selves; and that those which appear now void of
signification were formerly significant. These observa-
tions have in certain instances been extended, with
some plausibility, even to the syllables which are used
for purposes of inflection. Considerable ingenuity
Ancients
and Mo-
derns
compared.
G R A M M A. R. 5
Grammar. has been displayed in this sort of investigation by DE
\-/-/ BRoss Es, Court DE GEBELIN, TookE, and others;
by external causes, we may not improperly include Introduc-
sensation and emotion as modes of the passive prin- tory Sec-
and when we come to consider this part of our subject,
we shall certainly find them better guides than the
Ancients, who appear to have treated it with no very
reasonable neglect. w
It seems to follow from what has here been said,
that in order to study Grammar as a Science, a ge-
neral survey of the mental faculties should be premised
or presumed. This will afterwards lead us to a detailed
consideration of the Parts of speech, both in regard to
their separate properties, and also to their syntax or
union. Strictly speaking, the Pure Science of Gram-
mar ends here ; for as Voss IUs has observed, Science
is conversant with things eternal and invariable;
whereas Grammar, as generally understood, has no
immovable and unvarying essence, but relates to the
matter of Language, rather than to its form ; and hence
(as that writer contends) it ought rather to be called an
Art than a Science.
Preliminary View of the Human Mind with reference to
the Science of Grammar.
ciple, under the common name of Feeling. The states
of sensation, which are agreeable to our nature, we
properly call pleasure, those of an opposite kind we
call pain ; and the same names are naturally transfer-
red to those emotions of the Mind which seem analogous
to the respective sensations of the body. Thus the feel-
ing of guilt is called painful, and that of joy pleasant.
The pleasurable sensations and emotions, and their
real or supposed causes, are all called by the common
name of good, and their opposites by that of evil. The
expression of feeling is what constitutes in Language
the passive verb.
tion.
As we have called the passive principle, feeling ; Will.
so we call the active principle Will, or volition. It is
this principle which may truly be called the life of
the Human Mind ; it is this which forms and fashions
the mind; it is this which impels and governs the
man. The conscious Being, in his active state, has a
power: he says, I do this or that: and hence arises the
active verb. Hence also arises the pronoun : for the
very idea of an act involves the idea of a cause; and it
has been clearly enough shown by different writers, that
if the idea of a cause did not exist within the Mind, it
{}onscious- In the Mind of Man the consciousness of simple could never be suggested from without. ...
IlêSS, existence is the source and necessary condition of all The will, in its growth, becomes a Moral energy,
other powers; as in Language, the expression of that that is, it impels us to good, as good, and consequently
consciousness by the verb to be, is at the root of all to the greater good rather than to the less. To choose
other expression. the greater good is to do right, to choose the less good
But we are conscious of different states of existence, is to do wrong. Let Philosophers argue, as they please,
in some of which we act, and in others we are acted on Liberty and Necessity; let them reconcile, as they
upon : and thus in Language, a verb is a word which can, those high doctrines
signifies to do, or to suffer, as well as to be. No Lan- g *
guage, indeed, ever was, or ever could be, formed gº. . º
without such verbs ; but the case is different with re- * 2 J wiedge abs 3.
gard to Theories of Language, and Systems of Grammar. still the individual, from the first dawnings of Reason,
These may be, and have been constructed, on the distinguishes right from wrong, and knows that he is a
hypothesis, that the Mind of Man is a mere passive cause of the one, or of the other; and feels that the
recipient of mechanical impressions; a something which power which he exercises as a cause, is a talent for
may be impelled like a foot-ball, but which cannot which he is responsible. Thus is formed Conscience,
give to itself, or to any thing else, the slightest im- the light and guide of life. We have not now to dis-
pulse. On such a question as this, the only appeal cuss at length the nature and effects of this precious
lies to the common sense and daily experience of man- faculty : other and fitter occasions may be found for
kind ; and the result of that experience is clearly at- that investigation; but we cannot avoid noticing, that
tested by all Languages, living and dead—a species of as the ideas of right and wrong are seated not merely
evidence which is the less to be resisted, because it is in the Mind, but in the first and elementary rudiments
not the result of any systematic arrangement whatever. of the Mind, it is a dangerous and fatal error to repre-
Every Language in the World has grown up from the sent them as contrivances of Language, to say that
necessities of those who have used it, and not from “Right is no other than the past participle of the Latin
intention ; from accident, and not from theory; and verb regere,” and that “Wrong is merely the past tense
yet there is among them a universal agreement in their of the verb to wring.” This is part of the History of
Feeling. fundamental principles: those principles, then, are in- words : it is no part of their Philosophy.
disputably founded on the common constitution of the
Human Mind.
The Mind is undoubtedly passive in some respects.
If I open my eye to the light, I cannot choose but see :
if a sound strikes my ear, I cannot help hearing.
These, and many like states of existence, derived from
the bodily organs, are called sensations ; there are
other states, in which we are more or less passive,
derived from the Mind, and commonly called emotions.
When we come to analyze these latter, we shall easily
discover that we are not so entirely passive in their
reception, as is often supposed : nevertheless, as we
in both cases “suffer,” that is to say, are acted upon
Neither will nor feeling have in themselves any limit. Reason.
The stream of conscious being is, in itself, continuous.
What is it, then, that reduces the chaos of will and feel-
ing first into distinguishable elements, and then into
individual masses? It is the forming and shaping
power within us. It is the divine faculty, “ looking be-
fore and after,” to which, in its perfection, we give the
name of Reason. Reason holds, as it were, the ba-
lance between the passive and active powers of the
Mind. It is fed and nourished by the impressions of
the one: it grows and moves by the energy of the other.
It has several stages or degrees, of which the first is
Conception.
6 * G. R. A. M M A. R.
Grammar.
By conception, we mean that faculty which enables
-- the Mind to apprehend one portion of existence, sepa-
Conception, rately from all others.
In other words, the first act, or
exercise of the reasoning power is to conceive one object,
orthing, as one. Hence arises in Language the noun; for
“the noun is the name of a thing.” Here it is that almost
all the modern writers on Grammar have erred. They
seem to have considered no such power in the Mind to
be necessary, and no such act to be performed. They
seem to have supposed that things, or objects, affected
the Mind, as such, by their own power; and that the
Mind was quite passive in this respect. When we ex-
amine this fundamental part of their system, we find
the greatest possible confusion of terms. According
to one, the first elements of thought are ideas, another
calls them objects, a third sensations, and so forth. If
you ask what is meant by these respective terms, you
are still more bewildered. “An idea,” says one, “is
that which the Mind is applied about whilst thinking.”
A most vague and insignificant expression, then, it must
surely be ; and yet it has been justly observed, that
“vague and insignificant forms of speech and abuse of
Language have so long passed for mysteries of Science;
and hard and misapplied words, with little or no mean-
ing, have by prescription such a right to be mistaken
for deep learning and height of speculation, that it will
not be easy to persuade either those who speak or
those who hear them, that they are but the covers of
ignorance and hinderance of true knowledge.” All this
is eminently true of the abuse and misapplication of
the word idea, which had a perfectly distinct and spe-
cific meaning, until it was in an evil hour made “to
stand for whatsoever is the object of the understanding
when a man thinks,” or “whatever is meant by phan-
tasm, notion, species, or whatever it is which the Mind
can be employed about in thinking”—from that moment
the word idea became so extremely convenient to per-
sons, who did not much like the trouble of thinking, it
served as such a maid of all work, in the family of
Lady ALMA, the Mind, that nothing was either too
high or too low for it. “Seneca was not too heavy,
nor Plautus too light;” and persons, who, in the com-
mon phrase, “never had two ideas in their lives,”
would give you “ their ideas” on politics or the wea-
ther, on the flavour of venison, or the right of universal
suffrage, with equal facility and fluency
Some of these ideas, it has been said, are simple,
and some complex. In the former the Mind is passive,
in the latter there is an act of the Mind combining
several simple ideas into one complex one ; but this
distinction has been altogether denied, in more recent
times; and we have been told, that “it is as improper
to speak of a complex idea, as it would be to call a
constellation a complex star.” But be these ideas
simple, or complex; be they ideas of sensation, or
ideas of reflection ; ideas of mode, of substance, or of
relation, the great difficulty is to understand in every
case, how each idea exists as one ; how it is bounded,
limited, and set out in the Mind; and this, we say,
cannot be done in any case without an act of the Mind,
an exercise of the peculiar faculty which we call con-
ception.
What one set of writers say of ideas, another set say
of objects. “An object in general,” says Condillac,
“ is whatever is presented to the senses, or to the
Mind.” Happy definition | But still the question re-
turns: what constitutes one object P’ what is meant by
one presentation ? Is it the sensation, or thought,
which takes place in a minute, in a second, or in any
other portion of time? Is it the impression made on
one sense, or on one part of the organ of that sense ?
Is it the sensation of warmth, for instance, experienced
by the whole body; or that of light experienced by the
eye 2 Is it the impression made on the retina by a
house, by the door of the house, by the panel of the
door, or the pane of the window 2 Is it the altitude
of the building, or the colour of the bricks? These
questions are endless, and perfectly insoluble, if that
which makes an object one thing to the Mind be not an
act of the Mind itself; but if it be an act of the Mind,
then it follows, that with regard to the very first ma-
terials of our knowledge, the Mind is not passive, but
exercises some peculiar faculty; which faculty we call
conception. -
Condillac, indeed, admits, that objects are not dis-
tinguished but by remarking some one or other of them
particularly ; and this particular remarking he calls
Attention; from whence it may perhaps be concluded,
that the difference between him and us is a mere dif-
ference of words ; and that he means, by Attention,
nothing more nor less than what we mean by Concep-
tion. This, however, is an error; for Attention, ac-
cording to him, is a simple faculty, acting only in one
mode, and acting necessarily, from an external cause.
Thus he states, that the cause of Attention to sensible
objects, is an accidental direction of the organs; mani-
festly, therefore, according to him, the Mind is no less
passive in Attention than in sensation.
We say, on the contrary, that in Conception the
Mind acts. The word “to conceive,” in its origin, affords
an easy explanation of the mode of action. This word,
which is derived from con and capio, expresses the
action by which we take up together a portion of our
sensations, as it were water, in some vessel adapted to
contain a certain quantity; for we have before observed,
that sensation is in itself continuous, as an ocean, with-
out shore, or soundings: it does not divide itself into
separate portions, but is divided by the proper faculty
of the Mind. The faculty of Conception, like all other
faculties, operates by certain laws, in a certain direction,
and in a certain manner, for such is its constitution.
It cannot enable us to view things temporal under the
form of eternity, to conceive that a certain time occupies
a certain space; or that an emotion belongs to the
class of sensations; that jealousy, for instance, is red,
or green, or blue, or smooth, or rough, or square, or
triangular. These laws, which regulate the power of
conceiving thoughts, it will be necessary for a while to
consider. -
The first law that we shall notice, is that of extension.
We are so constituted, that we cannot conceive certain
objects otherwise than as occupying Space. The fa-
culty of conceiving them, therefore, presupposes in the
Mind a sense of space; but this sense has again its
necessary laws or modes of operation. In other words,
we cannot conceive Space but as extending in length
and breadth and thickness, and bounded by points and
lines, and surfaces. It is by applying these Laws to
certain objects that we conceive them to be more or
less extended, and to possess different shapes and
forms. To say that we get the idea of Space by the
sense of sight or touch, is to confound our notions of
Introduces
- tory Sec-
lon.
S-N-"
Space.
G R A M M A. R. 7
still, in order to become an element of Reason, it must Introduc-
exist, as one, in the Mind. Even the Conception of toy Sec-
many exists in the Mind as that of one multitude ; and OIl.
if that multitude be divided into distinct parts, so as to \-y-'
Grammar. sense, which imply an existence in Space; it is to reverse
<-- the order of knowledge; for if the Mind were originally
- unfurnished with a peculiar faculty, enabling, and
indeed compelling it to refer the sensations of sight
and touch to some part of Space, it could no more
acquire an idea of Space from those sensations, than
from the emotions of gratitude or fear. This peculiar
faculty, applied to the sensations of sight and touch, of
hearing, taste, and smell, enables us to conceive our
own bodily existence, and that of the external World.
According as we apply it more or less comprehensively,
we conceive the existence of objects larger or more
minute : and according as we exercise it with more or
less care and attention, the external forms and dispo-
sition of objects. appear to us more or less accurately
defined. It is not, therefore, the external object which
necessarily gives shape and form to the Conception; but
the Conception, which by its own act embraces a given
portion of space, and thus gives shape and form to the
external object. . . . -
Similar observations may be made on the law of
be numerically reckoned, the number, whatever it may
be, is still contemplated as one number. Simple Con-
ception indeed could never have advanced us beyond
the notion of an unit or integer ; it is by the aid of
the other reasoning faculties, which we shall hereafter
notice, that we are enabled to form the complex Con-
ceptions of number, and so to build up the whole Science
of Arithmetic.
Conceptions succeed each other indifferently, whe- Identity.
ther they are like or unlike; but the Mind can only
number them by classing them, and can only class
them by their similarity; which similarity, when com-
plete, is in the contemplation of the Mind Identity.
Much has been said of the source from which we
derive the notion of our own personal identity. Surely
if any thing is essential, not only to Reason, but to
Feeling, to Will, and even to Consciousness, it is this
Time.
duration, or Time. To say that time is a complex idea notion. When Descartes invented his famous reason-
gathered from reflexion on the train of other ideas, is ing, Cogito, ergo sum, he clearly assumed his personal
to forget that the very notion of a train is that of a identity: and it is utterly impossible for a Human Being
succession in time, and therefore presupposes what it to reason or think at all, without such an assumption.
is adduced to prove. There is nothing complex in the Even in madness, though the actual identity is often
nature of Time or duration, but it is a form under which confounded, though a man may fancy himself to be
we are necessarily forced to contemplate all things Alexander the Great, or even to be the Almighty, he
external to us, and some things within ourselves. It has before his Mind an imaginary identity: he thinks
is a Law of our nature, and so far as regards its peculiar and acts as one Being, and not as two: and again, in
objects, is inseparable from the Human Mind. But dreams, when we sometimes see ourselves dead, or
again, it is not the lapse of any particular portion of alive, yet the self which we contemplate is a mere
time which necessarily limits the duration of any object imaginary personage, with whom we have a strong
of our thoughts, for we can as easily think and speak sympathy, as we have with the hero of a romance.
of a century as of a second : it is the Mind which con- The contemplator always seems to think and act as a
ceives, as one object, the life of a man, or the gleam of separate individual, and never loses the deep sense of
the lightning, a long year of toil, or a brief moment of identity. - .
delight. - - We are next to inquire into the different kinds of Kinds of
Number, These then are the Laws of simple Conception. What- Conception thus formed; and we shall find that the Conception.
ever occupies a certain portion of time, or of space, or
of both, we consider as one thing, or one thought;
but things or thoughts succeed each other incessantly,
and by dividing sensation into units, we have done no
more than to divide the ocean into drops, or the sand
into grains. A further Law of Conception succeeds.
This faculty takes a more complex form. We distin-
guish Conceptions by their number; and hence, in all
Languages, the noun has a plural number as well as a
singular, in signification, and generally in form. But
as the plural is derived from the singular, so the
power of conceiving many depends on the power of
conceiving one. It has been justly observed by Mr.
Locke, that “there is no idea more simple than that
of unity, or one.”—“Every object our senses are em-
ployed about,” says he, “every idea in our under-
standings, every thought in our Minds brings this idea
along with it.” Now since this is the case, since no
object, no idea, no thought, ever is conceived in our
Minds without this impression of unity, why should we
imagine that any can be so conceived? And if it cam-
not be conceived without such impression, then must
we consider the power by which that impression is
produced as essential to the conception. Before we
can speak or think of any thing, we must first conceive
it to be one. This one may be finite or infinite; that
is, our conception may be perfect or imperfect—but
Ancients were right in dividing them into two, namely
substance and attribute; whence arise in Language the
substantive and adjective. It must be remembered
that we first conceive, as one thing or one thought, a
given portion of sensation, and that those sensations
in their simplest form are limited by the Laws of time
and space; but those Laws are always operating on
the Mind together, though not always with equal force.
Sensations which spread over a large extent of space
may occupy a short time, and those which continue
for a long time may lie within very narrow bounds of
space. Many parts of space too may be contemplated
in one moment of time, and many portions of time
may refer to the same point of space. Our first notion
of substance is personal, unless we should prefer saying
that the notion of substance is derived from that of per-
son; which might perhaps be a more Philosophical mode
of speaking; though the former more immediately ap-
plies to the common arrangements of Grammarians. We
refer all our states of being to a substance called self, to
which each man gives the name of I: and thus I feel and
know that, I am a cause of all the active states of my
being. By an inevitable necessity of my nature I am
led to believe that there must be a cause or causes
foreign to me of all the impressions made on me without
my own act. With respect to myself the Conceptions
which are limited by time and space give me the notions
8 - G R A M M A. R.
Grammar. of Matter and Motion as belonging to me, those which
S-N-" are not so himited give me the notion of Mind. To ex-
ternal causes, therefore, I attribute the same distinctions
of character: and hence the most general notion of ex-
ternal substance is that of a cause of the impressions
formed in me. But one cause often appears to be com-
mon to several different sensations. I therefore conclude
that it is one thing. I have, for instance, the sensations of
heat, and light, and colour, contemporaneously, and this
not once, but often ; and I conclude that there is some
common cause of all these sensations, to which cause
I give the name of Fire. The notion of substance, it is
said, is obscure ; it is no otherwise obscure, than as a
thinking and sentient being cannot sympathize with an
unthinking and insentient one. Obscure as it is said
to be by Philosophers, it is what the common bulk of
mankind consider as the very plainest and clearest of
all their notions. A common man is never troubled
with any doubts of the existence of the table or chair
that he sees before him, any more than he is of his
own personal identity. Others again think, that they
have a very clear motion of the existence of these ex-
ternal objects or substances: they can easily under-
stand how the Mind conceives the cause of a particular
sensation of heat, and a particular sensation of light, to
be one object, called fire ; and contemplates that object
as separate from the sensations produced by it ; but
they cannot understand how the Mind should conceive
as one thing, or thought, or one object of contempla-
tion, a common cause of all similar sensations. Yet
it is certain that men do, and ever have used words
in Language expressive of those common causes, and
that those words have always had the form of sub-
stantives. Much effort has been made to explain this
on the theory of abstraction. These notions have been
called abstract ideas; (a very improper use of the word
idea at least;) and it has been supposed that they
were formed by abstracting, or taking away from each
particular Conception, some circumstance of time or
place. Now it appears to us, that this is an operation
which is rarely, if ever, performed by the Mind. Cer-
tainly, the greater part of the Conceptions represented
to be so formed, may be shown to be produced in a
totally different manner. Thus the Conception of a
straight line, and the consequent Conception of straight-
ness in general, is certainly not formed by abstracting
from various lines, various inequalities; for if it were
so, every man would have a different notion of a straight
line from every other man, and every man would go
on abstracting, and consequently improving his Con
ception of straightness as long as he lived. Whereas,
in truth, the idea of a straight line, as soon as it is
once steadily contemplated in the Mind, is perfect,
and is equally so in all Minds. This could not be the
case, if all Minds did not act by some General Laws;
and since we are so constituted as to be able to reflect
on such Laws, we may separate those reflections from
the general mass of consciousness, as easily as we can
separate a particular sensation from the same mass ;
we may form of each, a Conception, a thought, as dis-
tinct from all other thoughts, as one external object is
conceived to be separate from all other external ob-
jects. The thought of a General Law as single, has no
reference to time or space. Even the Laws of Time and
Space are not supposed to be more or less Laws, or to
have a more or less real existence, at one time, or in Introduc-
one place than in another place, or at a different time.
It is indeed objected, that they have no real existence
at all ; that there is no truth but that of opinion, and
consequently, that “two persons may contradict each
other, and yet both speak truth ;” for such are the
precise words of Mr. Horne Tooke. (Vol. ii. p. 404.)
The same objection may be made with much more force
against the existence of the external World ; for the
learned and pious Bishop BERKELEY has fully shown,
that we have no assurance of the reality of matter or
motion, but that which depends on our instinctive Con-
ception of their existence, as causes of the changes
which we experience in ourselves. But as we are
utterly unable to believe, that there is no truth in our
own existence; and, as we find it hard to imagine, that
this “goodly frame, the earth,” this most “excellent
canopy, the air,” this “brave o'er hanging firmanent,”
this “majestical roof fretted with golden fires,” are all
fictions and non-entities; so it is difficult for us to ima-
gine, that Truth and Virtue, Beauty and Wisdom, Glory
and Happiness, are all empty names : we cannot well
believe that Time and Space are mere fictions of our
own Minds; and yet it is easier to believe this, than to
conceive their existence according to laws different
from those which we actually experience ; it is easier,
for instance, to conceive that there is no real existence
in Space, than that, if it exists, a straight line in space
is not the shortest that can lie between two given points,
or that a figure may be completely bounded by two
straight lines, or that the radii of a circle are unequal,
or that the three angles of a right-lined triangle are
greater or less than two right angles. Hence arises the
distinction of subjective and objective truth. The for-
mer we consider as existing in ourselves, the latter as
existing in objects out of ourselves ; the truth of a
mere opinion is subjective, the truth of the fact to
which that opinion relates is objective ; but if all Truth
were merely subjective, each man’s Mind would be the
only Universe, and it would be a solitary Universe, with-
out a Creator, without Time, or Space, or Matter, or
Motion, or Men, or Angels, or Heavens, or Earth, or
Virtue, or Vice, or Beginning, or Ending—one wild de-
lusion without even a framer of the monstrous spell !
Now since it is utterly impossible to believe this, either
deliberately or instinctively, it follows that there is some
objective truth, and that what a man tryeth, troweth,
or trusteth to (for these are all of the etymological
family of the word Truth) is in itself, more or less,
substantial and permanent. But if this be the case
with our conception of a stone, why not of a man?
And if of the motion of a stone, why not of the
thoughts of a man 2 And if of thoughts bounded by
the Laws of Time and Space, of Number and Identity,
of Good and Evil, why not of those Laws them-
selves? For the purposes of Grammar, it is hardly
necessary to press this argument ; for Language has
been made by men, according to their instinctive
opinions; and certainly the prevalent opinion has al-
ways been, that there is something which the Mind
contemplates, when it reasons on Man in general, as
well as when it reasons on Peter or John. It is pro-
bable that Sir Isaac Newton had some object before his
mind when he argued on light and colours, as well as
a lamp-lighter has, when he lights a lamp ; or as a
tory Sec-
tion.
G R A M M A. R. 9
Grammar. country lass has, when she buys a yard of blue or red rality, and are commonly called mouns of multitude, as Introduc-
\-N-' ribbon at a Fair. a troop, an army, a crowd. º Sec-
Conceptions, then, are either particular, general, or We have shown that a particular conception is Jº" 2
universal.
- formed by the Mind separating and sorting its sensa-
In strictness of speech nothing is particular, but that
Particular, tions and emotions according to certain necessary laws; General.
which occupies only one given portion of time, or of
space, or of both. Thus the emotion of fear at a
certain moment of time; the sensation of warmth at a
given moment, and in a certain part of the body; or
the sensation of brightness in a particular part of the
retina, are all particular conceptions; and it is some-
what remarkable in Language, that men (in early Ages,
and before they had much turned their thoughts to re-
flection) so entirely confounded the subjective and ob-
jective truth, both of sensations and emotions, that
they used the same word to denote both. A man,
for instance, would say indifferently, “I am hot,” or
“ the fire is hot.” So, in common parlance, we say
“ the bird fears the scarecrow,” but Shakspeare says:
We must not make a scareerow of the law,
Setting it up to fear the birds of prey. .
Nor is it only a simple sensation or emotion, of which
we may form a particular conception. We may cer-
tainly conceive as one thing, a substance; that is, many
sensations or emotions united in one common cause ;
whether that cause be active as a person, or passive
as a thing; for the notion of a person is founded on self,
as an active Being, and that of a thing on the same self,
as passive.
These, we say, are the only conceptions which, in
strictness of speech, are absolutely particular; but al-
most all writers call those particulars, which we find
to be identical; thus Peter or John is said to be a
particular individual, though the name, Peter, or John,
is given to an object which I have seen on many par-
ticular occasions, and only know to be identical by
reflection and comparison. In like manner, red is the
name of a colour impressed on my retina to-day and
yesterday, and which I know to be identical : and so
the word, to walk, implies an action which I perform
frequently and know to be the same on all occasions.
We dwell the more on this observation, because it
shows that those who strongly contend for the exist-
ence of nothing but particular objects, overlook the
fact, that what they call particulars are not such in
strictness of speech ; and that, if the only business of
the Mind were to receive impressions, (as Mr. Tooke
says it is,) we could never acquire even what they call a
particular idea or conception; we could never know
that the John of to-day was the same person as the John
of yesterday.
This latter species of particulars, however, is the first
element of Language. We invent signs, not to express
a single impression, but the same impression often
repeated; and these are of three kinds, the simple
sensation or simple quality producing it, which we call
an adjective ; the simple action, which we call a parti-
ciple; and the person or substance in which the cause
of sensation or of action resides, which we call a sub-
stantive.
To these particulars we may add the notion of num-
bers, either distinct or confused ; for the notion of
many objects or many qualities may still be viewed as
a particular notion: and hence arises, not only the
plural of nouns, but the singulars which imply plur
WOL. I.
and arranging them in certain forms more or less dis-
tinct. Thus a certain form is that of Peter; but the
same form applies nearly to John, the same nearly,
though with some other difference, to William ; and so
on. Now, when we contemplate this form as possibly
applicable to a variety of particulars, it constitutes
what we call a general conception ; and these general
conceptions, duly ordered and arranged one within the
other, form genera and species ; and of these, more, or
less distinct, opinion is chiefly formed.
But there is yet one higher step in the power of Universal.
conception, namely, the Universal. This is when we
contemplate the form itself in which our lower concep-
tions were cast. Thus, there is a certain law by which
the Mind can only conceive a straight line in a certain
manner, namely, as length, and as partaking in no de-
gree of curvature, nor interrupted, nor distorted in any
manner whatsoever. Now, the first line that we actually
conceive to be straight, is not exactly so, yet it ap-
proaches to the form in the Mind sufficiently to make
us give it the name of straight. The second, the third,
the fourth, and all successive lines, are perhaps equally
deficient; and, by comparing them with each other,
were there no common standard to refer them to, we
should never attain the knowledge of a simple straight
line. All the lines which we actually see, have breadth
together with their length, all have some curvature or
irregularity; but reflection shows us in the Mind, a line,
which is merely length without breadth, and which lies
evenly between its points. Of this, we are able to
make a distinct conception, to which, when we have
once attained, we find it entirely independent of time or
space, always the same, necessarily true in all its rela-
tions, equally applicable to all the particulars which fall
under it—a law of the Mind—in short, what was alone
and properly called by the Ancients—an idea. The
higher, the nobler, the purer these ideas are, the more
difficult is it for Man to conceive them. They are never
conceived without meditation and effort; and the
deepest meditation, the highest stretch of our faculties,
leaves us lost in admiration and awe at the great over-
powering idea of our Almighty Father.
Conceptions present themselves to our Minds, either
as accompanied, or not accompanied, with a sense of
objective reality. If they are not so accompanied, they
are mere creatures of the imagination : if they are so
accompanied, then, if the object producing them is
past, they are conceptions of memory, and if yet to
come, of eacpectation ; but, when the object is present,
the conception becomes a perception, whether it be of
an external thing, or of a general notion, or of an
idea.
We have hitherto spoken only of the faculty of con- Assertion.
ception, by which the Mind gives its thoughts their
separate forms ; but we have next to see them put into
action, and rendered, as it were, living and operative.
Thoughts and opinions come to us in the mass; and it
is by developing them into their constituent parts, that
we ourselves understand them ; but in order to com-
municate them to others, we must pursue the contrary
process; we must state the parts, and assert their
C
10
G R A M M A. R.
Grammar.
union. Assertion, then, is the faculty which we have
-v-' next to consider: it is, as it were, the uniting and
Affirmative
and nega-
tive.
Moods,
Tenses.
marrying together of two thoughts, and pronouncing
them to be one. Hence the word, which expresses
that function of the Mind, is called, by some writers,
the copula, or bond; but in common Graminars, the
verb ; and we rather adopt the latter term, because the
former may be apt to lead to the erroneous conclusion,
that the Mind in assertion, passively contemplates two
thoughts as united, whereas, it is active in declaring
that union, as it were, by its proper authority; an au-
thority, indeed, often exercised hastily and amiss, but
still the proper act of the Mind itself. Conception,
then, forms nouns, including under that term substan-
tives, adjectives, and even participles ; but these nouns
lie dead and inoperative to any purpose of reasoning,
till they are vivified by the verb, which pronounces their
existence to be a truth. Thus John, eaſisting, good,
loving, are all perfectly intelligible as conceptions of the
Mind; yet so long as they stand alone, we see not what
use is to be made of them in reasoning; but let us in-
troduce the verb, and a truth immediately flows from
the Mind, whence possibly some etymologists might
derive flâna, the verb, and reor, to think, from fiéw, I
flow. Thus we say, John exists, John is good, John
loves, and each of these assertions at once takes the
form of a truth, and becomes, as will be hereafter
shown, the germ and seed of other truths in the Mind.
To assertion belong affirmation and negation. We
declare, that conceptions exist, or that they do not
exist ; and the one of these excludes the other. A
thing cannot be, and not be at the same time ; but as
there are certain conceptions, which are the opposites
of each other, so affirming the one is denying the other.
To say that black is white, is therefore, in common par-
lance, to utter a gross and palpable untruth.
Neither affirmation nor negation, however, is always
positive. The Mind contemplates some truths as ac-
tual, that is to say, it conceives the subjective truth
within itself to be certainly agreeing with the objective
truth in the nature of things, and therefore pronounces
unhesitatingly and distinctly upon its existence; but of
other subjective truths it sees no objective counterpart,
and therefore pronounces them not actual, but probable,
or merely possible. On this distinction, in great mea-
sure, depends what is called the mood of verbs.
Again, we assert truths either with or without re-
ference to the time in which we speak. When we
speak with such reference, that is to say, when we
speak of particulars, we are necessarily compelled to
distinguish the present from the past and future ; and
hence the origin of tenses. When we assert any thing
of ideas, we speak of a truth ever present, and there-
fore we use the present tense in its purest form. Thus,
when we say John is good, we imply a possibility that
he might at some other time be bad; and when we say
John is writing, we do not imply a certainty that he was
not writing at some previous time, and will not be
writing at some future time; but when we say two
and two are four, we not only assert a truth of to-day,
or of this year, or of this century, but a truth which
must be ever present since we cannot conceive it ever
to have beginning or ending. This remark is sufficient
to show that those Grammarians are in error, who
made the signification of time a necessary characteristic
of the verb.
In whatever way we assert any thing, the assertion
is a declaring of some truth, real or supposed; it is a
propounding, or showing forth the existence of the
truth, or in the language of Logicians, it is enunciating
a proposition. This is not done by a peculiar word, as
for instance the word be ; but by the form of the word;
for the word be, in some of its forms, as, to be, and
being, is a simple conception; and so are the words
love, hate, walk, sing, and indeed all others which may
be used as verbs. Mr. Tooke, therefore, was very accu-
rate, as far as regards words, in saying that the verb
was “a moun, and something more;” but when toward
the end of his book, he came to consider what that
“something more” was, he found himself entirely at
a loss, and was forced to break off abruptly; since the
just solution of the difficulty, as we conceive, would
have overturned the whole system, which he had
laboured throughout two ponderous volumes to erect :
it would have shown the Mind of Man to be an active
intelligence, not only in forming conceptions, but in
uttering, declaring, propounding, asserting them to
be truths.
This discovery would have been still more fatal to
Mr. Tooke's Grammatical system, had it been more fully
developed ; for when we come to ask how, and in what
various ways, a truth, or to speak in the phrase of Logi-
cians, a judgment, is asserted, we shall find that this
depends entirely on the different kinds of conceptions;
and, as we have already seen, these kinds are produced
by different acts of the Mind ; whereas Mr. Tooke treats
them all as of one kind only, and all as received by the
Mind from passive impression.
We assert then either existence, or action. If the
former, we either assert it simply of a conception, as
“God exists;” or we assert it conjointly of two con-
ceptions, which are of a nature to exist together, as
the substance with its attribute, or the whole with all
its parts, or the universal with the particular. Thus
we say “God is good,” “two and two are four,” “gra-
titude is a virtue.” If we assert an action, we must
consider it either as proceeding from its cause, or as
received by its passive object, that is to say, we must
employ either the active or the passive verb ; and which-
ever we employ primarily, we must (if such be the na-
ture of the action) add the other secondarily. There
are, indeed, actions which rest in their causes; and the
verbs expressing these, whether active or passive, in
construction, are really of the kind called neuter, or
intransitive, such as, “to rejoice,” “to sing,” and the
Iike.
A truth asserted leads to a further truth, by that
faculty, which Shakspeare calls “discourse,” from the
ancient scholastic and accurate term discursus.
that beautiful and Philosophic passage—
Sure He that made us with such large discourse,
Looking before and after, gave us not
That capability and godlike reason,
To rust in us unused.
This faculty, for want of a better term, we shall call
deduction. It ari es from the comparison of truths;
and as that comparison refers to something common to
both the truths compared, the consequence or inference
to be drawn is always of the nature of a particular, un:
der some universal expressed or understood. Of the
forms of deduction, the most perfect is the syllogism; but
the whole force of the syllogism depends on the univer-
Introduc-
tory Sec-
tion.
The Mind
active in
assertion.
Existence
and action.
Deduction
Hence
G R A M M A. R.
11
we say, but in the first glance and motion of the Mind, Introduc-
as it were, they only appear in their secondary character, tº Sec-
as helps and expletives to the principal words in the 10Its
Sentence. • -V-
The passions must not be overlooked, in considering Passions.
the Mind in its relation to Language. It often happens
Grammar, sal conception which it involves. In the enthymeme,
S-N- which is an imperfect syllogism, the universal, though
not expressed, is understood. It is, therefore, clear,
that the modes by which one truth is deduced from an-
other, imply a power in the Mind beyond that of merely
The deduction may be made
receiving impressions.
from hypothetical premises. Hence arises a further
explanation of the use of moods in the verb. We
assert a truth, not as actual, but as possible, and the
consequence which we deduce becomes a contingency,
necessarily following from the premises, but not neces-
sarily true, because the premises themselves are not
necessarily so.
that an abruptness, a transposition, and that which
might be called an irregularity, if we referred only to
the operations of Reason, become appropriate, and even
necessary forms of speech, when the Mind is under the
influence of passion. The reasoning powers are then
disturbed and imperfect; the emotions become inordi-
nate, the will obtains a preternatural force. Hence
Review. Thus have we enumerated the three faculties which arises the interjection, which some Grammarians have
go to the making up of the reasoning power, and which refused to reckon among the Parts of speech ; but their
are conception, assertion, and deduction, answering to refusal is vain: so long as there are men with human
the simplex apprehensio, judicium, and discursus of passions and affections, there will be interjections in
the Logicians. All continued exercise of Reason re- their speech, words which stand out from the rest,
solves itself into a repeated exertion of these faculties; very significant of emotion though not of conception,
and the only difference is, that the truths produced by defying all rules of construction and arrangement, be-
one deduction serve to enlarge or improve the concep- cause such rules bear reference principally to the power
tions which are employed in framing other assertions of Reason, which is suspended or superseded, when-
and deductions. ever passion produces the animated and expressive in-
secondary Hitherto we have had occasion to notice only those terjection. Passion, too, has given birth to what we
Parts of operations of the Mind, as giving birth to the primary commonly (though not always very appropriately) call
*P* Parts of speech, the noun and verb, the substantive the imperative mood. When Esau says, “Bless me,
and adjective, the pronoun and the participle, which
are in most cultivated Languages distinguished from the
adverb, the conjunction, and the preposition, by being
subject to inflection or change of form, either in the
beginning, the middle, or the end of the words by
which they are expressed. This latter circumstance,
however, is merely accidental, and with respect to the
essential difference of the adverb, conjunction, and
preposition, from the other Parts of speech before men-
tioned, we must repeat what we have before stated,
that the Mind contemplates truths at first in the mass,
and then by reflection breaks down that mass into
certain portions which again are subdivisible ; so that
in asserting one truth, we cast as it were a rapid
glance over the subordinate branches of which it is
composed ; as in viewing the whole beauty and pro-
portion of the Apollo Belvidere, we see at once the
graceful turn of the head, the animated advance of the
arm, and the receding of the opposite foot; or as in
contemplating the agonized frame of the Laocoon, the
two sons with the folds of the serpents which twine
around them, occupy a secondary place in the imagi-
nation. When we come to develope these secondary
parts of the composition, we find in them the same
principles of unity and connection, as in the general
outline of the whole group : and so it is with the
subordinate parts of a sentence; which are, if we may
use the expression, truths within truths, assertions
within assertions. Thus even the long and flowing
sentences of Milton's prose are each reducible either to
an assertion, or at most to a deduction, as their ground-
work; but upon that groundwork are built many
other assertions, which are assumed, though not for-
mally stated as such. Each adverb, each conjunc-
tion, each preposition, contains such subordinate as-
sertion, and of course involves a conception ; it is
therefore true, that these Parts of speech ultimately
resolve themselves into nouns and verbs—ultimately,
even me also, O my father ſ” We feel the earnestness
of the prayer, widely different as it is from a com-
mand. Again, this same example shows us, that the
vocative case of the noun is of similar origin. “O my
father,” is a strong expression of passion; but it is
totally dissevered in construction from the enunciation of
any truth, and has nothing to do with any operation of
Reason. Many other forms and modes of speech take
their character from passion; as may be particularly
observed of the interrogative, so often the result of an
eager desire to know the very fact, which, it may be,
we fear and tremble to assert.
It is to be observed, that all the exercises of all the Conclusion,
human faculties may be clear or obscure, distinct or
confused. Our very consciousness may be that of
mere dotage, our feelings may be blunted, our will
wavering and undetermined, our conceptions vague,
our assertions doubtful, our deductions uncertain, our
passions a chaos. It has been elsewhere said, that
“the thousand nameless affections, and vague opinions,
and slight accidents which pass by us ‘like the idle
wind,” are gradations in the ascent from nothingness to
infinity; these dreams and shadows, and bubbles of
our nature, are a great part of its essence, and the
chief portion of its harmony, and gradually acquire
strength and firmness ; and pass, by no perceptible
steps, into rooted habits and distinctive characteristics.”
Still the channels in which the stream of Mind flows,
so long as it has any current, remain always the same :
the mental faculties which we exercise, so long as we
can exercise any, are subordinated to the same laws,
and display themselves in the same manner. Hence
speech is, in all nations, necessarily formed on the
same principles; and though no one Language was ever
constructed artificially, yet it is astonishing how dis-
tinctly all present the traces of the same mental powers,
operating, in the same manner, on materials so exceed-
ingly different.
12
G R A M M A. R.
Grammar.
UNIVERSAL GRAMMAR.
The general view which we have taken of the Human
\"N-' Mind, appeared to us to be indispensable toward a right
Gradations
of Science.
YWriters.
understanding of what we shall have to say of Gram-
mar, or the Science of Language ; for as we consider
Language to be a signifying or showing forth of the
Mind, it would have been impossible for us to have ren-
dered ourselves intelligible, in explaining the laws or
modes of signification, had we not first stated what we
understood to be the nature of the thing signified.
In different Languages there are some things acci-
dentally different, and some things essentially the same.
It has been owing to accidental circumstances in the
History of Mankind, for instance, that the name of
the Universal Creator, among the Jews, was Jehovah ;
that it is in France Dieu, and in English GoD ; and that
the Latin words locum tenems came to be changed into
the Italian word luogotemente, the French lieutenant,
and the English word, which we spell like the French,
but pronounce leftenant. It is also by accident, that
the word luogotenente signifies, in some parts of Italy,
the Civil Magistrate of a small community; that in
Erance and England the word lieutenant expresses
various ranks in the military and marine services; and
that in Ireland it is applied to the viceroy, or chief
representative of the Sovereign. On the other hand it is
owing to causes which exist more or less permanently
in Human Nature, that in the sounds uttered as Lan-
guage by an Esquimaux, a Hottentot, or a Chinese,
there are certain qualities common to them with the
eloquent voices of a Cicero or a Demosthenes. Though
their articulations vary in many respects, they all
articulate; and the nations that whistled like birds,
or hissed like serpents, never existed but in the inven-
tions of the same sort of travellers, as those who told
of Cynocephali and Cyclopes, and of men who sheltered
their whole body while they slept, by the shade of one
enormous foot. How far the laws of sound and ges-
ture are common to mankind, it is not possible, at
least it is not easy, to determine à priori; those laws,
therefore, we cannot consider in the light of Pure Science;
they form general Grammar, but not universal.
We come, however, in the contemplation of our sub-
ject, to a part of it, which is universally applicable,
and universally true. Cicero or Demosthenes, Plato
or Newton, Dante or Shakspeare, might express sub-
limer, bolder, clearer thoughts than men of a common
stamp, but they could only express them according to
the laws by which every Human Mind must necessarily
act in conceiving and uttering thought. Here then we
arrive at Universal Grammar, at the Pure Science,
which places this part of knowledge on an immovable
basis, renders it demonstrable and certain, and connects
it with that TRUTH, which is one and uniform through
all Ages, and which rashness and ignorance perpetually
assail, but can never subdue.
It is far from our intention to assert, that Universal
Grammar has hitherto been so successfully cultivated,
as to leave to future investigators no hope of improving
this Science. Its principles have certainly been no
where laid down with that happy and lucid order, which
has rendered Euclid's Elements, for above two thousand
years, a text book in Geometry. Much, however, has
been done. The ancient Greek and Latin writers have
traced all the principal paths of the labyrinth, and
elegant edifices of Science have been raised in modern
times by such authors as SANCTIUS, VossIUs, the writers
of Pont RoyAL, and the learned and amiable HARRIs.
The last of these writers, as being not only most familiar
to the English reader, but most rich in ancient autho-
rities confirmatory of his system, we shall follow as our
principal, though not sole guide at present,
“Those things which are first to Nature,” says
Harris, “are not first to Man. Nature begins from
causes, and thence descends to effects. Human per-
ceptions first open upon effects, and thence by slow
degrees ascend to causes.” And this is well illustrated
by Ammonius with reference to speech : “Even a child,”
says he, “knows how to put a sentence together, and to
say Socrates walketh ; but how to resolve this sentence
into a noun and a verb, and these again into syllables,
and syllables into letters, here he is at a loss.” Hence
we may see, that by the very constitution of our nature
the most complex things are most familiar to us, that
the most general laws, by the very reason that they are
most general, and most constantly in action, become
habitual to us without our reflecting upon, and con-
sequently without our understanding them. We con-
form to the complex and intricate laws of vision, we
judge of distances and magnitudes by the angles which
objects subtend, and yet during a great part of our lives
we have not the most distant suspicion that any such
things as angles exist, or that they are subtended on
the retina; nay, ninety-nine men out of a hundred, and
probably a much greater proportion, exercise the power
of vision throughout their whole lives, without so much
as wasting a thought on its laws. So it is in regard to
speech. All men, even the lowest, can speak their
Mother Tongue; yet how many of this multitude can
neither write nor read ; how many of those who read
know nothing even of the Grammar of their own Lan-
guage ; and how many who have been instructed so
far, have never studied Universal Grammar ! . In this
Science, as well as in all other things, the observation
which we have above made, holds true; namely, that
human perceptions open first upon effects, and thence
ascend to causes. Men first notice the practice of
speech, as the exercise of some natural faculty, which
proceeds, as it were, spontaneously from the wish of
communicating their thoughts and feelings. By and
by they observe that this faculty operates partly from
sudden impulses, and gives birth to expressions not
easily to be analyzed into any component parts, as
in the ejaculations of Philoctetes, which fill up many
lines in the Greek Tragedy representing his sufferings;
and that on the other hand, it is in far greater part the
result of thought, and distinguishable into portions
separately intelligible. Every discourse, however long,
consists of sentences. These are combinations of speech
Universal
Grammar.
\-N/~
Order of
study.
G R A M M A. R.
13
Grammar.
\-V-4
Sentences.
Enunciative
sentences.
which are obvious to all persons; and therefore, before
we proceed to analyze speech any further, it may be use-
ful to observe the different kinds of sentences; but our
analysis must not stop there; for it is equally obvious,
that sentences consist of words, and that every word has
some separate force or meaning. Here, however, the
power of dividing speech into significant portions ends;
for though words are made of syllables, and syllables of
letters, yet these two last subdivisions relate wholly to
the sound, and not to the signification. A syllable or a
letter may possibly be significant, as the English pro-
nouns I and Me; but then they become words, and are
so to be treated in the construction of a sentence.
Words, then, are the primary integers of significant
Language; but these may be distinguished according to
their separate properties and uses, into two or more
classes, which Grammarians call Parts of speech.
These Parts of speech, therefore, we shall consider sepa-
rately.
§ 1. Of Sentences.
A sentence is a number of words put together, and
obtaining, from their combination, a particular power of
enunciating some truth, real. or supposed, absolute or
conditional, or else of expressing some distinct passion,
together with its object. Sentences, therefore, are of
two kinds, according as they are directed to these two
different ends.
The enunciative sentence obtains its power of ex-
pressing fact or opinion, by the connection of the words
of which it is composed : for Aristotle observes, (what
indeed is self-evident,) that of those words which are
spoken without connection, there is no one either true
or false ; as for instance, “man”—“ white”—“ run-
meth”—“conquereth.” But let us put together only
these two words—
Jesus wept,
and we have recorded a Historical fact most affecting in
itself, and furnishing abundant food for deep and inter-
esting meditation.
When we read in SHAKSPEARE,
The quality of Mercy is not strained;
we immediately perceive the enunciation of a beautiful
truth, which is again presented under an expressive form
to the imagination by the following lines:
It droppeth as the gentle rain from heaven
Upon the place beneath.
So when Milton says:
w in the soul
Are many lesser faculties, which serve
Reason, as chief.
A truth respecting our intellectual (as the former re-
spected our moral) nature is distinctly asserted.
This kind of sentence may enumerate many particu-
lars, all bearing on one point of time, or referring to one
general idea ; such is the following picturesque delinea-
tion of what presented itself to young Orlando when in
pacing through the forest, “chewing the cud of sweet
and bitter fancy,” he threw his eye aside—
Under an oak, whose boughs were moss'd with age,
And high top bald, of dry antiquity,
A wretched ragged man, o'ergrown with hair,
Lay sleeping on his back; about his neck,
A green and gilded snake had wreath'd itself, .
Who, with her head, nimble in threats, approach'd
The opening of his mouth; but suddenly
Seeing Orlando, it unlimk’d itself,
And with indented glides, did slip away
Into a bush; under which bush's shade
A lioness, with udders all drawn dry,
Lay couching, head on ground with cat-like watch,
When that the sleeping man should stir.
Such also is the following argumentative sentence in
Bishop TAYLoR's Sermon on the Duties of the Tongue,
urging the Christian office of administering consolation
to the afflicted :
God hath given us speech, and the endearments of society, and
pleasantness of conversation, and powers of seasonable discourse,
arguments to allay the sorrow by abating our apprehensions; and
taking out the sting, or telling the periods of comfort, or exciting
hope, or urging a precept, and reconciling our affections, and reciting
promises, or telling stories of the Divine mercy, or changing it into
duty, or making the burden less by comparing it with greater, or by
proving it to be less than we deserve, and that it is so intended and
may become the instrument of virtue.
Sentences.
The enunciative sentence easily becomes interroga- Interroga-
tive.
be stated as beyond the sphere of the speaker's know-
ledge, or as being doubted by him, and desirable to be
known. This is commonly effected in Language by a
slight transposition of the words, sometimes by a mere
change of accentuation. As in Sterne's celebrated
Sermon, “We trust that we have a good conscience.”—
“Trust that we have a good conscience?” Again, by
transposing the lines above quoted, we make them in-
terrogations.
Is not the quality of Mercy strained P
Droppeth it as the gentle rain from heaven 2
But it is to be observed, that as some degree of emo-
tion is implied in the very nature of an interrogation,
so it is often used by the Poets, Orators, and others, to
give life and animation to their style, although no
doubt exists in their mind or that of their hearers;
and the matter which is questioned in point of form, is
meant to be asserted in point of fact. Thus when the
Poet says—
Who to dumb forgetfulness a prey,
This pleasing, anxious being e'er resign'd?
he means positively to assert that no one ever quitted
life with indifference. The humorous speech of Fal-
staff, when personating the King, illustrates our obser-
vation.
Shall the blessed sun of heaven prove a micher, and eat black-
berries P A question not to be asked. Shall the son of England
prove a thief and take purses P A question to be asked.
For the same fact which is simply asserted may tive.
Again, the enunciative sentence may be conditional or Condi-
contingent; that is, it may be placed in dependence on, tional.
or in counterbalance against some other truth ; as in
Macbeth—
If it were done, when 'tis done, then 'twere well
It were done quickly. —
Or in Hamlet—
Duller should'st thou be than the fat weed
That rots itself at ease by Lethe's stream,
Wouldst thou not stir in this."—
Or again in Macbeth, where the contingency takes place
in spite of obstacles which might be supposed capable of
preventing it:—
14
G R A M M A. R.
Grammar.
\-y-/
Passionate
Sentences,
Active and
passive.
Though Birnam wood be come to Dunsinane,
And thou oppos'd, being of no woman born,
Yet will I try the last. -
In all these and similar instances, the enunciation of
a truth is the immediate object in view ; but sentences
of another class owe their form and construction solely to
some passion, of which they indicate the object. And
it is to be observed, that the indication of an object of
passion is essential to the constituting such sentences as
these. Thus, when the Nurse, in Romeo and Juliet, on
finding her young lady dead, cries and laments, voci-
ferously, and the parents enter, asking “What noise is
here 2 What is the matter?” Her answers, “O
lamentable day !” “O heavy day!” are not sentences;
for though they plainly show the grief with which she is
agitated, they do not at all express the cause or object of
that grief. But when Hamlet cries—
Oh I that this too, too solid flesh would melt,
Thaw and resolve itself into a dew
we perceive a distinct expression of the wish to be de-
livered of life, as burthensome to him. The sentence is
as complete and Grammatical, and much more Poetic
than if the place of the interjection Oh! had been sup-
plied by a verb; for instead of an impassioned and
beautiful line, it would have been perfectly absurd, if
the Poet had said:
I wish that this too solid flesh would melt
We may observe that these passionate sentences, com-
bine quite as readily as the enunciative ones, with de-
pendent sentences, as “Oh ! that I had wings like a
dove | Then would I flee away and be at rest;” which
implies the same fact as the sentence “If I had wings
like a dove, I would flee away,” &c.
Sentences of the passionate kind either express a
passive feeling, as admiration and its contrary, or an
active volition, as desire and its contrary. Of the for-
mer kind, is that passage of the Apostle, “Oh the
depth of the riches both of the wisdom and know-
ledge of God!” and the line of Milton, comparing the
receptacle of the Fallen Spirits with their former happy
Seat—
Oh! how unlike the place from which they fell!
Those sentences which express desire and aversion are
commonly expressed by the mood called imperative;
but they as often imply humble supplication or mild
entreaty, as authoritative command. Thus the Poet
describes Adam gently calling on Eve to awake—
He, with voice
Mild as when Zephyrus on Flora breathes,
Her hand soft touching, whisper'd thus: awake
My fairest, my espous'd, my latest found,
Heav'n's last, best gift, my ever new delight,
Awake /
And again, when our first parents offer up in lowly
adoration their morning orisons—they say—
Hail universal Lord, be bounteous still
To give us only good!
But these emotions are widely different from others,
expressed in the same form of sentence: as when King
Henry says to Hotspur–
Send us your prisoners by the speediest means,
Or you shall hear from us in such a sort
As may displease you
Or when Juliet exclaims
Gallop apace, ye fi’ry-footed steeds.
To Phoebus' mansion
Or when Macbeth cries to the ghost of Banquo—
Avaunt / and quit my sight! Let the earth hide thee!
We have already had occasion to notice, that some
sentences are simple, and others complex. We have
only to add, that instances occur in which a sentence is
manifestly left imperfect, and that with great beauty, as
in the well-known line of Virgil :
Quos ego—sed motos prastat componere fluctus.
And so Satan first addresses Beelzebub, in the open
ing of the Paradise Lost:
If thou best he—but oh! how chang'd, how fallen:
In both these cases, the words, though not in them-
selves fully and clearly expressive of the thought which
we may suppose to be in the speaker's mind, are yet
not wholly unconnected, and therefore show at once,
that they are parts of sentences which, indeed, it would
be easy for the reader to fill up in his own imagina-
tion.
Mr. HARRIs distinguishes sentences into two classes,
as we have done above; only he gives them the names
of sentences of assertion, and sentences of volition.
Other writers have classed them somewhat differently,
but yet with reference to similar Principles. Thus Am-
monius states that there are four kinds of sentences
besides the enunciative, namely, the interrogative, the
optative, the deprecatory, and the imperative ; but
that in the enunciative alone is contained truth or
falsehood.
Sentences.
\-v-f
Imperfect
Sentences.
Harris.
We have observed, that sentences are composed of Aristotle.
words, of which latter every one has some meaning; an
this agrees with the definition of a sentence given by
Aristotle: AOI'OX 33 ºwvi) avv64T) amaavtuki), is via
Aépm kað’ avrò o muaivet Tu. We may remark also, that
these distinctions were familiar to the old Grammarians;
and hence Priscian observes, that the parts of a sentence
must be called parts with reference to the whole, so
that in a sentence in which the word vires occurs, we
must not divide it into two words, vi and res, though
these might be significant in another sentence; because
in the former case, they would have no signification with
reference to the whole sentence. But again, as sen-
tences are made up of words, there must be some rules
for constructing them, and these rules must depend on
the species of words which, as we have observed, are
commonly called by Grammarians, the Parts of speech ;
our next inquiry, therefore, must be, how those species
are to be distinguished, or by what rule they are to be
distributed into classes.
§ 2. Of the Parts of speech.
Some Principles of classification are better than others.
It is not sufficient that we comprehend all our motions
on a given subject under certain heads; but we must
be prepared to show, why we choose those heads rather
than others. If we are right in our notion of Pure
Science, it will guide us to the proper choice, among
these various modes of treating the same subject. It
will present to us one idea, which masters and directs
all the others, and will show us how the subordinate
ideas proceed from this common root. -
It is, however, necessary first to explain what we
mean by different classes of words. Take the following
Sentence.
Parts of
speech.
Classing of
words.
G R A M M A. R.
I5
Grammar.
S-2-’
Various
opinions.
The man that hath no music in himself,
And is not fill'd with concord of sweet sounds,
Is fit for treasons—
Here we know that various Grammatical writers call
the word the an article; man, music, concord, and
sounds, substantives, or nouns substantive ; no, sweet,
and fit, adjectives, or nouns adjective ; that, and him-
self, pronouns; hath and is, verbs; moved, a participle;
not, an adverb; and, a conjunction; in, with, and for,
prepositions.
The first question that occurs to us is, whether these
classes themselves are all recognised in all Languages,
and by all Grammarians? And a very little experience
will show us that they are not so. The same thing has
happened in Grammar, which has happened in all other
Sciences. Some authors have divided speech into two
parts, some into three, four, and so on to ten or twelve.
Others again have made their division depend on the
supposed utility of words; others on their variation;
others on the external objects to which they refer, and
others on the mental operations which they express.
On this point, it is worth while to hear what QUINC-
TILIAN says, in the IVth chapter of his Ist Book–
“On the number of the Parts of speech, there is but
little agreement. For the Ancients, amongst whom
were ARISTOTLE and THEoDECTEs, laid it down, that
there were only verbs and nouns, and combinatives,
(convinctiones,) intimating that there was in verbs the
force of speech, in mouns the matter, (because what we
speak is one thing, and what we speak about is ano-
ther,) and that the union of these was affected by the
combinatives, which I know most persons call conjunc-
tions; but I think the former word answers better to
the original Greek advěeguos. By degrees the Phi-
losophers, and particularly the Stoics, augmented the
number ; and first, they added to the combinative the
article, then the preposition. To the noun they added
the appellative, then the pronoun, and then the partici-
ple, being of a mixed nature with the verb; and finally
to the verb itself they subjoined the adverb. Our
(Latin) Language does not require articles, and there-
fore they are scattered among the other Parts of speech;
but we have added to the others the interjection. Some
writers of good repute, however, follow the doctrine
of the eight Parts of speech, as ARISTARCHUs, and in
our own day PALAMon, who have ranked the vocable,
or appellative under the moun, as one of its species;
whilst those who divide it from the noun, make nine
Parts. Again there are others who divide the vocable
from the appellative, calling by the former name all
bodies distinguishable by sight and touch, as a bed, or
a house, and by the latter what is not distinguishable
by one or both these means, as the wind, heaven,
virtue, God. These last-mentioned authors, too, add
what they call asseverations, as (the Latin) Heu 1 and
attractations, as (the Latin) fasceatim ; but these dis-
tinctions I cannot approve. As to the question whe-
ther or not the vocable or appellative should be called
wrpoo`myopia, and ranked under the noun, as it is a
matter of little moment, I leave it to the free judgment
of my readers.”
Although Quinctilian, who only touches on Grammar
incidentally, speaks of Aristotle as maintaining that
there were three Parts of speech, yet WARRO says truly
that Aristotle asserted there were two Parts of speech,
the verb and the noun. In fact, Aristotle, in his Book
rep; ºppºnveias, treats of those two alone; considering
that of them is made a perfect sentence, as “Socrates
philosophises:” and therefore PRISCIAN says, “ the
Parts of speech are, according to the Logicians, two,
viz. the noun and the verb, because those alone, con-
joined by their own force, make up a full speech, or
sentence ; but they called the other parts syncatagore-
matics, or consignificants.” Priscian himself, however,
maintained that there were eight Parts of speech ; and
he seems to have been implicitly followed for many
centuries; but, though it is of little consequence whether
we give the name of Parts to particular divisions or
subdivisions, it is of great importance to determine
on what Principle speech should be divided and sub-
divided. -
Recurring, therefore, to the sentence above quoted
from Shakspeare, we will inquire how the words can
be Grammatically distinguished : and many various
modes will readily present themselves:
Parts of
Speech.
1. It may be observed that some of the words admit Variable
of variation, and others do not.
Thus man may be and invari-
varied into man's and men: hath into have, hast, had, able.
and having : sweet into sweeter, and Sweetest, &c. and,
on the contrary, the words, the, in, and, not, &c. cannot
be altered. But this is manifestly not an essential
distinction, since it does not take place in the same
manner in all Languages; but, on the contrary, every
Language is distinguished, more or less, from every
other, by peculiar modes of varying its words. Thus
the Greek, Hebrew, Sanscrit, and Arabic Languages,
have a variation in some or all of their nouns to mark
the dual number, which is unknown to most other
Tongues. So the Greeks and Romans varied their
adjectives by the triple change of gender, number, and
case ; whereas the English never vary them in any of
those ways. If then the distinction of variable and in-
variable will not answer our purpose, let us look for
some one that is more essential.
2. Having considered in the former instance the Affective
sound of the word, we shall now take a distinction ****
which arises from its signification.
divides the Parts of speech into two classes, of which
he says, “the first includes the natural signs of senti-
ment, the other the arbitrary signs of ideas: the former
constitute the language of the heart, and may be called
affective; the latter belong to the language of the un-
derstanding, and are discursive.” It is manifest that
the Principle of this distinction is universal, because
all men must be influenced by sentiment and under-
standing, and all Languages must find some means of
distinguishing these different faculties in Language.
But the question is, whether this distinction is suf-
ficient to account for the different classes of words: and
most assuredly it is not so ; for although there are some
words which express only the objects of sentiment, and
others which express only the objects of knowledge,
yet there are many which express both together, and
many which directly express neither. Nor is it always
sufficient to use a word of one class in order to convey
either an emotion or a truth. These circumstances more
frequently depend upon the combination, than upon the
distinction of words.
SlWe
Thus M. BEAUzEE
3. Let us now come to a third distinction, that of the Object and
Port Royal Grammarians, who say, “ the greatest manner.
distinction of what passes in our Minds, is to say that
we may consider in it the objects of our thoughts, and
16
G R A M M A. R.
Grammar, the form or manner of our thoughts, of which latter the
•-y— principal is reasoning or judging ; but to this must be
Necessary
words and
abbrevia-
tions.
Principal
and acces-
SOry.
Noun and
verb.
added the other movements of the soul, as desire, com-
mand, interrogation, &c.” This, again, is a distinction
universally applicable to Language in point of significa-
tion: and when we come to apply it to existing Lan-
guages, it will be found sufficiently accurate.
4. But it has been observed, that this may be done
with more or less facility and despatch ; and that some
words are absolutely necessary for the communication
of thought, whilst others may be considered as abbre-
viations, in order to make the communication more
rapid and easy ; as a sledge may have been first con-
structed to draw along heavy goods, and may have
been afterwards placed on wheels to add celerity to the
motion. Such is the theory of Mr. HoRNE TookE, and
so far as we are here considering it, that theory is per-
fectly just. -
5. The words which are necessary for communicating
the thought in any given sentence with the utmost sim-
plicity, may well be called principals, and those which
only help to make out the thought more fully and dis-
tinctly may be called accessories. These are the terms
employed by Mr. HARRIs, and consequently his theory
so far coincides with that of Mr. Tooke. Mr. Harris,
however, adds, that the principals are significant by
themselves, and the accessories significant by relation:
whereas, Mr. Tooke says that the necessary words are
signs of things, and the abbreviations are signs of ne-
cessary words. We shall hereafter have occasion to
enter more at large into this part of his doctrine. It
is sufficient at present for us to observe, that that
doctrine does not interfere with the fundamental Prin-
ciple of classification in all Grammars which deserve
the name; that is to say, of all which have proceeded
on the signification of words, and not merely on their
sound.
Now, that principle, in whatever terms it is clothed
or expressed, is, that the moun and the verb are the
primary Parts of speech ; and that without them,
neither can a truth be enunciated, nor a passion be
expressed, in combination with its object. This Prin-
ciple is the most ancient. It boasts the support of the
greatest of Philosophers, of him, whom for many Ages
even Christianity recognised by the title of “the divine,”
as approaching the nearest of all Heathens to the divine
light of the Gospel. PLATO, in his Dialogue called
The Sophist, having most profoundly and unanswerably
argued on the nature of Truth, thus speaks of Lan-
guage: “We have in Language two kinds of manifes-
tation respecting existence, the one called mouns, the
other verbs. We call the manifestation of action a verb ;
but that sign of speech which is imposed on the agent
himself a noun. Therefore, of nouns alone, uttered in
any order, no sentence (or rational speech) can be com-
posed, neither can it be composed of verbs without
nouns; thus “goes,’ ‘ runs,’ ‘sleeps,’ and such other
words as signify action, even though they should all be
repeated in succession, would not make up a sentence.
And again, if any one should say ‘ lion,’ ‘ stag,”
‘ horse,” or should repeat the names of all the things
which do the actions before mentioned, still no sen-
tence would be made up by all this enumeration; for,
neither in the one way, nor in the other, do the words
spoken manifest any real action, or inaction, or declare
that any thing exists, or does not exist, until the verbs
are mixed with the nouns. Then, at length, the very
first interweaving of them together, makes a sentence,
however short; thus, if any one should say, “ Man
learns,' you would pronounce at once that it was a
sentence, though as short a one as possible; for then at
last, something is declared which either exists, or has
been done, or is doing, or will be done; and the
speaker does not merely name things, but limits, and
marks out their existence, by interweaving verbs with
nouns, and then, at last, we say “ he discourses, and
does not merely recite words.’” The only great name
that for nearly 2000 years was ever brought into com-
petition with Plato's, was that of his scholar ARISTOTLE:
but Aristotle also, as we have already seen, agreed with
Plato, in stating the noun and the verb as the two
primary Parts of speech, and indeed the only Parts
necessary to be considered in the formation of a simple
sentence. In other passages of his Works, looking at
the composition of Language in a more general point
of view, he enumerated three Parts, viz. the noun, the
verb, and the connective ; and, finally, in his Treatise
On Poetry, c. xx. he enumerates two Parts of speech as
significant, viz. the noun and verb ; and two as non-
significant, viz. the article and conjunction.
Parts of
Speech.
\-N-
Aristotle.
The doctrine that the noun and verb are the primary Previous
Parts of speech, is incontestable.
Grammarian, calls them the most animated ; and all
Grammarians concede to them, at least, the superiority
over all the other Parts of speech, in whatever manner
they choose to account for their preference.
not, however, inclined to adopt this, as the first step
in our methodical arrangement; because we conceive
that by approaching to the most general idea of speech,
we shall find it easier to reconcile the apparent dif-
ferences, and to correct the real errors of the different
Grammatical systems. We have already defined speech
to be the language of articulate sounds; and Language
to be any intentional mode of communicating the Mind.
Our most general idea of speech, therefore, is, that it is
any intentional mode of communicating the mind by
articulate sounds. Now keeping in view this idea, let
us see how it will apply to the doctrines of those
Grammarians whom we have already mentioned, in
respect to the mode of distributing speech into its
Parts.
We are
Apol,LoNIUS, the *
10Il.
When writers of any eminence advance a particular Combina-
doctrine, we may generally be persuaded, that it is not tion of
wholly destitute of foundation: although, from the na- theories.
tural partiality which men have for their own thoughts,
they may probably rank such doctrines higher than they
deserve. All the different theories which we have here
noticed are true, to a certain degree, and, by combining
them, together, we may perhaps attain to the best and
clearest view of our subject.
In the method which we are disposed to pursue, we
should say, that the Principle of M. BEAUZEE first
merits attention. There are words which are simply
affective, namely, interjections, which express no ope-
ration of Reason whatever; all other words are dis-
cursive, inasmuch as they may be employed in express-
ing the operations of Reason. Again, all words which
are employed in reasoning must be considered, with
reference to the sentence in which they are so employed,
either as principals or as accessories ; we say with re-
ference to the sentence in which they are employed ;
for it is here that a great error is often committed by
G R A M M A. R.
I7
Grammar. Grammarians.
They seem not to advert to the circum-
\-º-' stance that speech is an expression of the Mind, when
Divisions
of Gram-
m arians,
actually engaged in some operation. They treat words
as if they were corporeal substances, cast in a mould,
for use. Now, the very same words that are principals
in one sentence, may become accessories in the next.
The principal words in a sentence are of course meces-
sary for the communication of thought ; and thus we
combine the Principle of HARRIs with that of TookE.
We cannot, however, communicate what we do not
comprehend ; and in order to comprehend any thought,
we must first conceive an object, and then either assert
something respecting it, or express some emotion in
connection with it. Here, therefore, the theory of the
PoRT Roy AL Grammarians properly finds its place;
for they comprehend alike the assertion of a truth and
the expression of an emotion under the words, “the
manner of thinking.” With respect to the writers who
divide words, according as they are susceptible of
variation, or the contrary, although it is true that such
a quality exists in the words of most Languages, yet
we have shown that it cannot be taken into considera-
tion in treating of Universal Grammar, being a circum-
stance merely contingent and accidental. '
The result, therefore, of the preceding remarks, is,
that we consider speech as intended to communicate
either Passion or Reason ; when it communicates mere
Passion, without any precise object, it supplies the Part
of speech called the interjection ; when it communicates
Passion and at the same time indicates an object, it
indirectly reasons, and therefore requires the same
Parts of speech which are required in reasoning. Now
the Parts of speech required in reasoning are either
such as are necessary to form a simple sentence, or such
as serve for accessories, in order to give complexity to
sentences; but a simple sentence cannot be formed
without a noun and a verb, and is immediately formed
by putting a noun and a verb together. The noun and
the verb then are the necessary Parts of speech, the
former serving to name the conception, the latter to
supply in reasoning the assertion, or in passion the
emotion. There is, however, one observation very
important to be made with respect to the necessary
Parts of speech, namely, that every verb involves a
noun; that is to say, we cannot assert a truth, or ex-
press an emotion, which truth or emotion may not be
considered by the Mind as a conception. Thus, if we
say “God exists,” we excite in the Mind the two dis-
tinct conceptions of “God” and “Existence,” as much
as if we said, “God is in existence :” and so if we say
“Come Antony,” we excite the conception of coming,
as well as of Antony; but the difference is, that the
words “come” and “exists” are not presented to the
hearer as mere objects of thought, but as modes of
thinking about other objects, viz. “Antony” and
“God.”
Thus have we fixed the Principle on which the noun
and the verb are to be reckoned among the Parts of
speech ; and this Principle will readily enable us to clear
up several difficulties which occur in the subdivision of
these classes.
First, with respect to nouns; the old Grammarians in
general divided them into nouns substantive and nouns
adjective ; but R. Johnson, Harris, Lowth, and others,
consider the substantive alone as a noun ; and Harris
ranks the adjective with the verb, under the common
WOL. I.
name of attributive ; whilst Tooke, in consequence of Parts of
his singular notions respecting the Mind, asserts that
the adjective is literally and truly a substantive. That
author also contends that those words which “compose
the bulk of every Language,” and are commonly, though
improperly, called abstract nouns, are not even ne-
cessary Parts of speech, but abbreviations, or signs
of other words. The pronoun was originally considered
as a noun, and afterwards, though treated separately,
was still deemed a secondary sort of noun ; but Harris
distinguishes, in this respect, the pronoun personal
from the others, and considers only the former as a
noun, ranking the latter, together with the article,
among the accessory parts of speech. Lastly, the
participle, which was originally so called, because it
was thought to partake of the nature of a noun and of
a verb, (to be a noun when it formed the subject of a
proposition, and a verb when it formed the predicate,)
is wholly excluded by Harris from the class of mouns,
and referred to that of attributives: whilst Tooke (who,
however, does not explain what he means by a verb)
calls it the verb adjectived.
Speech.
Our Principle, on the other hand, will bring us back Nouns.
very nearly to the ancient distinction of nouns. For
a noun, in our view, is only the name of a conception,
or object of thought ; thus the “sun,” a “horse,” or a
“man,” is an object of thought, and as such may have
a name, which name is a noun. So “brightness,”
“strength,” “wisdom,” “thinking,” “moving,” “shin-
ing,” are objects of thought, and have names, and those
names also are nouns.
These nouns are considered substantively, when in Substan-
reasoning upon them, or asserting any thing of them, tive.
we make them the subject of'the assertion, and consider
them as that in which something else exists. Thus
“man,” as a thought, has its own peculiar relations to
other thoughts ; so have “wisdom,” “thinking,”
“strength,” “moving,” “brightness,” and “shining ;”
and all these, so considered, become nouns substantive.
But we may also contemplate each of these concep-Adjective.
tions only as existing in another object, as thinking or
wisdom in a man ; strength or moving in a horse ;
brightness or shining in the sun ; and this we say is
employing the same noun adjectively; because we are
forced to adjoin it to the substantive, in which alone
we contemplate it as existing. When we say, “a wise
man,” or a “thinking man,” we contemplate wisdom
or thought only as existing in that man ; so when we
say “the shining sun,” or “the bright sun,” “the strong
horse,” or “the moving horse,” we speak of the concep-
tions of shining or brightness, motion or rest, only
as modes of qualifying our use of the conceptions
sun and horse : and when we do this, the name of
the qualifying conception, is properly called a noun
adjective.
3.
The substantive conceptions which the Mind forms Pronoun.
represent the person communicating the thought, the
person to whom it is communicated, or some other
person or thing. Hence the Mind forms three classes
of conceptions; but a name being given to each of
these classes, stands for the class, a noun for many
nouns; and hence it is called the pronoun , and upon
the pronominal substantives depend the pronominal
adjectives. The article, which has been often treated
as a pronoun, represents the exercise of that faculty
of the Mind by which we distinguish the universal
D
18
G R A M M A. R.
Grammar.
conception from the particular. It seems, therefore, to
~~ be improperly ranked among the principal or necessary
Participle.
Verbs.
Accessory
Parts.
Simple sen-
tence.
Compli-
cated sell-
tence.
Parts of speech.
The participle is clearly a noum adjective, which
includes the idea of action, and consequently of time ;
for the “bright sun,” and the “shining sun,” differ but
little in signification, except that, in the latter, the sun is
considered as producing brightness by its own act. And
if the phrase be varied, and an assertion be introduced,
the assertive power depends not at all on the participle,
but on the verb, which must necessarily be added, as
the sun “ is bright,” the sun “is shining.”
With respect to the other principal or necessary Part
of speech, the verb, it is only material now to remark,
that those who confound it with the adjective and the
participle, overlook its peculiar function, which is that
of asserting; as the function of the noun is that of
naming. As to the separate classes of verbs, the verb
substantive, the transitive, the active, the passive, &c.
since these have not been treated of by Grammarians
as separate Parts of speech, it will not be necessary to
notice them in this part of our Work.
But the great dispute, especially in modern times,
has been with respect to the accessory Parts of speech,
the nature of which has been illustrated by a variety of
similes. They have been said to be like stones in the
summit or curve of an arch, or like the springs of a
vehicle, or like the flag of a ship, or the hair of a man,
or like the nails and cement uniting the wood and
stones of an edifice ; and hence some persons have
contended that they are only significant by relation ;
some that they are not Parts of speech ; and some that
they are not even words but particles.—Thus APULEIUs
says, “they are no more to be considered as Parts of
speech than the flag is to be considered a part of the
ship, or the hair a part of the man ; or, at least, in
the compacting and fitting together of a sentence, they
only perform the office of nails, or pitch, or mortar.”
PRIscIAN, however, an acute and intelligent Gram-
marian, observes, that if these words are not to be
considered as Parts of speech because they serve to
connect together others which are Parts, we must say
that the muscles and sinews of a man are no parts of
a man ; and he, therefore, concludes by declaring his
opinion, that the noun and verb are the principal and
chief Parts of speech, but that these others are the
subordinate and appendant Parts.
The decision of this and similar questions will he
easily made, if we only advert to the mental operations
which these accessory words express ; and in order to
explain this, we must first ask, what words in a sentence
are accessories 2 This question again is answered by
referring to what we have said of sentences. In a
simple sentence, all the words are principals. Thus
“Man is fit,” contains two nouns, which are the names
of two conceptions, viz. “man” and “fitness,” and the
assertion of their coincidence by the verb “is ;" and
moreover, since the conception of fitness is regarded
as existing not separately but in the other conception,
man, the word “fit” is an adjective and “man” is a
substantive. The same would be the case if the place
of the noun “man” were supplied by the pronoun
“he.” and that of the adjective “fit,” by the participle
•uited.
Such is the case when the sentence is simple ; but
we are next to consider how a simple sentence is ren-
dered complex; and this is no otherwise done than by
engrafting on it other sentences; but in these latter
the conceptions only are expressed, and the assertive
part is assumed or understood. Thus, if referring to
the passage before quoted from Shakspeare, we say
“Man is fit,” we may be asked, What is the fitness
or aptitude of which you are speaking? The answer
must be “it is treasonable.” And again, if we are asked,
What is the man of whom you make these assertions?
We may say “he is unmusical ; and suppressing the
assertions in the two secondary sentences we may form
of the whole one complex sentence, thus, “unmusical
men possess treasonable aptitudes.” -
In this first process of complication we find only
words capable of being used as principals, viz. nouns,
substantive or adjective ; pronouns, participles, and
verbs ; but suppose we again resolve these into their
constituent conceptions and assertions; suppose we
ask what do you mean when you speak of a treasonable
fitness, or aptitude 2 We may answer, we mean that
the fitness looks to treason ; treason is before the fit-
mess, (as its mark or object,) the fitness is for treason.
Here it is plain that the word “for” involves the con-
ception of foreness, (or objectiveness,) and applies that
conception to the other conception of treason : but it
does so still more rapidly and obscurely, than in the
cases before supposed ; and hence it is that in this
second process of complication we meet with words
which are no longer thought significant, and therefore
no longer called nouns or verbs, but articles, adverbs,
eonjunctions, and prepositions ; and these words are
the more numerous and frequent of occurrence, in pro-
portion as sentences are rendered more complex by
subdividing the primary truth into many others. Thus,
as the word “treasonable” may be supplied by the
words “for treasons,” so the word “unmusical” may
be supplied first by the words “hath no music in him-
self,” and secondly, by the words “is not moved with
concord of sweet sounds;” both which and many similar
modes of speech, consist of various aggregations of
sentences in which the subordinate assertions are as-
sumed by the Mind in the manner already shown.
The words which, by use, come to be most fre-
quently employed in any particular Language for these
secondary purposes, often lose their primary significa-
tion, and perhaps undergo some little change of sound ;
from which circumstances a great dispute has arisen
among Grammarians whether they are significant words
or not. Thus the preposition for, which, as we have
shown, conveys the conception of foreness, is nothing
more than the word fore in foremost, before, fore and
aft, and the like words and phrases; but by use, and
by the slight change which it has undergone, it has
come to lose the property of forming a principal part
in a sentence. These circumstances, however, it must
be observed, are merely accidental ; they may happen
to the same conception in one Language and not in
another ; and, therefore, they cannot form a just
Scientific criterion between the Parts of speech ; but
on the other hand, those Parts may, and must, be dis-
tinguished by the different operations of Mind which
they express ; and as we have seen that the operations,
expressed by the articles, adverbs, conjunctions, and
prepositions, are clearly distinguishable from those
expressed by the nouns, pronouns, verbs, and partici.
ples, inasmuch as they relate to a subordinate step in
JParts of
Speech.
Further
complica-
tion.
hange of
significa-
ion.
G R A M M A. R.
19
Grammar, the analysis of thought; so there can be no great diffi-
S--~' culty or impropriety in calling them accessories, with
reference to the others, which we call principals.
Etymology From what we have said, it will not appear strange,
of accessory
words.
Koerber.
Tooke.
that the accessory words should be for the most part
traceable to their origin as principals; that is to say,
that the Parts of speech last mentioned should in
general be found to have been once used (with little or
no difference of sound) as nouns and verbs. It has been
supposed that this was a new discovery of Mr. HoRNE
TookE's, and in many parts of his Work he seems to
have entertained that notion himself; how justly may
be seen from the very title of a little Treatise, by C.
Kor:RBER, printed at Jena in 1712, and called, Lericon
Particularum Ebraarum, vel potius Nominum et Ver-
borum, vulgó pro particulis habitorum. This writer
says, in his Preface, that his tutor Danzius taught that
“ most, if not all the separate particles, were in their
own nature nouns ;” that this was indeed a “new and
unheard of hypothesis;” but that on investigation the
reader would find reason to conclude universally (in
respect to the Hebrew Language at least) that “all the
separate particles are either nouns or verbs.” His words
are these : Particulae separatae si non omnes certé
pleraque suá maturâ sunt nomina—hanc thesin hac-
tenus novam et inauditam, &c. and again, Omnes omnimó
Ebraeorum particulas separatas aut nomina esse aut
verba.
Koerber illustrates his position by comparing the He-
brew particles with radical words, both in that and the
cognate Languages, particularly in the Arabic. Among
the instances which he gives, are the following, viz.
Juarta, near, being the same as Latus, side.
Praeter, beside or beyond. . Defectus, deficiency.
Terminus, boundary.
Inter, between . . . . . . ... . . . Distinctus, divided.
Post, after. . . . . . . . . . . . . . . . Tergum, the back.
Quoque, also. . . . . . . . . . . . . . Adde, add.
Wel, or . . . . . . . . . . . * * a s = • * Elige, choose.
He even explains the interjection Lo / as being identical
with the pronoun of the third person ; and suggests
that the termination of the accusative case is a noun,
signifying object.
Whether or not Mr. Tooke ever saw this little Treatise
by Koerber, or any other of similar import, is imma-
terial. His theory may, probably, have been a bond ſide
discovery, so far as regarded his own reflections, though
not one that was entirely new to the World. But he
seems to us to have connected with it a very material
error in Grammar, namely, that because a word was
once a noun, it always remained so, and consequently
that adverbs, conjunctions, &c. expressed no new or dif-
ferent operation of the Mind, and were not to be con-
sidered as separate Parts of speech, so far at least as
related to their signification. Had Mr. TookE been as
well acquainted with the writings of Plato, as he was
with those of the old English and Saxon authors, which
he studied with such meritorious industry, he would
hardly have fallen into this error; for he would have
perceived that speech received its forms from the Mind;
he would have acknowledged with that great Philoso-
pher, that “thought and speech are the same; only the
internal and silent discourse of the Mind with herself,
is called by us Atávota, thought, or cogitation ; but the
effusion of the Mind, through the lips, in articulate
sound, is called Adºos, or rational speech.” It is there-
fore the Mind that shapes the sentence into its principal
parts and accessories; it is the Mind which distributes
alike the principal and the accessory parts into subdivi-
sions, according as they are necessary to its own distin-
guishable operations.
Parts of
Speech.
S-N-a-’
Those ancient Grammarians who acknowledged only Ancients.
three Parts of speech, viz. the noun, verb, and conjunc-
tion, ranked some of the Parts which we here call acces-
sories under the principal Parts. Thus Apollonius of
Alexandria and Priscian rank the adverb under the
verb, and with them agrees Harris, who calls the adverb
a secondary attributive ; but Alexander Aphrodisiensis,
who is followed by Boethius, observes, that it is some-
times more properly referred to the class of nouns:
and so Tooke asserts some adverbs to be nouns and
some verbs. The preposition which was referred by
Dionysius and Priscian to the conjunctions, is on a
similar Principle included by Harris with the common
conjunction in the class of connectives; and Tooke
distributes both prepositions and conjunctions (in most
instances rightly, so far as their etymology is con-
cerned) among the verbs and nouns. Lastly, the article
appears to have most disturbed the Grammarians in
their arrangements; for Fabius says it was first reckoned
among conjunctions; and we have seen that, when
Aristotle divided speech into four Parts, he separated
the article from the conjunction, making of it a class
apart from the three other Parts of speech. Vossius
inclines to rank it among nouns, like a pronoun ; but
Harris having divided the accessory Parts of speech
into definitives and connectives, makes the article a
branch of the former. Tooke says that our article
the is the imperative mood of the Anglo-Saxon verb
thean, to take | Lastly, Scaliger says, the article does
not exist in Latin, is superfluous in Greek, and is, in
French, the idle instrument of a chattering people.
Since, in this diversity of opinions, we find no com-
mon view of any Principle which connects itself with
º ſº e
the idea of Language before laid down, we are com-
pelled to seek a new division. We say, therefore, that
the accessory Parts of speech represent operations of
the Mind, which from their frequent recurrence have
become habitual, and from their absolute necessity in
modifying other thoughts, must be found more or less
in all Languages. It is true that these operations are
not performed by all men with the same distinctness,
and therefore do not exist among all nations in the
same degree of perfection; and lastly, it is true that in
some Languages they are expressed by separate words,
and in other Languages by different inflections of the
same word. Hence a close connection is found be-
tween the prepositions of one Language and the cases
of another ; between the auxiliary verbs of one Lan-
guage and the tenses of another. Hence, too, the com-
parison of adjectives, always effected in Latin by dif-
ferent terminations, is sometimes effected in English
by adverbs prefixed to the adjective. In short, num-
berless illustrations of this remark will easily occur to
the recollection of any person at all acquainted with
different Languages, ancient or modern, barbarous or
refined.
Of the operations which we have described, one, and
that not the least essential, consists in determining
whether we view any given conception as an universal,
or a particular; and if as a particular, whether as a
New Princi-
ple propos-
d
Article.
p 2
20
G R A M M A R.
Grammar.
N-V-Z
Preposition.
certain, or an uncertain one ; and if certain, whether
of one known class, or another known class; and so
forth. Thus there is a certain conception of the Mind
expressed by the word “man:” but if we employ that
expression for the purpose of communicating the con-
ception, it is necessary that those who hear us should
know with what degree of particularity it is to be ap-
plied; for it would be one thing to say, that, according
to our idea of human nature, Man is universally bene-
volent; and another to say, that men in general are so ;
and a third to say that any individual man, under given
circumstances, is so ; and a fourth to say, that this or
that man is so. Of these different degrees of limitation
some may be marked by separate words; and of those
words, some may express a conception so distinct and
self-evident, as to be capable of forming a simple sen-
tence, in which case we should reckon them as prono-
minal adjectives, among the principal Parts of speech ;
as when we say, “this is good,” “that is bad,” the
words this and that are pronominal adjectives. But
since we cannot say, “the is good,” or, “a is good,”
and since these words the and a serve no other purpose
but to define and particularize some other conception,
and do not even perform this function completely,
without reference to some further conceptions, we may,
in those Languages in which they exist, reckon them
as a separate Part of speech, under the name of the
article.
The word preposition is badly chosen, as Vossius
observes, from its use (and even that use not without
exception) in the Latin Language; nevertheless, it has
become sufficiently intelligible to signify a class of
words which describe another sort of mental operation.
When one object is placed in a certain relation to ano-
ther object, whether it be a relation of time, of space,
of instrumentality, causation, or the like, the conception
of that relation serves as a bond to unite them in the
secondary parts of a sentence. That expression may
form part of a word, as “to overleap a fence ;” or it
Thus have we distributed words into various
ſ 1. used in enunciative sentences.
I. principal words.
1. primarily.
2. if expressive
WORDS «
2. accessory words.
adverb.)
The mental operations which these various classes of
words represent, are obviously distinct, but it by no
means follows thence that the words themselves are
so ; that a word which has been employed as a substan-
tive may not also be employed as a conjunction ; or
that the very sound by which we have expressed an
assertion may not be used as a preposition or an inter-
may constitute a separate word, as “ to leap over a
fence ;” and in the latter instance the word over is
called a preposition, which we therefore do not hesitate
to rank as a separate Part of speech.
As the preposition connects conceptions, the conjunc-
tion connects assertions; or, as it is commonly ex-
pressed, the preposition joins nouns, the conjunction
verbs, and consequently sentences. By connecting, in
both instances, we mean showing the relations, whether
of agreement or disagreement; and these also may be
expressed either in the form of the verb, or by means
of a separate particle: of which a sentence before quoted
affords an illustration—
Duller should'st thou be than the fat weed,
Wouldst thou not stir in this;–
where, if rendered into the more common expression,
“ if thou wouldst not stir,” the relation between stirring.
in the cause, and being dull, would be expressed by
the word if, to which we therefore give the name of a
conjunction. Hence it appears, that the conjunction may
not improperly be reckoned a distinct Part of speech,
since it expresses a distinct operation of the Mind.
More doubt may perhaps exist as to the adverb, a
class in which Grammarians have often confounded
words of very various effect and import, such as inter-
jections and conjunctions, Neither do we, in this in-
stance, any more than in those of the participle and
preposition, pay much regard to the etymology of the
word adverb; but we take it as a word in common use,
and applying to a large class of words which describe
operations of the Mind very distinguishable from those
which we have already considered. The adverb either
expresses a conception which serves to modify another
conception of quality or action ; or else it expresses a
conception of time, place, or the like, by which the
assertion itself is modified ; in either case it serves to
modify by its own force, and not, like the preposition,
as an intermediate bond between other conceptions.
classes according to the following Table:—
1. the moun, or name of a conception.
1. if expressive of substance (the substantive.)
of quality.
1. without action (the adjective.)
2. with action (the participle.)
2. secondarily (the pronoun.)
2. the verb, or expression of an assertion.
I. defining the extent of a conception as universal or particular (the article.)
2. expressing the relation of one substantive to another (the preposition.)
3. connecting one assertion with another (the conjunction.)
4. modifying either a conception of quality or action, or else an assertion (the
2. used either in passionate sentences, or as separate expressions of passion (the interjection.)
jection. In short, there is no reason why one word
should not successively travel through all the different
classes which we have here stated ; for we must ob-
serve, that words do not communicate thought by
their separate power and effect only, but infinitely more
so by their connection: and consequently the mode of
connecting the signs, and not the signs themselves,
Parts of
Speech
\-e-N-
Conjunc-
tion.
Adverb.
G R A M M A. R.
21
Grammar.
determines their place in any given class. The first
\-y-' exercise of the reasoning power, we have seen, is Con-
The noun.
Its origin.
ception ; and of all our mental operations, whether
relative to the external World, or to the laws of Mind
itself, conceptions may be formed; and to all the con-
ceptions which we form, names may be given ; and
those names are nouns: and therefore it is not sur-
prising that all other words, except interjections, should
be historically traceable to nouns as their origin; and
since Reason and Passion are so complicated in Man, we
must not wonder that a connection is often to be found
between interjections and nouns; or that the Latin
va, probably pronounced in ancient times wae, should
be the Scottish substantive wae, and our woe. Surely
this affords no proof, or shadow of a proof, that the
different uses of the same, or different words, do not
depend on the different exercise of the mental faculties;
but, on the contrary, it absolutely demonstrates the
necessity of some mental operation to distinguish be-
tween the different meanings, force, and effect of the
same sign, as employed on different occasions.
§ 3. Of Nouns.
Having thus settled the classes of words, we shall
attempt to explain them in order; and first we begin
with that which, according to all systems, stands first in
importance; that is to say, the noun.
“It is by the nouns,” says CourT DE GEBELIN,
“that we designate all the Beings which exist. We
render them known instantly by these means, as if they
were placed before our eyes. Thus, in the most solitary
retreat, in the most profound obscurity, we are able to
pass in review the universality of Beings, to represent
to ourselves our parents, our friends, all that we have
most dear, all that has struck us, all that may instruct
or amuse us ; and in pronouncing their names we may
reason on them with our associates. We thus keep a
register of all that is, and of all that we know ; even
of those things which we have not seen, but which have
been made known to us by means of their relation to
other things already known to us. Let us not be asto-
nished, then, that Man, who speaks of every thing, who
studies every thing, who takes note of every thing,
should have given names to all things that exist, to his
body and its different parts, to his soul, to his faculties,
to that prodigious number of Beings which cover the
earth or are hid in its bosom, which fill the waters, and
move in the air; that he gives names to the mountains,
the rivers, the rocks, the woods, the stars, to his dwell-
ings, to his fields, to the fruits on which he feeds, to
the instruments of all kinds with which he executes the
greatest labours, to all the Beings which compose his
society, or, that the memory of those illustrious per-
sons who deserve well of mankind by their benefactions
and their talents, is perpetuated by their names from
Age to Age. Man does more. He gives names to
objects not in existence, to multitudes of Beings, as if
they formed but a single individual, and often to the
qualities of objects, in order that he may be able to
speak of them in the same manner as he does of objects
really existing.”
This great power of the noun is to be attributed
solely to that faculty of the Mind by which it is formed:
and that power we have called Conception. Every act
of this power produces one thought, presents to our
view one object, more or less distinct. We conceive a
certain impression to which we give a name, be it “red”
or “white,” “John” or “Peter,” “man” or “woman,”
“ animal” or “vegetable,” “virtue” or “vice;” or what-
soever else we can distinguish from the mass of continued
consciousness which constitutes our Being.
We do not name every impression that we receive,
or every act that we perform. In truth, we do not
name any one separately and distinctly from all others.
It would be useless to do so, in a single instance : it
would be impossible to do so, in all. But we name
what often occurs to us. We have often a sensation
of colour; we call it “white :” we have often a feeling
of pleasure ; we call it “joyous :” we often see an object
which affects us with peculiar sentiments of regard or
aversion ; we call it “father” or “enemy :” we often
meditate on thoughts, which appear to us amiable or
the reverse: we call them “benevolence” or “ hatred.”
In this manner it is that our catalogue of names is
formed. -
Each of these thoughts or conceptions has its natural
and proper limits; but these we do not always very
accurately observe. No man confounds “red” with
“white,” but he confounds “whitish” with “reddish.”
A boy does not think his hoop square, but he knows
not whether it is circular, or elliptical. Thus it is, that
men do not agree in their opinions of many things, to
which they nevertheless agree in giving some common
names; otherwise it would be impossible for them to
communicate to each other any thing like the thoughts
or feelings which they respectively entertain.
Every noun, then, is the name of a class of similar,
or identical thoughts. Let us see how these classes
may themselves be classed. “Many Grammarians,” says
VossIUs, “ and among them some of the highest ce-
lebrity, first distribute the noun into proper and ap-
pellative, and then into substantive and adjective ; but
erroneously ; since even the proper noun is a substan-
tive, inasmuch as it subsists by itself in speech. But
let us seek our method from the Schools. Our great
Stagirite first divides to Šv (or that which is) into that
which subsists by itself, and is therefore called sub-
stance, and that which exists in another as in its sub-
ject, and is therefore called attribute. Afterwards he
proceeds to distinguish substance into primary and
secondary, the primary being an individual, the Se-
condary a genus or species. By parity of reason, there-
fore, we should divide the noun first into that which
subsists by itself in speech, and is called substantive,
and that which needs the addition of a substantive in
speech, and is called adjective, and afterwards we should
distribute the substantive into that which belongs to a
single thing, and is called proper, and that which
comprehends many, and is commonly called appellative.”
It is to be observed, that some ancient writers gave the
name of noun only to the substantive proper, and that
of vocable (vocabulum) to the appellative ; which latter
has been, in modern times, erroneously called an abstract
Il Ollºl.
We adopt the distribution of Vossius. We call
both substantives and adjectives mouns ; for both are
names of conceptions, and they are nothing more.
They do not imply any assertion respecting these con
ceptions; and herein they are clearly distinguished
from verbs. It is true that the adjective agrees with
the verb in expressing, not substance, but attribute ;
Nouns.
\-N/~~
Classifica-
tion of
IłOUllſ!.S.
22
G R A M M A. R.
Grammar.
and therefore it is, that Harris, and some other Gram-
<-2–2 marians, rank these two classes of words together under
Substantive
and adjec-
tive,
Not con-
vertible.
the title of attributives. We do not deny that this
arrangement is so far correct; but we say that it inter-
feres with the method which we conceive it most ad-
visable to pursue, as the most direct and Scientific. As
Vossius justly postpones the consideration of the classes
of substantives to the distinction between substance
and attribute ; so we postpone the consideration of the
assertion of an attribute, to the consideration of those
conceptions both of substance and of attribute, which
must necessarily precede all assertion. This we con-
ceive to be strictly the order of Science. Language is a
communication of the Mind ; the Mind, as far as it is
capable of communication, consists of thoughts and
feelings. Thoughts are formed by the reasoning power.
The reasoning power is divided into three faculties, Con-
ception, Assertion, and Deduction ; but Conception
necessarily precedes Assertion, because we cannot
assert that any thing exists until we know what that
thing is.
The noun, then, is the name of a conception : indeed
the English word noun is nothing but a corrupt pro-
nunciation of the French mom, which, like the Italian
mome, was again a corruption of the Latin momen, and
this latter was of common origin with the Greek évoua,
and answered exactly to our word name. It is of con-
sequence to observe, that the proper function of the
noun is to name, and nothing more ; for red is as much
the name of a certain colour, as Peter is the name of a
certain man, or England of a certain Country; and in
like manner virtue is as much the name of a certain
thought, as a ship is the name of a certain thing ; all
these, therefore, and whatever other words serve to
name any conception of the Mind, are nouns.
These conceptions, as has been repeatedly shown,
are either conceptions of substance, or conceptions of
attribute. This distinction, however profound it may
be, is nevertheless, and, perhaps, for that very reason,
so perfectly obvious in practice, that no man, however
ignorant, can possibly confound the kinds of conception
to which it relates. No man can imagine, that in the
phrase “a white horse,” the word “white” does not de-
note a quality belonging to the “horse;” or that in the
phrase “glorious victory,” the word “glorious” does not
denote a quality belonging to victory. No man, when
he says “the sun is shining,” thinks of the sun as an
attribute of shining; but, on the contrary, he considers
“shining” to be an energy, or property, or quality, or
attribute of the sun.
It has been contended that “the substantive and
adjective are frequently convertible without the smallest
change of meaning,” and in proof of this, it is asserted
that we may indifferently say “a perverse nature, or
a natural perversity:” now surely, although we would
not assert that the person advancing such an illus-
tration was altogether of “a perverse nature,” we
might without offence attribute his opinion, on this
particular point, to a little “natural perversity.” In
the one case, the friends of the person in question
would understand us to assert, that his whole mind
was tainted with the vices of obstinacy and self-
willedness, that he wilfully shut his eyes against the
Truth, and maintained opinions which he knew to
be wrong in Literature, in Philosophy, in Politics, and
in Religion—a description of his character, which
would naturally occasion them to take great offence.
In the other case, they would understand us to give S-N-
him credit for such reading and literary acquirements,
as might well have corrected what we look upon as
an error; and they could hardly take it amiss that
we attributed that error, rather to a slight defect, from
which the best natures are not wholly exempt, than to
gross ignorance, or total want of understanding. So
much for the particular expressions quoted as proof that
substantives and adjectives may be convertible without
the smallest change of meaning: on the other hand,
the well known instance of a “a chestnut horse,” and
a “horse chestnut,” affords a ludicrous example of a
change of meaning produced by such convertibility.
The fact is, that in all such instances, the views taken
by the Mind are different, according as it regards the
one conception, or the other as principal; just as the
man who is on the Eastern side of the street considers
the Western to be the opposite side ; whilst he who is
on the Western side thinks the same of the Eastern.
We may speak of a “religious life,” or of “vital reli-
gion.” In the one case, we are considering the con-
ception of “life” in the largest extent, as that which
must necessarily form the basis of our assertion, and
which may be differently viewed, according as it is
put in connection with the secondary conceptions of
religion, irreligion, business, pleasure, or the like : in
the other case, we take the conception of “religion”
as the most comprehensive object of thought, and then
limit it by the conception of life, or vitality. It is ob-
jected, that this limitation may as regularly be effected
by a substantive as by an adjective; and that “Man’s
life,” or “the life of Man” is exactly equivalent to
“human life ;” which we by no means deny ; but then
it must be observed, that the sentence takes a different
form, and instead of simple becomes complex; the in-
troduction of the casual termination s, in one instance,
and of the preposition of in the other, effecting such
complexity. Dr. WALLIS, indeed, in his valuable
English Grammar, first published in 1653, treats the
genitive “Man’s” as an adjective. He says, Adjectivum
possessivum fit & quovis substantivo (sive singulari, sive
plurali) addito S wt Man's nature, the nature of
Man, natura humana vel hominis ; men's nature, the
nature of men, natura humana vel homimum. But no
other Grammarian has adopted this notion, and the
Principle on which it rests, would equally go to prove that
all the oblique cases of substantives, in all Languages,
should be considered as adjectives ; for Mr. Tooke has
justly observed, that these cases cannot stand alone ;
although he has not noticed that this is owing to the
complexity of the sentences in which they are used.
The last-mentioned writer contends, that “the ad- Depend on
Nouns.
jective is equally and altogether as much the name different
of a thing, as the noun substantive.”
right; but if he means by thing, what, probably,
nineteen-twentieths of his readers suppose him to
mean, namely, an external substance, such as “a
horse,” or “a man,” or “the globe of the sun,” or “a
grain of the light dust of the balance,” he is as clearly
wrong. “Red” and “white,” “soft” and “hard,”
“good” and “bad,” “virtuous” and “wicked” do not
represent any such things as the latter ; but they do
represent conceptions of the Mind, some of which con-
ceptions may be considered as belonging exclusively to
If he means views of a
by thing, a conception of the Mind, he is perfectly conception.
G R A M M A. R. º
23
Grammar.
S-N-'
Substantive.
Kinds.
external bodies, others as belonging exclusively to mental
existence, and others as common to both. Mr. Tooke
says, he has “confuted the account given of the
adjective by Messrs. de Port Royal,” who “make sub-
stance and accident the foundation of the difference
between substantive and adjective;” but if so, he has
confuted an account given not only by Messrs. de Port
Royal, but by every Grammarian who preceded them
from the time of Aristotle ; and whatever respect we
may entertain for the abilities of Mr. Tooke, (which in
etymology were doubtless great,) we must a little he-
sitate to think that he alone was right, and so many
men of extensive reading, deep reflection, and sound
judgment, were all wrong. But how has he confuted
this doctrine 2 Why, truly, by showing that when a
conception is not regarded as a substance, it may be
regarded as an attribute ; and when it is not regarded
as an attribute, it may be regarded as a substance.
—“There is not any accident whatever,” says he,
“which has not a Grammatical substantive for its sign,
when it is not attributed ; nor is there any substance
whatever which may not have a Grammatical adjective
for its sign, when there is occasion to attribute it;” which
is pretty much like saying, there is not any captain
whatever who may not be degraded, and placed in the
ranks; nor any private soldier whatever who may not
be raised from the ranks and honoured with a captain's
commission ; and therefore there is no difference between
a captain and a private soldier. The premises are in-
contestable: the only fault is, that they have nothing
to do with the conclusion. We trust that in these
remarks we shall not be thought to have treated Mr.
Tooke with too much freedom. We are cautious not
to imitate his example, in calling the opinions which
he controverts “paltry jargon,” or in saying of him, as
he does of the learned and amiable Harris, that he
mistook “ fustian for Philosophy.” These expressions
prove nothing; but it is necessary to come to some
settled opinion on a question so essential to the Science
of Grammar, as whether there is any, and what dis-
tinction between substantives and adjectives; and on
this point, we trust, we have satisfactorily vindicated
the Principle laid down by Aristotle, and adopted by
all Grammarians from his time to that of Mr. Tooke.
The noun substantive, then, is the name of a concep-
tion, or thought, considered as possessing a substan-
tial, that is, independent existence; the noun adjective
is the name of a conception, or thought, considered
as a quality, or attribute of the former.
Of these the noun substantive, to which some writers,
as has been observed, give exclusively the name of
noun, first demands attention : and with respect to it
we shall notice first what is essential, and secondly what
is accidental.
The noun substantive differs essentially in kind and
in gradation ; it differs accidentally in number, gender,
and relation to other nouns or to verbs.
By a difference of kind, we mean that the noun sub-
stantive sometimes expresses a conception of corporeal
impression, and sometimes a conception of mental re-
flection. Conceptions of corporeal impression are ne-
cessarily particular; those of mental reflection are ne-
cessarily universal. By mental reflection we do not
mean the precise recollection of a given particular cor-
poreal impression ; for such recollection we consider to
fall under the same class as the original impression
itself; but we mean the reflection on colour as colour,
on goodness as goodness, on Man as Man, on being as S-V-2
being, and the like ; and thus we come to the ancient
definition of the noun, given by CHARISIUs and
DIoMEDEs, viz. Pars orationis significans rem corpo-
ralem, vel incorporalem.
It is objected, that there is no incorporeal thing Incorporeal.
existing; and as the noun is the name of a thing,
there can be no noun naming that which does not
exist. We answer first, we have nothing to do in this
place with any Metaphysical question as to the real
existence of objects answering to our mental concep-
tions. The only point that we are concerned to prove
is, that conceptions of mental reflection exist, as well
as conceptions of bodily impression; that distinct
thoughts exist, as well as distinct things; for if such
thoughts or conceptions exist, they must have names,
in order to be communicated, and such names will be
the very nouns in question. Now it is a curious re-
mark, which is made by Mr. Tooke, in his IId
volume, and which indeed had occurred to us many
years before the publication of that book, that “the
terminating k or g is the only difference between think
and thing.” Possibly that learned etymologist would
have been inclined to derive “think” from “thing,”
rather than “thing” from “think;” and possibly, as an
etymologist, that is as a Historian of Language, he
might have been right; but as a Philosophical Gram-
marian, he would certainly have been wrong ; for, let
us ask what it is that Language communicates ? Not
things certainly, but thoughts—thoughts of things, or
thoughts of thoughts. Now let us take any noun, for
instance the word “house.” We say, that this is the
name of a thing ; and we will admit that the person
using it had seen the thing, before he used the name;
but how came it to be a thing in the contemplation of
his Mind? How came he to form a conception of it?
We shall perhaps be answered, because he saw it.
But what is seeing? An affection of the nerves of
the eye. Now it never happens, when we see any one
thing distinctly, that it equally affects all the nerves of
the eye. Therefore, when the “house” was first seen,
other things were also seen. What was it that dis-
tinguished these different affections of the eye into
marks, signs, or thoughts of different things 2 What
was it that made the “house,” in particular, a thing, in
the contemplation of the Thinking Principle. Could such
an effect have been produced otherwise than by an act of
the Thinking Principle itself? And if this was an act
of the Thinking Principle, then the thought was parent
of the thing, so far, at least, as Grammar can have any
thing to do with it, namely, as capable of being known
to the Mind, and communicable by Language. Let us
pursue this investigation a little further. The word
“house” does not signify a thing seen only at one
moment of our lives; let us suppose, then, that we do
in fact see the same house several times; it must
necessarily happen, that we see it under very different
circumstances. As we approach to, or recede from it,
every step makes it affect the eye differently, both as to
form and colour. What is it that still makes us con-
sider the cause of these different impressions as one
thing 2 Plainly the Thinking Principle ; so that again,
and in a second degree, the thought is parent of the
thing; and be it observed, that it is not until after this
secondary process has been oftentimes repeated that
24
G R A M M A. R.
Grammar, we give the thing a name.
Now what are the acts of
S-v- the thinking principle, by which we form the concep-
Ideas.
Certainty.
tion of this external object as one thing 2 The apply-
ing to it certain laws of the Mind, which enable us to
say that it is “square,” or “circular,” the referring it
to certain laws of our Physical organization, which
enable us to call it “red,” or “white,” the comparing it
with other objects, so as to determine that it is “high or
low,” the dividing it into its parts and appendages, the
“walls” and the “ roof,” the “doors” and the “win-
dows,” and so forth. Thus, we see, that so simple a
thing as a house, cannot be conceived by the Mind,
unless the Mind has first conceived the ideas of
“square” or “circular,” “red” or “white,” high” or
* low.”
But these ideas are no Physical part or portion of
the corporeal object which we contemplate ; they can-
not be separated from it by any Physical means ; they
do not belong to it more than to any other object with
which they may happen to be associated; they are
therefore incorporeal things, thoughts, conceptions of
mental reflection. Hence it follows that the con-
ceptions of incorporeal things are, in the order of
Nature, prior to the conceptions of corporeal things.
And hence again it follows, that the former are not the
result of any abstraction from the latter, but on the
contrary, the latter are produced by combining together
the former. An abstract idea is therefore a contra-
dictory term ; and, consequently, an abstract moun is an
expression which we think it improper to adopt ; but
an idea, or universal conception, is one of the first and
most necessary conceptions of the Mind, and conse-
quently nouns expressing such conceptions are no less
essential to Language than names of corporeal objects.
They are also equally intelligible. Ask the most ig-
norant man his opinions of “sweetness and sourness,”
“black and white,” “virtue and vice,” and he will
reason on them quite as well as he will on any par-
ticular things or persons to which these qualities be-
long. Does any man ever say that the natural conse-
quence of “victory” is “defeat P* Does he argue
that there is no distinction between “red” and
“green 2° Does he contend that “ingratitude” is the
most acceptable return for “benevolence P’ Assur-
edly not. These terms stand for certain conceptions
in his Mind of which he may have a clear or an indis-
tinct consciousness, just as he may have a clear or an
indistinct recollection of any action that he has wit-
nessed, or of any person that he has seen ; but still
these conceptions are parts of the Mind communicable
by speech ; they bear names, and these names are sub-
stantives of the class under consideration.
It is again objected, that there can be no truth or
certainty in these thoughts, and consequently no pre-
cise meaning in the words by which they are signified,
inasmuch as there is no external standard to which
they can be referred. But where there are no means
of referring to the external standard, it is in fact no
standard at all. Now this must happen, in the great
majority of cases, with regard to corporeal conceptions.
No sooner have I seen “ Peter” or “John,” than he
may take his departure. Shall I then say he is a non-
entity ? And what has Truth or certainty to do with
external existence any more than with internal 2 We
do, in fact, attain greater certainty, and are more confi-
dently persuaded of Truth, in regard to some mental than
we possibly can be in regard to any corporeal concep-
tions. Mathematical demonstration is proverbially clear
and unquestionable ; but Mathematical demonstration is
carried on solely by means of ideal conceptions. If
men were to trust to Physical measurement, aided by
the very nicest instruments, they might be employed for
Ages before they could satisfy themselves that the three
angles of a right-lined triangle were universally equal
to two right angles.
Certain it is that all mental conceptions are of a
Nouns.
*N-"
Distinct-
nature to be apprehended with very different degrees of ness.
distinctness by different Minds applying to them dif.
ferent degrees of attention ; and it is as certain that
the words expressing them are often used loosely and
without much regard to their precise and literal signi-
fication. Thus, Mr. Locke has written two volumes,
principally relating to the word idea ; yet it would be
exceedingly difficult for any person to state what con-
ception Mr. Locke had of that word: and most certainly
he had not the conception which any one Philosopher
before his time ever attached to it. But this is a mere
proof of the abuse of terms, which affords no conclusion
at all against their use. If John happens to be called
Peter by mistake, this circumstance in no degree affects
his personal identity.
Again, it must be observed, that when an universal Union with
idea is coupled with a particular object, the idea may
exist in more or less intensity and vigour, according
to the peculiar mature of the object. “Whiteness”
may exist in snow more absolutely than in paper, and
in paper more than in ivory. “Virtue” may exist in
Peter more eminently than in John. A “square” may
be more truly formed in one mechanical instrument
than in another. To this circumstance is owing the
comparison of adjectives; but it does not affect the
nature of substantives. Whiteness is not less a sub-
stantive, when considered with reference to ivory or
paper, than with reference to snow ; and the virtue of
John, though less than that of Peter, equally belongs
to the universal idea of virtue.
Some confusion may perhaps have arisen from the
common use of the word “substantive,” as applied to
the names of mental and corporeal conceptions. By
“substance” we are apt to understand only material
or bodily substance, that which we can touch and
handle, and weigh and measure ; but this is merely a
verbal difficulty. “Substantive,” in the Grammatical
sense, means that which is considered as having an
independent and separate existence, and of which some-
thing may be affirmed or denied “ substantively,”
without reference to any other thing as its basis and
necessary support. This notion, then, of independent
existence is the real characteristic of all those words
which are called nouns substantive : it applies equally
to ideas and to bodies, to thoughts and to things. .
What we here call ideas are those mental concep-
tions, to which that name was originally given—con-
ceptions which, in the language of Plato, though they
run through the particular objects which participate
their nature, are separate from each individual—putav
ièéav Štú troXXèv, evos ékáarov kelpevov xwpts—Thus, as
he elsewhere says, the idea or mental conception of a
circle is different from every visible impression of a
circle; for the former is perfectly round, whereas each
of the latter has some part or other approaching to a
straight line. M. Condillac (who calls ideas, abstract
particular
objects.
Not mere
denomina-
tions.
G R A M M A. R. 25
Grammar. ideas says, that “abstract ideas are only denomina-
S->/~' tions.”
On this notion Mr. Tooke enlarges at great
length. His several chapters on abstraction, which
abound with much curious etymology, occupy above
400 quarto pages, in the course of which he is pleased
to inform his readers, that “heaven and hell” are
“ merely participles poetically embodied and substan-
tiated.” What practical inference is to be drawn from
this statement we know not; but we have carefully
endeavoured to understand Mr. Tooke's doctrine, so
far as it relates to the Grammatical explanation of the
(so called) abstract nouns. It appears to us, we own,
rather obscure, but perhaps it may be more satisfactory
to some of our readers ; and therefore we shall state
it as distinctly as we are able, in the following propo-
sitions:
1. The verb is the noun, and something more. (vol. ii.
p. 514.) -
2. The adjective is the noun, directed to be joined
to another noun. (vol. ii. p. 431.)
3. The participle is the verb adjectived, i.e. “it has
all that the noun adjective has, and for the same rea-
son, viz. for the purpose of adjection.” (vol. ii. p. 468.)
4. The abstract nouns “are generally participles or
adjectives used without any substantive to which they
can be joined.” (vol. ii. p. 17.)
The result of this seems to be, that when an abstract
noun is a participle (as Mr. Tooke says heaven is) it is
a moun, and something more, converted into a moun di-
rected to be joined to another moun, but used without any
noun to which it can be joined. How far this mode of
reasoning goes to show that there are not in the mind
any such ideas, as “whiteness,” “strength,” “virtue,”
and the like ; or that these words do not serve to com—
municate any thing but conceptions of solid, tangible,
corporeal substance, in an abbreviated form, must be
left to the determination of the judicious reader ; for
our own part, we cannot see that it tends much to en-
lighten what may be thought obscure, in the Works of
the ancient Grammarians; still less does it appear to
us to cast a doubt on those principles, which the Am-
cients have stated with great clearness and precision.
Before we quit this part of our subject, we should
notice, that Harris mentions three sorts or kinds of
substances; the natural, as “man ;” the artificial, as
“house;” and the abstract, as “whiteness.” The two
former fall under the class of corporeal conceptions, and
as no Grammatical distinction applies to them in prac-
tice, we think it unnecessary to enter particularly into
their consideration. The last kind are the same which
we have mentioned as denoting ideal or mental concep-
tions, and to which we think the word abstract inappli-
cable for the reasons already stated.
After considering the different kinds of substantives,
we come next to what we have called a difference of
gradation; and by this we mean that order or arrange-
ment of conceptions which classes them as genera,
species, and individuals. Although the ancient writers
have in general noticed only these three gradations,
yet it is easy to see that they may be multiplied indefi-
nitely. Thus we may say, “animal” is a genus,
“man” a species, “Alexander” an individual ; but we
may also divide the species man into white and black,
or King and subject, or Greek and Barbarian ; or we
may make “being” the genus, “created being” the first
species, “organized being” the second, “animal” the
WOL. I. -
third, and so downwards, in regular subordination, until
we come to the individual. Hence it appears, that the
only important distinction of substantives, in this respect,
is into words expressing individual things, and words
expressing classes more or less general , a distinction
answering to the old Grammatical terms nomen and
vocabulum, or momen proprium and momen appellati-
vum ; or, in the language of our modern Grammars,
nouns proper and common.
It has been truly observed by LocKE, that “it is Particular
impossible that every particular thing should have a
distinct peculiar name; for, the signification and use
of words depending on that connection which the
Mind makes between its internal operations and the
sounds which it uses as signs of them, it is necessary,
in the application of names to things, that the Mind
should have distinct conceptions of the things, and
retain also the particular name that belongs to every
one, with its peculiar appropriation to that conception.
But it is beyond the power of human capacity to frame
and retain distinct conceptions of all the particular
things we meet with ; every bird and beast men saw,
every tree and plant that affected the senses could not
find a place in the most capacious understanding. If it
be looked on as an instance of a prodigious memory,
that some Generals have been able to call every soldier
in their armies by his proper name, we may easily find
a reason why men have never attempted to give names
to each sheep in their flock, or each crow that flies over
their heads, much less to call every leaf of plants or
grain of sand that came in their way by a peculiar
name.” So far Locke, in which quotation we have only
taken the liberty to substitute for the word ideas, in one
place internal operations, and in two others conceptions.
The reasoning, however, is not affected by this change,
and it is such reasoning as must carry conviction to
every mind. We also agree fully with this writer, that
to name every particular thing, if possible, would be
useless for the purpose of communicating thought,
unless every man could first teach the whole of his
own endless vocabulary to every other man with whom
he conversed, or for whose information he wrote. And
again, supposing even this possible, it would not
conduce at all to Science ; for as Aristotle has said,
“ of particular things there is neither definition nor
demonstration, and consequently no Science, since all
definition is in its nature universal.”
Proper names are therefore comparatively few in
number. They serve to denote a very few of the im-
mense multitude of particular objects which fall under
our observation. Some of these, indeed, obtain a dis-
tinguished celebrity within a small circle. But the
Poet, the Orator, or the Historian, may raise them to a
prouder eminence. He may render them the symbols
or representatives of the classes to which they belong.
It is thus that “Alexander” becomes the synonyme of a
Conqueror, and “Cicero” of an Orator.
Even Proper names, however, have in general been
given to individuals from some quality or action not
strictly peculiar to them. Hence the old English
rhyme alluded to by VERSTEGAN, in relation to the
family name of Smith :
Whence cometh Smith, albe he knight or squire,
But from the Smith that smiteth at the fire *
And thus we see how words of individual import, as
|E
26 *. G R A M M A. R.
Grammar. well as conceptions of individual existence, arise from an angle as a genus, and the acute, the right, and the Nouns.
S-N-' ideas, that is from thoughts, not particular, but univer- obtuse angles as species.
sal. Nevertheless it must be admitted, that the uni- The nature and effect of these genera and species Their use in
versal idea is soon lost in the particular application. may be thus explained: all Truth, which is not intui- *5.
Few people reflect, that George originally signified, “a tive, must be discovered by reasoning; but reasoning
husbandman,” or that Charles and Andrew both signified is reducible, in all cases, to the syllogistic form. Now
“manly” or “strong,” the former from its Gothic, the a syllogism is a combination of propositions: and a
latter from its Grecian etymology. These names have proposition asserts either the agreement of a substance
now come to indicate individuals; and as even thus with its attribute, or of a genus with its species. The
a single word is not found to answer the purpose suffi- subject of the proposition is one conception, and the
ciently, we have the baptismal name and surname; as predicate is another. Each of these may be repre-
the Romans had the pranomen, the cognomen, and the sented by a noun substantive; but one of them (if not
agnomen. both) is necessarily common ; for the assertion that one
Nouns com" Beside these Proper names, all other substantives proper noun is another, e. g. that “John is William,”
II101l, are common, or what Locke calls general words, is no assertion at all, for any purposes of reasoning.
which he truly says are “the inventions and creatures Omitting, for the present, the consideration of those Genus and
of the understanding.” But the process of the under- propositions, which assert the agreement of the sub- *P*.
standing, in inventing and forming these words, he has stance with its attribute, let us consider those which
not accurately traced ; which, indeed, is not much to assert the agreement of a genus with its species, as
be wondered at ; since he proceeds solely on an incor- “ that man is an animal,” “that an isosceles is a tri-
rect, or at least an imperfect, maxim of the Schoolmen, angle,” or the like. If the conception “animal” in-
viz. Nihil est intellectu, quod non prius fuit in sensu; cludes the conception “man,” the proposition “man
the only rational meaning of which is, that we receive, is an animal” is true; and in like manner, if the con-
by means of our senses, the materials upon which ception “triangle” includes the conception “isos"
intellect operates, or by which it is first excited to the celes,” then the proposition “an isosceles is a triangle”
perception of Truth ; so that the maxim, as has been is true. But nobody who understands these concep-
well observed, ought in its perfect state to stand thus: tions can doubt the truth of the propositions. Why?
“Nihil est in intellectu, quod mon prius fuit in sensu, Because such is the nature of the idea “animal,” that
praeter ipsum intellectum.” Now the mode in which the it includes the idea “man:” that is to say, it not only
Understanding proceeds is easily to be discovered from applies to all the men that we ever have known, but to
the general aim and object of its process, which is to all that we ever can know, and to many other concep-
acquire some knowledge that may be useful, not only on tions besides; and in the same manner we may reason
one occasion, but on all similar occasions; to know concerning the ideas of “triangle” and “isosceles”
some truth which may not only apply to Peter or John, respectively.
but to all persons who resemble Peter or John ; but The genus, therefore, is an idea including the species;
this cannot be done, unless I have a common word not as a day includes an hour, or as a mile includes an
which implies that resemblance ; and the persons in inch, that is to say, as a given measurable part or por-
question cannot resemble each other but by relation to tion, but simply as being of more comprehensive appli-
some common conception, which does not necessarily eation, and therefore embracing all the particulars
belong to any one of them more than to any which the other embraces, and many more. Thus, let
other. That common conception, therefore, supplies us give what definition we please of the idea “man,”
the class-word, which renders the truth common. we shall find that the idea “animal” includes it, and
Thus Peter, James, and Andrew may be slaves; something more. If “man,” therefore, be considered
the conception of slavery, therefore, is common to as a species, “animal” will be a genus, or conception of
them all, and whatever is universally true of it, is a higher order; and it is simply on the principle of
true not only with relation to Peter, James, and one conception including or not including another, that
Andrew, but to all others who are, or have been, or may the whole doctrine of syllogistic reasoning depends.
be, in the state of life expressed by the word slave. From what has here been said, it is manifest, that the Individual,
Again, a slave and a free citizen agree in this, that distinction of genus, species, and individual, is properly
they are subjects ; a subject and a Sovereign in this, Logical. The first two classes, however, are necessarily
that they are men ; a man and a beast in this, that expressed by the nouns which we have called common :
they are animals. Now all these conceptions, to whilst the individual may either be expressed by a
wit, slavery, subjection, human nature, and animal proper name or by a common noun individualized (as
nature, are so many mental conceptions, or ideas, and will be hereafter shown) by the help of an article or
they are regularly subordinated, one to another, in a pronoun. With respect to the individual also, we have
certain gradation, according as they are viewed by the to observe that it is not necessarily indivisible; but on
Mind; which view is determined, not by any accidental the contrary there is a class of nouns called mouns of
impression received from the senses, but, on the con- multitude, each of which, though it represents a number
trary, by the general truth, of which the Understanding of Beings definite or indefinite, still represents them as
is in search. Thus, if I am in search of some truth one thing ; of this sort are the words, “an army,”
relative to the state of slavery: I may consider the “a regiment,” “a troop,” “a nation,” “a crowd,”
conception of slave as a genus, and divide it into the “a flock.” Those writers who have not well compre-
species of domestic, political, absolute, limited, and hended the distinction of genus and species, have
the like ; or if I wish to reason on animal nature, I sometimes explained the words representing them as
may regard animal as the genus, and man, beast, bird, mere nouns of multitude, that is to say, “as repre-
fish, &c. as species. In like manner, I may consider sentatives of many particular things,” instead of being
G R A M M A. R. 27
Grammar, representatives of an idea common to those particular
-- things. Ç
Thus have we shown that the noun, according to its
very slightest attention will show us that there is not Nouns.
merely unity, but multitude, or the idea of number in Number.
its most indistinct form; but in order to distinguish STN-7
essential distinctions, signifies a conception corporeal
or mental, and that it signifies it by a name proper or
common ; and such was also the doctrine of the An-
cients; for the same Grammarians who defined the
noun as “pars orationis significans rem corporalem vel
incorporalem,” add to that definition “proprié, commu-
7\iterve.”
But to these essential distinctions are to be added the
accidental ones, of which we have next to speak, viz.
number, gender, and relation, or case.
this multitude into given numbers, as twos, threes, or
fours, it will be necessary to refer each conception to
some other. Thus these two, John and Richard, are
tall; these three, Henry, William, and James, are
short; or these three, John, and Richard, and Henry,
stand in the first line ; these two, William and James,
stand in the second ; or the first, John, is counted on
the thumb; the second, Richard, on the fore finger ;
the third, Henry, on the middle finger; the fourth,
William, on the finger next beyond the middle; and
Number. Whatever is accidental may, or may not, be viewed the fifth, James, on the little finger. This last mode
in connection with that which is essential. Thus the con- of sorting and classing conceptions has been generally
ceptions or ideas of number, may or may not be viewed adopted by mankind, whence the Greek word traputrā-
in connection with other conceptions, as that of “man,” getv, “to reckon by fives,” was used generally for “to
or “whiteness,” or “sun,” or “star;” and if viewed in number.” Some Barbarous Tribes never went beyond
connection with any one of these, the complex concep- the use of one hand for this purpose; whereas the
tion may be expressed by a single word, or by two more cultivated nations employed both hands; and
words, as happens in regard to other combinations of this latter mode is the origin of our decimal system of
ideas; thus as “saint” is a single word, including the arithmetic, and explains why the numeral figures are
conceptions expressed by the two words, “ holy” and still called digits. g -
“man,” so the word “horses” includes the conceptions We have observed, that the first conception of number Plural
expressed by the words “horse” and “number.” is simply, that it is something beyond and different from *
Whence In order to understand when the conceptions of num- unity; that it is unity repeated, or multitude. Thus
derived, ber can, and when they cannot be added to other con-
have, to grow and multiply by contemplation.
ceptions, we must consider what the former are. For
this purpose we cannot, perhaps, refer our readers to
a more satisfactory or better authority than Plato's Epi-
momis, sometimes called the XIIIth Book on Laws ;
but the whole passage is too long to be extracted, and
we should do it injustice were we to exhibit it in an im-
perfect state. Suffice it to say, that Plato agrees with
Locke in asserting, that “ number is the simplest
and most universal idea,” for unity itself is in this
sense the origin of all our ideas of number. But
the latter Philosopher is by no means correct in
saying that “its modes are made by addition ;” for
we might as well say that they were made by
division, or by subtraction, or by multiplication ;
since addition is, equally with each of the others,
one of the powers of numbers, and presupposes the
idea which Locke imagines it to produce. He says,
“ by repeating this idea (viz. of unity) in our Minds,
and adding the repetitions together, we come by the
complex ideas of the modes of it. Thus by adding
one to one we have the complex idea of a couple.”
Very true, by adding; but not by simply repeating,
which is a totally different operation. John is one, and
Peter is one, and Henry is one ; but one is not two, or
three. What makes me then acquire the ideas of two
or three ? Certainly not the bare act of repeating one,
one, one; for children, and idiots, who cannot reckon
three, can do this ; and M. de la Condamine mentions
whole Tribes of Savages, who cannot reckon beyond
three, though certainly they could repeat one, two,
three, all the day long. There must, then, be some-
thing in the nature of the ideas of number without
which it would be impossible for us to “add one to
one,” and of Lourse to obtain “the complex idea of
number.” Now, this consists in the still more general
nature of all ideas, and in that power which they
Thus,
if we enumerate John, and Richard, and Henry, and
William, and James, and Edward, and so forth, the
far most nations have gone, in expressing, by one word,
the combination of number with any given conception ;
and this variation in the noun is called, by Gramma-
rians, the plural number. The plural number usually
differs from the singular in form, either by the use of a
word altogether different, as “pig and swine;” or by a
change of articulation, as “man and men ;” or by a syl-
lable added, as “horse and horses,” “ox and oxen;” but
as the variety of these forms proves that no one of them is
essentially necessary; so both experience and reflection
will show, that no change whatever is necessary, in the
noun itself, provided that some other word serves to
show us that the noun is used with reference to plurality;
thus in English we say “fifty sheep,” and “fifty head of
cattle ;” and so in Latin the genitive and dative cases
singular, and nominative and vocative plural of the first
declension, are identical.
The form in which the noun expresses unity of con- Dual
ception, is called the singular number : but it would number.
not be possible for nouns to have a separate inflection
for every separate conception of number, that could be
combined with them by the Mind. Therefore, they can-
not have separate forms for the dual, ternal, quaternal
numbers, and so on, ad infinitum ; but for some of these
numbers they may. Experience, indeed, has not shown
us that they have ever gone beyond the dual number in
this respect; and that has been dome by very few nations.
Some Grammarians have warmly agitated the question
whether the Latin Language has, or has not, a dual
number; and as this question may serve to illustrate,
in some degree, the Principles here advanced, we
shall advert to it, in that point of view. SCALIGER
says, Iones mon rectà fecere, qui dualem numerum
a plurali discerpsere: atque idcirco severiores AEoles
neque recepere, meque in Latinos transmisere; et nuga-
citas illa Ionum in multis temporibus verborum personas
aliquot mon potutt eruere in eo numero, in nominibus
autem pauculos casus expressere. “The Ionians acted
wrong in dividing the dual number from the plural ;
for this reason, the more severe Æolians neither re-
E 2
28
G R A M M A. R.
Grammar, ceived nor transmitted it to the Latins; and even this
S-N- Ionian trifling in many tenses of verbs, was unable to
make out a few of the persons in the dual number ;
and in the nouns they expressed a very few cases.”
Quinctilian, however, observes, that there were some
writers, in his time, who contended that the dual
number, in the third person plural of verbs, was pro-
perly marked by the termination e, as consedere, if two
persons sate together ; consederunt, if more than two ;
but, adds he, this rule is observed by none of our best
writers, quin e contrario “ Devemere locos;” et “Conti-
cuere omnes;” et “Consedere Duces” aperte nos doceant
nihil horum ad duos pertimere. Quid? Non Livius circa
initia statiºn primi libri “Tenuere” inquit “arcem
Sabini;” et moa, “ in adversum Romani subiere.” Sed
quem potius ego quam M. Tullium sequar, qui, in Oratore
“non reprehendo” ait “scripsere, scripserunt esse verius
sentio.” “On the contrary, the expressions • Devemere
locos' (AEm. I. and 6.) and “Conticuere omnes' (AEm. 2.)
and “Consedere Duces' (Ovid. Metam. 13.) may clearly
teach us that none of these verbs relate merely to two
persons or things. Does not Livy, almost at the very
beginning of his first Book, say, ‘Tenuere arcem Sabini;’
and shortly afterward, ‘ In adversum Romani subiere.”
But what authority need I follow in preference to that
of Cicero himself, who, in his Book De Oratore, says,
‘I do not blame those who write scripsere, but, for
my own part, I think scripserunt better.’” Wossius, too,
observes, that in the description of Africa by Sallust,
contained in his book on the Jugurthine war, we find,
in the course of a very few lines, the plurals posuere,
£nteriere, habuere, occupavere, miscuere, appellavere, ac-
cessere, corrupere, possedere, coegere, addidere, conces-
sere, condidere, and fuere; so that this supposed dis-
tinction in the third person of the verb appears to have
been quite imaginary. DoNATUS, however, a Gram-
marian so popular in the Middle Ages, that a “ Donat”
became the common term for an elementary book on
Grammar, argues more reasonably on the use of the
words ambo and duo. Numeri, says he, sunt duo,
singularis, ut hic sapiens, et pluralis, ut hi sapientes.
Est et dualis numerus, qui singulariter enunciari mon
potest, ut hi ambo, hi duo, “ There are two num-
bers, the singular, as hic sapiens, and the plural, as
hi sapientes. There is also a dual number which
cannot be expressed singularly, as hi ambo, both
these ; hi duo, these two.” Donatus is certainly right
in calling these expressions duals, since they relate to
the conception of two ; but for the same reason he
might call the expressions hi tres, illi tres, and the like,
ternals ; and so on, of any other numbers. This re-
mark, however, leads to the clear and easy solution of
the dispute among the Grammarians; since it shows
that each party was right in the different view that it
took of the subject. It is certain, on the one hand,
that the Latins could and did express the conception
which was expressed by the Greek dual; but it is
equally certain that they did not express it in the
same manner. Amongst the Ionian Greeks the idea
of two was expressed by a word which from long use
and habit had come to be employed as the terminating
syllable of any noun with which that idea was con-
nected. Amongst the AEolian Greeks, and their Latin
successors, the same idea of two was expressed by
words which never happened so to coalesce. Scaliger
on this, and some other occasions, reasons as if the
formation of different Dialects were a matter of pre-
meditation and study ; and, therefore, he calls the
Ionians trifiers, and describes the AEolians as more
grave and severe ; whereas it is certain that all Lam-
guages, in their early state, grow up without much
meditation or reflection, and that the cultivation and
polishing of its Language is one of the last results of a
nation’s civilization. Nor can this be otherwise ; for
ideas, which are the laws of Mind, develope themselves
in practice, and guide our mental operations, just as
animal laws direct our bodily actions, long before we
suspect either of them to exist. We walk, and dance,
and ride, according to the laws of gravitation ; we swim
by the Principles of hydrostatics; we form and express
thoughts by the laws of conception, assertion, and deduc-
tion ; but it is not until long after we have submitted
to those laws, that we begin to take cognizance of
them as distinct objects of thought; for the last opera-
tion of the human intellect is that by which it separates
itself from outward things, and discovers within its
own nature a world of beauty and order, which even
more than this wondrous body of Man with all its
curious apparatus, chemical and mechanical; more than
this terraqueous globe with its animal, and vegetable,
and mineral riches ; more than the Sun “ looking from
his sole dominion,” or even than the countless numbers
of the heavenly host peopling interminable space, dis-
covers to our finite comprehension the traces of that
Deity, who cannot be more fully revealed but by his
own divine word.
Nouns.
Number.
S-N-"
Thus it is, that in intellectual, as in moral specula- Absolute
tion, our simplest conceptions are most closely con- Truth,
nected with that absolute Truth, of which Mr. Tooke
altogether denies the existence. “Truth,” says he,
“supposes mankind : for whom, and by whom alone,
the word is formed. If no Man, no Truth. There is,
therefore, no such thing as eternal, immutable, ever-
lasting Truth ; unless mankind, such as they are at
present, be also eternal, immutable, and everlasting.
Two persons may contradict each other, and yet both
speak Truth.” This is not only not common sense, but
it is very bad Logic. The argument runs thus: A man
trowed or believed something to exist; he used the
word “troweth, troth, or truth,” to express this belief;
therefore no such thing existed. Again, two men be-
lieved that two different things existed ; they both used
the same word to express the same belief: therefore
the belief of both was equally well founded. Turn Mr.
Tooke's sentences how we will, they come to this sort
of reasoning. How is such a circumstance to be ac-
counted for, in a man of his acuteness? For that he
was acute, his single remark “that the verb includes
the moun, and something more,” incontestably proves.
But his extraordinary sophisms arise wholly from his
loose and hasty conception of the word thing ; which,
as he uses it, corresponds exactly to Condillac's
object, and to Locke's idea ; and really means
nothing ; that is to say, nothing certain, definite, or
intelligible.
That the Human Mind can embrace ETERNAL TRUTH,
in the widest sense of these terms, it would be folly and
madness to assert; but that none of the truths which
it is formed to comprehend are eternal, is a proposition,
to say the least of it, extremely bold. At all events, the
circumstance that men, “such as they are at present,”
may not be able clearly to comprehend a given truth,
Truth of
numbers.
G R A M M A. R. 29
Grammar, is certainly no proof of its falsehood. Suppose a child associated with the debauched and profligate Antony, Nouns.
s—v- does not well comprehend that two and two are four, and who at once flattered and subjugated the Roman Gender.
are they the less so 2 Now this is the case with all people, cannot receive a plural termination ; and for Ta-'
conceptions of number. We begin with unity, we pro- this reason, because the particular conception which it
ceed to multitude, we advance to numeration ; but the expresses cannot be associated with number ; since
elementary books of Arithmetic will teach us, that this there never was nor ever will be more than one such
last is the introduction to that Science by which Newton man; who therefore spoke Philosophically and truly,
brought down the old Divinities from their starry when he said—
thrones, and converted lovely Venus and potent Jove For always I am Caesar.
into silent monitors of the lapse of time, or friendly But if the word Caesar be used to express a different
guides of the adventurous navigator on a lonely ocean ; conception; if it mean something which is also found
that Science, by which Judicial Astrology was for ever whole and entire in Alexander, and Attila, and Jenghiz
confuted, and men learnt to gaze unmoved on the Comet, Khan, and Napoleon Bonaparte, then indeed “ the
which, as they once thought, Caesars” is a proper Grammatical form of speech ; be-
— From his horrid hair cause the noun is no longer a Proper name, but an
Shook pestilence and war. appellative. Then we may reason on the Caesars, as on
Such being the nature and power of the conceptions a class or species, and what we say of one will be equally
nected with of number, let us inquire how, and on what Principles true of another; but then the word, though the same in
other truths, it is that they are connected with other conceptions : sound, will be very different in signification ; and the
and here it will be seen that these Principles are founded reason which before prevented our adding to it the
in the essential nature of the noun, as universal and plural termination will no longer exist. §
particular; general, specific, and individual : for the Mr. Harris has mentioned various ways in which a How they
principal office of numbers is to apply Science to fact, Proper name may come to be used as an appellative. become
by distributing the genus into its species, and the The persons indicated by it may, as members of the P",
species into its individuals; number, therefore, is the same family, or from other accidental causes, happen
bond uniting the universal with the particular, the high- to bear the same name. Hence the expression of “the
est genus with the lowest individual, Eternal Truth with twelve Caesars,” to designate twelve Roman Emperors
momentary sensation. Therefore it is, that Plato says, who successively bore that name. Hence too the
éâtep aptèuðv čk Tijs &v6pwiriums pêoews ééNotºmu obic du Howards, Pelhams, and Montagues, “ because a race
Trote Tá ºpdvtuo, Yevolué0a. “If we were to take out or family is like a smaller sort of species;” so that
number from human nature, we should become void of the family name extends to the kindred, as the spe-
thought on every subject;” which he again illustrates cific name extends to the individuals. Again an-
by observing, that an animal which has not the distinct other cause which contributes to make Proper names
conceptions of two and three, or of even and odd, and plural, is the marked character of some individual
consequently, is quite ignorant of numeration, can who bears it, whether for eminent virtue, or for no-
never give any account of those things which he per- torious vice, or simply for any thing extraordinary and
ceives by sense and memory. singular in his conduct or opinions. It is thus that in
Distribute “The genus,” as Mr. Harris observes, “ is found speaking on the subject of Grammar, we might not
genus into whole and entire in each one of its species.” Thus improperly say, “these are the opinions of a Condillac "
species. the genus animal is found in the different species, man, referring to an author of some celebrity ; though, as
horse, and dog : that is to say, a man is an animal, a we think, of remarkable inaccuracy in his views of
horse is an animal, and a dog is an animal. By num- that subject. So the liberality of Horace’s patron and
bering the kinds, we find that the genus though one, is friend has made every patron of literature be called a
capable of being conceived as many, and therefore we Maecenas ; the odious cruelties of Nero have made his
can speak of many animals. Again, “the species may name a synonyme with the word tyrant : and on the
be found whole and entire in the individual.” Thus same principle Shylock, when he would express the
Socrates is a man, Plato is a man, Xenophon is a man; integrity and acuteness of the supposed young lawyer,
and by applying the conception of number to the species exclaims,
of man, we call them three men. The plural number, A Daniel come to judgment! Yea, a Daniel 1
therefore, belongs to genera and species: and accord- Gender, as an accidental distinction of nouns, has Gender.
ingly we find all Languages apply the plural number to given rise to much litigation among Grammarians.
words expressing genera and species, that is to say, to “Gender,” says Vossius, “is properly a distinction of
the words which we have called common, or appel- sex; but it is improperly attributed to those things
lative. which have not sex, and only follow the nature of things,
Proper But the case is totally different with Proper names, having sex, in so far as regards the agreement of substan-
Il:ll in tº S
strictly sin-
gular,
when strictly used as such ; for in that case they are
applied to individuals, and the individual is not found
whole and entire in the genus or species. The con-
ception of Caesar is not found whole and entire in the
genus animal, or in the species man, or in the class of
Romans, or of Conquerors, or of Generals, or of Soldiers,
or of Scholars. The word Caesar, therefore, when used
to express the very individual who passed the Rubicon,
and who spoke with so much affected liberality in be-
half of the traitors engaged in the Catilinarian con-
spiracy, and who doubted of a future state, and who
tive with adjective. Sex is properly expressed in refer-
ence to male and female, as Pythagorasand Theano; ager,
a field, therefore, is improperly called masculine; and
herba, a herb, is improperly called feminine. But animal
is neuter, because it is construed neither way.” Scaliger
says, that the Ancients improperly attributed sex to
words; and that with respect to the neuter gender, it
is absurd to attribute that to gender which is the ne-
gation of gender. Neither is it to be borne, says he,
that words should be called of the doubtful gender,
from the circumstance of their being sometimes used
30 G R A M M A. R.
Grammar, with a masculine and sometimes with a feminine con-
\-y-' struction. Mr. Harris, however, has, with some in-
genuity, endeavoured to assign reasons for the generic
accident, by no means capable of carrying us far to- Nouns.
ward the explanation of the Principles on which Gender.
Language in general was constructed. Harris, it must Yºr"
distinction of nouns. “Every substance,” says he, “is
male or female, or both male and female, or neither
one nor the other. So that with respect to sexes and
their negation, all substances conceivable are compre-
hended under this fourfold consideration.” Hence he
proceeds to consider Language as if it had been really
and intentionally formed with a view to this classifica-
tion of substances. As to the first and second class,
they are manifestly such as must on many occasions
require some mode of expression. The third is rare,
and its expression would in general be shunned. But
as to the fourth it must necessarily include by far the
greater portion of the objects of thought. In Languages
where the natural sexes alone are expressed by terms
corresponding to them, very little difficulty occurs in
this part of Grammar. In general, every noun denoting
a male animal is masculine ; every noun denoting a
female animal is feminine; and every noun denoting
neither the one nor the other is neuter. The only ex-
ception to this general rule, is an exception which is
founded in the Poetical part of our nature; and it hap-
pily serves to distinguish the Language of imagination
from that of reality. The instances to which we allude
are those in which the conception of a thing is raised to
the dignity of a person, where we dwell with such fond-
ness on our thoughts as to invest them, as it were, with
life and action. Virtue stands before us in the enchanting
form of a lovely female. Patience appears “gazing on
Kings’ graves, and smiling extremity out of act.”—So
Shakspeare says, -
The mortal Moon hath her eclipse endured.
But perhaps we cannot cite a finer instance of this
figurative use of gender than that which is so finely
employed in Milton's description of Satan—
His form had yet not lost
All her original brightness, nor appear'd
Less than archangel ruin'd.
But in Languages where the mere terminations of words
imply, or are supposed to imply, any or all of these
distinctions, it is no wonder that much confusion
arises in the various modes of explaining a circum-
stance so foreign to the general laws of thought.
“The Greek, Latin, and many of the modern Tongues,”
says Mr. Harris, “ have words, some masculine, some
feminine, (and those too in great multitudes,) which have
Mr. Har-
ris’s theory.
reference to substantives where sex never had existence.
To give one instance for many, mind is surely neither
male nor female; yet is voos, in Greek, masculine ; and
nvens, in Latin, feminine.” This learned Grammarian
could not but perceive that “ in some words these dis-
tinctions seemed owing to nothing else than to the
mere casual structure of the word itself;” but he was of
opinion that in other instances might be detected “a
more subtle kind of reasoning, which discerned even in
things without sex a distant analogy to that great dis-
tinction which, according to Milton, animates the
world !”
We are far from asserting that in particular instances
some such analogy may not have operated. Indeed
it appears to us to be of the nature of that Ima-
gination to which we owe the figurative language
above mentioned; but it could only have been a rare
be owned, expresses himself modestly enough, observ-
ing, “ that all such speculations are at best but conjec-
tures, and should therefore be received with candour
rather than scrutinized with rigour.” “Varro's words,
on a subject near akin,” says he, “are for their aptness
and elegance well worth attending: Non mediocres
enim tenebra in silvá ubi hac captanda, neque eó, qué
pervenire volumus senita trita, neque non in tramitibus
quadam objecta, qua, eumtem retinere possunt.” With
this allowance, we may therefore notice the gene-
ral Principle for which Harris contends, namely, that
“we may conceive such subjects to have been con-
sidered as masculine, which were conspicuous for the
attributes of imparting or communicating, or which
were, by nature, active, strong, and efficacious : and
that indiscriminately, whether to good or to bad, or
which had claim to eminence either laudable or
otherwise;” and again, that “the feminine were such
as were conspicuous for the attributes either of receiv-
ing, of containing, of producing, or of bringing forth,
or which had more of the passive in their nature than
of the active; or which were peculiarly beautiful and
amiable, or which had respect to such excesses as
were rather feminine than masculine.” Hence he
thinks it would be reasonable to consider as masculine
nouns, the “sun,” the sky,” the “ ocean,” “time,”
“death,” “sleep,” and “God;” and as feminines,
the “moon,” the “earth,” a “ship,” a “city,” a
“country,” and “ virtue.” But the question, as re-
spects the Science of Grammar, is not whether any or
all of these may not occasionally and accidentally be so
considered ; but whether there be any necessary cause
connecting in our Minds the conception of sex with
any of them. Now, there can be no other such cause
than personification, because sex is a personal distinc-
tion; but even that cause does not universally apply to
any of these conceptions. God, indeed, our Creator and
Preserver, we usually and properly regard as a person;
and then the reasoning of Mr. Harris is so far just, that
we cannot easily view the Supreme Being as a female ;
for even in those Heathen Mythologies which abound
with female Divinities, the chief and sovereign Deity is
always represented as masculine. But Harris himself
admits, what indeed the common experience of every
day sufficiently proves, that we often contemplate this
ineffable conception without any reference to sex, or
even to person, calling it “ Deity,” “Numen.” “To
6cºov.” It must be remembered, that personification
was more common among the Ancients than the Mo-
derns. The Greeks actually worshipped Sleep and
Death in the form of men : Virtue was portrayed be-
fore their eyes by the statue of a female. Nor must we
forget that many of these personifications have been
handed down to us from them by mere tradition and
the language of the Poets. Thus it is difficult for us,
who have seen Fame and Victory so often delineated
as females, on ancient medals, and in sculpture, who
read of them as such in Poetry, and know that Fama
and Victoria are nouns of feminine termination ;-it
is difficult for us when we do personify these airy
Beings, to figure them to ourselves as men, in a different
habit and form, with different accompaniments, and
expressed by words and sentences of a different cha-
º, .
Grammar.
\-y-Z
G R A M M A. R.
racter and construction. But there are comparatively
few things which we personify in our common prose :
and when we do so, the change of the form of words
from neuter to masculine or feminine, at once and
powerfully marks the transition of the Mind from cold
matter of fact to ardent Imagination. This, however, is
again an accidental circumstance appertaining to the
particular History of the English Language, and not
Gender of
Proper
Ila IIlêS.
Termina-
tions.
Union of
conceptions.
to the Philosophy of Language in general.
There is a curious difference of opinion between
SANCTIUS and HARRIs. The former writer asserts
“ that Proper names of men, cities, rivers, mountains,
and the like do not admit of Grammatical gender;”
Nomina propria hominum, urbium, fluviorum, mon-
tium, et cattera huffusmodi, genus grammaticum habere
mon posse: whereas the latter author says, “both num-
ber and gender appertain to words.-Number, in strict-
ness, descends no lower than to the last rank of species:
gender, on the contrary, stops not here, but descends
to every individual, however diversified.” This apparent
contradiction between two eminent writers is neverthe-
less easily reconciled. Harris attributes gender to
words as significant of the conceptions of the Mind.
Sanctius, on the other hand, following the authority of
Varro and Diomedes, considers Grammatical gender
as relating only to the termination or construction
of words. “Thus,” says Varro, “we do not call
those words masculine which signify male Beings,
but those before which are properly placed hic and
hi, and those feminine with which we can say hac and
had.” Sic itaque ea virilia dicinus, mon quae virum sig-
mificant, sed quibus praepomimus hic et hi: et sic mulie-
bria in quibus dicere possumus hac et ha. The reason
which this author assigns for his doctrine is suitable
enough to Grammar as an Art, but not as a Science.
Grammatica propositum non est singularum vocum sig-
mificationes explicare, sed usum. “The object of Gram-
mar is not to explain the significations of particular
words, but their wse.” Now, though the mere signifi-
cation of words is not the object of Grammar, the mode
of signification is so far from being an immaterial part
of that Science, that it is its sole foundation. There is
no doubt but that the expression or non-expression of
the distinction of sex in connection with other concep-
tions, must affect the relations of Language considered
as significant, and consequently must fall under the
Science of Grammar, according to the definition of it
which we have adopted. This expression is not essential
to all nouns, but it is an accident universally affecting
whole classes of nouns, and therefore demanding for its
application some rules of Universal Grammar.
Now those rules not only do not depend on the ter-
mination or other peculiarity in the sound of words, but
even in the Latin Language, as Wallis has observed,
sex is not so distinguished; for though the termination
wm is neuter, yet the words scortum, mancipium, ama-
sium, &c. are applied both to the male and female sex :
and so we find it even in Proper names, as Glycerium
imea, which Priscian notes as figurative.
Regarding only the Science of Grammar, as de-
pendent on the nature of thought, it is manifest, that
those conceptions which are of a nature to coalesce, in
Reason or Fancy, may be considered either distinctly or in
absolute union. Thus the conception of “number” and
that of “soldier” are absolutely united in the conception
of “army,” or “regiment,” or “troop ;” the conception
31
of “royalty” and that of “man” are absolutely united in
Nouns.
that of “king;” and so the conception of “sex” and that Gender.
of “child” are absolutely united in the words “boy” and
“girl.” This sort of union gives occasion to many classes
of words in most Languages, as “horse” and “mare,”
“ram” and “ewe;” “bull” and “cow ;” but there
is a second class in which the same distinction is ex-
pressed by the compound form of the word, as “shep-
herd” and “shepherdess,” “milliner” and “man-
milliner ;” and lastly, the sexual quality is often ex-
pressed by its proper adjective, as the “male and female
elephant,” the “male and female rhinoceros.”
There are some conceptions in which that of sex is Common
tacitly included, but may not be absolutely determinable, gender.
or may not require to be determined for the purpose of
communicating thought. Thus a “child” is either a
“boy” or a “girl;” but if we are reasoning on the
education of children generally, many thoughts may
occur to us which indifferently and equally relate to boys
and girls, and in expressing which we may therefore
use the neuter word “child.” And perhaps this con-
sideration alone would afford a sufficient answer to
those persons who contend, like Hobbes, that the ge-
neral word “man” is no more than the representation
of some one particular man in my memory or imagi-
mation : for if the word child in my thoughts represented
a boy, it could not represent a girl, and vice versd;
whereas we see in practice that it represents the two
contrary sexes at the same time, without the least dif-
ficulty, and serves the purposes of reasoning quite as
well, and oftentimes better than if we had employed
different words for the two sexes.
Lastly, there are conceptions, which in reality have
nothing to do with sex, but which, from various causes,
principally depending on Imagination or habit, we are
apt to consider in connection with notions of sex.
Thus the English sailor, who has contracted a sort of
affection for the tight vessel in which he has braved
the winds and waves; and who sees in her meat trim
and gallant tackling the elegance of female apparel, is
habitually led to speak of her as a female. Who has
not been electrified with the feeling expressed in the
old sea-song— -
She rights, she rights, boys—we're offshore!”
To a similar cause it is to be attributed that we can
hardly think of Britannia as a mailed warrior, “an
arm'd man for the battle,” or as a Sea-God wielding
his trident over the subject waves; but we see her,
Mike another Minerva, great in Arts and arms, circling
her brows at once with the olive and the laurel, cover-
ing the nations with her agis, and stretching out her
spear for their protection. If we speak of her domestic
greatness, it is as -
The nurse, the teeming womb of Royal Kings;
If we hament her errors, and her failings, we
Feel for her, as a lover, or a child.
Figurative
gender.
This is the language, not of mere plain unadorned Animated
Reason, but of Reason elevated and sublimed by Pas- style.
sion; yet does not this circumstance take it entirely out
of the domain of Grammar, viewed as teaching the ne-
cessary modes of communicating thought; for Passion
is a necessary part of our nature, and it necessarily
gives a hue and tinge to our conceptions, and forces
us to modify accordingly the forms of expression in
Language. Unhappy is the critic who knows nothing
32
G R A M M A. R.
Grammar. of this part of Grammar ; he will not only miss some
S-- of the finest beauties in the Poets, but if he attempt
Case,
Number of
CàSéS.
to correct what he thinks faulty, he will display, in the
most ridiculous light, his own want of taste. Mr.
Harris has finely exemplified this remark, by a quota-
tion from Milton— -
At his command th' uprooted hills retired
Each to his place: they heard his voice and went
Obsequious: Heav'n his wonted face renew’d,
And with fresh flow'rets hill and valley smil’d.
“Here,” says Harris, “all things are personified:
the hills hear, the valleys smile, and the face of heaven
is renewed. Suppose, then, the Poet had been neces-
sitated by the laws of his Language (or we may add by
the correction of the critic) to have said, Each hill
retir'd to its place. Heaven renewed its wonted face—
how prosaic and lifeless would these neuters have ap-
peared ; how detrimental to the prosopopoeia which he
was aiming to establish In this, therefore, he was
happy, that the Language in which he wrote imposed no
such necessity, and he was too wise a writer to impose
it on himself! 'Twere to be wished his correctors had
been as wise on their parts.” That they were not
always so wise we have a striking instance in the cele-
brated Bentley, who has taken upon himself to make a
vast number of alterations of this kind in Milton’s text.
Thus the great Poet in his picturesque description of
Creation, had written
The swan with arched neck
Between her white wings mantling proudly, rows
Her state with oary feet—
On which Dr. Bentley has the following note: “The
Swan her white wings and her state I wonder he
should make the swan of the feminine gender, contrary
to both Greek and Latin; always Kūkvos, cygnus.
Rather, therefore, his wings, his state.” This comes of
having learnt only the Greek and Latin Grammars,
and not knowing, even of these, the true foundations.
We pass now to the expression of the relations of
nouns to each other, which is effected by declension,
or case, if the relation and the conception coalesce in
one word, and by a preposition if in different words.
By this short statement we shall easily discover our
way among the disputes of Grammarians relative to
the cases of nouns. Declension is commonly used for
the variation of case; but Varro considers case as only
one mode of declension. His words are these : “of
words, as man and horse, there are four kinds of de-
clension; first nominal, as from equus comes equile;
secondly casual, as from equus comes equum ; thirdly
argumentative, as from albus comes albius ; and fourthly
diminuent, as from cista comes cistula.” We have,
however, at present, only to do with the second of these
modes.
It was long disputed what number of cases existed
in the Latin Language. These are thus enumerated and
explained by Priscian : “The first case is called the right,
or nominative case; for by this case, naming is effected;
as this man is called Homer, and that man Virgil.
The reason that it is sometimes called the right or
straight case is, that it is first formed naturally by
merely laying down the word, and then the other cases
formed by flexion from this, are called oblique. The
next is the genitive, which is also called by some the
possessive or paternal. The word genitive is either
derived from genus a race, because we signify by it the
tºmºgms
race to which any one belongs, as “he is of Priam's
race,” or from genero to generate, because from this
case are generated many other words and Parts of
speech, at least it is so in the Greek Language. Again
it is called possessive, because we signify possession
by this case, as ‘Priam's kingdom,” or the kingdom
possessed by Priam: whence possessive adjectives may
also be construed by this case: for what is ‘the
Priameian kingdom’ but ‘the kingdom of Priam,' or
‘Priam's kingdom ?’ It is called paternal for a similar
reason, because the father's name is thus expressed, as
‘Priam's son:’ and hence patronymic names may be
resolved into this case, as ‘Peleidan Achilles’ is the
same as Achilles the son of Peleus. The following
case is the dative, which some term the commendative.
I give a thing ‘to a man,’ or I recommend a person
‘to a man.’ Fourthly comes the accusative or causa-
tive : “I accuse a man,' or, ‘I (as a cause) make a thing."
The fifth case is the vocative or salutatory, as “O
AEneas l’ or “Hail AF meas' The ablative is also called
the comparative ; as “I take from Hector,” or “I am
stronger than Hector.' Each of these cases, more-
over, has many other different uses ; but they have
received their names from their most general and fami-
liar use, as we see happen in many other things.”
From this enumeration it is observable, that the sort
Nouns.
Case.
~~~~
Meaning of
of declension which the Ancients called case, not only the Word
expressed the relation of nouns to each other, but also
that which they bore to verbs, as agent or object; and
lastly, their use in the expression of passion, without
reference either to another noun or to a verb : in order
to explain the reasons of which it will be necessary to
observe, that the meaning of the word casus, which we
render case, is, properly, the falling or declining from
a perpendicular line. Thus, if the simple notion of
the moun be supposed to be expressed by an upright
straight lime, as in the letter I, the other cases may be
supposed to be expressed by lines obliquely declining
one way or the other, as in the letter V.
It was long disputed among the ancient Gramma-
rians, whether the nominative should, or should not,
be called a case. On the one hand, it was urged, that
conceptions are only expressed by speech, in some one
of the forms called cases, including the nominative ;
and that of these forms, the nominative expressing the
agent of the verb active, was the simplest, and was
therefore used whenever there was occasion simply to
name a thing or person. Thus we should not say,
that the name of the person slain by Marcus Brutus,
was Caesaris, or Caesari, but Caesar. Those, on the con-
trary, who called it a case, contended that every ex-
pression of a conception in speech, was a declension, or
falling away from the simple conception in the Mind,
which taken by itself does not imply either action, or pas-
sion, or relation. Thus, before I can assert any thing
whatsoever of Caesar, I must form the conception or
thought of “Caesar,” as a person; but when I put that
thought to another, when I mention the wife “of Caesar,”
or the friends who were faithful “to Caesar,” or those
who revolted “from Caesar;” or assert that “Caesar
conquered,” or that “Caesar was killed ;” or express
a feeling of any sort by the exclamation “O Caesar”—
on these and all such occasions my conception deelines
from its original simplicity, and consequently my ex-
pression should be said to decline, or fall away from
the pure noun. They added, moreover, that it was not
C81SC.
Nomina-
tive.
G R A M M A R.
33
Grammar,
always the simplest form of the noun, but was some-
S-N-2 times more distant from the radical, and therefore
Accusative
and abla-
tive.
the agent to every verb in a simple sentence.
more deserving of the appellation of oblique than some
other cases; as, for instance; the vocative or ablative,
which latter some writers have considered as the
primary and original case of the noun.
Since the notion of action implies the notion of an
agent, there must be a form of the noun which denotes
The
action, however, may be represented as proceeding
from the agent, or as feceived by the object. On the
former supposition, it becomes a verb active, and the
nominative case is the form of the noun which denotes
the agent. . On the latter supposition, it becomes a
verb passive; and the nominative case is the form of
the noun which denotes the object. Thus, “Caesar
fights,” “Caesar is killed,” are two simple sentences,
in both of which Caesar is the nominative case. In the
former, the word Caesar signifies the agent that fights;
in the latter, the same word Caesar signifies the object
that is killed. In both instances the nominative is
essential to the completion of the sentence; for when
we speak of fighting, as proceeding from an agent, we
must necessarily express that agent; and when we
speak of being killed, as received by an object, we
must express the object. Hence the trivial rule, that
the nominative answers to the question who, or what ;
as “Caesar fights.” Who fights? — Caesar. “Caesar is
killed.” Who is killed ?–Caesar. It is justly observed
by Harris, that the character of the nominative may
be learnt by its verb. The action implied in the verb
“fights,” shows the nominative “Caesar" to be an
active efficient cause. The suffering implied in the
words “ is killed,” shows the nominative “Caesar" to
be a passive subject. There are some beings which
may be considered in both these lights; as Caesar is
active in the one instance, and passive in the other.
But there are others which cannot, except figuratively,
be considered otherwise than as passive, and, conse-
quently, can only become nominatives to passive verbs;
as we may say, “the house is built;" but we cannot
say, “the house builds.”
. The nominative is the most essential of all cases;
and it has therefore been described as “ that case
without which there can be no regular and perfect
sentence.” With respect to those sentences in which we
make the positive it serve for a nominative, and which
the Latins used without any nominative at all, as pluit,
“it rains;" twedet me, “It wearies me,” or “I am wea-
ried;” these are imperfect sentences, which we shall
hereafter consider separately. In all other instances,
although it may not be necessary to express the object
to which an action is directed, or the agent from which
a suffering proceeds, yet the converse is absolutely ne-
cessary: thus, when we say, “William builds,” it is not
necessary to add “a house,” or “a palace;” but if we
say “builds a house,”or “builds a palace;” it is neces-
sary to prefix the name of the builder.
ln order, however, to extend and enlarge a sentence,
it often becomes necessary to state the object of a verb
active, or the agent of a verb passive. Hence arises
the necessity for two other cases, which have been
called the accusative and the ablative. When we say
there is a necessity for such cases, it will be understood,
from what we have before observed, that we do not
contend for the necessity of any particular termina-
VO L. I. / >
tions, or inflections, or prepositions, or arrangement of Chap. T.
words, to mark these varieties of case; we only mean, Sºº-
that it is necessary, that by some means or other, the
noun, which indicates the conception, should be placed
in such or such a relation to the verb which constitutes
the assertion. It may happen, and, in point of fact, it
does happen in some languages, that there are no in-
flections of case ; but there are means in all languages
of determining when a noun is the object of an active,
or the agent of a passive verb. It has, indeed, been dis-
puted, whether the cases of nouns should be reckoned
according to the relation in which they stand to other
words, or according to the diversity of their inflections;
nor are there wanting names of high repute on either
side of this question. Sanctius contends, that there is
a natural partition of cases, according to the relations
which they imply, and, consequently, that there
must necessarily be the same number of cases, which
he estimates to be six, in all languages. Vossius ob-
jects to this reasoning, and alleges, that if the cases of
nouns were to be reckoned by the relations which they
bear to other words, they must be endless. This con-
test, like many others, has arisen from confounding
Universal Grammar with Particular. The difference of
inflection, or position, belongs to the latter; that of
signification to the former. True it is, that the rela-
tions of nouns to other nouns and to verbs are infi-
nite; but yet they are distinguishable into certain great
classes; and whether these classes ought or ought not
to be called cases is a mere verbal dispute. We shall
so designate them, for the sake of convenience; at the
same time, it must be understood that our arrange-
ment is not intended to interfere with the Grammar of
any particular language, in which the cases are ar-
ranged according to their inflections.
In our sense of the word case, then, the nominative,
that is, the agent of the active, or object of the passive.
verb, may be called the primary case; and the second-
ary cases are the accusative and the ablative, in so far
as they perform the functions above noticed. These
two cases, it is to be observed, are respectively conver-
tible with the nominative, by a change of the verb from
active to passive; for “James loves John” is convertible
with “John is loved by James;" the accusative of the
first being the nominative of the second, and the nomi-
native of the first being the ablative of the second.
So the matter stands in the simpler combinations of Dative, &c. º
thought; but let us consider what is to be done, if in
one and the same sentence we wish to express not only
the agent and object of any action, but also the end to
which the action is directed ; the cause on account of
which it happens, or the instrument, mode, and cir-
cumstances of its performance. For these purposes, it
is necessary, that the conception of such end, or cause,
or instrument, &c. should be expressed by a noun;
and that some means should be adopted to show
whether the noun was meant to stand in the relation of
end, cause, or instrument, or in any other relation to
the verb. It is, as Vossius justly observes, quite im-
possible that any language should have separate in-
flections for all these relations, and therefore some of
them are, in most languages, represented by separate
words, or particles, commonly called prepositions; but
others are often expressed by inflections, the number
and diversity of which vary exceedingly in different
languages. Thus, in the Sanscrit, there are separate
. F
34 G R A M M A. R. -
Grammar. inflections to signify the end, the instrument, the
These are the only distinct uses of the noun which Chap. T.
S-N-2 source, and the situation, answering to our prepositions
it appears necessary to consider under the head of S-S-
- Genitive. Thus have we noticed three classes or degrees of the Latin ablative “ oculo captus,” and to a case in
relation in which the noun may stand to the verb; but Sanscrit, which expresses the cause or instrument, but
it may also be related to another noun, as depending neither the location, nor the derivation, although both
on, or belonging to it. Thus the words “Priam's these latter equally demand the ablative in Latin.
kingdom,” “the son of William,” mark a dependence The dative equally varies. In Greek it answers some-
of “son” on “ William,” and of “ kingdom” on times to the Latin ablative, as gºv ()éð, “ cum Deo;”
“Priam.” This relation is expressed by a separate and sometimes to the Latin accusative, as årt rô képêet,
inflection in Greek, Latin, English, and many other “ob lucrum.” So the English vocative, is sometimes
languages; and it is commonly called the genitive case. expressd by a Latin accusative, as, O, caecas hominum
Now the use of the genitive case in nouns substantive mentes! “O, blind understandings of men ſ” and some-
differs but little from the use of an adjective. It ex- times by a Greek genitive, as Túc àvatéstác “O, im-
presses one conception, as dependent on another, and pudence " Numberless instances of a like kind might
the expression of the latter serves to individualise and be adduced; but these are sufficient to show, that
specify the former. ‘The dependent conception is, however convenient it may be, in the Grammar of any
therefore, in fact, a mere attribute of the other, and particular language, to distinguish the cases of noums
consequently the genitive is easily convertible into an by their terminations, yet this is a method totally in-
adjective. Thus Baot)\eoc 2xmirrpov, regis sceptrum, consistent with those distinctions of signification on
the king's sceptre, are easily converted into 2km"rrpov which alone Universal Grammar can be founded.
Bagixtkov, sceptrum regium, the kingly sceptre. For We have said that the moun adjective is the name of Adjective,
the same reason we find that in some languages, the a conception or thought, considered as a quality or
Chinese for example, the adjective is in no manner dis- attribute of another conception. In order to explaim
tinguished from the genitive or possessive case of a sub- this definition, it will be proper to advert to the nature
stantive ; for it is said, that the word had signifies good- of a simple enunciative sentence or logical proposition,
mess, and gin signifies man; but had gin is a good which consists of a subject, a copula, and a predicate.
man, or man of goodness; and gin hao is human The subject, or that concerning which something is
goodness, or the goodness of man. Hence, too, we asserted, is always a noun substantive; the predicate
see why Wallis considers the English genitive case as may be a noun adjective. Thus, in the sentence
a possessive adjective; e.g. “the king's court,” aula ‘John is tall,” the subject is “John,” which is also a
regia, where he differs from all other English gram- noun substantive; the predicate is “ tall,” which is
marians, in calling the word “king's" an adjective. also a noun adjective. Complex sentences are re-
On the other hand, Lowth reckons the words mine and solvable into more simple ones: and where adjectives.
thine, which are usually called adjectives, as the pos- are used, so as to render a sentence complex, they are
sessive cases of me and thee. It is, perhaps, from always resolvable into the predicate of a logical pro-
a similar cause that Dr. Jonathan Edwards asserts position. Thus, if it be said, that “a wise man is.
the Muhhekaneew or Mohegan Indians to have no ad- cautious,” this sentence is resolvable into the two
jectives at all in their language; a fact on which Mr. simple sentences “a man is cautious,” and “ that man
Horne Tooke lays great stress, but which, in reality, is wise,” and in each of these the adjective is the
proves nothing as to the signification of language, predicate of the proposition. *- - -
whatever it may do as to its forms or inflections. The corollaries to be drawn from this statement are.
Vocative. It seems hardly necessary to distinguish the voca- several. In the first place, whenever the name of a
“to,” “by,” “from,” and “in.” In the Latin lan-
guage, a particular inflection is used to signify the end
to which an action is directed, and the case known by
that inflection is called the dative ; because verbs of
giving usually require the expression not only of the
thing given, but of the person to whom the gift is
made, and whose convenience or benefit is the end to
which the gift is destined. In order to express the
other relations above noticed, the Latin language avails
itself of the accusative or ablative inflection, either
alone or with a preposition.
tive case by any particular inflection. Indeed, we find
the terminations of the nominative and accusative
equally employed in Latin as exclamatory: and it is
said that the Sanscrit grammarians do not allow the
vocative to be a case. Yet, when we are speaking of
the different relations in which a noun may stand to
other words in a sentence, it is impossible to overlook
its use in those sentences where it stands forth pro-
minently as the object addressed or invoked. Thus,
in the first ode of Horace, we find two verses almost
wholly occupied with vocatives:
Mecanas, atavis edite regibus,
O et praesidium, et dulce decus meum !
relation or case; but we must observe, that the cases,
as distinguished in different languages, either by in-
flection, or by being joined with certain prepositions,
do not by any means agree with the classes of relation
here noticed. In the Greek idiom, the genitive ter-
mination- sometimes answers to an English accusative,
as ºrivo Tá úðaroc, I drink water ; sometimes to the
Latin ablative, as avrī āyaśāv &troötöðvat kaka, “ mala
pro bonis reddere;” and sometimes to the Latin accusa-
tive, as 'Avºp &vr' àvööc iro, “vir contra virum eat.”
The English genitive, “blind of an eye,” answers to
conception is employed as the subject of a proposition,
Thus, the conception expressed.
it is not an adjective.
by the words “good" and “goodness” is the same;
but if we predicate any thing of this conception; if,
for instance, we say “goodness is amiable,” the word.
And
goodness must necessarily be a substantive.
this does not depend on the form of the word; for if
the idiom of our language allowed us to say “good is
amiable,” or “ the good is amiable,” the word “good”
would be as much a substantive as “goodness.”
Hence it follows, that the distinction between a
substantive and an adjective does not necessarily de-
pend on any difference between the conceptions which
G R A M M A R.
35
its name (7) to riches in general; and particularly (8) Chap. I.
Grammar, they express, but between the different modes in which
to money. Hence were denominated all kinds of pay- Sºrrº-
S->' those conceptions are contemplated by the mind.
If
we contemplate goodness as a separate idea, if we
assert any thing of that idea, if we make it the subject
of any proposition, then it is a substantive; but if we
predicate it of any thing else, if we consider it only as
a quality of that thing, then it is an adjective.
Hence, again, it will follow, that an adjective and
a substantive cannot be convertible, without wholly
changing the meaning of the proposition in which they
are employed. Thus, to say that “envy is criminal,”
and that “criminality is envious,” are two propositions
entirely different.
. It is equally a rule of Universal and of Particular
Grammar, that an adjective cannot stand alone, but
must be joined with its substantive; which, is in truth,
no more than saying, that a predicate must necessarily
refer to some subject. Mr. Tooke, however, controverts
this rule, though it is certainly as old as the words ad-
jective and substantive. He objects that the rule
equally applies to the oblique cases of nouns substantive,
and that therefore “the inability to stand alone in a
sentence is not the distinguishing mark of an adjec-
tive;” but, though it were not a distinguishing mark, it
might yet be a rule common to all adjectives. How-
ever, the real intent of the rule is to distinguish adjec-
tives from the substantives with which they are used ;
and that in the most simple sentences; and with re-
ference not to their form or inflection, but to their sig-
nification. Thus, if we say “a golden” is valuable,”
the sense is incomplete, and the adjective “golden"
requires the addition of a substantive, as, for instance,
“ring,” to render it intelligible. On the contrary, if
we say “gold is valuable,” the sentence is perfect. Mr.
Tooke contends that “the adjectives golden, brazen,
silken, uttered by themselves, convey to the hearer's
mind, and denote the same things as gold, brass, and
silk.” The short answer to this is, that it is contrary to
common sense and experience to confound these terms
together ; and nobody ever does so who understands
the English language in the slightest degree. But if
we wish to trace the source of Mr. Tooke's error, we
must examine more particularly his expressions. First,
what does he mean by “uttered by themselves 7”
Words uttered by themselves are like syllables or let-
ters uttered by themselves. They are the mere ele-
ments of discourse. Their proper force and effect in
rational speech must depend on their connection with
each other. Again, what is meant by “denoting the
same things?” In so far as they are both of the same
origin, there is doubtless a common conception to
which they both bear relation; but it does not follow
that they both bear the same relation to it. A nume-
rous tribe of words derived from, or connected with,
this term, gold, is to be found in the different European
languages. Is it to be said that they all “convey to
the hearer's mind and denote the same things?” Let
us see how this can possibly be made out. From (1)
the splendour of the rising or setting sun, was denomi-
nated (2) the yellow colour resembling that splendour.
From the name of that colour, was derived (3) that of the
jaundice, which rendered the whole body yellow, and (4)
that of the gall, which produced the jaundice. From
yellow also came (5) the name given to the yolk of an
egg. And again, from this colour came (6) the name of
gold. Gold, being the most precious of metals, gave
ments, whether (9) voluntary gifts, or (10) offerings,
or (11) tribute, or (12) rent, or (13) fines; as well as
(14) debts due on any of these accounts. In process of
time, certain societies were formed and maintaified by
regular payments from each member, and these societies
received their name (15) from this circumstance. The
name was afterwards extended to societies (16) or fel-
lowships in general; and it occasioned the peculiar
designation of a well-known building (17) in London.
Fines in ancient times were. applied, in the nature of
punishment, to almost all crimes; and hence their name
came to signify (18) punishments in general; and par-
ticularly a barbarous mutilation (19) often used as a
punishment. Lastly, the general term for punishment
was naturally applied to the criminality (20) by which
the punishment was occasioned.
We have traced in the margin" these progressive
changes of signification, as they are to be found in
the Maeso-Gothic; Anglo-Saxon; Alamannic; Lom-
bardian; Precopian; Greek; Latin, old, middle, and
barbarous; Suevian ; Swedish ; Islandic ; Russian ;
German ; Dutch; Welsh ; Italian ; old and modern
French, and old and modern English. Every change
of application is occasioned by a new operation of the
mind. The sound of the word conveys a new thought,
similar indeed to the preceding, and having reference
to the same conception, but placing it in a new light.
It would be absurd to say, that the thought remained
the same through all these different uses; and it is
equally incorrect to say, that it remains the same after
any one step. There is as real, though not as great
a difference between “gold” and “golden,” as there
is between “a guilder" and “Guild-hall.” If Mr.
* 1. Gr. Yaxa. (Hesych.) -
Dut. Geel. Ger. Gelb. Russ. Gelto.
2. Suev. Gel. Swed. Gael.
Isl. Gulur. Lat. Gilvus, helvus, gulbus, galbeus, galbinus. Ital.
Giallo. Fr. Jaulne, jaune. A. S. Geolu, geolewe. Eng. Yellow.
3. Ger. Gel-sucht. Dut. Geel-zugt. Russ. Geltukla. Fr. Jaul-,
misse, jaunisse. Eng. Jaundice.
4. Russ. Geltchy. Eng. Gall.
5. Russ. Geltoky. Fr. Jaume. Eng. Yelk, yolk.
6. Isl. Gull. M. Goth. Gulth. Praecop. Goltz. A. S. Gold.
Dut. Goud. (N. B. Wachter derives gold from gel-od, yellow sub-
stance.) -
7. Wel. Golwd. -- -
8. Ger. Gelt. Dut. Geld, (hence guilder, &c.)
9. Ger. Gilt. (Lomb. Launchild, a mutual gift).
10. A. S. Gyld. (Godgyld and deofulgyld, offerings to God aid
offerings to devils). -
11. Isl. M. Goth. and A. S. Gild. (Danegyld, tribute to Danes;
Wodegild, tax on woods; Hormegyld, tax on horned cattle; whence
the family name of Hornyold, still subsisting.)
12. Isſ. Aakergield, rent of a field. Giltbar aaker, a field pro-
ducing rent. -
13. Barb. Lat. Geldum, gildum,
the fine for killing a man. - -
14. Alaman. Gult, a debt. Gelter, a debtor or creditor.
15. A. S. Gild. Barb. Lat. Gelda, gilda, gildonia, (whence Me-
nage derives the expression, courir leguilledou).
16. Diobol-gelde, the devil's fellowship. (See EccARD). A. S.
Frythgyld, the society of confederates. The dean of Guild, an
officer well known in Scotland, &c.
17. Guild-hali. – .
18. Alaman. Gilten, to suffer punishment.
19. Eng. Geld, gelding. Germ. Geltze,
Gallda, (Gieltºfte, aries castratus).
20. A. S. Gylt, agyltan, gyltend, gyltig. Eng. Guilt, guilty.
N. B. It is remarkable that an analogy similar to that which exists
in the above articles, 1, 2, 3, and 6, is found in the Latin words
Aurora, Aurcus, Aurigo, and Aurum. -
Isl, Gield. A. S. Wergeld,
porca castrata. Isl.
F 2
36
G R A M M A. R.
origin, are so far from denoting the same conceptions,
that they are often used in direct opposition to each
other. “Is this gold 2—No, it is only gilt.” So gold and
golden are not the same. They both, indeed, refer to
the same conception; but they refer to it in different
ways. In the one instance, the conception (namely
gold) is the verything of which we are speaking; it is
the logical subject of the proposition; the mind looks
at it, as it were, directly; as when Bassanio says,
Thou gaudy gold,
Hard food for Midas—l will none of thee.
Whereas, in the other case, it is noticed but inci-
dentally, as a thought passing over, and giving a mo-
mentary tinge to another thought, but differing from it
as the light in which we view a substance differs from
the substance itself. 'So the same Bassanio, in the
same scene, speaking of his mistress's portrait, says,
here in her hair,
The painter plays the spider, and hath woven
A golden mesh to intrap the hearts of men.
If is very true that these secondary thoughts, which
are expressed by adjectives, may be brought more dis-
tinctly before the mind, and treated as substantives in
connection with other substantives. It is thus, that
instead of “a virtuous man,” we may say “a man of
virtue;” but though there appears, in this instance, very
little difference of meaning, yet, on analysing the two
expressions, we shall find that a new and distinct ope-
ration of the mind is performed, which operation is
here expressed by the word “ of.” We do not merely,
as in the case of the words “virtuous man,” con-
template the conception of “man” as a substance,
and that of “virtue" as a quality belonging to the in-
dividual in question; but we contemplate “man” as
having a substantial existence, and “virtue" as having
an existence capable of coalescing with man; and
further, we contemplate the actual union of these two
thoughts, as expressed by the word “ of.” Slight,
therefore, as the difference of meaning is between the
words “a man of virtue and a virtuous man,” yet the
grammatical difference is not to be overlooked: and
the best proof of this will be to consider how totally
the style of any author would be altered if we were
always to change the genitive case of the substantive
into an adjective, and vice versä. Suppose that, in-
stead of the line— -
& “The quality of mercy is not strained,”
we were to say, “the merciful quality is not a quality
of compulsion,” we should certainly not augment the
force and beauty of the language; and we should as
certainly change the flow and current of the thought;
we should alter the Grammar without improving the
poetry. f -
From what has been already said, we may perceive
the absurdity of asserting that “adjectives, though con-
venient abbreviations, are not necessary to language,”
and still more, that “the Mohegans have no adjectives
in their language;” for though this latter fact is vouched
by “Dr. Jonathan Edwards, D. D. pastor of a church
in Newhaven; and communicated to the Connecticut
Society of Arts and Sciences, and published by Josiah
Meigs,” yet it amounts to nothing else but that the
Mohegans cannot distinguish subject from predicate,
or substance from quality; and if so, they must be ut-
Grammar. Tooke were right, to gild a thing would be to convert
S->~ it into gold: whereas these words, though of the same
terly destitute of the faculty of reason, which we sup-
pose neither Dr. Edwards, nor Mr. Meigs, nor Mr.
Tooke, intended to assert. - |
It is a common rule, that the adjective should agree
with its substantive in gender, number, and case, from
whence, perhaps, it might at first sight be inferred,
that gender, number, and case, properly belong as well
to the adjective as to the substantive. This, however, is
not the fact: the adjective simply expresses a quality;
but it must of necessity be connected in language with
its substantive, and that connection is effected in many
languages by a similarity of inflection; and as the in-
flections of the substantive express gender, or number,
or case, those of the adjective often follow a similar
rule of construction. This construction, it is obvious,
is a matter belonging only to particular, and not to
Universal Grammar. It may exist in one language
and not in another; and, in fact, there are languages,
(our own for example) in which all these variations
are wholly unknown. -
&
Chap. I.
On the contrary, the variation of degree is one which Degrees of
belongs, in an especial manner, to certain adjectives,
but not at all to substantives; and where there are
variations of degree, they may be compared together,
whence arise, what are technically called by gram-
marians, the degrees of comparison.
Substantives cannot be compared, as such, in point
of degree; for that would be to suppose that the nature
of substantial existence was variable; and that one
existing thing was more truly existing than another,
which is absurd. “A mountain,” says Harris, “can-
not be said more to be, or to eatist, than a molehill; but
the more and less must be sought for in their quanti-
ties. In like manner, when we refer many individuals
to one species, the lion A cannot be called more a lion
than the lion B; but, if more any thing, he is more
fierce, more speedy, or exceeding in some such attri-
bute.
nus, a crocodile is not more an animal than a lizard is,
nor a tiger more than a cat; but, if any thing, they are
more bulky, more strong, &c.; the excess, as before,
being derived from their attributes. So true is that
saying of the acute Stagyrite, ek’ &y strièéxotro à éoria
Tô pa)\ov & rô ºrrow ; substance is not susceptible of
more and less." Sanctius, referring to this same pas-
sage of Aristotle, observes, that we may hence infer
that comparatives cannot be drawn from nouns sub-
stantive. “ Hence,” adds he, “they are deceived,
who reckon the words sener, juvenis, adolescens, infans,
&c. as substantives, for they are altogether adjectives.
Nor is it to be objected, that Plautus has made from
panus the comparative panior; for he does not there
mean to express the substantial existence of the Car-
thaginian, but his cunning, as if he had said callidior ;
for the Carthaginians were reputed to be a very cun-
ning people. So the writer who used the word Nero-
nior, from Nero, meant only to signify an excess o
cruelty.” .
As substantives in general admit not of degree; so
there are some adjectives which equally exclude either
intention or remission. Thus Scaliger justly observes,
that the word “medius” can neither be heightened
nor lowered in degree; and that the same may be
said of “hodiernus,” and of many other adjectives.
On this topic Mr. Harris thus expresses himself: “As
there are some attributes which admit of comparison,
So again, in referring many species to one ge-
comparison,
G R A M M A. R.
37
Grammar. so there are others which admit of none. Such, for
--~ example, are those which denote that quality of bodies
* arising from their figure; as when we say a circular
may compare it with some assumed motion of the Chap. I.
quality in general. S-º-Nºse.”
The positive is the simple expression of the quality: Positive.
table, a quadrangular court, a conical piece of metal,
&c. The reason is, that a million of things partici-
pating the same figure, participate it equally. To say,
therefore, that while A and B are both quadrangular,
A is more or less quadrangular than B, is absurd.
The same holds true in all attributives denoting defi-
nite qualities, whether contiguous or discrete, whether
absolute or relative. Thus, the two-foot rule A, can-
not be more a two-foot rule than any other of the same
length. 'Twenty lions cannot be more twenty than
twenty flies. If A and B be both triple or quadruple
of C, they cannot be more triple or more quadruple
one, than the other. The reason of all this is, that
there can be no comparison without intension and re-
mission; there can be no intension and remission in
things always definite: and such are the attributes,
which we have last mentioned.” This reasoning, which,
as far as it goes, is very just, seems nevertheless to re-
quire some further development. What is here meant
by “things always definite?" Plainly, what we have
already called ideas, and those clearly conceived. The
idea of a circle, when clearly conceived, is a thing
always definite. By the generality of men it is clearly
conceived; and consequently they would think it ab-
surd to say, that one table was more circular than an-
other; but those who have not a distinct idea of a
circle would not perceive the absurdity of the expres-
sion. To them, circularity would appear capable of
intension and remission; and therefore they would
conclude, that this quality admitted of comparison as
much as sweetness or sourness, hardness or softness,
heat or cold. Hence we find in language such words
as round, which expresses the idea of circularity in a
vague and indistinct manner; and these words are
commonly used in the comparative as well as in the
positive degree. For the same reason, all words signi-
fying bodily sensation are capable of comparison; for
though we agree generally in the meaning which we
attribute to them, yet there is no definite idea to which
any one of them can be distinctly referred. Men em-
ploy the terms “hot, cold, white, black, green,” &c. so as
to convey to each other's mind certain general notions,
but not to communicate precise and distinct ideas,
like those expressed by the words, “ square,” or “tri-
angle.” Again, in moral qualities there is usually the
same indistinctness. We say, one man is braver or
wiser than another; because we possess no absolute
standard of bravery or wisdom. If we possessed such
a standard, we should simply say, that each of the two
was either brave or not brave, wise or unwise. There
is no more common comparison in all language than
between that which is good and that which is better;
yet the pure idea of goodness presented to us by the
Christian religion excludes all comparison——“There is
none good but one, that is GoD.”
We have observed that where there are variations of
degree, these variations may be compared together.
Grammarians have fixed three degrees of comparison;
the positive, the comparative, and the superlative.
It seems material to observe, that the comparison
here referred to is of two kinds. We may either coin-
pare a quality, as existing in any given substance, with
the same quality as existing in other substances, or we
and Harris säys, it is improperly called a degree of
comparison; but in this he seems to be wrong; for it
is that form in which the comparison of equal degrees
of the same quality is expressed, either affirmatively
or negatively. Thus we say, in the positive degree,
“Scipio was as brave as Caesar,” “Cicero was not so
eloquent as Demosthenes.” -
The comparative expresses the intension or remis- Compara-
sion of any quality in one substance, compared with tive.
the same quality in some one other substance, as
“Cicero was more eloquent than Brutus;” “Antony
was less virtuous than Cicero.” Hence it is manifest,
that there are, properly speaking, two kinds of the
comparative degree, one expressing the more, and the
other the less of the quality compared. Languages in
general have employed a peculiar inflection only to
express the former; but the latter is in its nature no
less capable of expression; and both belong to those
distinctions which constitute Universal Grammar. It
Ås to be remarked, that the comparative, though it ex-
cludes the relative positive, does not necessarily in-
clude the absolute positive. If we say “John is wiser
than James,” we exclude the assertion, that “James
is as wise as John;” but we do not necessarily include
the assertion either that “John is wise,” or that
“James is wise.” All that may really be intended by
the affirmative, is a negation of the negative. It may
only be meant to assert that “John is less unwise than
James.”
The superlative expresses the intention or remission Superlative,
of a quality in one thing or person, compared with all
the others that are contemplated at the same time.
There must be more than two objects compared, but
the number compared may be indefinite : we may say,
Octavius was the most prudent of the triumvirate;
Homer was the most admirable of poets; Solomon was
the wisest of men. . In other respects, what we have
observed of the comparative, applies equally to the
superlative, which may properly be considered as ex-
pressing the most or the least of the quality in ques-
tion, but which does not, any more than the compa-
rative, necessarily include the absolute positive. Of
this remark, the common proverb, “Bad is the best,”
affords a sufficient illustration.
Hitherto, we have only spoken of the comparison of
qualities existing in one subject with those existing in
another; but the comparison may be made with a
general conception of the quality: and here also
may be three similar degrees. Where the quality is
supposed to be of the general or average standard, we
use the positive; where we mean to express simply an
excess beyond that standard, we use the comparative.
Thus Virgil says: - -
* Tristior, et lacrymis oculos suffusa nitentes:
and Horace,
} Rusticior paullo est.
Lastly, where we mean to express a high degree of
eminence in the quality of which we speak, we use the
superlative, as vir doctissimus, vir fortissimus, a most
learned man, a very brave man; that is to say, not
the bravest or most learned of all men that ever ex-
isted, or of any given number of men; but a man pos-
3S G R A M M A. R.
employs these pronouns in a secondary sense, as if Chap. I.
Grammar sessing the quality of learning or bravery in a degree
- they expressed a quality instead of a substance; but S-
S-N-2 far beyond the common standard. -
It is of small consequence to inquire whether all
these forms of speech together are properly named de-
grees of comparison, and equally immaterial whether the
particular names, positive, comparative, and superlative,
are well chosen to designate each degree. Many emi-
ment grammarians have contended on these points.
Vossius objects to the name positive, because the two
other degrees are equally positive, that is, equally lay
down their respective significations, whence the Greeks
called the superlative hyperthetic, from riSévau, to lay
down. Not more appropriate, says he, is the name of
the comparative degree, since comparison is applied to
many words, both nouns and adverbs, which are not of
this degree, as the adjectives, like, unlike, double; and
among adverbs, equally, similiter, &c. Moreover, com-
parison is effected no less by the superlative than by
the comparative : for it would be equally a comparison
if I were to say, speaking of Varro, Nigidius, and
Cicero, “ Varro is the most learned of the three ;” as
if I were to say, speaking of Varro and Nigidius only,
‘‘Varro is the more learned of the two.” Lastly, the
word superlative is not well chosen, since it merely sig-
nifies preference, or the raising one thing above ano-
ther: and in this sense the comparative itself is a
superlative; for in saying, “Warro is more learned
than Nigidius,” I prefer, or raise Varro above Nigidius
in regard to learning.
For similar, reasons, Scaliger proposed new names
for the three degrees. The first he called the aërist,
or indefinite ; the second, the hyperthetic, or exceeding;
and the third, the acrothetic, or highest degree. Quin-
tilian and others call the positive the absolute degree;
others call, it the simple, and so forth ; but none of
these names having come into general use, we think it
more convenient to hold to those which are commonly
received; not considering the choice of a name as very
important, compared with the accuracy of a distinction;
and that the, three variations of adjectives in degree
are essential to Grammar, we have already sufficiently
proved. - - :
- It is of more consequence to note, that intension
and remission not being confined to adjectives, the
degrees of comparison are not confined to them, but
are common also to certain verbs, participles, and ad-
verbs; in short, to the whole class of attributives (as
they are called by Harris), provided that, in significa-
tion, they import qualities which may be increased or
diminished. Thus, as the adjective “amiable" admits
of the comparative and superlative “more amiable,”
and “, most amiable;” so we may use the expressions
“ more loving,” “, most loving;” “to love well,” “to
love better;”:# to love more,” “to love most of all.”
These indications of degree, however, have been rarely
expressed by inflection, except in adjectives; and this
seems to be the true reason why the degrees of com-
parison have often, but inaccurately, been considered
by grammarians as belonging to adjectives alone. It
is scarcely worth while to occupy attention with such
words as auroraroc, used by Aristophanes; or ipsissi-
mus, employed by Plautus. Some critics, indeed, have
seriously adduced these as examples of comparison in
pronouns, as if I could be more I, or he more he in
reality; whereas it is plainly seen, that the eomic
writer, by a natural boldness in-the use of language,
not as if a man could be more or less himself without
losing his personal identity.
We come now to consider the two great classes, into Kinds of
which adjectives may be divided; and these, as we have adjectives,
before observed, depend on their expressing, or not
expressing action. Thus, if we say “a four-footed
animal,” although the quality of being four-footed has
reference in this instance, to action, as its final end ;
yet, as it does not express action (for a table or a chair
may also be four-footed), this is an adjective of the first-
mentioned kind. . On the other hand, if we say “ a
moving animal,” we clearly express that action is really
taking place: this, therefore, is an adjective of the second
kind. Now, of these two kinds, the former are exclu-
sively called adjectives by the majority of gramma-
rians; but the latter are as commonly called parti-
ciples; and we adopt these distinctive terms from an
unwillingness to alter the received nomenclature of
grammatical science; but at the same time, we wish it
to be clearly understood, that both the adjective and
participle of the common grammarians fall under the
definition which we have above given of the word
adjective in its largest sense. .
Of the adjective simple, or unmixed with any idea
of action, little remains for us to observe; but before
we proceed to the consideration of the participle, it
may be proper to notice a large class of adjectives,
which, though they do not express action, yet bear
reference to it. Such are those words expressive of
the capability or habit of action, which Mr. Tooke, in
his eager desire for singularity, has thought fit to class
among the participles. There is great hazard when
a writer chooses to treat all his predecessors with con-
tempt, that he may ehance to fall into very gross errors
himself. Mr. Tooke has confounded, in his new scheme
of participles, the verbal adjectives, gerunds, and par-
ticiples of former writers; and, at the same time, has
laid down no clear definition of his own to guide us
out of the labyrinth. What is more, he has adopted as
participles the verbal adjectives in bilis, icus, and icus, .
and excluded those in ar, arius, bundus, icius, &c. which
seem quite as much entitled to the same distinction.
Upon a full consideration of all these different kinds
of adjectives, there seems to be no reason for classing
them apart from the simple adjective, and as little for
confounding them with the participle. *
They ought not to be separated from the simple ad-
jective, because they do; in fact, express only a simple
quality; and it is difficult, if notimpossible, to draw a
line between qualities which are originally derived from
action, and qualities not so derived. Let us take, for
instance; the word falsus, false.
rived from falla, which expresses the act of failing or
deceiving; yet, by a transition of meaning, it comes to
signify simply that which is not true. In like manner,
many of the words which Mr. Tooke treats as par-
tieiples have been really introduced into the English
language as simple adjectives, without the least re-
ference to the action, which their radicals expressed in
other languages. Take; for instance, the word “pal-
pable.” We commonly say “it is palpably false,”
“the truth is palpable,” &c.; yet, perhaps, few per-
sons, when they use these phrases, entertain any notion
of feeling and handling the truth or falsehood in ques-
-º-
No doubt this is de-
• G R A M M A. R.
39
Grammar, tion, though palpare, to feel or handle, is the undoubted
*
S-> origin of this word. The same may be said of “duc-
Participle.
tile,” “frail,” “sensible,” “noble,” and many other
English adjectives, which have not the slightest pre-
tence to be considered as participles.
If the mere derivation from a verb is to entitle a word
to be called a participle, we should have numerous
classes both of substantives and adjectives so dis-
tinguished; for if ductilis be a participle, because it is
derived from duco, so is audar, because it is derived
from audeo; ridiculus, because it is derived from rideo;
and a thousand other adjectives. Nay, we may add
to this list the substantives derived from verbs, if the
mere derivation is to be a test of the grammatical use.
Thus, we may say, that pistrinum, a bakehouse, is a
participle of pinso, to bake; juramentum, an oath, of
juro, to swear; judicium, a judgment, of judico, to
judge, &c.
The truth seems to be, that in this, as in number-
less other instances, Mr. Tooke has mistaken the his-
tory of language for its philosophy. Because the word
noble is derived from nosco, to know, therefore he calls
it a participle of that verb . At this rate, all the parts
of speech must become an inextricable mass of confu-
sion; for, historically speaking, each is derived from
the other, and there cannot be any rule which gives
any one the precedence. If we look to the significa-
tion, all is clear.
action, or it does not. If it does not, it is a simple
adjective; and the circumstance of its referring to the
habit or capacity for action cannot alter its character.
The words “forcible” and “culpable” relate originally
to the actions of forcing and blaming; but they relate
to them only as the ground-work of an existing quality,
and not as being really in action, or as having been so,
or to be so, at any given time. These considerations
will probably suffice to clear away all the difficulties
which Mr. Tooke has raised respecting what he calls
the participles of the potential mood active, the poten-
tial mood passive, the official mood passive, and the
future active. They are all, as used in the English
language, simple substantives, or simple adjectives:
and to rank them among participles, would not only
be to oppose the great majority of writers who have
treated of these subjects, but to confound all reason-
able principles relating to this part of Grammar.
We come, then, to that part of speech which is com-
monly denominated the participle. The origin of this
name is well known. “Partem capit a nomine, partem a
verbo.” But this is an explanation which is merely ap-
plied to the learned languages. The definition of Vos-
sius is, participium est vow variabilis per casus significans
rem cum tempore. Here, too, we see nothing of Univer-
sal Grammar. The being variable by cases is a mere
accident of certain languages. The signifying a thing,
with time, depends indeed on more general principles,
and these it is necessary to examine.
What is meant, in this part of the definition, by
“signifying a thing,” we need not, perhaps, make mat-
ter of dispute. We will assume, that it means, in the
language which we have adopted, “naming a concep-
tion.” The participle simply names ; it does not assert.
The words “loving, moving, reading, thinking,” &c.
assert nothing respecting these acts; they merely name
the acts, or rather they name the conceptions, as in
action. It is said that the participle is ranked among
Either a given adjective expresses
...”
nouns when it constitutes the subject of a logical pro- Chap. I.
position; and among verbs when it forms the predicate; \*Y*
but this is not accurate; a participle, as such, can
never form the subject of a proposition. The example
given is, Militat omnis amans, TIác & påv troXspići; but
in this instance amans is a mere adjective, agreeing
with homo understood; and it is the same in the Greek.
On the other hand, when the participle is a predicate,
as Socrates est loquens, it fills the proper office of an ad-
jective; and is not to be treated as a verb, at least in
the sense which we have attached to the latter term. -
The adsignification of time is proper to the participle,
inasmuch as time is essential to action. This point,
however, Mr. Tooke contests, upon the ground, that
the Latin participles, present, past, and future, are not
confined to the times from which they respectively re-
ceive their designations. Proficiscens is a participle of
the present tense; yet Cicero says, alful proficiscons,
thus connecting time present with time past. So pro-
feckuro tibi dedi literas, connecting the past with the
future : and again, quos spero societate victoriae tecum
copulatos fore ; where spero is present, copulatos past,
and fore future. None of these examples, however,
prove any thing against the expression of time by the
participles, but merely that time is contemplated in
various lights by the mind in one and the same sen-
tence. Thus, in the phrase abfui proficisceus, the first
word relates to the time of speaking, and the second to
the time of acting. The going was present, when the
absence (which is now past) was present. Again, dedi
refers to a time past; but when that time was present,
the departure (expressed in profecturo) was future.
A thousand such cases as these would lead to no in-
ference whatsoever against the expression of time by
the participle. **. - ! .
It is necessary to observe, however, that words
which express time, express it in two ways, either as
simple existence, or as relative to the different portions
of duration. Thus, when we say “justice is at all
times mercy,” the present is a mere expression of ex-
istence, a present continuous. So when we say “the
sun rises every day,” we speak of an act habitually pre-
sent. It is the nature of the human mind to be able'
thus to contemplate duration; but this in no degree
interferes with, still less contradicts, the view which
we take of different portions of time, as past, present,
and future, with relation to each other. The assertion,
for instance, that the sun rises every day, does not at
all clash with the other assertion, that the sun rises at
this moment. In both cases time is referred to ; a
certain portion of time is designated in the one case,
which coincides with the general assertion in the other;
and, in fact, the difference between the two assertions
does not depend on the verb itself, but on the accom-º
panying words “every day” and “this moment.”
In these respects the verb and participle agree. The
participle is an adjective so far participating the nature
of the verb as to signify action, and it cannot signify
action without the capability of adsignifying time.
Particular languages may or may not have separate
words adapted by inflection to signify the different por-
tions of time in a participial form. In truth, the notion
of time is in all such cases a new element in the com-
pound conception, which compound conception may be
expressed by one word or by several. The complexity
of conception may go still further. It may include the
40 ... * G R A M M A. R.
Grammar distinctions of active and passive, of absolute and con- comparison, it is clear that participles, as well as other Chap. I.
J - ditional; and, in short, of all those which we shall adjectives, when they express qualities capable of in- \-
have to consider when we come to treat of the verb.
Hence we see, that languages may have as great a
variety of participles as they may of moods and tenses;
and it does not seem of the nature of language alto-
gether to exclude participles from the parts of speech ;
for Mr. Harris is perfectly right in saying, that if we
take away the assertion from a verb, there will remain
a participle. Of course he is speaking of the signifi-
cation, and not of the sound, and therefore Mr. Tooke's
ridicule of this passage is entirely misplaced. It is an
observation, as old as Aristotle, that the words
“Socrates speaks" are equal in signification to the
words “Socrates is speaking;” but it is evident that
the assertive part of this sentence consists entirely in
the word “ is ;" which word being taken away, the
word “speaking” still expresses a quality of Socrates,
and expresses that quality in action, and is therefore
a participle. And so it will happen with every verb,
as is instanced by Harris in the verbs yoépét ypápov,
“writeth,” “writing.” Tooke misrepresents Harris
as saying, that, by removing et and eth, he takes away
the assertion; whence he concludes, that Harris sup-
posed the assertion to be implied in those syllables;
but Harris says nothing about taking away et and eth.
He says what is very true, that the words ypápet and
writeth imply assertions, and that in the words ypápaſy
and writing, the assertion is taken away, and yet there
remains the same time and the same attribute ; which
expressions of time and attribute, without assertion,
constitute a participle. :
It has been laid down as a rule by some writers,
that there can be no participles but what are derived
from verbs; and hence they deny that such words as
togatus, galeatus, &c. are to be called participles. Au-
gustinus Saturnius, who treats particularly of this
point, calls them, by way of distinction, participials.
It is manifest, however, that this is a distinction alto-
gether nugatory, in regard to Universal Grammar.
When Othello says -
My demerits may speak umbonnetted,
he uses exactly the same form of speech, as if he had
said uncovered, and the one word is as truly a participle
as the other; although there may be no authority for
the use of the verb “to bonnet.” Uncovered and un-
bonnetted equally express a quality, with reference to
an action of past time, viz. the removing the cover or
bonnet from the head; and it is by this signification,
and not by their etymology, that the part of speech to
which they belong is to be determined.
We must not be surprised to find, that participles of
different classes pass into each other. Many active
participles come to have a passive signification. The
word evidens, which was originally active, is found with
a passive meaning, from whence our common adjective,
evident, is derived. This is a circumstance not pecu-
liar to participles; for when we come to treat more at
large of those transitions of meaning, which are the
ground-work of etymological science, it will be found
that they apply to every part of speech indifferently.
Men cannot always find a separate term to express each
distinct shade of thought, and they naturally avail.
themselves of those expressions which come the nearest
to their meaning.
From what has before been said on the subject of
tention and remission, may admit the three degrees of
comparison: thus we may say amantior as well as
durior, amantissimus as well as durissimus. It matters
not, that in some languages the idiom will not allow
of expressing the degrees of comparison by inflection;
that, for example, in English we cannot say lovinger,
or lovingest ; this is a mere accident of the particular
language, depending principally on circumstances con-
nected with its sound; and it is to be observed, that
however barbarous such words as lovinger or lovingest
might sound to the ear, yet they would be perfectly
intelligible to the mind: there would be nothing absurd
or contradictory in the combination of the thoughts;
for the same combination is effected by the words
“more loving," and “most loving;" and in all lan-
guages there must be means more or less concise, or
circuitous to express such combinations.
We have seen how the conception of a quality consi-
dered alone, and rendered the subject of assertion, be-
comes a noun substantive; and this applies, in principle,
as well to those qualities which are expressed by parti-
ciples, as to those which are expressed by other ad-
jectives. Whether the same or a different word shall
be employed for this purpose is, again, a matter of par-
ticular idiom. In English, we use the very same word
for both purposes. Thus, “ singing,” “dancing,” &c.
may be used in construction as adjectives, or as sub-
stantives of the sort commonly called abstract. We
may say “a singing man,” “a dancing woman;" or we
may say, “singing is an accomplishment,” “dancing is a
recreation,” &c. In Latin, the idiom is different: cautans,
saltans, &c. can only be used in the former of these two
ways; but, nevertheless, a similar principle is observa-
ble in the use of what are called gerunds and supines.
Scaliger gives the following account of the gerund:
“From these (participles) our ancestors chose certain Gerund.
tenses, by means of which they might imitate those
Greek terms Ackršov, paxmréoy, &c. but with a more
ample and extensive use. These they called gerunds,
assigning them to three cases, pugnandi, pugmando,
pugnandum ; of which, the second preserved the power
of a participle, but so much the more aptly as the verbs
were excelled by the participles. For, as the cause of
action is more plainly shown by saying “cardens vul-
neravi,” than by saying cecidi, and better still by saying
“quia carderem vulneravi," the whole of this is expressed
by the gerund “cadendo vulneravi.” Moreover, in many
things the form and the end are the same; but the end
is partly out of us, as the ship is a thing out of the ship-
builder; and partly within us, in our minds, as is that
which is called an idea, by which we are impelled to the
external end. Now both of these they very skilfully
expressed; for both pugnandi and pugnandum signify
the end. Thus I may say, pugnandi causá equum ascendi,
I mounted my horse for the purpose of fighting ; or
pugnandum ester equo, I must fight (or the fighting must
be) on horseback.” “Hence it appears that these
(gerunds) are participles, differing little from other par-
ticiples, either in nature, or use, or even in form.”
Again he observes: “ some writers have called these
gerunds from their use participial nouns; for they are
neither pure nouns, since they govern a case; nor are
they pure participles, since, with a passive voice, they
bear an active signification.” h
G R A M M A. R.
41
Grammar.
Sºzº.”
\
© The same author thus speaks of the supine. “Nearly
Similar is the explanation to be given of the supines;
but these latter express the same meaning more forci.
bly. Thus, eo ad pugnandum signifies a future action;
*9 Pugnatum expresses the future so as to be quite ab-
solute.” “Hence it signifies activity with actives, and
passiveness with passives: eofactum injuriam, or injuria
-mäki factum itur; but indeed it always savours, in
some degree, of passiveness; for it does not so much
mean eo ut facian, as it means eo ut hoc fiat ; as if
one were to say, I am going indeed for the purpose of
doing so and so, but I hope it is already done; and
like Sosia's speech, Dictum puta, “ suppose it said.”
“Since, therefore, the end (or aim) of an action was to
be thus signified, the other extreme was not improperly
expressed by a different word.” Hence Scaliger ex-
plains the different use of the supines in um and u, the
latter of which he regards as a sort of ablative case.
“There is equally a movement,” says he, “from and
Pronoun.
to an object; and therefore we rightly say venatu
wenio, as we do venatum, cado.” He goes at length
into these considerations, opposing in some measure
what, other grammarians had said of the supine in u ;
but these questions are beside our present object: and
, all that is peoessary for us here is to show the chain
of connection which unites the participle, as an adjec-
five, Qn the one hand with the noun substantive, and
ºn the 9ther with the gerunds, supines, and infinitive
Hitherto we have considered the noun only in its
primary use, whether as substantive or adjective: we
have now to regard it in a secondary light, under the
common grammatical designation of a pronoun.
The name of the pronoun is sufficiently descriptive
of its use, which is to stand in the place of another
noun. The necessity for such words in language is
obvious; but as it has been well and briefly explained
by Mr. Harris, we shall adopt that learned author's
words. “Every object which presents itself to the
..senses, or the intellect, is either then perceived for the
first time, or else is recognised as having been per-
ceived before. In the former case it is called an ob-
ject rmg ºrpºrnº yuágstoc of the first knowledge or ac-
quaintance; in the latter it is called an object rmg
Ševrépacyvågsøg of the second knowledge or acquaint-
ance. Now as all conversation passes between parti-
culars or individuals, these will often happen to be
reciprocally objects rmg ºrpºrng Yugosac, that is to
say, till that instant unacquainted with each other.
What then is to be done? How shall the speaker
address the other when he knows not his name 7 or
how explain himself by his own name, of which the
other is wholly ignorant? Nouns, as they have been
described, cannot answer the purpose. The first ex-
pedient upon this occasion seems to have been 3éičic,
that is, pointing, or indication by the finger or hand,
some traces of which are still to be observed as a part
of that action, which naturally attends our speaking.
But the authors of language were not content with
this: they invented a race of words to supply this
pointing; which words, as they always stood for sub-
stantives, or nouns, were characterised by the name of
ºrwypig, or pronouns.”. So far Mr. Harris. His
observations, indeed, apply in strictness only to the
personal prongun; but upon similar principles rests
"the necessity for the other classes of pronouns, as will
W 0 L, I,
invented the pronoun thou.
easily appear when we come to consider them sepa- Chap. I.
As the noun is divided into substantive and adjective,
$o the pronoun, its representative, exhibits the same
diversity. If it be necessary to have a word repre-
senting a whole class of substantives, it is equally ne-
cessary that the quality which consists in belonging to
that class should be represented. If I, or you, or he,
be to be expressed, mine, or yours, or his, is to be ex-
pressed also, - -
We begin, therefore, with the pronoun substantive:
and of this we shall consider, first, the distinctions
which relate to it as a member of a simple proposition;
and, secondly, those which relate to it more generally.
Considered as the subject of a simple proposition,
we have to notice in the pronoun not only the distinc-
tions of number, gender, and case, which are common
to it with the noun, but also the further and peculiar
distinction of person. The noun substantive being the
name of a conception, that is of a thing, or of a person,
does not specify whether that thing or person is the
speaker, or is spoken of, or spoken to. One of these
three characters it must needs sustain: and in the in-
tercourses of speech that character is soon distin-
guished: and here also the statement of Harris is pe-
culiarly clear and satisfactory.
“Suppose the parties conversing,” says he “to be First per-
wholly unacquainted, neither name nor countenance on son.
either side known; and the subject of the conversation
to be the speaker himself. Here, to supply, the place
of pointing, by a word of equal power, they furnished
the speaker with the pronoun I. “I write, I say, I de-
sire,’ &c.; and as the speaker is always principal with
reason, the pronoun of the first person.”
respect to his own discourse, they called this, for that
-- of the conversation to Second
be the party addressed. Here, for similar reasons, they person.
* Thou writest,’ ‘ thou
walkest,’ &c.; and as the party addressed is next in
dignity to the speaker, or at least comes next to him,
with reference to the discourse, this pronoun they
therefore called the pronoun of the second person.”
“. Lastly, suppose the subject of the conversation Third per-
neither the speaker, nor the party addressed, but some son,
third object, different from both : here they provided
another pronoun, he, she, or it, which, in distinction
from the two former, was called the pronoun of the
“Again, suppose the subject of th
third person.” “And thus it was that pronouns came
to be distinguished by their respective persons."
The description of the different persons here given is
taken from PRIscIAN, who took it from Apollon IUs:
Personae pronominum sunt tres, prima, secunda, tertia.
Prima est cum ipsa, quae loquitur, de se pronuntiat; se-
cunda, cum de ed pronuntiat ad quam directo sermone
loquitur ; tertia, cum de ed quae nec loquitur, nec ad se
directum accipit sermonem, I. xii. p. 940. THEopore
GAZA gives the same distinctions: Ilpºrov (ºrpágwrov,
- . . gº - gº sº Af ſe ^ y - r - -- - sº
sc.) ºff repi Šavrā ºppáčei & Aéyºv' &vrépov, d, repi ră,
. t … ** - * - - • * t p • º
ºrpèc'òy 5 Ådyoc. rpárov & Tepi répe. Gaz. Gram. l. iv.
p. 152.
This account of persons is far preferable to the
common one, which makes the first the speaker, the
second the party addressed, and the third the subject;
for though the first and second be, as commonly de-
scribed, one the speaker, the other the party addressed;
yet, till they become subjects of the discourse, they
G.
42
G R A M M A. R.
of number; nor, indeed, is it easy to conceive a language Chap. I.
so constructed as to have pronouns without such a S-
Grammar, have no existence, Again, as to the third person's
S-TY-- being the subject, this is a character which it shares
Number,
in common with both the other persons, and which can
never therefore be called a peculiarity of its own. To
explain by an instance or two : When Æneas begins
the narrative of his adventures, the second person im-
mediately appears, because he makes Dido, whom he
addresses, the immediate subject of his discourse.
Infandum, Regina, jubes renovare dolorem.
From hence forward for 1,500 verses (though she be
all that time the party addressed) we hear nothing fur-
ther of this second person, a variety of other subjects
filling up the narrative. In the mean time the first
person may be seen every where, because the speaker
is every where himself the subject: they were, indeed,
events, as he says, - -
Quaeque ipse miserrima widi,
Et quorum pars magna fui,
Not that the second person does not often occur in the
course of this narrative; but then it is always by a
figure of speech, when those who, by their absence, are,
in fact so many third persons, are converted into se-
cond persons, by being introduced as present.
When we read Euclid, we find neither first person
nor second in any part of the whole work. The rea-
son is, that neither the speaker nor the party addressed
(in which light we may always view the writer and his
reader) can possibly become the subject of pure ma-
thematics. - -
... It follows, from what has here been said, that the
pronoun is strictly a necessary part of speech; for
though, as standing in the place of other nouns, it may
be considered a mere abbreviation of discourse, yet cir-
cumstances often occur in which such abbreviations
become indispensible. It is clear that discourse could
not be intelligibly carried on where the parties were
not known to each other by name, and did not also
know by name each individual of whom they might
speak, unless there were some means of distinguish-
ing them otherwise than by their separate and in-
dividual names, which means are really supplied by
the pronoun. - - - -
It has been observed, that notwithstanding the se-
parate characteristics of each person, there may be a
coalescence of the pronouns of different persons; but
this is subject to certain restrictions. The pronoun of
the first or second person may easily coalesce with the
third ; but the first and second cannot coalesce with
each other. For example, we may say (and the dif-
ference of idiom in different languages does not affect
these expressions), “I am he,” or, “thou art he:” or,
as in the text, “art thou he that should come, or do
we look for another ?” But we cannot say, “I am
thou,” nor “ thou art I:” the reason is, there is no
absurdity for the speaker to be the subject also of the
discourse; as when we say, “I am he;” or for the
person addressed, as when we say, “thou art he?”, but
for the same person, in the same circumstances, to be.
at once the speaker and the party, addressed is impos-
sible; and, consequently, so is the coalescence of the
first and second person. - -
Since the pronoun stands in the place of a noun,
and since number, as we have seen, is a conception
which may be combined in general with mouns, it
follows that the pronoun may have the distinctions
distinction. As to the first person, it is clear that
there may be many speakers at once of the same
sentiment, or, what comes to the same thing, one may
deliver the common sentiment of many, and in their
name; for the same reason, therefore, that the pronoun
I is necessary, the pronoun we is so too. Again, the sin-
gular thou has the plural you, because a speech may be
spoken to many, as well as to one: and the singular he
has the plural they, because the subject of discourse
often includes many things or persons at once.
The pronoun is also susceptible of the distinction of Gender.
gender, because the noun which it represents is so.
A difference, however, has been said to exist in this
respect between the pronouns of different persons:
and the reasoning thereon is plausible. It is certainly
true that the pronouns of the first and second
person, both in the dead and living languages, have
no distinct inflection expressing their gender; and
the reason for this is alleged to be that the speaker
and hearer being generally present to each other, it
would have been superfluous to have marked a distinc-
tion by art, which from nature, and even dress, was
commonly apparent on both sides. “Demonstratio
ipsa," says Priscian, “secum genus ostendit.” However,
it is by no means true that the pronoun of the first
and second person have no gender. They have not,
indeed, in any known language, inflections distinguish-
ing them in point of gender, but they always take, in
construction, the gender of the noun which they re-
present. Thus Dido, •
cui me moribundam deseris hospes 2
And Mercury addressing Æneas,
Tu nunc Carthaginis altae
Fundamenta locas, pulchramque wrorius urbem
Exstruis? - -
It is agreed on all hands that the pronouns of the
third person must almost of necessity receive the dis-
tinctions of gender in all languages. These pronouns
are called in Arabic the pronoun of the absentee, and, in
fact, they usually refer to persons or things which being
absent require to be distinguished, as to gender, &c.
by some expression in the discourse.
It is further to
be observed, that the pronouns of the first and second
person each apply only to certain known and present
individuals; whereas, the pronouns of the third person
may, in the course of one and the same speech, refer to
a great diversity of objects, requiring to be distinguished
by their respective genders. “The utility of this dis-
tinction,” says Harris, “may be better found in suppo-
sing it away.” Suppose, for example, we should read
in history these words: he caused him to destroy him—
and that we were to be informed that the he, which is
here thrice repeated, stood each time for something
different, that is to say, for a man, for a woman, and
for a city, whose names were Alerander, Thais, and
Persepolis. Taking the pronoun in this manner, divested
of its gender, how would it appear which was destroyed,
which was the destroyer, and which was the cause of
the destruction ? But there are no such doubts when
we hear the genders distinguished; when, instead of
the ambiguous sentence, “ He caused him to destroy
him,” we are told, with the proper distinction, that
“She caused him to destroy it.” Then we know with
certainty what before we knew not, viz. that the pro-
G. R. A. M. M. A. R.
43
words “and it,” we substitute the subjunctive pro- Chap. I.
Grammar. moter was a woman; that her instrument was the hero;
\-ev-, and that the subject of their cruelty was the unfor-
Case.
tunate city.
Case is a distinction which we have already observed
to be not essential to the noun, but only accidental. It
therefore is to be ranked among the accidents of the
pronoun; yet, so frequent is the occasion to use pro-
nouns, that many of them, especially those which are
particularly denominated personal, have the variations
of case, even in languages which vary their nouns in
this respect very little or not at all. When a person
speaks of himself as the performer of any action, he
seems naturally led to adopt a different phraseology
from that which he employs in speaking of the action
as done toward him ; and hence the difference between
I and me, thou and thee, runs throughout far the greater
number of known languages. After all, Universal
Grammar only furnishes the reason for this difference
when it exists, but does not prove its existence to be
necessary. There may be languages of which the pro-
nouns have no cases; but where they have cases, the
same function is performed by each case in the pronoun
as in the noun. - -
Substantive pronouns have been distinguished, and,
as it seems, with sufficient accuracy, into prepositive
and subjunctive. By prepositive are meant all those
which are capable of introducing or leading a sentence
without having reference, at least for the purposes of
construction, to any thing previous. We insert these
words, “at least for the purposes of construction,” be-
cause in truth all but the pronouns of the first and
second person must refer to some person or thing pre-
viously indicated. When we say “ he reigned,” or
“she lived,” we presume that the persons included by
he and she are previously known. These pronouns,
however, may introduce or lead sentences which do not
depend on any previous sentence in point of construc-
tion. . But it is not so with the other class of pronouns,
viz. the subjunctive. These cannot introduce an ori-
ginal sentence, but only serve to subjoin one to some
other which is previous. The principal subjunctive
pronouns in English are who and which, and sometimes
that. It does not seem essential to the constitution of
a language that there should always be such pronouns
as these; for they may always be resolved into another
pronoun and a conjunction; and consequently by such
other pronoun and conjunction their place may always
be supplied. Let us take the example given by Harris.
We will suppose that it is desired to combine into one
sentence the two following propositions: -
• 1. “Light is a body.”
- 2. “Light moves rapidly.”
Here it is obvious that the use of the noun light, in
the second proposition, may be supplied by the pro-
noun it, as thus: -
“Light is a body :
It moves rapidly.” --
This slight change, however, leaves the two proposi-
tions still distinct: let us then connect them by the
conjunction and ; thus: -
“Light is a body;
And it moves rapidly.”
Here is a connection of the two propositions, yet still not
so much dependence of the latter on the former, not
so intimate a union therefore of the parts, as if, for the
noun which ; thus:
“ Light is a body, which moves rapidly.”
Accordingly, we see that in the punctuation, which
most accurately represents the proper mode of reading
the passage, we gradually diminish the interval be-
tween the two propositions, from a period to a comma.
Of the nature of the subjunctive pronoun is the
interrogative: and therefore we very commonly find
the same word performing these two functions. Thus,
in English, the subjunctives who and which, are
used as interrogatives, though with a remarkable.
difference in their application. As subjunctives, in
modern use at least, who is applied to persons, and
which to things. As interrogatives they are both ap-
plied to persons, but who indefinitely, and which defi-
nitely. Thus, the question, “Who will go up with
me to Ramoth-gilead 7” is indefinitely proposeed to all
who may hear the question: but when our Saviour
says, “Which of you, with taking thought, can add
to his stature one cubit?” the interrogation is indivi-
dual, as appears from the partitive form of the worls
“which of you ;” that is to say, “what one among
you all.” These applications of particular words are
indeed matters of peculiar idiom; but the distinctions
of signification to which they relate properly belong to
the science of which we are treating. -
The interrogative pronouns are necessarily of a re-
lative nature, and on that account were ranked by the
stoics under the head of the article; but as they do
in fact stand for, and represent nouns, they are pro-
perly called pronouns. On interrogatives in general,
Vossius has the following just observation :-" It ap-
pears to me, that the matter stands thus: there are
two principal classes of words, the noun and the verb;
and, therefore, to one or other of these every interroga-
tion must refer. For, if I ask who, which, what, how many,
I inquire concerning some moun; but if I ask where,
whence, whither, when, how often, I inquire concerning
some verb. As, therefore, the words which are subsi-
diary to the verb are called adverbs, so the words
which refer to the noun should be called pronouns.”
Of all the substantive pronouns, those only which
directly and simply represent the three persons of a dis-
course, as above explained, that is to say, the subject
of the discourse, whether that be the speaker, the per-
son spoken to, or the person or thing spoken of; these
three classes alone, we say, are º called pronouns
personal. Some grammarians seem to have inaccu-
rately supposed, that all but the personal pronouns of
the first and second person were to be considered as
belonging to the third person. This, however, is inac-
curate, at least with respect to the relatives, who,
which, that, as may be observed in those lines of the
old song: - * a t
What! you, that loved!
And I, that loved!
Shall we begin to wrangle?
Where the relative that is of the second person in the
first line, and of the first person in the second line: and
if translated into Latin it must be rendered, not tu quae
amabat, and ego qui amabat, but tu quae amabas, and
ego qui amabam. - -
We shall not here go into a detailed consideration
of the various distinctions which different authors have
G 2
44
G R A M M A. R.
Grammar.
>~~
Adjective
pronouns.
made in the other classes of pronouns, the demonstra-
tive, the distributive, &c. It may suffice to say, that
their number and variety in any one language must, in
a great measure, depend on the classification of concep-
tions, which had become habitual among the early
formers of that particular language. Thus we cannot
in English express, without periphrasis, the Latin pro-
nouns qualis, quântus, &c. any more than we can the
adverbs quoties, qualiter, &c. Nor must it be forgotten,
that many of these pronouns pass into different classes
according as they are used in particular passages.
“Sunter istis,” says Vossius, “ quae pro diverso, vel usu
vel respectu, ad diversas pertineant classes.
This latter remark applies not only to the various
uses of substantive pronouns, but to their transitions
Almost all pronouns,
from adjective to substantive.
except the first and second personals, are clearly ad-
jectives in origin; but we cannot admit that they con-
tinue to be such when they stand by themselves, or as
Lowth rather singularly expresses it, “ seem to stand
by themselves.” It is true, that in such cases, they
often have “ some substantive belonging to them,
either referred to or understood;” but this only proves'
that they are pronouns. Whether we say “this is
good,” “it is good,” or “he is good,” there is always
some noun referred to, or understood : and the words
it and he “seem to stand by themselves,” just as much
as the word “ this” does. So in the phrases “one is
apt to think,” and “I am apt to think,” the words one,
and I equally “seem to stand alone,” that is to say,
they equally do stand alone. They perform the func-
tion of naming an object, so far as it is necessary to be
named; and they name it not as a quality of another
object, but as possessing a substantive existence in
itself. The words this, that, who, which, all, none, and
many of a similar kind, are therefore (in our view of
them) substantive pronouns when they stand alone,
but adjective pronouns when they are joined to a noun
substantive. When Antony says
- This—this was the unkindest cut of all.
We consider the word this to be a substantive pro-
noun. It may, indeed, be explained by transposition,
as if it were, “this cut was the unkindest of all;” but
such is not the order of the thoughts: and, in fact, the
particular wound inflicted by Brutus had been before
described at some length, but the noun cut had not
been used: and supposing that, for dramatic effect,
the line had been broken off at the word “was,” it
would have been impossible to say that the pronoun
this had any specific reference to this particular noun
cut, as we may easily perceive by so reading the
passage.
See, what a rent the envious Casca made
Through this the well-beloved Brutus stabb’d;
And as he pluck'd his cursed steel away,
Mark how the blood of Caesar followed it,
As rushing out of doors, to be resolv’d,
If Brutus so unkindly knock'd, or no :
For Brutus, as you know, was Caesar's angel. .
Judge, O ye gods, how dearly Caesar lov’d him
This—this was
If the passage had thus broken off, the pronoun
this would have rather seemed to refer to the whole
narrative of the share which Brutus had taken in the
transaction; that narrative presenting to the mind one
complete and definite conception. &
A passage in Othello will further illustrate our mean- Chap. I.
ing. Iago pretends to caution Othello against suffering S-v-
his mind to encourage any
honour: - - - -
—O, beware, my lord, of jealousy
It is a green-eyed monster which doth flake
The meat it feeds on, &c. &c.
After he has pursued this strain of reasoning for
some time, Othello, interrupting him, exclaims with
surprise, º
suspicion against his wife's
Why, why is this 2
Evidently meaning, Why do you act thus 2 Why do
you talk of jealousy to me, who am not at all disposed
to be jealous ! The word this cannot here be said to
refer to any one noun that precedes, or to any one
noun that follows it; and it is therefore most mani-
festly used with the force and effect of a substantive.
On the contrary, it is clearly used as an adjective,
in a subsequent passage, where Othello, speaking of
Iago, says— - - -
This homest creature, doubtless,
Sees and knows more, much more than he unfolds.
Whether the same or different words shall be em-
ployed to express the substantival and adjectival form
of pronouns is matter of idiom. Thus, a language
may, or may not, have different forms for the personal
and possessive pronouns. Lowth considers the word
mine as the possessive case of the personal I; but the
English substantive mine (if a substantive it be) answers
to the Latin meus, which is certainly an adjective. On
the other hand, the Latin mi, which is commonly called
the vocative singular of meus, seems to be the same
word with mihi, the dative case of Ego ; for it is used
in connection with plurals as well as singulars, and
with masculines, feminines, and neuters indiscrimi-
nately. Thus we have in Plautus, mi homines ; and in
Petronius, mi hospites; and in Apuleius, mi sidus, mi
parens, mi herilis (Sc. filia), mi conjuw, &c.; and in a
passage of Tibullus, the different manuscripts have,
some mi dulcis anus, and some mihi dulcis anus ; in all
which instances, the dative mihi seems to be intended
to be used in that manner which grammarians often,
though incorrectly, call redundant; and describe, as
adopted, nullá necessitatis, sed potius festivitatis causá.
There are many other idioms relative to the use of
pronouns which it is not here necessary to consider,
such as the combination of the adjective own and the
substantive self with the pronouns my, thy, &c. in Eng-
lish; and the subjoining the syllables met cunque, &c.
to certain pronouns in Latin, as ipsemet, quicumque, &c.
which are usually accompanied with some correspond-
ing change in the force of the original pronouns.
The qualities from which different classes of pro-
nouns take their common grammatical designations, as
distributive, definitive, &c. may in general be viewed as
existing in the objects, and both the object and the
quality may be set forth together, as in comfmon sub-
stantives and adjectives. Thus the quality of alter-
nation, if we may so speak, is expressed in English by
the word either, and the quality of diversity by the
word other, and these may doubtless be united with
their proper substantives in the same manner as any
other adjective may. Thus we say, “ take either
horse,” “choose another man;” and in these and si-
G R A M M A. R.
45
Grammar,
have no etymological affinity to the words one and two. Chap. I.
milar passages the words either and other are to be
>~~ considered as pronominal adjectives. . . . -
Numerals.
The connection between the pronoun and the article
has always been admitted to be very close and in-
timate; and therefore many authors rank some of these
pronouns, especially the definitives, among the articles.
Harris is of this opinion, and he cites in support of it
the authority of several ancient grammarians. We do
not pretend to decide very dogmatically on this point;
but, upon the whole, we are disposed to follow the
great majority of writers, in confining the designation
of article to those words which perform the simple
function of individualising conceptions; nor can we
think it right to reject altogether the pronominal ad-
jectives, which must be the case if we were to adopt
Harris's criterion: “the genuine pronoun always stands
by itself, assuming the power of a noun, and supply-
ing its place; the genuine article never stands by itself,
but appears at all times associated to something else,
requiring a noun for its support as much as attribu-
tives or adjectives.” It does not appear to us correct
to say that the pronominal adjectives do not stand for
other nouns. They seem to stand for the names of
various different conceptions which are principally used
for the purpose of distributing our conceptions. The
words this and that, for instance, adjectively used,
answer to the adjectives near and distant. -
After all, it might, perhaps, have been better if the
personal pronouns alone had received the name of pro-
noun; and the words which we are now considering
had been arranged in a class between the personals
and the article, for they seem to hold a middle place
between both; but as we consider it safest not to dis-
turb a long settled order of things, we extend the name
of pronoun to all these different classes.
There is one set of words which seem to belong to
the class of definitive pronouns, but which yet de-
mand a consideration apart. We mean the numerals.
We have heretofore shown the fundamental importance
of the conceptions of number. Those conceptions must
have names, and when the names are used to express
the mere ideas of number, as when we say, “one and
one are two,” they may be considered as nouns; in the
same manner as the words line, point, angle, which are
also names of ideas, are considered. But when these
nouns aré used with an express or , tacit reference to
some other noun, they become pronouns, either sub-
stantive or adjective. When we say, “two men are
wiser than one,” or “many men are wiser than one,”
the numeral “ two" seems as much a pronoun adjective
as the word “many.” And again, if speaking of men,
we say, “two are wiser than one,” the word two
appears to be a pronoun substantive. -
Numerals are commonly divided into cardinal and
ordinal: we have hitherto spoken of the former, that
is to say, of the names given to our distinct ideas of
number, simply as distinguishing them from each other,
as one, two, three, &c.; but these same conceptions,
viewed with reference to order, form in the mind a
class of secondary conceptions, which are treated as
qualities of the substances to which they belong.
Hence originate such words as first, second, third,
fourth, &c. These may be called pronominal adjectives.
The ordinal numbers are in general derived from the
cardinal numbers, but not necessarily so; for in many,
perhaps in most languages, the words first and second,
In English, the word first is properly forest, or fore-
most, and is connected with the prepositions for and
before ; just as our comparative and superlative further
and furthest, improperly written, in modern times,
farther and farthest, are derived from forth. Of the
numerals, and of definitive pronouns in general, we
shall have occasion to speak again when we treat of
the article, which is in fact only the definitive pronoun
adjective in a new and peculiar form. -
§ 4. Of verbs.
The verb expresses that faculty of the human mind
by which we assert that any thing exists or does not
exist: and as all existence is either contemplated by
the mind simply as existence, or as existence in one of
its two distinguishable states—action or passion,
therefore the common definition of the verb is suffi-
ciently accurate, viz.: “that the verb is a word which
signifies to do, to suffer, or to be.” Yet we must ob-
serve that the essence of the verb does not consist in
the mere signification or naming of existence, or of
action, or of passion; because so far as that goes the
verb is a mere noun; but what Mr. Tooke has ob-
served is strictly true in language, viz. that “the verb
is a noun and something more.” He has not been
pleased to tell his readers what that something more
really is : and he affects a sort of mystery respecting
it which is peculiarly out of place in a work of science;
but nothing can be more obvious or less controvertible
than that this something more, which is the true cha-
racteristic of the verb, is the power of assertion.
It is by this peculiarity alone that the verb is distin-
guished from the noun, as a very few familiar instances
will demonstrate. It often happens in language that
the very same identical word, the same in orthography,
in pronunciation, and in accent, is both noun and verb.
How then can we determine when it is one, and when
it is the other ? Very simply, and very infallibly.
When it involves an assertion it is a verb; when it does
not it is a noun. The word love, in English, is one of
the words which we have just described. It is impos-
sible to tell, a priori, whether it will be a noun or a verb
in any particular discourse. We must wait to see how
it is used, and then all doubt will vanish, Thus it is a
noun in those exquisite lines—
— Love is not love,
Which alters when it alteration finds,
Or bends, with the remover to remove;
Oh no It is an ever fixed mark.
That looks on tempests and is never shaken.
And again, it is a verb, in the speech of the crafty
Richard to his unsuspecting brother.
- I do love thee so,
That I will shortly send thy soul to heaven.
Against the doctrine that assertion is the peculiar Objections.
office of verbs, various objections have been urged.
First, it has been said that we may assert, without the
express use of verbs: and this is true; but then the
assertion is an act of the mind, not expressed, but, as
grammarians say, understood. The verb is wanting;
but its place is not supplied by any other part of speech,
such as a noun, pronoun, conjunction, or the like.
Now, whether any particular operation of the mind
may or may not be understood, without being expressed
in speech, is pretty much a patter of habit, and there-
46 G R A M M A. R.
Grammar. fore forms the peculiar idioms of different languages;
S-TY--" but in Universal Grammar we have to regard the
operation of the mind itself, whether expressed by one
or more words, or to be collected from inflection, re-
lative position, accentuation, or any other mode of
signification. -
Let us consider a few examples. In the Hebrew
language the verb is often omitted. Thus, in the 3d
chapter of Exodus (ver. 2.), “the bush burned with fire,
and the bush not consumed,” i. e. was not consumed.
Again (ver. 4.), “God called unto him out of the bush,
and said, Moses, Moses! And he said, here I,” i. e.
here am I. And again (ver. 6.), “Moreover he said,
I the God of thy father, the God of Abraham, the
God of Isaac, and the God of Jacob,” i. e. I am the God
of thy father, &c. So it is in the Greek language.
Thus in St. Mark's Gospel, chapter the 10th, verse 18,
&ösic &ya60c, si pu) # to 6 Océc, “No one good, except
one, God,” i. e. No one is good,” &c. Again, in St.
Luke's, 6th chapter, verses 20 and 21, Makáptor of
wroxów, Makáptot ot retvövrec viv, Makāpiot & k\atovreć
avöv–“Blessed the poor, blessed the hungry, blessed
the weepers,” i. e. Blessed are the poor, blessed are
the hungry, blessed are the weepers. The same idiom
occurs in Latin. Thus in the parallel passages to those
above cited, “ Nemo bonus, nisi unus Deus,” i. e.
Nemo est bonus, &c. And again, “Beati pauperes,
beati qui nunc esuritis, beati qui nunc fletis,” i. e.
Beati estis pauperes, &c. The French language also
admits a similar phraseology: thus,
Heureux celui, qui dés ses jeunes ans
S’est tenu loin du conseil des mechans;
i. e. heureux est celui.
Nor is our own language a stranger to the same con-
struction. Thus in Milton's beautiful description of our
first parents:
In their looks divine,
The image of their glorious Maker shone,
Truth, wisdom, sanctitude severe and pure,
Severe but in true filial freedom placed ;
Whence true authority in men; though both
Not equal, as their sex not equal seem'd ;
For contemplation he, and valour form'd ;
For softness she, and sweet attractive grace.
i.e. whence true authority is in men; both were not
equal; he was form'd for contemplation; she was
form'd for softness, &c.
Now, in all these cases, the mind performs the act
of asserting; in the words of Plato it manifests some
action, and declares that something exists; and this
manifestation or declaration is not contained in the
nouns themselves, which do nothing more than name
the conception; thus, when we say “nemo bonus,” the
assertion is neither included in memo, nor in bonus, for
these are mere names of conceptions. Nemo is the
subject; bonus is the predicate; but neither of them
includes the copula. The two terms are not connected
by any thing which either of them contains, but their
connection is inferred by the mind from their juxtapo-
sition. But the question, which we have here to con-
sider, does not relate to verbs not expressed, but to
verbs expressed; and universally where the verb is
£xpressed, it imports assertion either simple or modi-
fied, either direct or implied.
A second objection to that account of the verb
which we adopt is, that connection and not assertion is
the distinguishing characteristic of verbs. It is true
is
that the verb connects; but it does more, it declares Chap. I.
the co-existence of the connected conceptions as parts S-->
of one assertion. The conjunction also connects, but
it does not predicate one thing of another, or make up
one proposition of two distinct terms. Thus, if we say
“he is good,” the conceptions expressed by the words
he and good, that is to say, the conceptions of a par-
ticular man and of goodness, are not only connected,
but the one is asserted to exist in the other, and to be
a quality belonging to it. Otherwise is it in the speech
of the duke of Buckingham wishing happiness and
honour to his sovereign Henry VIII.
May he five
Longer than I have time to tell his years!
Ever belov'd, and loving may his rule be
And when old Time shall lead him to his end,
Goodness and he fill up one monument!
Here the same conceptions, viz. those of a parti-
cular man and of goodness are connected, but the
one is not asserted of the other, and they make up no
intelligible meaning when taken together, without the
further aid of a verb. We cannot assert without con-
necting our thoughts; for to assert is to declare some
one thing of some other thing, which cannot be done
without connecting those things together in the mind;
and therefore it is that connection is always one cha-
racteristic of the verb; but it is a secondary charac-
teristic, being involved in its more important function
that of asserting, declaring, or manifesting real exist-
€IlC6. Y, *
Thirdly, the verb being ranked with the adjective
and participle, under the general head of attributives, it
has by some been considered that attribution, that is
to say, the expression of a quality, or the denoting of
the predicate in a proposition, is the proper function
of a verb: but again we must remark, that this is but
an accidental circumstance applying to some verbs,
and applying to them not as verbs, but in regard to
the nouns which they involve. Thus, when we say,
“Cicero spoke,” the verb spoke includes the name of
an act, viz. speech, or speaking, which, at a certain
time, belonged to Cicero, and which is predicated of
him as having so belonged ; but this name is a noum,
and if expressed simply in connection with Cicero, as
Cicero speech, or Cicero speaking, it produces no in-
telligible meaning: and therefore, in order to convert
it into a verb, a power of assertion must be given to
it, which is done either by a distinct word, as “Cicero
was speaking,” or, by a peculiar inflection of the same
word, as “Cicero spoke.” “All those attributives,”
says Harris, “which have this complex power of de-
noting both an attribute and an assertion make the
species of words which grammarians call verbs. If
we resolve this complex power into its distinct parts,
and take the attribute alone, without the assertion,
then have we participles.”—From this statement it is
manifest that the assertion is that which constitutes
the true characteristic of the verb ; and that the attri-
bute which it expresses is not essential to it, but may
appear under a different form, and constitute another
part of speech. *
To be significant of time, or, as it has been ex-
pressed, to be nota rei sub tempore, is still less the
characteristic of the verb, than those other circum-
stances are which we have been considering ; for ex-
istence may be contemplated without any reference to
G R A M M A. R.
47
must, on the contrary, be considered to be a noun, Chap. I.
Grammar, the lapse of time, as when we say “two and two are -
either by itself, or else as involved in a verb; whereas S-TYN-"
S->~~ four.” We cannot, indeed, assert any thing without a
declaration of existence, and the existence of all in-
dividual things is referable to time. Time, therefore,
is a necessary adjunct of all such assertion, and con-
sequently of the verbs by which it is effected; but
even in these instances the signification of time is but
secondary : it is the assertion, that is, the manifesta-
tion, or declaration that the truth is so, or so, which
constitutes the appropriate function of the verb.
One more objection which we shall notice is, that
the infinitive mood asserts nothing, and consequently
that assertion cannot be essential to verbs. To which
we reply, that the infinitive is not properly a verb, but
rather, as some of the ancient grammarians called it,
'Ovopia Åmparuköv, a verbal noun; or "Ovopia fińparoc,
the verb's noun. Hence it follows, that in English we
may often use indifferently the participial noun, or the
infinitive, as “singing,” or “to sing;” “parting,” or
“to part,” &c. -
— Parting is such sweet sorrow, *
- That I could say good night, till it were morrow.
Where the sense would be unaltered if it were ex-
pressed thus:
— To part is such sweet sorrow. -
Thus, too, in the Latin language, Priscian remarks,
that “currere est cursus, and scribere, scriptura ; and
legere est lectio :" and he enforces this remark by ob-
serving of infinitives, “itaque frequenter et nominibus
adjunguntur, & aliis casualibus, more nominum ; ut
Persius:
Sed pulchrum est digito monstrari & dicier hic est.
The stoics, indeed, as Harris informs us, “ had this
infinitive in such esteem, that they held this alone to be
the genuine ſimpia, or verb, a name which they denied to all
the other modes. Their reasoning was, they considered
the true verbal character to be contained simple and
unmixed in the infinitive only. Thus, the infinitives
Tepitrarév, “ambulare,” “to walk,” mean simply that
energy and nothing more. The other modes, besides
expressing this energy, superadd other affections which
Thus, ambulo and
55
respect persons and circumstances.
ambula mean not simply to walk, but mean “I walk,
and “walk thou,” and hence they are all of them re-
solvable into the infinitive, as their prototype, together
with some sentence or word expressive of their proper
character. Ambulo, “I walk,” that is, indico me ambu-
lare, “I declare myself to walk;” ambula, “walk thou,”
that is, impero te ambulare, “I command thee to walk;
and so with the modes of every other species. Take
away, therefore, the assertion, the command, or whatever
else gives a character to one of these modes, and there
remains nothing more than the mere infinitive, which,
as Priscian says, significat ipsam rem quam continet
verbum.” To all this reasoning it is sufficient to answer,
that if the stoics refused the appellation of Émpa to all
moods but the infinitive, they clearly did not mean by
the word Émpia that distinction which is commonly de-
signated by the term verb : and in truth it appears
that they meant by it the predicate of a proposition,
and nothing more : “thus Ammonius says, "rāorav
ºppvny karmyopapuevov Čpov čv Trporaget rotégav PHMA
ka)\étabat, “ that every word forming the predicate in
a preposition was called a verb.” In the view that we
have taken of Grammar, the predicate of a proposition
ence; such is the verb to be, in its purest form.
the copula of the proposition is the true verb, either
alone or combined with the predicate. In the sentence,
“Socrates teaches,” the copula, that is to say, the
essential part of the verb, is involved in the word
“ teaches.” In the sentence, “Socrates is teaching,”
it is expressed separately by the word “is;” and con-
versely in the word “teaches,” the predicate is ex-
pressed in combination with the copula; and in the
word “teaching” it is expressed alone.
What has been already said will easily lead us to a Different
kinds of
'erbs.
division of verbs into their different kinds; for they
either express the simple copula of a logical proposition,
or they express the copula in connection with a predi-
cate. In the former case, the verb is called by gram-
marians a verb substantice, and simply affirms exist-
In
the other case, the verb expresses being, together with
some attribute of action or passion; and as the name of
such attribute is properly a noun, all such verbs in-
clude a noun. We have said that the verb to be, in
its purest form, is the verb substantive; by which we
W
mean this verb, when it merely answers the purpose of
asserting, and has a separate subject and predicate, as,
“Socrates is wise,” “Socrates is reading,” &c. Other
words as well as the word is may be used in the same
manner, if it becomes idiomatical to give them this
simple effect: such was the use in Greek of the verbs
intápxet, tréNet, ytyveral, &c.; and, on the other hand,
the verb substantive is may be used more emphatically
to assert existence, as “God is,” i. e. “God exists,”
or “is existing.” : -
The nature of the verb substantive is thus explained Verb sub-
by Harris: “Previously to every possible attribute, stant*
whatever a thing may be, whether black or white,
square or round, wise or eloquent, writing or thinking,
it must first of necessity exist, before it can possibly be
any thing else. For existence may be considered as
an universal genus, to which all things, of all kinds,
are at all times to be referred. The verbs, therefore,
which denote it, claim precedence of all others, as
being essential to the very being of every proposition
in which they may still be found either expressed or by
implication; expressed, as when we say “the sun is
bright;" by implication, as when we say “the sun
rises,” which means, when resolved, “ the sun is
rising.” “ Now all existence is either absolute or
qualified; absolute, as when we say “B is ;” qualified,
as when we say “B is an animal;” “B is round,”
“black,” &c. With respect to this difference, the
verb is can by itself express absolute existence, but
never the qualified without subjoining the particular
form ; because the forms of existence being in number
infinite, if the particular form be not expressed we
cannot know which is intended. And hence it follows,
that when is only serves to subjoin some such form, it
has little more force than that of a mere assertion.
It is under the same character that it becomes a latent
part in every other verb, by expressing that assertion
which is one of their essentials.”
Beside the verb substantive, all other verbs imply verbs of
action, and these are commonly distinguished into action.
active, passive, and neuter. It is matter of idiom
whether these different classes shall be expressed by
different inflections or not; but the distinction of the
48 G R A M M A R,
Grammar,
classes themselves is in the nature of the human
\_v^2 mind, and must therefore have some correspondent
expression in language.
neuters as a branch of the latter.
agree in this, that they reciprocally suppose a separate
agent and object, whilst the neuter verb supposes an
action terminating with the agent. In the active
verb the action is considered as passing from the agent
to the object, and consequently the object takes the
lead in the sentence, as, “John loves Mary:” in the
passive verb the action is considered as received by the
object from the agent, and consequently the object
takes the lead in the sentence, as, “ Mary is loved by
John.”
This difference, as we have already had occa-
sion to advert to it in treating of cases, needs no further
- explanation here.
- numerous classes of action which terminate in them-
The neuter verb includes all those
selves, as, “to sleep,” “to walk,” to stand.” Some
persons reckon the verb substantive among neuters;
but it seems better to distinguish it altogether as we
have done from verbs of action, and to treat the
{t will be observed
that by action we do not mean simply motion, but also
rest, or the privation of motion. Thus, “to stop,”
“to cease,” “ to die,” are not less acts than “to
walk,” “to fly,” “to live,” “to wound,” or “to kill:”
in short, whatever imports any diversity in the states
the verb does not merely name those states, but asserts
for modifications of being; and we need not repeat, that
, them to be really existing at some period of time.
Other dis-.
tinctions.
Various other distinctions of verbs occur in gramma-
.tical works, but they seem all to be merely subordi-
nate to those which we have noticed, or else expla-
natory of them. Thus the verbs transitive and intran-
sitive are, in other words, active and neuter; for the
verb active is considered as passing over from the agent
to the object, whilst the neuter is considered as not
passing over.
Those who speak of actives-intransitive,
seem to confound the true distinction between the ac-
tive and neuter; thus they call the verb to sleep a
neuter, and to walk, an active intransitive, probably
because more physical activity is shown in walking
than in sleeping; but it is not the quantity or degree of
, action that makes the difference between these classes
of verbs, but the simple consideration whether they
When we say a
have or have not a separate object.
separate object, we do not mean an object necessarily
distinct from the agent; for there is a class of verbs
called reflectives, in some languages, in which the agent
is its own object; but these verbs are truly actives.
When a person says, Je me flatte, “I flatter myself.”
the verb flatte expresses an action as proceeding from
the agent Je to the object me. So in the Latin, Ego-
met mí ignosco, “I pardon myself,” ignosco expresses an
action as proceeding from the agent Ego to the object
mihi. An accurate examination of the operations of
the mind in such cases will convince us that we really
distinguish the self, or being, with whom the action
originates, and in whom it terminates, into two parts,
or at least view it in two lights.
ters or pardons is viewed as active, the being which is
flattered or pardoned is viewed, as passive. This power
of self-contemplation is the origin of the ancient fable
of Narcissus; it is the foundation of that moral rule
which the philosophers of antiquity considered to be
‘divine, - . . .
E coelo, descendit yºu areavlov.
Active and passive verbs.
The being which flat- .
And, Socrates very finely distinguishes between the chap. I
physical, and moral power of contemplation by remark- Sºrº-
ing, that the eye, which sees every thing else, cannot
see itself; whereas, there is no created object which the
human mind can or ought so much and so profoundly
to contemplate as its own existence and energies.
It is material to observe, that the quality of neuter
or active is not necessarily appropriated to any parti-
cular verb ; but that a neuter, by a slight change of
signification, may often pass into an active, and vice
versá. Thus the Latin verb abstineo, “I abstain,” is com-
monly used as a neuter; but even in the best writers
we find it employed as an active : Cicero says, absti-
mere manus ; and Livy says, Romano bello fortuna Aler-
andrum abstinuit. We cannot translate these passages
literally into English, “to abstain the hands," and
“Fortune abstained Alexander from a Roman war;"
but the reason of this is, that the active or neuter use
of particular verbs is a mere matter of idiom. In
English, as in most other languages, custom has con-
fined certain verbs to the one class, and certain others
to the other class; but there is generally a number of
verbs which are used both in an active and neuter sig-
nification, the construction alone determining of which
kind they are. - .
It is again noticeable, that verbs usually neuter have
often one particular construction in which they as-
sume an active form. This happens where the accusa-
tive which follows the verb is in substance the very
same conception which the verb itself expresses, as
“ to live a life;” or where it forms a species of which
that conception is the genius, as “to dance a minuet,”
that is, to dance a dance of the species called a minuet.
For a similar reason we use such expressions as “to
walk a mile,” “to ride a race,” “to swear an oath.”
It is only by a bold poetic licence, that Timon, addres-
sing the courtezans, says: -
I know you’ll swear, terribly swear,
Into strong shudders, and to heav'nly agues,
Th’ immortal gods that hear you
The expression “to swear the gods,” is employing a
neuter verb in an active sense unknown to the general
idiom of the English language, and only justified by
that energy of feeling with which the all-powerful poet .
has invested the dramatic character of Timon.
In the distinctions of verbs, as in most other parts
of Grammar, we find grammarians. continually Con-
founding signification with form. Thus they say there
are five classes of verbs in the Latin language: 1. Verbs
ending in o, which also admit or ; these they call actice.
2. Verbs ending in o, which do not admit or ; these
they call neuter. 3. Verbs ending in or, which are also
used in o ; these they call passive. 4, 5. Verbs end-
ing in or, which are not used in o ; these they call
common, or deponent.
Vossius justly blames this division; but his own
method is not wholly free from censure; for though he
properly begins with the triple distinction of significa-
tion, according as the verbs express doing, suffering,
or being, he proceeds to subjoin to this a fourfold dis-
tinction in point of form, observing that verbs are either
biform (ending in o and or), and these are active and
passive; or else they are uniform, ending in o only if
neuter, and, in or only if deponent, or common. By the
word deponent are meant those which have laid aside
G R A M M A. R. 49
interfere with the primary grammatical classes; and Chap. I.
Grammar, the passive signification properly belonging to the ter-
rather belong to the richness of a language, than to its S-2-2
S-N-2 mination or ; as in Virgil,
v Pictis bellantur Amazones armis;
so in Plautus,
Adeunt, consistunt, copwlantur dexteras.
By the word common, are meant those which, though
used actively by some writers, retain also a passive
signification as employed by other writers of great
weight and authority. Thus complector is generally
used with an active signification; but it is passive in
the speech of Cicero for Roscius—“Quo uno maleficio,
scelera omnia complewa esse videantur.” The middle
verb in Greek has sometimes the effect of the Latin
deponent, that is to say, it has a passive form with an
active signification; but in other instances it is rather
of the nature of a reflective verb, producing a sort of
mixed sense between the active and the passive. “The
mixed sense,” says KUSTER, “ consists in this, that the
action of such middle verbs does not pass over to
another object, but is reflected back on the agent, so
that the same being becomes both agent and patient;
and this, whether he directly suffer any thing from
himself, or order, direct, or permit it to be done to
him by another.” Thus émetyev in the active is to
urge or impel another; but £irst yeaðat in the middle
form is to urge or Impel one's self, that is, to make
haste. Hence it happens that the same word in the
active and middle forms has two distinct, and, in
some measure, contrary senses, as Öaveirat is to lend,
but Savewaa.offat is to borrow ; and it is remark-
able that our common English verb borrow an-
ciently signified both to lend and to give a pledge for
that which was lent, and hence to be plighted or
married to a person. Thus Wachter, says “ Borg,
mutuum, auf borg geben mutuo dare, auf borg nemen
mutuo accipere. Proprie quidem est mutuo datum,
à borgen mutuo dare ; mox etiam mutuo acceptum,
quia dare et accipere sunt correlata et in notione
debiti et crediti conveniunt.” Again, “Borgen, mutuo
dare, dare in creditum. Belgis borgen, Anglis burrow.
Ab hoc significatu habent Anglosaxones borgiend Faene-
rator. And further, “Borgen, mutuo accipere accipere
in creditum. Anglosax; borgan, borgian.” The old
Scottish ballad speaking of Tam Lane, or Tom Linn,
who was carried away by the faries, and married to a
lady of the fairy court, says:
She that has borrowed young Tam Lane
Has gotten a stately groom.
Thus we see that the principle, which in one lan-
guage gives different meanings to the same form of
speech, founds in other languages a distinction of
meaning between different forms of the same word.
We have thought it necessary to take this short
notice of the classes of verbs last mentioned, both
because the terms deponent, common, middle, &c. are of
frequent occurrence in grammatical writers; and more
particularly because some of the very best gramma-
rians have endeavoured to unite in one common sys-
tem these distinctions of form, with the distinctions of
signification, an attempt which cannot but be prejudi-
cial to scientific clearness and accuracy; inasmuch as
it confounds Universal Grammar with Particular; and
thus forms a system which properly belongs to neither.
There are again other distinctions which relate in-
deed to the signification of verbs; but which do not
VOL. I.
necessary construction. Such are the Latin inceptive
verbs in sco, as albesco, tumesco, the Greek verbs of habit,
in tºw, pi\tirriča, ; the Hebrew verbs called by some
writers intensive, and many others in most languages.
Verbs of this kind are generally derived from other
verbs, but sometimes from nouns, as calesco, horresco,
splendesco, from the verbs caleo, horreo, and splendeo ;
noctesco and rureso, from the nouns now and rus. Of the
Latin verbs in sco, it has been disputed whether they
can or cannot properly admit the expression of past
time; but Vossius satisfactorily proves that they may,
by adverting to their proper signification, which is not
merely inchoative but also continuative. “Hence,’
says he, “as the philosophers teach that all motion is
produced by succession, there must be in it a beginning,
a middle, and an end; and it is one thing to have per-
fected the beginning, another to have proceeded to the
middle, and another to have reached the end; and he who
says that he did at a certain time begin a movement, only
means to assert that such beginning was perfected,
and not the whole motion.” Many various classes of
verbs may be things distinguished by various shades
of derivative signification. They do not simply assert
the conception involved in them to exist, but to exist
under some particular modification. Thus we have
seen that the Latin verbs in sco, imply the inchoation
and continuation of an action. Verbs in to, so, alo and
co, are called frequentatives, or iteratives; as pensito,
from pendo ; tracto, from traho ; vendito from vendo :
but it has been observed, that they often imply, in a
secondary sense, not the repetition of an action so,
much as its greater violence; and may therefore be
called intensive or augmentative. Thus, rapto, derived
from rapio, is used by Virgil to signify not only the
repeated, but the violent dragging of Hector's body
in triumph round Troy--
Ter circumu Hliacos raptaverat Hectora muros.
On the other hand, they are sometimes taken to signify
a weaker degree of the same action ; as TURNEBUS
observes---‘‘There are many words, which, by learned
grammarians, are reckoned to be of a frequentatiye
form, and which plainly exhibit the appearance of that
form; but which if they are narrowly inspected, and
if we observe the manner in which they are used by
the best authors, should rather be called desideratives.
I will enumerate a few of them, which may afford to
the studious sufficient specimens to direct their
search for others of the same kind.” “Capto is not, “I
take frequently,’ but ‘ I endeavour to take,’ as capto
caºnam, 'capto benevolentiam. Vendifo is not ‘ I sell fre-
quently, but ‘ I desire to sell;’ as in Cicero (de
Arusp. Resp.) atque ei sese, cui fotus cenierat, etian
cobis inspectantibus tenditaret, that is, se ei vendere
vellet ; and so in Plautus, lingua venditaria is not “a
tongue which sells, but ‘which wishes to sell, as the
parasite says his own was. Dormito is not ‘I sleep
often, but ‘ I am nodding, or napping,' as in Plautus
(Amphitr.) te dormitare aiebas: and so in the gospel
of St. Matthew (chap. XXV. V. 5. Évvcačav Trägat kat
£ká0svčov, “they all slumbered and slept), the word
#vvcačav is elegantly rendered by the translator dormi-
tarunt ; because they who are ready to fall asleep can-
not keep their heads upright. Ostento is not ‘ I show
frcquently,’ ‘ but I wish to show.’ Munito is used by
II
50
Ú R A M M A. R.
Grammar. Cicero (pro Roscio) in the sense of munire cupio.
S->~ fine, there are many other words which might be cited;
In
but it is sufficient, to have pointed out the class, as it
were, and to have afforded a specimen of them to the
studious.”
Slight shades of distinction are to be observed in
the use of these and similar words:. nor does the same
termination always express the same modification of
the original thought. Thus the termination so in viso,
has a desiderative force, in pulso a frequentative, for
the former is I go to see, the latter is I knock or push
frequently; and in like manner verso, as used by Horace,
is I turn over frequently:
Vos exemplaria Graeca.
Nocturna versate manu, versate diurna.
Of the termination sso, different commentators speak
differently. Thus Virgil:
Hand mora, continuo matris praecepta facessit.
On which passage Servius observes, that facesso is a fre-
quentative verb, inasmuch as there were many victims
sacrificed ; on the other hand, Nonius and Donatus
both explain facesso to signify simply the same as facio;
but in reality it has an intensive force, and signifies
more than the simple verb, though not necessarily a
repetition of the same act. Thus, in the passage just
cited from Virgil, facessit obviously means setting about
the business that was commanded, with diligence and
anxiety. The termination co is noted as having in
general a weakening force; for claudico is 1 halt a
little, and the difference between nigrantem colorem,
and nigricantem colorem, if any, is that the latter is less
strongly inclining to black. Critics have observed a
difference between those verbs which express only the
simple desire to do an act, and those which express
together with the desire the actual engagement in it:
the latter kind they call desideratives; but the former
they distinguish as merely meditatives. Thus facesso,
as we have seen, is a desiderative; but most of the
verbs in rio are meditatives; for esurio rather implies
a negation of the act of eating, and is only I hunger, or
have a desire to eat, without any gratification of that
desire. But here, too, we perceive that the termina-
tion is not a sure guide to the use of the word, for
scaturio and ligurio imply the performance of the re-
spective actions, and not merely the desire or medita-
tion of them ; as in Horace, -
Si quis eum servum patinam qui tollere jussus
Semesos pisces tepidumque ligurrierit jus,
In cruce suffigat.
Lastly, the termination lo or llo, generally serves to
diminish, as murmurillo, I murmur gently, from mur-
muro; sorbillo, I sip drop by drop, from sorbeo ; can-
tillo, I hum a tune, or sing in an under voice, from
cano, and the like.
In most languages there are negative or oppositive
verbs, as volo and molo in Latin; to do and undo in
English; fier and meſier in French, &c. There are also
in various languages, as in Persian, Sanscrit, &c. causal
verbs formed by a peculiar inflection, whereas in some
other languages the simple and causative meaning are
found in the same word. Thus it is probable that our
verbs to lie and to lay, though recently distinguished
in use, and indeed supposed to be derived from two
different Anglo-Saxon roots, were both of the same
origin; for Wachter explains the ancient German word
lage situs, sedes, campus; and observes that it agrees Chap. I.
with the Latin locus, hence ligen in the first sense is to S-S-S-
lie, or occupy a certain lage; and legen in the second-
ary sense is to cause to lie, to cause to occupy a lage.
In like manner our common verbs to fell and to fall
are the same. “To fall timber” is an expression still
used in many parts of England, and it signifies to fell,
or cause to fall. So we say to bleed a person, for to
make him bleed. - -
The words which we have been considering, as dis-
tinguished by grammarians into so many classes of
verbs, inceptive, desiderative, frequentative, negative,
causal, &c. are all derivatives; and derivative words
are, in fact, compounds; that is, they unite the name
of one conception with that which serves as the name
of another, as the name albus, white, is united with the
termination esco, which serves as the name of growth;
so that albesco is, literally, I grow white. But we have
seen that what is effected in one language by the deri-
vative verb is effected in another by the simple verb.
The thought expressed is, in both cases, the same;
but the mode of expression varies; and the variations
are properly matter of particular, and not of Universal
Grammar.
After having thus reviewed the different kinds of
verbs, we come to the considerations which regard all
these kinds alike, and which are usually ranked by
grammarians under the heads of mood, tense, person,
number, and, in some languages, gender. -
The Mood of a verb is that manner in which its Mood.
assertive power is exhibited, and which depends on
the state of mind in which the speaker may be placed
with relation to the assertion. Hence grammarians
have sometimes defined the mood to be a certain incli-
nation of the mind shown in speech. Thus PRIscIAN
says, Modi sunt diversae inclinationes animi, quas varia
consequitur declinatio verbi. The latter circumstance,
however, belongs not to Universal Grammar. Whether
the different moods have or have not different forms of
declension, or conjugation, depends on the idiom of
the particular language; but whatever variations the
verb may have in point of form, it must necessarily be
susceptible of those varieties, in point of signification,
which properly belong to its assertive power.
Grammarians differ widely as to the number, and
no less as to the names of the moods. ScALIGER says,
that mood is not necessary to verbs; and SANCTIUS
contends that it does not relate to the nature of the
verb, and therefore is not an attribute of verbs: non
attingit verbi naturam, ideo verborum attributum non est ;
on which passage PER1zoNIU's very justly observes,
that great as the merit of Sanctius was in many parts
of his work yet he had in others, and particularly in
what regarded the moods of verbs, been misled by an
excessive desire of novelty and change. It is very true,
as observed by Sanctius, that the great mass of gram-
matical writers are so extremely discordant in their
opinions respecting this part of the science of which
they treat, that they have left us scarcely any thing
on it which can be said to be established by general
consent. Some make only three modes, others four, five,
six, and even eight. Again, some call these affections
of the verb moods; others call them divisions, qualities,
states, species, &c.; and as to the various appellations
of each mood we have the personative and impersona-
tive, the indicative, declarative, definitive, modus fini-
G R A M M A. R.
51
Grammar. endi, modus fatendi, the rogative, interrogative, requi-
S-'sitive, percontative, assertive, enunciative, vocative,
precative, deprecative, responsive, concessive, permis-
sive, promissive, adhortative, optative, dubitative, im-
perative, mandative, conjunctive, subjunctive, adjunc-
tive, potential, participial, infinitive, and probably
many others. .
In this confusion of terms and of notions, it is ab-
solutely necessary to adopt some distinct principle
which may guide us through the labyrinth; and that
principle, we apprehend, will be easily and intelligibly
supplied by adverting to the peculiar function of the
verb itself, namely, assertion. It must be observed,
that we use this term, in its largest sense, for the
manifestation of some distinct perception or volition;
and we consider, that in every such manifestation an
assertion is either expressed or implied. Portia, ad-
dressing Brutus, says, -
---. Dear, my lord,
Make me acquainted with your cause of grief.
And again, she says,
Upon my knees
I charge you, by my once commended beauty, .
By all your vows of love, and that great vow
Which did incorporate and make us one,
That you unfold to me, yourself, your half,
Why you are heavy. ' -
In both these instances she asserts her earnest de-
mand to be made acquainted with the secret cause of
that sick offence which she perceived to exist, not in
her husband's health, but in his mind. In the one in-
stance, however, the demand is expressly asserted by
the words “I charge you that you unfold:” in the
other it is implied, with no less clearness, by the words
“ make me acquainted.” Whether, therefore, the as-
sertion be express or implied, the verb is that part of
the sentence by which it is manifested; the verb ani-
mates the sentence, connects the passion with its
object, or the object with its predicate.
Again, Caesar in describing Cassius, first asserts po-
sitively what he had observed in his outward appear-
ance, and then hypothetically what might be supposed
to pass in his mind:
Yond Cassius has a lean and hungry look;
Seldom he smiles, and smiles in such a sort,
As if he mock'd himself, and scorn’d his spirit,
That could be mov'd to smile at any thing.
And so, referring to Antony's expression, “fear him
not,” Caesar asserts positively that he does not fear
º but puts a case hypothetically, in which he might
O SO =
I fear him not;
Yet if my name were liable to fear,
I do not know the man I should avoid,
So much as that spare Cassius. -
Having thus explained what we mean by the term
assertion, we proceed to apply that principle to the
doctrine of moods. “. .
Assertion, then, takes place either in an enunciative
sentence, or in a passionate sentence: in the former it
is express; in the latter it is implied. By express as-
sertion a truth is enunciated, absolutely if the sentence
be simple, but conditionally, in the dependent branch
of a sentence which is complex. By implied assertion,
in like manner, a passion is connected with the object
either absolutely or conditionally: in the one case the
desire or aversion is positive, in the otherit is qualified
by some consideration of circumstances. These four
Chap. I.
kinds of assertion supply us with four correspondent S->
thoods of the verb, namely, the indicative, the conjunc-
tive, the imperative, and the optative. It has been
contended, that there are two moods in which assertion
does not take place, namely, the interrogative and the
infinitive ; but these we are not inclined to reckon as
separate moods, for reasons which will hereafter be
stated. Of the other four moods we proceed to take
notice in the order above-mentioned.
If we simply declare or indicate something to be or
not to be, this constitutes the mood called by most
grammarians the indicative, but by some the declara-
tive, enunciative, &c.
“he died,” “we shall rejoice,” are all simple asser-
tions of fact, some of which do, and some do not re-
late to passions of the mind, but which do not neces-
sarily imply any passion in the enunciation. Some of
them too may in reality be contingent, or doubtful,
and may be dependent on the truth or falsehood of
other assertions; but as they are not so enunciated,
but on the contrary are declared positively and simply,
they belong to the indicative mood. It is to be ob-
served that the indicative, from its very nature, is ca-
pable of being united with the conjunctive, as well as
of standing alone. An assertion does not necessarily
become the less positive for being coupled with another,
although that other may be doubtful or contingent.
Thus, “I love,” “I walk,”
Indicative,
When a fact is asserted not as actual but merely as Conjunc-
possible, or contingent, the form of words by which tive.
such assertion is expressed in any particular language,
may perhaps be the same as if the assertion were more
positive; yet the context will show, that the verb is no
longer in the indicative mood. The mood adapted to
such contingent assertion has received various appella-
tions, of which we consider the conjunctive to be the
most appropriate, inasmuch as the contingency is
usually marked by a conjunction (such as if, though,
that, ercept, until, &c.) which connects the dependent
sentence with its principal.
There are various methods of thus connecting sem-
tences; but they may be distinguished into two great
classes. In one class an uncertain sentence is con-
nected with a certain one; in the other, both sentences
are uncertain; that is to say, in the former case, a
conjunctive is dependent on an indicative; in the latter,
both sentences are conjunctive. Some grammarians
make this distinction the ground of a distinction of
moods, calling the contingent assertion, in the first case,
subjunctive, because it is subjoined to the indicative;
and in the other case potential, because it states a po-
tential, and not an actual existence. It seems, how-
ever, unnecessary thus to multiply moods; first, be-
cause no language (that we know of) has assigned
separate forms to the potential and subjunctive; and,
secondly, because if we were to proceed this length,
there is no reason why we should not go much further,
and call every possible variation of contingency a sepa-
rate mood. Of these we shall here notice some in-
stances easily distinguishable in point of principle.
1. “ Ut jugulent homines surgunt de nocte latrones.”
Here jugulent is in the conjunctive, as indicating the
end and object of the rising.
2. “ Peter said unto him, though I should die with thee, yet will
I not deny thee,”
H 2
52
G R A M M A. R.
Grammar.
Here “I should die” is mentioned as a motive to de-
\-'nial, but an insufficient one.
3. “ Si fractus illabatwº orbis,
Impavidum ferient ruinae.”
Here, in like manner, illabatur is in the conjunctive,
as expressing a fact which might be the cause of fear
to ordinary minds, but which is not so to the just and
stedfast-minded man; and the conjunction si in the
one case is equivalent to though in the other, both of
them having the proper force of our expression “even
if.”
4. “Except a man be born of water, and of the spirit, he cannot
enter into the kingdom of God.”
Here the conjunctive be born, is placed in opposition
to the indicative “cannot enter;” so that if the one be
in the negative, the other must be so too, and vice
versd; for the implication is, that if a man be born of
water and of the spirit, he can enter into the kingdom
of God. Accordingly, the Greek conjunctions in this
and the preceding example are directly opposed to
each other: in No. 3, the word used in the Greek text
is Kāv, that is, Kai éâv; but in No. 4 it is £av på.
5. “ Caementis licet occupes
Tyrrhenum omne tuis & mare Apulicum,
InOIl aill Iſlum niletu
Non mortis laqueis expedies caput.”
Here the condition differs from that of No. 2, in
being a fact of present time; and on the other hand
the indicative non earpedies differs from the indicative
feries in No. 3, by being in the negative.
6. “ The sceptre shall not depart from Judah, nor a lawgiver
from between his feet, until Shiloh come.”
Here both the facts are future, but the conditional
one is the term or boundary of the other.
7. — “ tacitus pasci si posset Corvus, haberet
Plus dapis.”
In all the preceding instances one assertion is abso-
lute; but here it is neither asserted that the crow can
feed in silence nor that it has more food; both parts of
the sentence, therefore, are contingent, and conse-
quently, both are in the conjunctive mood.
8. If it were done when 'tis done, then 'twere well
It were done quickly.
A
Here is also one contingent, namely, 'twere well, de-
pending on another contingent, if it were done; and
on each we see a further contingency also depends.
These eight examples are sufficient to show that
the varieties of contingent assertions are too various
to be considered and treated as so many distinct
moods of the verb. The six first are of the kind called,
by some writers, subjunctive; the two last are of the
kind called, in contradistinction from the subjunctive,
potential; but as they are all equally conjunctive, it
suffices to give them that name; and, indeed, it is a
more correct and systematic distribution of the gram-
matical nomenclature so to do; for the proper correla-
tive to the term indicative is not subjunctive or po-
tential, but some term which comprehends them both ;
as, for instance, the term conjunctive. The indicative
asserts simply: the conjunctive asserts with modifica-
tion : if the one is a mood, so is the other; but if the
conjunctive is a mood, then its subdivisions cannot be
properly so called; but they should rather be called
sub-moods, if it were necessary to give them any pecu-
liar denomination.
The effect of passion is to break in upon and dis- Chap. I.
turb the regular processes of reasoning. Reasoning is \le^^-
conducted by express assertion, absolute or conditional. Imperative.
Passion goes at once to its object, assuming it as the
consequence of an implied assertion. Thus, if the fact
be that I desire a person to go to any place, it is not
necessary that I should distinctly state my desire in the
indicative, and his going in the conjunctive; but by the
natural impulse of my feelings—feelings which lan-
guage conveys as clearly as it does the more gradual
processes of thought—I say in a mood different both
from the indicative and the conjunctive—go / Now,
this mood, from its frequent use in giving commands
to inferiors, has been called the imperative, and that
name, as being the most general, we shall adopt. Some
writers have distinguished from the imperative, the
precative, the deprecative, the permissive, the adhor-
tative, &c.; but so far as language is concerned, these
are but different applications of the same mood: the
operation is the same in communicating the object of
the passion and implying the assertion that such pas-
sion exists. A few examples may serve to explain our
meaning : , '
1. Let there be light, said God; and forthwith light
Ethereal, first of things, quintessence pure,
Sprung from the deep ; and from her native east
To journey through the airy gloom began. Milton.
2, Fear and piety, .
Religion to the gods, peace, justice, truth,
Domestic awe, night-rest, and neighbourhood,
Instruction, manners, mysteries, and trades,
Degrees, observances, customs, and laws,
Decline to your confounding contraries 1 - - -
And let confusion live 1 - Shakespeare.
3. Help me Lysander help me! Do thy best,
To pluck this crawling serpent from my breast!
Ay me for pity. What a dream was here ! Id.
4, Go, but be mod’rate in your food - -
A chicken too might do me good. Gay.
In the first of these examples, we have an instance
of the highest imperative, that which proceeds from the
Almighty Power, to whose command all things created
and uncreated are subject; and who, in Milton's fine
paraphrase of the first chapter of Genesis, is described
as calling into existence the hitherto uncreated essence
of light. The second example is deprecative, or rather
imprecative, in which Timon calls down on his worth-
less fellow-citizens the natural consequences of their
profligacy. The third is precative, in which poor de-
serted Hermia, waking from a terrific dream, calls for
help from her faithless lover Lysander. The last is
permissive, in which the old dying fox, after a long
harangue to dissuade the younger members of his com-
munity from pursuing their usual trade of rapine, at
length permits them to go out on a similar excursion.
Now, in all these varieties of the imperative mood,
the grammatical process, both of thought and expres-
sion, is the same. In all of them the assertion of de-
sire or aversion on the part of the speaker is clearly
implied. The sense is, “I command that there be
light”—“I wish that confusion may prevail”—“I pray
you to help me”—“I permit you to go;” but it is un-
necessary to express those various assertions, because
they are all implied in the imperative moods, and
without those moods they could not be so implied.
The imperative animates the passionate sentence, as
the indicative or conjunctive animates the enunciative
G. R. A M M A. R. * 53
Optative.
Grammar. sentence. It converts the name of an object of passion,
S*Y* or will, into a manifestation that such object exists;
just as the indicative or conjunctive converts the name
of an object of perception or thought into an assertion
that it is really existing. The original text, “God
said let there be light, and there was light,” affords
a plain example of this operation in both ways. The
conceptions in both, are two; namely, eaistence and
light. Each of these, without the verb, would remain
a mere noun. The word “light” does so remain; but
“existence” by becoming a verb, exhibits itself first in
the imperative as an object of volition, and then in the
indicative as an object of perception. In the one case
it implies an assertion of the Divine Will that light
should exist; in the other it expresses an assertion
that light did exist.
We should not be inclined to separate the optative
mood from the imperative, were it not that various lan-
guages, and particularly the Greek, distinguish it by
a separate inflection. The difference between these
two moods appears to be rather a difference of degree
than of kind; for we cannot agree with Scaliger, who says
(lib. iv. c. 144.) differunt, quod imperativus respicit per-
sonam inferiorem, optativus potentiorem: “they differ in
this, that the imperative regards an inferior person, the
optative a superior.” This difference is altogether acci-
dental. Moreover, it makes no provision for the com-
mon case of wishes expressed between equals; and
again, how are we to determine whether a request is
addressed to a person in one character rather than
another ? Or why should we not have moods to desig-
nate the different degrees of superiority and inferiority ?
The fact seems to be, that the more distant and in-
direct union of the will with its object, has given rise, in
some languages, to a peculiar form of the verb, gene-
rally called the optative mood. Yet even this distinc-
tion does not appear to be very accurately observed in
practice, for we sometimes see the optative used where
the imperative might have been more naturally ex-
pected. Thus, in the Electra of Sophocles, when
Orestes is forcing Ægisthus into the palace, to kill him
in the apartment where he had murdered Agamemnon,
he says to his reluctant victim,
Xoſéñig &y staro arov ráxei Aáyoy yºg &
Nüy Éciv &y&y, &AA&aig Jux'ſ; wégi.
Go in, without delay, for now the strife
Is not for useless words, but for thy life.
Where the optative Xopótc undoubtedly expresses a
pretty strong volition that Ægisthus should do what
he was equally unwilling to perform.
The common distinction between the optative and
the imperative is nearly expressed by the English use of
the auxiliaries “may” and “let.” Thus, the following
passage in the Hymn to Sabrina is an example of the
optative : .
Virgin daughter of Locrine
Sprung of old Anchises’ line,
May thy brimmed waves, for this,
Their full tribute never miss,
From a thousand petty rills
That tunible down the snowy hills
Summer drouth, or singed air,
Never scorch thy tresses fair
Nor wet October's torrent flood
Thy molten crystal fill with mud :
May thy billows roll ashore
The beryl, and the golden ore!
May thy lofty head be crown'd
With many a tow'r and terras round 2
These are matters not within the power or control Chapºt
of the speaker, and which he, therefore, can only wish.
On the contrary, when the speaker can command the
execution of his wishes, he uses the word let, as the
king, in Hamlet,
Let all the battlements their ord'nance fire-—
- Give me the cups,
And let the kettle to the trumpet speak,
The trumpet to the cannoneer without,
The cannon to the heavens.
Is is observed by Vossius, that the Latin optative is
no other than the conjunctive ; and, indeed, the form
is the same in both ; for we say, utinam amen, or clim
amem ; utinam amarem, or clim amarem ; utinam ama-
verim, or clim amaverim ; utinam amatissem, or clim
amavissem. And so in the passive voice, utinam ama-
rer, or cilm amarer ; utinam amer, or cilm amer; utinam
amatus sim, or cilm amatus sim, &c. The mood, how-
ever, is not to be determined by the form, but by the
signification; for it often happens that particular lan-
guages do not possess distinct forms for the different
moods: and where they do, the form of one mood
is frequently used with the force of another. This
even takes place in the Greek language, which pos-
sesses the richest abundance of inflections in its verbs.
The Greek indicative is often used for the subjunctive
and optative, and that through almost all its tenses,
as VIGER has shown at large in his celebrated treatise
on Greek idioms: and in return the optative, especially
in the Attic dialect, is used for the indicative.
Many authors contend for a mood which they call Interroga-
interrogative : and it must be admitted that the act of "".
the mind, in asking, is different from that which it
performs in indicating, or stating conditionally, or
commanding, or wishing. Yet it is unnecessary to
constitute, on that account, a separate mood of the
verb; for the interrogative is no other than the indica-
tive, with such accentuation or transposition of words,
as to show the doubt of the speaker, and sometimes
with an interrogative particle prefixed. The question
is, as it were, the answer anticipated; but the answer,
if complete, must necessarily be in the indicative mood,
and, consequently, so must the question be. Thus:
“Did Brutus kill Caesar 7”—“ Brutus did kill Caesar.”
“How many years reigned Augustus 7”—“Augustus
reigned forty-four years.” Varro, indeed, speaks of the
moods of asking and answering as different, but this
is true only with reference to the whole state of mind
expressed in the respective sentences, and not with
reference to the particular form of the verb, which in
both instances must necessarily be the indicative mood.
Hence Apollonius says, "Hys ºv Tpokespuéun öpictkh
#yk\toric Tºv #ykeſpuévny karápaguárošáAAega, pedicarat
rä ka)\é, Sat ôpicuºh, Lāvarxmpa,0éto a ë ràc karapáorewc,
itrocpépet éic rô cival épicuch—“the indicative mood, of
which we speak, by laying aside that assertion, which
by its nature it implies, quits the name of indicative;
when it re-assumes the assertion, it returns again to its
indicative character.”
It only remains to consider that which, as Vossius Infinitive.
observes, not only the semidoctum vulgus, but even some
of the scientissimi, have called the infinitive mood. We,
however, are so far from ranking it among the moods,
that we do not acknowledge it to be a verb at all;
but consider it, as we have already stated, to be more
properly called a verbal noun.
54
G R A M M A. R.
Grammar.
^sºvº
“cupio videre tui,
Two principal grounds are alleged for reckoning the
infinitive among moods, first that it is expressive of
time, and, secondly, that it governs nouns, in construc-
tion, like a verb. As to the first of these reasons, it
can only be valid in the opinion of those who adopt
the definition of a verb, as being nota rei sub tempore,
which definition we have already shown to be inexact.
Time is an element which enters many ways into our
conceptions, but the parts of speech are not determined
by the nature of the conceptions expressed, but by
the manner of expressing them; and, as we have often
repeated, there are two principal modes of expression,
that is to say, naming our conceptions, and asserting,
or manifesting their existence. Now the infinitives,
“ to love,” aimer, amare, “ to have loved,” avoir aimé,
amavisse, assert nothing by themselves, either as to the
conception of love, or as to the conception of time in
which the action of loving took place, they express both
only in the way of notation, or naming, and not in the way
of declaration; and therefore, in so far as either of these
conceptions is concerned, the infinitive must remain in
the class of nouns. As to construction, it is clear that this
is merely a question of Particular Grammar. To say
generally, that the infinitive governs a noun which fol-
lows or precedes it, is only to say, that it causes such
noun to be in some case; but this is also effected by
another noun; and therefore the mere circumstance of
a change of case is in itself no test of the nominal or
verbal character of the infinitive. The particular case
in which the governed noun is remains to be considered,
and that is to be ascertained, not by its termination, or
inflection, or accompanying particle, but by its signi-
fication. Now, as to its signification, if the governed
noun be not the object or the agent of some action or
existence asserted, the case in which it is does not imply
that the governing word is a verb. Hence the phrase
I desire “the sight of thee,” is exactly similar to the
phrase “I desire to see thee.” The words “sight”
and “see,” neither of them assert that the action of
seeing takes place, and consequently the words “thee”
and “ of thee” are not either of them the agent or the
object of any such assertion : and we cannot conceive
any reason, in the signification of the words, which
should have prevented the Latin idiom from being
” as well as “cupio visum tui ;” for, in
fact, “ videre” and “visum” are alike names of the
action of seeing; they alike express the object of the
verb “cupio;” in other words, they are nouns, and it
is matter of idiom whether the relation which they
bear to the following noun should be expressed by the
termination e or ui.
We have before observed, that Priscian says currere
is cursus ; and we have shown that, in English, “to
part” is “parting;” there are, therefore, three kinds of
verbal nouns, which in various idioms are differently
interchangeable, namely, those which are called by
various writers the infinitive mood, the abstract noun,
and the participle (including the gerund and the su-
pine). This will appear from a comparison of the
idioms of almost all languages. We are told, that in
Galic the present participle and the verbal noun are the
same; and again, that the infinitive is formed by the
dative of the present participle. In the Ethiopic Gram-
mar, Ludolf says, Infinitivus sapissime nominascit; and
again, cum affiris ba et la, Latiné per gerundia in do et
dum exprimi potest. In Bengalese, too, the infinitive
answers to the verbal noun; and the first gerund sup- Chap. I.
plies the place of the English infinitive, when two verbs S-
come together. From these and many similar observa-
tions we may infer, that there are various classes of
nouns substantive and adjective derived from (or ra-
ther connected with) all verbs ; but that these nouns
relate solely to the noun, which, as we have stated, is
involved in every verb, and not to the part of the verb
on which its verbal character essentially depends.
These nouns may be thus classed :
1. Verbal adjectives (commonly so called), which ex-
press the conception in the form of an attribute, as the
Latin verbals in bilis, &c. of which Mr. Tooke makes a
class of participles, and which do not involve the no-
tion of time. -
2. Participles (commonly so called), which agree with
the former, except that they involve the notion of time.
3. Abstract nouns (commonly so called), which ex-
press the conception in the form of a substantive, as
the Latin nouns in io, &c. which do not involve the
notion of time. -
4. Infinitives (commonly called infinitive moods),
which agree with the former, except that they involve
the motion of time, .
Now it happens in most languages, that distinct
forms are wanting for some of these four classes of
nouns, or that the forms are reciprocally used for each
other. Hence “ he learns to sing,” or “he learns sing-
ing,” are used in English indifferently; and so “ he
learns singing,” and “he is singing,” are equally con-
sistent with our idiom.
We have said that the infinitive involves the notion
of time; and this we conceive is the proper distinction
between currere and cursus, when they are distinguish-
able; for we may say festinat currere, but not (in the
same sense at least) festinat cursum. It is only when
currere does not involve the notion of time that the
remark of Priscian become strictly accurate; and when
this happens, then, in fact, the word currere belongs to
the third, and not to the fourth class of words above-
mentioned.
In respect to the expression of time by infinitives,
a distinction is to be observed analogous to the dis-
tinction which we have before noticed between the
verb substantive and the verb of action. If an indivi-
dual fact is meant to be referred to, then, as this fact
must necessarily occur at some given time, the time in
question is expressed by the infinitive; and it is then
only that we give it the name of infinitive. Thus,
GéAw, 0éAw pixiigat means, I desire to love at this mo-
ment; whereas XaAerov rô puſh pixiiaat means, the state
of not loving is hard at all times. In the former case,
pixiigat is strictly an infinitive, and should not be rem-
dered into Latin by the accusative amorem, but by
amare. In the latter case piń pi\ijoat is strictly equiva-
lent to a noun of action, and consequently is used, in
the Greek idiom, with the article rô.
Whether we call infinitives nouns, or verbs, the pro-
priety of the name infinitive is very evident from the
observation of Vossius: Ut finitum est nomen, tum philo-
sophus, tum plurativus philosophi; quippe illo unus, hoc multi
significantur: at contra infinitum est sui, quia utriusque est
numeri, item Græcum Četva, quoet ille et illi denotatur; sic
finitum verbum est audio, ac facio, ut quo certus numerus
designatur; infinita autem sunt audire, agere, ut quae deft-
ciant numeris ac personis, et undique sunt indefinita ac
G R A M M A. R.
55
Grammar. indeterminata. “As the noun philosophus is finite, both
S-2 in the singular and in the plural philosophi, since the
former signifies one person, and the other many; but
on the other hand the word sui is infinitive, because it
is both singular and plural ; and in like manner the
Greek word &etva is infinitive, because it denotes both
him and them ; so the verbs audio and facio are finite,
as designing a certain number; but audire and agere,
which express no certain number or person, and are in
every way indefinite and indeterminate, are called
infinitives.”
It is to be observed, that the Latin nouns in io seem
properly to have been definites; that is to say, that
they originally signified only a certain number of acts,
and not action in general, as actio meant a singular
exercise of the active power, and actiones several such
exercises; but in a secondary use of the word actio, it
came to be employed for such exercise generally; and
in this secondary use it is properly an infinitive, and
coefficient with agere. The Greeks, it is well known,
though they did not give their infinitive moods the
terminations of case, like other nouns, yet distinguished
them by the articles of the different cases; as rô ypſi-
peiv, rā Ypdºpetv, ev Tó ypcipew. This construction is
unknown to the Latin; for we cannot say hoc amare,
higus amare, &c. nor ad amare, ab amare, the place of
which latter phrases is supplied by the gerunds, as ad
amandum, ab amando. And again : in English, it is
only by a forced imitation of the Greek idiom, totally
unsuitable to the genius of our language, that Spenser
says—
For not to have been dipp'd in Lethe's lake
Could save the son of Thetis from to die.
And this Hellenism is the less excusable, as we have
actually an infinitive which admits of being used with
the preposition: for the proper and intelligible English
construction would have been—
Could save the son of Thetis from dying >
whereas the usual opposition between the prepositions
“from" and “to" renders the phraseology of the poet
intolerably harsh and inconsistent. Nor does it ap-
pear that Harris, who seems to approve of this line of
Spenser's, is much more accurate in another example,
viz. “he did it to be rich” where, he says, we must
supply by an ellipsis the preposition for, as “he did it
for to be rich.” Certainly this is a provincial way of
speaking, but it is a mere rustic pleonasm. In French,
pour s'enrichir is proper, because the infinitive s'enrichir
has not in itself the objective mark; but in English,
where that mark is supplied by the preposition to, a
similar mark in the word for, is altogether superfluous.
We have thought it necessary to dwell the longer on
the consideration of the infinitive, because in rejecting
it not only from the moods but from the verbs, we cer-
tainly deviate, more than we are generally disposed to
do, from the path pursued by the great majority of
grammatical writers. Yet that this deviation is justified
by high authority, we have before shown, in stating that
many of the ancients (and those, as Harris says, “ the
best grammarians") have called the infinitive 3vopa
fimparuköv, or övoua fiñparoc: and with these agrees Pris-
cian, in the following passage, “a constructione quoque
wim REI verborum, id est, NoMINIs, quod significat ipsan
rem, habere INFINITI v UM possumus dignoscere.” “From
the construction, too, we may perceive, that the infinitive
has the force of the thing, of the verb, that is to say
of the noun, which signifies the thing itself.”
here called the thing, (or substance) of the verb, is
what we have called the conception, the mere name of
which is a noun. Thus, “I die” expresses the con-
ception of dying, but it not only names that con-
ception, it asserts the thing to exist, with reference
to a certain person; whereas “to die” expresses the
conception, that is to say, names the thing, and
does nothing more: it does not manifest the existence
of the thing as an object either of perception or vo-
lition; it does not assert that any person is dying,
or has died, or will die, or may die; neither does it
evince any desire that such an event should occur, either
positively or conditionally. “Take away the assertion,
the command, or whatever else gives a character to any
one of the other modes,” says Harris, “ and there re-
mains nothing more than the infinitive.” Take away
from the other modes, say we, whatever gives them the
verbal character, and there remains the noun. Whether
we call this noun a verbal noun, or a participial noun, or
simply an infinitive, is immaterial; provided we clearly
understand, that it belongs not to the class of verbs, but
to that of nouns, and that its nature does not depend
on its form; since, in English, the words death, to die,
and dying, may all be used as infinitives; and, when so
used, are generally convertible into each other, without
any change of meaning. Lastly, we may observe, that
as the participle is a verbal adjective, so the infinitive is
a verbal substantive. The former can supply only the
predicate of a proposition, as “I am walking;” the
latter may form the subject, as, “walking is pleasant,”
“ to walk is pleasant;” in which two latter sentences
the words “walking” and “to walk” are both infini-
tives, and must be translated into Latin by the word
ambulare, and not by the word ambulans. This consi-
deration º: it the more remarkable, that Harris
should inzline to rank the infinitive among the moods of
the verb, since he himself had classed the verb among
attributives, all of which, as he observes, “are, from
their very nature, the predicates in a proposition.”
The second peculiarity of the verb consists in its
tenses. The word Tense plainly shows that our chief
grammarians, in the early periods of grammatical study
in England, were Frenchmen; for it comes from the Latin,
tempus, through the French thus, tempus, temps, tems,
tense. Tense, therefore, originally and properly means
the expression of time in combination with the assertion
of existence; but this must not be taken to be the sole
effect of the tense in particular languages, as we shall
presently perceive. In order, however, to comprehend
this subject fully, we must begin, as Harris judiciously
does, by considering existence according as it is muta-
ble or immutable. We are well aware that, in the
proud and insolent ignorance of modern philosophy, we
shall be told that there is no such thing as immutable
existence; that men's minds are made up, as the
bodies are, of a certain small dust, which is perpetually
whirling about, and taking various forms and arrange-
ments, some of which it pleases every man to call true,
and others false; that this latter circumstance, how-
ever, is a mere delusion of the individual's mind, mentis
gratissimus error; that when the man dies, his notions,
their truth and their falsehood, their wisdom and their
folly, all die with him; and though some truths wear
better than others, and keep in fashion for twenty or
Chap. I.
What is \º-
Tense.
5ſ;
G R A M M A. R. *
Granmar, thirty centuries, while the greater part of our notions
S-> do not last longer than the small ephemeral insects of
Present,
the Nile, yet that in the end they all sink into one
common Lethe. & -
animae quibus altera fato
Corpora debentur.
The opposite philosophy to this, although stigma-
tized as “a metaphysical jargon and a false morality,
which can only be dissipated by etymology,” we feel
ourselves constrained to adopt, from the utter repug-
nance of the former to any thing like common sense
or intelligibility. We cannot conceive that the objects
of intellection and science are mutable in any possible
number of years, or in any imaginable conjuncture of
circumstances. We cannot, for instance, believe that
in a square the diagonal ever was, or will be, or can
be, commensurable with one of the sides. These two
magnitudes are not incommensurable because Euclid
happened to think so, or because his doctrine on the
subject has prevailed for above two thousand years.
Their incommensurability is a truth as independent of
that lapse of time, as any two things can possibly be
of each other. The opposite to it cannot be conceived
by the human mind. The existence of this truth,
therefore, is justly styled immutable.
Of such immutable existence the Present tense is
usually considered the proper exponent, because, in most
languages, it is among the simple forms of the verb,
and in particular has no distinct mark of time about it.
There is no reason, a priori, that there should not be a
separate inflection of the verb to distinguish perpetual,
absolute, immutable existence, from that which is pre-
dicated with reference to some certain time; but as no
language that we know of has adopted any such form,
and as absolute existence is naturally contemplated
under the form of a time perpetually present, it is
sufficient for us to consider this as one of the uses of
the present tense.
The other use of the present tense depends on the
nature of mutable existence. Now, mutable objects
exist in time. When, therefore, we declare them to
exist, that is, whenever we employ a verb active, or
passive, or neuter, we must declare them to exist in
some time. But time is distinguishable as to its
periods into present, past, and future; and as to its
continuity into perfect or imperfect; and though the
present, from its nature, must be definite and positive,
yet the other two periods may be stated indefinitely
and with relation to some different time. From these
sources, and from the differences of mood already no-
ticed, may be derived all the tenses, which appear in
use, in different languages. And first, as to the Present,
considered as marking a certain portion of time, it is
manifest that we may consider as present to us a greater
or less portion of time. Time flows on continuously, and
has in itself no stops or periods, but the mind dwells on
certain portions, and gives them a distinct expression
in language. The names of these portions are various,
as an age, a year, a day, an hour, a moment; but the
assertion of their existence is a collateral incident to
the verb. It has been well shown by Mr. Harris that
the present time, strictly speaking, is not cognizable by
any human faculty; for it is
6 vs Like the lightning, which doth cease to be,
Ere one can say it lightens. -
“Let us suppose,” says he, “for example, the lines Chap. I.
AB BC—"
B
A C
“I say, that the point B, is the end of the line AB,
and the beginning of the line BC. In the same man-
ner let us suppose AB BC to represent certain times,
and let B be a now, or instant, which they include; the
first of them is necessarily past time, as being previous
to it; the other is necessarily future, as being subse-
quent.” Hence he concludes, that time present has at
best but a shadowy and imaginative existence; and, of
course, as sensation refers only to time present, that
sensible existence is itself altogether imperceptible,
eluding the steady grasp of thought, and approaching
to absolute nonentity. This will, doubtless, appear
strange to the modern philosophers, who hold that
sensible existence is the only existence; but let them
meditate on what they mean by the words now, or
instant, or moment; let them consider how difficult it
is to arrest the fleeting progress of time, and fasten it
down to the periods indicated by those terms; and
they will, perhaps, perceive that their motions are not
quite so clear as they have hitherto fondly imagined.
We will assume, that in the above diagram the per-
fect present is correctly indicated by the point B. At
that moment, I open my eyes and I contemplate, at one
view, a large theatre crowded with numerous happy
faces, with splendour, and beauty, with the diversities
of age and sex, and condition, with mirth and gravity,
and all the passions, which, though not meant to be
brought into public, could not entirely be thrown off
and left at home, like an unvalued garment. Or, per-
chance, I am on a proud hill-top, from whence, at one
glimpse, I behold mountains and vallies spread in rich
perspective before me, with the near cottages, and the
distant town, and, beyond all, the remote and hazy
ocean. I see the variegated foliage, and the ripening
corn, the clouds of heaven sailing high in air, the
rustics at their labour, and the little vagrant boy pick-
ing daisies at my feet, and delighting in his idleness.
Witheut any time for reflection, without a thought of
the successive action of the machinery in this grand
landscape, I say, “I see” all this, at the present mo-
ment, and I enunciate it in the present tense perfect.
But if I wish to express a continuous action, if, for
instance, I mean to describe myself as remaining for
some time in contemplation of the scenes just de-
scribed, I am compelled to change my expression, and
to adopt the present tense imperfect. In that case, I
say “I am contemplating,” “I am beholding:" and
the diagram before drawn will not them so well express
the time intended to be described as the following one:
B

A C
Here, the present time, designated by the letter B,
extends indefinitely toward A and C, embracing a seg-
G R A M M A. R. 57
Caesar's leaping into the Rubicon, or of the first shot Chap. I.
Grammar, ment, the whole of which is viewed by the mind as
which was fired at the commencement of the thirty Sº-
\-e-Z being at once present to its contemplation, though
Past.
without any definite boundary on either side. The
English language easily distinguishes this sort of pre-
sent tense from the other, by the use of the verb to be
and the participle present; but in most other languages
the present perfect and the present imperfect have one
and the same form, and can only be distinguished by
the context.
We have seen that the present imperfect implies
something of the past, and something of the future.
Modern philosophy is very well satisfied to pass over
all the difficulties which occur in regard to the nature
of time. We are told, “ that we have our motion of
succession and duration from this original, viz. from
reflection on the train of ideas which we find to ap-
pear one after another in our own minds,” and that
“ time is duration set out by measures.” This is
surely any thing but reasoning. First, it is assumed
that there is a train of ideas which constantly succeed
each other in every man's understanding.” Each of
these ideas then must either occupy an indivisible
point of time, or it must have some distinguishable
duration. In the former case we cannot at all under-
stand how reflection on many indivisible points should
afford us the notion of any continuous quantity. In
the latter case there would be no occasion to reflect
on a train; for the reflection on a single idea would
present to us the notion of duration in itself. But
what are these ideas; and how do they march in
train 7 Are they all of equal duration? If so, or
if not, what is it that determines the duration of
each 2 Is it not the voluntary act of the mind?—Again:
is there no interval in the train 7 Aliquando dormitat
Homerus, was an old remark; and we suspect that it
applies even to the most lively and active minds of the
modern philosophical school. On the hypothesis above
stated, it would seem that before a man could have
any notion of duration, and consequently of time, he
must have formed in his own mind thoughts of a cer-
tain duration; these thoughts must have succeeded each
other in a distinguishable order, he must have been
fully aware of that succession, and he must afterwards
have made it the subject of reflection. But this state-
ment is absurd; for on what is he to reflect? On a suc-
eession which would not present any notion of duration
unless it involved that notion in the first instance; nor
would the succession of any two or more ideas produce
a notion of duration if the thoughts themselves, or the
interval between them, did not involve it. The truth
is, that the idea of duration, or time, is not to be made
up out of any other elements, but is an original law,
and first element of thought in the human mind. We
perceive duration of time just as we perceive extension
of space, because it is one of the necessary forms under
which alone we can contemplate existence. Whilst we
are contemplating the indivisible moment which con-
stitutes the perfect present it has already melted into
the imperfect present; and if we attempt to seize it
again, it has already become the past; its distinction
is then fully marked ; for the past is presented to us
by memory, as the present is by sensation. -
The past has its perfect and its imperfect, its de-
finite and its indefinite, its positive and its relative.
We may speak of an action which was performed on a
given day, at a given hour, and a given minute; as of
WOL. I.
years' war: or we may speak of an action in which a
person was occupied, and which was going on at the
time to which we refer. Thus the ancient artists in-
scribed their works with the word faciebat, to indicate
that they did not put them out of hand, as finished and
perfect, but that they had been for some time engaged
making them, and would have carried further their at-
tempts toward perfection, had time and circumstances
permitted. Thus, too, Syrus in the Heautontimoru-
menos, describing the employment in which he found
Antiphila and her servants employed, says,
Texentem telam studiosé ipsam offendimus:
- Anus
Subtemen nebat: praeterea una ancillula
Erat : ea teacebat una.
Again, we may speak of the past time definitely,
fixing the epoch when it happened, as,
That day he overcame the Nervii.
Or indefinitely declaring that the act of which we are
speaking is past, but not ascertaining whether the time
of its performance was near or distant; as,
Thou art the ruins of the noblest man
That ever lived in the tide of times.
Lastly, the past time may be mentioned simply as past
at the present moment, or as past at some time
preceding the present; and these two tenses may
be reciprocally distinguished as positive and relative.
Thus, in the positive, Macbeth says,
I have liv'd long enough : my way of life
Is fallen into the sere, the yellow leaf.
In the relative, Thyrsis (the attendant spirit), in the
Masque of Comus, says,
This ev'ning late, by then the chewing flocks
Had ta'en their supper on the sav'ry herb
Of knot-grass dew-besprent, and were in fold,
I sate me down to watch.
As the past time exists in memory, so the Future Future,
exists in imagination. Such is the nature of man, or
he would be unable to attain “that large discourse,
looking before and after,” which the poet truly assigns
to him. The conception of duration may be supposed
to exist in a being which had only the perception of
the present and the past; but to render that concep-
tion operative and useful, to convert it into an accurate
idea of time, it is necessary that the notion of futurity
should be superadded. It is a mistake to say that the
present impression is distinguished from the memory of
what is past by superior vividness and strength. It
often happens that things present
Pass by us, like the idle wind
Which we regard not;
whilst objects of memory so fully occupy our attention,
that, like Hamlet, we think we see them “in the mind's
eye.” Still we see them (whilst we possess our reason-
ing faculties) not as present, but as past, with a spe-
cific difference of perception. The perception of the
future, as such, is also specifically different from either
of the others. Reason and reflection alone could not
explain to us the necessity of such a distinction, be-
cause it is an element of reason, so far as that faculty
applies to events occurring in time. It would be as
correct to say, that by reasoning on the nature of light
and colours, we come to discover the existence of red
l
58 G R A M M A. R.
Grammar. and green, as to say, that by reasoning on duration, but “you shall go” implies the volition of the speaker. Chap. I.
It is a striking proof how much nicety and difficulty Sº-
S-2- we come to discover that there is a past, a present, and
a future.
When we treat of past, present, and future, we treat
of them with reference to some particular moment;
for as time is perpetually flowing on, that which was
future yesterday is to day present, and that which was
present yesterday is to day past. The particular mo-
ment which thus characterises the time, is that in
which the speaker or writer is addressing himself to
his hearers or readers. We have seen, however, that
that moment is not always referred to as indivisible,
but sometimes as capable of extension and indefinite
continuance. So it was observed to be in the present
and past; and so it is in the future. A person may say,
“I shall mount my horse; and he may say, “I shall
be an hour riding from London to Richmond.” In the
former instance the tense may be called the future
perfect; in the latter the future imperfect. Again, the
future may be definite ; as, “I shall mount at six
o'clock;” or, indefinite, as “I shall ride some time in
the course of the day.” Lastly, it may be positive,
considering the act only as future at the moment of
speaking, which is the case with all the preceding ex-
amples, or relative, considering the act as not to take
place till after some other which is also future. Thus,
a person may say, “I shall have mounted my horse be-
fore the clock has struck;” or “ I shall have been
riding an hour when I reach the next milestone.”
These distinctions refer properly to time. There are
others which refer to the contingency of the act, or
to its frequency and habitual performance; these
seem to draw their distinctive character properly from
the mood, or kind of verb, and therefore, we think
them not so much tenses as modifications of the
tenses already named. Somewhat more of doubt may,
perhaps, be allowable with respect to those forms
of speech which imply either the immediate intention
to begin an act, or its recent completion. Of the
first class are “I am about to write,” “I was beginning
to write,” “I shall begin to write:” and of the second
class “Je viens d'ecrire, “ I have just written;” Je
venois decrire, “I had just written;" "Eaopat yeypapác,
“I shall have done writing.” Yet though these forms
of speech serve to mark given periods of time, and
therefore may be called tenses; yet they seem to go
somewhat further, by including other notions not
strictly referrable to time. At all events, there must
be a limit to the combinations, which are distinguish-
ed as tenses. Time is capable of endless divisions,
and language would be infinitely minute in all its
ramifications, if it provided a separate inflection for
all those separate modifications of thought. It is true,
that idioms vary in nothing more than in the varieties
of tense, for which they provide. Some are very
meagre; others luxuriant; some are strictly confined
to differences of time; others mix up, with these, a
variety of other considerations. Thus the English
language marks a distinction unknown, we believe, to
any other language, between the future of choice and
the future of necessity: and what is remarkable, that
distinction varies with the different persons of the
tense. “I shall go” implies no particular volition,
nor indeed any thing but the certainty of the event.
“I will go” implies absolute volition. On the other
hand, “you will go” implies no volition of any person,
there is in the peculiar use of the tenses of verbs, that
scarcely a single Scottish writer, however eminent, will
be found to have accurately observed the distinctions
of “shall” and “will" throughout all his compositions.
The reason is, that the writers in question have from
infancy become accustomed to the Scottish idiom, and
idiom is much less a matter of reasoning than of habit.
A critical examination of the idioms regarded as most
elegant, will show them to abound with the same
pleonasms and ellipses, which are commonly consi-
dered as marks of rusticity in the language of the
people. The English idiom above-mentioned, how-
ever, is of very simple explication. It refers primarily
to the will of the speaker. If, therefore, he says “I
will,” it is to be understood that so far as his power
extends, the action is to be performed; but if he says
“I shall," inasmuch as he indicates no volition of his
own, nothing further is to be inferred but the futurity
of the action. Again, if he says “you shall go,” he
“shall go,” he intimates a necessity; for the original
meaning of shall is that which is necessary, and must, or
at least, ought to be done, from the Maeso-Gothic skal.”
But this necessity, being declared by the speaker, re-
lates to his will alone. Thus, in Coriolanus:
It is a mind
That shall remain a poison where it is,
Not poison any further. -
CoRIola NUs. Shall remain 2 -
Hear you this Triton of the minnows? Mark you
His absolute shall?
SICINIUs.
On the other hand, when the speaker says “you
will go,” “he will go,” he intimates no will of his own;
and, therefore, nothing is understood but the futurity of
the action. The proper force and effect, therefore, of
the two English futures may be thus expressed:
1. Future compulsory. “I will go," i. e. it is my will
to go. “Thou shalt go,” i. e. it is my will to compel
thee to go. “He shall go," i. e. it is my will to com-
pel him to go. -
2. Future not compulsory. “I shall go," i. e. there is
some cause compelling me to go, independently of my
will. “Thou wilt go,” i. e. there is some cause com-
pelling thee to go, independently of my will. “He will
go," i.e. there is some cause compelling him to go, in-
dependently of my will. .
The same reasoning of course applies to the plural
number as to the singular; and, consequently, “we
will go,” “ ye shall go,” “they shall go,” belong to
the first kind of future; and “we shall go,” “ye will
go,” “they will go,” belong to the second. What we
have here called the future compulsory has sometimes
a merely permissive force, sometimes a promissive,
and sometimes it is used in the manner of an impera-
tive mood, as “Thou shalt not steal,” “Thou shalt
do no murder,” for “steal not,” “ murder not;” and
this idiom is found both in the Greek and Latin: Eogabe
ev vputeic re)\etot, Ye shall be therefore perfect, i. e. Be
ye therefore perfect, St. Matt. ch. v. v. 48. And so
Horace : Inter cuncta leges et percunctabere doctos.
Lib. i. Epist. 18.
To return from this digression, we may observe,
that though various circumstances, of the nature of
* See JUNIUs ad vocen. Also WACHTER, schuld, schuldig.
G R A M M A. R.
5
Ö
Grammar, those which we have already pointed out, do, in fact,
<>~ enter into the composition of tenses in various lan-
guages; yet they do not properly belong to the scien-
tific division of tenses in Universal Grammar, which
ought to regard only distinctions of time, and not these
beyond a certain degree of minuteness and complexity.
Where the divisions of time are very minute or com-
}lex, their expression rather forms a sentence than a
word. It is more than the mind can easily grasp or
communicate in one combined form, and which there-
fore to be understood requires to be analysed into dif-
ferent words. -
In a subject which has undergone such various
treatment by grammarians, as the distribution of tenses,
we are far from arrogating to our own method any very
superior merit; still less do we recommend the name
which we have given to each tense as the best calcu-
lated to express its distinctive character. Instead of the
perfect and imperfect, some writers use the terms abso-
lute and continuous ; and what we have called positive
and relative, corresponds nearly with the perfectum
and the plusquam perfectum, the futurum, and paulo
post futurum. - - -
The arrangement proposed by the learned Mr. Harris,
though differing considerably from that which we have
suggested, is, we must acknowledge, entitled to great
1. Denoting time absolutely and indefinitely:
attention: and, therefore, withoutgoing into all his rea- Chap. I.
sonings in favour of it (for which we refer to the 7th S-Tº-
chapter of the 1st book of Hermes), we think it right
to state its general outline. . .
“Tenses,” he observes, “are used to mark present,
past, and future time, either indefinitely, without refer-
ence to any beginning, middle, or end; or else defi
nitely, in reference to such distinctions.” -
“If indefinitely, then have we three tenses, called
aorists (so called from the Greek aopuzov, undefined, or
unlimited), viz. an aorist of the present, an aorist of
the past, and an aorist of the future.”
* If definitely, then have we nine other tenses, viz.
three to mark the beginnings of the present, past, and
future respectively, three to denote their middles, and
three to denote their ends.”
“ The three first of these nine tenses we call the
inceptive present, the inceptive past, and the inceptive
future: the three next the middle present, the middle
past, and the middle future; and the three last the
completive present, the completive past, and the com-
pletive future.” - -
“And thus there are in all twelve tenses, of which
three denote time absolutely, and nine denote time
under its respective distinctions.”
1. Aorist of the present, ypcipw, scribo, I write;
2. Aorist of the past, #ypalpa, scripsi, I wrote;
3. Aorist of the future, Ypapto, scribam, I shall write.
2. Denoting time under the respective distinctions of inception, continuity, and completion.
1. Denoting inception :
2. Denoting continuance:
3. Denoting completion:
1. Inceptive present, piéN\o ypcipeiv, scripturus sum, I am about to write;
2. Inceptive past, Éplex)\ov ypdqely, scripturus eram, I was beginning to write;
3. Inceptive future, plex)\haw ypcipew, scripturus ero, I shall be beginning to write.
1. Extended present, Tvyxciva, ypſipov, scribo, or scribens sum, I am writing;
2. Extended past, Éypapov, or ērāyxavov ypdqov, scribebam, I was writing;
3. Extended future, egopiat Ypaptov, scribens ero, I shall be writing.
1. Completive present, yeypaſpa, scripsi, I have written;
2. Completive past, #ysyptiqety, scripseram, I had done writing;
3 Completive future, coopiat yeypaſſwg, scripsero, I shall have done writing.
Whatever arrangement we adopt, we shall certainly
riot find it fully followed out in many languages; for
while some have great varieties of inflection or con-
struction to express the different times, others have
fewer; and yet it may happen that the idiom, which
upon the whole is the least rich in tenses, is more
minute than all the others in some one particular dis-
tinction.
On the combination of tense with mood, much judi-
cious criticism is to be found in various grammarians,
and particularly in the work last quoted, the Hermes
of Mr. Harris, who has collected not only his own ob-
servations, but those of the philosophers of successive
ages; herein evincing a modesty the more admirable,
when it is contrasted with the too prevalent vanity of
the present day, by which every Tyro in science and
literature is led to believe himself a luminary arising
to enlighten and vivify a benighted world. These
self-complacent gentlemen often succeed in drawing
round themselves a little circle of admirers; and in
that case their contempt of all who preceded them in
their own particular line of study is usually unbounded.
It may, perhaps, be useful to observe, that such over-
weening presumption, as it proceeds on a great mistake
in point of fact, so it indicates a narrowness of mind
extremely inconsistent with true genius, or the power
of permanently benefiting and delighting mankind.
Let us hear Milton, that noble ornament of modern
poetry, speaking of his predecessors, even the nost
ancient :
— O sad virgin, that thy power
Might raise Musæus from his bower.
And elsewhere :
Nor sometimes forget
Those other two equal’d with me in fate
(So were I equal'd with them in renown !)
iłlind THAMYRIs, and blind MAEoNIDEs.
And again :
AEolian charms, and Dorian lyric odes,
And his who gave them breath, but higher sang,
Blind Melesigenes, thence HomeR call’d.
On the other hand, we are certainly taught a very
I 2
60 G R A M M A R.
Grammar. different mode of estimating ancient and modern poets
4 * of an universal language, is not beyond the bounds of Chap. I.
~~~" by the too well known philosopher of Sans Souci,
rational hope, or expectation, and if ever attained, S-
Ah! dans ces jours, ou notre heureux destin
Nous a fourni, pour effacer Homere,
Un Apollon plus vif, et plus brillant;
Comment peut ou, en possedant Volt AIRE,
Avec dédain, regretter un instant
Ce viewa bavard 2
It would be somewhat curious to form a list of the
modern writers who have been characterised by their
admirers, or by themselves (which is still more fre-
quently the case), as being absolute inventors in the
different branches of science and literature: and the
best commentary on such a list would be another,
somewhat more difficult indeed to make out, which
should contain the discoveries, or even improvements,
for which the world is really indebted to these, its sup-
posed enlighteners and guides. In Grammar, we have
been told that a certain writer of recent date dispel-
led, “by a single electric flash of genius,” the obscurity
which hung over the whole science. It is the duty of
the encyclopaedist to correct such errors in point of
fact, and to expose such absurdity in point of opinion.
In physical science there may be discoveries which go
to alter much of our general reasoning on all subjects con-
nected with those discoveries. Substances altogether un-
known, organisations never before suspected to exist, may
be rendered obvious by experiment. But in the sciences
which depend on a knowledge of the human mind, it
is altogether weak and absurd to suppose that any
such cause of sudden and total improvement can exist.
By industry and attention, we may, perhaps, be enabled
to methodise some portions of every such science better,
or even to correct, in some degree, their general arrange-
ment; but we cannot possibly find in them any one
topic which has not been admirably handled by some
philosopher, ancient or modern ; and as to the great
leading systematic principles on which they respec-
tively depend, these will generally be found to have
been established from the highest antiquity. The illus-
trations of Particular Grammar, it is true, are of the
nature of physical science, for they depend on the
comparison of numerous dialects, ancient and modern,
some of which are to this day unknown to the civilised
and studious world, and others remain in a great mea-
sure buried in the dust of antiquarian records. The
etymologist, therefore, may possibly discover some
facts affording an important clue to discoveries beyond
the attainment of Plato or Aristotle; as, for instance,
those which may hereafter explain the confusion of
languages, or the dispersion of the different families of
mankind over the face of the earth; nor are we at all
inclined to undervalue the etymological studies of
modern writers, and particularly of the late Mr. Tooke;
but it is material to observe, that whatever they are,
they belong less to the philosophy of language than to
its history. Again, that part of Grammar which relates
universally to what we have called the matter of lan-
guage, that is, to the construction and use of the organs
employed to effect a communication of the mind, is evi-
dently physical and of course follows the common
laws of physical science. In this, therefore, we may
possibly look for discoveries, affecting in a very
great measure the whole system of such communi-
cation. In this view, the formation of a common
alphabet for all nations, or of a real character, or even
not yet been done.
may be the result of some great, and perhaps sudden,
improvement in this part of grammatical science; nor
while we are speaking on this subject, should we neglect
the opportunity of paying a deserved tribute of respect
to the memory of that excellent man, Bishop Wilkins,
whose Essay towards a Real Character, and a Philosophical
Language, first published in 1668, is beyond compare
the most ingenious work of the kind which has ever
fallen under our observation. But the pure science of
Grammar, however it may lend its aid to any of the
discoveries here spoken of, cannot receive from them
any great or important improvement; for its principles,
as we have abundantly shown, are founded in the
operations of the human mind, and certainly the
human mind was understood, and all its principal
functions developed and explained by the philosophers
of ancient Greece and Rome, with far more clearness,
depth, and precision, than they have been by any writer
in France or England within the last fifty years. The an-
cient grammarians were formed in the schools of ancient
philosophy, and were themselves philosophers of no mean
rank. Such a person was Apollon I Us, of Alexandria,
surnamed AvakóAoc, or “the difficult,” whose four books
Trept Xvvráčewc, “ on Syntax,” are considered to be the
most philosophical of any extant on the Greek lan-
guage. He himself says he composed them, puera
Trdanc depiósiac, “with all possible accuracy.” PRIS-
cIAN, who professes to make him his chief guide, says
of his dissertations, quid Apollonii scrupulosis quaestioni-
bus enucleatius possit inveniri ? The celebrated THEo-
DoRE GAZ A confesses that he owes to him almost
every thing. The learned THoMA's LIN Act. R follows
him, as it were step by step. And lastly, Harris, who
quotes him liberally throughout the whole of Hermes,
declares him to be “ one of the acutest authors that’
ever wrote on the subject of Grammar.” In thus
tracing the literary genealogy of grammatical authori-
ties, we at once prove their present title to respect,
and show that it could not have subsisted through so
many centuries, if it had not been originally founded
in superior talent and ability. When, therefore, we
find an author like Apollonius employing much learning
on the illustration of the tenses, and their combination
with the different moods, we are not to be persuaded
that such speculations are wholly trifling or useless to
those who would obtain a perfect acquaintance with
the science of Grammar.
Now Apollonius observing on the connection which
we have already noticed between the future tense and
the imperative mood, satisfactorily explains why in most
languages there is not a distinct form for the future
tense of that mood. The reason is that all imperatives
are in their nature futures; for thus argues Apollo-
nius: 'Earl Yap pun yuyouévouci pº yeyováow Tiposačic'
td & pun yuápſeva i pº yeyovóra, Éturmêetórmra è? #xovra
£ic Tô Đoreo6at McNAovrog ási. “A command has respect
to those things which either are not doing or have
But those things which being not
now doing, or º; not yet been done, have a natu-
ral aptitude to exist hereafter, may be properly said to
appertain to the future.” And again, he says, "Atravra
rd trposakrika šykespiévny #xet rºw répéA\ovroc &differty—
oxe66v Yap #v to p &i rö, Örvpavvokrovno ac rupiaq60, Tºp
ripamóngeral, kard rāv xpóve Évvotav' rii śskMost 3inx-
-, *.* ...tº,
- - - - *:::::: tº r
** = , = , sº-
G R A M M A R.
only commands something which has not yet been done; Chap. I.
but it forbids also that which is now doing in a slow S-
Grammar, Aaxöc, kaðū rô pew rposakrtköv, rö 8: 8ptsuków. “All im-
\-Y-' peratives have a disposition within them which regards
the future. With regard to time, therefore, it is the same
thing to say, Let him that kills a tyrant be honoured, as
to say, He that kills a tyrant shall be honoured; the dif-
ference being only in the mood, inasmuch as the one is
imperative, the other indicative.” So Priscian shows
the connection of the imperative with the future.—
“Imperativus vero praesens, et futurum (tempus) natural;
quédam necessitate videtur posse accipere. Ea enim impe-
ramus, quae vel in praesenti statim volumus fieri, sine aliquá
dilatione, vel in futuro.” “The imperative (mood) seems
to receive the present and the future (tense) by a cer-
tain natural necessity; for, we command those things
which we wish to be done, either immediately at pre-
sent, without any delay, or in future.” From this
reasoning, it is plain that the present tense of the im-
perative mood is a present inceptive, looking necessa-
rily to a continuance or completion in futurity. It is
really present only to the speaker, but as to the person
addressed, it is a future, either immediate or prospec-
tive. Thus, when Lear cautions Kent not to interfere
between him and his anger to Cordelia, the will and
the act are closely conjoined:
Come not between the dragon and his wrath ! -
But when he imprecates curses on his unnatural and
cruel daughters, the object of his prayer is one which
cannot take effect till a far distant period, and which
may continue for a long course of years:
If she must teem,
Create her child of spleen, that it may live
And be a thwart, dismatur'd torment to her;
Let it stamp wrinkles on her brow of youth,
With cadent tears fret channels in her cheeks,
Turn all her mother's pains and benefits
To laughter and contempt.
In the nature of things there is no reason why any
particular idiom should not have a distinct form of
imperative for the proximate and distant future; ex-
cept that in general usage, the gradations might be so
minute as not easily to be distinguishable ; and that as
some degree of futurity is necessarily implied in every
present command, any fixed barrier, separating the
nearer from the more distant, and assigning one form
of tense to the one, and another to the other, must be
purely arbitrary.
From what has been said it might perhaps be inferred
that the imperative mood could not in any case admit
of combination with a past time; but this would be
incorrect, for the mind can throw itself forward, as it
were, into futurity, and so command an action to be
past. We cannot by our will alter that which is past
at the moment of our speaking, but we can command
that at a future moment it shall have been dome : and it
is thus that Apollonius distinguishes between the im-
peratives present, and the imperatives past in Greek.
Thus in explaining the difference force of a karréral rac
apºreMec, “set about digging the vines,” and akdºldra,
rac duréXec, “get the vines dug,” he says the first is
spoken Éic trapdraaw, by way of extension or allowance
of time for the work, the other étc avvrexévoortv, with a
view to immediate completion. Andelse where explain-
ing the difference of these tenses akatre and aktipov, he
says of the latter & Hävov rô pº yewópevov trpogdorge,
dANd kal rô yivöpisvov čv trapardoel drayopévet, “it not
**
6]
*
and tedious progress.” Therefore, if a person is
writing slowly, to say to him, ypápé, “write,” would
be unmeaning; for that he is already doing : but to
say, Ypſilov, “get your writing done,” would be, in
fact, to forbid that tedious mode of writing which he
was pursuing. In this explanation of the imperative
past tenses, Apollonius is followed by Priscian, who
says, “Apud Graecos etiam practeriti temporis sunt impera-
tiva ; quamvis ipsa quoque ad futuri temporis sensum per-
tineant ; ut &vegx9ſira Trv\a, aperta sit porta. Videtur
enim imperare ut in futuro tempore sit praeteritum ; ut si
dicam, aper; nunc portam, ut crastino sit aperta. “The
Greeks even possessed imperatives of past time;
although these also belong to a sense of future time;
as, civetpx6āra, TüAa, “let the door be opened:’ for
this expression seems to command that at a future
time the action may be past; as if I were to say,
open the door now, in order that it may be open to-
morrow.” And the inference which he draws from
this reasoning is not less remarkable nor less correct.
“Ergo nos quoque possumus in passivis, vel in aliis passi-
vam declinationem habentibus, uti praterito tempore im-
perativi, conjungentes participium practeriti cum verbo
imperativo praesentis, cel futuri temporis ; ut amatus sit,
vel esto, reptAhaffw: doctus sit, vel esto, ösötödyffa : clausus
sit, vel esto, kek}\éto 6a).” “Therefore, even in passives,
or in words having a passive conjugation, we may use
a past tense imperative, by joining the participle past
with the imperative verb of present or future time; as
amatus sit, or esto requxñoffw; doctus sit, or esto èeów-
3dyffa ; clausus sit, or esto kek}\étaffw.” It is objected
that these are not tenses but combinations of words;
to which Vossius justly replies that such combinations
are uniformly admitted to be tenses in the indicative
and subjunctive moods; and consequently they may
be so in the imperative. Either, therefore, says he,
we should always reject those periphrastic modes
of expression from among the tenses, or we should
allow this diversity of tense to the imperative. In
many languages, and particularly in the English, to
adopt the former alternative would be to say, that
our language was almost wholly destitute of tenses;
but we, who have all along regarded grammatical dis-
tinctions, principally with reference to signification,
must certainly admit, that the modification of the as-
sertion, in regard to time, whether it be effected by a
change of accentuation or quantity in the syllable, or
by a syllable prefixed, interposed, or adjoined; or,
lastly, by some combination of distinct words, is to be
regarded as a tense. We are not ignorant that, in all
our English compound tenses, the auxiliary verb ori-
ginally performed a more leading part in the combina-
tion, and the verb now considered as principal was
used in the infinitive, being regarded, in the common
grammatical phrase, as “ the latter of twe verbs.”
Thus Chaucer,
Quoth then Creseide wol ye don o thing 2
That is, “will you do one thing?
And so, &
Thou shouldest never out this grove pace,
That thou ne shouldest dien of mine hond.
That is, “shouldest die.”


62 G R A M M A R.
Grammar. But to the general purposes of grammatical science it.
S-r- is of littke import how the tense came to be originally
Person.
formed.
modern use, has even laid aside its infinitive termina-
tion, in order to coalesce, as it were, more intimately
with the other element of the tense thus formed by
their combination. - *
It is true that all our auxiliaries do not simply
signify time. Indeed none of them do so properly ;
for have, the auxiliary of past, time, properly signifies
possession ; because we cannot properly be said to
possess an act until it is past; so, will implies futurity,
because volition regards only that which is not yet
in being. In like manner, may, can, must, &c. do not
in themselves imply time, except with reference to the
conjunctive mood. Hence Vossius has observed, that
what is commonly called the present conjunctive has in
some instances a future import; as, when Cicero says,
in one of his epistles to Atticus, “Est mihi praecipua
causa manendi; de qua utinam aliquando tecum loquar.”
“I have a particular reason for staying here, concerning
which I hope I may some time or other talk to you;”
where utinam loquar “I hope I may talk” relates entirely
to a future time.
numerous and minute remarks of many learned critics
on the mixed or variable times which are expressed by
all the conjunctive tenses. Suffice it to say, that the
combination of any mood which implies contingency
or futurity, with a tense, referring to present or past time,
must necessarily affect the expression of time, and,
consequently, that in this respect, the tenses of the
indicative must differ from the analogous tenses in any
other mood. As, therefore, in nouns, the term gender,
originally used to express the mere distinction of sex,
has been applied in use to distinguish large classes
of words from each other, with reference only to their
terminations; so in verbs the word tense, originally
meaning the expression of time alone, has been also
used in most Grammars to express that idea in com-
bination with the others which we have noticed.
We come next to a quality usually attributed to the
verb, but certainly not necessary to be combined with
it in the same word, namely, Person. The difference
of person peculiarly belongs to the pronoun, and has
been sufficiently explained in treating of that part of
speech.
expressed by a pronoun. This is universally the case
in the Chinese, for the verb being alike in all the
persons, it would be impossible to distinguish one
from the other without the addition of some other
word. The three persons singular of the present tense
run thus: - …”
- Ngo Ngai, I love;
Ni Ngai, Thou lovest;
Ta Ngai, He loves.
And the same occurs in the other tenses, and in the
plural number.
In English we find it partially the case; for though
in the singular we have three distinctions of person in
the present, as “I love,” “thou lovest,” “he loves,”
and two in the past, as “I loved,” “ thou lovedst,”
yet in all other parts (with the exception of the irregular
to be) the verb remains unaltered. Nor does this
arise from any peculiarity in the original genius of our
language, for the more ancient dialects from which it
It suffices, that at present the former verb
acts merely as auxiliary to the latter, which indeed, in
It is needless here to follow the
In many languages the Person is necessarily
is derived, abounded with personal terminations. Now Chap. I.
these terminations, it is very manifest, were, in their S-º'-
origin, nothing more than the pronouns themselves,
which, in process of time coalesced with the expression
of conception, assertion, and time, and so formed words,
signifying at once all these different circumstances,
together with the additional distinction of person. -
The English language is chiefly derived from two
sources, the Anglo-Saxon and the Latin, of which the
former is related to the Maeso-Gothic, and the latter
to the Greek: and it is remarkable that all these four
dialects bear a great resemblance in the manner in
which they express the persons of the verb; as will
appear by inspection of the following table of the
present tense: - - - 4 g
Gothic. Saxon. Latin. Greek.
1st person a e eo o
Sing. }: tº ºn tº G º G ais est €S eis
( 3d . . . . . . aith ath et ei
1st person a. [Il tº 9 tº . . . . OII] G [l
Dual. }; tº g º 'º tº e OS tº @ e ... eton
3d . . . . . . atS e = e tº º ºs eton
- | 1st person a. [Y]. ath emus omen
Plural. 2d . . . . . . aith ath etis ete
3d. ... . . . . . and ath ent | on ti"
The similarity continues through the other tenses; and
in all it is manifest, that the personal termination is
the personal pronoun. We mention this circumstance,
connected rather with the etymology than with the
philosophy of language, partly in illustration of the
general doctrine of personality in verbs, and partly to
account for some circumstances which have given oc-
casion for dispute, on this subject, among grammatical
writers. Thus, for instance, we see why, in the Greek
and Latin tongues, the two principal pronouns, that
is to say, those of the first and second person, ego and
tu, are never used but for emphasis, or else, where the
verb is omitted. For the former reason, Virgil says,
Nos patriam fugimus, tu, Tityre, lentus in winbrá,
Formosam resonare doces Amaryllida sylvas.
For the latter reason, Juvenal thus expresses him-
self: - . *
Semper EGo auditor tantum, nunquamne reponam P
It was necessary for Virgil, to express, emphatically,
the opposition between the different lot of the two
shepherds: and, therefore, though this opposition
would, to a certain degree, have been manifested by
the mere words patriam fugimus, and doces resonare ;
yet, for poetic effect, it became necessary to add the
emphatic words mos and tu. In like manner, the Atys
of Catullus exclaims, in the extremity of passionate
regret,
IZoo gymnasi fui flos, EGo eram decus olei !
In the line above quoted from Juvenal, we see that
there is a necessity to express ego before auditor, be-
cause the verb ero is wanting; but there is no neces-
sity for expressing it before reponam, because it is
involved in the termination of that word. The same
thing, indeed, is true of the third person, so far as
respects merely the pronoun; for the verbal termina-
tion et, at, or it, is undoubtedly the same as the pro-
noun id, or iste; and therefore the pronoun of the third
* Onti is the more ancient, ousi the more modern termination
of this person, in Greek.
t; R A M M A. R. - 63
Grammar, person is never expressed but for the sake of distinc-
Jºv-, tion or emphasis, any more than those of the first and
Imper-
sonals.
second persons. Thus, Virgil says,
-tº-4 -
- Amplcrus nati CYTHEREA petivit;
Arma sub adversá posuit radiantia quercu.
ILLE Deaf domis et tanto lactus honore
Frpleri nequit, atque oculos per singula volvit,
Miraturque, interque manus et brachia versat.
Here ILLE is necessarily expressed to distinguish the
agent of the verb nequit from Cytherea, the agent of
the verb petivit; but that distinction being once made,
the verbs volcit, miratur, and versat, are employed
without a nominative expressed. - -
Again, the same author says,
Arcades his oris genus a Pallante profectum
‘Delegere locuma, et posuere in montibus urbem.
HI bellum assidué ducunt cum gente Latina;
IIos castris adhibe socios. e
Where H1 in the nominative, and Hos in the accusa-
tive, are used emphatically; and the former without
necessity, so far as mere intelligibility is concerned ;
for the verb ducunt alone would have sufficiently indi-
cated that the Arcadians were the persons who warred
against the Latins. ~.
Some verbs are called impersonal, a name which
only seems to mean that they are not usually conju-
gated with distinction of persons, but remain always
in the form of the third person. If they had no other
peculiarity than that from which their name is derived,
it might not be necessary to notice them in a treatise
on Universal Grammar; but, in truth, they are con-
structed on a principle different from that which
has been already explained in reference to person.
The impersonals are of two kinds, active and neuter.
By active we mean those which require an object, as
“it grieves me,” “it pains me,” miseret me, decet me,
&c.; by neuter we mean those verbs of which the
action terminates in itself, as “it rains,” “it snows,” “it
is hot,” “ it is cold;" the Latin pluit, the French
il fait chaud, the Italianfa freddo, the German es donnert,
esſriert, &c. In all these instances the verb contains a
mere assertion of the existence of the conception; but
does not indicate any agent. These verbs have been
sometimes explained as agreeing with a nominative
implied in them: thus pudet is said to be a verb agree-
ing with the implied nominative pudor, as if the meaning
were, “shame shames me;” but this is perhaps rather
a formal than a substantial explanation. Pudet in
reality contains, and does not merely imply the noun
pudor: it expresses the same conception as the noun,
and asserts its existence. It is therefore rather of the
nature of a verb substantive, than of a verb active ;
and though, in some idioms, a nominative is expressed,
yet in reality that nominative is superfluous, or, at
most, is only introduced to keep up the general
analogy of the language. The nominative it in the
English language, and il in French, have no dis-
tinct reference to any conception. They are pronouns,
which do not stand for any noun. If any one
should say, “It rains,” we cannot, as in the common
case, where a distinct nominative is expressed, ask
“what rains 7” for the answer would only be it ; and
if we were then to ask, “what is it?" we must be
left without any answer. Hence, in translation, the
nominative it is often lost. We do do not say, in
Latin, Hoc pluit; nor in Greek, TOYTO xp); nor in
Italian, EGLI fa frcddo. The proper motion of an im- Chap. I.
personal verb, therefore, is, that it expresses action S_TYT-'
without expressing an agent. Many such forms exist
in language. The French on dit, is of the nature of an
impersonal; so are the English “they say;” the Ita-
lian si dice; the Spanish se cuenta; the English “me-
thinks;” the German mich diinkt; the Portuguese basta,
parece, contem, succede, &c.
Where the object of an impersonal is expressed, as
“it grieves me,” the sense may be rendered by a pas-
sive verb, of which that object is the nominative, as,
“I am grieved;” and, on the other hand, the Latin
language admits of passive impersonals, followed by
a dative or ablative, which are equivalent to personal
verbs active; as in Livy, Romam frequenter migratum
est a parentibus raptarum ; for parentes raptarum mi-
graverunt. Where the impersonal is the former of two
verbs (according to the common mode of speaking),
the latter being in the infinitive mood, the proper con-
struction is to regard the infinitive as a noun form-
ing the nominative to the verb, which, consequently, is
not an impersonal, but a personal. Thus, in the sen-
tence, Dulce et decorum est pro patrid mori; when,
rendered into English, “It is sweet and seemly to die
for our country;” the nominative it does not properly
render the verb is an impersonal, because it relates to
a definite conception, which is afterwards expressed,
and which renders the verb personal. Hence, in all
such sentences, the word it is superfluous, snd may be
got rid of by mere transposition; thus, “to die for one's
country is sweet and seemly;” or, it may be said to
answer to the emphatic word that, if the sentence were
turned as follows: “To die for one's country, that is
sweet and seemly.”
It has been contended that many of the Latin imper-
sonals are not really so, because they may be used as
personals. Thus Horace repeatedly uses decet in the
plural, as, -
- Tristia maºstum
Vultum verba decent,
So Ovid, -
Nec dominain nota” dedecuere comae.
In these instances, however, the verbs really become
personals: and as we have before seen that the same
verb is often of different kinds, being sometimes used
as an active, and sometimes as a neuter, so there is
nothing but the idiom of a particular language to pre-
vent the same verb from being used sometimes as a
personal, and sometimes as an impersonal. The im-
personal neuter may, in like manner, be used as an
active; for, as Scaliger has observed, we may say
pluit sanguinem et lapides, and, indeed, pluit is even used
with a nominative (Gen. xix. v. 24.), Dominus pluit
super Sodomam et Gomorrham sulphur et ignem.
The expression of Number is another accidental Number.
quality of the verb, which belongs to it not as a verb,
but in so far as it may be combined with the expression
of person. It is, therefore, like the same quality in
the adjective, a mere method of connecting it in con-
struction with the noun substantive, or pronoun which
forms its nominative. Accordingly, it applies to verbs
in the same manner as it does to nouns and pronouns.
When they admit a dual number, as in Sanscrit, Arabic,
and Greek, the verb admits the same; when they do
not, it has only a singular and a plural. Indeed, the
matter could not well be otherwise, since we have

64
G R A M M A. R.
Grammar. seen that the personal terminations of the verb are
S->~' really the pronouns themselves coalescing with it.
Gender.
Compari-
50ſl.
The verb is equally said to be in the singular or
plural, whether it has or has not distinct terminations
appropriated to those different numbers; we call
“I love” singular, and “we love” plural; but it is
manifest, that in all such instances the expression of
number exists only in the pronoun, and is imputed to
the verb by grammarians quite gratuitously. These
are questions of Particular Grammar: all that can be
laid down on the subject, as a rule of Universal Gram-
mar, is, that as on the one hand there is nothing in
the peculiar nature of the verb which involves the idea
of number, so there is nothing in the idea of number
which can prevent it from being combined with the verb,
where the genius of the language permits such a union.
Since the verb, by means of its connection with the
pronoun, admits person and number, there is no reason
why it should not also admit Gender ; and, in fact, this
distinction obtains in the Arabic, the Ethiopic, and
some other languages. It is, however, rare; and as
gender properly belongs only to nouns, or pronouns
substantive, with respect to which it has been already
discussed, we need not here pursue the investigation.
Some writers contend, that the verb, as expressing
an attribute, is capable of comparison; nor does it
'appear that this can be gainsaid, if we regard only the
attributive nature of the verb. There are, indeed, cer-
tain attributes, as has been already observed, which
are not intensive; and these of course cannot admit
degrees of comparison; neither can the assertive power
be compared: for the verb must either assert a thing
to exist or not to exist. On the other hand, verbs
may be compounded with conceptions implying com-
parison, as “to outdo,” “to overtake,” subesse, su-
peresse, &c. They may too, in general, be compared by
means of the adverbs of comparison, more, most, less,
least, &c.; but we are not aware that it has been at-
tempted, in any language, to combine in one and the
same word the assertive power with the comparative.
It is not easy to conceive any form of verb which in
itself would express the degrees of comparison; and
the reason probably is, that though the mere qualities of
substance may be simply intensive, yet actions are inten-
sive in various modes, as well as in various degrees. Of
different substances, concerning which whiteness can be
predicated, some may be more and some less white; but
of different beings concerning which the act of walking
may be predicated, all equally walk, though one walks
more, another less; one faster, another slower, &c. :
and so of mental action, several persons love, but one
loves more warmly, another more violently, another
more purely; so that there is not in actions, as there
is in qualities, a simple scale of elevation and de-
pression; and, consequently, the mere comparison of
more and less would not answer all the purposes of
language, as applied to the verb, though it does as
applied to the adjective. For this reason participles,
when they are compared, lose their participial power;
for sapientior and potentior do not express acts, but
habits, or fixed qualities, and therefore answer to the
English adjectives “wiser,” and “more powerful.”
Thus have we seen, that though the proper force
and effect of the verb–that on which its essential
character depends, is assertion, yet it is capable of
uniting therewith, and in fact does so unite, not only
the conception, which Priscian calls the res of the verb, chap. 1.
but the expression of mood, tense, person, number, Sºv-
and even gender. “Observe,” says the President
DES BRosses, “how, in one single word, so loaded
with accessory ideas, every thing is marked, every
idea has its member, and the analogical formulas are
preserved throughout on the plan first laid down.”
Elsewhere he adds, “All this composition is the work,
not of a deeply-meditated combination, nor of a well-
reasoned philosophy, but of the metaphysics of in-
stinct.” The Goths, the Saxons, the Greeks, and the
Latins, in forming the schemes of conjugation above
noticed, were probably impelled by principles in the
human mind, the very existence of which they hardly
suspected. Similar principles have operated, but with
endless diversity of application, in the formation of all
the various dialects which have been spoken in ancient
and modern times, by nations the most barbarous and
the most civilized; and it is the development and
explication of these ever-operative principles' which
forms the proper object of the science of Universal
Grammar.
§ 5. Of articles.
Having explained the uses of the principal parts of Articles.
speech, we come now to consider the accessories. The
principal parts, as we have already stated, are those
which are necessary for communicating thought in a
simple sentence: and the communication of thought
requires the naming of some conception, and the asser-
tion of its existence as an object either of perception
or of volition. Conceptions are named by the noun:
they are asserted to exist by the verb; but it often
becomes desirable to modify either the name, or the
assertion, or the union of both. How is this to be
done? We have seen certain modifications incorpo-
rated with the noun by its cases, and numbers, and
genders; with the verb by its moods, tenses, and per-
sons; with the adjective by its degrees of comparison;
and with the participle, gerund, supine, and infinitive,
by their marks of time, relation, &c. The same, or
similar effects, may be produced by separate words;
and what must those separate words be 7 Nouns, or
verbs, which, appearing in subordinate characters, are
no longer to be considered as such.
We wish to modify a conception; how can we do
it but by another conception? We wish to modify an
assertion ; how can we do it but by another assertion?
It is therefore plain, that the accessory words must
have had originally the character of principals; that is
to say, they must have been either nouns or verbs.
This is a truth extremely obvious in itself; and of
which it plainly appears, that many grammarians have
been fully aware; but there is another truth, which
seems to have been less apprehended, namely, that these
subordinate and accessory words act a very different
part from that which they sustained as principals in a
sentence. The mind dwells on them more slightly ;
they express a more transient operation of the intellect.
In process of time some of them come to lose
their original meaning, and to be only significant as
modifying other nouns and verbs. It cannot be de-
nied, that this is a fact. It cannot be denied that the
words and, the, with, and the like, have no distinct
meaning, at present, in our language, except that
which depends on their association and connection
G R A M M A. R. 65
Grammar, with other words.
The Etymologist may succeed, or
*w- he may not succeed, in his attempts to trace these
Particles.
non-significant words to the significant words from
which they are derived; but whether he be successful
or unsuccessful, the fact will be no less certain, that in
their secondary use they lose their primary character
and signification ; they are no longer nouns or verbs,
but inferior Parts of speech.
These inferior Parts of speech have been called par-
ticles: and, as such, are sometimes distinguished from
words, and sometimes treated only as a separate class
of words. To explain and account for them seems to
have given much trouble to many Grammatical and
Philosophical writers; and after all, the subject has been
often left in a state of great confusion. Lockſ, in
his IId volume, has a short and somewhat vague
chapter on particles, from which we may infer that he
considered mouns to be the names of thoughts, or, as
he expresses it, of ideas. All other words, he thought,
served to connect ideas. The principal of these (which
we call the verb) he calls the mark of affirming or de-
nying; and he says, “the words whereby the Mind
signifies what connection it gives to the several affirma-
tions and negations that it unites in one continued
reasoning or narration are called particles.” Elsewhere
he says of these particles, “they are not truly by
themselves names of any ideas;” and again, “they
are all marks of some action or intimation of the Mind,
and therefore, to understand them rightly, the several
views, postures, stands, turns, limitations, and excep-
tions, and several other thoughts of the Mind, for which
we have either none or very deficient names, are dili-
gently to be studied.” The confusion which occurs in
these passages between ideas, thoughts, and actions of
the Mind, leaves Locke's real meaning very much
in the dark; but it seems as if he thought that the
particles (in some instances, at least) could not be de-
rived from nouns, inasmuch as they signified some
thoughts, which had never been expressed by means of
Il OUlſlS.
HoogeveeN states the general doctrine of particles
very briefly. He says, particulas in suá infantiá
fuisse vel verba, vel nomina, vel ea nominibus formata
adverbia. “The particles were, in their infancy, either
verbs or nouns, or adverbs formed from nouns.” Ipsa
veró, QUATENUS PARTICULE, per se solaº spectatae, nihil
significant. “They themselves, as particles, considered
alone, signify nothing.” And again, in defining the
particle, he says, particulam esse voculam, ea nomine
vel verbo natam. “ The particle is a small word de-
rived from a noun or a verb.” Had Mr. Tooke properly
reflected on these passages, which he quotes from
Hoogeveen, he would have found them to contain all
that was valuable in his own system, without the errors
into which he has fallen.
The term particle is, perhaps, not well chosen to in-
clude the inferior Parts of speech ; nor do Grammarians
agree as to the extent of its signification. Locke only
describes it as including “prepositions and conjunc-
tions, &c.;” leaving it to his reader's judgment to deter-
mine what classes of words fall under the et catera :
SCALIGER says, ut omittam particulas minores, cujus-
7modi sunt praepositiones, conjunctiones, interjectiones :
and Hoogeveen, as we see above, seeins to distinguish
the particle from the adverb ; whilst other Grammarians
WOI. I.
include in it all indeclinable words, and even the Article,
which in Greek is declinable. It is not, however, ne-
cessary, that we should adopt either this, or any other
generic term, to express the Parts of speech of which it
remains for us to treat; but we shall proceed to consi-
der them separately, in succession ; and first we shall
treat of the Article.
Articles.
\-y-'
The proper office of this Part of speech is to reduce a Qffice of the
noun-substantive from a general to a particular signifi- *
cation. We have already observed, in speaking of
nouns, that by far the greater part of them must be
what Mr. Locke calls general terms, that is to say,
names common to many conceptions. We cannot give
a distinct name to every distinct object that we per-
ceive, or to every distinct thought, which passes
through the Mind; nor are these thoughts, or even
these objects so entirely distinct to human conception
as many persons are apt to imagine. If I see a horse
to-day, and another horse to-morrow, the conceptions
which I form of these different objects are indeed dif-
ferent in some respects; but in others they agree. The
one horse may be black, and the other white; but they
are both quadrupeds. The word horse is a noun,
expressing the conception which I form of the points
in which they agree. But this word applies to a class
of conceptions, and it is necessary that I should possess
some means of expressing the individuals of that class.
Now those means are afforded by adding the Article to
the noun. To illustrate what we mean, let us take a
general term ; for instance, the word Man. The con-
ception expressed by this word alone, is one which
rticle.
exists in several other conceptions, as in that which I
form of “ Peter,” or of “James,” or of “John.” Peter,
therefore, is a word expressing the general conception,
“Man,” together with something peculiar to a certain
individual; and the same may be said of James and
John; but it must frequently happen, that the proper
name Peter, or James, or John, is unknown to us.
How, then, are we to express our conception of any one
of them 2 To each the term “Man” belongs; but it
belongs to each equally ; and therefore it does not
distinguish the individual from his class, or one indi-
vidual from another. If, therefore, we use this term
“Man,” we must also employ some other means of
showing that we mean by it this, or that man ; or at
least some one man, as distinguished from the con-
ception of “Man” in general. Now, these means are
afforded by the Article ; and they are afforded in two
different ways: we either speak of the general term
simply, as applicable to a notion of individuality, or else
with relation to some particular circumstance which we
know belongs only to an individual. In the former
case we may be said to enumerate, in the latter to
demonstrate, the person or thing intended. In the one
we say positively “a man,” in the other we say rela-
tively “the man.”
Hence arise two classes of Articles.
called the indefinite and the definite ; but it has been
justly observed by HARRIs, that they both define, only
the latter defines more perfectly than the former. It
would, perhaps, be more appropriate to call the one
positivāşand the other relative, or the one numeral, and
the other demonstrative. We shall adopt the first two
of these designations, merely for convenience ; but we
consider the names by which it may be thought fit to
K
They have been Two classes.
66
G R A M M A. R.
of the Article, it must be answered affirmatively ; and Articles.
this is a question which, as it relates to the operations S-V-'
of the Mind, properly falls within the scope of pure
Grammatical Science. h
Even though a particular Language may have no Gradations
class of words called Articles, the persons speaking ºf concep-
Grammar. designate the different classes of words, as compara-
\- tively unimportant. The most material object with us
is to establish the classification itself on clear and in-
telligible principles.
Grammarians have disputed whether the Article be,
or be not a necessary Part of speech. Before this ques-
Whether
necessary.
tion can be properly answered, it must be clearly stated.
Mr. Tooke says, “in all Languages there are only two
sorts of words which are necessary for the communi-
cation of our thoughts; and these are, 1. noun, and
2. verb ;” and he adds, that he uses the words noun
and verb “ in their common acceptation.” It would
seem from this, that he meant to describe the Article as
unnecessary; for in common acceptation it is certainly
not considered to be identical, either with the noun,
or with the verb. However, he afterwards describes it
as “necessary for the communication of thought,” and
even “denies its absence from the Latin, or from any
other Language.” We have already adverted to the
doctrines of the ancient writers, who considered the
noun and the verb as the only, or, at least, as the
principal and more distinguished Parts of speech ; but
they who reasoned thus, either included the Article
among the syncatagoremata, that is, consignificant words,
or else denied its necessity, and even its existence, in
some Languages, particularly in the Latin. Noster
sermo, says QUINCTILIAN, Articulos mon desiderat.
Articulos, says PRISCIAN, quibus nos caremus—Ar-
ticulos integros in nostrá non invenimus Linguá. And
so SCALIGER, Articulus nobis nullus, et Graecis super-
fluus. And VossIUs, Articulum, quem Fabio teste
Latinus sermo non desiderat, imó, me judice, plane ig-
morat. From these authorities, and indeed from a very
slight inspection of the Language itself, it is clear, that
the Latin had no separate words answering to the
Articles of the English and other Languages; nor is
it less clear, that the Greek had only the relative
Article 6, 7), Tö, and was entirely destitute of our po-
sitive Article. Mr. Tooke is undoubtedly right in
inferring, from the necessity of general terms, the
necessity of the Article ; if we thereby understand the
necessity of some means to apply general terms to their
individual instances. He is, however, wrong in sup-
posing that this purpose is always effected either by a
distinct word, or by some prefix or termination added
to words: nor is the ingenious, but fanciful Court DE
GEBELIN less erroneous in asserting that the Article
was supplied in Latin by the termination ; for the ter-
mination in no manner whatsoever defined whether
the word was to be taken in a more or less general
acceptation. It indicated the case, the number, and
the Grammatical gender; but it did nothing else.
Homo signified “Man” in general, or “a man,” or
“the man” before spoken of; and the termination
afforded no help toward determining in which of these
three senses the word was to be taken in any parti-
cular passage. This was to be discovered in Latin, as
in some other Languages, merely by the context. If,
therefore, the question, whether the Article be neces-
sary, mean whether a separate class of words perform-
ing the function of the Article be necessary, it must be
resolved in the negative ; because no such class is to
be found in the Latin and some other Languages. If,
on the other hand, it mean whether in all Languages
there must be some mode of performing the function
that Language must certainly distinguish, in their con-
ceptions, the general from the individual. In treating
of the noun, we have already spoken of the different
gradations of conception ; but it is necessary that we
should here advert again to the grounds of this dis-
tinction. The inattentive observer of internal objects
believes that their forms are always impressed dis-
tinctly on the eye; and that every superficies is bounded
by a visible outline. A more reflecting and more ac-
curate Philosophy teaches us, that even in contemplating
the objects which we most admire, Imagination does
much more than mere sensible impression toward sup-
plying us with a knowledge of their forms; and, that,
in a sense not merely Poetical,
We half create the wondrous world we see.
In like manner, the inattentive observer of the opera-
tions of Mind, as they relate to Language, is apt to
suppose that all his thoughts or conceptions are definite
and distinct; and consequently, that the words which
serve to name these thoughts are so too; but this is
far from being the case. Let us consider each of the
three classes of conception before noticed, viz., the con-
ception of a particular object, that of a general notion
applicable to many particulars, and that of an idea or
wniversal truth. The first and last of these are in them-
selves perfectly definite. No man can have two dis-
tinct ideas of “virtue,” considered absolutely and in
the primary signification of the word: and the same
may be said of “squareness,” “power,” “duration,”
“space,” “wisdom,” &c., &c. In like manner we
cannot have two distinct conceptions of a particular
person or thing, and therefore, when we know its proper
name, as “George,” “Louis,” “London,” “Paris,”
“Alexander,” “Bucephalus,” “Europe,” “Guildhall,”
&c., &c., it is unnecessary to prefix thereto any other
word for the sake of more clearly showing the indivi-
duality of our conception.
Hence we see the reason why neither Proper names
nor universal terms do of necessity require to be used
with an Article, either positive or relative. The idiom
of a particular Language may, indeed, sanction such a
construction ; but this depends on separate considera-
tions, to which we shall hereafter advert. Generally
speaking, such idioms as the following cannot be neces-
sary to intelligibility in any Language: “the George
reigns in the England,” or “a Guildhall is situated in
a London;” or, “ the virtue produces the happiness;”
or “an Alexander aimed at a glory;” and the reason is
obvious; because it is not necessary to define or dis-
tinguish, in such sentences, one George from another
George, one England from another England, one virtue
from another virtue, &c. -
But the remaining class of conceptions, though ge- General
10Il.
neral in their nature, serve to communicate the greater terms.
part of our knowledge respecting particular objects.
We have often no other conception of the individual
than that he belongs to such or such a species. We
know the man only by his profession, the soldier only
by his regiment, the officer only by his rank. Hence
G R A M M A. R.
67
Grammar, the great use of general terms in all Languages; and
~~ hence too, the necessity for individualizing them, either
tacitly in the Mind, or expressly in Language. When
this process of individualization is effected by a separate
word, we call that word an Article ; and thus we say,
that it is necessary to add the Article “a” or “the” to
the general term “man,” in order to designate an indi-
vidual of the human species. * - -
It is to be observed, that, in a secondary sense, all
words of the other two classes may be considered and
treated as general terms; and, consequently, may re-
quire the use of the Article to individualize them. For,
first, the idea expressed by an universal term, such as
“virtue,” “truth,” and the like, may be considered as
existing separately in each subordinate conception of
quality, action, &c. in which it is involved. If we
speak of virtue simply, as opposed to vice, or in any
other manner which regards the pure idea of virtue,
without any modification, it is an universal term which
needs not the aid of an Article; but if we speak of
those subordinate ideas, such as justice, prudence,
temperance, fortitude, in each of which the higher
idea of virtue is involved, as the conception of Man is
in the conception of Peter or John, we may consider
the word virtue, in a secondary sense, as applicable to
each of them separately, and therefore may call each
“a virtue,” or “the virtue.” And not only does this
apply to subordinate conceptions of the same kind and
nature as their superior, but sometimes to others, in
which that superior is equally involved. The conception
of injustice is of the same kind and nature as the con-
ception of vice. They are both ideas, both universal,
both regard qualities of the Mind ; but the conception
of an unjust action partakes, though in a remoter de-
gree, of both these ideas, and therefore it is sometimes
called “an injustice,” or “a vice.” Thus Hamlet, on
Horatio's saying that he is not acquainted with Osrick,
replies, “Thy state is the more gracious, for 'tis a vice
to know him.” And so Bassanio, urging the Duke to
wrest the law to his authority, exclaims,
. To do a great right, do a little wrong.
It is only in this secondary sense that such words as
virtue and vice, right and wrong, can be employed in
the plural number; and hence arises in all Languages
a vast class of general terms, which unhappily are but
too often perverted in use. The idea of crime does not
always agree with our conceptions of crimes; and we
often find an opposition between the notions of right
and rights, honour and honours.
Secondly, a Proper name, which, in its primary sense,
designates only an individual man, may be made to
stand for a conception common to many other indivi-
duals; because we can suppose, however contrary it
may be to fact, that there is a class of men, each pos-
sessing those qualities and powers which make up all
that we know of a certain individual. Thus the word
SHAKSPEARE primarily means that wonderful Poet
who wrote Hamlet and the Midsummer Night's Dream,
who could portray the characters of Othello and
Falstaff, Richard II. and Richard III., and who as
much excelled every writer of his day in the sweetness
and facility of his language, as he did in richness of
imagination and in profound knowledge of the human
heart. It is in vain to expect another Being so endowed
to arise before the return of the fancied Platonic year;
and yet we may suppose a whole club of such drama-
tists, like the “cluster of wits” in Queen Anne's Articles.
time; we may imagine one from every Country under ~~
heaven; and therefore we may talk of “a French
Shakspeare,” or “a German Shakspeare,” “the Shak-
speare of Tennessee,” or “the Shakspeare of Tom-
buctoo.”
The words which answer the purpose of indivi- Articles
dualizing general terms, in the two modes above de-
scribed, were originally pronominal adjectives. In some
instances they have undergone a change of form, by
becoming Articles; in others, they remain unchanged.
The French le and un, are the Latin ille and unus; the
English the and a are the Anglo-Saxon that and ane.
Hence, it is not surprising, that many Grammarians
comprehend, under a common designation, the demon-
strative pronoun and the Article. Such was the doctrine
of the Stoics, some of whom gave to both these kinds of
words the common name of Article, calling our pronoun
the definite Article; and our Article, the indefinite Ar-
ticle; whilst others considered both as pronouns, and
only denominated our Articles, Articular pronouns.
Articulis autem pronomina communerantes, says Pris-
cian, finitos, et Articulos appellabant ; ipsos autem
Articulos, infinitos Articulos dicebant; vel ut alii dicunt,
Articulos connumerabant pronominibus, et Articularia
eos pronomina vocabant.
There are, however, some marked differences be- Difference
tween the pronominal adjective and the Article, which, *.*
we think, justify us in considering the latter as a sepa-
rate Part of speech.
In our own Language, the same words which act as
pronominal adjectives may also be used substantively;
and, in particular, the words that and one are some-
times to be considered as substantive pronouns, as
when we say, “that which I love,” “one whom I re-
spect ;” but we cannot, in like manner, say, “the
which I love,” “a whom I respect.” This distinction,
however, depends on the idiom of the English Lan-
guage, and, therefore, will not afford a discriminating
characteristic between the separate Parts of speech in
Universal Grammar.
The case is different, when we come to consider the
manner in which the pronominal adjective and the
Article respectively affect the meaning of a general
term. They both individualize it: but the Article per-
forms this function simply; the pronominal adjective
does more ; it marks some special opposition between
different individuals. When we say, “the man is
good,” there is no opposition implied in the word “the,”
although there may be in each of the other words.
We may say, for instance,
1. “The man is good; but the boy is bad.”
2. “The man is good; but he was bad.”
3. “The man is good; but he is not wise.”
On the contrary, when we say “that man is good,” we
imply no opposition to the other words in the sentence,
but only to the word “that.” We intimate not only
that there is a particular individual who is good, but
also that there is some other, who is not good. This
distinction is strongly marked in Latin by the prono-
minalºjectives hic and ille; as when Ovid says,
- dissimiles HIC vir, et ILLE puer.
Where the English Article the is used, the Latins, who
have no such Article, do not supply its place by the pro-
nominal adjective, but use the noun alone, as
68 G R A M M A. R.
Grammar. “Blessed is the man that walketh not in the counsel of the ungodly.”
~~' Beatus vir, qui mon abiit in consilio impiorum; and
The.
not Beatus ILLE vir.
It is manifest, that the act of the Mind is very dif-
ferent in the two cases of which we have spoken.
Simply to individualize, is a more transient operation
than to individualize and at the same time to contrast.
Hence, the word the is less susceptible of accentuation
than the word that. It resembles, in this respect, those
Greek pronouns which are called enclitic. When the
oblique cases of the personal pronouns, in that Language,
were used by way of contradistinction, they were
strongly accented, and were called by Grammarians
&p6otovovačvat, uprightly accented ; but when they were
merely subjoined to verbs, without any emphasis being
placed on them, they were called 'Eqicxvtukai, that is,
leaning, or inclining. Thus the Greeks had, in the first
person, Euod, 'Ego', 'Ewe, for contradistinction, and
Moo, Moi, Mē, for enclitics; whence Apollonius pro-
poses, instead of the common reading, in the beginning
of the Iliad—
IIzºz & Pool xúrairs—
to read
IIzīāz 3 #zo, Adozu're-
For it is plain, argues he, that a distinction is in-
tended by the Poet between the words ‘Yºu’v and 'Euoi ;
and therefore the enclitic ao? is improper. The Prin-
ciple in the Human Mind, which converts the contra-
distinctive pronoun into an enclitic, is no other than
the eager desire of hastening toward the object of its
wishes—
Semper ad eventum festinat ;
and the same Principle it is, which converts the demon-
strative pronoun into an Article. Instead of “this
horse,” or “that horse,” we say “thè horse:” shorten-
ing the Article in pronunciation, because we dwell but
little upon it in thought. In the Anglo-Saxon Lan-
guage, the word that appears to have been shortened
into the ; and we have retained the longer word for our
pronoun, whilst we use the shorter for our Article.
When Mr. Tooke asserts that the word the is the
imperative of the verb thean, it does not appear that he
throws any great additional light on the subject. It
may, however, be curious to observe how he wrests an
etymology, to support his theory. “That,” says he,
“in the Anglo-Saxon theast, i. e. thead, theat, means
taken, assumed.” Now, the i. e. here plays a notable
part. The fact is, that there is a Saxon verb thean,
which properly means “to do,” or “prosper.” “Ill
mote he the,” in old English, is, “Ill may he do,” or
“ prosper.” And there is a Saxon pronoun that, an-
swering to our “ that.” It is not very clear that these
two words have any other connection than what Mr.
Tooke ingeniously supplies by id est. The Gothic
verb thiham, which Mr. Tooke also cites on this occa-
sion, (vol. ii. p. 59,) is our verb to take; and seems to
form a third element in this etymological medley. We
are not much advanced in the knowledge of Articles,
by being told that the verbs to do, to prosper, or to take,
have some similarity in sound to the pronoun that ; and
yet this is all we learn from Mr. Tooke. As to the
verb “to the,” it seems to be the origin of our old
English word thews; as in Hamlet—
Nature's crescent does not grow alone
In thews and bulk.
And so Falstaff says,
Care I for the limbs, the theius, the stature, bulk, and big sem-
blance of a man P
Again, the word that, in old German, signifies an
“act” or “deed,” and is derived, by WACHTER, from
the verb thun, which is nothing but our old English
doen, to do. It is possible that all these words may
have some etymological affinity to each other; but if
the connection were more clearly made out than it really
is, it would throw but little light on the true Gramma-
tical force of our Article.
Articles.
\-V-2
Much of the general reasoning which we have applied A.
to the relative Article the, is equally applicable to the
numeral Article a, or an. In French, the word um,
“one,” is spelt in the same way as the Article un, “a,
or an,” but it is pronounced more slightly. In English
the word has been not only abbreviated in point of quan-
tity, but changed in articulation, from “one” to “a.”
The mental operation, however, is the same in both in-
stances. The conception of one is expressed, not in
opposition to that of two, three, or any other conception
of number, but as distinguished from all the other indi-
viduals of the same class.
In the Scottish Dialect, ane was retained as an Article
to a late period; thus N1col, BURNE, in his “Dis-
putation,” A. D. 1581, says, “Tertullian provis, that
Christ had ane treu body, and treu blude.” And on
the other hand, in the old English, the numeral pro-
noun one was sometimes abbreviated to o, as we read in
Chaucer—
Sithe thus of two contraries is o lore;
aud so in the more ancient MS. Poem of the Man in
the Moon— :
He hath his o foot his other to foren ;
but it was still accented as a separate word; whereas
the Article a (as we have before observed of the other
Article the) is passed over hastily in pronunciation, as
a mere prefix to the general term, which it serves to
individualize. Again, the numeral pronoun one (like
the relative that) is capable of being used alone, which
the Article a or an is not. We may say, “one seeks
fame, another riches, and a third, the wisest of the
three, content;” but if we use the Article, we must add
its substantive, as “a man should seek content, rather
than fame or riches.”
Since it has appeared that all Languages do not em-
ploy separate words to perform the office of the Article,
it may be thought that those words when so employed
in any Language are always superfluous; but this would
be a great error. Articles add much to the clearness,
the strength, and the beauty of a Language: and to be
perfectly furnished with them it is necessary to possess
both positive and relative Articles. The Latin Language
had neither: the Greek had only the latter of the two;
but most of the modern European Languages have both.
It follows, that in this respect the Latin was less per-
fect than the Greek, and the Greek than either the
French or the English ; and Scaliger was, therefore,
wrong in denying the use of this Part of speech alto-
gether : Articulus, says he, nobis nullus, et Graecis
superfluus; and his sarcasm on the French nation was
somewhat misapplied, when he called the Article otiosum
loquacissimae gentis instrumentum.
Yet it must be allowed, that in many European Lan-
guages, and in none more frequently than in the French,
Not super-
fluous.
Sometimes
so used.
G R A M M A. R. 69
when we speak of “the King,” “the Queen,” “ the Adverbs.
Grammar. instances occur in which the Article is employed super-
Prince Regent,” meaning the King of England, the \-N-7
~~~~ fluously. This circumstance is, for the most part, attri-
butable to an elliptical mode of speech, which is suffi-
ciently capricious. In English, we generally prefix the
relative Article to the names of our rivers, but seldom
to those of our mountains. We say, “the Thames,”
“ the Tweed ;” i. e. the river Thames, the river Tweed;
but we never say a Thames, a Tweed: nor do we say
the Snowdon, the Skiddaw ; or, a Snowdon, a Skiddaw.
In French, the superfluous use of the relative Article is
very frequent ; but it is to be explained on the same
Principle of ellipsis. Il seroit à souhaiter, says Condillac,
qu'on supprimât l'Article toutes les fois que les noms sont
suffisamment déterminés par la nature de la chose, ou
par les circonstances; le discours en seroit plus vif,
Mais la grande habitude, que nous mous en sommes faite,
me le permet pas: et ce n'est que dans des proverbes plus
anciens que cette habitude, que mous mous faissons un loi
de la supprimer. On dit • Pauvreté n'est pas vice, au
lieu de dire, La pauvreté n'est pas un vice. “It is to
be wished that the Article were suppressed whenever the
noun is sufficiently determined by the nature of the
thing, or by the circumstances; the style would thereby
be rendered the more lively. But the great habit that
we have acquired of using it, does not permit this
change; and it is only in old proverbs, more ancient
than this habit, that we make a rule of suppressing it.
We say, Pauvreté n'est pas vice, instead of saying, La
pauvreté n'est pas un vice.” It is here to be observed,
that the proverbial expression, which Condillac seems
to recommend, is as much defective as the common
expression which he blames is redundant. The Article
la before pauvreté is superfluous, and originates in an
ellipsis of some word answering to “state” or “condi-
tion ;” so that “the poverty,” means “the condition
of poverty:” but, on the other hand, the word vice
properly demands the Article un ; for it is not meant to
deny that poverty is the idea of vice, which nobody
would have asserted ; but to deny that poverty is one
of those states which necessarily include the idea of
vice. The most accurate and Philosophical mode of
expressing this sentence would therefore be, if the idiom
of the Language permitted it, Pauvreté n'est pas un
vice; answering exactly to the English idiom in such
phrases.
As the French often employ the Article redundantly
with an universal term, and with the names of places,
so the Italians employ it with the names of persons:
Il Tasso, La Catalami, meaning “ the famous Poet
Tasso,” “the celebrated singer Catalani.” It is ob-
vious that these expressions are to be accounted for on
the same Principle of ellipsis already explaimed. The
Article in all such cases does not in reality serve to
modify the Proper name expressed, but the general term
understood.
There is a particular use of the relative Article, with
a general term, which, as it tends to individualize, in
a special and peculiar manner, should not be passed
without notice. Certain individuals, having obtained
celebrity for their peculiar excellences, have been de-
nominated from this circumstance, as 5 troumrijs, the
Poet, means Homer; 6 pittep, the Orator, Demosthenes;
6 6.e0}\dºyos, the Theologian, St. Gregory Nazianzen;
3 yew Ypſiq}os, the Geographer, Strabo ; 6 Aetzvooroº'atms,
Athenaeus, author of the Work entitled The Feast of
the Sophists; but this is no more than we daily practise,
Queen of England, and the Prince Regent of England;
just as we hear in private families and narrow circles
of society, of “the captain,” “the doctor,” “the parson.”
“the squire,” &c. the particular application of which
general terms is settled, as it were, by a common under-
standing among the parties; since each of the individuals
thus honourably distinguished has his little sphere of
celebrity, and
Is talk'd of, far and near, at home.
Plurima ejusdem farinae, says Viger, ubique obvia.
We do not think it necessary to enter at length into Other dis-
those distinctions of the Article, which do not coincide tinctions.
with our definition of this Part of speech. Such is the
distinction often found in the Greek Grammarians be-
tween the prepositive and subjunctive Articles. The
prepositive, viz. 6, #, rö, is what we have called the
relative Article: the subjunctive, viz. 8s, j, 6, is what
we have called the subjunctive pronoun. The latter.
it is manifest, has no effect whatever in individualizing
a general term, because it is only employed in a de
pendent sentence, with reference to a term which must
have been individualized in the prior or leading sentence,
The learned HICKEs, in that invaluable Work the
Thesaurus Linguarum Septentrionalium, suggests that
the Anglo-Saxon sum, which answers nearly to the Latin
quidam, should be considered as an indefinite Article.
It appears to us rather to belong to the class of pro-
nouns ; yet in this and some other instances the two
classes of words approach very nearly together ;
And thin partitions do their bounds divide.
§ 6. Of Adverbs.
Before we enter on the consideration of the prepo-
sition and conjunction, we find it convenient to treat of
the Adverb, which, in our Language, and probably in
most others, furnishes the greater part of the words
employed in the other two classes. Mr. TookE mo-
destly observes, that “neither Harris, nor any other
Grammarian, seems to have any clear notion of the
nature and character of the Adverb ;” and then he pro-
ceeds to give us his own motions, not of the Adverb in
general, but of a number of Adverbs in particular,
from which, and from what he had before said of the
conjunctions and prepositions, he leaves his reader to
collect that knowledge which, in his opinion, no Gram-
marian beside himself had ever acquired. As this
does not appear to be a very fair way of treating the
Grammatical student, we shall endeavour to pursue a
more satisfactory method, even at the hazard of adopt-
ing, from the ancient Grammarians, some of those
notions which appear to Mr. Tooke so obscure.
The Adverb was originally so called, because it was
added to the verb, to modify its force and meaning; hence
the Greek writers defined it thus: 'Entgºm/wa éatt uépos
Adye āk\vrov, Širi to fiſſua Tiju ävapopäv éxov.—“The
Adverb is an indeclinable Part of speech, having relation
to the verb.” The question of its being indeclinable or
not, is unimportant in our present investigation, since
this circumstance depends on the idiom of a particular
Language; but the relation which the Adverb bears to
the verb depends on the Science of Universal Grammar:
and this relation is stated by most of the ancient
Grammarians as the peculiar property of the Adverb.
DoNATUS makes it the only characteristic of this Part

70 G R A M M A. R.
of speech : Adverbium est pars orationis, qua, adjecta may be thought of this reasoning, it clearly corrobo- Adverbs.
rates the fact, that the Adverb is employed to modify \-y-
Grammar.
~~~
Modifica.
tion.
verbo significationem ejus aut complet, aut mutat, aut
minuit. “The Adverb is a part of speech, which be-
ing added to the verb, either completes, or diminishes,
or alters its signification.” WossIUs, however, observes,
that the Adverb is added not only to verbs, but to nouns
and participles; and consequently, that its name must
be understood to have been given to it, not from the
use to which it is always applied, but from that for which
it most generally serves. Non solis adjicitur verbis,
sed etiam nominibus et participiis : nomen igitur accepit
mon ereo quod semper, sed quod plurimum fit. By the
word, nouns, Vossius, as he afterwards explains it,
means adjectives, both nominal, pronominal, and par-
ticipial. “We say,” adds he, “bené disserens, as well as
bené dicere, and bené doctus.” And so we may say, pror-
sils meus, propemodium suus, et magis nostras, as well as,
prorsils amicus, propemodièm liber, magis Romanus, &c.
For want of a clear and intelligible definition of the
Adverb, some writers have undoubtedly exposed them-
selves to the sarcasm of Tooke, who thus translates a
sentence of SERVIUs: Omnis pars orationis, “every
word,” quando desinit esse quod est, “when a Gramma-
rian knows not what to make of it,” migrat in Adver-
bium, “he calls an Adverb.” And, indeed, among the
twenty-one classes of Adverbs which are enumerated
by CHARISIUs, there are some which ought rather to
be called interjections, as the pretended Adverb of invo-
cation, Heus / that of answering, item, that of wish-
ing, utinam, and that of showing, ecce. Nay, even
nouns and pronouns were sometimes reckoned among
Adverbs; as quanti datur, eras Romae, &c. &c.
It is impossible to avoid these errors, unless we first
establish a definition of the Adverb, to which, as a test,
the various classes of words comprehended by different
Grammarians under this common designation may be
applied. We are aware of Scaliger's remark, Nihil in-
felicius Grammatico definitore; but the task which we
have undertaken obliges us to state, as clearly as we
can, what we consider to be the function properly dis-
tinguishing the Adverb from all other Parts of speech.
The Adverb, then, is a word added to a perfect sentence,
for the purpose of modifying primarily an adjective, or a
verb, or secondarily another Adverb. We shall first con-
sider the purpose for which it is used, then the sentence
to which it is added, and, lastly, the sort of word which
may be so employed. -
I. It is used to modify an adjective, or a verb, or
another Adverb. All these words, it is well known,
are called by Harris attributives: and therefore he aptly
denominates the Adverb “an attributive of a secondary
order,” or “an attributive of an attributive.” Harris,
indeed, argues that the Greek word 'ETupºu'a is of the
same force and meaning as these phrases, inasmuch as
the word ‘Pſiua is used by many writers to signify not only
what is commonly called a verb, but also what are called
adjectives, participles, &c. Thus AMMONIUs says, catá
tà Totò o muawduevov, to uév KAAOX, cat AIKAIOX, ka?
&ra Towaota ‘PHMATA Aéreo 6al, cal oilk 'ONOMATA.
—“According to this signification, (that is of denoting
the attributes of substance and the predicates in proposi-
tions,) the words fair, just, and the like, are called
verbs and not nouns.” And so PRISCIAN, speaking of
the Stoics, says, Participium connumerantes verbis, PAR-
TICIPIALE VERBUM vocant. “Reckoning the participle
among verbs, they call it a participial verb.” Whatever
the adjective and the verb. On the other hand, the
Adverb is not employed to modify the substantive; be-
cause that is the function of the adjective, or of the
article. Let us then consider, first, the Parts of speech
which are primarily modified by the Adverb.
1. The adjective. Under this term we comprehend Qf adjec-
the adjective simple, or proper, the participle, or parti-
cipial adjective, and the pronominal adjective. It is
manifest that all the attributes which these various
classes of words express are capable of modification.
Thus, a house which is “ lofty,” may be “surprisingly
lofty,” or “very lofty,” or “moderately lofty;” or some
one may assert that it is “not lofty.” And in like
manner we may speak of “a remarkably intelligent
youth,” “an over indulgent parent,” “a truly affec-
tionate friend.” So, when we use a participle, or a pro-
nominal adjective, we may modify it by the aid of an
Adverb, as “much obliged,” “greatly indebted,”
“ wholly yours,” “absolutely mine,” “nobly born,”
“well bred,” “highly gifted,” “universally respected,”
“ little moved,” “ less affected,” “not so energetic,”
“equally judicious,” “how admirable !” “thus far.”
“no further.” In all these instances, it is obvious,
that the attribute expressed by the adjective undergoes
some modification from the Adverb. In truth, we form
a double conception, as, first a conception of lofliness
with reference to the house, and secondly a conception
of surprise with reference to the loftiness; so that the
sentence “the house is surprisingly lofty” resolves
itself into these other two sentences, “the house is
loſty” and “ the loftiness is surprising.” Mr. Harris,
therefore, had great reason to call the Adverb an attri-
butive of an attributive; for, in the latter of these two
sentences, we find the word “surprising” represents an
attribute of that loftiness, which, in the prior sentence,
was considered as an attribute of the house. It is not
the house altogether which excites surprise, but only
its quality of loftiness. A house may be both lofty and
surprising, without being surprisingly lofty.
The instances which we have hitherto noticed, may Compara-
When we tive.
be called those of positive modification.
say a house is “surprisingly lofty,” we do not compare
its loftiness with that of any other house; but if we
have occasion to make that comparison, we resort to
another class of Adverbs, and say it is “more lofty,”
or “less lofty,” or “equally lofty,” or “as lofty,” or
“the most lofty,” or “the least lofty;” in short, we ex-
ercise that mental operation which has been already
described in treating of the comparison of adjectives;
only the degrees of comparison are expressed by
Adverbs, instead of being incorporated in the same
word with the attribute compared. Nor is this all.
We may compare different attributes of the same sub-
stance, as well as different substances in regard to the
same attribute. We may consider the house as being
“ more lofty than convenient;” or as being “equally
convenient and lofty.” It is manifest, that in all cases
of comparative modification, the Adverb cannot be
employed simply or singly. It is then of a relative
nature, being necessarily joined in construction, either
with some other word, or inflection of a word in the
same sentence; which words, or inflections, when they
serve to modify adjectives or verbs, we also consider to
be of the nature of Adverbs.
^a
G R A M M A. R.
71
sentence, it is. Scarcely necessary to explain the enun- Adverbs.
Grammar. 2. The verb. It must be remembered, that the verb
S-N-' asserts or manifests existence, either simply or toge-
Of verbs.
Secondary
l! Stº.
Sentences
modified.
ther with some attribute of action or passion.
The
Adverb, therefore, may either modify the attribute in-
volved in the verb, or it may modify the mere assertion
of existence. When it modifies the attribute, its ope-
ration is exactly similar to what we have described, in
regard to the adjective. “He runs swiftly” is of the
same import as “he is running swiftly;” and the word
swiftly modifies the verb runs, and the participle run-
ming, in the very same manner. The case is somewhat
different when the Adverb modifies the assertion of
existence; and this it does whenever it expresses any
limitation of the time, place, circumstances, or actual
occurrence of the fact. Thus the words, “ now,” “then,”
“when,” “always,” “never,” &c., modify the assertion
in point of time. If I say that a certain event “happens
now,” my assertion is limited to the present time; if I
say it “happened yesterday,” the assertion is limited to
a certain time past. The assertion, that it “always
happens,” contradicts the opposite assertion, that it does
“not always happen,” and a fortiori the assertion that it
“ never happens.” So, with respect to place, the asser-
tion that a fact occurred here, or there, is no assertion,
with regard to what may have happened elsewhere.
Again, the occurrence of any event may be certain or
doubtful, actual or contingent; and we may therefore
say, “it will perhaps happen,” “it may possibly take
place,” “it is certainly the case,” “it really occurred,”
&c. As to the variety of circumstances attending
different transactions, which may be expressed by
Adverbs, they are beyond enumeration. The event in
question may occur aboard, or ashore, aloft, or below,
abroad, or at home, the ship may be cut adrift ; the
army may be afoot 3 it may be marehing homewards, the
battle may cease awhile, it may be begun anew, it may
terminate successfully, &c., &c., &c.
Such being the primary uses of the Adverb, it is easy
to conceive that the secondary use is similar. As the
adjective modifies the substantive, and the Adverb
modifies the adjective, so may a second Adverb be ap-
plied to the former with the same power of modification.
As the word admirably may be prefixed to good, so may
very be prefixed to them both together ; and we may
say “a very admirably good discourse;” in which, and
the like instances, the analysis is similar to what we
have before stated. The discourse is good, the good-
ness is admirable, the admiration is extreme.
II. We have next to consider the sort of sentence to
which an Adverb is added, and the manner in which
the addition is effected.
First, we say, the Adverb is added to a perfect sen-
tence ; and by a perfect sentence we here mean one
which either enunciates some truth, or expresses some
passion with its object. Therefore, even to a simple
imperative the Adverb may be added, since a perfect
sense is expressed without it, and its addition only
serves to modify the verb. Thus the word “fly '' is,
in effect, a perfect sentence, for it implies an agent and
an act, and it couples the conception of the act of flying
with the conception of the person addressed, if not in
the perception of the speaker, at least in his volition.
To this sentence, therefore, an Adverb may be added
consistently with our definition, and we may say “fly
quickly l’ After this explanation of the passionate
ciative.
there can be no difficulty: thus, when Macbeth says,
After life's fitful fever he sleeps well.
there can be no difficulty in understanding that the
Adverb well modifies the verb sleeps. A question,
however may arise where the verb merely expressses
existence; as, in the line just quoted, if the expression
had been “he is well,” it might be questioned whether
the word well was an Adverb or an adjective. A similar
remark may be made on such expressions as “he is
asleep,” “he is awake,” &c. It is true that in the Eng-
lish Language these and many other such words have
an Adverbial form, and cannot be employed in immediate
Connection with substantives, as “a well man,” an
“ asleep man,” or “an awake man:” yet where they
thus form the predicates of verbs, they are in effect
adjectives. “He is well” corresponds exactly with
“he is healthy”—“he is asleep” with “he is sleeping”
—“he is awake” with “he is waking:” and in a ques-
tion of Universal Grammar, the idiomatic form of the
words cannot at all decide the question.
When we say the sentence must be perfect, we mean
it must be perfect in the Mind; in expression a part or
even the whole of it may be understood. A part is
understood when the Mind evidently supplies what is
necessary to complete the sentence, as in the animated
lines of WALTER Scott—
—On Stanley !—On!—
Were the last words of Marmion,
Here the Adverb on manifestly refers to some verb
understood in the Mind, such as “march,” “drive,”
“rush,” or the like. The verb is suppressed, because
it is indifferent to the speaker : the Adverb is expressed,
because it is of the utmost importance—because to the
thoughts and feelings of the dying warrior the mode of
getting at the enemy was totally immaterial; but to
get at them by some means or other was his most eager
wish. The whole of the sentence is understood, when
the adverb is responsive: as, “Will you come? Yes.”—
“When will you come 2 Presently.”—“How often did
he come 2 Once.”—For these answers mean, “I will
come certainly”—“I will come presently”—“He came
once.”—And consequently the Adverbs, yes, presently,
and once, are to be taken as modifying the verb “will
come” and “did come,” respectively.
III. We have next to inquire what sorts of words Words em-
may be employed, as Adverbs, to modify adjectives and ployed.
verbs: and in reality the proper answer is—all sorts.
For the expression of Servius, though ridiculed by
Tooke, is literally true: Omnis pars orationis migrat in
Adverbium. “Every Part of speech is capable of being
converted into an Adverb.”
1. From what has already been said, it is manifest
that an adjective may be used Adverbially. Let us
suppose that it is necessary to enunciate these three
propositions successively.
1. A certain quantity exists.
2. That quantity is large.
3. That largeness is sufficient.
We have here three conceptions, viz., quantity, large-
ness, and sufficiency. The first is only considered as a
substance ; the Second is considered as an attribute in
one instance, and as a substance in the other; and the
When the verb expresses action or passion, \-
72 G R A M M A. R.
Grammar, third is only considered as an attribute. is perfectly arbitrary: it cannot possibly be supported Adverbs.'
by History, and we do not see the least ground for it S-
Now, if we
S-V-' unite these three sentences in one, and say there is
M uch *
‘quality at all.
“ a sufficiently large quantity,” we, in fact, convert the
adjective “sufficient” into an Adverb, In some in-
stances this difference in the employment of the word,
is attended with a correspondent change in the form ; as
in English the adjective sufficient is changed into
the adverb sufficiently ; but this neither prevails in all
Languages nor in all Adverbs of the same Language ;
and is, indeed, a circumstance often appearing to be
perfectly accidental, or capricious. Again, the adjec-
tives thus employed sometimes remain unchanged in
form, but lose in practice their adjectival use, either
partially or altogether. These circumstances, it is true,
depend on the idioms of particular Languages; but it
is not the less important to notice some of them, be-
cause there is no more common source of error among
Grammarians, than the confounding of what is universal
in Language with what is particular, the Scientific rule
with the accidental exception. This will appear from
many instances in the class of words now under our
consideration, namely, the adjectives proper, when used
as Adverbs ; and in order to consider them the more
distinctly, we shall notice first the simple, and then the
compound words.
Much, very, enough, fain, lief, scarce, stark, and se-
veral other words, more or less frequently employed as
Adverbs, were originally simple, uncompounded adjec-
tives. They have all some peculiarities in their use,
the notice of which may serve to illustrate the present
investigation.
Much is employed Adverbially before a participle, or
after a verb; and, though in modern use, we do not
give it the regular adjectival construction, as “a much
quantity,” “a much portion,” &c.; yet, this was an-
ciently and still is provincially dome with its derivative
muchel, muckle, or mickle. Mr. Tooke, who says that
this word much has “exceedingly gravelled all our
Etymologists,” derives it from the Anglo-Saxon verb
mawan, “to mow,” of which, says he, the regular
praeterperfect is mow, and the past participle mowen.
“Omit the participial termination em,” continues he,
“ and there will remain mow, which means simply that
which is mown ; and, as the hay, &c. which was mown,
was put together in a heap, hence, figuratively, mowe
was used in Anglo-Saxon to denote any heap ; and this
participle, or substantive, call it which you please—for
however classed, it is still the same word, and has the
same signification—was pronounced, and therefore
written ma, mo, &c., which, being regularly compared
gave ma, maer, maest, mo, more, most, &c.; and much
is merely the diminutive of mo, passing through the gra-
dual changes of mokel, mykel, mochill, muchell, moche,
much.” Such is the substance of an etymological dis-
quisition, in the course of which Mr. Tooke takes upon
him to speak with great contempt of Junius, Wormius,
Skinner, and Johnson, and pretends to remove all those
difficulties which have so “exceedingly gravelled” other
|Etymologists | The leading Principle in this disquisition
is a very extraordinary one. Mr. Tooke assumes that in
the formation of Language, the conceptions of distinct
action must necessarily have obtained a name before
those of quality. Indeed it is not very clear that he
conceives mankind ever to have acquired conceptions of
However, the fundamental assumption
in any rational system of Philosophy. We may ob-
serve, that the reasoning relative to the words more and
most would be at least equally satisfactory if its
order were exactly reversed, and the premises made the
conclusion. These words more and most, we might say,
are the comparative and superlative of the old word mo,
which was an adjective signifying “much :” when much
of any thing, therefore, was heaped together, it was
called mo; and consequently a mowe was a “heap ;”
but as hay, when it is cut down, is, in the very act of
cutting, heaped together, to cut hay was called to mow,
and the hay that was cut was said to be mowed. These
opposite trains of reasoning agree in this, that names
must necessarily be supposed to have been given to the
conceptions of the Human Mind, in some one certain
order, that is to say, either proceeding from the more
general to the more particular, or the contrary. We do
not know that this can be positively asserted ; but, if it
may be so, still we should incline against Mr. Tooke's Ety-
mology. According to him, our rude ancestors could
not have known whether a thing was much or little,
until after they had invented the art of making hay,
had regularly conjugated their verbs, added the parti-
cipial termination en, taken it away again, and com-
pounded the word (thus unnecessarily prolonged and
curtailed) with a syllable implying diminution; and after
all they could never alter the signification of the word;
but if they talked of much money, or much wisdom,
much acuteness, or much absurdity, the word much
would only signify a heap of hay ! So much for his
theory: as to his facts, we believe it would be exceed-
ingly difficult to discover where or when ma was used
for a hay-mow, or a barley-mow ; and when we come
to derive mokel, muchel, or michil, from mo, we shall be
“exceedingly gravelled” to account for the unlucky k
and ch which happen to be inserted before the syllable
said to be expressive of diminution.
That there may be some affinity between mo and
much is possible; but it is very improbable that much
should be an abbreviation of muchel. On the contrary
muchíl is, in all probability, derived from much. At
least, it is certain, that we find much, or mich, as early
as we do muchil. WACHTER, speaking of these words,
says, Simplicissimum est MICH quod in antiquissimis
dialectis ponitur pro magno et multo. “The most sim-
ple is mich, which, in the most ancient dialects, signifies
great and much.” Thus, in the old Persian, mih was
great, mihter greater, mihtras greatest; whence the
Sun was called mithras. The aspirate h was easily
converted into the guttural ch, and the palative k or g.
Hence the Greek ué), in páyas; and the Latin mag,
in magnus, magister, &c.; and as that which is great
is usually powerful, we have an infinite number of
words from this radical signifying power, as the Maeso-
Gothic and Anglo-Saxon magan, to be able, which
supplies our auxiliaries may and might, the old German
7machen, and Anglo-Saxon makan, to make, &c., &c.
Again WACHTER, speaking of the ancient word mich,
says, Posted invaluit MICHEL, eodem sensu. “After-
wards michel came into use, in the same sense.”
Hence the Gothic mikils, the Anglo-Saxon micel, the
Alamannic mihhil, the Islandic mikill, and, possibly,
the Greek peºãM). Nor does it at all appear that the
G R A M M A. R.
73
* \
Grammar. final syllable el or le is meant to express diminution;
v_\,-, muchel is no more the diminutive of “ much,” than
handle of “hand,” or spindle of “spin;” but much and
muchel are used eodem sensu, and so were anciently lite
and litel. .
It is at least certain, that much is to be found in
English as early as muchel, and that these two words
seem to be used indifferently by our most ancient writers.
The modern English Language is founded on the Anglo-
Normannic, of which the two earliest specimens referred
to by HICKEs are the Life of St. Margaret and the
Description of Cokaygne.” In the former of these we
find—
- Tho ho couthe of wisdom ho hatede muche sunne.
× :* :: 2k 2k
And yeld here servise, ofte mid muchele wowe,
In the latter,
Undir heuen his lond iwisse
Of so mochil ioi ant blisse
# : ; º 2:
The yung monkes, euerich dai,
Aftir met goth to plai:
Nis ther hauk no fule so swifte,
Bettir fleing bi the lifte,
Than the monkes heigh of mode,
With har sleuis anthar hode,
Whan the abbot seeth ham flee
That he holt for moch glee.
The date of these Poems is not positively fixed, but
they were certainly anterior to Edward I. That monarch
died in 1307; and among the Harleian MSS. (No.
2253, fol. 72.) is an Elegy, apparently written imme-
diately after his death, and consequently before the
time of Chaucer or Gower, in which are these lines:
The pope to is chambre wende
For dol ne mighte he speke na more
But after cardinals he sende
That muche couthen of Cristes lore.
With respect to the two great Poets themselves,
CHAUCER and GoweR, the former seems generally to
prefer the word much ; but the latter uses it indiffer-
* The Description of Cokaygneis a rude, satirical Poem, probably
written about the year 1200, in ridicule of the Monastic life: and it
is curious, as affording the etymology of the modern term cockney.
From the Latin, coquina, a kitchen, came the French words coquin
and cocagne. Coquin was originally coquinus, an attendant in the
kitchen, a turnspit, and thence came to signify any other mean, worth-
less person. Cocagne was the luxury of the kitchen. Hence, to this
day, among the amusements of the common people in France, at
public feasts and rejoicings, it is usual to erect a mast called the
Måt de cocagne, at the top of which are placed roast meats, and
other delicacies, as prizes for those who can most quickly reach them
by climbing. The land of Cocagne, therefore, is an imaginary land
of luxury, which the author of the Poem above quoted places “far in
see biwest Spayngne,” i.e. “far in the sea to the westward of Spain,”
the supposed situation of the great island Atlantis, the Hesperian
Gardens, and other fancied scenes of happiness, beyond the reach of
navigation, as then practised. The metropolis of England being
considered, by the rude inhabitants of the country parts, as a seat of
mere luxury and idleness, afterwards received, in contempt, this
name of cokayné, corrupted by them into cokeney, as appears by a
scoffing rhyme of one of the old barons—
Were I in my castle of Bungay,
Beside the river of Waveney,
I would not care for the king of Cokeney.
And it is somewhat amusing to trace in the satirical Description of
Cokaygne, the origin of the puerile story of roasted pigs running
about the streets of London, crying “come eat me.”
The gees irostid on the spitte,
Fleegh to that abbai God hit wot,
And gredith gees, al hote, al hot.
WOL, I,
ently with muchel. Thus, in The Testament of Love Adverbs.
(book ii.) “Moche folk at ones mowen not togider S-N-"
moche thereof have ;” i.e. “Many persons at once should
not have too much thereof, viz. of riches.” And again
(book iii.) “Opinion is while a thinge is in non cer-
tayne, as thus : yf the son be so mokel as men wenen.”
“Opinion is while a thing remains in uncertainty, as
whether the sun be so large as men suppose.” In the
Romance of Kyng Alisaunder, which was probably sub-
sequent to the time of Chaucer, we find—
With muche ost he is comyng
zk 2: 2:
Dieu mercy, to mychel harme
Many knighth there gan hym arme.
In that of Octovian Imperator, about the same age,
Ther n'as nother old ne yynge
So mochell of strength.
And in the Lyfe of Ipomydon, (Harl. MSS. 2252.) also
of the same period,
Hye and low louyd hym alle,
Moche honoure to hym was falle.
From all these authorities, it is very clear that much is
the name of a conception of greatness in quantity,
quality, number or power; and that when this con-
ception is viewed as the attribute of any substance, the
word much is an adjective; when as the modification of
an act or quality, it is an Adverb.
Very is correctly stated by Mr. Tooke to be the Latin Very.
adjective verus, “true,” changed, in old French and
old English, into veray, which, in modern French, is
vrai. The adjectival use of this word still remains in
the Liturgy of the Church of England, “very God of
very God.” Chaucer uses it as an adjective both in
the positive and comparative degree. Thus, in his
translation of Boethius, On the Consolation of Philo-
sophy (b. iv.) “It is clere and open that thilke sen-
tence of Plato is very and sothe.” And again, (b. iii.)
“which that is a more verie thinge.” From the same
word veray we have our compound adverbs verily and
verament, of which the latter, though now obsolete, was
once in Poetical use. Thus, in the above-quoted
Romance of Kyng Alisaunder, published by Mr. Weber,
from MSS. in the Lincoln's Inn and Bodleian Libraries:
By the steorres and by the firmament
He him taughte verrament.
:: ºk 2: 2% ºe
Ther ros soche cry verrement
No scholde mon yhere the thondur dent.
That an adjective primarily signifying “true,” should,
in a secondary sense, form an Adverb expressing emi-
nence of degree, as applied to all other qualities, is not
surprising; for a thing that is very good or bad, may
be said cat' éoxmv, to be truly good or bad. The Ita-
lians express the same modification of qualities by
molto, “much,” the French by fort, “strong,” the
Latins by multum, “ much,” and valdé, from validus,
“strong:” and our ancestors by a variety of attributes,
as swythe, sothfast, right, full, strong, well, &c.
Swythe may possibly be the adjective swift ; but is Swythe.
more probably from the Gothic sive, sicut; as sooth is
from the Gothic so, haec. We still use sooth Poetically
for “truth;” as in Lear—
In good sooth, or in sincere verity.
And in Macbeth—
If I say sooth, I must report they were
As cannons overcharged.
L
74 G R A M M A. R.
Grammar. In the Geste of Kyng Horn, (Harl. MSS. 2253. fol.
\-'83.) which WARToN says is the most ancient English
Strong, which we only use as an adjective at present, Adverbs.
seems to have been anciently adopted in the Norman- *-a-
metrical Romance, we find—
Tueye feren he hadde
That he with him ladde
Alle riche menne somes
Ant alle suythe feyre gomes.
We must observe that Warton was a very inaccurate
transcriber, and therefore is not to be relied on as au-
thority for any minute peculiarities of diction or ortho-
graphy; but we have, in general, corrected his quota-
tions, by the original manuscripts, and cite them from
the latter, with such variation only, as is necessary to
render them legible in the English character, changing
the Saxon th and w for modern letters, and filling up
the contractions, which would only serve to puzzle a
mere English reader.
In Kyng Alisaunder, this word often occurs, as
He smot the hors, and in he leop:
Hit was swithe brod and deop.
So in the Ballad on the defeat of the French by the
Flemings, at the Battle of Bruges, A. D. 1301, (Harl.
MSS. 2253. fol. 73. b.)—
Sire Jakes de Seint Poul yherde hou it was,
Sixtene hundred of horsmen asemblede othe gras,
He wende toward Bruges, pas purpas,
With swithe gret mounde.
Saxon Adverbially, as a translation of the French fort
and the Latin valde.
Thus, in the Geste of Kyng Horn—
Horn, quoth heo, wellonge -
Y haue loued the stronge. -
Well is derived from the Anglo-Saxon substantive
wela, “well-being,” or “felicity.” In that Language
the Adverb was wel ; in the Maeso-Gothic it was waila ;
in the Alamannic wuola; in the Islandic vel; in the
Dutch wel; and in the German wohl. Of the substan-
tive use of this word, we have an instance in the De-
scription of Cokaygne—
Ther nis lond undir heuenriche
Of wel, of godnis, hit iliche.
In the present day we rarely use it to modify adjectives
proper, or numerals, but those constructions are com-
mon in the old writers. We have just quoted the in-
stance of wellonge, i. e. very long. In the Ballad on
the Battle of Bruges, before mentioned, we have wel
7muchele, i. e. very great :
Sire Jakes ascapede by a coynte gyn
Out at one posterne ther the men sold wyn
Out of the ſyhte hom to ys yn
In wel muchele drede.
In Syr Launfal—
Sºthºast. Sothfast is the same adjective sooth, compounded (as with Attour ther was a bacheler
in the word stedfast) with fast, i. e. firm. Hence it And haddeybe well many a yer
was similar both in meaning and use to Swythe. In a Launfal for soth he hyght.
sort of Dramatic Poem, probably of the XIIIth or Again, in the Description of Cokaygme—
XIVth century, on Christ's Descent into Hell, (Harl. 5 Wendith meklich hom to drink
MSS. 2253. f. 55. b.) are these lines: Ant geth to har collacione
And so wes seyde to Habraham, A wel fair processione.
That wes sothfast holy man. In the Prologue to the Canterbury Tales—
Again, in the Pricke of Conscience (see Warton, v. i. That night was come into that hostelry
p. 258.) it is used adjectively— . Wel nine and twenty in a company,
Thou mercyfull and gracious god is, Chaucer also has the compound weleful, i. e. full of
* Thou rightwis, and thou solº/at º ... felicity. “O weléful were mankind if thilke loue that
Right. Right, the Latin rectus, we still use adverbially in gouerneth the heuen gouerned their corages.”
the titles “right honourable,” “right reverend,”
“right worshipful,” &c. The ancient usage was more
general.
In the Geste of Kyng Horn—
Athulf quoth he, ryht anon,
Thou shalt with me to boure gone.
In the Romance of Syr Launfal—
Her manteles were of grene felwet
Ybordured with gold ryght well ysette.
In Chaucer’s Clerke’s Tale—
Therys right at the west syde of Ytaly,
Down at the rote of Vesulus the colde
A lusty plaine.
Full, sometimes used Adverbially at the present day,
was much more frequently so in former times.
In Chaucer's Franklein’s Tale—
Listenythe of a knyght of tyme full olde.
In the MS. Poem of St. Jerome (Cotton MSS. Calig.
A, 2.)—
Seynte Jerome was a full good clerke.
In the Geste of Kyng Horn it is superadded to another
adjective used adverbially—
The leuerokes that beth cuth
Lightith adun to manis muth
Idight in stu ful swithe wel.
ary only by the adjective “
The word enough is explained in BAILEY's Diction-
sufficient.” It is, indeed,
used adjectively after the verb to be, as “that is
enough,” i. e. “that is sufficient ;” but we cannot em-
ploy it as we do the word “sufficient” in immediate
connection with a substantive; we cannot say “an
enough quantity,” as we do “a sufficient quantity.”
For this no other reason can be given than established
usage,
Quem penes arbitrium est, et jus, et norma loquendi.
This same adjective is used Adverbially, with-
out any change of form ; but again, custom obliges
us to place it after the adjective which it modifies,
and not before it, as is usual with other Adverbs. We
say “very large,” “pretty large,” “too large,” “suffi-
ciently large,” but we must say, “large enough.” The
accidental variation of arrangement, however, in no
degree affects the Grammatical character of the word,
which is decided by its signification and use, not by its
form or position. The Etymologists have thrown little
light on this word. Mr. TookE supposes it to be “the
past participle genoged, multiplicatum, manifold, of the
verb genogan, multiplicare.” It may, perhaps, be
doubted, whether there ever was such a word as
genoged, with the signification of multiplied; but if
Well.
1.nough.
G R A M M A. R.
75
Grammar, there was, how does this circumstance explain the
S-' Grammatical use of our present Adverb enough 2 What
Fain,
has the conception of sufficiency, conveyed by this
Adverb, to do with multiplication, any more than with
division. A single thrust through the body may be
quite enough to dispatch a man, and if it be not, he
will hardly wish it multiplied. Dr. Johnson's obser-
vations also on this word are rather singular. “It is
not easy,” says he, “to determine whether this word be
an adjective or Adverb,” as if it must, of necessity, be
always one, or always the other; and yet he afterwards
says, (which is equally erroneous,) that “after the verb
to have, or to be, it may be accounted a substantive.”
Add to this his suggestion, that when enough is an
adjective, “ enow is its plural ||"—although, in his
Grammar, he had said, that English adjectives were
indeclinable, “having neither case, gender, nor num-
ber”—and of course no plural. JUNIUs says, induc-
tus orthographid, quam praeclarae antiquitatis monu-
7mentum nobis eachibet, libens dedurerim ENough &
Gothico GANAH; et GANAH a Yavow, lactitid afficio, vo-
luptatem affero ; quod nihil aqué miseros mortales ea-
hilaret, quâm rerum omnium satietas. “Induced by
the orthography which the monument of illustrious an-
tiquity (the Coder Argenteus of Upsal) exhibits, I should
willingly derive enough from the Gothic ganah ; and
ganah from Yavdw, “I exhilarate or give pleasure;’ since
nothing so much exhilarates miserable mortals, as to have
enough of every thing.” Lastly, the Rev. Mr. LEMON,
in his English Orthography, derives enough from ikavös,
“sufficient in quantity or quality,” and adds, “indeed
our word enough undoubtedly wears a very Gothic ap-
pearance; but still is derived from the Greek.” Of
such etymologies, and such reasoning on them, it is
time to cry enough 1 The plain fact is, that the word
enough is the Anglo-Saxon word genoh, or yenogh,
having precisely its present meaning; and that this
word had some affinity with the Maeso-Gothic ganah,
the Frankish ginuagi, the old German ginuoh and
kamuht, the modern German genug, and the Dutch
genoeg, all words of the same signification, and all
descended, as WACHTER conjectures, from a more an-
cient Teutonic word, nog, which HELVIGIUs derives
from the Hebrew anag, “to delight.” However this
may be, these words are connected with a great number
of others, all bearing some relation, more or less dis-
tinct, to the conception of “sufficiency,” as the German
genuge, “ plenty,” genugen, vergmügen, gnug thun, “to
satisfy;” genau, “exact,” &c. &c.; nor is there any rea-
son to believe that our rude ancestors could not form a
conception of what was “enough,” quite as easily as a
conception of what was “multiplied,” and give a name
to the former as easily as to the latter. Now, such
name, when used substantively, would be a noun
substantive ; when used as the attribute predicated
directly or indirectly of any substance, it would be an
adjective ; and when used to modify the conception of
any attribute, it would be the Adverb enough, which we
are at present examining. -
Fain, says Mr. Tooke, is a participle: and then he
gives three examples, in each of which it has merely the
force of an adjective proper, which it still retains in
the Scottish name of a well-known tune, “I’ll mak
ye be fain to follow me,” i.e. “I’ll make you be glad
to follow me.” This word is used substantively in Kyng
Alisaunder:
Now quyk, sire, and smel,
Doryng alle thy bellis,
And do thy seolf thyn fayn
Thy folk al to ordeyne.
Lief also, Mr. Tooke contends, is a participle.
is not so; because it expresses no particular action,
but an habitual quality. Participles often make this
transition. Thus, the word “innocent” is, literally,
“doing no harm;” yet, in common parlance, it ex-
presses a certain Moral state of being, a freedom from
guilt. It would be as rational to say that love was a
participle, as lief, for they are both equally connected
with the Anglo-Saxon verb luftan, “to love.” The
general conception which prevails through these and a
great number of derivative words in the Northern Lan-
guages, is found in the old German lieb, which Wachter
explains to be bonum, quod omnes appetunt, sive sit
homestum et natura, conveniens, sive delectabile tantum.
“Good, that which all desire, whether as being honour-
able, and well suited to the nature of Man, or as merely
delightful.” Hence lieb, amatus, carus, dilectus, ami
cus; in which senses, he says, it occurs in all the Dia-
lects. Thus the passage in St. Mark’s Gospel, “Thou
art my beloved son,” is rendered in the Gothic, Thu is
sunus meins sa liuba. Mr. Tooke properly says, it
“always means beloved.” but beloved differs, in modern
use, from loved ; for as we do not use the verb to belove,
but, to love; so beloved, though a participle in form,
has the force and effect of an adjective proper. Leove
is thus used in the Poem on Christ’s Descent into Hell,
where Eve says to Christ,
Knou me Louerd icham Eue
Ich ant Adam the were so leoue.
In the comparative, it occurs in the Prologue to Kyng
Alisaunder, where the Poet says, there are many per-
SOIlS
That hadde levere a ribaudye -
Than to here of God other of Seinte Marie.
Gower has it in the superlative:
Three pointes, which I fynde
Ben leuest unto mans kynde
The first of hem it is delite
The two ben worship and profite.
In the Romant of the Rose, it is used for the beloved
person :
His leefe a rosen chapelet
Had made and on his heed it set.
It is also found in composition, as leflich, which is the
modern word “lovely,” leofman, which is Shakspeare's
word “leman,” leofsum, “amiable,” &c. In short, the
word leof, in all its forms, is no other than the word
love, which our ancestors used adjectively, whilst we
use it only as a substantive and as a verb. No One
thinks of saying that the substantive love is formed by
adding to the verb love the participial termination ed,
and then taking it away again ; nor is there any greater
reason for supposing this operation to take place with
the adjective.
Scarce and stark are admitted by TookE to be ad-
jectives, and their Adverbial use is equally well esta-
blished. Stark, indeed, is now seldom used as an
adjective, and only in combination with a very few ad-
jectives as an Adverb; but these are merely the acci-
dents of idiom. There are, as has been already
observed, several other simple adjectives which, either
in ancient or modern use, are employed as Adverbs;
Adverbs.
S-N-"
It Lief.
Scarce.
Stark.
L 2
76
G R A M M A. R.
Grammar, but we have already specified instances enough of these,
*~~”
Adverbs in
dy
and must now proceed to the compounds.
The first and most numerous class are those ter-
minating in ly, the greater part of which are only em-
ployed at present as Adverbs; while the same words,
in a simple form, without the termination, are used
adjectively. Thus we have in modern use the adjectives
“wise,” “grateful,” “judicious,” and the Adverbs
“wisely,” “gratefully,” “judiciously.” Hence some
persons, from an injudicious desire of precision, apply
what they suppose to be a distinctive mark of the Ad-
verb to words which do not require it, such as well and
ill ; for which they say welly and illy. Welly, indeed,
is provincial in the North of England, in the peculiar
sense of well nigh as fully is in Scotland, in connec.
tion with comparative adjectives, as, “fully more,”
“fully better,” &c. Ly is an abbreviation of the adjec-
tive like ; and the words wisely, gratefully, judiciously,
&c. are the compound adjectives wiselike, gratefullike,
judiciouslike, &c. The termination lyk or lich is com-
mon in old English. Thus, in Kyng Alisaunder, we
have the adjectives eorthliche, (earthly, mortal,) ferliche,
(strange, wonderful,) and the Adverbs gentiliche, (gently,)
sikerlyk, (securely, certainly,) theofliche, (like a thief,)
quykliche, (quickly,) stilliche, (quietly,) skarschliche,
(scarcely,) aperteliche, (openly.)
So, in Syr Launfal,
He gaf gyftys largelyche,
Gold, and siluer, and clodes ryche.
And again, in the same Poem—
The lady was brygt as blosme on brere,
With eyen gray, with louelych chere.
This word lovelych is the identical word leflich before
mentioned, and which occurs in one of the most ancient
love-songs now existing in English, composed probably
about the year 1200. The song begins, “Blow North-
erne Wynd,” and the lover describes his mistress
With lokkes lefliche and longe.
CHAUCER writes our word early, erliche ; as in the
Knight's Tale.
An tellin her erliche and late.
In the Description of Cokaygme we have already seen
the Adverb meklich (meekly.) In the Geste of Kyng
Horn we find evenliche (evenly, straightly) used as an
Adverb :
Thou art fair & eke strong,
& eke euenliche long.
This termination, therefore, is not less pure and dis-
tinguishable in the old English than it is, as Mr. Tooke
observes, in the sister Languages—German, Dutch,
Danish, and Swedish, in which it is written lich, lyk,
lig, liga. In the Anglo-Saxon we find it used both ad-
jectively and Adverbially, as in the translation of Bede's
Ecclesiastical History, (book iii. c. 3.) “tha liftgendam
stanas thatre cyricean, of eorthlicum sellum, to than
heofomlicum timbre, gebaer /’ “the living stones of the
Church, from earthly seats, to the heavenly building, it
bore.” And again, (loc. cit.) “tha cyricean wundorlice
heola & rihte :” “the Church he wondrously held and
ruled.” The simple adjective “like” is, in the Anglo-
Saxon, lic, which also signifies “the body.” In Maeso-
Gothic leiks is “like ;” and leik is the body: whence
the Scottish word lyke-wake, corrupted into late-wake,
signifies “the watching of a dead body.” That the
name of the conception of “body” should be trans-
ferred to the conception of “likeness,” is not at all Adverbs.
surprising; for what is so like any person or thing, as
the very body of that thing, or of that person? Hence,
SHAKSPEARE, meaning to intimate that the use of the
Drama is to represent the exact likeness of living man-
ners, says, it is “to show the very age and body of the
time, its form, and pressure;” as if he had said, “the
Drama holds up a mirror to the present time, exhibits
its age of manhood or decrepitude, represents its very
body, the shape which it bears, and the impression
which it produces on the mind of the observer, as a
seal does on wax, or a statue on the plaster from
which a cast is to be taken.” Neither is it surprising
that the adjective “like” should enter into composition
with a great number of other adjectives; for if any
attribute could not be exactly predicated of a particular
substance, something like that attribute might be so; if
a person or thing could not be said to possess exactly a
certain quality, it might be said to possess a quality
similar, or nearly the same ; if it was not great it might
be greatlike ; if not good, goodlike, &c. Thus the pro-
nominal adjectives such, each, which, were formed from
compounds literally signifying so-like, one-like, and
what-like.
1. In the Maeso-Gothic swa is “so,” and swa leik is
“such.” In the Anglo-Saxon it is contracted to swylc,
in the Old English to swylke and swiche, and thence to
sich and such. And the same is found in the cognate
Languages: in the old Frankish and Alamannic, it is
solich, sulich ; in the Dutch, zulk ; in the Swedish, slyk;
and in the modern German, solehe.
In the Romance of Richard Coer de Lion, we have
Kyng Alysaundre ne Charlemayn
Hadde neuer swylke a route.
And Chaucer says,
In swiche a gise as I you tellen shal.
2. The words ilk and ilka are to be found in our old
writers, and still exist in the Scottish Dialect. Ilk was
sometimes written iliche, and has been abbreviated to
each. The following lines occur in a satirical Poem
entitled Syr Peni, or Narracio de Domino Denario :
(MS. Cotton. Galb. E. 9.)
Dukes, erles, and ilk barowne
To serue him er thai ful boune
Both biday and nyght.
In another part of the same Poem are these lines:
He may by both heuyn and hell
And ilka thing that es to sell
In erth has he swilk grace;
where we see swilk used for “such,” and ilka for
“every,” as it is by BURNs, in his Twa Dogs—
His honest sonsie, baws’nt face
Ay gat him friends in ilka place.
3. Which is, in the Anglo-Saxon, hwile ; in the
Maeso-Gothic, hweleiks ; from huvas, or hºwe, “whom,”
and leiks, “like.” In the Alamannic it is huwielich ; in
the Danish, huilk ; in the Dutch welke ; in the German,
welche. The word whilk, anciently written quhilk, was
common in Scotland to a late period, and perhaps still
exists in some remote parts of the Country. It is uni-
formly used in the Work of N1col, BURNE, before quoted:
as “I micht produce monie siclyk places, quhilk I never
hard zit cited be zou ;” that is, “I might produce many
such places, (of Scripture,) which I never heard yet cited
by you.”
G R A M M A. R.
77
4. Agreeing with these is the old English thilke, still
ing, than Pedants do, by the marrow rules with which Adverbs.
they are conversant, adhere to no such distinction. Thus \-N-"
Grammar.
v.se/~ retained in the Wiltshire Dialect, and pronounced thik,
Prefix a.
for “that.” Thus SPENSER, in his May, says,
Our blonket liveries been all too sad,
For thiſke same season, when all is yelad
In pleasance. 4
CHAUCER, in his translation of Boethius, says, “Cer-
es yet liveth in good point thilk precious honour of
mankind.”
And in the Poem on Christ's Descent into Hell are
these lines :
The smale fendes that bueth nout stronge
He shulen among men yonge
Thilke that nulleth ageyne hem stonde
Ichulle he habben hem in honde.
That is, “the small fiends that are not strong shall go
among mankind, and those persons who will not stand
against them, I am willing they should have in hand.”
Thus have we traced a substantive (signifying body)
through its transitions, first into an adjective proper,
(like,) thence as part of the compound adjectives proper
and pronominal, (lovelike and solike,) and, lastly, into
the termination (ly,) which we still use both in adjec-
tives and Adverbs, though with idiomatic differences in
respect to particular words, some being only considered
as belonging to the one class, and some to the other.
Thus, goodly, though not much used in the present day,
and rather as an Adverb than an adjective, is employed
by SHAKSPEARE in the latter character, through all its
degrees of comparison :
I. In Hamlet—
I saw him once, he was a goodly king.
2. In All's Well that Ends Well—
If he were honester he were much goodlier.
3. In King Henry VIII.-
She is the goodliest woman that ever lay by man.
So the word kindly is commonly considered to be an
Adverb, but BURNs uses it as an adjective, in Poor
Maillie's Elegy:
Thro' a' the toun she trotted by him;
A lang half-mile she cou’d descry him;
Wi’ kindly bleat, when she did spy him,
She ran wi' speed.
On the other hand, the word lonely is treated in the
English Dialect as an adjective ; but BURNS, in the
same Poem, employs it Adverbially:
Our bardie, lanely, keeps the Spence
. Sin’ Maillie's dead.
Godly, lovely, portly, and some other such words, are
employed exclusively, in modern times, as adjectives ;
but it is observable that godly has obtained by custom a
different meaning from the identical adjective godlike.
We have, too, some of these words in one form of com-
position, and not in its correspondent compound. Thus
we say ungainly for awkward; though the word gainly,
formerly in use, has become obsolete. Dr. HENRY
MoRE, a very learned writer of the XVIIth century,
says, “She laid her child, as gainly as she could, in
some fresh leaves and grass.” (Conj. Cabal.)
A mistake similar to that which we have noticed in
regard to the termination ly, also prevails with reference
to the prefix a, which is considered by some persons as
necessary to distinguish Adverbs from their adjectives,
as aloud from loud; but the Poets, who commonly
judge of Language more correctly, by a delicacy of feel-
MILTON describes the “civil suited morn”—
kercheft in a comely cloud
While rocking winds are piping loud—
not “loudly,” nor “aloud.” In fact, this prefix is of
different origin in different Adverbs, and is more or less
essential in modern use, according to the diversity of its
original signification.
1. It is corrupted from the Saxon participial prefix
ge or ye; as adrift, that is, driven.
2. It stands in the place of the prepositions in or on;
as alive, anciently written on lyve, i. e. in life, or in a
living state.
3. It was formerly expressed by the preposition of;
as anew, anciently written of new, as we now say of
late.
4. It is the positive article a as awhile, i. e. a time.
5. It is part of the pronominal adjective all ; as alone,
anciently written all one, i. e. absolutely one.
6. It is the French preposition à, as adieu, which,
however, is rather to be ranked among interjections.
7. It appears to be merely superfluous, as alike, an-
ciently written iliche, for like.
We shall consider the participles, substantives, &c.
hereafter; for the present, we mean to direct our atten-
tion only to those Adverbs with the prefix a which ap-
pear to be directly formed from adjectives proper, as,
aloud, from “ loud ;” anew, from “new ;” abroad, from
“broad.”
Aloud, anew, and abroad were anciently written “on Aloud.
loud,” “ of new,” and “on broad,” corresponding to
the expressions still current, “on high,” “ of late,” &c.
Thus, in the Poetical History of Sir William Wallace,
the Scottish author of which seems to have lived not
long after our great English Poet Chaucer, we read,
On loud he speir'd what art thou?
GAwiN Doug LAs, another Scottish Poet, in his spi-
rited translation of the AEmeid, which was completed in
1513, has these lines:
The battellis were adjonit now of new.
And again—
his baner quhite as floure
In sing of battell did on brede display.
It may be thought that the expressions “of new,”
“om broad,” “on loud,” and the like, are elliptical ;
and that a substantive is always understood, as “of new
beginning,” “on broad expanse,” &c.; but what we
mean by a substantive understood, is a conception pre-
sent to the Mind, though not expressed in Language.
Now, in the Adverbs “amew” and “abroad,” or, in
their equivalent phrases “ of new,” “on broad,” there
are no conceptions present to the Mind but those of
newness and breadth, except that of the connection be-
tween these conceptions and the verbs which they are
intended to modify. The words new and broad, there-
fore, notwithstanding their adjectival form, are rather
to be considered as substantives. They name the re-
spective conceptions, not as attributes of a fancied
“beginning” or “expanse,” but as general terms,
which may serve, with the aid of a preposition, to
indicate some circumstance or modification of the action
expressed in the verb. The Port-Royal, Grammarians
observe, that the greater part of Adverbs are only in-
tended to express, in one word, what could not other-
78
G R A M M A. R.
drifan, to drive. To be turned adrift, is to be put in Adverbs.
the state of a vessel driven about by the winds and S-V-
Grammar, wise be marked except by a preposition and a noun
~~~~ substantive, as sapienter for cum sapientid : hodie for
in hoc die; and this observation applies to the class
before us. To display a banner “broadly,” “on broad,”
or “abroad,” is to display it “in breadth;” to begin a
battle “newly,” “ of new,” or “anew,” is to begin it
“with newness,” compared with the former beginning.
And this force of the expression may frequently be
illustrated by comparing it with its converse, as “on
high” with “the earth,” “abroad” with “at home.”
Nor should we hesitate to explain thus even the plural
iyriatows in the angelic doxology (St. Luke, ch. ii. 14.)
86ta év i \riotous Geig, kal &ri Yās épìvn—“Glory to
God in the highest, and on Earth peace ;” for, as 86;a is
opposed to éipſium, so is du tyriatows to èTi ſãs ; and it
signifies “in the heights,” or rather in “the heights of
heights;” as in the 148th Psalm, “Praise Him ye
heavens of heavens.” Again, it may receive further
illustration from some equivalent modes of expression,
as, at large, written by Chaucer, at thi large:
Thou walkest now at Thebes at thi large.
LoNGLAND, in his celebrated Vision of Piers Plouhman,
written about 1350, instead of the Adverb alone, uses
the expression mine one :
And thus I went wide wher walking mine one.
{ *
A mode of expression not dissimilar to my 'lane, which
is still used with the same meaning in some parts of
Scotland.
waves, without a pilot or a helm. This conception of
driving, considered absolutely, forms the substantive
drift, which we apply Physically to the snow driven
along the ground by the wind, or to the sand driven
along the channel of a river by the stream. Intellectu-
ally, it is applied to the tendency of the arguments in a
train of reasoning: as in Shakspeare * -
What is the course and drift of your compact?
That is to say, whither do they drive? The word
adrift, therefore, may have originally been ydrived, as
Mr. Tooke seems to suppose ; or it may have been on
drift, that is, “in the state of driving ;” but in either
case, it presents the notion of a state or quality pro-
duced by action. Aghast seems to be aghasted, that is
affrighted, as one who has seen a ghost. It is from the
Anglo-Saxon gast, a ghost. Ago is the participle ygo,
gone ; as in Chaucer :
A clerke ther was of Oxenforde also
That unto logike hadde long yºo.
Asunder certainly bears some sort of reference to
the participle of the verb sondrian, which may also have
some connection with the substantive sond, the sand;
but it is also to be observed that, in many of the North-
ern Dialects, the general conception of separation, or
being apart from other things, is expressed by words of
this radical. In the Codea, Argenteus we have sun-
Askew. It may be doubted whether the words askew, askance, dro siponiam seinaim,” apart from his disciples”—
Askance and awry are taken immediately from adjectives or from bi the warth sundro, “when he was alone”—aftadia
Awry, participles. In respect to the first, Mr. Tooke seems to sundro, “he went apart.” In the Anglo-Saxon Sunder
have quoted Gower erroneously: spraec is “a private or separate conference.” Sundor land
And with that worde all sodenly is any separate and distinct tract of land possessing
She passeth as it were askie, peculiar privileges, (whence the modern name of Sun-
All cleane out of the ladies sight. derland,) sundor gyfe, a privilege, or peculiar grant.—
Askie is “into the sky,” tenues vanescit in auras, and So the Pharisees are called sundor halgan as affecting a
does not appear to bear any relation to askew, which singular and peculiar sanctity. Considering, therefore,
is connected with our word skewer, or as the vulgar, that this word sunder, or, as we express it, sundry, has
more consistently with its etymology, pronounce it, so distinct an adjectival force, it seems rather more pro-
skiver, an instrument used to “twist” and “wrest” bable that the word asunder was originally formed from
meat into a shape fit for the table ; from the Danish the adjective than from the participle, and was probably
skidºv, wry, crooked, oblique, and skiaver, to “twist,” expressed on sunder, from “sunder,” as on newe, from
to “wrest,” or force awry. Our word shy is probably “new.” -
of this origin. Shy is in German scheu, whence the Afret was the participle of the verb to fret, or to
verb scheuen ; in Frankish scuvan, in Dutch schuwen, Jreight, Thus in the Romant of the Rose—
in Italian schifare, in French esquiver, and in English For round environ her crounet -
to eschew, all having reference probably to the Greek Was fulle of rich stonis afret ;
>katēs, the “left hand,” inasmuch as the left hand has which may either mean in fret-work, or freighted,
always been the mark of interiority, that which was loaded with jewels. The former, viz. fretwork, seems
turned from, or eschewed : “ the sheep were on the to be taken from the act of gnawing or eating, as “a
right hand and the goats on the left. The word scaff moth fretting a garment,” whence “eating cares,” edaces
£eems to be of the same origin. Thus, in Kyng Ali- cura, are said to fret the mind : and Chaucer has
saunder, we find,
Alisaunder lookid ashoif The sow fYetting the chyld right in the cradil,
As he no gef nought thereof; There are two old German verbs fressen and fretten.
where it seems difficult to determine whether we should The former is our verb to fret, the Maeso-Gothic fretan,
understand, “Alexander look'd scoffingly,” or “Alex- Anglo-Saxon fraten, Dutch wreten, Frankish frezan, all
ander look’d askew.” signifying to eat or devour. The other verb is from the
Participles, 2. It is not only the adjective proper which serves old German word fret, a load or burden, restrained in
to modify other adjectives or verbs. The participle
performs the same office, and in the same manner.
An Adverb may be said to be derived from a participle,
when it expresses a quality or circumstance produced
by the action which the participle denotes. Thus
adrift is an Adverb, which may be said to be directly
or indirectly taken from the past participle of the verb
French to the lading of a ship, whence our substantive
freight. In a very rude specimen of the satirical talents
of the XIIIth century, (Harl. MSS. No. 2253, fol.
124. b.) the author, reviling the ribalds, or idle, disorderly
persons of his day, says,
The dueil huem afretye
Rau other aroste.
G R A M M A. R. 79
Grammar. Atwist is evidently from the past participle of the
S-N-2 verb twisan, which Mr. Tooke properly deduces from
twy, two ; but it is somewhat extraordinary that this
\ very instance should not have shown him the error of
his motion, that words in their Grammatical transitions
from one Part of speech to another, undergo no change
of signification. And it is the more remarkable, be-
cause WALLIs, of whose Grammar Tooke speaks with
some respect, has given three curious stanzas of his
own composing, on the word twist, with a view of
showing the variety of significations which may be
expressed by English words of similar origin:
When a twister, a twisting will twist him a twist,
For the twisting his twist he three twines doth intwist;
But if one of the twists of the twist doth untwist,
The twine that untwisteth, untwisteth the twist.
2.
Untwirling the twine that untwisteth between,
He twirls with his twister the two in a twine;
Then twice having twisted the twines of the twine,
He twitcheth the twine he had twined in twain.
The twain, that in twining before in the twine
As twins were intwisted, he now doth untwine,
'Twixt the twain intertwisting a twine more between
He twirling his twister makes a twist of the twine.
The proof that these words, alliterative as they are
in sound, and identical in origin, do nevertheless express
a great variety of conceptions, is very ingeniously given,
by exhibiting them in a Latin translation, in which the
same care is taken to avoid similitude of expression, as
in the former case to observe it.
1.
Quum restiarius aliquis, conficiendistorquendo funibus
jam occupatus, vult sibi funem tortilem contorquendo
conficere; quo hunc sibi tortilem funem torquendo con-
ficiat, tria contortu apta filamenta complicando invicem
associat; verüm si ea contortis illis in fune filamentis
wnum forté se explicando complicationi eximat ; hoc
ita se ea:plicando dissocians filamentum, funem torsione
factum detorquendo resolvit. .
2.
Ille autem celeriter evolvendo reteazens intermedium
illud quod se explicando dissociaverat filamentum, ver-
sorio suo torsionis instrumento, duo reliqua celeri volvems
turbine contorquet, funiculum ea binis filamentis inde
conficiens. Tum veró, quum jam secundá vice torquendo
convolverat funiculi bi-chordis bina filamenta; quem
er binis filamentis torquendo concinnaverat funiculum
raptim divellendo dirimit.
3.
Tandem, qua torquendo pridem in funiculo bimembri
jilamenta duo, tanquam gemellos, und consociaverat tor-
quendo, jam detorquendo dissociat; et binis illis filamen-
tum adhuc aliud intermedium interserendo comsocians,
versorium ille suum gyro celeri fortiter versando, ex funi-
culo bimembri plurimembrem torquendo conficit funem.
The participles hitherto mentiomed have the form of
past time; but we also, though less frequently, see
those which have the form of present time used in like
manner Adverbially; as “stark staring mad,” “roaring
drunk,” and, in Shakspeare, more elegantly, “lovirg
jealous.” -
I would have thee gone,
And yet no further than a wanton's bird,
Who lets it hop a little from her hand,
S. -
Like a poor prisoner in his twisted gyves, *
And with a silk thread pulls it back again,
So loving jealous of his liberty.
But in all these cases the specific notion of time
does not attach to the participle. When it becomes
an Adverb, it loses that property; because it either
modifies a verb, and then the time is expressed in the
verb itself, or it modifies an adjective, and then there is
no expression of time necessary.
3. The numeral pronouns supply a class of Adverbs,
which are not very abundant in any Language. Verbs
of action represent conceptions which may be often
repeated. If it be meant to limit the action to a single
instance, the conception of the number one must be
expressed, and so of any other number, and to this is
added, either expressly, or, at least, in the Mind, the
conception of time. Thus we say, “he marched sic
times through Spain;” “he conquered more than twenty
times in pitched battles;” “he was twelve times crowned
with laurel.” In most Languages, it is unnecessary to
express the conception of time in connection with the
lower numbers, the numerals themselves supplying an
inflection, by which that conception is perfectly under-
stood. Thus are produced our Adverbs once, twice,
thrice, which are no other than the old genitives, onis,
twyis, threyis. The Latin Language is more felicitous
in this respect; it has decies, vicies, centies, and millies to
express ten times, twenty times, a hundred times, and
a thousand times. -
In a Poem of the time of Henry VI. entitled, “How
the wyse man taght hys son,” (Harl. MSS. 1596.) is
the line
For and thy wyfe may onys aspye.
In Kyng Alisaunder,
Twyes is somer in that londe.
:: % º:
Ye haveth him twyes overcome.
With respect to the Adverb, once, however, it is to be
noted, that as one is not always opposed to two or three,
or any specific number, but sometimes merely to many;
so once does not always signify “at one time,” as op-
posed to two, three, or any other number of times, but
merely “ at some time” different from the present.
Thus, when WoRDsworth says of Venice,
Once did she hold the gorgeous East in fee,
he means to contrast the greatness of a former time
with the degradation of the present. As if he had
said, although at this present time she lies so low, there
was one other period, at least, in her History, which
presented a far different picture. At that time she
was rich and great, famous and powerful—
— Now lies she there,
And none so poor to do her reverence.
Nor is this signification confined to the time past.
Once equally means some uncertain time as applied to
the future. Thus, in the Merry Wives of Windsor—
I pray thee, once to night, give my sweet Nan this ring.
Nearly the same effect is given in Latin to the Adverb
olim, which means some one point of time, either past
or future; and seems to have the same connection with
the relative article, as our word once has with the posi-
tive; for olim appears to be derived from olle, which the
early Romans used for ille, and which, in the plural,
was written oloe, as in the Royal Law: Si parentis puer
verberit, ast ODoE plorassint.
Adverbs.
\-N-7
Numerals.
80 *
G R A M M A. R.
Grammar.
The numerals hitherto spoken of are those called
"…—N- cardinal; but the ordinals also supply a certain class
Demon-
strative
pronouns.
of Adverbs, as thirdly, fourthly, fifthly, &c. which are
formed from the adjectives third, fourth, fifth, &c. by
adding the termination ly, before explained. In the
Latin Language, the correspondent words tertid, quartô,
&c. are manifestly the adjectives tertius, quartus, &c.
with the termination of the ablative case. In English,
too, we use the adjective first, Adverbially, without any
alteration. It is a circumstance worthy of note in the
History of Language, that the first two of the ordinal
numbers generally appear not to be taken from the
names of the cardinal numbers; thus we do not say in
English the oneth, the twoth, nor in Latin unitus, duitus,
nor in Greek évotós, òvitös ; but in these Languages
respectively, first, second, primus, secundus, ºrpiºtos,
ëe0tepos : and when we look to the etymology of these
words, we shall be inclined to suspect that they are
in their origin simpler, and therefore, perhaps, earlier
than the adjectives taken from the ordinal numbers.
The word first is manifestly the superlative of fore, the
first, being, of course, the for-est, or that which is before
all others. Of this word, however, we shall have oc-
casion to speak more at length when we come to con-
sider the preposition and conjunction for. The Latin
primus is in like manner the superlative of the old
word pri. Scaliger, speaking of the word primus, says,
superlativum est; nam PRI vetus vor fuit, sicut NI :
postea latione vocali fusāe sunt NE, PRAE, unde Adver-
bium, PRIDEM ; comparativum, PRIUS ; superlativum,
PRIMUM ; nam ab Adverbio, pridem, primum qui ducunt,
errant. And elsewhere, ea DI factum est DE; sicut
ea. PRI, PRAE ; et sicut ex N1, NE. Vossius observes,
that prae was connected with an old adjective praes,
present, that is, before the persons assembled ; for
when the names were called over at the public meetings,
each individual answered praes. The Greek Tptºros is
in like manner the superlative of mpſ), which is found
in various shapes, but most simply in the preposition
trpo, answering exactly to the Latin prae, before, either
with regard to time or place, and secondarily as to order,
or what we call preference. The word Tp(S, indeed,
is used for the first dawn of day; but this appears to
be merely a contraction from 7pwº, which, however, is
undoubtedly connected with trpo ; nor can there be
much doubt that the three radicals to which we have
alluded, viz. pri, pro, and for, have all one common
origin.
The demonstrative pronouns, with which we rank the
subjunctive, form, in most Languages, a large class of
Adverbs, the construction of which is elliptical. The
words here and there, hence and thence, hic and illic,
hinc and illine, for instance, are manifestly in their
origin demonstrative pronouns, equivalent to the words
this and that ; but, by use, they have come to signify
“at this place,” “at that place;” “from this place,”
“from that place;” the substantive “place” being
clearly understood by the Mind. Neither can it be
doubted that the Latin Adverbs qu'um and quo are the
subjunctive pronoun qui, with the terminations of the
accusative and ablative case; which word qui is pro-
bably the same in origin with the Gothic huo, the
Saxon hwa, the Scottish quha, and the English who.
It happens, that the English Language is not per-
fectly systematic in regard to the pronouns which it
has adopted for Adverbial purposes; and the same may
be said of most other Languages. We have the simple
Adverbs just mentioned, which form three distinct
classes, with reference to place, distinguishing the
place where we are, from another definite place, and
supplying an interrogative for the place which we know
not, which interrogative is also a subjunctive. ...
The first of these is here, the second there, and the
third where. It happens too, with regard to place, that
each of these three forms has three varieties to express
“at a place,” “to a place,” and “from a place;” and
all these are variously compounded with several other
words or particles, fore, ever, soever, &c. Some of the
words which form Adverbs of place, also become Adverbs
of time, manner, cause, &c.; but these latter ideas have
some few Adverbs which are peculiar to themselves,
agreeing, nevertheless, in principle and derivation, with
the Adverbs of place. Hence may be formed the follow-
ing Table of the simple Adverbs of this kind:
here . . . . . there . . . . . . . . . . . . where?
Place & hence . . . . . . • thence . . . . . . . . . . whence?
hither . . . . . . . thither . . . . . . . . . . . whither?
Time . . . . . . . . . . . . . . . then . . . . . . . . . . . . when 2
Manner . . . . . . . . . . . . thus . . . . . . . . . . . . . how P
Cause © e º ºs e s e º O e s s e º s v tº e º e º a s tº e º o º why?
these Adverbs, have not always been thus accurately
distinguished. In our old Language, we shall find the
prepositive forms here and there often interchanged
with the subjunctive or interrogative form where; yet
it is clearly evident that these distinctions must have
always existed in point of signification, however inac-
curately or imperfectly expressed.
The word here is not only used in its simple form,
but in a variety of compounds, as, hereafter, hereabout,
hereat, hereby, herein, hereinto, hereof, hereon, hereupon,
hereto, hereunto, heretofore, herewith, heirfoir, heirintill,
&c. In the simple form it is principally confined to the
signification of “this place;” whereas, in the compounds
it generally signifies “this time,” “this thing,” “this
event,” or the like. The cognate word hier, in German,
does not follow exactly the same variations of meaning.
Both in its simple and compound forms it principally re-
fers to place, as hieran, hieraus, hierdurch, hierein, hier-
inmen, hierober, hierunter, &c.; and so, heran, herbey,
herein, &c.; though some compounds are more general
in their application, as, hierum, hiervon, hierzu. In both
Languages, however, it is manifest that the word here,
hier, for her, intrinsically signifies no more than the
word this ; and that the other significations, such as
“place,” “time,” “event,” “reason,” or the like, are sup-
plied by the Mind, according to the context. The words
heirfoir and heirintill, being of the old Scottish Dialect,
now obsolete, it may be proper to explain them by some
instances. In the Scottish Act of Parliament, A. D. 1493,
the King (James IV.) recites the inconveniences of
alienating the Royal domains, thus, “Sen it is leuit
and permittit be the constitutiounis and ordinancis of
lawis ciuile and canon, that persounis constitute in
youtheid and tender age quhilks argreitlie dampnageit
and skaithit in thair heritage be imprudent alienatiounis,
&c. may at thair perfectioun of age mak reuocatioun,
&c.; Heirfoir, we, James, be the grace of God, King of
Scottis, &c. reuoks, reducis, cassis, and annuls, all
infeftments,” &c. In this example, the word heirfoir is
simply “for this,” the word “cause” or “reason”
being understood, Again, in the Act of 1554, “like
Adverbs.
\-N-2
Here,
x G R A M M A. R.
81
'Grammar, as and all the hieast and maist wailyeable thing is of the
S-N-" premissis had bene expressit heirintill,” where the word
heirintill signifies “ in this,” or “ within this,” viz.
“writing,” or “statute.” We cite the words of these
Acts from the careful copies of the original documents
lately printed by order of Government, which present a
very valuable record of the state of the Scottish Lam-
guage from 1424 to 1592; and in which collection we
also find many other Adverbial compounds of the word
heir, as heirof, heirupone, heirtofoir, heirafter, heiranent,
&c. in all which, heir signifies this, although, in some
instances, it is applied exclusively to place ; in others,
to time ; and, in a third class, to this time, this place,
this thing, &c. indifferently. &
The pronouns are among the simplest, and probably
the most ancient words in all Languages; and hence we
p
must not be surprised to find some difficulty in tracing
the pronominal Adverbs to their proper origin. How-
ever, it can hardly be doubted that the elements of the
word here are to be discovered in he and er, which
occur in many of the Northern Languages, as signifying
this person or these persons, this thing or these things,
so that the radical conception is what we express by the
word this. First, the element he occurs, in Anglo-
Saxon and old English, in the words signifying he, she,
they, and their respective cases. The Anglo-Saxon
pronoun personal is he, heo, hi, he, she, they ; and
the very word here occurs for the genitive plural, as
heom does for them. The same, or similar words are
frequent in old English writers. In the Vision of Piers
Plouhman—
Hermets on a heape with hoked staues
Wenten to Walsingham, and her wenches after.
3: 2é 3% 2}: % tº: -
Cokes and her knaues cryden, hote pyes, hote:
That is, “ their wenches,” and “ their knaves,” or
“boys.”
In Chaucer's Parson's tale, “ Certes this vertue
makith folk vndertake hard and greuous things by her
own will ;” that is, “ their own.” In an ancient Ballad,
probably of the XIIIth century, beginning “In Mayhit
muryeth,” (Harl. MSS. 2253. fo. 71.)—
Ynot non so freoh flour
Ase ledies that beth bryht in bour,
With loue who mihte hem bynde :
That is, “I know no flower so fresh as ladies who are
bright in bowers, to those who may bind them with
love.” In a dialogue between a body and a spirit, of
the same date, (Ibid. fo. 57.) “ he wolleth” occurs for
“ they will.” This word was sometimes written heo, as,
in a satirical Poem against the Ecclesiastical lawyers,
(Ibid. fo. 71.)—
Heo shulen in helle on an hok
Honge there fore.
And sometimes hi, as in another manuscript in the
Harleian collection, (No. 2277, fo. 195.)—
Tho hi dude here pelrynage in holiestedes faste, -
So that among the Sarazyns ynome hi were atte laste:
That is, “they did their pilgrimage, so that they were
taken at last.” -
In the Lai le frain, which is a translation from the
Norman-French of the celebrated Poetess Marie, we
have he and hye, for “she ;” and him for “her:”
The maiden abode no lengore,
Pot yede hir to the chirche dore;
WOL. I.
ER swo sem ungar meyar eru.
O Lord, he seyd, Jesu
*: *i; × -
Hye loked vp, and by bir seighe
An asche, by hir, fair and heighe.
:: :: 2: 2:
A litel maiden childe ich founde,
In the holwe assche therout,
And a pel him about.
Crist, &c.
2: -
The other element, er, is found in the modern German
er, he, and in the Islandic er, am, is, and who ; as in
the Edda of Snorro, Feyma heiter su kona ER offam.
“Feyma is called the
woman who modest is, as the young maidens are.” In
the Frankish and Alamannic the demonstrative and re-
lative pronouns of the third person are er, her, and ir.
Thus, in the Frankish of OTFRID the Monk, ER gibot
then uuinton, “ He commanded the winds :” in that of
TATIAN, ER quam in sin eigan, “ He came to his
own.” In the Alamannic of Isidore, Dhaz IR. Jhesus
wuardh chimennt, “That he Jesus was named.” These
two elements, then, viz. he and er, are identical in sig-
nification ; and are only redoubled for the sake of
emphasis, which is a habit common to Barbarous na-
tions, and to the illiterate in all Countries. Hence it is,
that the French have their ce-ci and ce-la, and even
ce-lui-ci and ce-lui-la ; and that our own rustics com-
monly say this here, that there, thick there, &c. From
this source undoubtedly come the Gothic, Anglo-Saxon,
Danish, and Islandic her, the Frankish and Alamannic
hier, hiar, hiera, the modern German and Dutch hier,
and the English here, all used to signify, “ at this
place,” although the simple and radical meaning of
them all is simply “this.” The various explanations
which are given to the Adverb here by Dr. Johnson
only serve to show that the conception of a distinct
and particular place is no necessary constituent in the
meaning of the word. Thus here is opposed to a future
time, as well as to a different place, by BACON, in his
advice to Williers: “ you shall be happy here and more
happy hereafter;” which might be paraphrased “ in
this life and in a life after this”—“ in this world, and in
Adverbs.
\-N-f
a world after this”—“ in this state of existence, and in .
a state of existence after this,” always retaining, how-
ever, the conception expressed by the word this. So
when the words here and there are explained by Johnson
“ dispersedly; in one place and another;” as in another
extract from Bacon : “I would have in the heath some
thickets made only of sweet-brier, and honey-suckle,
and some wild vine amongst; and the ground set with
violets; for these are sweet, and prosper in the shade;
and these to be in the heath here and there, not in
order.” The words here and there are still to be ex-
plained this and that ; for the Imagination forms con-
ceptions of places separate from each other, although
quite indeterminately as to any specific external situa-
tion, and even as to number, except that the place
signified by the word here is an Imagination separate
from that expressed by the word there. The indistinct
process of the Imagination, therefore, in the passage
above cited, may be explained by supposing an indi-
vidual carelessly wandering over the ground which is
to be ornamented, and occasionally stopping to say, I
will have a thicket planted in this place and another in
that place. The same expression occurs in a beautiful
Sonnet by Shakspeare—
Alas! 'tis true, I have gone here and there;
M
82 G R A M M A. R.
Grammar, which corresponds with the expression “ranged,” in
\-y– the preceding verses-
- As easie might I from my selfe depart,
all the kinge's liegis be vnharmyt & vnscaithit of the Adverbs.
said house & of thaim that inhabits theirin fra hyn S-N-2
furth.” - - -
As from my soule, which in thy brest doth lye:
That is my home of loue. If I haue rang’d,
Like him that trauels, I returne againe.
Here and there are doubtless used indefinitely in such
phrases; but not more indefinitely than the pronouns
this and that might themselves be used, as in the Song,
This way, or that way, or which way you will;
and in DRAYTON's pleasing description of a winter even-
ing's chat with his friend—
Now talk of this, and then discours'd of that,
Spoke our own verses twixt ourselves, &c.
Nay, even the pronoun personal is sometimes used
with the same uncertainty of application; as in Chaucer's
spirited description of a tournament, in the Knyght's
Tale—
He rolleth under foote, as dothe a ball,
He foyneth on his feet with a tronchoun,
And he hurleth with his horse adoun,
He through the body is hurt and sith ytake.
In none of which instances is there any certain ante-
cedent to the word he , and yet it stands first for one
man, then for another, then for a third, and lastly for a
fourth.
Hence and hither may be considered as cases of the
word here; but perhaps it would be more accurate to
treat these three words as different compounds of the
element he, with er, am, and der. Hence is the Anglo-
Saxon hedman, and the Frankish hina. It seems to be
connected with the Islandic ham, he, and hin, it; and
with the syllabic him, which, in various German com-
pounds, signifies “from this place,” “from this time,”
“ at this time,” “ to that place,” &c.; and which is
used alone to signify any thing that is “gone hence;”
“ lost,” or “ annihilated ;” as in the Leonore of
BöRGER—
O mutter, mutter, hin ist hin!
Werlohren ist verlohren :
So they say er ist him for “he is dead:” hinrichten is
to execute justice on any one, to put him to death ;
hindag is “ this day;” hinfort, “ henceforth,” “from
this time forth ;” which is also expressed forthin. Im-
?merhim is an exclamation answering to our “let it go,”
and meaning “be it ever thus, I care not ;” as, er mag
immerhim schreyen, “he may bawl as long as he likes.”
So himauf and hinab, “ above and below ;” himein
and hinaus, “within and without,” mean respectively
above this place, below this place, within this place,
out of this place. Hinfahren is to go away, to go from
this place; and, in the Frankish, hinafahrt is “death.”
Our English word hence, in old writings, is hen, han,
him, and hennes. In the Romance of The Sewyn Sages,
we find, *
A fend he is, in kinde of man;
Binde him, sire, and lede han.
Chaucer, in the Knyght's Tale, says,
The fires whiche on min auter brenne
Shal declaren er that thou go henne
This auenture of loue.
So in Christ's Descent into Hell—
Bring vs of this lothe lond
Louerd henne into thyn hond.
In the Scottish Act of Parliament, A. D. 1438, “ that
Hither is the Anglo-Saxon and Gothic hidre. In the
old English too it was often written with a d, as in
Chaucer's Monk's Tale— . . .
And if you list to herken kiderward.
So in two manuscript Poems in the British Museum,
(Harl. MSS. 2253, fo. 64, and fo. 124.)—
Herketh hideward, and beoth stille.
º 2: * 26
Herkneth hideward horsmen
A tidyng ichou telle. - -
And, in the Poem on Christ's Descent into Hell, Satan
SayS,
Ne may non me worse do,
Then ich haue had hiderto. g
There, thence, thither, are manifestly constructed on There.
the same principles, and applied in the same manner
as here, hence, and hither; and as we suppose the first
element of here to be he, so we suppose the first element
of there to be the, which, in the Anglo-Saxon, was pre-
fixed as an article to substantives in all cases, and in
both numbers; and which appears in various Dialects
under the forms of thei, thy, tho, tha, all relating to
the pronoun that. Thei is the Gothic conjunction
“ that.” Thy in the old English compound forthy,
signifies “for that,” viz. cause. Tho is explained by
Junius, qui, illi, and tunc, viz. “ that person,” in the
plural ; and “ that place” used Adverbially ; and he
adds, that the Anglo-Saxon tha admits all these signifi-
cations.
Tho, for “then,” (see Warton, vol. i. p. 161.)—
The messengers tho home went.
Tho, for “when,” (Harl. MSS. 2253. fol. 37)—
Tho Jhesu was to hell ygan.
Tha, for “those,” (The Seuyn Sages, v. 3901)—
Al tha wordes ful well he knew,
He was so ferd him changed hew.
Thae, for “ those.” See the second volume of The An-
tiquary, (one of the recent Novels which so accurately
delineate the manners and Language of Scotland,) p.
297–
Thae's your landward and burrowstown notions.
Tho, for “those,” (Harl. MSS. 2253. fo. 55, 56.)—
Parmafey ich hold myne
All tho that bueth her ynne.
There seems to be compounded of the and er; as
here, of he and er ; but however this may be, there
manifestly agrees with the German der, which is a
demonstrative and relative pronoun, as well as an
article, and consequently answers to our the, this, and
who. In like manner, the Anglo-Saxon thatre or thatr
formed the genitive of the article, and also the de-
monstrative and relative adverb; as in the 4th chapter
of Joshua, “ Nyman twelf stanas on middan thatre ea,
thar tha Sacerdas stodon, & habban forth mid edw, to
eowre wicstowe, & wurpan hig thatr.” “Take twelve
stones from (the) midst (of) the water, where the priests
stood; and have (them) forth with you, to your abiding-
place, and cast them (down) there;” in which passage
we see thatre and thar, answering to the, where, and
there successively. So in the old English, there is
G R A M M A. R.
83
Grammar, often used in two connected sentences, for there and
conjunctively, occurs in a rude old English Poem before Adverbs.
quoted, (Harl. MSS. 2253. fo. 71.)— \-y-' -
J.-- where; as in Chaucer's Wife of Bath's Tale—
There as wont to walken was an elfe,
There walketh now the limitour himself.
It might not unreasonably be surmised, that where
the operations of the Mind are so distinct, as those in-
dicated by a demonstrative and a subjunctive pronoun
or Adverb are, they would necessarily require ex-
pressions equally different; but a careful attention to
the History of Language will show us that it differs very
widely in this respect from its Philosophy. It is for
want of having sufficiently considered this circumstance
that we find Grammarians so often at a loss to account
for different idioms, and giving reasons for them which
are purely imaginary, not to say absurd. It is, no
doubt, a great excellence in a Language, to mark, by
distinct expressions, the distinct operations of the Mind,
and the more nicely this is done, the more accurate and
expressive does a Language become; but this is generally
the result of time and of an undefinable sense of in-
convenience, which induces men to inflect and vary
words, as it were, insensibly, and to assign to the
various inflections, though of similar origin, different
effects. In no Language, however, has this Principle
been carried into full operation; and hence we see the
different meanings of a word, and the different Parts
of speech which it constitutes, passing into each other
by gradations, which, at first sight, it is not always
easy to explain. Thus, in Greek, the subjunctive pro-
noun, or, as some call it, the subjunctive article, 8s, is
sometimes said to be used for the prepositive à ; some-
times for tt's interrogatively; and sometimes for dvºrds.
, Again, "Oates sometimes answers to the Latin relative
quis, and sometimes to quisquis. The Adverb "Otrov,
besides the common signification “where,” answers to
“whither;” and in argument, to “since;” and in de-
scription, to “in this place,” or “in that place.” So,
8te, “when,” signifies also “since,” like the Latin cum :
and the examples of this kind are infinite. We shall
not, therefore, be surprised to find considerable diversity
from the modern idiom in the following, and many
similar instances:
Ther is used for the, that or them ; as, in The Seuyn
Sages, therewhile for the while :
Therwhile, sire, that I tolde this tale,
Thi some mighte tholie dethes bale. ,
GAwIN Douglas has “thare above”
that ;” and “tharon” for “ on them.”
In the old Scottish Dialect thir was used for these,
or them ; as in the Act of 1424, “thir ar taxis ordaynt
throu the counsaile of Parliament.” So in DUNBAR’s
Goldin Terge, written about a century afterward—
Full lustiely thir ladyis all in feir
Enterit into this park of maist pleseir.
× -k sk *k *
And every ane of thir in grene arrayt
And harp and lute full mirreyly they playt.
In the same Dialect we find thairto and thairfra,
thairfoir and thairefter, tharapone, thairuntill, &c.
Chaucer uses therto in the sense of “ moreover,” or
“in addition to that,” as in the Rime of Sir Thopas—
He couthe hunt at the wildé dere
And ride an hauking forby the riuere
With grey goshauke on honde:
Therto he was a good archere.
for “above
Therefore, which, in modern times, is commonly used
Heo shulen in helle on an hok
Honge there fore.
In short, comparing the different authorities, ancient
and modern, we find that the word there, however va-
riously spelled, did not originally relate to place exclu-
sively, but was equally applied to time, to persons, and
to events: and the same may be said of thence and
thither. Thenceforth, which we use with reference to
time, agrees with the old Scottish phrase fra thin furth,
as in the following passage in the Act of 1503, which
is, on many accounts, worthy of notice :
“It is statute and ordanit that fra thin furth na baroun, fre-
haldar, nor vassal, quhilkis ar within ane hundreth merks of this
extent that now is, be compellit to cum personaly to the parliament,
bot gif it be that our souerane Lord write speciale for thame. And
sa (sal) no be unlawit for thair persons, and thai send thair pro-
curatours to answer for thane, with the baronis of the schire, or the
maist famous personis. And all that ar aboue the extent of ane
hundreth merks to cum to the parliament, vnder the pane of the
auld vnlaw.”
Thither was, in the Anglo-Saxon and old English,
thider, as in the Poem often quoted, (Harl. MSS. 2253.
fo. 55.)—
God for his moder loue,
Let us neuer thider come.
And as they had hideward for “hitherward,” or “toward
this place,” so they had theſlerwart for “thitherward,”
or “toward that place :” as in the ludicrous Poem
called The Huntyng of the Hare:
Thei toke no hede thealerwart,
But euery dogge on Oder start.
Where, whence, and whither.—These words have also Where,
a similar analogy, together with this further peculiarity,
that they serve indifferently for interrogatives and sub-
junctives. Thus in the interrogative : -
They continually say unto me, where is thy God?—Psal. xlii. 3.
And he said, Hagar, Sarai’s maid, whence camest thou; and
whither wilt thou go?—Gen. xvi. 8.
And again in the subjunctive—
Let no man know where ye be.—Jer. xxxvi. 19.
I wist not whence they were-Josh. ii. 4.
He went out, not knowing whither he went.—Heb. xi. 8.
We have already seen that the subjunctive force of the
word where was not peculiar to it, but was sometimes
expressed by the word there. We do not find this to be
the case in English with the interrogative force of the
same word ; but in Greek the relative pronoun T's is also
an interrogative ; as in St. Mark's Gospel, ch. ii. ver. 6, 7 :
"Ha'av 8é TINEX tſºv ºpappatétov ćice? ka0;juevo, ka?
ôtaxonºuevot ev Taºs kapātavs divtſºv: TI' obros of Tw
AuX6, BAao'ºmpilas; TIX 65uatat dºuévat àuaprčas, et aſ
cis 6 €eds;–“ But there were certain of the Scribes
sitting there, and reasoning in their hearts, why doth
this man thus speak blasphemies 2 Who can forgive sins,
but God only ”—Hence it is clear, that the interroga-
tive effect of a word does not require a peculiar form, any
more than the subjunctive. So the Latin quidam, which
means “a certain person,” and aliquis, which means
“some one,” are reciprocally connected with the inter-
rogative quis, and the subjunctive qui. SCALIGER was
of opinion that the Latin quis and qui were the Greek
kač 6s and cat 3: and TookE, probably thinking to im-
prove on this etymology, has only gone further in error.
He says, “As ut (originally written uti) is nothing but
ôté; Sois quod (anciently written quodde) merely cat 67t.—
M 2
84
G R A M M A. R.
Grammar.
S-/-/
Quodde, tuas laudes culpas nil proficis hilum. -
- - LUCILIUs.
“Qu in Latin being sounded not as the English, but
as the French pronounce qu, that is, as the Greek K;
kai, by a change of the character, not of the sound,
became the Latin que, used only enclitically indeed in
modern Latin. Hence kal 6th became in Latin qu'otti,
quoddi, quodde, quod.”—The only foundation for all
these conjectures seems to be, that in the very nature of
a subjunctive pronoun something equivalent to a con-
junction is implied; and as to the assertions respecting
the Roman pronunciation they are perfectly gratuitous.
It is not very probable that the ancient pronunciation
of qu was the same as of K ; on the contrary, it more
probably resembled that of x, or rather of the Gothic
G), which our Anglo-Saxon ancestors expressed by hw,
the old Scottish writers by quh, and we by wh. Sca-
liger and Tooke forgot, that if their explanation might
be thought to account for the subjunctive pronoun, or
conjunction, it left the interrogative pronouns and Ad-
verbs quite unexplained; and the fact seems to be, that
the Latin Language originally agreed with the Gothic
and other Northern Languages in employing the arti-
culation marked by the AEolic digamma, where the
softer Greek Dialects omitted that articulation; thus the
Greek 6vos was the Latin vinum and Gothic weim ;
the Greek 6 was the Latin va, and Gothic wai; and
lastly, the Greek aspirated pronouns j, 6, were the Latin
qua, quo, and the Gothic hua, hwo. -
It is manifest that where did not originally refer to
place alone, any more than here or there did ; but, like
those words, was originally a pronoun signifying this or
that ; for in its composite forms it often signifies no
more than those pronouns, the substantive to which it
refers being usually expressed, but sometimes under-
stood. Thus we have whereabout, for “about which
business”—
Let no man know any thing of the business whereabout I send
thee.—l Sam. xxi, 2.
Whereto, for “to which thing”—
It shall prosper in the thing whereto I sent it.—Isaiah lv. 11.
Jºhereby, for “by which name”—
There is none other name under heaven given among men
whereby we must be saved.—Acts iv. 12.
Mherefore, for “for which cause”—
What is the cause wherefore ye are come 2—Acts x, 21.
All these compounds may be employed interrogatively,
(and indeed the subjunctive use of some of them has
at present become rather obsolete,) but in this form also
they are not necessarily significant of “ place.”—Thus
whereby is used for “by what means?”—
Zacharias said unto the angel, whereby shall I know this 2—Luke
i. 18.
Wherefore, for “for what reason?”—
Now he is dead wherefore should I fast?—2 Sam. xii. 23.
It is to be observed, however, that there are certain
Adverbs compounded with where, which cannot be used
interrogatively, such as whereas, wherever, wheresoever ;
but the reason is that in these, as well as in whensoever,
whithersoever, &c. the pronouns as and so, and the
word ever, necessarily give them a relative force and
effect :
Have ye not spoken a lying divination, whereas ye say, The Lord
saith it *—Ezek. xiii. 7.
Ye have the poor with you always: and whensoever ye will ye
may do them good.—Mark xiv. 7.
The Lord preserved David whithersoever he went.—2 Sam. viii.6.
It would be impossible to express these passages in-
terrogatively, “whereas say ye?” “whensoever will
ye?” “whithersoever did he go?” not on account of the
meaning of the words “where,” “when,” or “whi-
ther,” but of the others with which they are com-
pounded. -
In these compounds, the particles or words as and
so seem to have been originally used superfluously, as
the particle or word that was in many similar combina-
tions. Hence, on the one hand, we have where for
whereas ; and on the other, we have where and that for
where: and, in like manner, we find many such ex-
pressions as how that, which that, &c. Where for
whereas, occurs in the preambles of many old Statutes.
In a remarkable document existing among the Rolls of
Parliament, A. D. 1461, we find it so used. The do-
cument to which we refer is called Cedula formam actits
in se continens, and was exhibited in the first Parlia-
ment summoned by King Edward IV. After reciting
many alleged crimes, on the part of Henry VI. and his
followers, it contains a judgment, or law of attainder,
against the latter, and of forfeiture of the Duchy of
Lancaster against Henry. Of the recitals, some are
introduced by the word forasmoch, and others by the
word where : thus, “Forasmoch as Henry Duc of So-
mersett purpossing ymaginyng & compassing, of ex-
treme & insatiate malice & violence to distroy the
Right Noble and famous Prynce of wurthy memorie
Richard late Duc of Yorke, Fader to our Liege & So-
verayne Lord Kyng Edward the fourth, & in his lyf
very King, in right, of the Reame of England, &c.
and also Thomas Courteney late Erle of Devonshire,
&c. &c. (naming various persons) with grete despite &
cruell violence horrible & unmanly Tyrannye murdred
the seid right noble Prynce Duc of York; and where
also Henry Duc of Exceslre, Henry Duc of Somersett,
&c. &c. (naming the same and other persons) rered
warre ayenst the same King Edward thir right wise
true & naturall liege Lord, &c. It be declared and
adjudged by the assent & advis of the Lordes Spiri-
tuelz & Temporelz & Commyns,” &c. &c. In the
more ancient Parliamentary records, which were in
French or Latin, preambles of this kind were intro-
duced by the old French word come, or by the Latin
cum, both which words are the ancient quom from qui,
who. -
Where that, in CHAUCER's Knyght's Tale (see Harl.
MSS. 7335.)—
Duk Theseus him leet out of prison
Fferly to goon wheer that him list al;
and in DUNBAR's Goldin Terge—
Full lustily thir ladies, all in feir,
Enterit into this park of maist pleseir,
Quhair that I lay heilit with leivs rank.
:: *: : 2k &
Then crap I throw the brenches & drew neir
Qhuair that I was richt suddenly affrayit.
How that (Harl. MSS. 7333. fol. 147. b.)—
How that the foule fende asseylithe the soule.
Which that (Harl. MSS. 7333, fol. 203.)—
Mvsing vpon the restlees besinesse
Which that this troubly worlde hath ay in honde.
Adverbs.
\-y->
G R A M M A R. 85
From what has been said, it is abundantly clear that Adverbs.
the Adverbs here, there, where, hence, thence, whence, *N-”
Grammar. So is, in like manner, compounded with where, who,
\-y-' what ; as in the English whereso and whoso, and the
Scottish quha sa, which mean respectively “whereso-
ever” and “whosoever:”
1. And redde wherso thou be, or ellis songe.
CHAUCER. Troilus.
2. He inclos’d
Knowledge of good and evil in this tree,
That whoso eats thereof forthwith attains
Wisdom. MILTON. Par. Lost.
3. It is ordanyt, that all men busk thane to be archaris fra thai
be xii yeris of eilde. And quha sa vsis not the said archary the
lorde of the lande sal raiss of him a wedder.
- Scottish Act of Parl. 1424.
Mor is it extraordinary.that the words that, so, and
as should be used in a similar manner; for, as Mr.
Tooke has justly observed, “as is an article, and
means the same as it, that, or which.” And again,
“ the German so, and the English so, though in one
Language it is called an Adverb, and in the other an
article, or a pronoun, are yet both of them derived
from the Gothic article so or sa, and have, in both
Languages, retained the original meaning, viz. it, that.”
But on these words we shall presently have occasion to
make some further remarks. - -
Where is also used with the pronominal adjectives
any, every, mo, but still adverbially, as in the common
expressions anywhere, everywhere, nowhere ; and being
thus limited to some determinate signification in respect
of place, it is neither subjunctive nor interrogative :
Those subterraneous waters were universal, as a dissolution of the
exterior earth could not be made anywhere but it would fall into
waters. BURNET. Theory of the Earth.
'Tis nowhere to be found, or everywhere. Pope.
In the old English it was even used with a simple
adjective, as wide-wher.
And thus I went wide-wher walking mine one.
LoNGLAND. Piers Plou.
Whence is sometimes found, in the old English, un-
necessarily cumulated, as it were, with thence ; nor is
this any thing more than we have already observed to
be common in the formation of pronouns and prono-
minal Adverbs in all Languages, as ce and ceci in
French, ita and itaque in Latin, &c.
Thus, in the Romance of Syr Ypotis (see Warton,
vol. i. p. 208)—
The emperour, with milde chere,
Askede him whethence he come were.
And the same may be observed of theqence in the Ro-
mance of Alisaunder (see Warton, vol. i. p. 309)—
Thedince so ondrace with his ost.
In the West of England, to this day, we find that the
country people use for hence and thence, the words
hereance and thereance, which are manifestly similar and
unnecessary cumulations of expression.
Whither is confounded with ward in our old writers
as well as hither and thither; but though the latter two
are noticed by Johnson, the first is not so :
A puissant and mighty pow'r
Of gallow-glasses and stout kernes
Is marching hitherward in proud array.
SHAKSPEARE.
By quick instinctive motion, up I sprung
As thitherward endeavouring. MILTON. Par. Lost."
Who so wolde myghte ride
Whiderwardis so they wold.
Hen, P.I. -
Fomance of K. Alisaunder,
and every female bird höna, from hon, she.”
hither, thither, and whither, although in their modern
and uncompounded use they principally express a con-
ception of “place,” yet did not really include the name
of any such conception in their original signification,
but were the mere pronouns he, this, and what, di-
versely compounded, and assigned by use to separate
and distinct significations.
The very same is to be observed of the Adverbs Then
and When, which we have above noted, as principally
signifying time. We have not, indeed, the word Hen
for “at this time,” though it occurs in old English for
hence, i. e. from this place. Thus, in the scoffing
Ballad made on the defeat of Henry III. at Lewes, in
1264, and which, from its temour, must have been com-
posed very soon after the event, we find the following
lines:
He hath robbed Engelond the mores ant the fenne
The gold ant the selver ant yboren henne.
Hamm, in the Islandic, is “he,” and hun is “she ;”
and STIERNHELM, (Gloss. Ulph. Goth. p. 85,) speaking
of the Gothic word hama, as in hama hrukida, “the
cock crew,” (Matth. xxvi. 74), says, Omnis avis mascula
dicitur HANA, ab HAN, ille, et femina HöNA, ab HoN,
filla ; “every male bird is called hana, from han, he ,
Hence
we may infer that the element en was compounded in
some of the Northern Dialects, as we have already seen
that er was, viz. with he, the, and who, producing
hem, then, and when, as well as here, there, and where,
all of them originally pronouns, and all used in a re-
stricted sense by an ellipsis of the words time, place,
&c. as Adverbs.
In the Gothic, Than is both “then” and “when,”
and yuthan is used for “now.” Than is also used for
autem, 3é, “but ;” and it is manifestly nothing more
than the article or pronoun thana, or thanei, answering
to the Greek Tov or ov, as Seimon THANA haitaman Ze-
loten, Xiuwwa TO'N cavočuevov ZnAwajv, “Simon, who
(was) called Zelotes,” (Luke vi. 15): THANEI wildedum,
*ON #9eXov, “whom they would,” (Matth. xxvii. 15.)
Thon, for “those,” is still used in many parts of Scot-
land; thynfurth we have seen in the old Dialect of that
Country, for “thenceforth,” which, in the Parliamentary
Articles of 1461 above quoted, is written “thensforth :”
and as henne was used in old English for “ hence,” so
thenne was used for thence, i. e. from that place; as in
Christ's Descent into Hell ;
Nas non so holy prophete,
Seththe Adam & Eue the appel ete,
Ant he were at this worldes syne,
That he me moste to helle pyne :
Ne shulde he neuer thenne come,
Nere Jesu Crist Godes some.
When is the Gothic hºwan, which is used for the Latin
quando, quoniam, quantum, quam, and is manifestly the
same as hujana, quem, “whom ;” as HWANA sokeith,
“whom seek ye?” (John xviii. 4.) As the Gothic than
and hwan, and the old English there and where were
often used convertibly, so were then and when ; and in
the Harleian MSS. (No. 2253, fo. 55. b.) we find the
for when :
The he com there, thoseide he.
It will not be necessary to use much argument in Why.
proof of the identity of origin between Why and the
86 G R A M M A. R.
How,
Grammar, words before mentioned, where, whem, &c.; it is mani-
~~' festly only another form of the pronoun who. In modern
usage we do not oppose thy (in the sense of this cause)
to why ; but this mode of expression occurs in the old
words forthy and withthy. Forthy occurs in the Scot-
tish Act of 1424, in the two senses of “because” and
“therefore.” So in BARBour's Bruce—
But God that most is of all might
Preserved thame in his forsight
To venge the harm and the contrair
That those fell folk and pantener
Did to simple folk and worthy,
That couth not help themselven ; forthy
They were like to the Maccabeis.
The same author seems to use nought for thy in the
sense of “nevertheless,” as
And nought for thy, thocht they be feil,
God may richt weil our werdes deil.
-k st 2k :r 2}:
And not for thy thair faes then were
Ay twa for ane that they had there.
So he uses with thy for “provided,” or “on this condi-
tion”—
And I sal be in your helping
With thy ye give me all the lond
That ye have nou into your hond.
In all which instances thy is simply this, viz. cause,
reason, or condition, all which substantives are under-
stood by the sort of ellipsis already explained,
How is simply the pronoun who, or huja, sometimes
written in old English ho; as in the Harleian MS.
No. 2277. fo. 1.-
Seinte Marie day in Leynte, among
Alle other dayes gode
Is ryt forto holde heghe
Ho so him vnderstode.
And as we have seen the pronoun that, and the Adverb
as, used convertibly, so we find how in the old Scottish
Dialect used where we should emply so, or as ; e. g.
housone, for “so soon as”—
That housone ony truble, questioun, or causis happynnis to be
movit—than incontinent it salbe lesum, &c.
Scottish Acts, A. D. 1554,
We have thus traced, at some length, the English
Adverbs of place, time, &c. which are in truth no other
than the demonstrative and subjunctive pronouns,
appropriated by custom to certain distinct significations;
but though the particular applications are matter of
mere idiom, and vary, as we have seen, considerably in
the same Country at different periods; yet in most, if
not all Languages, the same general Principle is to be
traced. In most, if not all, the words which are em-
ployed as Adverbs of time, place, manner, and cause,
are pronouns with little or no variation of form.
In Latin, from the pronouns is, ea, id, come the Ad-
verbs ibi, alibi, ibidem, inde, proinde, ita, itaque, ideo,
iccireo, eo, adeo, eorsum, uspiam, musquam, &c. From
hic, haec, hoc, come hinc, huc, adhuc, huccine, horsum,
hodie, antehac, posthac, hacpropter, &c. From ille, illa,
illud, come illic, illico, illuc, illino, olim, &c. From qui,
qua, quod, come quo, quoque, quam, quando, quia,
quamvis, quare, quin, quidem, cum, cur, and probably
wbi, alicubi, ubivis, &c.
Ibi, says MARTINIUS, in his Lericon Philologicum,
A. D. 1655, is from is, as āvrö6, from divros ; and
ibidem is from ibi and idem. The same author observes,
that huc was anciently written hoc, as in the VIIIth Adverbs.
To which VossIUs \-N-0
AEmeid, Hoc tumc ignipotens, &c.
adds, that ad huc meant ad hoc, (subauditur tempus ;)
and that they also used hac for hatc. Whence antehac
and posthac signified respectively ante hac (tempora)
and post hac (tempora.) GIFFANIUs, in his Index to
Lucretius, observes, that for hinc and illinc, the Ancients
used him and illim. VossIUs notices the ancient quor,
for cur, as quoi for cui; quoigue for cuique ; quoiusque
for cujusque; and quoiguam for cuiquam.
Ubi appears to have been formerly cuibi, or cubi, for
so it is found in the compound alicubi; but cuibi must
have been written in the most ancient Latin quoibi ;
for, in the Laws of the Twelve Tables, we find quoi,
quoius, quoium, and quom, instead of the more modern
cui, cujus, cujum, and cum. Ibi and ubi, therefore,
were merely is and qui compounded with the particle bi,
which was, perhaps, of similar origin with the Gothic
bi and the English by. We must not omit, however, to
notice that the distinction between the relative and
interrogative force of the word ubi was accurately marked
by the accent. UBI interrogativum, says Martinius,
penultimam acuit, ut, UBI est Pamphilus 2 Relativum
gravatur, ut, Saevus UBI AEacidae telo facet Hector. Sic,
UNDE, QUANDo, et similia interrogativa penultimam
acuunt, relativa gravant. It was also repeated for the
sake of emphasis, as ubi ubi, for ubicumque; an idiom
similar to that of the Anglo-Saxon tha tha, quampri-
7mum, thor thar, quo in loco, &c.
It is needless to trace the pronominal Adverbs in
Greek; but it may be somewhat curious to observe the
same Principle in the Persian Language, in which the
pronouns are een, this; aun, that ; ke, who ; che, which.
From een, “this,” are derived eenjá, “here,” eerist,
“ hither.”
From aun, “that;” &njá, “there;” dinsü, “thither;”
angāh, “then.” -
From ke, “who;” cu or cujá, “where,” “whither.”
Erom che, “which ;” chum, “how, or when?” chend,
“how many?" chera, “wherefore ?” hemchun, “so as,”
&c. (See Sir William Jones's Persian Grammar; and
compare pages 32 and 33 with 93, 94, 95, and 96.)
The pronominal Adverbs which we have just consi- So.
dered serve principally to modify the verb; for when Also,
we say “this is here, and that is there,” the words here
and there serve to modify the assertion; and the same
may be observed of the phrases “to come hither,” “to
go thither,” &c.; but there are some other Adverbs
which are derived from pronouns, and of which the
principal use is to modify adjectives. Such are the words
so, as, than, &c. We have already noticed the pro
nominal origin of so and as, which are both synonymous
with it or that. As, in the German, is written es, and
forms the pronoun it. That, in the Scottish colloquial
Dialect, is sometimes used for so, as in The Antiquary,
(vol. ii. p. 281,) “that muckle,” for “so much.” These
words so and as had respectively their compounds all-so
and all-es, which latter was the old English als. So and
also are the Scottish swa and alsua, which occur in the
Act of 1424. Richtswa occurs in the Act of 1478; and
swa furth, i. e. “so on,” in that of 1491.
Alles was formerly used where we should use also,
as in the Romance of the Kyng of Tars, (see Warton,
v. i. p. 191)—
And alles I swere withouten fayle.
Mr. Tooke has correctly explained this word alles, als,
G R A M M A R. 87
Else,
Than.
Grammar. to be all-as, and to correspond with the words all that,
wev-, as in the following instance:
Glidis away undir the fomy seis . . .
Als swift as ganze, or fedderit arrowis fleis.
GAWIN Douglas.
i. e. “glides away with all that swiftness that arrows
fly with.” &
So in Robert DE BRUNNE, an English writer: (circ.
A. D. 1300 :)
Richard als suithe did raise his engyns.
In the Scottish Act of Parliament, 1493, alswell, for
‘ as well,” or “all as well.” Als, in the sense of also,
very frequently occurs in our old writers. Thus, in
BARBour's Bruce, which was written about A. D., 1375,
we have,
And the gode Lord als of Douglas.
2. sº 2k :: º:
He might have seen, that had been there,
A folk, that merry was & glad,
For their vict'ry; and als they had
A lord so sweet & debonair. -
Again, in the before-mentioned English Poem, entitled
The Pricke of Conscience—
Andals he yaf him a fre wille.
It would seem that the word elles, or els, is some-
times to be considered as identical with alles, or als;
and sometimes to be derived from the old German el,
alius, alienus, peregrinus, which WACHTER calls Wor
Celtica et primitiva, qua, Graecis effertur axNos, et
Latinis alius. HENIschius, in his Thesaurus of the
German Language, explains el alius, jemand el, alter,
quispiam, somebody else, niemand el, memo alius, no-
body else. The compounds and derivatives of this word
are found in all the Northern Languages, as in Welsh,
aliwn alius, alon alieni, alltad alienigena, alltudo in
exilium pellere, allwlad alienigena, &c.; in Gothic
aljath alio, aliorsum, aljathro aliunde, aljakunja alieni-
gena, &c.; in Frankish allasuara alio ; in Alamannic
allesuanan aliunde; in Anglo-Saxon elles alias, alioquin,
elles-hwar aliorsum, altheodig exterus, peregrinus,
eltheodisce men peregrini, elreordig barbarus; in Islandic
ella alius; in provincial German al-fanz aliena loquens,
el-gotze, idolum peregrinum, ellend terra aliena, būff-el
bosperegrinus. To which we may add the Scottish el-
ritch, strange, of a foreign Country, for ritch is from
Tyk, a kingdom, or dominion.
Mr. Tooke derives this word else from a-lesan, an
Anglo-Saxon verb, of which he says it is the impera-
tive, and that it signifies dimitte hoc, or hoc dimisso.
The derivation is not very probable; but he expresses
the most violent indignation at its having been ques-
tioned by some anonymous critic; as if an error in con-
jectural etymology were a matter of moral turpitude,
and inferred absolute infamy to a man’s character. In
reality few errors can be more innocent—a circumstance
peculiarly fortunate to Mr. Tooke; for among many
ingenious conjectures he has certainly ventured on some
that are perfectly erroneous.
Than has been already explained under the word
them : for it seems to have escaped the notice of most
English Grammarians that these two words are perfectly
identical, and indeed have not been generally distin-
guished in use much more than a century. Thus in
Shakspeare's Sonnets (A. D. 1609)—
*Tis better to be vile then vile esteemed,
When not to be receiues reproach of being ;
and in Milton's Paradise Lost (edit. 1669)— Adverbs.
– Native of heav'n for other place \-y-
None can then heav'n such glorious shapes contain.'
So we have thane for at that time in the Harleian MS.
No. 7333. f. 14. b. :—“This balade made Geffrey
Chaunciers the laureall poete of Albion, and sent it to
his souerain lorde Kynge Richard the Secounde, thane
being in his castell of Windesore.”
Thus, which is similar to so, is the word this. As in Thus,
the Ist Sermon of LATIMER, A. D. 1562: “He hath
lain this long at great costes and charges, and canne not
once haue hys matter come to the hearynge.”
5. If there be a doubt whether any one particular class verbs
of words can be used Adverbially, that doubt must tº
apply to the Verbs. In English, the words to which this
doubt applies are either of uncertain etymology, or else
their use is rather conjunctional or interjectional than
Adverbial. -
Yet has been considered as the imperative mood of Yet,
the Anglo-Saxon verb gytan, or getan, to get ; but
it is not very evident how this imperative can be applied
to the different senses in which the word yet is used.
It is differently written in our old manuscripts, gyt,
Jite, yet, yut, yit, but generally with the Saxon letter
which answers to our g or y, (consonant,) and which,
from the similarity of its form to z, is printed as that
letter in old Scottish books.
It sometimes relates simply to time, and would seem
to be connected with the Gothic ya, now, as “ is
he not yet arrived?” i. e. is he not arrived at this late
hour 2—Where it is to be observed that the corre-
spondent word in French is encore, which clearly ex-
presses the conception of time; for encore is the Italian
amcora, which Menage derives (perhaps not quite cor-
rectly) from hanc horam ; but which is certainly from
the Latin hora, the hour, or time. In this sense, yet is
used nearly in the same manner as the adjective Adverb
still, as
He yef of the holy cross sum that ther yut is.
RoBERT OF GLoucestER, 296.
Sometimes yet has the force of moreover—
Gyt he presented him the spere. WARTON, i. 94.
Yite I do yow mo to witte. Har/. MS. 913.
Sometimes of also—
The slear of himselfe yet sawe I there.
CHAUCER. Kn. Tale.
Sometimes of nevertheless—
Alas that he yet shulde dye. Elegy on Edw. I
So in the striking passage of Macbeth—
Though Birnem wood be come to Dunsinane,
And thou oppos'd, being of no woman born,
Yet will I try the last.
Where yet is used for also, moreover, or nevertheless,
it is properly to be considered as a conjunction; but
the distinction between a conjunction and a relative
Adverb is not always easy to be drawn.
Yes and No, if considered as Adverbs, must be taken Yes.
to modify the verb contained in the interrogative sen-
tence to which they form the answer. They are com-
monly ranked by Grammarians as belonging to this Part
of speech ; but perhaps it might be more proper to
consider them as interjections. Whether or not in Eng-
lish they are verbs, may be doubted. The French
word oui undoubtedly is the participle “heard;” the
Italian si is probably sit, “ be it so; and Mr. Tooke
88 G R A M M A. R.
Grammar. labours to derive our yes from the French ayez, “ have
S-N- it,” “enjoy it.” This is not the happiest of his etymo-
It is not very clear that the word dyez was used in Adverbs.
French before yes was used in English ; since it ap- \-N-
logies, at least it is not one of the best supported ; for
he quotes CHAUCER's Romant of the Rose very much
at random, in support of his conjecture :
And after, on the daunce went
LARGESSE, that set al her entent
For to ben honorable and fre;
Of Alexander's kynne was she ;
Her most joye was yuis,
Whan that she yafe, and sayd HAUE THIs. º -
Where Mr. Tooke says, “Which might, with equal
propriety, have been translated—
When she gave, and said Yes.”
The most frigid critic could not well have missed the
spirit of his author more completely. Largesse, or
liberality, is personified, like another Timon, scattering
her gifts on all sides, and not waiting for any demand
to which she might answer yes. So we find, from the
admirable Scenes with Lucullus and Lucius, that Timon
had been in the habit of surprising them with unex-
pected presents:
LUGULLUs. One of Lord Timon's men 2–A gift, I warrant. Why,
this hits right: I dreamt of a silver bason and ewer to-night, Flami-
nius, honest Flaminius, you are very respectively welcome, sir. (Fill
me some wine.) And how does that honourable, complete, and
free-hearted gentleman of Athens, thy very bountiful good lord and
master P
FLAM. His health is well, sir.
Lucul. I am right glad his health is well, sir—and what hast
thou there, under thy cloak, pretty Flaminius P
. . 26 sk 2. 2: sk 2k
SERV. May it please your honour, my lord hath sent—
LUCIU's. Ha! What hath he sent P I am so much endeared to
that lord: he is ever a sending. How shall I thank him, thinkst
thou?—And what hath he sent now * -
In like manner Largesse set all her pleasure in free,
spontaneous, and unexpected acts of bounty, with the
munificence of a mighty monarch, another Alexander,
surprising those whom she benefited by the sudden
exclamation, “Have this l’’
If our yes were derived from ayez, we should
find the latter word used in that sense, in some
of the French Dialects ; but this circumstance no-
where occurs ; and it can hardly be doubted, but
that yes includes, or is derived from the word yea.
Junius, indeed, explains yes as a contraction of yea is :
which etymology, if right, affords an explanation of
what Tooke calls Sir Thomas More’s “ridiculous dis-
tinction” between yea and yes. More says, that if a
question be framed affirmatively, the answer, if affirm-
ative also, should be by the word yea; if framed
negatively, by the word yes. Thus he supposes one
person to ask Tyndal the translator, if his book is
worthy to be burned, and another to ask him if his
book is not worthy to be burned. To the first, he says,
the answer should be yea, and to the other yes ; and he
appeals for this distinction to the then common use
and practice, in which a man of such eminence in the
profession of the Law, and of such frequent attendance
about the King's person and Court, could hardly be mis-
taken. If More then was right, yea meant simply
“true,” or “so,” i.e. “it is as you say ;” but yes sig-
nified “true it is,” or “so it is,” rejecting the negative
which had been introduced into the question ; in other
words it signified, “it is as you mean, but not as you
say ;” for the questioner, in both cases, is understood to
intend the same assertion, though the expressions are
Opposite.
pears to be a corruption of avez , which was taken
from havez, or habez, part of the very ancient verb haben,
of which the radical hab, in the sense of our word have,
was common to the Latin with all the Gothic Lan-
guages; for the Latin verb was habere, the Maeso-
Gothic haban, the Anglo-Saxon habban and habban, the
Frankish, Alamannic, and modern German haben, the
Islandic hafa, the Danish haſne, the Swedish hafwa,
the Dutch hebben ; and it even seems to have been used
in one Dialect of the Greek Language, for Hesychius
and Phavorinus prove that dBeus was used for éxets,
particularly by the Pamphyliams, and from this root an
infinity of nouns are derived in the Northern Languages.
It would therefore require some diligence of investi-
gation, to discover at what period in the History of the
Frankish, or French Language, the distinctive b or v of
the radical word was dropped in the imperative ayez;
and it could not have been long after that period, if at all,
that the imperative was converted, by common use, into
an Adverb among the French; and again, at a much later
period that this Adverb was adopted from the Norman-
French into the Norman-Saxon, whence it must have
descended to the modern English ; not one of the steps
in which progress has Mr. Tooke attempted to verify:
and if he had, in all probability his labour would not
have led to any confirmation of his conjectural etymology
of the word yes. -
Again he suggests that Yes and Yea are of very
different origin, the one being from the French verb
avoir, the other from some Northern verb (he does not
exactly determine which) that signifies “to own.” Now
verbs also of this signification are very numerous, as
well as the adjectives and substantives derived from
them. Thus the Gothic verb is aigan, the Anglo-
Saxon agam, whence our verb to owe is derived ; the
Islandic eiga, the Swedish aga, the Alamannic eigan,
and with these probably the Greek éxetv has some af.
finity. Nor is the adjective less general, with the sense
of own, proprius. In Gothic it is aigin, in Anglo-
Saxon agen, whence the old Scottish awin, and old
English Owen, the Alamannic eigan, the Danish eget,
the Islandic eyga, and the Dutch eygen. It does not,
however, happen in these Languages generally, that the
affirmative Adverb, or interjection, has the form of any
part of the verb, or indeed much resemblance to it.
Our yea is undoubtedly the Gothic ya, yai, which, with
very little change, pervades most of the Northern
Dialects, being in Welsh ie, in Armoric, Dutch, German,
and Swedish, ja, (where the j is pronounced as y,) and
in Anglo-Saxon ia, ya, yaº, yea. Of this word ya, the
origin is much doubted by etymologists. Some derive
it from the Hebrew Jah, Jehovah; but as we cannot
think that the Hebrews would ever have profaned the
name of the Almighty, by thus introducing it into their
most common and trivial discourse; so it is still less
probable that the nations, who knew not Jehovah,
should have done so, except from imitation of the
Hebrews; and this etymology, if true, would present
a singular contradiction to the words of CHRIST in the
Gothic translation of the Gospels. Our Saviour com-
mands His disciples not to swear at all; but, in their
common discourse, to use simple affirmations or ne-
gations. Whereas, on the hypothesis above mentioned,
the Gothic text siy wattra izwar ya, ya, (Matth. v. 37,)
Yea.
-Hyº -- ºr "
: * ~,
G R A M M A. R. 89
yea to this—namely, that God hath forbidden you to Adverbs.
Grammar. ought to be rendered, “let your word be, by Jehovah' -
eat of every tree? - N-y-Z
^*” by Jehovah [" It seems most probable that ya was
originally of similar origin with the Latin word sic,
which was used for the same purpose. Thus, in
Terence, we find—Itane ais Phanium relictam solam P
SIG. Daturne illa hodie Pamphilo nuptum 2 SIc EST.
Quid narras 2 SIG EST FACTUM. In which three
different examples, we see the affirmative Adverb gra-
dually brought back, as it were, to its pronominal
origin; for the last answer might as well have been
ita est factum, or id est factum. -
The Latin sic, so, and si, if, were manifestly of simi
lar origin with se, himself, which in the dative is si-bi.
and with the verb sit, which was anciently written
si-et.
In the Gothic, we shall, in like manner, perceive a
connection between ya and the pronouns and Adverbs
of pronominal origin, so, it, this, and that :
Ya-ins (ille) “ this man,”
Ya-ind (illuc) “to that place,”
Ya-thau–(forsan) “it may be so,”
Ya-u (si) “be it, that,”
Yu— (jam) “at this time.”
Besides the mere expression of acquiescence in a
question or demand, yea has, in its modern use, a par-
ticular force which answers to the Latin imo; and imo,
it is to be observed, is really the pronoun im, which
occurs constantly for eum in the remaining fragments
of the Laws of the Twelve Tables; as, si IM aliquips
occisit, joure casus esto, where MACROBIUs says: ab
eo quod est Is, non EUM, casu accusativo, sed IM dire-
runt. In this sense of the word yea, MILTON says,
They durst abide
Jehovah thund'ring out of Sion, thron’d
Between the cherubim—yea, often plac'd
Within His Sanctuary itself their shrines.
It is somewhat remarkable, in the English idiom, that
the word may (the antipodes, as one would think, of yea)
is used in the very same sense as that which we have
just described. Thus DRYDEN says, “This allay of
Ovid's writings is sufficiently recompensed by his other
excellencies ; nay, this very fault is not without its
beauties.” What is still more singular, BEN Jonson
uses both yea and may with the same augmentative force
in one and the same sentence: “A good man always
profits by his endeavour; yea, when he is absent ; nay,
when dead, by his example and memory.” In all these
passages, yeaseems still to bear its relation to the pronoun
this; for the meaning is, “they durst abide Jehovah
thundering out of Sion; this they did and often more.”
“A good man profits by his endeavours; this he does
when present, and even when absent:” and the word
may only serves still further to complete the same sense;
for, in the instances above quoted, the meaning is, “the
allay of Ovid's writings is accompanied by other excel-
lences: this is the case, and not only this, but the very
fault has its beauties.” “A good man profits us by his
endeavours when absent: this he does, and not only
this, but even when he is dead, we profit by his exam-
ple and his memory.” -.
There is still one more use of yea, which confirms
our view of its import ; as in the 3d chapter of Genesis
—“ Yea 2 Hath God said, ye shall not eat of every
tree in the garden?” Here the word yea has an inter-
rogative force; and means “is this so 2° Do you say
WOL. I.
In fine, the conception always expressed by yea is
that of true and affirmative existence. Hence Dr.
HAMMOND, explaining the passage “all the promises of
God in him are yea and amen,” (2 Cor. i. 20,) says,
“ that is, they are verIFIED, which is the importance of
yea; and confirmed, which is meant by amen.” Now
the conception of positive existence, as applied to a
particular thing or event, is expressed by the words
“this is ;” and if there be an ellipsis of either word, the
same conception may be expressed by the other word.
In this view of the subject, it seems not unreasonable
to conclude that the word ya may have been originally
either a pronoun, or a part of the verb of existence ;
and it is to be remembered, that in many, perhaps in
all Languages, the verb of existence is merely expressed
by a pronoun.
Ay appears to be merely yea, a little varied in pro-
nunciation. Dr. JoHNson, indeed, suggests that it
may be derived from the Latin aio; but words in ge-
neral are not transferred from one Language to another,
so as to come into common use, without leaving some
traces of their gradual progress. The Latin terms
which have been incorporated with our colloquial dis-
course, have been received either through the medium
of the French, or else have been technical terms, chiefly
of the Law ; and in either case it is generally easy to
discover the gradations by which they have come to
form a part of our Language. Now there is no such
proof of the transition of the Latin verb aio into the
English ay, but much to render it improbable. Ay has
some slight differences of application from yea, as yea
has from yes; but this is no more remarkable than the
different force and effect which, as we have already
seen, is given in different cases to the same word, yea.
Thus, in the following passage from Shakspeare's Henry
VI., ay expresses somewhat more of passionate and
proud reproof, than if the word yea were employed:
Remember it; and let it make thee crest-fall’n;
Ay, and abate this thy abortive pride.
As yea appears to have been corrupted into ay, so was
ay into I, but without any variation of meaning :
Hath Romeo slain himself? Say thou but I;
And that bare vowel, I, shall poison more
Than the death-darting eye of cockatrice.
Romeo and Juliet.
The other Adverb aye, always, (for it is a totally
different word,) we shall have occasion to consider it
hereafter.
Nay and no have some differences in use, but they Nay.
JUNIUs indeed No.
are probably the same word in origin.
suggests, that may is from the two Saxon words ne-ia,
“not yes;” but there is no proof that the Saxons, or
any other nation, ever used this strange periphrasis to
express a conception which is so universal and primary
in the Human Mind; being, as it were, the bound and
limit of all other conceptions. The following are the
remarks of the President DE BROSSES on this subject:
“Man, in order to communicate his perceptions, has
occasion to express, not only existing objects, and the
manner of their existence, but also in what manner
they do not exist. And so with regard to feelings, he
has occasion to make known whether they are agree-
able to his will, or not agreeable to it. It is necessary
N
Ay.
90
G R A M M A. R.
. No man is always wise. Adverbs.
Man is not wise always. \-y-
Grammar. then, that besides the different radicals serving to ex-
S-N-" press positive ideas, and different classes of objects, he
should have another radical, which may serve to ex-
press a negative idea; appropriated merely to indicate,
that what he describes is not in what he wishes to
describe. One single radical will always suffice for that
effect, to whatever object it may be applied. Negation
being an absolute and privative sensation, a mere
counter-assertion, it is quite enough that we have one
vocal sign, one organic articulation, to advertise the
hearer, that what we say is not in the subject of which
we speak. . The negative feeling being one which con-
tains in itself a positive and contrary volition, it is not
difficult for a man to express it by a gesture, or, what
is the same thing, by a single movement of the organ
of speech.” The learned President proceeds to show,
that in the formation of many Languages, mankind had
chosen the nasal articulation for the expression of
what he calls the sentiment negatif. This is at least so
far true, that the general conception of negation is
expressed in the Latin, and most of the Northern Lan-
guages, by the syllables na me, ni, no, &c. Ne, says
WACHTER, particula negandi vetustissima, a Scythis in
Persia, Greció, et Septemtriome proseminata ; qua, Persis
effertur NEH, Graeci vi et vé in compositis, sicut Latinis
NE and NI, Gothis, NI, NIH, NE ; Anglo-Saxonibus
NA, NE ; Francis et Alamannis NI ; Anglis No ; Suecis
NEY ; Sorab. NE ; in compositis. He also justly ob-
serves of the letter n, that in many compounds it is an
abbreviation of me, mi, &c. and as such has a negative
power; as in the German nichts, niemand, niemal,
nimmer, and many others, of which the list might
be extended to an immense length, were we to include
all the European Languages. Nor is it only in the
distinct compounds, such as ever, never, one, mone,
volo, nolo, ullus, nullus, &c., that this effect is dis-
cernible, but also in some terms which conversely ex-
press positive and negative conceptions, as light, might,
lua, now, &c. Without entering deeply into those
Metaphysical speculations on the rà èv and the to hi) ºv,
for which Mr. Tooke so much ridicules Lord Mon-
BoDDo, and without pretending to decide the disputed
points respecting positive and negative ideas, positive
and negative quantities, and the like, it is sufficient
for us to observe, that every child, in the first glim-
mering of Reason, must necessarily form a conception
of negation ; and that it does in fact acquire, among
its first articulate sounds, the sound which expresses
that conception. The child has as distinct a concep-
tion that its nurse is not present, or that its food is
not agreeable to its palate, as it has of the opposite
circumstances. It may perhaps be urged, that this
negative conception is in its very nature adjectival;
that it can only be applied in the manner of an attri-
bute to some other conception which is of a substan-
tive nature. Il est impossible, says DE, BRoss Es,
de former un Nom absolument privatif, c'est & dire
wne locution, qui me contienne pas une idée vraiment
positive. Be it so ; but at least the adjectival con-
ception may be applied, in the manner of all other
conceptions of the same class, to modify substantives,
adjectives, verbs, and Adverbs; thus we may apply
the negative words or particles no, not, and un,
to modify the substantive mam, the verb is, the ad-
jective wise, or the Adverb always, in the following
phrases: wº
Man is always unwise. . . . . -
Man is never wise (i.e. always not wise.)
Whether there be any thing in the nature of the
nasal organ, which peculiarly fits it, as De Brosses sup-
poses, for the expression of conceptions of doubt and
privation, may, perhaps, be reasonably questioned.
Negative terms are found in many Languages to which
this remark certainly cannot apply. However, the ne-
gatives in Latin and in the Gothic Languages, gene-
rally have the nasal articulation variously combined; nor
do these various combinations necessarily give a distinct
force to the word. The Latin me, mon, and mec, were an-
ciently confounded, and so were the English me, no, not,
nor. In a fragment of the Laws of Numa Pompilius,
preserved by FULVIUS URSINUs, we find nei for me.
Sei Hominem folminis occisit, im Sopera genua NE1 tolito. .
Again, in a fragment of the first Tribunician Law, nec
is used for me— A.
Sei quis aliuta farsit cum pequnia familiag sacerestod: sei quis im
occisit paricida NEC estod.
Again, in the Laws of the Twelve Tables—
Patris familias quei en do testato moritor quoigue souos heres NEC
escit,
In old English me was used for not and for nor.
1. For not in the Harleian MS. 2253. fo. 70. b.-
Ne mai no lewed lued libben iu londe.
2. For not in the Prophecy of Thomas de Essedowne,
in the same volume, fo. 127—
Whenne shall this be 2
Nouther in thine tyme, ne in myn.
No was used in the same two senses.
1. For not in the Romance of Alisaunder—
Alisaunder and his folk alle
No had noght passed theo halvendall.
2. For mor, in the Description of Cokaygne—r
Ther nis halle, bure, no bench. -
On the other hand, mor, in the old Scottish Dialect,
was used for than :
The fell strong traytour Donald Owyr
º Mair falset had nor udir four. DUNBAR.
Compleitly, mair sweitly
Scho fridound flat and schairp,
Nor Muses, that uses
To pin Apollo's harp.
ALEx. Montgomery, circ. 1597.
The particle me, which forms part of our modern
words, none, never, &c., was anciently incorporated with
many verbs, as, I not, for “I ne wot,” or “ know
not;” I muste, for “I me wist;” I nabbe, for “ I ne
have ;” I mulle, for “I me will :” I molde, for “ I ne
would :” it mis, it mas, it mere, for “it me is,” “it me
was,” “it me were.”
The hors vanisheth I not in what manere. -
CHAUCER. Squ. Tale.
In all this wurhliche won
A burde of blod and of bon
Neuer yete ynuste non
Lussomore in londe,
Harl, MS. 2253, fo. 72.
Uch a srewe wol hire shrude
Tha he nabbe nout a smok, &c. Ibid. fo. 61, b.
I nul soffre that no more, Ibid. fo, 55. b.
Gº R: A M M A. R. 91
Grammar.
}Jouble ne-
gative.
Ado.
Together.
Whil God wes on érthe
And wondrede wyde
What was the reson
Why he nolde ryde P
For he molde no grome
To go by ys syde. .
* Harl, MS, 2253, fol. 124. b.
Ther nis loride vntler heuenriche. - •
(Id. No. 913.
that he nas wenemyd anon. -
Lyf of Seint Patrik.
Wymmen were the best thing
That shup our heye heune kyng
- Yeffeole false mere. Harl, MS. 2253. fol. 71.
It is sufficient for the general purposes of communi-
cating thought, that the negative conception should be
once expressed in a simple sentence; but we generally
find it redoubled in old English, a circumstance de-
rived from the Anglo-Saxon idiom, as, Ne om ic na
Crist, “I am not the Christ.” (John i. 20.) The same
idiom prevails in the modern French, although it was
not always observed in that Language at an earlier
period. In the XVIth century they said, l'habit NE
faict le moyne: at present the same proverb is expressed
thus, l'habit NE fait PAs le moine. It is difficult to
account for the reduplication of the negative upon any
other Principle than that of the eager desire, which we
commonly see in Barbarous and ignorant People, to give
utterance to their strong feelings and imperfect con-
ceptions, and which usually leads to much tautology
in their discourse. This genuine result of Barbarism,
however, has been sometimes mistaken for a proof of
extraordinary learning ; and critics have dignified it
with the title of an Archaism, a Hellenism, or some such
pompous appellation. “The editor of Chaucer,” says
HICKEs, “ knowing nothing of antiquity, asserts that
the Poet imitated the Greeks in using two negatives to
express negation more vehemently ; whereas Chaucer
was entirely ignorant of the Greek Language, and only
used the two negatives according to the prevailing cus-
tom of his own times, when the Language had not yet
lost its Saxonisms, as, “I me said none ill.” In the
Saxon writers, indeed, three and even four successive
negatives are sometimes to be found, as, NE yeseah
NEFRE NAN man God ; “ no man ever saw God.”
(John i. 18.) And again, NE NAN NE dorste of than
dage hyme NAN thing mare aziyear. “and no man
durst from that day forth ask him any more questions.”
(Matth. xxii. 46.) It is to be observed, however, that
some of the best of those writers, and particularly the
Royal translator of Bede’s Ecclesiastical History, gene.
rally employ but a single negative; and such also is the
uniform style of that venerable monument of Gothic
literature, the Coder Argenteus. -
There are some Adverbs which have a very obvious
affinity with verbs, such as ado, together, &c. but which
it would, nevertheless, be somewhat difficult to trace
directly to any particular part of the verb. Ado is well
known in English from the name of the popular drama,
Much Ado about Nothing. In the Scottish Dialect too
it is very ancient. In the Preface to Gawin Douglas's
translation of the AEmeid we find the expression, “it has
nathing ado therwith.”
Together has a manifest relation to the verb gather,
which, however, we now use with some diversity of
meaning. The Adverb and the verb rather seem to
refer to some common origin, which does not exist
in English, but appears in a more simple form in . Adverbs.
Dutch, in which gade is a consort, as een duyfen hadre S->
gade, “a dove and her mate ;” gadeloos, Anatchless ;
gadelyk, sortable, &c. The word gathering, which was
formerly used in English for a meeting, or assemblage,
has fallen into disuse; but was anciently in very general
acceptation; as in BARBOUR-
And the kyng than a parlament
Gart sett theraftir hastily
And thider summond he in hy
The barouns of hys roialtie
And to the lord the Bruce sent he
Bidding to come to that gathering.
In the Scottish Acts of 1592 the word togidder occurs;
but in more recent compositions it is spelt, as it is in
fact pronounced in Scotland, thegither. Thus in the
well-known Song descriptive of the connubial affection
of an old married couple:
John Anderson, my jo, John, we clomb the hill thegither.
In some of the old Romances the words to and geder are
written separately, as if the latter were considered as
the plural of gede, answering to the Dutch gade. (See
Warton, i. 100) —
To gederschal sit at the mete,
The correspondent expression ºn fere is, in like manner,
derived from the Anglo-Saxon foera, and old English
jere, a companion; as in the Geste of King Horn—
Tueye feren he hadde
That he with him ladde.
The Scottish Dialect employed the verb to effeir, and
the participle effeiring, thus in the Act of 1587, “Orda-
nis lettrez to be direct heiropone, gif neid beis, in forme
as effeiris :” and again, “The elvand, the pund trois,
& the stame proportional & effeiring.” Barbour uses
the word with some slight difference in the signification :
Sheriffs & baillies made he then
And all kind other officeirs
That for to govern land effeirs.
Another expression nearly correspondent to together
was the Adverbial phrase all samyn, or in Samyn, an-
swering to the Latin insimul, and to the French ensemble.
GAw1N Douglas employs both togidder and all samyn
in the same passage: - -
Togidder with the principallis of younkeris
The Sobir senatouris & pure officiaris
All samyn kest encense.
In the Romance of Syr Laumfal—
To daunce they wente alle yn same.
In that of Octouian Imperator—
The emperour with barouns yn same
Rood to Parys.
BARBOUR employs the double Adverb twasum Samyn,
i. e. two together :
That was in an euill place,
That so strait and so narrow was,
That twasum samen might not ride.
The word samen is the English pronoun same : it is
now probably obsolete in Scotland, but was the legal
language of 1592, as appears by the Acts of that year,
and also by ALEXANDER MontgomeRY's Tale of the
Cherrie and the Slae, composed about the same time:
Lyk as befoir we did submit
Sae we repeit the samyn zit.
6. The last class of separate words which we have Substan-
to notice as used Adverbially are substantives. It is tives.
N 2
92 G R A M M A. R.
weile in German used substantively for a space of time, Adverbs.
Grammar. manifest that substantives may be used in the forma- -
as in German es ist eine gute weile, “it is a good while,” \-'
S-N-' tion of compound words to express the attributes of
While.
attributes. Thus stone, in its primary sense, is a sub-
stantive, and blind is an adjective ; but in the com-
pound stone-blind, the former part of the word modifies
the latter, as much as if we were to say “a stony, or
stonelike blindness.” In like manner, substantives
standing alone may be taken Adverbially, as modifying
either a verb or an adjective. The latter mode is the
less common in modern English, but it occurs not un-
frequently in the older Dialects: the former mode is
common in most Languages. The Adverbial use of the
substantive to modify a verb, somewhat resembles the
ablative absolute of the Latin Grammarians. It ex-
presses a conception simply, without asserting it to
exist or not to exist. The construction is consequently
elliptical, and the sense may always be more fully ex-
pressed by adding the assertion. Thus, in the text
“I will sing praise to my God while I have my being,”
(Psal. civ. 33.) the word while, which was originally a
substantive signifying time, becomes an Adverb, by
the absolute mode of expressing it. The passage is
literally “I will sing praise to my God, time I have
my being,” i. e. “during the time;” and the three fol-
lowing propositions are included in the whole passage
as co-existent:
I will sing ;
I shall have my being ;
Time will endure.
Nothing but use and the convenience of discourse
has assigned their peculiar Adverbial force to substan-
tives thus employed. The conception of time, for in-
stance, may be employed, as in the above case, simply
to mark continuance, or to mark continuance from a
certain point, or to a certain point. Thus in the
text “There was a great earthquake, such as was not
since men were upon the earth, so mighty an earth-
quake and so great;” (Revel. xvi. 18.) the word since,
which is also a noun signifying time, may be rendered
“from the time that.” And again, in the text “I
will not leave thee until I have done that which I have
spoken to thee of,” (Genes. xxviii. 15.) the word until
may be rendered “to the time that.” Until, indeed,
is not a noun signifying time, as while and since are ;
but the word while is often used for it in our provincial
Dialects, and occurs in many of our old compositions.
Thus in the Scottish Act of Parliament, 1587, the
enactment is ordained to last “Ay, and quhill His
Hienes nixt parliament.” So in Alexander Montgomery:
Cum se now, in me now
The butterflie and candill
And as scho flies quhyl scho be fyrt.
Of until and since we shall speak more particularly
among the prepositions. The substantives used as
Adverbs of time in English are while, tide, sithe, time,
and season. - -
While is the Gothic and Anglo-Saxon hwila, and
Alamannic uuila, time, or a certain space of time, which
seems to be of the same (origin as our wheel, in the
Anglo-Saxon hweol, Danish and Swedish hiul, Islandic
hiool, and Dutch wiel, which are derived, by J. Davies,
from the Welsh chuyl, turning, and seem to have some
aſfinity with the Latin volvo, and Gothic walwyan, to
rell; nor is there any more apt or more common sym-
bol of time than the continual rolling of a wheel. Be
this as it may, we find the word while in English and
or “a long time.” So in the relation of the meeting
of Joseph with his father Jacob, (Gen. xlvi. 29.) “he fell
on his neck, and wept on his neck a good while.” We
have seen this word used in the two senses of “whilst”
and “ until :” it is also used in the Scottish Dialect for
“sometimes,” as in the well-known anecdote of an
English traveller, who had been confined at a village in
Scotland several days together by the rain, and who, at
length, losing his patience, asked the landlord pettishly,
“What! does it rain here always?” To which the
other replied with a smile, “Hoot, na! it snaws whyles.”
The word awhile is commonly used Adverbially for “a
short time ;” as Samuel said to Saul, “Stand thou still
awhile that I may show thee the word of God.” (1 Sam.
ix. 27.) The same idiom occurs in the Goldin Terge of
DUNBAR :
Acquentance new embrasit me a quhyle,
And favourt me till men micht gae a myle,
Syne tukhir lief, I saw hir nevir mair.
In a very ancient English Love-song whyle is used in this
sense without the article. (Harl. MSS. 2253. fol. 63. b.)
Betere is tholien whyle sore
Then mournen euermore.
It is somewhat remarkable that though in the German
Language the substantive weile is not used Adverbially
in the same senses as while is in English, yet it has
the same Adverbial, or rather conjunctional sense that
we give in matters of reasoning to since, which word,
as we have observed, also signifies “time.” Thus the
German weil implies the consequence or dependence of
one fact on another, as WELL ers verlanget, so soll ers
habem ; “since he desires it, he shall have it.”
The compounds of while still in use, such as mean-
while, erewhile, require no explanation. They plainly
express the conception of time, and signify “in the
meantime,” “ sometime before,” &c. Erewhile was
anciently written whilere, and so we find in the different
old Dialects whilom and umquhill, which both agree with
the old word sometime for “formerly.” “The whiles”
occurs in old writings for meanwhile ; as in Kyng
Alisaunder—
Alisaundre is in his lond
And hath some a newe sonde.
From a cité in the Est
That nul no Phelippes heste.
Thider he wendith with gret pres,
This storily cities for to dres.
The whiles, herith a cas.
A riche baroun in Grece was, &c.
Whiles was used at no great distance of time where we
now use while or whilst ; as in SHAKSPEARE's Much
Ado about Nothing—
What we have we prize not to the worth
PWhiles we enjoy it.
The same idiom also prevailed in Scotland-
The bramble grows althoet it be obscure, -
Quhylis mountane cederis tholes the bousteous winds,
And myld plebyan spirits may lief secure,
Quhylis michty tempestis toss imperial mynds.
Montgomery.
Mr. Tooke conceives that whilst and amidst are mere
corruptions, and that we should write them as formerly,
whiles, and amiddes; but it would seem that there was
some particular reason for the final t, because in the
common Scottish Dialect of the present day it is found
G R A M M A R. 93
Grammar, in the word alongst. Possibly the expressions originally
S-v- were “ on long is it; on mid is it;” “ while is it.”
Stound.
Tide,
In the Morale Proverbes of Crystyne, printed by Cax-
ton, A. D. 1478, we find the expression long saison for
“a long while,” or “a long time:” -
A temperat man cold from hast asseured
May not lightly long saison be miseured.
So in the Dictes and Sayings of Philosophers, printed
1477, “There was that season in my company a wor-
shipful gentleman called Lewis de Bretaylles.”
Stound, which is from stond, occurs adverbially in the
sense of time; as in Octouian Imperator—
Men blamede the bochere of stoundys
For his some.
This, which we should now express oftentimes, was an-
ciently expressed also ofte sithes; as in CHAUCER's
Troilus and Cressida—
And such he was I proued ofte sithes.
Sumwhile occurs in Kyng Alisaunder—
There woned sumwhile Kyng Appolyn.
In the Lay le Freine, published by Mr. Weber, we find
therwhiles :
- The abbesse hir in conseyl toke
To tellen hir hyenought forsoke
Hou hye was founden in althing
And tok hir the cloth and the ring
And bad hir kepe it in that stede ;
And therwiles she lived so sche dede.
The Scottish umquhill appears in the Act of 1455,
“James umquhill Erle of Dowglas.” In the Act of
1540 we find both “wmquhile James Coluile,” and
“Archibald sumtime Erle of Anguis.” Sumtyme, an-
swering to olim, occurs in Montgom ERY's Cherrie and
Slae: -
Then furth I drew that double dart
Qhuilk sumtyme schot his mother.
Our word tide is connected with the word sithe be-
fore mentioned by the German zeit, (pronounced tseit,)
for on the one hand it is tseit, tide, dropping the initial
s; and on the other it is tseit, sithe, dropping the
initial t , and in both cases changing the final t into its
approximate articulation, viz. in the one instance d, in
the other th: We do not use tide in modern English
Adverbially; but in German the word seit is used in the
sense of “since,” or “from that time.” In the differ-
ent Northern Languages this word appears in various
forms, and with many analogous significations. In the
Alamannic Glossaries we find citi, “times;” whence
probably comes the Latin cito, quickly. In the Frankish,
vomna alten ziTIN, “from old times;” tho sih thin ZIT bi-
brahta, “when the time was brought near;” in modern
Dutch, in voorige TYDEN, “ in former times;” by onzen
TYD, “in our time,” &c.
called, in Frankish, citi and ziti, and in Anglo-Saxon,
tida; as in Gloss. Keron. fora einera ZITI, before one
o'clock; and in the Anglo-Saxon Gospels, hu ne synt
twelf TIDA tha's dages 2 “are there not twelve hours
in the day ?” In modern German they say welche ZEIT?
for “what's o'clock?” hochzeit, a marriage festival, or
any other festival; in which latter sense the expression
runs through a great variety of Dialects, as the Frank-
ish hoho ziti, the Alamannic hohzit, the Swedish hogtyd
and hogtyds dag, the Dutch hooghtiid, the Anglo-Saxon
heah-tid, and the old English high tide and hock-tide.
In German, too, the separate words hohe zeit are used
as we use “high time; as, es ist hohe ZEIT, “it is
The hours of the day are
high time” (that such a thing were done.) So they Adverbs,
say bey zeit, as we do Adverbially betimes; bey zEITEN -/~
wieder commen, is “ to come back in good time,” von
ZEIT zu zEIT, “from time to time,” essenszeit, “ dinner-
time,” &c. In this last sense, where we say church-
time, the Dutch say kerk-tyd; and where we say bed-
time, our Saxon ancestors said bed-tid. So underniid
was the hour of nine o'clock in the forenoom, when the
wndernsang was sung in churches, and when individuals
were accustomed to take the meal called in Gothic
wndaurnimat, and in Anglo-Saxon simply undern.
Hence, in the Romance of Syr Lizunfal–
In hys chamber he hyld hym stille
All that vndern tyde.
The German zeit is also a season or “time of the
year;” vier zeiten, “the four seasons.” The Dutch
tyd is “opportunity,” “ convenient time,” “leisure,
“sufficient time.” Of the same origin are our moort-
tide, Whitsuntide, and the tide, or periodical time of the
sea's ebb and flow. -
Let him hear the cry in the morning, and the shouting at noon-
tide.—Jer. xx. 16.
Noon-tide repast, or afternoon repose.
MILTON. Par. Lost.
And behold, at evening-tide trouble; and before the morning he
is not l—Isaiah, xvii. 14.
In the Romance of Kyng Alisaunder, long tydes means
a long while (several days, as it should seem by the
context)—
They reste heom longe tydes
And wel ofte on ryver rydes.
Hence our verb to betide, or happen at a certain time,
which, by BARBOUR, is written simply tide—
But ye trusted unto lawtie,
As simple folk, but malwitie,
And wist not what shuld after tide.
Hence the substantive tidings, what happens at a certain
time, and, in a secondary sense, what is reported to have
happened.
Hence, too, the adjective tidy, of which the first sense
is seasonable, happening in due time—
If weather be fair and tidie, thy grain
Make speedilie carriage for fear of a rain.
TUsse R.
So the Islandic tidugur, tempestivus; the German ad-
verb zeitig, maturely, in good time ; (answering to the
Scottish timeous, and timeously ;) the German substan-
tive unzeit, an inconvenient time, with its adjective
wnzeitig, unseasonable; unzeitige geburt, “an untimely
birth,” and of the same construction as our untidy.
Thus we have seen in different Languages the con- Ever. Aye.
nection and interchanged use of those substantives which
furnish a large class of the Adverbs of time. There is
another class also relating to time, derived from a source
common to most of the Northern Languages, viz. the
Adverbs ever and aye, with the compounds of the
former, as evermore, never, nevermore, &c. Ever is the
Latin avum; as aye is the Greek áldov; and that avum
and dubv are the same word no one can doubt, who re-
members that in the Latin of the early Ages a was
written ai, and um was written om ; and that the mo-
dern Latin v was the Æolic digamma q, or our w
which, in fact, is the abbreviated articulation of the
vowel sound 00, as our y is the abbreviated articulation
94 G R A M M A R.; ,
Grammar of our vowel soundee. Thus the ancient Romans would
refer to a third meaning, in which those opposites con- Adverbs:
\-v- have written avum aiqom or aidon ; for we find rocon
cur; for of opposites, as Aristotle has observed, there S-ka-
for rogum in the Laws of Numa Pompilius. VELIUS
LoNGUs says, qua nos per E, antiqui per AI scriptita-
verunt; and MARIUS. VICTORINUs to the same effect,
a syllabam quidem, more Graecorum, per ai scribunt.
Om for um occurs constantly in the Laws of the Twelve
Tables, as devortion, eorom, finiom, &c. In the frag-
ments of the Laws of Numa Pompilius we read acnom
for agnum. . . . . . . .
. Pelea: Asam Junonis nei tagito. Sei tagit, Junon?
crimebos demiseis AcNoM feminam cabdito. -
The AEolic digamma is described by Dionysius o
Halicarnassus, in the Ist book of his Antiquities, where
he says that the ancient Grecians used a letter, which
was āq Tep Yūpa Śutta’s éti piùav ćp0)u èrtſevºlvöuevov
rats TAaqtats, “like a gamma with two (horizontal)
lines united to one perpendicular ;” and the examples
which he gives are FeNévy for “EXevn, Fåvaš for dvaš,
Foſkos for oticos, and Fav).p for &vijp. The AEolians
employed this letter to express a sort of aspiration either
at the beginning or in the middle of words; and as
they said opus (or owis) for 6ts, and wrov (or owon) for
duov, so it is probable that they said apwu for dutºv. In
ancient Latin inscriptions the F is inverted, as DIGIAI
for Divac. - -
The Latins not only introduced the articulation w,
in order to separate two vowels, but also the aspiration
h, as in cohors for coors, from cöorior ; aheneus for
acneus, mihi for miſſ, &c.
If it be thought necessary to seek a common radical
for these words avum and dubv, it may probably be
found in the ancient av or ab, . which seems to have
very generally signified the flowing of a river; which,
like the rolling of a wheel, has been in all times con-
sidered as a symbol of time. Etiam hodiernis Persis,
says BAxTER, in his Glossarium Antiquitatum Britan-
nicarum, (ad voc. ABALLABA,) AB pro aqua est, quam
et veteres nostri Av, SAV, et TAv appellavere; and again,
(ad voc. ABONA,) momen suum sortita est ab ipso flu-
mine, quod Britannis plurativo numero dicitur Avon,
et antiquá scripturâ ABON. Hence, aben, in old Ger-
man, is to fall, to decline, and der abend is the evening,
the falling or declining of the Sun: and the Helvetic
Swiss, as PICTORIUs asserts, use the verb with reference
to the decline of life, as ich aben fast, “I decline, or
draw fast to my end.” 3%
However this may be, there can be little doubt but
that the Anglo-Saxon afre, whence our Adverb ever is
lineally descended, was of the same origin with the
Latin substantives avum and aetas, which latter is only
a derivative of the former, being written in the Laws of
the Twelve Tables avitas.
AEfr, ever, e'er are used to denote time in its gene-
ral continuity; and consequently to denote eternal du-
ration, of which we have no other, or at least no better
conception, than of time, in continuity unlimited. The
same contraction e'er, spelt in Gothic and old Scotch
air, in Anglo-Saxon ar, in Frankish er, and in modern
English ere, denotes time, in its inception, or the time
immediately preceding the event or period of which we
speak; and this word, in its compounds, erliche, early,
also signifies time incipient, but not prior to the period
in question. In general it may be regarded as a rule
in etymology, that where the simple and compound
word have two meanings apparently opposite, they both
is the same Science: we reason in the same manner,
though to contrary results, on positive and negative
quantity, on lights and shades, on vice and virtue.'
There can be no doubt that erliche is derived from er.
It signifies a conception, like the conception expressed
by er; but for that very reason it differs from-e, , be-
cause, according to the scholastic rule, simile nori est
idem ; yet, on the other hand, as similarity approaches.
to identity, and as the limits are not always accurately
distinguishable or distinguished, it is not always easy
to decide, whether in Language, two terms like er and
early, do or do not absolutely exclude each other's
meaning; or even whether one word, like er, may not
embrace two meanings, excluding each other in their
different application to facts. Thus, in the Gospel of
St. Mark, are the two following passages: Kai rput
évvvyov Miav čvaords āśńM6e (ch. i. ver. 35.)—Kal Atav
Tpwi Tàs putas oaftBārwv épxovtat éiri to avnuetov čvarei-
Mavros toū j\lov. (ch. xvi. ver, 2.) It is plain that the
exact points of time here spoken of, with relation to the
diurnal revolution of the Earth, are different; and if we
assume a moment immediately preceding the elevation
of any part of the Sun's disk above the visible horizon,
the time referred to in the first passage will be before
such moment, and that referred to in the latter will be
after it; and consequently the conception of the one will
be as opposite to the conception of the other in this re-
spect, as before is to after. Nevertheless, they are both,
expressed in the Gothic translation by the word air, the
first being AIR uhtwon usstandands, the other, filu. AIR
this dagis : in the first instance, the Anglo-Saxon version
has swithe ER arisende : in the second, the Anglo-Saxon
has swythe ER daºye, and the Frankish, ER themo
liohte ; and comparing together these different uses of
the words air, ar, er; it is impossible not to perceive
that they sometimes stand for our word ere, and some-
times for our early. In the modern English version,
the two passages are correctly distinguished thus:
“in the morning, rising up a great while before day,
he went out”—and “very early in the morning, the
first day of the week, they came unto the sepulchre, at
the rising of the Sun.”
In our ancient writers, or is frequently used in a Ere,
similar sense with ere; but it may be doubted, whether
this be the same word differently spelt, or a contraction
of before. However this be, we find it both alone, and
followed by ere and ever; which may possibly be, a
Imere reduplication for the sake of greater emphasis,
as we have already seen in various examples. .
The various uses of these words, air, er, or, ar, ere,
or ere, and or ever, will appear from the following quo-
tations : - .
BARBour, in his introductory verses, uses air :
Old stories that men redis
Represents to thame the deidis
Of stalwart folk that lived air.
*: 3's 32 ::
He shuld that arbitry disclair
Of thir twa that I tald of air.
In the metrical Chronicle of England, composed in the
reign of Edward II., (see RITson's Metrical Romances,
v. iii. p. 337,) we find er, -
-- This lond was cleped Albyon
Er then Bruyt from Troye com.
G R A M M A R. 95
Eft.
rammar. In Octouian Imperator, ere and er-, .
They that were ere than agaste º 3
* Tho hadde game. . . . . . . . .
# , , , is ºk
' ' ) •
* That day Clement was made a keysht - -
For his er dedeswys and wyght.
In Richard Coer de Lion, or—
He it is, my dedly foo:
He schal abeyen it, or he goo.
In Kyng Alisaunder, ar and or—
No schal he twyes seo the sonne
Ar he have him perforce yixonne
3. : ºk 2: * .
For Alisaunder wol or night
Breke the castel doun ryght.
In Macbeth, or ere—
The deadman's knell
Is there scarce asked for whom ; and good men's lives
Expire before the flowers in their caps, -
Dying or ere they sicken.
In the Book of Daniel (ch. vi. ver. 24) or ever—
The lions brake all their bones in pieces, or ever they came to the
bottom of the den,
Erliche and erst, the compounds of ar, form first
adjectives, and then Adverbs, both retaining an exclusive
reference to time. The Adverb erliche occurs in CHAU-
cER's Knyght's Tale: -
And tellen her erliche and late.
Erst is the superlative of ar, being the Anglo-Saxon
arista, primus; and it is used in the senses of early
time, past or future, i. e. “formerly,” “soon.”
In the Romance of Sir Guy (see Warton, i. 170.) it
means “at any former time,” “before :”
Suche one had lie never erst seene.
In SPENSER's Fairy Queen, “at erst” is used for “at
the earliest future time,” “as soon as possible :”
Sir Knight, if knight thou be,
Abandon this forestalled place at erst.
Erewhile and whilere, are the same compound in two
different forms, but with a single meaning, viz. “a time
preceding the present, usually at no great distance,” as
in SHAKSPEARE's As You Like It :
That young swain that you saw here but erewhile ;
and in the Tempest—
Let us be jocund. Will you troul the catch
You taught me but whilere.
Of the other compounds from ar, viz. erelong, ere-
now ; and of those from ever, as, evermore, mever, never-
more, forever, &c. it is unnecessary to speak.
Ayforth is used by Barbour, as a derivative from aye,
ever, always;
To put in writ a sothfast story
That it last ayforth in memory.
Of the same origin with the Saxon af, and Latin
avum, seem to be the Gothic aiw and aiwa ; the Danish
evig; the Dutch eewig and eeuwe ; the German ewig ;
the Frankish and Alamannic evo and ewic ; the old
Danish or Runic aeff, aftsaga, aftntyr, &c.
Whether we ought to refer to the same origin the
Anglo-Saxon aft and old English eft, may perhaps be
doubted; but the fact of their common origin seems
not improbable. The words aft and afr certainly re-
semble each other in sound, and both relate to a com-
mon conception, viz. that of time.
time.
CHAUCER uses eft in the sense of a second time:
Were I unbound, also mote I the,
I wolde neuer eft come in the snare.
:k 2: , ºr 2k *:
For thee have I begon a gamen plaie
Which that I neuer doen shal eft for other
Altho he were a thousand fold my brother.
GAwrN Douglas uses it a little differently, in the
sense of “a short time afterwards.” Thus, in describ-
ing the snake, which, after devouring the offerings on
the altar, glided back into the earth, (AEn. 5. l. 92.) he
says, - -
And but mair harm in the graif entérit eſt.
In this latter sense the word eft is used by SPENSER :
Eft, through the thick they heard one rudely rush,
With hoise whereof, he from his lofty steed
Down fell to ground, and crept into a bush.
SPENSER also uses in this latter sense the compound
eftsoons : - -
Eftsoons the nymphs which now had flowers their fill
Run all in haste to see that silver brood.
Upon the whole, it appears that the Adverbs which
relate to time generally, are all traceable with more or
less distinctness to mouns, that is, to names anciently
given in various Dialects to the general conception of
The case is still plainer when we come to the
particular divisions of time, such as morning, evening,
day, night, week, month, year.
Tomorrow, our Adverb, which answers to the Latin Tomorrow.
cras, signifying the next day to that on which we speak,
is simply “the morning,” and in the present Scottish
Dialect is expressed “the morn;” as, “wul ye gang
til the kirk the morn ?”—“will you go to church to-
morrow 2° Morwe and dawe, in old English, meant
morning and day, from the old German morg and tag,
the final g being of an obscure sound between our y
and w. The morning is in Gothic maurgin, Alamannic
amorgan, Isl., morgun, Danish and Dutch mergen, modern
German morgen, and Anglo-Saxon marigen, mergen,
7morgen. WACHTER says, that in the ancient com-
putation of time the evening being reckoned first, the
morning came from that circumstance to signify the
future day. Whether this was the reason or not, the
fact is certain that most of the Northern Nations did so
use the word morning ; and hence we have the expres-
sions amorwe, amorrow, on morrow, by the morrowe,
tomorrow.
LYDGATE has “the morwe” for “the morning,” in
his Poem on the Virgin Mary (Harl. MSS. 2255. fol.
88.)—
Atween midnight and the fresh morwe gray.
CHAUCER, in the same sense uses morrow—
The merrie lark the messager of daie
Salueth in her songe the morrow gray.
RoBERT of GLoucestER uses amorwe, for “on the
following morning”—
Tho the kynges men muste amorwe wer he was bicome,
In the Proces of the Seuyn Sages it is used in the same
SellSe—
Amorewe themperour gan rise
And clothed him in riche gise.
So CHAUCER, in the Knyght's Tale—
And thus thei been departed till amorrow,
In Octouian Imperator we find amorn for the next day—
, Adverbs.
Sºº-y-Z


96
G R A M M.A .R.
Grammar.
Today.
Ambrn the emperoure, yn ire,
Sente aboute, in hys empyre,
After many a ryche syre
To deme her dome.
The folk tho cam from each a schyre
Ryght ynto Rome.
In Richard Coer de Lion occur, in this sense, on morwe
and on the morwe—
On morwe they .#. to ryde
With her hoost to Taburet.
* 2: 2: is .
On the morwe, withouten fayle, .
The cyte they gunne for to assayle.
CHAUCER seems to use on morrow for “in the morn-
ing,” as opposed to “in the evening,” in the Plowman's
Tale—
To worship God men would whate,
Both on even and on morrow,
Such harlotry men would hate.
So he says, “morrow milke” for “morning milk”—
An anelace and gipsere all of silke
Hing at his girdle white as morrow milke;
and in the same Prologue—
Well loued he by the morrow a soppe in wine.
Our present word morning seems to have been formed
as a participle from a verb to morrow, or to morwen,
whence we have the old words morrowing and morwen-
ing.
In DUNBAR's Goldin Terge—
Sweit were the vapouris saft the morrowing.
Hailsum the vail, depaynt with flowris ying;
and in CHAUCER's Troilus and Cressida—
Bright was the sonne and clere the morwening.
The word morn too seems to have been brought into
common use in Scotland, at least before the year 1449,
since it occurs in the Act of Parliament of that year:
The first of the thre tymis begynmande on the morn nixt after the
sheref court.
As we have morrow from morgen, so we have sorrow
from sorgen, the modern German word being derived
from the old German sorg, macror, tristitia, which
doubtless originates in the Frankish ser and our sore.
In one of the Harleian Manuscripts (No. 2251. f. 298.)
we find
To tell my sorwe my wittes bien all bare;
and in Octouian Imperator—
There was many a wepyng eye,
And greet sorwe of ham that hytseye.
The origin of the prefix to in our modern words to-
morrow, tonight, today, &c. will be considered when
we come to the prepositions. As to the distinction
adopted in modern times between the two words morrow
and morning, it is no more than what occurs in a variety
of cases; as in the instance just mentioned of sore and
sorrow ; where the former word, at least in its substan-
tive sense, is applied to a bodily disease, the latter only
to a mental affliction.
The Adverb today is of the same class with tomorrow.
Anciently we had the Adverb aday for “in the day-
time;” as in Syr Launfal (Cotton. MSS. Calig. A. 2.
fol. 39.)—
Aday whan hyt is lygt.
Of which expression we at present retain a trace in the
colloquial phrase now adays. In the same Poem the
substantive days is written dawes. The opening lines
alſe
Le doughty Artour's dawes
That held Engelond yn good lawes,
The substantive name of the conception, Day, was
easily converted into a verb, as in the very old Pastoral
Ballad (Harl. MSS. 2253. fol. 71. b.)—
In May hit muryeth when hit dawes.
The present participle of this verb’ occurs in the old
Scottish Song, the tune of which is said to have been
played by the troops of King Robert Bruce, in march-
ing to battle : -
Landlady count the lawing - -
The day is near the dawing
The cocks are at the crawing. -
But the participle is written dawening, as from the verb
dawen, or dawn, in Kyng Alisaunder :
In the cole dawenyng -
Wende we forth in althyng;
Then mowe we, God hit wote,
Resten our bestis in the hote.
In the time of Shakspeare, the substantive dawning
appears to have been most common ; as in King
Henry V— , -
Alas poor Harry of England he longs not
For the Dawning, as we do ; -
and in Cymbeline—
Swift, swift, ye dragons of the night ! that dawning
May bare its raven eye—
In more modern times the substantive use has come to
be confined to the word dawn.
Adverb:
The Adverb tonight presents in itselfnothing remark- Tonight.
able ; but it suggests an observation on the Latin
Adverb of the same signification. Cicero uses the ex-
pression noctu an interdiu, “by night or by day;” but
that neither this nor the ablative termination is neces-
sary to give the noun an Adverbial force, is evident
from the circumstance, that in the Laws of the Twelve
Tables, the simple nominative noa is used for “by
night:”
Que Nox fortom farsit, sei im aliquips occisit, joure caesos estod.
The Adverb anights was formerly in common use; as
in SHAKSPEARE's As You Like It—
CLowN—I remember when I was in love, I broke my sword
upon a stone ; and bid him take that for coming unights to Jane
Smile.
And, in like manner, an even was sometimes used for
“in the evening ;” as in The Seuyn Sages—
An even late the emperour
Was browt to bed with honour.
The substantive e'en for even is still retained in the com-
mon salutation of the Scottish peasantry, “gude en;”
but as we have changed morrow to morning, so we have
even to evening. The Germans, on the contrary, retain
morgen and abend. These circumstances appear to be
perfectly accidental; for whilst we have adopted the
participial termination in these two instances, we have
unaccountably rejected it from the word dawning.
The common people, in many parts of the country,
still use the Adverbial expressions to week, to month, and
to year, which are otherwise obsolete. Some copies of
Chaucer have this last expression in the IIId Book of
the Troilus and Cressida (v. 242.)—
Whan I the saw so languishing to yere.
But this is possibly an error of the transcriber.
Some Adverbs of time, which are probably derived
from substantives, are also Adverbs of place ; but, in
general, we mean to consider the Adverbs of place
G R A M M A R. 97
Deal.
Grammar, among the prepositions, since the same words are
-v- almost invariably employed for these two purposes.
Thus we equally say “John walks before,” in which
phrase “ before” is an adverb; and “John walks
before Peter,” in which phrase “before” is a preposi-
tion. So we say “ Peter walks behind,” or “ Peter
walks behind John ” and a similar observation applies
to the words about, above, below, &c.; hence, the old
jest, that a man beating his wife in an upper chamber
is a man of perfect integrity, “ because he is above,
doing a bad action;” or (with a slight variation of
expression) because he is above doing a bad action.”
The substantive Deal is often employed in the na-
ture of an adverb of quantity. Thus in Saint Mark's
gospel, c. x. v. 48.
He cried the more a great deal, thou Son of David, have mercy
In ancient times, this word, deal, entered into nume-
rous adverbs and adverbial phrases. We had halven-
dall, thriddendale, somdele, everydeale, a full great
dele, a thousand deale, &c. -
In Kyng Alisaunder we have both halvendall and
thriddendale.
Alisaunder and his folk alle,
No hadde nought passed theo halvendall.
3. $ 3.
The knighttes sloden on heighe brymme
And lepen into the cees arme :
That was bothe reuthe and harme.
Swithe wightlych hybigynne
The thriddendale, and fair swimme.
In CHAUCER, very frequently, somedele—
A goodwife also there was, beside Bathe;
But she was somedele defe, and that was scathe.
: ; § 3. *
The rule of Sainct Maure, and of Saint Benet,
Bicause that it was old and somalele streit,
This ilke monke did letten old things passe.
In the Romaunt of the Rose, he thus uses a thousand
dele, and euerydeale in the same passage:
Richesse a robe of purple on had
, Ne trow not that I lie or mad,
For in this world is none it liche,
Ne by a thousand deale so riche,
Ne none so faire, for it full weale
With orfraies laied was euerydeale.
Again in the Prologue to the Canterbury Tales, de-
scribing the Physician, he says,
He kept his Pacient a full great dell,
In houres, by his majike naturell.
Our word deal is the Anglo-Saxon substantive dal,
and Gothic dail, the Dutch deel, the Frankish and
Alamannic teil, and modern German theil, all which
signify a part or division. It is the same word with
our dale, because in hilly regions “ the dales” form
the great natural divisions of the country; and it is
also the Gaelic dal, a farm, or division of land occu-
pied by one tenant. As the verbs dailjan, dalan,
deelen, teilen, and theilen, corresponding with the
abovementioned substantives in the different northern
dialects, all mean to divide, so some others signifying
to divide by cutting, are reasonably believed to be of
the same origin, particularly the barbarous Latin ta-
liare, which is the origin of the Italian tagliare, and
the French tailler, from which last come our English
WOL. I.
substantive taylor (tailleur) and the proper names Tel- Adverbs.
fair (taille-fer), Tallboys (taille-bois), Talbot (taille- S-TYT’
bote), &c.
The old adverb afyn, appears to be the French sub-Afyn.
stantive fin “the end;” but in the following passage
from Syr Launfal (Cotton MSS. Calig. A. 2. fol. 35.b.)
it seems to be used as an adverb of quantity in the
sense of “sufficiently,” “to a sufficient end.”
Mete and drink they hadde afyn,
Pyement, Clare, and Reynysh wyn,
And elles greet wondyr hyt wer.
Trop, which in modern French is used to signify Trop.
the excess of quantity or quality, answering to our
adverb too, was in old French used for the adverb
much, as “ ceste aide eust este moult grant, et trop
plus que aides de fait de monnoye.” It is the Italian
adverb troppo, from the old barbarous Latin substan-
tives troppus and troppa, which last was a corruption
of turba. In the Alamannic laws (Tit, 72. s. 1.), “si,
in TRoPPo de jumentis illam ductricem aliquis invola-
verit,” “ if in a troop of beasts of burden, any person
steals the leading animal.” From troppus came the
French troupeau, as from turba came the old French
torbe, and old English turbe ; as “ToBBE dez cercieles,'
“ a turbe of teles.” (See the ancient manuscript en-
titled Femina, quoted by Hickes, v. i. p. 154.)
The substantives guise, or wise, way, and gate, fur- Guise, wise,
nish a variety of adverbs principally relating to the Wº, Bºº.
manner of doing an action.
Guise and wise are the same worb, being both iden-
tical with the Anglo-Saxon wise, Dutch wyse, German
weise, Italian and Spanish guisa, and French guise ;
the mode or manner of a thing's existence, that by
which it shows itself to us, or makes itself known.
Hence we find the German verb weisen, Dutch wyzen,
and Swedish wysa, to show, and the German verb
wissen, Frankish and Alamannic wizzen, Gothic and
Anglo-Saxon witan, Dutch weten, Swedish weta, and
Islandic vita to know: and probably from this source
are the Latin video and visus.
CHAUCER, who followed chiefly the French pronun-
ciation, uses the word guise— -
In swiche a guise as I you tellen shal.
BAR Bour, whose dialect was more purely Gothic,
says wise.
He sware that he shuld vengeaunce ta
Of Bruce, that had presumed sa,
Against him for to brawl or rise,
Or to conspire on sic a wise.
Hence we have the adverbs likewise, otherwise, (which
is the Alamannic andarwis), &c. which are sometimes
expressed in a substantive form, as “in like wise,”
“ in no wise”—“ in what wise"—on this wise.”
In likewyse it is statut be the haill parliament.
- Scottish Act, A. D. 1424.
Whosoever shall not receive the kingdom of God as a little
child shall in no wise enter therein.
St. Luke, c. xviii. v. 16.
For therein is taught how and in what wyse
Men vertues shulde use and vices despise.
MS. circa, A. D. 1480.
When Sir Edward the mighty king
Had on this wise don his liking
Of John the Baliol
BARBour. Book i. v.:189. ..
O ; : . .
98
G R A M M A. R.
Grammar. The modern English adjective righteous, and the
S-V-' Scottish legal term wrongous, with their derivative
adverbs, are originally from this source. Righteous
is the Anglo-Saxon rightwise, and it is used by BAR-
Bour, for right, lawful—
And that he cam to make homage
To him as to his rightwisk
# * * * *ghtwis kyng
Lawtie to love is no folly -
Through lawtie live men rightwisely.
In the Scottish acts, A. D. I.425, occurs wrangwisly—
Swa that the causis litigiousis and pleyis be not wrangwisly pro-
longit.
Gate is identical with gait, and means going ; hence
the gate of a city or dwelling is that through which
men go; the gait of an individual is his manner of
going; and in the old adverbs algates, the word gates,
means the modes of going on.
O thou, my love, O thou my hate
For the mote I be dede algate.
Gower, Confessio Amantis.
Barbour has “how gate”—“ this gate”—“many
gates,” &c.
He told him hailly all his state,
And what he was, and als how gate
The Clifford held his heritage.
:* 3. * #
This gate lived they, in sic thirlage,
Bailth puir and thay of hie peerage.
:* i. #. % p
For knowlege of manie estates,
May whiles avail ful many gates,
It is not surprising, that way should be used adver-
bially in the same sense as gate; since they both ori-
ginally signify a passage, or road by which we reach
our destined object. In modern English, indeed, we
apply the adverb always to time, but this is evidently
a secondary meaning. In the old Scottish writers we
meet with the phrases “ on woman ways”—“ on
Buchan ways,” &c.
In some satirical verses by one CLERK, a contem-
porary of Dunbar's, are these lines, ridiculing the
affected dress of a great man's servant—
With velvet bord about his threid-bare coit,
On woman ways weil tyit about his waist.
In the verses of ALEXANDER Scot, in praise of the
month of May :-
In May men of amours suld gae
To serve their ladies and nae mae
Sen thair relief in ladies lyes,
For sum may cum in favour sae
To kiss their luve on Buchan ways.
Our common adverb away seems to have been for-
merly written on way, and thence owai, as in the Seuyn
Sages, v. 1181.
The maister was owai inome
The Emprour was to chaumbre icome.
In Italian the simple noun via is used, as andate via,
go away.—So in Launcelot Gobbo's laughable soli-
loquy, in the Merchant of Venice—
Certainly, my conscience will serve me to run from this Jew
my master. The fiend is at mine elbow, and tempts me—via Ž
says the fiend—away ! says the fiend!
Kind which we now use only for “ sort” or “ spe-
cies,” was formerly nature, a signification which it
too few (supporters) against so many enemies.
many might easily bring three (persons) to death.”
long retained in the English idiom: as in Hamlet's Adverbs.
answer to the king :- \-N-
KING. But now, my cousin Hamlet—and my son—
HAMLET. A little more than kin, and less than kind.
That is—“ cousin and son a close affinity, in-
deed l—something more than a common relationship,
and yet something repugnant to nature—as he after-
wards intimates to the queen.
QUEEN. Have you forgot me 2
HAMLET. No, by the rood, not so :
You are the Queen—your husband’s brother's wife;
And—would it were not so—you are my mother.
Kin and kind, though thus used in contradistinction
by Shakspeare, were originally the same word, and
doubtless of the same origin with the Greek Yevos,
and Latin genus, connected with which are many large
classes of words in most of the northern languages.
In the sense of “ sort” or “ species,” it gave occasion
to the Scottish phrases allkin, or all kind, no kind, what
kind, &c. In the modern colloquial dialect of that
country the expression “allkin kinds of things” is not
uncommon. The other expressions occur frequently
in BARBOUR :—
But God that is of maist poustie
Reserved to his majestie
For to knaw in his prescience,
Of all kind time the first movence.
#. $ * 3.
But thay would upon no kind wise,
Ishe, to assail them in fighting,
Till cuſed wer the nobil king.
:* * #
The King Robert wist he was there,
And what kind chiftains with him were.
Besides the kind, or nature of an action, we may
advert to a variety of circumstances expressed by ab-
stract nouns, as wonder, ease, need, abundance, order,
chance, fellowship, &c. &c, and all these nouns may
take an adverbial construction.
The word wonder, has been used as an adverb, in Wonder.
different forms, as wonder, wonderly, wondrously.
Thus CHAUCER in the Romaunt of the Rose—
Such light sprang out of the stone,
That Richesse wondir bright shone,
Bothe her hedde and all her face,
And eke about her all the place.
BARB.our uses wondirly—
But wondirly hard things befel
To him, or he to state was brought.
“ Eath,” says JUNIUs, “idem est cum easie facilis;” Ease.
and easie he derives from the Gothic azets, whence
also the French aise.
In that early romance, the Geste of Kyng Horn, we
find the word ethe for easily,
The Kyng hade to fewe
Ageyn so monie Schrewe.
So fele myghten ethe
Bringe thre to dethe.
That is, “the King (Allof, the father of Horn) had
So
JoHN DE TREVISA, one of our earliest English prose
writers, has the following passage in his translation of
a Latin sermon of Radulf Bishop of Armagh, about
A, D. 1387.
“In my tyme in the Universite of Oxenford, were thritty thou-
sand scolers at ones, and now beth unnethe sixe thousand.”
G R A M M A. R.
99
Grammar.
If this were a solitary instance of the word, we
~~' might perhaps suppose it to be of the same origin as
Need.
Abundance.
Order,
Chance.
beneath and to signify “ under six thousand ;” but
numberless other instances show that it means
“ hardly six thousand.” Thus in CHAUCER's Canter-
bury Tales
The glorious sceptre and real majeste
That hadde the King Nabuchodomosor,
With tonge unethes may descrivid be.
So in the old ballads of The Huntyng of the Hare—
Sum theifondleyd on the grownd,
Althei wer wel my swomand,
Unethe thei had heir lyfe.
Need is from the Gothic nauth, Anglo-Saxon neod,
Alamannic not, Danish noed, Dutch nood, all implying
necessity, hard compulsion, or want. Hence our col-
loquial adverb needs, in the proverb “ needs must,
when the devil drives.” The Dutch proverb says,
nood breekt wet, for “ necessity has no law.”
In BARB.our we find needlings—
they that were arrested then
Were of their taking wonder wo;
But needlings it behoved so.
From the substantive, abundance, we have in mo-
dern English use only the adverb abundantly; but the
idea has perhaps in other times and countries given
origin to more than one adverb.
Mr. TookE. contends that the word asseth, in Chau-
cer's Romaunt of the Rose, signifies enough, sufficient,
in the following passage, applied to a miser—
Yet neuer shal make rychesse,
Asseth unto hys gredynesse.
Where URRY explains asseth to mean assent; and
interprets the passage thus; “ riches (here personi-
fied as a deity) shall not assent to the miser's greedi-
ness;” whereas Mr. Tooke more probably understands
it to signify, “ that riches will never give sufficiency,
or content to the miser's greediness;” in conformity
with the preceding lines—
Rychesse ryche ne maketh nought,
Hym that on treasour sette his thought;
For rychesse stonte in suffysaunce.
It remains to be considered whether asseth, in this
sense, comes from the French assez, or from the
Gothic word azets, above noticed. Tooke's argument
of forth from fors, as asseth from assez, is conclusive
neither way; for as forth does not come from fors, so
possibly asseth may not come from assez. Our law-
term assets is certainly the French adverb reconverted
into a noun, and it shows the origin of the word to
have been the Latin ad satis.
M. Court DE GEBELIN ingeniously traces another
French adverb to a source signifying abundance. Sou-
vent, often, he says, is from the Italian sovente, and
that from the Latin Sape, which he derives from the
Hebrew shepo abundance; and supposes that the En-
glish sheep may be from the same source, as implying
that in which the wealth of early ages almost exclu-
sively consisted; much in the same manner as pecu-
nia is derived from pecus.
The modern adverb orderly is expressed by GAwiN
Douglas, perordoure.
He had do schaw the credence that they brocht,
Perordou're albale thare answere faland nocht.
And in another place, he says—
Rowpand attanis adew, quhen all is done,
Ilkane perordoure.—
Adverbs.
The opposite idea, that of chance, has been ex-Chance.
pressed adverbially in various ways, as perchance,
percase, casually, peradventure, perhaps, mayhap,
haply. - -
SHAKSPEARE uses perchance, with a remarkable
diversity in the two following passages :-
Viola. Perchance he is not drown'd :—what think you, Sailors?
CAPT. It is per chance, that you yourself were sav’d.
Why Mr. Tooke derives chance from escheoir, rather
than from cheoir, the old French verb corrupted from
cadere, it is not easy to discover.
BAcon uses percase—
A virtuous man will be virtuous in solitudine and not only in
theatro ; though percase it will be more strong by glory and fame.
This word, Tooke observes, was anciently written
parcas, and it certainly was formed by the two French
words par cas, answering to the Latin per casum.
In the Scottish dialect we find in case used anciently
in the sense of lest, and the same use continues pro-
vincially to this day.
Thus ALExANDER MonTGoMERY says—
He hit the yron quhyle it was het,
In case it sould grow cauld.
Peradventure, (anciently peraunter and paraunter,)
is the French par avanture.
In the romance of The Lyfe of Ipomydon, we find
paraunter.
Tomorrow when I the duke see,
Paraunter in suche plyte I may bee,
That I wille the bataille take.
It also occurs in the forms of inaunter, inaventure,
be adventure, &c.
Thus GAwiN Doug LAs—
Quhen thyne allame musing as thou Salga,
Be aventure besyde ane water bra.
Perhaps, mayhap, haply, as well as the adjective
happy, and its compounds, are from the word happen,
anciently written hap, which was used both as a verb
and as a noun.
Gower uses hap as a substantive :-
The happes ouer mannes hede
Ben honged with a tender threde.
In the ballad of Octouian Imperator we find the sub-
stantive unhap.
He slogh the xii. dusepers of Fraunce,
Thys was unhap and hard chaunce,
To all Crystendome.
In CHAUCER we find uphap for perhaps, or upon
hap :—
Thou seekest rewarde of folkes smale wordes, and of vayne
praysynges. Trewely, therein thou lesest the guerdon of vertue,
and lesest the grettest valoure of conscyence, and uphap thy re-
nome everlastyng.
Z'est. of Loue.
Mr. Tooke seems to be in error in reckoning the
anomalous expression hab nab among the derivatives
from hap : it is rather from hab, the root of the Latin
habeo, and of the verbs habon, haben, &c. to be found
in all the Teutonic languages.
LILLY in his Euphues employs this expression adver-
bially— Y
O 2
100 G R A M M A. R.
Grammar. Philautus determined, had nab, to send his letters. sal, Anglo-Saxon saul, Alamannic sela, and Gothic Adverbs.
S-V- In the present day it is used rather as an interjec- sailºgla, the soul. * , tº g . - \-N--
tion on challenging a person to drink a glass of wine: We have traced the adverbial termination ly to the
and seems to have been originally a mark of discard- substantive leik, body; and therefore it is not sur-
ing ceremony. Hab ne hab Have it or not, as prising to find ilke (which is only the word leik in
you will; a form of speaking not unlike the vulgar another form) employed as we now use the word
will he, mill he, as in Hamlet :— *: C - - -
CLowN. Give me leave. Here lies the water. Good. Here UIS UHAUCER-
stands the man. Good. If the man go to this water and drown This ilke worthy knight had been also
himself, it is will he, mill he, he goes; but if the water come to Sometime with the Lord of Palatie.
him, and drown him, he drowns not himself. So in the Scottish de of desi ti th tº ſº
tº ſe So in the Scottish mode of designating the princi-
Fellowship. In speaking of the adverb together, we have al- pal family of a name, “Macpherson of that ilk,” is
SameneSS.
Self.
&
i.
g
g
g
g
ef
r iſ
i
g
:
g
g
#.
ready noticed the substantive fere, and the pronoun
same as used to imply fellowship. But it may be
worth while to trace these words still further. In
LYE's Junius we find “ Fere, vet. Angl. socius, D. S.
foera :” and the word is retained, in the same sense,
in the admirable and well known Scottish song of
Auld lang syne—
An' gie's a hond, my trusty feir,
An' here's a hond to thine.
An' we'll tak a right gude-willie waght
For auld lang syne.
In the romance of Octouian Imperator, we find in
fere used for in company—
Clement fleygh, and hys wyf yn fere
Into Gascoyne as ye mowe here.
In the ballad of “ A contrauerse bytwene a Louer and
a Jaye,” printed by Wynkyn de Worde, are these
lines—
The foules to here
Was myne entente,
Syngynge in fere
On bowes bente.
BAR Bour, describing the three traitors who attacked
King Robert Bruce, has these lines—
he perceived that in hy
By their effeir, and their having,
That they lov’d him in no kind thing.
And again— *
He said, you ought to shame, pardie,
Since I am one, and ye are three,
For to shoot at me upon feir.
As we have seen together and in same used synony-
mously in the same passage; so we may find pas-
sages which employ in fere and in same synonymously.
Thus in the romance of Richard Coer de Lion—
To Westemenstre they wente in fere,
Lordyngs and Ladyys that ther were—
And aftyr mete, in hyyng
Spak Kyng Henry our kyng,
To the Kyng that sat in same,
Leve Sire, what is thy name 2
In same corresponds exactly with the French en-
semble, as may be observed in the instance before
quoted, and also in the Lay le Fraine, which was evi-
dently a close translation from the French—
Le Codre and her mother there
Yn same unto the bour gan fare.
From close association, to identity, the transi-
tion is easy. As fere is connected with same, so is
same with self: and perhaps it may not be hazarding
too much to say that same is to be found in the sub-
stantive form in the Greek awaa, body; and self in
the German seele, Dutch ziel, Swedish sidel, Islandic
Macpherson of the same, or Macpherson of Macpher-
SOIl.
WAcHTER observes that the terminating particle
sam in German, which is our some, is synonymous with
lich (our ly); and that the German writers use pro-
miscuously friedsam and friedlich, for peaceful. So in
old English we find loathly and loathsome, lovely, and
lovesome. The Goths and Germans both compound
sam with leik, but in the inverse order, the former
using samaleiko, the latter gleichsam, adverbially for
“ as, like as, almost.”
The following words and particles, in various lan-
guages, seem to be connected with our English same
and some. -
In Greek, besides the substantive awaa, we find the
preposition ovv, or ovu, and (as the sibilant articula-
tion easily passes into the rough breathing) the ad-
jective 6aos, and the adverbs ºus and āua, with the
compounds of all these ; oriupaxos, one who fights in
the same cause; ovurá0eta, a feeling of the same kind,
avuºtevta, an agreement of the same sounds; Guotos,
like, or approaching to the same ; ousatos, of the same
essence, buoteatos, of an essence like, or approaching
to the same ; duáētos, the adverbial noun of āua,
duačpváðes, Hamadryades, nymphs who were born and
perished at the same time with the trees.
In Persian, the particle hem, agreeing nearly with
the Greek áua in sound, and entirely in sense, forms,
when prefixed to nouns, a class of compounds im-
plying society and intimacy, as hemdshiyan of the same
nest ; hemdheng of the same inclination; hemkhābeh of
the same sleeping place. - -
It might, perhaps, be thought too great a refine-
ment of speculation to suggest that in Latin homo was
connected with the Greek Čuos, as sum, sim, similis,
simul, semel, were with a vu ; but that these latter
agree in origin with our English word same cannot
reasonably be doubted.
In Moeso-Gothic we find sums, unus, aliquis, quidam;
samo, ipsum ; Saman, simul, una, pariter ; samalaud,
aequalia; Samaleiko, similiter; Samaleikos, convenientia,
&c.
In Anglo-Saxon, samman, to collect ; sibsum, pacific;
langsum, tedious ; Sam-wyrkan, to co-operate, &c.
In Frankish, liepsam, lovely; leidsam, loathsome ;
sama, as, in like manner, zisamane, together.
In Islandic, samfara, a society; samlag, marriage.
In Danish, samle.
In Dutch, samen, tzamen, together; samt, with ;
tzamenbinden, to bind together; tzamenhang, a series,
or connection; and numberless other compounds with
tzamen.
In German, sammt, with ; tammtlich, altogether,
G R A M M A. R. 10]
seltkalauffa, is raro occurrentia, seltsan, insolitum ; in Adverbs.
Grammar. sammlung, a collection; zajammen, together ; zusam-
Swedish saillsam, rare.
S-N-" menbinden, to bind together; and numberless other
compounds with zusammen ; also langsam, slow; lang-
samkeit, slowness.
In French, ensemble together; rassembler to collect
together, &c. -
In modern as well as ancient provincial Scottish
and English the termination some is used in many
compounds, otherwise obsolete, as winsome, grusome,
lissome, foursum, threttiesum, luvesome, wilsome, &c.
Lissome, a Wiltshire word, is an abbreviation of
lithesome, from the Anglo-Saxon lithe, pliant, and lith,
a joint; Islandic leda, to bend; old Scottish lidder,
flexible.
ALEXANDER Scot, in his “Justing and Debate,”
has—
Thou art mair large of lyth and lim,
Nor I am be sic thrie.
GAw1N Douglas—
His smottrit habit over his schulderis lidder,
Hang pevagely knyt with ane knot togidder.
DAVID LINDSAY uses the word threttiesum—
Thir currish coffes that sails owre sune,
And threttiesum about a pack.
ALEXANDER Scot has foursum—
For were ye foursum in a flock,
I compt ye not a leik.
LINDSAY also has wilsome—
He leaves his saul mae gude commend,
But walks a wilsome way, I wiss.
ALEXANDER MonTGoMERY uses lovesum—
Quha wald haif tyr’t to heir that tune,
Quhilk, birds corroborate ay abune,
With lays of luvesum larks 2
In the ancient MS. No. 2253, of the Harleian col-
lection, we find lossom—
The mone mandeth her bleo,
The lilie is lossom to seo.
And in another poem of the same collection—
With lossum chere he on me loh.
In the romance of Syr Launfal—
Sche had a croune upon her molde
Of ryche stones and of golde,
That lossom lemede lygt.
‘In the ancient ballad, “Blow Northerne Wynd,”
which was probably composed about A. D. 1200.—
A burde of blod and of bon,
Never yete y muste non,
Dussomore in londe.
Self may be traced in like manner through various
dialects, as the Moeso-Gothic silba, self. Frankish
and Alamannic selbo, self. Anglo-Saxon sylf, self.
Islandic sialf, self. German selbst, self or same, which
Wachter explains as selbist ipsissimus. And so in
compounds, as the Anglo-Saxon sylf-myrth, and Ger-
man selbst mord, self murder; the Islandic sialfvit-
ringur, self-taught; the Alamannic selpuuillin, self-
willed. It is not to be doubted but that self or selb is
allied to seld or selt ; and that both are from the more
radical sel, or sol, implying individuality.
In the Anglo-Saxon we find seld, rarus, with its
comparative seldor, superlative seldost, and compounds,
as seldhwanue, &c. In the Alamannic and Frankish
We find Shakspeare using both self and seld, in
modes now obsolete ; thus—
If I might in intreaties find success,
As seld I have the chance.
Troilus and Cressida.
seld shown Flamens
Do press among the popular throngs.
Coriolanus.
Being over full of self-affairs, my mind
Did lose it.
Midsummer Night’s Dream.
I would not have your free and noble nature,
Out of self-bounty, be abused.
Othello.
Shakspeare's compound seld-shown, is similar in
form to the old word selcouth, which we have disused
though we retain uncouth.
Selkouthe occurs in Kyng Alisaunder –
Thise men han selkouthe wyues
And childern bot ones in all her lyues.
And again, shortly afterward—
Now is this a selkouthe game.
Selcouth and uncouth are both from couth, which
seems to have some obscure connection with the
Gothic qithan, and Anglo-Saxon cuethan, to say ; but
in signification it means knew, or known.
Thus CHAUCER says of the Squire's Yeoman—
Of wood craft wel couth he al the usage.
Selcouth therefore is “ seldom known,” and un-
couth, “unknown ;” and the latter word shortened to
unco', forms in the Scottish dialect an adverb signify-
ing “extremely,” “prodigiously,” “strangely,” as
in Burns's “Address to the unco' guid, or rigidly
righteous.” Unco's also are used, in the same dialect,
substantively, for “ news,” and also for “strangers;”
and in old English uncouth is used as an adjective for
“foreign” or “strange.” Thus the romance of Ri-
chard Coer de Lion describes Henry receiving the King
of Antioch, and his daughter, on their arrival in
England—
Kyng Henry lyght in hyyng,
And grette fayr that uncouth kyng,
And that fayr lady alsoo,
Welcome be ye me alle too.
The French adverb guere, or gueres, furnishes a re-
markable instance of a substantive used adverbially,
if the derivation of this word by M. Cour. DE GEBE-
LIN, (who explains it as synonymous with our word
wares,) be correct.
The Dictionnaire de l'Academie, thus describes the
word, in its adverbial form—
GUERE, EREs, adv. Pas beaucoup, peu. Il ne s'employe jamais
qu'avec la negative : il n'y a guere de gens tout 3-fait desinter-
essez : il n'y a gueres de bonne foy dans le monde. On le met quel-
quefois dans le sens de presque point: et alons on le joint tousjours
avec que. Il n'y a guere que luy qui fust capable de faire cela,
c'est à dire, il n'y a presque que luy.
Gueres, then, adverbially used, signifies, according
to the academicians, “ not much,” “but little,” “al-
most none.” Certainly, these meanings are at a great
distance from wares, goods, or merchandize ; and yet
it is highly probable that M. Cour de Gebelin's deri-
vation is correct.
In the first place, it is not gueres alone, but negueres
Gueres.
102 G R A M M A. R.
Lastly, we may observe that the old French adverb Adverbs,
Grammar, which signifies pas beaucoup. We have already spoken
nagueres, (explained in the Dictionnaire de l'Academie to
\—y—/ of double negatives properly so called ; and we may
further observe, that in some idioms three, or even
four negatives are often accumulated on each other
without altering the general effect of the sentence.
In the following example from CHAUCER, there are
four in succession :-
He never yet no vilanie ne sayde,
In alle his lif unto no manere wight.
Instances of a like kind are to be met with both
in Greek and Latin writers. “ Notandum est,” says
VIGER, “ plures negationes interdum vehementius
negare ; ut Oiyêe pºv ćywye uſ) 8xt Tooto Touffoatpºt, Ne-
que ego id unquam fecerim. Quod Tullius ipse, cum
alibi, tum etiam de Finibus iii. cap. 15, imitatus est,
dum ait ; Quanquam negent, nec virtutes, nec vitia cres-
cere.” And HoogeveeN adds, “ Scitè admodium qua-
tuor negativa conjungit Plato, in Parm. prope finem.
"Ott TäNAa Tův um &vtwv 86evi 88aañ 88autºs 88eputav
icouvuvéav ćxet. Quoniam alia cum eorum, qua non sunt,
aliquo nullibi ullo modo aliquod commercium habent. But
the union of ne and gueres depends on totally different
principles ; and gueres has only come to be consi-
dered as a negative particle in French, from the long
habit of using it only in conjunction with a negative.
The same is the case with pas and point; pas being
the Latin passus, a step ; and point, the Latin punc-
tum, a point. Ne pas, therefore, is literally not a step,
ne point, not a point; and it is remarkable that we
use the word jot, signifying the Greek iota, or He-
brew jod, (which last is little more than a point in
writing,) nearly in the same manner. So in SHAK-
SPEAR E-
This nor hurts him, nor profits you, a jot.
And in like manner we use, not a whit, or no whit,
as in HookER—
The motive cause of doing it is not in ourselves, but carrieth us,
as if the wind should drive a feather in the air, we no whit further-
ing that whereby we are driven. *
Whit, then, or jot, may as well be called a negative
as gueres, or pas or point.
Secondly, ne gueres is literally “no abundance ;”
and M. de Gebelin traces three gradations of meaning
in the word gueres, viz. 1. Exchange ; 2. Things ex-
changeable, or commodities; and 3. Abundance of com-
modities, or abundance simply. Upon this last tran-
sition we may observe that the vulgar in the present
day use the word lots exactly in the same way; for
they not only say “lots of goods,” but “ lots of fun.”
As to the preceding steps in the derivation, WACHTER
considers the Teutonic waren, custodire, to be connected
with the Greek éptov, custodia, and also with the
barbarous Latin warenna, a warren.
JUNIUs explains ware, merx, mercimonium, A. S.
wara, B. wagre, Su, wara ; and, he adds, “ potest vox
desumpta videri ex waeren, cum curá custodire; quod
mercimonia sollicitè custodiantur.” MENAGE in his
Origines de la Langue Francoise, says of the word guerite.
“On pronongoit anciennement garite. Les Espagnols
disent aussi garita. Il y a apparence qu'ils ont pris ce
mot de nous, et que nous l'avons pris de l'Allemand,
ou du Flaman, waeren, ou bewaren, qui signifie garder,
sauver, conserver.” It is scarcely necessary to add
that the French initial gu is the Saxon w; as in guespe,
wasp ; Guillaume, William ; guichet, wicket, &c.
signify iln'y a pas long temps,) is only another form of
this same substantive, Wares, restricted to the signi-
fication of time ; as in the example, Cet homme qui
nagueres estoit les delices de la cour. “ This man who,
not long since, was the delight of the court.” In this
sentence, nagueres is an abbreviation of the sentence,
Il n'y a gueres de tems, “ there is but a little time.”
Various parts of the body afford (as Tooke observes Parts of the
particularly of the hand and foot,) a variety of allu-body.
sions, and adverbial expressions in all languages.
Thus we have headlong, chiefly topsy-turvy, vis-à-vis,
face to face, at eye, by eye, at the eye, near hand, hond
habbing, maintenant, à genoua, aside, aback, astride, a
foot, on foot, foot-to-foot, foot-hot, pedetentim, &c.
To begin with the head. We will not insist on the Head.
derivation of the Latin apud, from caput, according to
some etymologists; but caput was certainly the ori-
ginal of the Italian capo, which is our cape, (or head-
land,) and the French chef, “On a fait chef de capo,”
says Menage, “ comme chen de cane, qu'on a depuis
prononcé chien.” Hence from caput comes pocket-
handkerchief, thus, chef the head, couvre-chef, kever-
chefe, kerchief, a napkin for covering the head or
neck; handkerchief, the same napkin carried in the
hand; and pocket handkerchief put into the pocket ;
whence the Scotch often say pocket-napkin.
In the romance of Richard Coer de Lion.
The #ever-chefes he toke on honde,
And thought in that ylke while,
To slee the Lyon with some guile.
De son chef is an adverbial phrase in French, an-
swering to our colloquial expression, of his own head.
‘‘ On dit aussi de son chef,” says the Dictionnaire de
l'Academie, “ pour dire de luy mesme, de son mouve-
ment, de son authorité. Il a fait cela de son chef, sans
en avoir ordre. Chef, the head, being taken for the
whole person, the tenant in chief was one who held
lands of another directly in his own person. “ All
tenures being derived, or supposed to be derived from
the king,” says BLACKSTONE, “those that held immedi-
ately under him, in right of his crown and dignity
were called tenants in capite, or in chief.” . From the
same origin is the old law term chivage, or chevage.
“ Villeines,” says Sir Edward CokE, use to pay their
lords an acknowledgement of their bondage, for their
several heads; and thereupon it is called chevage, che-
vagium of the French word chef, as it were the service
of the head. Of which Bracton saith chivagium dici-
tur recognitio, in signum subjectionis et dominii de
capite suo.” Thus our poll-tax, and polling at ekec-
tions, are from the poll taken for the whole head,
and that for the person. The adjective chief is some-
times used by the poets adverbially, but we more
commonly say chiefly. Chiefest is used by MILTON.—
But first, and chiefest, with thee bring,
Him that yon soars on golden wing,
Guiding the fiery-wheeled throne,
The cherub, Contemplation. -
The French had an old adverb derechef, for une
autre fois, de nouveau ; but it is now obsolete. ME-
NAGE says, “ il vient de derecapo composé de ces trois
mots, de re capo.”
Of the word headlong, Johnson says, (with little
G R A M M A R.
103
Aur trousses. Façon de parler du style familier, pour dire à la Adverbs.
Grammar, consideration of grammatical principle,) “it is often
*~~' doubtful whether this word be adjective or adverb;”
Eye.
Foot.
and thereupon he cites from SHAKSPEARE an instance
on which one would think no grammarian could have
doubted for a moment—
I’ll look no more
Lest my brain turn, and the deficient sight
Topple down headlong,
Headlong here applies most emphatically to the
verb topple, in the manner specified by Donatus; ad-
jecta verbo significationem ejus complet.
The modern adverb, “ visibly,” supplies the place of
several adverbial phrases, relating to the eye, in an-
cient authors.
Thus CHAUCER uses at eye—
This maiest thou understand and seen at eye.
GoweR has at the eye—
The thing so open is at the eye
That every man it may behold.
In the romance of King Alisaunder, we find by
eyghe-
Theo two barouns he kneow by eyghe,
The foot supplies various adverbs and adverbial
phrases, as a foot, pedetentim, and the remarkable ex-
pression foot hot, occurring frequently in Chaucer,
Gower, Gawin Douglas, &c.
A foot is obviously the same as on foot, which oc-
curs in the tale of The Seuyn Sages, in rather a sin-
gular passage—
A childe thai had bytwix tham two
The fayrest that on fote myght go.
The Latin adverb pedetentim, is thus explained by
Voss IU's.
Pedetentim, quasi pede tentando. Cato dierum dictarum de
Consulatu suo, Eam ego viam pedetentim tentabam. Est apud
Charisium in II.
“ Foothot,” says Mr. Tooke, “ means immediately,
instantaneously,” and so far he is undoubtedly right;
but whethet hot, means, as he supposes, heated, or as
WAR.ToN suggests, hit against the ground, that is,
stamped, may be matter of doubt. “In the twinkling
of an eye,” “ in the space of a look,” “ at a glance,”
are expressions used to express the shortest possible
lapse of time: and “ a stamp of the foot,” may well
be supposed to convey a similar idea of brief duration.
DUNBAR, in his Goldin Terge, has the following
lines :-
And suddenlie, in the space of a luke,
All was hyne went, ther was but wilderness;
Ther was nae mair but bird, and bank, and bruke.
In twinckling of an ee, to schip they went.
E vestigio is a well known Latin phrase for con-
festim, properanter, &c.; thus Cicero, giving an ac-
count of the assassination of Marcellus says, “e vesti-
gio, eo sum profectus.” By a similar analogy we say
one misfortune treads upon the heels of another : and
thus in Timon of Athens, the Poet answers the Painter's
question :-
PAIN. Sir, when comes your book forth 2
PoET. Upon the heels of my presentment.
In this sense the French colloquially use aur trousses,
or a ses trousses, Thus in the Dictionnaire de l'Acade-
77276 :-
poursuite.
On dit aussi estre awar trousses de quelqu'un, pour dire estre tou-
jours à sa suite, soit pour l'espionner soit de quelque autre maniere
qui l'incommode.
The good old Bishop LATIMER, who, it must be
confessed, was more remarkable for piety in his ser-
mons, than for elegance of style, uses “in their tailes.”
I will be a suter to your Grace that ye wil geve your Bishops
charge, ere they goo home, vpon theyr allegiaunce, to loke better
to theyr flocke—and send your visitours in their tailes.
Fote hot is generally accompanied with some other
expression serving still more clearly to shew the idea
of quickness, which it is meant to convey.
Thus Gower—
And forthwith, all amone, fote hote,
He stale the cowe.
So CHAUCER—
And Custaunce han they taken anon, fote hot.
And GAwiN Douglas—
The self, stound, amid the preis, fute hote,
Lucagus enteris into his chariote.
The same idea is expressed by the phrase in a trice,
for which Gower uses as who saith treis.
All sodenly, as who saith treis,
Wher that he stode in his paleis,
He toke him from the men's sight;
Was none of them so ware, that might
Set eie wher he becom.
It is therefore probable that a trice meant no more
than the time of crying “ thrice l’’-a common signal
for starting in a race, launching a vessel, &c. after
once, and twice have been called out as notes of pre-
paration.
For near, in point of time, we find nearhand used,
though rather as a preposition than an adverb, in the
Scottish Act of Parliament, A. D. 1429, “gif it be
mere hande the Witsonday or Martyn mes, the seysing
salbe gevin to the party contrary.” This is not very
unlike the French use of maintenant for “ now.” Hond-
habbing, which is a more exact translation of mainte-
nant, is used in a different sense in the abovementioned
tale of The Seuyn Sages —
'Th’ Emperour saide, I fond hire to rent,
‘Hire her and hire face ischent;
'And who is founde hond-habbing
"Hit nis non nede of witnessing.
Hondhabbing, or hand-habend, is a law term of Saxon
origin, corresponding with the Norman term mainour,
or manner; and they are both applied to a thief taken
flagrante delicto, with the goods stolen in his hand :
see Leg. Hen. I. c. 59 ; Bracton, lib. 3. tract. 2. cap. 8.
&c. “One mode of prosecution, by the common
law, without any previous finding by a jury,” says
JAcoB, “ was when a thief was taken with the main-
our, that is with the thing stolen upon him, in manu.”
The French adverb maintenant, which is literally the
same as hond-habbing, being formed of main, the
hand; and tenant, holding ; has come to be restricted
by use to the signification of “ now ;” that is to say,
the time, which we hold, as it were, by the hand;
opposed to that which we have suffered to escape;
for the word maintenant, ‘‘ now,” is used in contra-
distinction to autrefois, “ formerly.”
We use the expression at hand, as the French do à
la main, and the Germans, bey der hand, to signify a
Je lui mettray un prevost aux trousses, a ses trousses. S-N-
104
G R A M M A. R.
Grammar.
ditated oration “ an extempore speech.”
thing that is near, or within reach. Thus SHAKSPEARE
in the first part of HENRY IV.
GADSHILL. What, ho Chamberlain
CHAM. At hand, quoth pickpurse.
GADSHILL. That's even as fair as at hand, quoth the
Chamberlain.
In Latin we find ad manum, and sub manu, differing
from each other, if at all, only by slight shades of
meaning, as they both do from in promptu, and er
tempore, which two latter we have naturalised as a
substantive and adjective ; for we call an unpreme-
ditated epigram “ an impromptu ;” and an unpreme-
** Ad manum
esse,” says STEPHANUs, “est aliquid ita in promptu
esse, ut quasi manu teneatur.” Thus Livy, “adde
quod Romanis ad manum domi supplementum esset.
We find the phrase sub manu employed by PLANcus in
a letter to Cicero, “ Vocontii sub manu ut essent, per
quorum loca mihi fideliter pateret iter.” This, as
MANUTIUS observes, is a Grecian mode of speaking ;
for LUCIAN says, "Ott öv Tpwtov bro Tiju Xeºpa &A0m,
“ quod primum sub manu venerit.” The Greeks also
have the adverb 7poxeipws, answering nearly to our
phrase “ out of hand.” In some parts of the West of
England, the adjective handy is used as an adverb or
preposition with reference to place, as “he lives handy
Warminster;” or, “ he lives handy.”
The Grecians used 8td. Xeupwu, as Cicero does de
manu in manum, which the French have literally co-
pied in their phrase de main en main, and we in ours,
‘‘ from hand to hand.” The Dictionnaire de l'Acade-
mie exemplifies this by the following sentence, “ C'est
une tradition que nos ancestres nous ont laissée de main
en main.”
We say, “ to have a work in hand;” the Germans
say, “ unter den handen haben.” We also say “to
take it in hand;” they say “vor die hand nehmen.”
En un tourne-main is an adverbial phrase in French,
to signify a very brief space of time, not longer than
is necessary to turn the hand : “ c'est un esprit incon-
stant : il change en un tourne-main.”
The allusion to the hand seems to be altogether su-
perfluous, in our adverbs beforehand and behindhand.
Thus in ARBUTHNoT's History of John Bull—
When the lawyers brought extravagant bills, Sir Roger used to
bargain beforehand to cut off a quarter of a yard in any part of
the bill.
The subtantive use of the word forehand is more
emphatic ; as in King Henry the Fifth's fine soli-
loquy—
And but for ceremony, such a wretch,
Winding up days with toil, and nights with sleep,
Had the forehand and vantage of a king.
Dr. Johnson accuses Shakspeare of a licentious use
of the adverb behind hand, as an adjective ; but the
Face, front.
truth is that the mighty poet knew and felt the powers
of the English language much better than his critic.—
and these thy offices
So rarely kind, are as interpreters
Of my behind hand slackness.
Winter’s Tale.
The face, and front, or forehead, furnish many ad-
verbs and adverbial phrases in various languages.
Shakspeare has a front for “in front,” “indirect oppo-
sition to the face,” as in Falstaff's inimitable narrative
of his pretended combat:—
These four came all a front, and mainly thrust at me.
made me no more ado, but took all their seven points in my tar-
get, thus.
The Latin primá facie has become naturalised in
English style; so that we even speak of “a prima
jacie case.” Quintilian has primd fronte, lib. vii. c. 2.
“ dura primâ fronte quaesto.” In the same sense we
say “ at the first blush, this question appears difficult.”
We have also copied the Greek adverbial phrase
Tpdawzrov Tpos ºrpoo wrov, in our “face to face ;” as
in St. PAUL's First Epistle to the Corinthians, ch. xiii.
ver. 12. “ For now we see through a glass, darkly;
but then face to face;” whence the French adverb and
preposition vis-à-vis is also taken ; for vis in old
French was a face. Thus MARoT says in his Temple
of Cupid :-
Car en ce lieu Vn grand prince je vis,
Et vne dame excellente de Vis.
The Italian adverb dirimpetto, expresses the same Breast.
idea of direct opposition, but refers to the breast,
petto (from the Latin pectus) instead of the face. Our
adverb abreast is employed in a different sense, which
however is not very happily explained by Dr. JoHN-
son. He says—
ABREAST, adv. [see BREAST.] Side by side; in such a position
that the breasts may bear against the same line.
And then he quotes, as an illustration of this idea,
the animated exhortation of Ulysses to Achilles—
Take the instant way;
For honour travels in a strait so narrow,
That one but goes abreast.
Surely Ulysses did not mean to advise Achilles to
advance “side by side” with any other warrior; but
on the contrary to keep the path in which but one
could travel, and particularly not to suffer Ajax to
advance “in the same line” with himself.
In petto has been adopted as an adverb from the
Italian language into the English ; but only in a
figurative sense. We say, “I have a scheme in pett
to attain this object;" that is, I have it in reserve, un-
known to my adversary.
I Adverbs.
The French sometimes use à genoua, ‘‘ on the Knees.
knees,” in a figurative sense. “ Je vous le demande à
genoux,” says the Dictionnaire de l'Academie, “ sig-
nifie aussi, demander avec un grand empressement.”
Our wellknown adverbs aside, aback, ahead, &c. Side.
scarcely need further notice, than merely to show
their analogy to the class of adverbs and adverbial
phrases of which we are now treating. -
CLERK, the Scottish poet, in his satire on Pride,
describing the dress of a proud serving-man, men-
tions—
His hat on syde set up for ony haist.
And so GAwin Douglas—
Now bendis he up his burdoun with ane mynt,
On syde he bradis for to eschew the dynt.
In FALconER's Marine Dictionary we find the fol- Back
lowing explanation of the technical meaning of the
adverb aback, as applied to the sails of a ship:—
A back, coeffé, the situation of the sails when their surfaces are
flatted against the masts by the force of the wind. The sails are
said to be taken aback when they are brought into this situation
either by a sudden change of the wind, or by an alteration in the
ship's course: they are laid aback to effect an immediate retreat
without turning to the right or left.
G R A M M A. R. 105
Grammār.
The simple noun back is also used adverbially, of
S-v-' which Dr. Johnson has given a variety of examples,
Topsy-
turvy.
Upside-
down.
diversifying their supposed signification according to
the context; as, 1. to the place from which one came;
2. backward from the present station ; 3. behind, not
coming forward ; 4. toward things past; 5. again, in
return; and 6, again, a second time; but in reality
the force of the word back is in all these instances very
nearly, if not altogether identical ; having the same
analogy to the time, place, or other circumstances
spoken of, as the back has to the other parts of the
body, in their general position. -
Lo! the Lord hath kept thee back from honour.
Numbers xxiv. 11.
But where they are, and why they come not back,
Is now the labour of my thoughts.
MILTON.
In the first instance, honour is represented, as it
were, before the person; but he is prevented from
advancing toward it ; in the other instance, the indi-
viduals in question have gone forward, and are ex-
pected to return; and in both cases the situation
expressed by the adverb back is that in which the
backs of the persons were originally placed.
Where we use the simple substantive back, adver-
bially, the adverb arere from the French arriere, was
formerly employed; as in Richard Coer de Lion—
Kyng Richard bethought hym thoo,
And gan to crye, “Turne arere,
Every man with his banere.”
From back we form the compound adverb back-
wards, as from fore we do forwards ; and these words,
backwards and forwards are directly opposed to each
other in signification, as they are in etymology.
Topsyturvy, and upsidedown, are adverbs perfectly
familiar and intelligible in modern colloquial usage;
but somewhat obscured by the learned labours of
etymologists. SKINNER suggests that topsyturvy is
“quasi tops in turves, i.e. vertices seu capita in ces-
pite.” LYE says, “ Topsy-turvy, inverso ordine.
Haud scio an sint a top, fastigium, et Isl. tyrva, ob-
ruere. BARB.our uses the phrase top o'er tail.
And when the king his hounds has seen,
These men assailyie their master sa,
They lap to one, and can him ta
Right by the neck full sturdily,
Till top o'er tail they gart him fly.
Upsidedown is so written by SPENCER, RALEGH, and
other writers of the age of Queen Elizabeth ; but
some older authors write it upsodown, which Tooke
(for what reason does not appear) considers the more
proper form of the word.
In the romance of the The Seuyn Sages we meet
with this phrase several times repeated—
Bitwene the adder and the grehound,
The cradel turnd up so down on ground.
* $ #. #
The cradel and the child thai found,
Up so down upon the ground.
$ * # *
Of the adder he fond mani tronsoun,
And the cradel up so down.
So Gower :-
—If the lawe be forelore,
Withouten execucion,
It maketh a londe turne up so downe.
Correspondent with the English upsidedown, or
WOL. I,
upsodown are the Italian sossopra, and the French Adverbs.
sans dessus dessous, which the Dictionnaire de l'Aca- S-N-'
demie thus explains— z
SANS DEssus DEssous. Façon de parler du style familière,
qui signifie qu'une chose est tellement bouleversée, qu'on ne re-
connoist plus ni le dessus, ni le dessous. On dit aussi sans devant
derrière, pour dire qu'on ne reconnoist plus ce qui doit estre der-
rière, mi ce qui doit estre devant.”
MENAGE says, “il faut escrire sens dessus dessous,
comme on escrit en tout sens, de ce sens lä, &c. Sens
c'est-a-dire face, visage, situation, posture, &c.
Others again say it should be written c'en dessus
dessous, as being taken from the old phrase ce que
dessus dessous, used by CoMMINEs the historian, “ De
tous costez ay veu la maison de Bourgogne honnorée,
et puis tout en un coup choir ce que dessus dessous.”
Ahead is principally used as a sea-term ; as in
DRYDEN–
And now the mighty. Centaur seems to lead,
And now the speedy Dolphin gets ahead.
Our mariners, indeed, appear to have had a special Prefix, a.
affection for this prefix, a ; for they have a vast va-
riety of adverbial expressions, in which it is employed,
as aboard, ashore, ahull, apeek, atrip, aweigh, abaft,
aloft, aftoat, astern, alee, aloof, alongside, alongshore,
amidships, athwartships, &c. &c. all of which are fully
explained in the work before referred to, FALconER's
Marine Dictionary. Many other adverbs there are,
ancient and modern, beginning with the same prefix,
besides those already noticed; as aswom, alive, aftre,
ablaze, aloud, asleep, aroune, alove, abroad, alength, &c.
In the romance of Amis and Amiloun, occcurs aswon Aswon.
for “ in a swoon.” * -
He loked opon his scholder bare,
And seighe his grimly woundethare,
As Amoraunt gan him say.
He fel as won to the grounde,
And oft he seyd, Allas, that stounde,
That euer he bode that day.
So in the romance of The Seuyn Sages—
The Leuedi when sche herde this,
21swome sche fil adoun I wis.
In Octouian Imperator, we have on lyue, for “alive.” Alive.
Her some bygan to the and thryue,
And wax the fayryste chylde on lyue.
So in CHAUCER's Troilus—
By God, quoth he, that wol I tel as bliue,
For prouder woman is there none on liue.
So likewise in a MS. ballad written about the time
of HENRY VI. entitled “How a Merchande dyd hys
Wufe betray.”—
Y thanke hyt God, for so y may,
That evyry skapyd on lyve away. -
In “ A mery Geste of the Frere and the Boye, em-
prynted at London, by Wynkyn de Worde,” we find
“ thy lyve,” used for “thy life.” v. 86.
That shall last the, all thy lyve.
CHAUCER has “hir live,” for “ in their lives.”—
They were ful glad to excusen hem, ful blive,
Of thing the which they neuer agilt hir live.
In another passage he extends this adverbial phrase
to a greater length' “ time of al here lyues.”
Ne neuer shul, time of al here lyues.
The adverb blive, which occurs above, is thus noticed
P :
106
G R A M M A. R.
Grammar, in LYE's Junius, “ belief, belife, belive, blive, confestim,
S-N-" protinus, statim, extemplo : a Norm. Saxonico bilive,
Afire,
ablaze.
Alond.
Asleep.
Aroune.
Alove.
Alength.
Home.
de quo nihil certi habeo quod dicam.”
little doubt, however, but that it is from the substan-
tive, “ life,” and signifies in a quick and lively
Iſla IIIler.
It occurs in the romance of The Seuin Sages.
His owin Lady he toke byliue,
And gaf the Knyght until his wiue.
The same adverb is found in the ballad on the
Battle of Bruges, beforementioned. -
Thenne seide the Kyng Phelip lustneth nou to me,
Myn Eorles ant my Barouns gentil ant fre,
Goth faccheth me the traytours y bounde to my kne,
Hastifliche ant blyue. -
GAwiN Douglas uses in fyre, for our modern ad-
verbs a fire —
Turnus seyes the Troianis in grete yre,
And althare schippis and nauy set in fire.
In like manner, Gower employs the expression on
blaze for our colloquial adverb ablaze —
That casten fire and flam aboute
Both at mouth and at nase,
So that thei setten all on blase.
In the romance of Octouian Imperator is alond, for
** to land.”
The Kyng of Masydonye com ryde
With hys ost along that tyde,
jº, 3. #
The Kyng of Greece herde that cry,
To lond he rowede ryght hastyly.
In Amis and Amiloun also occurs in slepe, for the
modern “asleep.”
The Knight that was so hende and fre,
Welfair he leyd him vnder a tre,
And fell in slepe that tide.
In Richard Coer de Lion, aroume for “ aside.”
Alle that was ther tho hym beheeld,
Hou he rod as he wer wood,
Aroume he hovyd, and withstood.
CHAUCER, in the Testament of Love, uses the adverb
aloue, for in love.—
Wo is hym that is aloue !
We have before mentioned on brede, which is used
by CHAUCER and Douglas for abroad, or on breadth ;
similar to this is the adverb alengthe, used by NicoLLs,
whose work was published in 1550, under the follow-
ing title. “The hystory writtone by Thucides the
Athenyan, translated oute of Frenche into the Englysh
language, by Thomas Nicolls, Citezeine and Golde-
smyth, of London.” In fo. 118, a, is the following
passage, “They dyd take a greate piece of timber
and made it hollowe—afterwardes they fastened yt
wyth yrone at bothe endes, and also alengthe.”
The substantive home is used adverbially in En-
glish both in its simple sense of a place of residence,
as “to go home ;” and in the figurative meaning of
completion, which SHAKSPEARE seems particularly
fond of giving to it—
No further halting. Satisfy me home
What is become of her
Cymbeline.
It confirms me home,
This is Pisanio's deed.
Ibid.
There seems.
He charges home my unprovided body.
Dear,
Wearthy good rapier bare, and put it home,
. Othello.
In the simple sense of a dwelling, our adverb home,
answers to the Greek adverb oucače, and to the Latin
accusative domum, as the word heim does in the Ger-
man compound heimgehen, “to go to our place of re-
sidence.” But though the nouns house and home,
may in certain cases be applied indifferently to the
same edifice, yet we not only do not use the word
house adverbially, as we do home, but we affix a dif-
ferent idea to it when used substantively, with the
preposition “to.” This peculiarity of idiom cannot
be better exemplified than by a circumstance which
occurred to a German nobleman, who not long since
visited London. Nach hause gehen, in German, and
aller a la maison, in French, both signify “ to go
home,” the foreigner, therefore, returning from a
visit, thought that he could not err in ordering his
coachman to go “to the house;” but as the latter had
been accustomed to drive some of his former masters
to “the House of Commons,” which alone he knew
by the distinctive name of “ the house,” he accord-
ingly proceeded thither, instead of conveying the no-
bleman to his own residence. -
As “ home” answers to the Latin domum, so “at
home,” answers to domi; for as Vossius observes of
“ domi focique,” in Terence, (Eun. Act IV, scen. 7,)
“ dubium non est quin sint genitivi adverbialiter po-
sitivi.” DoNATUs, indeed, goes further ; for he calls
not only these genitives, but even accusatives and ab-
latives, adverbs. “ Roma Romam, Romá,” says he,
“ sunt adverbia loci, quae imprudentes putant nomina.
In loco, ut sum Roma: ; de loco, ut Romá venio; ad
locum, ut Romam pergo. And with this very learned
grammarian agrees SERVIUS. DIoMEDEs, in like man-
ner, calls vili and carö “ a-stimationis adverbia;” and
others call forte, fortuna, nihil, casu, militia, belli, &c.,
adverbs; which doctrine is strenuously resisted by
VossIUs in his first book De Analogid. It is not here
necessary to examine this dispute very minutely; but
we may observe that the distinction between an ad-
verb, and a genitive case used adverbially is not
made out by VossIUs with that clearness for which
his grammatical writings in general are remarkable.
It may be allowed that where a noun substantive or
adjective is joined with another, either expressed or
necessarily understood, it should rather be considered
as making a part of an adverbial phrase than as an
adverb. Thus sponte sua, domi suae, or mane primo,
may be regarded respectively as clauses in a sentence;
but sponte, or domi, or mane, alone may be called ad-
verbs ; and such is the distinction drawn by that ex-
cellent grammarian PRIscIAN.
Of the Latin adverbs, palam, and clam, Mr. TookE
quotes, with some approbation, the etymology given
by M. L'Eveque, who derives them from the Sclavonic
pole, “ the earth,” and kolami, ‘‘ wooden stakes.”
This derivation seems farfetched; yet it is not impos-
sible that some affinity may have existed between the
radical sounds of the Sclavonic and ancient Latin
languages. Certain it is, that clam was originally
written calim, as in the law of the Twelve Tables,
quei CALIM endo urbe now coit coiverit kapital estod.
This was the law against secret societies which Por-
Adverbs.
\-N-
Domi,
Palan.
Clam.
G R A M M A. R. J07
CHAPMAN, the most poetical of all translators of Adverbs.
Grammar. cius Latro charged Catiline with having violated.
Homer, abounds in such epithets, as gold-helm’d, V-V-
J– Calim, “secretly, obscurely,” had evidently a relation
Compound
adverbs.
to caligo, obscurity, or cloudy darkness, and caligo
may possibly have been derived from cala, a wooden
log or stake, which thrown moist on the fire would
produce a thick smoke :—
—lacrimoso non sine fumo,
Udos cum foliis ramos urente camino.
. The word cala is thus explained by SERVIUS ;
‘‘ calas dicebant majores nostrifustes, quos portabant
servi sequentes dominos ad proclium. Unde etiam
calones dicebantur Nam consuetudo erat militis
Romani, ut ipse Sibi arma portaret et vallum. Vallum
autem dicebant calas. Sic Lucilius, -
Scinde calam, ut caleas:
i. e. O puer, frange fustes, et fac focum.” The deriva-
tion of palam it is not so easy to trace. It signifies
“ openly, publicly,” as in VIRGIL–
Ipsa palam fari omnipotens Saturnia jussit.
And it may have some relation to the verb palo “to
wander about,” as in SULPICIUS.
Sic nostri palare senes dicuntur.
Or to Pales the rural goddess invoked by VIRGIL.
Te quoque magna Pales.
Or to palus, a marsh, or palus, a pale or stake.
Possibly all these words, though differing in the
quantity of their first syllable, which in some is short
and in others long, may have had an indistinct con-
nection ; but be this as it may, we can scarcely doubt
that the adverb palam was derived from some noun in
the old Latin language, and was indeed that noun in
an antiquated form. It must be observed too that it
was not used merely as an adverb, but as a preposi-
tion. Thus LIvy says “ palam populo.” CIcERo
“ palam hoc ordine,” and HoRACE “te palam,” which
last example proves the error of Calepin and others,
who thought that coram was “ in presence of one
person,” and palam. “ in presence of many.” Clam
also was used as a preposition, as in TERENCE, “ Haec
clam me omnia;” nor was this all; for it was some-
times used adjectivally, by the same author. “Si
sperat fore clam:” in which manner also the corre-
spondent Greek adverb kpup8m v, was sometimes em-
ployed, as by DEMost HENES, Où Yùp & kpi}8qv Čstiv ºff
Wrijºjos, Mºjaei Tois 6eois. “Suffragium, etsi obscurum
est, Deos tamen latere non potest.”
We have noticed the adverbial force of substantives
used in the formation of compound adjectives ; parti-
cularly of the substantive stone, which in forming the
compound adjective stone-blind serves to modify the
adjective, blind. The English language is not very
rich in compounds; yet some of this kind occur par-
ticularly in our old writers, and in the proverbial and
trivial expressions of the vulgar. Thus bolt-upright,
is as upright and straight as a bolt, the old word for
an arrow. So SHAKSPEARE uses the compounds death-
practised, for “practised in death ;” tongue-tied for
“ restrained from speaking;” wreckfull, for “ full of
wrecks,” &c. -
With this ungracious paper strike the sight
Of the death-practised Duke.
mind-master, town-guard, forcefull, oar-bound, &c.
Mars, most strong; gold-helm’d; making chariots crack;
Never without a shield cast on thy back;
Mind-master, town-guard, with darts never driven;
Strong-handed, all-arm’d, fort and fence of heaven;
Father of victory !
Hymn to Mars.
Alcides, force-fullest of all the brood
Of men ;
Hymn to Hercules.
Chuse two and fifty youths, of all the best
To use an oar; all which see straight imprest,
And in their oar-bound seats.
Odyss. b. 8.
In the ballad of The Huntyng of the Hare, is ston-
styll.—
Jac Wade has a dogge wyll pull,
He hymselvue wyll take a Bull
And holde hym ston-styll.
In the Scottish Act of Parliament, A. D. 1587, en-
titled “ Mesaris and wechtis and the just quantitie
thereof,” the word rewl-richt, (i. e. as straight as a
ruler) occurs in the directions for making the Firlot
measure. “That the mouth be reyngit about with
a circle of girth of irne, inwith and outwith, haveing
a croce irne bar passing ovir fra the ane syd to the
wther, thrie squarit ane edge doun and a plane syde
quhilk sall gang rewll richt with the edge of the
firlot.”
Adverbs themselves may be in like manner com-
pounded. “ Ut in aliis classibus,” says VossIUs, “ ita
quoque in adverbiis, compositorum alia fiunt e duobus,
ut perdiu, abhinc, alia è pluribus ut forsitan. Nam, ut
forsit ex fors et sit, quasi forte sit ; ae forsan ex fors
et an, quod et in fortassean; ita forsitan ex tribus istis
fors, sit, an. And thus it is in English. We have to-
gether formed of to and gather; and we have altogether
formed of all, to, and gather. So in French tout à
fait, “altogether,” from tout, d, and fait; in Italian
nondimeno, “ nevertheless,” from non, di, and meno,
&c.; in German vielleicht, “perhaps,” from viel much,
and leicht easily; nimmermehr, “nevermore,” from
nie, immer, and mehr, &c.
In forming compounds of this nature, all parts of
speech (except interjections) are employed. “ Nulla
est vocum classis, says VossIUs, “ ea qua non adverbium
componatur.” Thus a composite adverb may be formed
in any of the following ways:—
1. From a pronoun and substantive, as quare from
qud and re.
2. From an adjective and substantive, as postridie,
from postero and die.
3. From an adverb, substantive, and adjective, as
nudiustertius from nunc, dies, and tertius.
4. From a substantive and verb, as pedetentim from
pede and tentare.
5. From a participle and substantive, as perendie
from peremptd and die,
6. From an adverb and adjective, as nimirum, from
ne and mirum.
7. From a preposition and substantive, as obviam,
from ob and viam. - -
My tongue-tied muse in mamen holds her still Lear, 8. From a pronoun and adverb, as alibi, from alio
- Sonnet 85. and ibi. © a tº
Against the wreckfull siege of batt’ring days. 9. From a pronoun and preposition, as adhuc, from
- Sonnet 65. ad and hoc.
P 2
108
G R A M M A. R.
Grammar.
10. From two verbs, as scilicet, from scire and
S-V-' licet.
Quare.
Dies.
Hotlie.
11. From two adverbs, as etiamnum, from etiam
and nunc.
12. From an adverb and a verb, as deinceps, from
dein and capio. . -
13. From a preposition and adverb, as abhinc, from
ab-and himc.
14. From a conjunction and adverb, as etiam from et
and jam.
VossIUS, not improperly, ranks among compound
adverbs those which are formed from other words, by
the addition of an adverbial particle, like our prefix,
a, or termination, ly; as tantisper, from tantus and
per; quandoque, from quando and que, &c. So we
find not only scienter, from sciens and ter, but even
Catiliniter, from Catilina ; not only jucunde, from ju-
cundus, but Tullianè, from Tullius.
It may be worth while to examine more particularly
some of these compounds. :
To begin with the first, quare. This adverb, consi-
dered in its origin and derivatives, will aptly illustrate
the transition from a distinctly significant phrase, to
an indistinctly significant, or consignificant ; or, as it
has even been termed, insignificant word or particle
The entire phrase is quá de re, as in PLAUTUs :—
AN. Nimia nos socordia tenuit.
, AD. Quá de re, obsecro *
AN. Quia jam non dudum ante lucem ad a dem Veneris
WeIllſll U.S.
Qud de re, shortened, in familiar discourse, to quá
ze signified “ for what thing”—“ for what cause"—
“ wherefore,” or, as it is expressed in the Scottish
idiom, “what for;” as “what for did you not come,
when you were called 3” i. e. why did you not come
The separate words qua and re, having by long use
been melted together in pronunciation, as quare, this
latter word, in old French, became quar, and in more
modern French car; but the last mentioned word,
even in the 16th century, had travelled so far from its
source, that the learned H. STEPHANUs did not recog-
mise in it the Latin quare, but thought it was derived
from the Greek qāp. MENAGE has justly corrected
this error in his Origines de la langue Francoise, under
the word car. “Henry Estienne et autres,” says he,
“le deriuent de Yáp. Il vient de quare, et c'est pour-
quoy vous trouwerez escrit quar dansles anciens liures.
On pronongoit, il n'y a pas encore long-temps, care,
cando, camobrem, canquan au lieu de quare, quando,
quamobrem, quamguam.”
It is at first sight as difficult to trace the Italian
adverb oggi, the French adverb jadis, and the English
substantive journalist all to the Latin dies; and yet no
etymologies are more certain than these three.
From hoc dte, by the mere rapidity of pronuncia-
tion, came the Latin hodie; and this, by an imperfect
attempt on the part of the barbarians, to imitate the
Roman articulation, was easily changed into hoggi,
pronounced as an Englishman would pronounce hodje.
Thus ANNIBAL CARo, in his verses on the death of
Francesco Molza, A. D. 1544.
E questo el monte ond'ê c’hoggi si scorga,
La gioria de le muse
and the modern Italians have softened this word into
oggi.
From dies also the Romans formed the adjectives Adverbs.
diurnus and diurnalis, “ daily;” and these in the
corrupt Latinity of the later ages, received secon- Journal.
dary meanings. “ Diurnum pro die dixit infima Lati-
nitas,” says SALMASIUs, “ et diurnale mensuram agri
quae uno die posset arari." The Italians from diur-
num, in the secondary sense of a day, made giorno,
which the French shortened into jour; and diurnale,
in like manner, produced giornale, journal, journalist.
Again from the Latin dies came the adverb diu, and Jadis.
from jam dies came jamdiu. The Italians altered jam.
into gia, and the French into ja, and hence jamdiu
became jadis.
Hora, another Latin word in constant use to mark Hora.
the lapse of time, has also undergone very considera-
ble changes. In the Norman French of the 13th
century, we find the word onkore. It occurs in a
letter from Perres de Mounfort, (Peter de Montford,)
1 Rym. Faed. p. i. fo. 339, ed. 1816, giving an account
of some successes which he had obtained over the
Welsh, in 1256. He first states the occurrences of
the Thursday next after St. Matthew's day; and then
continues, “ E onkore le lundi siwant ;” where the Encore.
word onkore is what was anciently written in French
ancore and now encore. In Italian it is now spelt an-
cora, and was formerly anch'ora, “ iterum,” “again,”
‘‘ once more.”
MENAGE, and Cour. DE GEBELIN, derive it from the
Latin phrase in hanchoram, or hanchoram; but this is
not quite satisfactory. ‘Qpos, as we learn from Herodo-
tus, was an Egyptian name of the sun, the great mea-
surer of time ; from whence, probably, the Greek tipa
came to signify “time,” in general, or a certain portion
of time, as “a season,” a day,” an “ hour.” And so in
Latin “ Hora,” says R. STEPHANUs, “ Tempus signi-
ficat, h. e. quancunque aeternitatis partem, sive an-
num, sive diurnum, sive nocturnum spatium complec-
tens. Item partes ipsae, in quas distinctus est dies,
similiter hora vocantur.” From the Latin hora came
the Italian hora, ora, or, which was used not only as a
substantive signifying a certain portion of the day,
but as an adverb signifying “ now” “ at this hour,”
“ at this point of time.”
Thus PETRARcA—
Ma ben veggi’ hor, si come al popol tutto,
Favola fui gran tempo. -
Hence it was redoubled, with a relative force con-
necting different parts of a sentence, and signifying
“ at one time,” and “ at another time;” as “ now,”
in the following line of Pope—
Now high, now low, now master up, now miss.
Thus MACHIAvELLI in the first book of his Istorie
Fiorentine, says, “Vedendosi l'Imperatore assalire da
tante parti, per aver meno nemici, comincio, ora con i
Vandali, ora con i Franchi, a fare accordi:" that is,
he began to make treaties, at this time, with the Van-
dals; at that time with the Franks. Hence also ora
was used conjunctionally, as connecting one link in a
chain of reasoning with all that has preceded it ;
agreeing also in this respect with our word “ now.”
Thus in SouTH's sermons : “The other great and un-
doing mischief, which befals men is by their being mis-
represented. Now, by calling evil good, a man is
misrepresented,” where the word “ now” may be para-
G. R. A. M M A. R.
109
*
Grammar. phrased, “ at this point of my discourse ;” as “I
S-N-" have already shown you the major proposition, namely,
Qorest.
Adverbial
phrases.
that all misrepresentation is mischief; now, at this
period of my discourse, I show you the minor proposi-
tion, namely; that to call good evil is misrepresenta-
tion ; and after I have shown you the major, and
minor, you can easily come to the conclusion your-
selves, namely, that to call good evil is to do mischief.
Hence the authors of the Dictionnaire de l'Academie,
say, “Or est une particule qui sert a lier une propo-
sition a une autre, comme la mineure d'un argument
a la majeure. Le sage est heureua - or est il que Socrate
est sage, donc Socrate est heureur.” In ancora, therefore,
the word ora itself includes the meaning of in hanc
horam ; and to this is prefixed the Italian adverb
anche ‘‘ also,” which seems to be a corruption of the
Latin etiamque ; as in BoccAcIo, “ Anche dite voi,
che voi vi sforzerete, e di che?” Encore, therefore,
is literally “ also now ;” “ we have heard the song
lately, let us also hear it now ;” “ we have heard it
once, let us hear it again.” Hora also appears in the
French alors from the Italian allora, which is the
Latin ad illam horam ; and in the Spanish agora or
ahora, which is the Latin hóc hord. The French de-
sormais is de hora magis; we find it written in the
abovementioned letter of Perres de Mounfort, desor-
emes, “Pour quei je vous pre e requer—ke hom mette
consul coment la terre seit desoremes defendue.”
We had formerly a remarkable adverb from this
source in our old law French, viz, qorest, for so it is
written in the Statute of Westminster, 22 Edw. IV.
A. D. 1482, “en temps del victorious reigne nostre dit
Seignur le Roy qorest.” This was a corruption of qui
or est, “ who now is.” If the word qorest had re-
mained in use, and its etymology had been unknown,
it might perhaps have prevented an ingenious legal
objection, said to have been taken in behalf of a pri-
soner,who was indicted on a statute passed in the reign
of GEoRGE II. but was not brought to trial until that of
GEoRGE III, when it was argued (in arrest of judg-
ment, or otherwise,) that the indictment charged the
prisoner with violating a statute alleged to have been
passed in the reign of “our lord the king that now is,”
whereas in fact no such statute had been passed in
that reign. Whether this was a real occurrence, or a
fiction, it served to supply the humourous genius of
Foote with another jest which also turned on the
peculiar use of the adverbs employed. He introduced
a character boasting of the skill with which he had
escaped from a charge of perjury “We were in-
dicted,” says he, “ for committing perjury now, but
we proved that we committed it then. If they had
indicted us for committing perjury now and then, it
would have gone hard with us.” This adverbial
phrase, “ now and then,” is perfectly idiomatical in
English, and there is perhaps no expression exactly
corresponding to it, in any other language. The Ita-
lian talvolta, and the French de tems en tems, are some-
what similar to it in signification, but with neither of
them is it quite identical.
From what has been said of compound adverbs, it
will have been seen, that the greater part of them were
originally, short phrases, or clauses added to a perfect
sentence, for the purpose of modifying the adjective, or
verb, which it contained. The office of such a phrase
is, therefore, exactly the same as the office of an ad-
verb, and thence we call it, as Mr. TookE does, an ad-
verbial phrase. Two corollaries follow from this V-y—’
remark, both of which we have seen illustrated in
the preceding examples; first, that a distinctly signifi-
cant adverbial phrase may degenerate, in length of
time, to an indistinctly significant adverb; and, se-
condly, that the adverbs of one language, or idiom,
may be supplied by analogous adverbial phrases in
another.
An adverbial phrase, which occurs frequently in our For the
old writers, has greatly puzzled most of their com- nonce.
mentators—the phrase “for the nonce.” As it is used
by SHAKSPEARE in two instances, it would seem
merely to signify “ for the occasion,” “to serve the
present turn.”
I have cases of buckram for the nonce, to immask our noted
outward garments.
First Part of Hen. IV.
—When in your motion you are hot,
And that he calls for drink, I’ll have prepared him
A chalice for the nonce. -
Hamlet.
Yet, perhaps, even here, a sort of ironical senti-
ment of admiration at the importance of the occasion
is meant to be expressed, implying really a contempt
for the parties concerned ; and this is more clearly
the meaning in another instance.
This is a riddling merchant for the nonce /
First Part of Hen. VI.
Admiration appears to be expressed, but not iron-
ically, by CHAUCER, in the Romant of the Rose.—
- But he were konning for the nones,
That coud devise all the stones
That in that circle shewen clere,
It is a wonder thing to here.
In the Canterbury Tales, on the contrary, he seems
to use it with some mixture of the ridiculous :-
The miller was a stout carle for the nones,
Full big he was of braune and eke of bones.
And again, the Host, ironically praising the Monk,
says to him—
As to my dome,
Thou art a maister when thou art at home—
And therewithal of braune and eke of bones,
A right wel faring person for the nones.
In describing the Cook, it is doubtful whether he
means to express any thing more than that this per-
sonage was engaged for the purpose of exercising his
art in case of need :—
A coke they had with them for the nones,
To boil the chickens with the maribones.
Here the phrase might perhaps have been supplied,
had the rhyme permitted it, with the other phrase of
‘‘ for nede,” which CHAUCER elsewhere uses :-
The stone so clere was and so bright,
That also sone as it was night,
Men mighte seen to go for mede,
A mile or two in length and brede.
LIDGATE evidently uses for the nones, in the simple
sense of “for the purpose.”
Her young Sonne she tooke,
Tender and greene both of flesh and bones,
To certaine men ordained for the nones,
Fro point to point, in all manner thing,
To execute the bidding of the king.
I 10
G R A M M A. R.
Grammar. However the writers already quoted may differ in
S-V-' their application of this phrase, still there is no doubt
but that they all understood it, and all applied it ac-
cording to the just analogies of language; but this was
not the case with SPENSER, who in the following pas-
sage applies it in a manner wholly arbitrary and licen-
tious : e
I saw a wolf
Nursing two whelps: I saw her little ones
In wanton dalliance the teat to crave, -
While she her neck wreathed from them for the nonce.
Mr. Tooke justly observes that Spenser is no autho-
rity for the right use of the English language. The
reason is not to be sought in any want of genius,
taste, knowledge, or feeling ; for in all these this
great poet deservedly ranked high ; but he had
adopted (probably from his great and deserved admi-
ration of Chaucer) the erroneous ambition of writing
in an antiquated dialect, and hence his language was
often that of no age, ancient, or modern.
In the ballad of “ The Huntyng of the Hare,” we
meet with “ in the nownes,” which seems to be used
in a sense rather different, and not very intelligible;
though probably of the same origin with “ for the
nonce."—
The course Y wold that ye had sene,
In the nownes ye had me the coppe gene;
For therof had Y mede.
The derivation of the word nonce, or nones, is as ob-
scure as the exact meaning of the phrase appears
doubtful. “NoNCE, n. s. (says Dr. JoHNson.) The
original of this word is uncertain. SKINNER imagines
it to come from own or once; or from nutz, German,
need or use.” These two derivations may both be
thrown aside as mere conjectures, destitute not only
of proof but of probability. - -
TYR whit suggests as its origin the Latin pronunc ;
but pro nunc is hardly to be called a Latin phrase:
and from pro nunc to for the nunc, and then for the
nonce, are barely possible transitions. JUNIUs says it
may be from the French word noiance, “ atque ita
for the nonce tantundem significabit Anglis ac si dice-
rent quia mihi sic libet, vel ob hoc solium, ut ei in-
commodem.” But this meaning does not seem ap-
plicable to any examples of the phrase now extant.
Nonce, the French denomination of the Pope's nuncio,
may possibly have led to a phrase of somewhat simi-
lar import, for as the nuncio had often powers little
short of royal, he must have appeared to the com-
mon people as a sort of king or prince ; and as we
say, “ this is a dish for a king,” so they might say
this is “ a cook for the nonce,”—“ a cook for the
nuncio.” He is “a stout churl fit to wait upon the
nuncio,”—“ a stout carle for the nonce.” There is a
curious passage in BALE's Acts of English Votaries,
(A. D. 1550,) retailing the scandal of a former writer
on Thomas a Becket, which seems to give some co-
lour to this explanation. “ In the towne of Stafforde
was a lustye minion, a trulle for the nonce, a pece for
a prince. Betwixte this wanton damsel, or primerose
pearlesse, and Becket the chancellor, went store of
presentes, and of loue tokens plenty.”
It must be confessed, however, that this explana-
tion will not suit several of the instances which occur
in old writers; and it is besides observable that the
more ancient orthography was nones, from which
nonce was probably a corruption. . Now nones is the Adverbs.
name of a fixed time of the day, viz. the ninth hour, -
at which time a certain religious service was always
performed. “NoNE,” says the Dictionnaire de l'Aca-
demie, “ se dit aussi de celles, des, sept heures canon-
iales, qui se chantent, ou quise recitent après Sexte.
(L'Ecriture dit que N. S. fut crucifié à Seate, et qu'il
rendit l'Esprit a l'heure de None.) Oil en estes vous de
votre Breviaire? - J'en suis a None. Après Seate on dit
None; et puis vespres.” Hence it is possible that a pro-
verb may have originated among the clergy and clerical
students, then a very numerous body, that such a one
was always ready for the nones; and this may have
been metaphorically applied to any thing done in due
time, and with a special regard to any fixed purpose.
It is somewhat in favour of this etymology, that our
word noon, mid-day, anciently written none, is be-
lieved to be of the same origin. “ NoNE, n. s.” says
JoHNSON, “ non, Saxon; nawn, Welsh; none, Erse.
Supposed to be derived from nona, Latin, the ninth
hour, at which their cana, or chief meal, was eaten ;
whence the other nations called the time of their din-
ner or chief meal, though earlier in the day, by the
SãIſle Ila. Iſle, - - -
Mr. Tyrwhit endeavoured to help his derivation of Anon.
for the nones, from pro nunc, by deriving anon from
ad nunc ; but amon is probably, as suggested by JU-
NIUS, in one, (minute, understood.) It occurs in the
ballad on the Battle of Bruges.— -
Tho the kyng of Fraunce yherde this, anon
Assemblede he is dousse pers eueruchon.
So CHAUCER, according to the Harleian MS.
1758, fo. 68.
Our oost vp on his stiropes stood a non.
It is somewhat differently written in Syr Launfal.
No.
Wha they had sowpere the day was gon,
They wente to bedde, and that anoon.
So in the Harleian MS. 7333, fo. 150.
And a noon the knyght cride to his seruantis, &c.
LIDGATE also writes it in the
MS. 2278, fo. 45.
Wherupon the kyng gan caste anoon.
In “ The Proces of the Seuyn Sages,” the MS. of
which is transcribed in the Scottish orthography, it
is written onane.
The sext maister rase vponane,
The fairest man of than ilkane.
same manner, Harl.
And in like manner GAwiN Douglas—
Thus sayand scho the bing ascendis on ane.
To revert to the phrase “for the nones,” it is in For the
form, though not in signification, like another phrase, maistry.
“for the maistry,” which occurs in CHAUCER :—
- A monke there was, fayre for the maistry.
This phrase is also found in the rude old ballad of
the Mon in the Mone, and seems to signify “in a mas-
terlike manner,” “in a superior degree:”—
We shule preye the hayward hom to vrhous,
And maken hym at heyse for the maystry.
There is also some analogy to the phrase “for the
nones,” in the Latin pro tempore, the French a propos,
the Italian a posta, &c.
G R A M M A. R.
1 || 1
Grammar.
French ad-
verbs with
tle
A l'écart.
A l'encan,
A l'abri.
The French have many adverbial phrases beginning
with a, such as à l'écart, à l'encan, a l'abri, &c.; the
origin of some of which is sufficiently obvious, but of
others less so.
A l'écart, answers to our adverb aside; as, ille mena
a l'écart, sous préterte de promenade, “ he drew him
aside under pretence of a walk.” Ecart is also used
in French as a substantive, to which we have no
single word corresponding, as son cheval fit un écart,
“ his horse started aside.” This word was formerly
written escart, and may probably have been more an-
ciently escarpt, answering nearly to our expression
“ a sharp turn.” Thus MENAGE derives escarpe (as un
rocher escarpé) from the German scharf, formerly scarf,
in English, sharp; and in Anglo-Saxon scearp; and
elsewhere speaking of the word escarpins, he says, it
is taken from the Italian scarpini, “ d'où nous auons
fait escarpins, en mettant, a nostre ordinaire, un e de-
uant 1's. - -
The French a l'encan, is a mere corruption of the
Italian all'incanto, “by auction;” and incanto is so
called from cantare, the price offered for the article
being cried out aloud, or (as our sailors say) sung out ;
whence this mode of sale is called in Scotland “ pub-
lic roup,” agreeing with the German rufen to call
aloud ; Swedish roop, clamour; Gothic hropjan, to cry
out; Dutch roepen, to call; roep, a call, &c. In the
north of England, too, roopy is hoarse, (from crying
out,) and a cold, (which makes a person hoarse,) is
called a roop. In certain parts of the country a sale
by auction is termed “a sale at public outcry.”
A l'abri is a French phrase of which the Dictionnaire
de l'Academie gives the following explanation. A
L'ABRI, façon de parler adv, a couvert, se mettre à l'abri
de la pluye, du vent, du mauvais temps, de la tempeste.”
And again, “ ABRI, s. m. lieu oil l'on se peut mettre
a couvert du vent, de la pluye, de l'ardeur du soleil,”
&c. “ABRI, se dit aussi fig, de quelque lieu que ce
Soit oil l'on est en Seureté, et généralement de tout
ce qui nous met hors de danger.” “On dit fig. se
mettre à l'abri de la persécution.” On the origin of this
word etymologists differ, and the way in which they
differ serves to illustrate the true and false genius of
etymology. PIERRE PITHou, a very learned old
French lawyer, in his valuable treatise on the Counts
of Champagne, derives the name of the country of
Brie, in France, from abri ; and that from arbre, be-
cause that which is under cover of a tree is à l'abri,
protected from the rain, wind, and sun ; and MENAGE,
catching at this ingenious notion, fills up from his
own imagination the steps by which the supposed de-
rivation has proceeded. From the Latin arbor, pro-
nounced albor, and thence alberus, says he, came the
Italian albero ; and from alberus came albericus, albri-
cus, which the Spaniards pronounced abrigo, and which
CovaFRUVIAS explains reparo contra las inclemencias del
cielo, particolarmente contra el frio. Now, the fault of
this reasoning is, that it is almost entirely conjectural;
and conjectural etymology is like conjectural criticism,
which ought only to be indulged in very sparingly,
and under the control of a most sound and experi-
enced judgment. There is no doubt but that BENT-
LEY was a man of prodigious learning; but a more
ridiculous book was never published than his edition
of Milton's Paradise Lost, in consequence of the ab-
surd latitude of conjectural criticism, which he allowed
himself in the notes. Among Etymologists some
Adverbs.
most ingenious men, such as Cour DE GEBELIN and \-y-
WHITER, may be taxed with this infirmity, nor is ME-
NAGE entirely exempt from it, though his work con-
tains abundance of sound information on language.
The other and more judicious derivation of abri is
from the Latin apricus. Aperio was to lay open, as in
Livy, “ quum calescente sole dispulsa nebula aperuisset
diem,” and in PLINY, “Quem (florem) noctu com-
primens, aperire incipit solis ortu.” Hence, (as Ser-
vius observes), Aperilis, or Aprilis, was the month
which opened the earth in spring. The old Latins, in
like manner, called places open to the Sun aperica,
“ Aprica loca dicuntur,” says SALMASIUs, quae oppor-
tune Solem accipiunt, quasi aperica, quod Soli aperta
sint, nam apericum veteres dixere.” Hence Virgil
by this epistle describes old men fond of sunning
themselves.—
_Aprici meminisse senes.
And in like manner he applies the same epithet to
the sea-birds delighting to sun themselves on the
open rock in summer time :-
Ex procul in pelago saxum—
apricis statio tutissima mergis.
Now, those places which were distinguished as
open to the sun, were generally sheltered from cold
biting winds; and it was with reference to this circum-
stance that they were so called ; for apricus was a
word of the winter or spring, but not of the summer.
“ Est sciendum,” says R. Stephanus, “ apricum non
dici nisi respectu frigoris. Nam in aestivo calore nihil
propriè apricum dicitur.” Whatever, therefore, was
sheltered either from cold, wind, rain, or even from the
extreme heat of the Sun, came to be called apricum,
and this word shortened, as is common in French
words derived from the Latin, formed apri, or abri.
Some of the French adverbial phrases beginning
with ā, have been adopted, as words, into the Eng-
lish language. Such are our colloquial adverbs ala-
mode, and apropos. Others have furnished us with
adverbial phrases, such as, at random. The substan-
tive randonnée still remains in French as a term of the
chace. “ Le lievre fut pris à la troisieme randonnée,”
and the word randon was formerly in use. MENAGE
says, “RANDoN. S'enfuir à grand randon: l'origine de
ce mot ne m'est pas connue. Du Substantif randon,
on a fait le verbe randomner, pour s'enfuir rapidement.”
From the French randonner came the verb to randy,
used in the west of England in the peculiar sense of
taking the part of a candidate at an election in a noisy
and riotous manner. The adjective randy is also used
in the north of England, and in Scotland, by the vul-
gar, to signify riotous, noisy, obstreperous. See
GRose's Provincial Glossary. The word randon was
early introduced into the English language; for it oc-
curs in the Description of Cokaygne,
The monkes ligtith nogt adun
.Ac furre fleeth in o randun.
In the romance of Richard Coer de Lion, we find
“ with gret randoun."—
Hys brothir come to that bekyr,
Upon a stede, with gret randoun,
He thoughte to bere Kyng Richard doun.
BARBour uses the expression “ into a randoun."—
At random.
1 12
G R A M M A. R.
Grammar.
S-N-"
Sir Aymer then, but mairabade,
With all the folk he with him had,
Ished enforcedly to the fight,
And rode into a randoun right.
Hickes derives randon from the Frankish rent dun
a torrent, compounded of rennan, “ to run,” and dun
“ down.” In the abovementioned Description of Cok-
aygne, the word rent occurs signifying the running of
a Stream.—
Ther beth iiii. willis in the abbei
Of tracle and halwei
Of baum and ek piement
Euer ernend to rigt rent.
The Gothic and its derivative languages often use
rennen and rinnen in the sense of flowing; and to this
origin WACHTER is inclined to attribute the name of
the Rhine. “ Huc etiam,” says he, “ spectat, multo-
rum judicio, Rhenus ''
Spick and Span is an adverbial expression, which
at present has descended to the vulgar, but which was
currently used by many of our best authors from
Chaucer to Swift. Mr. Tooke has rather dogmatically
laid it down, with some contempt for those who may
differ from him in opinion, that the proper significa-
tion of spick and span new, is “ shining new from the
warehouse.” The way in which he makes this out is
rather curious. Spyker, he says, is a warehouse in
Dutch, and spange is any thing shining in German.
The Dutch use the phrase spick-spelder-nieuw ; and
the Germans use the phrase spanneu ; and there-
fore by taking spick from the Dutch and span from
the German, we may ascertain the meaning of the
English spick and span. We cannot say, that this
appears to us a very satisfactory mode of illustrating
our own language. Dr. JoHNson (though no great
etymologist) seems in this instance to have proceeded
more rationally, in looking to the English words spike
and span as likely to throw some light on the subject.
We doubt, however, whether he is altogether right in
saying that spick and span new is a metaphor originally
taken from cloth, and signifying “ newly extended on
the spikes or tenters.” Perhaps it will be found that
the two expressions span new, and spick and span new
are of different origin. It is true, that spannan in
Anglo-Saxon is to stretch, and from thence comes
our verb to span ; the participle of this latter, how-
ever, is not span but spanned ; as in SHAKSPEARE–
My life is spann’d already,
I am the shadow of poor Buckingham.
But the word span, spon, or spun, was the participle
of the verb to spin ; as in the memorable old distich
of the friends of equality—
When Adam delved and Eve span,
Who was then the gentleman 2
Span-newe, therefore was litterally newly spun ; and
so it appears to have been used by CHAUCER—
- Troylus
Was never ful to speke of this matere,
And for to praysen unto Pandarus,
The bounte of his righte lady dere;
This tale was aye span-newe to begyne.
It is still more clear that such is the meaning of
spon-neowe, in the romance of Kyng Alisaunder; where
the king instead of punishing the Persian who at-
tempted his life, sends him away with honours and
rewards— .*
Spick and
Span.
Adverbs.
\-N-
Richeliche he doth him schrede,
In spon-neowe knyghtis wede, -
And sette him on an hygh corsour,
And gaf him muche of his tresour.
Spick and Span, or more properly Spike and Span,
was more probably taken from the lances in use infor-
mer times, of which the spike was made of iron, and
the span or part grasped in the hand, was made of
wood. Of course a lance which was new, both in
spike and span, was considered as most valuable.
The idea of newness is expressed in various ways
by the people of different countries, as by the Dutch
spick-spelder-nieuw, according to Mr. Tooke, “ new
from the warehouse and the loom;” by the Germans
span-neu, spannagel-neu, funkelneu, and funkelnagelneu,
by the Danes funckelnye, and by the Swedes (as Mr.
Tooke says) spitt-spangandny. ADELUNG does not
agree with Tooke in considering span in span-neu, and
span-nagel-neu, to signify shining ; but he thinks its
meaning doubtful. He however elsewhere observes
that spinnen (in the past tense ich spann, and by the
vulgar ich sponn) is a very old word, being found in
the Gothic, Anglo-Saxon, Frankish, Islandic, Swedish
and English, and being derived, as he is inclined to
think, from the Greek araetv. This, therefore, may
perhaps have been the origin of span-new in German
as in English ; while spann-nagel-new may have been
derived from spanne, “ a span,” the measure of the
outstretched fingers, and “nagel,” the finger-nail;
so that it would imply newness “in part and whole,”
“ in span and nail,” ad unguem.
The labour of the Smith appears to have suggested
the metaphors of funkel new, i.e. sparkling new ; as
it certainly did our fire-new, and brand-new, or brent-
7761).
Thus SHAKSPEARE-
Fire new.
Brand new.
£)
Despight thy victor sword, and fire-new fortune. '
JCear.
Your fire-new stamp of honor is scarce current.
Richard III.
And BURNs, in his incomparable tale of Tain
o'Shanter—
Warlocks and witches in a dance—
Nae cotillion breaf-new frae France,
But horn pipes, jigs, strathspeys, and reels,
Put life and mettle in their heels.
The adverb, or adverbial expression, pell-mell is ra- Pell-mell.
ther curiously treated by JoHNSoN, who designates it
a noun substantive, and in proof of his assertion cites two
passages, in which it has an adverbial, and one in which
it has an adjectival construction. “ Pell-mell,” says
he, “n. s. [pesle mesle, Fr.] confusedly; tumultu-
ously; one among other.” -
When we have dash'd them to the ground,
Then defie each other; and pell-mell
Make work upon ourselves.
Shakspeare's King John.
Never yet did insurrection want
Such moody beggars, starving for a time,
Of pellmell havock and confusion.
Henry IV.
He knew when to fall on, pellmell,
To fall back, and retreat, as well.
Hudibras.
So much for Johnson and his examples. How
long “ confusedly,” or “ tumultuously,” have been
G R A M M A. R.
1 13
Grammar. nouns substantive, we know not ; but clear it is, that
-V-2 in such phrases as “ to make work on ourselves, pell-
mell,” or “ to fall on, pell-mell, the word pell-mell,
performs the function of an adverb in modifying the
verbs “ make,” and “fall,” respectively; and it is
equally clear, that in the phrase “a time of pell-mell
havock,” the same word performs the function of an
adjective, in qualifying the substantive “ havock.”
Johnson is certainly right in deriving this word
from the French pesle mesle, now written péle-méle;
but the previous history of the word it is not quite so
easy to trace. -
The origin of the latter syllable, however, is much
clearer, than that of the former. We will begin with
the Greek verb utoryw to mix, which is supposed to
agree with the Hebrew missech. The roots utory and
Auš, appear to be the same (just as our English verb
ask, pronounced by the vulgar ar, is the Anglo-Saxon
acs in the verb acsian,) and they also appear in the
simpler form uty; large classes of words involving
the same idea, being derived from each of these
sources, as Muayāyketa, Mao'ymtetºw, Mayoóð), Mta-
yovduos, Mišarēpēa, MuñéNAques, Mušis, Mušoëwn, Miya,
Mºyas, IIoMyutyńs, Affyvpoptºs, &c.; but the most re-
markable, with reference to our present inquiry, is
MuayóAas a tumult, “ nimirum,” says the Lexicogra-
pher, “ex confluxu et mistione multitudinis.”
The same roots, misc, and mir, are of early date in
the Latin language, appearing in the words misceo,
nictus, mirtio, mirtia, &c.
The Frankish miskenti, mixing; and duruhmiste,
thoroughly-mixt; are of this origin, as are the Ar-
moric misgu, and many Sclavonic words in various
dialects.
In the modern German, we have mischen, to mix;
mischung, mixture, &c. In the modern Italian, are
mischiare, mischiato, mischianza, mischiamento, mischio,
mischiatura, and likewise mistura, mistione, mistianza,
mistiato, all conveying the idea of mixture. It is also
observable that the older Italian writers use mischia
and mistia, for a quarrel. In Spanish we have marto,
"mistion, mistura, misturar, all signifying to mix. In
Portuguese mestura, mesturar, mesturado, mestico, mis-
turada, miaºto. In French mirte, miction, miationer.
In English to mix, mixture, &c.
But we must now revert to the Latin. In that lan-
guage derivatives in ellus, illus, olus, ulus, &c. were
Common; as misellus, from miser; tenellus, from tener;
lapillus, from lapis; sciolus, from scio; tepidulus, from
tepeo. In like manner we find, in the times of clas-
sical antiquity, miscellus, and miscellio, from misceo;
as “vites miscellae,” explained to be “ quae in omni
agro conveniunt, quod caeteris ubique utiliter miscean-
tur.” CATo, de R. R. cap. vi. and so “ Uvae miscellae,”
“ quibus praeliganeum vinum fit, quod operarii bi-
bant.” Ibid. cap. xxiii. According to FESTUs, “ Mis-
celliones appellantur, qui non certae sunt sententiae,
sed variorum mirtorumque judiciorum.”
JoANNES DE JANUA, a writer of the middle ages, uses
this word miscellio for one “ qui novit artem miscendi
varios cibos.”
In the barbarous Latin of those times we also find
misculare, of which we have the following instances—
Nulla persona audeat pillum misculare cum aliquà laná.”
Stat. Riper, cap. 225.
WOL. I.
misculasse.
Edict. Pistense, cap. 23.
Deistis rapinis nihil vos debeatis misculare.
Aurum vel argentum adulterasse, vel
. Hincmar,
And hence was formed misculatio:—
Per ipsam misculationem. -
Placit. sub Carol. Mag.
From miscel or miscul, to mell, or mel, the transi-
tion was easy; thus, l. miscel; 2. miscl; 3. mescl:
4. misel; 5. mesel; 6. misl; 7. mesl; 8. medl - 9. mell
or mel ; of all which there remain instances in the
barbarous Latin of the middle ages, forming words
which convey the idea of mixture, under various mo-
difications; such as mirt corn, mixture or community
of geods, a mixt colour, a disease which rendered the
skin of a mirt colour; a mob of people confusedly
mirt together; a disturbance, or fight, in which the
parties were confusedly mirt, &c. -
1. Miscel, as “Miscella,” and “Miscelantia.”
Miscella, mixt and confused riots, or disturbances,
“ Si miscella in villā fortè facta erunt.” Charta Theob.
Com. Campaniae, A. D. 1200. -
Miscelantia (erroneously written miscedantia) is also
a mixing of people in a tumult—
Quod aliqua persona non debeat currere, cum armis, ad aliquam
rixam, miscelantiam, vel rumorem. .
Stat, crimin. Riper, cap. 175.
2. Miscl, as “ miscla,” and “ misclantia.”
Miscla occurs in the same sense as miscella, and in
the same charter.
Misclantia, is used, like miscelantia, to signify a riot
“ Consules teneantur denuntiare omnes rixas et mis-
clantias.” Stat. Mantua, c. 17.
3. Mescl, as “ mescla,” “ mesclania,” “ mescle-
lana,” “mesclatus.” -
Mescla, in an account dated A. D. 1322, is used to
signify “ mirt grain.”
Mesclania, in a charter of the year 1244, occurs,
with the same sense of “miat grain.”
Mesclelana seems doubtful; it may be of the same
meaning as mesclania; or it may be intended to sig-
nify “mirt wool.” “ Pannorum falsorum et falsae
Meselelande.” Chart. Libert. Cast. Nov. de Arrio, A. D.
1356. -
Mesclatus pannus,” in a charter of the year 1329, is
“ cloth of a mixt colour.” -
4. Misel as “misellus,” and “misellaria.”
Misellus, a leper. This word occurs in very old
writings, particularly in a charter of the year II65,
and also in Matthew of Paris, under the year 1254.
Some authors suppose it to be the classical word mi-
sellus, used by Cicero as a diminutive of miser; but
this would not account for its peculiar application to
the disease of leprosy, whereas its agreement with
the old French mesel and the English measles in re-
ferring to a disease which gives the skin a mixed co-
lour, leaves little doubt that its origin is from mis-
cellus ; more especially when we consider that though
we, for convenience, here arrange the barbarous Latin
words with the classical, before we come to the Ita-
lian and Old French, yet in reality the two latter were
in most instances the medium of corruption, which led
to the Latin of the middle ages.
Misellaria. An hospital built for lepers. This word
is common in old records, particularly in a charter of
the year 1245.
Q
Adverbs.
\-> -
114
G R A M M A. R.
5. Mesel, as “ Meselia,” and “Meselaria.”
~~~ Meselia is explained “bonorum mobilium commu-
nitas inter conjuges.”
1267.
Meselaria is the same word as miselarua already no-
ticed. “Lego pro remedio animae mea: Centum
Libras Turonenses Monasteriis, et Ecelesiis, Hospita-
libus, meselariis, capellanis et pauperibus in civitate
Tolosanā.” Chart. A. D. 1281. -
6. Misl., as, ‘‘ mislata.”
Mislata is used for “ a tumult.”
lebon. A. D. 1083.
7. Mesl, as, “meslea,”—“mesleia,”—“mesleare,”
“ mesleiare,”—“ mesleta.”
Meslea, “ a tumult.” Chart. A. D. 1293.
Mesleia this word also occurs for “ a tumult,” in
many ancient charters, e. gr. “Si homo episcopi
fecerit mesleiam in terrà comitis.” Chart. A. D. 1206.
So in Chart. A. D. 1207, 1224, &c. In one instance it
is erroneously written merleiam.
Mesleare, “ to mix”—“ monetas prohibitas cum
bonà monetà mesleare.” Edict. Phil. Pulchr. A. D.
1329.
Mesleiare, “ to quarrel,” to “raise disturbances"—
“si infra claustrum serviens rixando, vel mesleiando
aliquem percusserit.” Chart. A. D. 1206.
Mesleta is used for riots or tumults in the old Sici-
lian Constitutions. -
8. Medl, as ‘‘ medletum.” -
Medletum an affray or tumultuous quarrel. “Cog-
noscere de medletis,” to hold plea of affrays, or tumul-
tuous quarrels. GLANVIL, l. i. c. 2.
9. Mell, or mel, as “ melleia,” “ meleia,’’ ‘‘ mel-
leya,” “meleare,” “ melleta,” “melliator.”
Melleia often occurs for a tumult, particularly in
Stat. Eccl. Meldens. -
Meleia is used indifferently with mesleia for a tu-
mult, in the charter of 1206, before quoted, as “si ad
mesleiam applegiatus sit—si ad meleiam aplegiatus non
fuerit.”
Calida melleia, or calida melleya, a tumult while the
blood is warm—this word occurs in many instances.
Vide Tabular. S. Genov. Paris, A. D. 1241. Chart.
Phil., iii. Reg. Franc. Chart. A. D. 1352, &c.
Meleare to riot or make disturbance “rixando vel
meleando.” Chart. A. D. 1206.
Melleta occurs in the old laws of Scotland for
“ affrays.”—“Ad vicecomites etiam pertinet, propter
defectum dominorum, cognoscere de melletis, dever-
beribus, et de plagis.” Regiam majestatem, 1. i.
c. 3. s. 7.
Melliatores are common brawlers; Stat. Coll. Cornub.
A. D. 1380. -
In the modern languages, we find numerous ana-
logies to the words already quoted from the barbarous
Latin.
From misculare, come the Italian mescolare; mesco-
lamento, mescolante, mescolanza, mescolata, mescolata-
mente, mescolato, and mescolatura. Also the Spanish
mescla, mesclar, mesclado, mescladura, and the Portu-
guese mesclar, mesclado. g
It is also worth while to note the Italian mislea,
which, like the barbarous Latin mesleia signifies a
tumult or conflict “onde si comincio una grande
zuffa e mislea.” Giov. Villani.
The various dialects of the French language, how-
Regist. Parlam. Paris, A. D.
Wide Concil. Lil-
ever, will more clearly point out the connection of . Adverbs.
these terms with our present adverb.
In a charter of BERNARD DE LA Tour, in the pro-
vincial dialect of Auvergne, A. D. 1270, mescla, is used,
like the barbarous Latin miscla, for a tumult. “E si
i a mescla, e om i trai glasi nudament, per la mescla.”
“And if there be a riot, and men draw their swords
nakedly during the riot.”
Mesclaigne, like mesclania above cited, signifies
“ mixt grain,” “une quarte de mesclaigne.” Reg.
cens. Dom. de Nereux, A. D. 1418.
Meslée is used for a crowd, or mixed and confused
number of persons, in a letter written A. D. 1479,
“une meslée de gens, qui estoient assemblez au lieu
de Semur.” -
The Dictionnaire de l'Academie says of this word,
“il se dit proprement d'un combat opiniastré, ou
deux troupes de gens se meslent, l'espée à la main,
1'une contre l'autre. Rude meslée, sanglante meslée, se
jetter dans la meslée. Il se dit aussi d'une batterie de
plusieurs particuliers : il y a une grande bagarre, une
grand meslée, dans la rue. Il a perdu son chapeau dans
la meslée. Il se dit aussi d'une contestation aigre
entre plusieurs personnes. Comme je vis que la dispute
s'echauffoit, je me tiray de la meslée.”
But though the substantive meslée is thus chiefly
confined to quarrelling or fighting, the verb mesler is
applied to almost any sort of mixture, as mesler des
grains ensemble, mesler des couleurs, mesler l'eau avec le
vin, &c. &c.; in short it is, as MENAGE justly observes,
merely the Italian verb mescolare abbreviated.
The old adjectives meslius, meslieuw, take their
meaning from the substantive meslée : the nouns mesil
and mesel, take theirs from the verb mesler.
Meslius is an old French word for quarrelsome,
riotous; as in Le Doctrinal 2–
Li hom qui par coustume est meslius.
Meslieuw has the same signification, or rather is the
same word varying only in orthography. “ Icellui
Guerard, qui estoit homme merveilleux meslieux et
rioteux.” M. S. Letter, A. D. 1432. *
Mesil is “mixt grain.” “ Le carge de mesil xiii.
den.” Pedag. Bapal.
Mesel, a leper, leprous. “Oindre le visage du Seig-
neur, qui estoit mesel,” M. S. Letter, A. D. 1408. “Li
mesel ne poent estre heirs a nului.” Anc. Const. Nor-
mand. This severity of the law against lepers was
not peculiar to Normandy. Great part of the romance
of Amis and Amiloun turns upon this circumstance ;
and Mr. Weber, the learned editor of that romance,
has collected in the notes some curious information
respecting the laws relative to lepers; especially from
a MS. in the French Royal Library (No. 8407,) where
it is said “que home ne pot sa femme lessier que por
fornication, et por lepre non, et mesel se poent ma-
rier.” The fate of “False Creseide,” as related by
CHAUCER, also illustrates this subject ; and CHAUCER
employs the word mesel for a leper.
Mezellerie is an old French word for a hospital of
lepers.
Mézeline is described in Restaut's Dictionary, as
“ sorte d’étoffe mélée de soie et de laine.”
Mesteil is doubtless of a similar origin. It is said
in the Dictionnaire de l'Academie, to be “ Froment et
seigle meslez ensemble.”
G R A M M A R.
115
S-N-1 nifying to riot, quarrel, or cause to quarrel.
^
Meller agrees with meleare abovementioned, in sig-
Thus in
a letter dated A. D. 1427, is the following passage :-
“Pour ce que icellui Wairon, qui estoit parent au
suppliant, l’avoit mellé envers le Seigneur Du Bos.”
Melleys is the same as meslius, or meslieuw ; e. gr. in
a letter, A. D. 1375, “ Jehan Fenin, qui estoit homs
rioteux, et felons, et melleys.” -
Mellif agrees in signification with the preceding.
“Si aucun des dits chappelains est mellif ne rioteux.”
Chart. Joan. Duc. Brit. A. D. 1433.
Finally mellée, or mélée, is the same as meslée, the
Italian misléa, the Auvergnese mescla, and the barba-
rous Latin meleia, melleia, meslea, miscla, and miscella,
signifying a closehanded battle, or tumult, in which the
different parties are confusedly mixed together, and
fight, as we say, pellmell.
Thus in the Roman De Vacces—
Tel vient sain a mellée qui au departir saigne.
Hence caude mellée was the literal translation of
calida melleia. Philipe de Beaumanoir says “ quant
caudes mellées sourdent entre gentilshommes.”
This latter term was early adopted into the juris-
prudence of Scotland. The following passage occurs
in the laws of King Robert II. A. D. 1372, “ homici.
dium ex calore iracundiae, videlicet chaudemelle.” In
the English law, as Glanvill had written medletum,
where the Scottish lawyers had written melletum, so
for chaudemelle was written chaudemedley, which has
since been corrupted into chancemedley.
Indeed our meddle and medley are only the French
mesler and meslée, changing, as Sir Edward Coke ob-
serves, the s into a d.
Upon the whole, then, it is clear that the syllable
mell in the adverb pellmell is derived from the Greek
pugnu, and signifies a melée, or mict contest.
But what is the signification of pell; or has it any
signification 2 -
It might perhaps appear at first sight not impro-
bable to derive this syllable from pela, pellir, pelain,
or pila. -
Pela is a barbarous Latin word, from the old Latin
pala, a stake, and it is the origin of a word spelt very
variously in French, paelle, paele, pelle, pèle, used in
modern language for a shovel, either of wood or iron,
but probably in more ancient times for a plain wooden
stake.
Pellir is used in a charter of the year 1411, for to
drive away with such a stake or shovel “pela co-
ere.
É Pelain is explained by CARPENTIER, the continuator
of DU CANGE, to signify “ clades, strages, defaite, de-
route, in Gest. Brit. apud Marten, tom. iii. anecd.
col. 1465.—
Ceci leur fist a crespelain
Ou illes mist entel pelain.
Pila is “ a ball,” from whence comes our word a
pile or heap, and pilagium, which is used in a record
of the Chambre des Comptes at Paris, A. D. 1310, and
which CAR PENTIER explains “ servitii genus, messem
nempe, seu foenum in pilam sive struem ordinare.”
. The word pillocellus occurs in a MS. of the year
1354, and is explained by CARPENTIER, pila lusoria ;
but in the passage cited by him, it seems more probably
to signify a racquet, and may therefore possibly have
been written pillomellus; perhaps in the French of Adverbs.
that day pile-malle, from pila and malleus.
According to these etymologies pellmell must either
have signified a contest with staves, or a contest fol-
lowed by defeat ; or else it must have been a meta-
phor wholly borrowed from the tennis court; but
these are at best ingenious conjectures, and we are
inclined to think that pell was merely added to mell,
for the sake of the sound, and to strengthen the con-
ception of confusion already expressed in the word
melée, by describing a “ confusion worse con-
founded.”
Certain it is, that this principle of the iteration of
sound, with a trifling variety of articulation, in order
to augment the force of the expression, enters very
largely into the formation of words and phrases, in all
countries, especially among the common people, and
more particularly where the conception to be ex-
pressed, though accompanied with strong feeling, is in
itself vague, obscure, and confused. Whatever, there-
fore, may be thought of the application of this prin-
ciple to the adverb pellmell, it is of great consequence
to the proper understanding of grammar, that the
principle itself should be carefully considered ; nor is
it any objection to such consideration, that the prac-
tice in question originates with the vulgar and igno-
rant. On the contrary, this very circumstance throws
an additional light on the science of language; for it
is not only in the formation of such words as the one
under consideration ; but in the general frame and
construction of all languages, that we may find
reason to attribute a great influence to the strong
feelings and imperfect conceptions of the ignorant,
the vulgar, and the barbarian ; and moreover, even in
the class of expressions which we are more particu-
larly examining, there is a force and a suitableness,
which eventually makes them force their way upwards
in society, until they become equally familiar and in-
telligible to high and low, to the coarse and to the
refined. This is owing chiefly to orators and poets,
who (if they are truly such) will not address them-
selves solely to morbous sensibilities, or pedantic
judgments, and therefore will not ask whether an ex-
pression has been branded as obsolete or trivial by the
magisterial asterisk of a lexicographer; but whether
it will carry conviction and enthusiasm to the mind
of the hearer or reader. The great LUTHER some-
where recommends to one who would know all the
powers and energies of the German language to listen
to it as spoken by the mother in the house, and the
dealer in the market. BURNs, the delightful poet
BURNS, would never have attained that immortality
which is insured by his “ Twa Dogs,” and his “ Tam
o'Shanter,” if he had confined himself to such book-
language as the “Verses to Miss C , a very young
Lady.” -
Beauteous rose-bud, young and gay,
Blooming on thy early May,
Never may'st thou, lovely flow'r,
Chilly shrink in sleety show'r?
&c. &c. all in the same strain.
That the alliterative formation of words by the
vulgar is not confined to England or France, but is
natural to such persons in all countries, we may learn
from a curious little story which occurs in Eton's
Q 2
116
G R A M M A R.
Grammar. Survey of the Turkish Empire.
“An Arab, who had
-—y—’ let out his camel to a man, to travel to Damascus,
Misch -
Inasch.
Schick-
schnack.
helter-skelter, hum-drum, hurly-burly,
complained to a kadi, on the road, that the camel was
overloaded. The other bribed the kadi; ‘What has
he loaded it with ?' asks the cadi. The Arab answers,
* With cahué (coffee) and mahué;’ i.e. coffee et catera,
(changing the first letter into m makes a kind of gib-
berish word, which signifies et catera.) * Sugar and
mugar, pots and mots, sacks and macks,' thus going
through every article the camel was loaded with. “In
short,' concludes the complainant, ‘ he has loaded it
twice as much as he ought.”—“ Then,’ says the kadi,
* let him load the cahué, and leave the mahué, the su-
gar, and leave the mugar, the pots, and leave the mots,
the sacks, and leave the macks;' and so on, to the end
of all the articles enumerated ; and as the poor Arab
had told every article, and only added et cattera, ac-
cording to the Arab custom, the camel took up the
same loading as it had before.”
The learned and laborious ADELUNG has collected
several instances of words similar to our pell mell in
form, and probably in the mode of their original con-
struction, both in German and other languages; such
as the German mischmasch, (answering to BURNs's
miastic-martie, to the Low Saxon and Danish misk-
mask, and to the French micmac,) also schnicksnack,
wischwasch, zick-zack, wirr-warr, fick-facken ; the Low
Saxon hink-hanken, tick-tacken, &c. -
To these we may add in English chit-chat, ding-
dong, dingle-dangle, fiddle-faddle, giff-gaff, handy-dandy,
namby-pamby, pit-a-pat, prittle-prattle, riff-raff, see-saw,
skimble-skamble, slip-slop, snip-snap, tag-rag, tittle-
tattle, &c. - -
There are also many expressions, which if not
formed by mere alliteration, seem to be retained in
use chiefly by that quality in their construction, such
as rigmarole, hocus-pocus, hugger-mugger, &c.
“It is a property,’” says ADELUNG, “ of the com-
mon, or vulgar German language, and of its cognate
dialects, to form a kind of intensive or frequentative
words, by a repetition of the same sound.” And else-
where he observes, that “ in the Low Saxon dialect,
particularly, this is customary,” and that “in doubling
the syllable they generally change its vowel;" but in
High German such words are rare.
Mischmash, in German, is a heap of things thrown
together without taste or order; from mischen, to
mix. The French in borrowing from it their word
micmac, have given it the secondary sense of inten-
tional confusion and obscurity.
The Dictionnaire de l'Academie defines micma”, “ in-
trigue, manigance, pratique secrete pour mesnager
quelque interest illicite. Il y eut bien du micmac dans
cette affaire; on ne connoist rien à tout ce micmac.” It
is, like most words of this kind, stigmatised as a low
expression.
Schnickschnack is a kind of strange, foolish chatter-
ing, or jargon, from schnack, chattering. In Dutch
sink is to sob, and snack is a droll, chattering fellow.
These words are doubtless formed by imitation of the
sound, described, as the common word Sniggering, for
suppressed laughter, as in English. To the same
origin are we to ascribe the provincial word sneck, the
latch of a door; whence a sneck-up, was a thievish
vagabond, who watched his opportunity to lift the
knick-knack,
sneck up. and steal into a house for the sake of pilfer- Adverbs.
ing. Thus Sir Toby Belch calls Malvolio, in derision,
“sneck-up;” Falstaff says of Prince Henry, “ the
prince is a Jack, a sneck-up ;” and Mr. STEvens has
collected many other instances of this cant term of
reproach from various old plays.
Wischwash seems to differ but little from the former, Wisch-
being derived from waschen or schwassen, to babble or Wasch,
talk idly.
Zickzack, is the origin of the French ziczac, and of Zic
our zigzag ; and they all signify a line continued back-
wards and forwards from point to point. Its origin
is clearly the German zacken, a point, or pointed sub-
stance, as the points in the branches of a stag's horn;
and so eiszacken, an icicle or pointed piece of ice. And
this word agrees with the Dutch, tak, a bough ; the
Swedish tagg, the Islandic taggar, the French dague,
and daguet, and the English tack, tag, dag, jag, &c.
Our word tack, is used for a small pointed nail, for
fastening things together with nails; and also for the
action of a ship in going from point to point.
Our old word takil, and the Welsh tacel, a pointed
arrow, was a derivative of tack. Hence CHAUCER,
The takil smote, and in it went.
In Islandic, tag is the point of a lance.
This word tag, Mr. Tooke says, is in English the
participle of tian vincire ; but he is wrong, it is a
point of metal put to the end of a string, and to tag
with rhyme, is to point a line with rhyme. Dag is the
very same word, it was an ancient name for a dagger
or short pointed sword, called in French dague, whence
the old French verb daguer was to stab with the point
of the dagger; and daguet was a young stag (called
by Shakspeare a pricket) when the points of his horns
first begin to shoot. In Italian and Spanish, a pointed
short sword or dagger is daga, in Dutch dagge, in
German degen. Skinner calls a pointed piece of cloth
a dag, from the Anglo-Saxon dag, sparsum pendens,
and in Dutch the pointed end of a rope is called een
endye dagg'. GRose says a pointed spade is called in
Norfolk and Essex a dag-prick. With dag also agrees
jagg which signifies a point, and jagged, cut into
points.
Wirrwarr is a confusion of many things whirling
round, as it were, in confused circles, and clashing
together. LEssING seems to have adopted it from
the Low Saxon dialect. “ Salmasius macht ilber diese
stelle einen trefflichen Wirrwarr.” “ Salmasius
makes, on this place, a fierce confusion.” Wirren, the
origin of this word is our whirl, and the Latin gyrare :
in its first sense signifying to turn round in a circle,
and thence to confuse, or disturb the state and order
of things. Thus in the Frankish of OTFRID uuio er iz
allaz uuirrit quomodo omnia perturbet,
Fickfacken is a trivial word in Low Saxon, signify- Fick-
ing to run about idly here and there without any par- facken.
ticular object, or to employ one's self in idle tricks.
Adelung supposes it to come from facheln, as if it
were a metaphor taken from the motion of a fan; but
it seems rather from fach, which is probably the same
in origin as our pack, meaning a portion, quantity,
division, &c. “Das schlagt nicht in mein fach.”—
“That does not fall to my lot, it is no business of
mine, I have nothing to do with it.” The Gothic
fahan, and Frankish fahen, are explained by WACHTER,
G R A M M A. R.
117
Grammar.
v-
Chit-chat.
“ capere, quocumque modo, manu, mente, ambitu,
spatio,” and he considers it to be the same word
which in other dialects is pronounced fangen. From
fahen come fashig capax, anſahen, ordiri auffahen ex-
cipere, empfahen vel entfahen accipere, umfahen, am-
plecti, unterfahen conari, and lastly the above word
fach which is the Anglo-Saxon fac, as in stowe fac,
spatium loci, lytel-fac modicum temporis, twegra daga
fac biduum. In Lower Saxon it answers in compo-
sition with numerals to the Latin plea, and our fold,
as einfach, simplex, zweiſfach, duplex, vielfach, mul-
tiplex, &c. In the Scottish dialect the word feck is
still retained in the sense of quantity; as “will it
rain to day There'll be nae feck.” The verb fa'
seems also to be the old fahen or unterfahen, as in
BURNs—
A king may mak’a belted knight,
A lord, a duke, an’ a’ that;
But an honest man’s abune his might;
Gude faith, he manna fa’ that.
Hinklanken, in Low Saxon, is to go hopping along
lamely, with one leg shorter than the other, from
hinken to halt. e
Ticklacken, in the same dialect, is to touch gently
and often, from ticken which is connected with the
Gothic tekan, and the old Latin tigere, to touch. From
this comes trictrac (backgammon) which MENAGE
says the French anciently pronounced tictac.
Besides the preceding, several other words of a si-
milar construction are cursorily mentioned by Adelung;
as the Low Saxon titeltateln, wibbelwabbeln, tiesketauske
or zieskezaaske; the Swedish pickpack, willerwalla,
dingldangl, and the Islandic fimbulfambe, to which we
may add the French charivari.
We now come to the English words of this kind.—
Chitchat, Dr. Johnson says is “ corrupted by re-
duplication from chat, and is a word only used in
ludicrous conversation ; as in the Spectator, No. 560,
“I am a member of a female society, who call our-
selves the chitchat club.” It is true that most of these
words are originally trivial, and many of them ludi-
crous; but when they find their way into books of
such classic celebrity in our language as the Spectator,
it is surely necessary that the student of language
should understand by what means they got there,
upon what principles they were formed, and to what
class of words they are properly to be referred in
grammatical arrangement. Now the only part of
this word which was originally significant is chat;
but even of chat the origin is unsatisfactorily ex-
plained by Johnson, who, though entitled to the
highest praise for industry as a lexicographer, was
perfectly ignorant of the history of the English and
other modern languages. Thus he suggests that
chat may be from “ the French achat, a purchase, or
cheapening, on account of the prate usually produced
in a bargain.” He might as reasonably have derived
it from the French chat, a cat; because many old
women chatter to their cats. Long after the word
chat was in common use in England, the French
word, now spelt achat, was spelt achapt, and the verb
acheter was achepter, or achapter, being derived, as
some suppose, from the barbarous Latin adeaptare :
put at all events agreeing with the German kaufen,
Dutch koopen, Scottish to coff, Anglo Saxon ceapan, or
aceapan to buy, ceap, cheap, ceapman a dealer or
chapman, ceapstow, forum mercatorum, Chepstow, in
Wales.
as Chipping Norton, Chipping Ongar, Chippenham, &c.
SKINNER derives chat and chatter from the French
cacqueter; but this latter, as well as the Italian chi-
acchierare, and chiacchillare, and the Latin cachinnare,
may rather be compared with our verb cackle; whereas
chatter, as when the teeth chatter with the cold, is
more analogous to the German zittern, to tremble ;
with which also agrees our word twitter. All these,
however, are instances of the onomatopoeia, or for-
mation of significant words by the mere imitation of
sound. The insignificant syllable chit was subse-
quently prefixed to chat, as we suppose pell to have
been to mell, zick to zack, and tick to tack, from an
indistinct wish to give it an intensive, or frequentative
force.
Ding-dong, “a word (says JoHNson) by which the
sound of bells is imitated.” It is singular, that the
learned lexicographer should call this word a noun
substantive, and cite as an example, from SHAKs-
PEARE- -
Let us all ring Fancy's knell.
Ding, dong, bell
In this instance, ding-dong is manifestly a mere
interjection. It is, however, sometimes used adver-
bially, as when it is said “ they went to fighting ding-
dong.” To ding, is to strike or beat, from the Anglo-
Saxon dyngian ; and dong, dung, or dang, are used in
different dialects as the past participle of this verb.
Thus there is an old Scottish song in praise of the
town of Dunse, entitled “Dunse dings a’,” i. e. Dunse
beats or excells all other places.
entitled “Jenny dang the weaver;” that is, she beat or
overcame the weaver. A Yorkshire lad, who had come
to London as a servant, was one day asked by his
master what had occasioned some water to be spilt
on the carpet. He replied, in his provincial dialect,
“I dung doon turn ;” meaning, I accidentally
knocked down the tea urn.
Mr. TookE justly observes, that the substantive
dung, manure, is this participle dung : and he quotes,
among other authorities, Sir THoMAS MoRE, who
spells it dong. “All other thynges in respecte of it I
repeite (as Sainct Paule saith) for dong.” Ding-dong
therefore is no more than “ strike stroke.”
Dingle-dangle expresses in English, as it is said to do
in Swedish, a swinging or oscillating motion, from
the verb dangle, which SKINNER supposes to have
been originally hangle, from hang. If so, it was pro-
bably formed, like the words we have been consider-
ing, hangle-dangle. In a modern comedy an uncle
reproaching his extravagant nephew, says, “I shall
see thee go off, just at twelve o'clock, dingle-dangle.”
Fiddle-faddle. Dr. Johnson quotes this word both
in its substantival and adjectival use.—
She said that their grandfather had a horse shot at Edgehill,
and their uncle was at the siege of Buda, with abundance of fiddle-
faddle of the same nature.
Spectator, No. 299.
She was a troublesome, fiddle-faddle, old woman.
Arbuthnot.
The history of this word is curious. According to
CICERo, faith between man and man was called fides,
from fio to be. “ Fundamentum est autem justitiae
fides ; id est dictorum conventorumque constantia et
Adverbs.
Hence many names of places in England, N-V-
Ding-dong.
There is also a song
Dingle-
dangle.
Fiddle-
faddle.
1 18
G R A M M A. R.
l
Grammar, veritas. Ex quo (quanquam hoc videbitur fortasse
\-N-" cuipiam durius, tamen ut audeamus imitari Stoicos,
qui studiose exquirunt unde verba sint ducta) creda-
mus, quia fiat quod dictum est, appellatam fidem.”
Again, a harp was called fides, according to FESTUs,
on account of the truth of its tones, “fides, genus
citharaº, dicta quod tantum inter se chordae ejus,
quantum inter homines fides, concordent.” The di-
minutive of fides gave fidicula, which our Anglo-Saxon
ancestors called fithele, the Germans fidel, and we
fiddle. In modern times, however, the more digni-
fied name of this instrument in German is violine, and
in English violin ; and some degree of contempt is at-
tached to the word fiddle, both as a noun and a verb :
in its primary sense it expresses an inferior instru-
ment and a vulgar performance; in its secondary
sense, to fiddle, is in the words of Dr. Johnson, “ to
trifle, to shift the hands often and do nothing ; like
a fellow that plays upon a fiddle.” To convey this
latter idea, the more forcibly, the word is repeated,
with the mere change of a vowel. SKINNER seems
anxious to discover some separate meaning for the
word faddle, which he thinks may be from the French
fade, and Latin fatuus; or from the German faden, a
thread ; so that “ a fiddle-faddle person” would be
either a fiddle-foolish person, or a fiddlestring person;
which etymologies are equally superfluous and inap-
propriate.
Giffe-gaffe, is formed from the Anglo-Saxon gifan,
to give ; as ding dong is from dingan. This expres-
sion, now obsolete, occurs in one of Bishop LATIMER's
Sermons, published in 1562. “Somewhat was geuen
to them before, and they must neades geue somewhat
againe ; for giffe-gaffe was a good felow.”
Handydandy. This word also, it pleases Dr. JoHN-
son to call a noun-substantive. It may be so used,
no doubt ; but in the instance which he cites from
SHAKSPEARE, it is an interjection.—
See how yond justice rails upon yond simple thief! Hark in
thine ear! Change places, and handy dandy / which is the justice 2
which is the thief ?”
Giffe-gaffe.
Handy-
dandy.
Lear.
Helter-skelter, Dr. Johnson who admits this to be
an adverb, explains it, “ in a hurry, without order, tu-
multuously.” In fact, it combines these notions with
something of inconsiderate eagerness, whether occa-
sioned by fear, as when a troop of men are said to fly
helter-skelter, or by a desire to reach a particular ob-
ject, as when Pistol hastens to carry to Sir John Fal-
staff the glad tidings of Prince Henry's accession to
the throne :-
Sir John, I am thy Pistol, and thy friend;
And helter-skelter have I rode to England,
And tidings do I bring
Helter-
skelter.
4e
SKINNER, in his anxiety to make sense of every part
of this expression has given two etymologies which
make nonsense of the whole. He thinks it may
either be derived from the Anglo-Saxon healster
sceado, “ the darkness of hell;” or from the Dutch
heel-ter-schetter, which he thinks is “all dispersed or
shattered to pieces.” The real origin of the word,
however, is obscure. If we suppose the principal
meaning to be in the first part, it may possibly come
from the Islandic hilldr pugna; if in the latter part,
it may be from the German schalten, to thrust for-
ward; or from skale, which in the dialect of the
north of England, means “ to scatter and throw Adverbs.
abroad as molehills are when levelled;” or from skeyl S-N-2
which in the same dialect is to push on one side, to
OVerturn. -
Humdrum. It seems to be admitted that there is Humdrum.
no origin for this word, but the interjection hum !
which is explained to be “a sound implying doubt or
deliberation;” it forms, however, first an adjective,
and then an adverb ; as “I was talking with an old,
humdrum fellow,” Spectator; and again—
Shall we, quoth she, stand still, hum-drum ;
And see stout bruin overthrown 2 -
Hudibras.
Hurlyburly. Dr. "Johnson has recorded an absurd Hurly-
etymology of this word, from the names of two fa- "y.
milies, Hurleigh and Burleigh. The word hurl, or
hurley, signifies a tumult, from the French hurler, to
howl like wolves or dogs; and to this the word burly
appears to have been added, as a mere reduplication.
When the hurly-burly's done,
When the battle's lost and won—
That will be ere set of sun.
Macbeth.
Methinks, I see this hurly all on foot.
A. John.
He, in the same hurl, murdering such as he thought would
withstand his desire, was chosen king.
Knolles.
Knickknack. In this word, which is chiefly used Knick-
as a substantive, the syllable knick is only prefixed to knack.
knack for the sake of the sound, and to give a slight
degree of intensity to the meaning. The word knack
is reasonably enough derived from the Anglo-Saxon
cnawan, to know ; and is explained “ a little machine,
a petty contrivance, a toy.”
Knaves, who in full assemblies have the knack
Of turning lies to truth, and white to black.
Dryden.
When I was young, I was wont
To load my she with knacks. I would have ransack'd
The Pedlar's silken treasury, and have pour'd it
To her acceptance.
Winter’s Tale.
Namby-pamby. This word seems to be of modern Namby-
fabrication, and is particularly intended to describe pamby.
that style of poetry which affects the infantine sim-
plicity of the nursery. It would perhaps be difficult
to trace any part of it to a significant origin.
Pit-a-pat. This expression also Dr. Johnson calls Pit-a-pat.
a substantive ; and gives the following example—
A lion meets him, and the fox's heart
Went pit-a-pat.
I'Estrange.
Here put-a-pat is clearly an adverb; as it is in the
Beggar's Opera.
As when a good housewife sees a rat
In her trap, in the morning taken,
With pleasure her heart goes pit-a-pat,
In revenge for the loss of her bacon.
This expression is not derived from the French pas-
a-pas, (with which it has nothing to do, either in
meaning or etymology,) nor “from the French patte-
patte,” which it is apprehended, never was a French
phrase; but the verb to pat is “ to strike lightly, to
tap,” and a pat is “a light quick blow, a tap ;” the
word being, no doubt, made from the sound. It is
G R A M M A. R.
I 19
Grammar, true that Casaubon learnedly deduces it from the
S-N-' Greek Atavrāv; but this is an etymology, which we
Prittle-
prattle.
Riff-raff.
Skimble-
skamble.
Slipslop.
Snip-snap.
need not trouble ourselves to refute. Pat marks the
strong blow, in the beating of the heart; and pit is
prefixed to it, to express the weaker blow, which
forms the alternation.
Prittle-prattle. As prattle is a diminutive of prat,
agreeing with the Dutch praten, and possibly derived
from the Latin praedicare; so prittle prefixed to prattle
makes a further diminutive, and is particularly ap-
plied to the early attempts of children to talk.
Riff-raff. We have the verb to raff, to huddle up,
and take away hastily without distinction. CAREw
says, “ their causes and effects I thus raff up toge-
ther;” and a rafe or raff, in the provincial dialect of
the midland counties of England, is “a low fellow,”
probably from comparison with dirt and other matters
thus carelessly swept away. To the word raff, in
this signification riff being prefixed, augments the
feeling of contempt, whilst it applies the expression
more loosely to a whole class of people. Raff is no
doubt connected with reave, of which rafte is the old
past tense :-
O trust, O faith, O depe assurance'
Who hath me rafte Creseyde 2
And Mr. TookE does not err much in saying, that
riff-raff is identical with rof, the past participle of
the Anglo-Saxon reaftan ; but he is entirely mistaken
in ascribing the adjective rough to the same origin;
for rough is the German rauh from ragen, eminere,
prominere; whereas reaftan agrees with the German
raffen and rappen the classic Latin rapere, the barba-
rous Latin reffare, &c.
See-saw. The significant syllable here is saw, and
the word see saw is meant to express a motion similar
to that of sawing; see being merely prefixed for the
sake of adding force to it. Pope uses it as a noun,
and Arbuthnot forms a verb from it.—
His wit all see-saw, between that and this.
POPE.
Sometimes they were like to pull John over; then it went all of
a sudden again on John's side : So they went see-sawing up and
down.
ARBUTHNOT.
Skimble-skamble, is formed, as Johnson observes,
“ by reduplication from scamble. Thus SHAKSPEARE
makes Hotspur ridicule the pretended prodigies and
portents of Glendower—
A couching lion, and a ramping cat,
And such a deal of skimble skamble stuff,
As puts me from my faith.
Scrabbling, scrambling, scambling, shambling, are all
words expressive of an awkward, struggling, or shuf-
fling motion.
Slipslop. This is, in like manner, said by Johnson
to be formed by reduplication of slop. He expounds
it “bad liquor;” but since the days of Fielding it
has come generally to signify the incorrect and un-
grammatical language of chambermaids, from the
character of Mrs. Slipslop, in Tom Jones.
Snip-snap. Tart dialogue, in which each party
snaps, as it were, at the other's argument before it is
finished.—
Dennis and dissonance, and captious art,
And snip-snap short, and interruption smart.
POPE.
Tag-rag. This word is in signification very similar
to riff-raff. Dr. Johnson does not make a separate
Adverbs.
word of it, but places it among his examples of the Tag-rag.
use of the word tag, which, he says, signifies any
thing paltry and mean; but why tag should have
that signification, it is not easy to guess; certainly
not from the etymology which he gives of it; for he
derives it from the Islandic, tag, the point of a lance.
The leading conception in the compound tag-rag is
undoubtedly that expressed by the word rag ; and tag
seems to be prefixed to it merely for the sound. Casca
speaking with the utmost contempt of the Roman po-
pulace, whom he calls “ the rabblement,” and the
“ common herd,” and ridicules for their “chopped
hands,” and “sweaty nightcaps,” goes on to speak
thus of their conduct towards Caesar:—
If the tag-rag people did not clap him and hiss him, according
as he pleased and displeased them, as they use to do the players in
the theatre, I am no true man.
Tittle-tattle. This is properly described by Dr.
Johnson, “a word formed from tattle by reduplica-
tion. Idle talk, prattle, empty gabble.”
Of every idle tittletattle that went about, Jack was suspected for
the author.
ARBUTHNOT's Hist. of J. Bull.
You are full in your tittletattlings of Cupid.
SIR P. SIDNEY.
We have sufficiently shown that this mode of form-
ing words is common to many languages; that it is
of considerable antiquity in our own language; and
that, so early at least as the age of Queen Elizabeth,
words so formed were adopted into the style of the
best authors; not indeed as conveying any distinct-
ness of impression, or dignity of sentiment, but as
appropriate and suitable to the subject before them,
and to the feelings with which they wished it to be
regarded.
The pleasure derived from alliteration is one of the
earliest and simplest of the mere pleasures of sound in
language. Hence alliteration appears to have pre-
ceded rhyme, in the rude attempts at poetry, which
were made by Gur Saxon ancestors; and even after
rhyme was introduced into English verse, the ballads
and popular poems of the day were full of alliterative
expressions. In one of those poems already quoted,
(Harl. MS. 2253, fo. 124,) we find an expression,
which seems to be the origin of our trivial word rig-
"marole. The poem in question begins thus:–
Of rybaudzy ryme
Ant rede o my rolle. -
That is, “ of ribalds (or idle, disorderly persons,) I
rhyme, and read out of my roll.” The accounts, re-
cords, and other long and tedious writings of that day
were usually preserved on rolls; therefore a “read-
o'-my-roll” story would be an apt expression for a
long, tedious story ; and the vulgar would easily cor-
rupt read o' my roll into rigmarole.
Tittle-
tattle.
Rigmarole,
Hocus-pocus, is a vague word for juggling and Hocus-
cheating.
Thus BUTLER says—
For Justice, though she's painted blind,
Is to the weaker side inclin'd,
Like charity; else right and wrong
Could never hold it out so long;
pocus.
120
G R A M M A R.
Grammar.
And, like blind Fortune, with a sleight,
Conveys men's interest and right,
From Stiles's pocket into Nokes',
As easily as Hocus-pocus.
The expression “is corrupted,” as Dr. Johnson
says, “from some words that had once a meaning and
which cannot now be discovered.” The suggestion
of TILLoTson is probably the right one. At the time
of the Reformation, many jests, and some of them
grossly profane, were made on the rites of the
Roman Catholic church ; and the priests who cele-
brated the holy mysteries were treated as no better
than jugglers. Thus, in a Scottish poem of that pe—
riod, beginning “The Paip, that Pagane full of pride,”
we find the following passage—
Thay sillie Freiris, mony Yeiris,
With babbling bleirit our ee.
Hay Trix! Tryme go Trix .
Under the grenewod Trie.
The words hoc est corpus, employed with reference
to the doctrine of transubstantiation, were very likely
to have been turned into ridicule by the opponents of
that doctrine, and from hoc est corpus, corrupted by
vulgar pronunciation, may have been formed hocus
pocus. JUNIUs derives the expression from the Welsh
word, hocced, a trick, and the English word poke, a
bag; but it is neither probable that a juggler's bag
would obtain the mixt Welsh and English name
hocced-poke, nor, if it did, that the Latin termination us .
would be substituted for the second, and added to the
third syllable.
SKINNER, with more learning than judgment, de-
rives hocus-pocus from quassare and fodicare. “ To-
tum enim istiusmodi artificum mysterium,” says he,
“ in eo consistit, ut pilas vel sphaerulas, in vasculis
seu pyxidibus quassent, et digitis quam celerrime
motis, res immissas Surripiant.” From quassare, he
derives the French hocher, and from fodicare the
French pocher; which, he says, is “ digito extrudere
et quasi effodere;” but though hoche-poche in French
might possibly convey the idea of shaking a bag and
thrusting the fingers into it, we have not met with
that word so used ; still less can we suppose it to
have been Latinised, in termination, if derived from
this origin. The French hocher, to shake, is the
Dutch hutsen, or hutselen, from whence come our
huddle and hustle. The Dutch have the word hutspot,
for a dish made of meat cut into small pieces, and
shaken in the pot, with vegetables, &c. whilst it is
dressing. The French also have hochepot, and the
Scotch have hotch-potch, with the same meaning. The
French hochepot, signifying some kind of cookery, is
used by Chaucer; and it was adopted in a figurative
sense into the terms of our law, at least as early
as the year 1474; for at that time Sir THoMAS LIT-
TLETon wrote his Commentaries, in the third book
of which, (sect. 267,) occurs this passage, “En cel
case le baron ne le feme avera riens, pur lour purpar-
tie de le dit remnant; sinon que ils voile mitter lour
terres, dones en frankmarriage, en hotchpot ovesque le
remnant de la terre.”—“ Et ill semble, que cest parol,
hotchpot, est, en English, a pudding ; car en tiel pud-
ding nest communement mies un chose tantsolement,
mes un chose ovesque auters choses ensemble.”
CokE, however, observes, “ in English we use to say,
hodgepodge.” But as none of these derivations from
hutsen or hocher have any relation, in point of mean- Adverbs.
ing, to hocus-pocus, so neither can they at all serve to S-N-
explain the manner in which that word acquired the
Latin termination us; which circumstance becomes
perfectly intelligible, if we adopt Tillotson's suggestion
as true.
Hugger-mugger.
from L'Estrange's fables : “ There's a distinction
betwixt what is done openly and barefaced, and a
thing that's done in huggermugger, under a seal of
secrecy and concealment.” Johnson explains it “se-
crecy, bye-place ;” but it does not appear to have so
much to do with the place where, as with the manner
in which things are concealed; and it seems to allude
to hugging things up close to prevent their being
seen. The conjectural etymologies of this expression
are exceedingly various. SKINNER derives it from
the Dutch hugghen, which, he says, signifies to ob-
serve, and the Danish morcker, darkness; an etymo-
logy alike improbable and inappropriate. JoHNsoN
says it is “ corrupted perhaps from huger morcker, a
hug in the dark,” in what language hug er morcker
has this signification he does not mention, nor does
any phrase correspondent to the English hugger-mug-
ger, appear to have ever become proverbial in any
other language. The Spanish affords the nearest ap-
proach, to the separate parts of this expression; for
hogar is a chimney corner, and muger is a woman; and
if we could suppose hugger mugger to be taken from
that language it might refer to the notion of a woman
cowering in the chimney corner; but as nothing can
be more delusive than to be guided in etymology by
mere similarity of sound, we may safely reject this
derivation of the phrase in question. Some persons
have supposed hugger-mugger to be derived from the
old-English word hoker; because Sir THoMAs MoRE,
(it is said,) uses the word hoker-moker; but it is not
very clear that he meant by it what we mean by hug-
ger-mugger; and if he did, no great stress is to be laid
on a casual variation of orthography in that age,
when spelling had nothing like fixed rules. The
word hoker, had no reference in point of meaning,
to the idea conveyed by the word hugger-mugger ;
for it signified peevish, froward, and was probably
taken from the French hocher la tête to shake the head
at any thing in sign of contempt.
Thus CHAUCER in the Reve's Tale, describing the
Miller's Wife :-
She was as digne as water in a diche,
And as full of hoker and of besmare,
As though that a Lady should her spare
What for her kinred, and her mortelry
That she had lerned in the nonnery.
And the same idea is still more fully expressed in
the Lay le Freine :
Than was the leuedi of the hous,
A proude dame, and an enviedus,
Hokerfulliche missegging,
Squeymous and eke scorning.
The last etymology that we shall mention is from
the Dutch title, Hoog Moogende, (High Mightinesses,)
given to the States General, and much ridiculed by
some of our English writers; as in Hudibras—
But I have sent him for a token
To your Low-country Hogen Mogen
This word implies a clandestine Hugger-
way of doing things, as in the following example mugger.
G R A M M A. R.
12]
Adverbial
phrases.
Grammar. It has been supposed that hugger-mugger, corrupted
--"
from Hogen Mogen, was meant in derision of the secret
transactions of their Mightinesses; but, it is probable
that the former word was known in English before
the latter; and upon the whole it seems most pro-
bable that hugger is a mere intensive form of hug, and
that mugger is a reduplication of sound with a slight
variation, which, as we have already seen, is so com-
mon in cases of this kind.
The same disposition toward alliteration appears
in some of our quaint proverbial phrases, where the
words are distinct, as in “ tit for tat ;” and also in some
passages of our comic writers. Thus in the Taming
of a Shrew; Petruchio, in his feigned anger against the
Tailor, exclaims—
What’s this? a sleeve 2 'tis like a demi-cannon,
What! up and down carv'd like an apple tart:
Here's snip and nip, and cut, and slish and slash /
So Parson Evans says to his friend, Justice Shal-
low :-
It were a goot motion, if we leave our pribbles and prabbles, and
desire a marriage between Master Abraham and Mistress Anne
Page.
We have observed that the primary use of the ad-
verb is to modify adjectives or verbs, and its secon-
dary use to modify adverbs. The same may be said
of adverbial phrases, and generally of whatever stands
in the place of an adverb. Thus we may say “this
$ 9
happened afterwards,” or “ this happened long after-
wards,” or “ this happened many days afterwards,” or
“ this happened not many days afterwards.” In the
first case the adverb afterwards modifies the verb
“ happened;” in all the other cases the same adverb
afterwards is modified, first, by the adjective long used
adverbially, then by the adjective and substantive
many days forming an adverbial phrase, or standing in
the place of an adverb; and lastly, by the adverb,
adjective, and substantive, not many days, which in
like manner may be said to form an adverbial phrase,
or to stand in the place of an adverb. So in Lord
BERNERs's translation of FRoiss ART, executed by
command of King HENRY VIII. and printed in his
reign, the following passage occurs, fo. cxcix. b.
“Nowe the Duke of Berrey commaundeth me the con-
trary; for he chargeth me incontynent his letters sene,
that I shulde reyse the syege.” In this passage in-
contynent is an adverb modifying the verb reyse; and
the letters sene is a phrase, (similar in construction to
the Latin ablative absolute, as it is termed, visis epis-
tolis,) which modifies the adverb incontynent, a word
at that time used where we should say immediately.
Thus in the romance of The Foure Sonnes of Aimon,
printed in 1554, we find—
Now up Ogyer, and you Duke Naymes, light on horseback
incontinent.
Adverbial phrases are in another point of view ma-
terial to the consideration of adverbs properly so
called. By comparing different languages we not
only find, that a certain phrase in one language cor-
responds to a different phrase in another language;
but that phrases in the one correspond to words in
the other. Thus in comparing the French with the
Italian we not only find such expressions as a chaudes
larmes, answering to a dirotte lagrime, or tout-à-coup,
to di primo lancio; or à gorge deployée, to alla smascel-
WOL. I.
lata; but we also find a tatons rendered by tentone, a
peu près, by quasi, &c. &c.
Adverbs.
We have now exhausted the considerations arising Recapitula-
out of our definition of the adverb. We said, first, tion.
that an adverb was a word used for the purpose of mo-
dification ; and we showed how it modified primarily
an adjective or a verb, and secondarily another ad-
verb. Secondly, we said, that for this purpose it was
added to a perfect sentence,” and we distinguished be-
tween a sentence perfect both in the mind and ex-
pression of the speaker, and a sentence perfect in the
conception, but broken short in the utterance. And
thirdly, we explained what sort of word might be used
for the purpose of such modification. Under this
head we showed that the adverb might be a simple or
compound word, and we instanced adjectives, partici-
ples present and past ; pronouns, numerical and de-
monstrative; verbs and substantives, all of which have
been used as adverbs, and indeed constitute the mass
of the words commonly known by that designation.
We showed also that compound adverbs might be
formed of all the other parts of speech; and, lastly, we
noticed a variety of adverbial phrases, or words de-
rived from such phrases, which, in the construction
of sentences, supply the place, and perform the func-
tion of adverbs. In the course of these investigations
it has been rendered most manifest that phrases often
become words, and that of words it is the use and not
the form, which entitles them to be considered as ad-
verbs. If a substantive be employed adverbially it is
equally an adverb whether it have or have not previ-
ously undergone any inflection. Now, in the passage
quoted from the laws of the Twelve Tables, is as
much an adverb as noctu, quoted from Cicero.
It may be proper, however, before we close the
chapter of adverbs to advert to some few considera-
tions, which though they have no particular reference
to any part of the definition above given, have occu-
pied much of the attention paid by other writers to
this part of speech. -
In works professedly treating of grammar, it has
not been uncommon to distribute adverbs into classes
according to their signification. Thus the very learned
and admirable HICKEs, (a name never to be men-
tioned without veneration,) enumerates in the Anglo-
Saxon language no less than 28 different kinds of
adverbs ; viz. I. of time; 2. place; 3. exhorting ;
4. dissuading ; 5. excepting ; 6. denying ; 7. affirm-
ing ; 8. wishing; 9. doubting ; 10. diversity; 11.
distance ; 12. Quantity; 13, separation; 14. situa-
tion; 15. transition ; 16. comparison ; 17. augmen-
tation; 18. remission; 19. congregating; 20. quality;
21. manner; 22. likeness; 23. opposition; 24, order;
25. demonstrating ; 26. interrogating; 27. number;
and 28. cause. It is almost needless to observe that
this sort of enumeration is infinite ; for there is
scarcely a conception of the human mind which may
not be applied adverbially, and even form a class of
adverbs. HARRIs has only spoken particularly of ad-
verbs of intension, remission, comparison, time, place,
motion, and interrogation ; but he has quoted a pas-
sage from THEodor E GAZA, which is more to the pur-
pose; for that acute grammarian justly observes that
the readiest way to reduce the infinitude of adverbs,
(considered according to the conceptions signified by
them,) is to refer them by classes to the ten logical
R
Other
writers.
Classifica-
tion.
122
G R A M M A. R.
natural order, in any system, as where the adverb is Adverbs.
treated of before the participle, which was done by ~~
Grammar. predicaments, existence, quality, quantity, relation,
\-…~" &c. &c.
Such a classification, however, though it may be
useful to the memory, is no essential part of the
office of a grammarian, because there is no difference
in grammatical use between an adverb of one of these
classes, and an adverb of another such class; between
an adverb of time, for instance, and an adverb of
place; an adverb of quantity, and an adverb of qua-
lity; or if any such difference exist in a particular
language, it depends on the idiomatical peculiarities
of that language, and not on any essential principles
of universal grammar.
DoNATUS SERVIUs, and some others; or after the
preposition, which was the order of PRIscIAN, who
therein followed ApolloNIUs. We trust it will be
found in the sequel, that the order which we have
adopted from DIoMEDEs and Vossius, is the most na-
tural and the best, namely 1. adverb, 2. preposition,
3. conjunction, and 4. interjection.
From the consideration of classes of words, we
come to that of words singly ; and among these we
find frequent instances of the confusion before alluded
to ; adverbs are treated as being other parts of speech;
Confound- A more important consideration is this, that ad- and other parts of speech are treated as being adverbs.
ed with verbs are often confounded with other parts of speech, It is not surprising, that where a noun retains its
...i. by writers of no mean reputation; and this happens form unchanged, the adverbial character, which it
in two ways ; for 1st. the whole class of adverbs may
be confounded with other classes; or 2dly. particular
words, whether adverbs, or others, may be confounded
with classes to which they do not belong.
BEN Jonson says, “ Prepositions are a peculiar
kind of adverbs, and ought to be referred thither.”
CARAMUEL says, “Interjectio posset ad adverbium re-
duci; sed quia majoribus nostris placuit illam distin-
guere non est cur in re tam tenui haereamus.”—“ In-
terjectiones,” says VossIUs, “ a Graecis ad adverbia
referuntur, atque eos sequitur etiam BoETHIUs.” It is
clear from the definition of an adverb, which we have
given, that a preposition can no more be considered
as a peculiar kind of adverb, than a substantive can
be considered as a peculiar kind of adjective or verb;
for the proper function of the preposition is to mo-
dify a conception of substance; and the proper func-
tion of the adverb is to modify a conception of attri-
bute, either alone, or combined with an assertion ;
but the part of speech which names a conception of
substance is the noun substantive ; the part of speech
which names a conception of attribute is a noun adjec-
tive ; and the part of speech which asserts is the verb.
Again, as to interjections, they do not serve to mo-
dify either noun or verb; but on the contrary are in-
terjected, as it were, between different nouns or verbs,
and as VossIUs says, “ citra verbi opem, sententiam
complent;” for though, as we have said, the inter-
jection may, both in signification and construction,
supply the place of a verb, in certain instances; as in
the passage, “O ! that I had wings like a dove,”
where the interjection O ! supplies the place of the
verb “ I wish;” yet this, in no respect, modifies the
signification of the verb “ had,” but merely affects its
construction in the sentence.
If, indeed, with certain of the Greek philosophers,
we were to admit only three parts of speech, the
noun, the verb, and the combinative, it might at first
sight appear somewhat doubtful under which head
the words which we have termed adverbs, should
properly fall; for some of them, as we have seen, are
in origin nouns, and others verbs; but in that case
we ought not to look so much to their origin, as to
their use; and, therefore, we should class them
among verbs; for by verbs the philosophers, here
alluded to, really meant what Harris calls attributives;
and the adverb is, as he has justly said, the attributive
of an attributive.
. It adds something to the confusion of the classes of
words, if they are placed out of their common and
acquires in construction, should be sometimes over-
looked. Among the adverbs which we have cited,
some e. gr. wonder, are now used only as substan-
tives; others e. gr. right, full, &c. are now rarely
used but as adjectives ; and as substantives and ad-
jectives respectively they would probably be treated
by all those persons, who do not reflect that it is the use
of a word in a particular sentence that determines the
part of speech to which, in that sentence, it belongs.
We have seen Dr. JoHNson, a scholar certainly of
great acquirements, designating as nouns subtan-
tive, such words as pell-mell, ding-dong, handy-dandy,
pit-a-pat, and see-saw, when in the very examples
which he quoted they were used as adverbs; and
this is the more remarkable because he designates
other words, of the very same formation and use, ad-
verbs ; e. gr. helter-skelter, which certainly approaches
as nearly to pell-mell, in its grammatical use, as it
does in the mode of its formation, and in its general
import. -
On the other hand, the term adverb is that which
almost all grammarians apply to an indeclinable word
when they either are at a loss to ascertain its proper
use, or do not give themselves time to reflect on the
matter. The acute and ingenious DE BRossEs calls
the French chez an adverb, which is most manifestly
a preposition, for chez moi, and apud me, are phrases
exactly similar in construction. Even the learned
VossIUs calls the Latin mecastor an adverb, and
R. STEPHANUs terms it “jurandi adverbium.” Now
mecastor is from the Greek ua, and Castor, the name
of a deity, and it is literally, ‘‘ by Castor,” an oath
used as a common expletive in conversation. Thus
we find in Terence, “Salve, mecastor, Parmeno;”
where mecastor cannot by any ingenuity be made to
modify the verb salve, or indeed any other word ;
but is truly and properly an interjection, which all
words of the same kind must be, such as Gadso l which
though Mr. TookE distinctly calls an oath, yet he
preposterously reckons among the adverbs. Gadso
and 'Odso were abbreviations of “ by God it is so;”
or “ is it so, by God?” for men happily shrink from
their own profaneness, and rather reduce their words
to unmeaning exclamations, than advert seriously to
their original import. As to the obscene Italian
expression to which Tooke alludes, it had probably
nothing to do with the interjection Gadso, however
it may have furnished a hint to the unpolished satire
of Ben Jonson, in the passage quoted from one of his
plays. -
G R A M M A. R.
123
Grammar.
CHARIsrus, out of the twenty-one classes of ad- to place such words as nevertheless, which Dr. John- Preposi-
Q-y-' verbs, that he enumerates, mentions three, which are
clearly interjections; namely those which he calls
adverbs of wishing, as utinam; of answering, as hem 1
and of showing as ecce 1 This last mentioned word
is sometimes used redundantly with the similar
word en, as in APULEIUs, “ En, ecce, prolatam coram
exhibeo,” where Vossius, reckoning it among ad-
verbs, nevertheless adds, with his accustomed saga-
city, “ nisi hac (adverbia demonstrativa) potius in-
terjectioni accensentur.” Mr. TookE, however, falls
into the common error, and enumerates among ad-
verbs the plain interjection lo 1 which is (as he him-
self observes) the imperative of the verb to look. This
consideration alone should have taught him that lo!
could not be used with an adverbial construction ;
and the same may be said of halt 1 and fie / which he
nevertheless includes in his list of adverbs. Halt 1 is
the imperative of the German halten, and was pro-
bably transmitted to us directly from the French,
who borrowed it from the Italians, and they from
the Germans.
The Anglo-Saxon healdan, and our verb to hold,
are indeed the same verb with the German halten, but
from them we could never have formed in the impe-
rative halt 1 terminating with a t ; although our old
writers used halt, as the past tense of those verbs.
Had such been our derivation of the present word,
halt it would probably have been more extensive in
its application ; but its confinement to the purposes
of the military art, shows that it was received from a
foreign nation, with that distinct application.
As to fie 1 the imperative of the Gothic and Anglo-
Saxon verb fian, to hate; from whence comes fiand,
the fiend, the enemy of mankind, it is surely as ge-
nuine an interjection as proh ( or va. 1 or any other
word of that class. *
Mr. TookE too, calls “ prithee” an adverb. It is
the phrase, “I pray thee,” shortened, and used as an
interjection ; and it never did or could serve as an ad-
verb in modifying either a verb, an adjective, or ano-
ther adverb. By a similar error some ancient writers
reckoned the verb amabo among adverbs, but CAELIUS
CALEAGNINUs expunged it from that class ; and
rightly so, as Vossius remarks.
Thus, too, DonATUs called qualso an adverb. The
truth is that such verbs as quaeso and amabo, thrown
into a sentence interjectionally, and not connected
with any other word in the construction of the sen-
tence do not differ, as to grammatical principle, from
pure interjections, and therefore may be referred to
that part of speech; but cannot be regarded as ad-
verbs without great impropriety.
The interjections heus! and utinam, have also been
reckoned among adverbs : and even the pronouns
compounded with a preposition, as mecum, nobiscum, and
the like, the error of which is ably pointed out by
VossIUs in his first book De Analogid, cap. 2.
There is, perhaps, some nicety in determining whe-
ther certain words are more properly to be reckoned
adverbs or conjunctions. Thus primö, deinde, denique,
and such like words, are called adverbs, and some-
times not improperly so ; but when they serve to
combine together sentences, and to show the rela-
tion of the verbs to each other, they ought to be
deemed conjunctions. In this class we are inclined
son, and after him TookE, call an adverb.
Thus in the following passage from Lord BAcon —
Many of our men were gone to land, and our ships ready to de-
part; nevertheless the admiral with such ships only as could sud-
denly be put in readiness made forth towards them.
Nevertheless answers exactly to 'yet, which is dis-
tinctly stated to be a conjunction both by Johnson
and Tooke. Nay Johnson, in explaining the word
get, thus expresses himself—
YET conjunct (gyt, get, geta, Saxon.) Nevertheless, notwith-
standing, however.
And in the sentence above quoted the sense would
be exactly the same, whether we should say—
Though many of our men were gone to land, the admiral put
forth.
Or—
Many of our men were gone to land, yet the admiral put
forth.
Or—
Many of our men were gone to land, nevertheless the admiral
put forth. #
Upon the whole, it will be seen, in these and simi-
lar instances, that the conjunction is an adverb and
something more. It is an adverb, inasmuch as it serves
to modify the verb, with which it is immediately con-
nected ; but it is something more, inasmuch as it
shows a relation between that verb and another, and
connects together the sentences to which those verbs
belong.
§ Of Prepositions.
tions.
\-V-Z
We now come to a class of words, best known in Name.
modern times by the name of prepositions, though
they have by some writers been more appropriately
termed adnomina, or adnouns. As our object, how-
ever, is to change as little as possible received terms
and modes of reasoning, we shall adopt the generic
word preposition, for the part of speech, which we
have at present to consider.
In the Greek and Latin languages, the words thus Errors
distinguished were most commonly (though with respecting-
some exceptions) placed immediately before the sub-
stantives to which they referred ; and they were
subject to few variations in point of form. These
circumstances, as will presently be shown, were
merely accidental or idiomatical, but they were un-
fortunately selected by some grammarians as essential
to the preposition; and hence originated the well-
known definition prapositio est pars orationis invaria-
bilis, quae praepomitur aliis dictionibus. Some of the
Greek grammarians, considering that prepositions
connected words, as conjunctions did sentences,
ranked both the preposition and conjunction under
the common head of Xiw8éopos, or the connective, and
the stoics adding this circumstance to the ordinary
position of the preposition, in a sentence, called this
part of speech Xvvéeguos Tſpoëetukos. Another acci-
dental peculiarity of most of the words which were
used as prepositions, in Greek and Latin, as well as
in some modern languages, was that their original and
peculiar meaning had, in process of time, become
obscure ; and from hence some persons were led to
R 2
124
G R A M M A. R.
Grammar.
S-N-2 own.
Definition.
think that these words had no signification of their
The learned HARRIs gives the following defi-
nition, “A preposition is a part of speech devoid itself
of signification, but so formed as to unite two words, that
are significant, and that refuse to coalesce, or unite of
themselves. CAMPANELLA also says of the preposition
per se non significant ; and Hoogeveen says, “ Per
se posita et solitaria nihil significat.” Under the same
impression, the Port Royal grammarians say, “On a
eu recours, dans toutes les langues, ā une autre invention,
qui a 6té d'inventer de petits mots pour étre mis avant les
noms, ce qui les a fait appeller prépositions. And M. de
BRossEs says, “Je n'ai pas trouvé qu'il fut possible
d'assigner la cause de leur origine; tellement que j'en
crois la formation purement arbitraire.”
Now, in all this there was certainly much inaccu-
racy of reasoning. As to the position of these sort of
words in a sentence, even in Latin, the preposition
tenus was always placed after the noun which it go-
verned ; so Plautus uses erga, after a pronoun, as in
'mederga, for erga me; and cum is employed in like
manner in the common expressions mecum, tecum, no-
biscum, vobiscum. These and other examples of a like
kind induced some authors to make a class of post-
positive prepositions. “ Dantur etiam,” says CARAMUEL,
“Postpositiones, quae praepositiones postpositiva solent
dici;” but there are languages in which all the pre-
positions, if we may so speak, are postpositive.
Dr. JAULT, speaking of the Turkish and Hungarian
tongues, says, “Les prépositions de ces deux langues,
aussi bien que de la Georgienne, se mettoient toujours
après leur regime.” And HALHED in his grammar of
the Bengal language, says, “ the noun in regimine,
with a preposition, should properly be in the posses-
sive case, and prior in position.”
It is not surprising that Mr. TookE should ridicule
these postpositive prepositions, and nonsignificant words
which communicate signification to other words; but
unfortunately he only substitutes worse errors of his
own, when he asserts that prepositions are always
names of real objects, and do not shew different ope-
rations of the mind.
The real character and office of the preposition have
been stated with a nearer approach to accuracy by
Bishop WILKINs and Vossius; but neither of them
seems to have given a full and satisfactory definition
of this part of speech. WILKINs says, “ Prepositions
are such particles whose proper office it is to join in-
tegral with integral on the same side of the copula,
signifying some respect of cause, place, time, or other
circumstance, either positively or privately.” WossIUs
says, praepositio est vox per quam adjungitur verbo
nomen, locum, tempus, aut caussam significans, seu
positive seu privative.”
It suited Wilkins's scheme of universal grammar to
call the preposition a particle, but however appro-
priate this may be to a theoretical view of language,
such as it never did, and probably never will exist,
it does not suit our view of those philosophical prin-
ciples on which the actual use of speech among men
depends. On the other hand, as Wilkins includes
under the term integral both the noun and the verb,
he is in this respect more accurate than Vossius, for
the preposition does not merely join a noun to a verb,
but sometimes to another noun.
We, therefore, with that diffidence which becomes
all persons who endeavour in any degree to clear the
path of science, shall propose the following definition
of a preposition ; a preposition is a word employed in a
compler sentence to express the relation in which a sub-
stantive stands to a verb, or to another substantive.
Saul was before David.
He speaks concerning the law.
The Duke of Wellington liberated Spain.
Caesar, with his army, extinguished freedom in Rome.
Justice is nobler than unlicensed force.
In these examples the same function is performed
in the construction of the respective sentences, by the
words before, concerning, of, with, and in ; but it is per-
formed in somewhat a different manner. ,
1. The preposition before, expresses the relation of
priority, in which the substantive Saul, stands to the
substantive David, the mere verb of existence inter-
vening. -
2. The preposition of expresses the relation of ap-
purtenance, in which the substantive duke, stands to
the substantive Wellington, no verb intervening. .
3. The preposition concerning, expresses the rela-
tion of subject to action, in which relation the Substan-
tive law stands to the verb speaks. -
Preposi-
tions.
\-N/~
4. The preposition with, expresses the relation of
means to action, in which the substantive army, stands
to the verb, eactinguished.
5. The preposition in, expresses the relation of place
in which the substantive Rome, stands to the same
verb, eactinguished. :
I. We say, that the preposition is always employed
in a compler sentence; for as the noun and verb make
up one proposition, and the noun, verb and adverb
two, so the noun, verb, and preposition, with the
noun which follows, or is governed by the preposi-
tion, make up three propositions. Thus “ John
walks before,” is a sentence involving these two
propositions—
John is walking.
John is before.
But “John walks before Peter,” is a sentence in-
volving these three propositions—
John is walking.
John is before.
Peter is behind.
In like manner the sentence “ the Duke of Wel-
lington conquered,” may be resolved into these three
propositions—
The Duke conquered.
He belonged to a certain town.
The town (to which he belonged) was Wellington.
And thus we may always resolve a sentence into its
separate propositions, by expressing in a distinct form
the conception implied by the preposition, and con-
necting it successively with the two terms related to
each other.
II. The origin and use of prepositions may best be
considered, by adverting to the three different modes
in which the particular relation of a substantive to a
verb, or to another substantive, may be expressed in
language, namely, by a combination of words, by a
Complexity
of sentence.
Origin and
UISE.
G R A M M A. R.
125
Grammar, single word, or by the declension of a word. , ,
S-N-" A combination of words constitutes a phrase, or claus
Substanti-
val phrases.
Stead.
Cause.
Fault.
in a sentence, which may be introduced solely to ex-
press the relation conveyed in a different language, or
mode of writing, by a single preposition. Thus in the
letter which Hotspur reads in King Henry IV. part 1.
“I could be well contented to be there in respect of
the love I bear your house,” the words “ in respect of
the love,” may be rendered in Latin “ propter amo-
rem ;” or may be turned in English “ for the love.”
Let us, therefore, first consider how phrases of this
kind are formed.
1. We may place under the head of substantival
phrases, all those in which the conception of the rela-
tion meant to be expressed is given in the form of a
substantive. Such are the phrases, “ in respect of,”
“ per rispetto di,” “ in consideration of,” “a cause
de,” “permancanza di,” &c. &c.
Now these words respect, rispetto, consideration,
cause, and mancanza, retain in English, French, and Ita-
lian, respectively, their separate use as substantives;
and the same may be said of the expression more
common in Scotland than in England, “ in place of ;"
but the phrase corresponding to this last, viz. “instead
of,” exhibits a noun, which, in the sense of “ place,”
has become obsolete. Accordingly, Dr. JoHNson, in
his Dictionary, has the following articles :-
STEAD, n. s. 1. Place. Obsolete.
Fly, therefore, fly this fearful stead anon,
Lest thy fool hardize work thy sad confusion.
- Fairy Queen.
Instead of. Prep. [a word formed by the coalition
of in and stead place.]
I. In room of ; place of.
Vary the form of speech, and instead of the word church make
it a question in politics, whether the monument be in danger.
SWIFT.
Here, we see, is some little confusion ; inasmuch as
Johnson has not very clearly explained whether he
considers the two words in and stead, or the three
words in, stead, and of, to have coalesced into one
word, and formed one preposition. It may, therefore,
be more advisable to call all such expressions prepo-
sitional phrases.
It is easy to conceive, that the noun stead might
have been used alone, with the same force and effect
as we now use the whole phrase instead of; for, in
fact, the word statt, which is only a variety of pro-
nunciation, is so used in the German language, as
statt meiner, “ instead of me :” and in a manner not
very dissimilar, we ourselves use the Latin noun vice,
especially in the official notices of appointment to
rank or office, as, “X. Y. to be captain by purchase,
vice T. B. promoted.”
“ Because of,” answers to the French prepositional
phrase, à cause de, and to the Italian per rispetto di.
Dr. Johnson says of the word because, “it has, in
some sort, the force of a preposition ; but because it
is compounded of a noun, has of after it.”—
Infancy demands aliment such as lengthens fibres without break-
ing, because of the state of accretion.”
' ARBUTHNOT, on Aliment.
The substantive faute in French is employed in the
formation of a prepositional phrase, both with and
of.”
without the preposition a preceding it; as “il est *
mort, faute de secours,”—“ à faute de lui rendre foi
et hommage, il fera saisir le bien.” So in low col-
loquial English, we use the expression “for fault
of;” as in the Merry Wives of Windsor—
QUICKLY. Peter Simple, you say your name is 2
SIMPLE. Ay, for fault of a better.
And in Italian the substantive mancanza, is em-
ployed in a similar phrase : “ non fugia fatto, che
per mancanza di fede, o di memoria.” (Lettere di
G. DELLA CASA.)
In the spite of is a prepositional phrase occurring in
Bishop LATIMER's sermons :-
A gentlewoman came to me, and tolde me that a great man
kepeth certayne landes of hers from her, and wyll be her tenaunte
in the spyte of her tethe.
This phrase is shortened by some of the poets to
spite of Thus Rowe– -
For thy lov’d sake, spite of my boding fears,
I'll meet the danger which ambition brings.
The substantive spite signifies malice, rancour,
hatred, malignity, malevolence; but the prepositional
phrases “spite of,” “ in spite of,” and “in the spite of.”
are often used, as JoHNson observes, without any ma-
lignity of meaning ; for words, in the course of time,
obtain, in some instances, a greater latitude, and in
others a closer restriction, of meaning; and in the
present case there is a transition from the idea of that
opposition which arises from malignity, to the more
comprehensive idea of forcible opposition in general.
It is somewhat doubtful whether the substantive
despite, and the prepositional phrases, despite of, and
in despite of, are not of different origin from the pre-
ceding. Spite is certainly connected with the Dutch
spyt, spite, vexation; and in that language are the
phrases my te spyt “in spite of me,” and spyt zyn
bakkus, in spite of his teeth ; but the Dutch spyt
enters into the composition of several other words, as
spytig, spiteful, fretful, vexatious, Spytigheyd, fretful-
ness, spytiglik, spitefully; and they say dat is spytig,
for “ that is vexatious,” “ that is a pity.” The no-
tion conveyed by all these words is analogous to the
sense of being pricked or wounded by a pointed in-
strument, and it is doubtless connected with our
word spit, and with the German spitze, which signi-
fies any substance terminating in a sharp point.
Hence spiz, according to WACHTER, is “acutus, acu-
minatus;” spizzi stechun, in Frankish, is pointed
stakes, “ Dicitur allegorice,” adds Wachter, “de in-
genio acuto, sed callido, maligno, et ad decipiendum
nato. Inde spiz-kopf caput astutum, spizbube, fur
vafer,” &c.
Spite.
\-N-"
Despite, on the other hand, is from the French Despite.
depit, formerly spelt despit, which MENAGE derives
from dispectus, (he must mean despectus,) despised.
From despectus was formed the Italian dispetto, as in
the prepositional phrase per dispetto di, “in contempt
Thus BoccAcro says, “ Che ne dobbiam fare
altro, se non torgli que' panni, ed impiccarlo, per dis-
petto degli Orsini, a una di queste querce.” The French
depit or despit, is explained in the Dictionnaire de l'Aca-
demnie, “fascherie, chagrin meslé de colère ;” and it is
added, “On dit, en depit de luy, pour dire malgré
luy;” but in an earlier period of the French language
126 G R A M M A. R.
\-V-' not anger, but contempt.
Gré.
Grammar, the prevalent idea conveyed by the word despit was
Thus in a poem on the Game
of Chess, the earliest on the subject now extant, hav-
ing been transcribed in the 13th century, (MS.
Cotton. Cleop. b. ix. 1.) we find the following pas-
Sage:–
Mes vme gentz sount ke endespit,
Vnt les-giuspartiz, e prisent petit,
Purceo q' poi enseiuent ou nient.
i. e. “ but there is one kind of people who have in
contempt games (of chess) and prize them little ; be-
cause they know little or nothing about them.”
And from the lines immediately following it appears
that the obsolete verb despire was exactly our verb
“ to despise.”
Mes ceo net pas a dreit iugement,
De despire ceo du’t neu seit lauerité.
i. e. “But this is not (according) to right judgment,
to despise that of which one knows not the truth.”
ShaksPEARE appears to have felt the true meaning
of the word despite, as implying, from its Latin origin,
contempt, when he makes Coriolanus exclaim to the
tribune, Sicinius—
Thou wretch despite o'erwhelm thee!
The French substantive gré, gave rise to our obso-
lete preposition mauger, (for so it is spelt in Bishop
Latimer's sermons,) and it will be worth while, first,
to trace the growth of this substantive from the Latin
adjective gratus, and then to observe how it was em-
ployed in various prepositional phrases, and those
phrases ultimately melted down into a single word,
so as to form a clear and genuine preposition.
From the classical Latin adjective gratus, agreeable,
were formed the barbarous Latin substantives gratus,
and gradus, signifying that which is agreeable to a
person, or conformable to his free will ; as in the
following instances:—
Idem feodum a manu monachorum alienare non possumus, nisi
grato et voluntate Ducis Burgundiae.
Chart. A. D. 1197.
Tu qui meus es, quomodo teneas hoc quod ego non dedi tibi
extra meo gratu ?
Vet. Chart, ap. Beslium, p. 392.
Ipse autem de suo gradu respondit quod in illud scriptum non
intraret.
Capit. Carol. Calv, tit. 24.
From these substantives come the barbarous Latin
verbs grato et grator, “ to agree or grant freely,” and
the adverb gratanter, “willingly.”
From the same source came also the Italian sub-
stantive grado, free will, approbation, thankfulness,
as in DANTE :-
Ma poichè pur almondo furivolta,
Contra suo grado, e contra buon usanza,
Non fu dal vel del cuor giammai disciolta.
And in BoccAcIo—
Niuma ragion vuole, che grado si senta del non ricevuto bene-
ficio.
So we find a grado, and a grande grado, used in an
adverbial manner, for “ agreeably,” “ very agree-
ably.”
Tanto bene, e s a grado comincio a servire ad Egano, che egli
gli pose amore.
BoccAcio.
Fatto era, quanto egli aveva comandato, a grande grado
piacere disanta Chiesa.
M. WiLLANI.
Di grado, and di proprio grado, are also used, in an
adverbial manner, for “willingly,” “ spontaneously.”
Che difendesse la sua franchezza, e libertà, eche non si mettesse
di grado in servitudine ; perocchè maggior vituperio è sostenere
Servitudine di proprio grado, che per forza.
Volgar. Pist. Senes. 95.
From the Italian grado proceeded the old French
greit, grez, and grá.
Car ilh s'estoient tos bin wardeis, sans avoir mal greit de nulle
des parties.
HEMRICURTIUS, de bel. Leod. c. 38.
Tos furent lié de sa venue;
Grez, et mercez lui out rendue.
MS. Poeme : Guer de Troie.
Gré, in more modern French, is explained “bonne,
franche volonté, qu'on a de faire quelque chose ;” as
“il y est allé de son gré, de son plein gré;” “ils ont
contracté ensemble de gre a grè ;” “ ille fera bon
gré, mal gré. Savoir gre, is “ to be satisfied with" a
person's conduct, to be obliged to him for it : lui sa-
voir un gré infini, “to be infinitely obliged to him.”
Thus, in a letter written by order of the King of
FRANCE, in 1814, to the author of certain political
works, it is said, “ Sa majesté vous sachant un gré in-
fini de la manière dont vous avez pris, dans des temps
difficiles, la défense de ses justes droits,” &c. and
these phrases appear to be imitated from the Italian
so grado, as in BoccACIo—
Signori, di city, che iersera vi fu fatto, so io grado alla fortuna.
Faire gre, in old French, was to do what is agree-
able to right and justice, as to satisfy a debt, a tax,
or a reckoning.
Seil avient que uns hom fesist semonrre un autre pardevant le
justiche por dete, et cil, de qui on se clameroit, ne seroit mie de
le quemugne, si connoissoit le dete, il seroit tantost a 2 sols et
demi, et se li convarroit faire son gré s'il avoit de coi; et s'il
desconnoissoit la dete, il en demoueroit quites.
Usat. MSS. Civ. Ambian.
Icellui Guillaume compta et fist gre à l’oste de l'escot de lui, et
de ses compaignons.
MS. Letter, A. D. 1395.
This expression is imitated by CHAUCER in his Mer-
chant's Second Tale, v. 1326.
And he myght be take he shuld do me gre.
From the substantive grè came the old French greer,
to agree to, grant, or approve :-
Toutes les choses dessus dittes il grgerent, roerent, ratefierent,
et accorderent.
Chart. A. D. 1323.
In the same sense was used agréer, whence came
the barbarous Latin agreamentum, “ an agreement,”
which Rastall whimsically expounds aggregatio men-
tium.
From gre came also the old French word engrés for
willing, ready, well disposed.
Soions engrés, soions engrant,
De lui servir et jour et nuit'
MS. Mirac. B. M. V. lib, 2.
The word grew, anciently used in Valentia for a
marriage gift freely made by the husband to the wife,
* Preposi-
G R A M M A. R.
127
Grammar.
\-N-"
appears to agree with the old French greit. (See Fon-
TANELLA de pact, Nupt. f. 2. cl. 7. gl. 1.)
GAwiN Douglas has adopted the French gre into
the Scottish language, in the sense of a prize; as—
The bull was the price and gre of thare dereyne.
But Junius erroneously derives this from the French
degré, the origin of which is the Latin gradus, a step.
Having thus traced the simple word gratus, “agree-
able,” through its derivatives, we have next to view
it compounded with malé, “badly.”
Malo grato is used by MATTHEw of PARIs, in two
passages, with a slight difference of construction.
Under the date of the year 1245, he says, “Liberta-
tem ecclesiae, quam ipse nunquam auxit, Sed magnifici
antecessores ejus malograto suo, stabilierunt.” Under
1252, he thus relates the quarrel of King John with
his brother “ Cui ait electus — Domino Deo vos
commendo. At Rex, et ego te diabolo vivo. Et ego
te malo grato Dei et ejus sanctorum,” &c. Here the
adjective and substantive appear to be used separately;
but they are combined into one word in malegratibus
dentium which occurs in a MS. of the year 1350.
“ Galteronus ira motus dicit ad supplicantem plurima
verba injuriosa, quod malegratibus dentium ipsius Sup-
plicantis, ipse bene solveret simbolum suum.”
So, in Italian we find, a mal mio grado, a mio mal
grado, a mal grado di lui, mal mio grado, mal suo grado,
nalgrado di voi, &c.
La casa oscura, e muta, e molto trista meritiene, riceve, a mal
7mio grado.
BoccAcio.
Il di sequente passarono il fosso, a mal grado della forza de'
Pisani.
M. VILLANI.
Che chi possendo star cadde tra via
Degno é, che mal suo grado a terra giaccia.
PETRARCA.
In like manner mal and grá are combined in French.
These two words appear to be used as an adjective
and substantive in the Roman de Rou.
Guert out si le conseil troublé,
Que puis n'i out home escouté,
Qui de faire pais ait parlé,
Qui des plus riches n'ait mal gre.
But they seem rather to form a compound substan-
tive, in the following passage of a MS. letter dated
A. D. 140l.
Guillemette Quesnel jeune femme non mariée, pour ce qu’elle
estoit ensainte, et grosse d’enfant—doubtant le malgré de ses
amis, &c.
Malgré became maugré by the general tendency of
the French to corrupt al into au, as alter auter, autre;
wltra, outre; thus mau is used for mal in the old pro-
verb, “a mau chat, mau rat,” meaning “ two knaves
well met.” So in the compounds maudire, to curse ;
maudisson, a curse, opposed to benisson, a blessing ;
as in the Scottish dialect malison is to benison; mau-
mené ill used mauffait, a goblin ; maugréer, to revile,
rail upon, and show ill will to.
CHAUCER frequently uses maugre as a preposition.
Thus in the Knight's Tale :-
And I will loue her maugre all thy might.
In BARBouſt we find the same word spelt magre:—
Through him I trow my land to win,
Magre the Clifford, and his kin.
|
Lastly, in Bishop LATIMER's Sermons, it is spelt
nauger.— -
God worketh wonderfully, he hath preserued it mauger theyr
heartes.
The English substantive, time, and the French Time.
temps are used in prepositional phrases, more or less, Term.
ample or abbreviated. Thus, in the statute 1 Ric. III.
c. 7., which was enacted A. D. 1483, and remains on
record both in the French and English languages of
that day, we have “ the meane tyme” where we
should now use “ in the mean time,” “all plees the
meane tyme to cesse ;” in the French copy “ toutz
plees le meane temps de cesser.” In another part the
phrase is fuller “ en le mesme temps toutz pleez ces-
sent ;” “ in the same tyme all plees cesse.” And
elsewhere we have “ al temps de le dit fine levez;”
“ at the tyme of the seid fyne levied.”
But in another passage, the words tyme and temps
are respectively used without either preposition or
article preceding them, “ saving to every persone
such right, &c. as they have to or in the seid londes,
&c. tyme of such fyne ingrossed.”—“ Sauvant a chas-
cune persone autielk droit, &c. queux ils ount au ou
en les ditz terres temps dutiel fine engrosse.” The
word term is also used in the same absolute way, in
the first chapter of the statutes made in this year,
(the earliest statutes on record in the English lan-
guage,) “ne leses à terme de vie ou des ans, ne an-
nuiteez grauntez à ascune persong ou perSonez pur
leur service pur terme de leur vies,” which in the En-
glish MS. copy runs thus, “Nor leses terme of lyff or
of yeres, nor annuites graunted to eny personne or
persones for their service, terme of their lyfes.”
From the French substantive tour comes the old Tour.
word entour, which is used both as part of the prepo-
sitional phrase à l'entour de, and also alone, as the
mere preposition “about.”
An ode of RonsaRD, imitated from Anacreon, begins
thus— -
Le petit enfant, Amour,
Cueilloit des fleurs, ā l'entour
D'une ruche, oil les avettes,
Font leur petites logettes.
In the letter of PERREs DE MoUNFort, before
quoted, we have entour, where in modern French
environ would be used, the former preposition having
become obsolete though the verbs entourer and en-
vironner, are alike in use. “Defendimes le givez del
ewe de Osk—jekes au Samadi entour oure de midy.”
“We defended the fords of the river Esk, until Satur-
day about the hour of noon.”
2. Adjectives may be used in the same sort of prepo-Adjectival
Thus MILTON, in his “ Essay on the phrases.
sitional phrases.
reason of Church Government,” says, “If the course of
judicature to a political censorship seem either tedious
or too contentious, much more may it to the disci-
pline of the church, whose definitive decrees are to be
speedy, but the execution of rigour slow, contrary to
what in legal proceedings is most usual.”
This adjective, contrary, we find used preposition-
ally in the Scottish acts of Parliament, both in the
phrase “ in.contrar the command,” and also in the
separate word “ contrare,” as in the act of 1554,
128
G R A M M A. R.
Grammar.
Salvus,
Long.
“ contrare the priuilegis of oure crowne.” In the
latter instance it answers precisely to the French pre-
position contre, and therefore is equally entitled to be
ranked in that class. In old French there was also
the preposition encontre, which now exists only as a
substantive, signifying an adventure : nor is the verb
encontrer at present in use, though the substantive
rencontre, and the verb rencontrer both are so; and
though in English we retain encounter and rencounter,
both as substantives and as verbs. It is probably
from rencounter that we originally took the expression
of running counter to ; as in LocKE—
He thinks it brave at his first setting out to signalize himself in
running counter to all the rules of virtue.
Where, as the words counter to, perform the func-
tion of a preposition, they may justly be described as
a prepositional phrase.
The Latin adjective salvus, when placed in the ab-
lative case absolute, may be considered as used pre-
positionally, and has in fact given rise to the Italian
salvo, the French sauf, and the old English saufe, all
which may be regarded as real prepositions.
CICERo, in a letter to P. Lentulus, the proconsul,
describing his success in a debate against the tribunes
of the people, thus speaks—
Quod ad popularem rationem attinet, hoc videmur esse conse-
cuti, ut ne quid agi cum populo, aut salvis auspiciis, aut salvis
legibus, aut denique sine vi, possit.
where we see, that in the construction of the sentence,
salvis and sine, have the very same effect ; for agere
salvis auspiciis, and agere salvis legibus, and agere sine
vi, describe three modes of action, in which the rela-
tion of the substantives auspiciis, legibus, and vi, to the
verb agere is expressed by an intervening word, in the
nature of a preposition.
In the vocabolario degli Academici della Crusca, we
find salvo thus described, “ SALvo. Avverb. che talora
si adopera in forza di prepositione; e vale eccettuato,
fuorche, se non,” and among other examples given
is the following, “Rendégli la signoria di Lombardia,
salvo la Marca Trivigiana.”
In the Dictionnaire de l'Academie Francoise, it is
said, “ SAUF se met quelque fois par manière de pré-
position, et signifie Sans blesser, sans intéresser, Sans
donner atteinte; sauf votre honneur,” &c. And again,
“SAUF signifie quelquefois hormis, excepté, a la re-
serve de; illuy a cede tout son bien, sauf ses rentes.º.
Gower has adopted this word sauf into English
poetry with a conjunctional force :—
Saufe only, that I crie and bidde,
I am in tristesse all amidde.
The word long is employed in English preposition-
ally, as we shall presently show ; but not always in
its adjectival sense. The English adjective long, is
from the Latin adjective longus, signifying length
either of space or time. It does not appear that
longus was ever employed prepositionally, although it
may perhaps be justly said that longe was so, in such
phrases as “ longe gentium,” which CICERo employs
in writing to Atticus.-
Scribendum aliquid ad te fuit—non quo me aliquidjuvare posses,
quippe res est in manibus; tu autemabes, longe gentium.”
The Italian lungo, however, which is only this same
adjective longus differently pronounced, is universally Preposi-
reckoned among prepositions.
LUNGo. Preposiz. Rasente, Accosto; e si usa per lo più col
quarto caso. Lat. Juarta, prope. -
Vocab. Della Crusca.
Già eravam dalla selva rimossi—
Quando ’ncontrammo d’anime una schiera,
Che venia lungo l'argine.
DANTE.
The French use long substantively in the preposi-
tional phrases “ le long de,” “ du long de,” and “ au
long de,” and this both with respect to space and
time; as il a jeune tout le long du caréme; allez tout
du long de l'eau, &c.
They also appear to have formed their adverb and
preposition loin, formerly written loing, from loinquo,
a corruption of the Italian longinquo, which was the
Latin adjective longinquus, derived from longus, as
propinquus was from the old word propus, mentioned
by VossIUs.
In old and modern English we have the following
words, which it will be convenient to consider toge-
ther, endlong, along, to belong, and to long. Mr.
TookE treats of them at some length ; but not satis-
factorily. Along, to which he ascribes only one origin,
appears to have had two, viz. on long, i.e. on length;
and gelang, i.e. belonging, or appertaining to. When
Mr. TookE observes that the Anglo-Saxon lengian is
“ to make long,” he merely proves that long in longus
and leng in lengian, were originally the same word,
which is by no means extraordinary; for the radicals
leng, lang, lag, lank, are found in most of the northern
dialects, expressing a variety of conceptions all con-
nected either with the idea of length, or else with the
more general idea of position; for lagen, “to lay,” and
langen, “ to stretch out,” appear to have been words
of the same or similar origin. Hence we have
1. The Gothic lagg, Anglo-Saxon lang, lang, long,
Frankish and Alamannic lang, lanc ; modern German
and Scottish lang, Islandic langr, all signifying that
which is extended in length, either of space or time.
2. The Alamannic alangaz, alonges, et alongi, totum,
ex integro; “Dictio figurata,” says WACHTER, “ quà
longus ponitur pro non-interruptus, quia integrum con-
tinuo simile est.
3. The Frankish gilengen, to prolong.
4. The German langsam, slow, tedious from length
of time.
5. The Frankish langen, to draw or stretch out in
length ; lang, plaustrum ; the German belangen, tra-
here in forum, accusare, &c.
6. The German verlangen, desiderare, and the En-
glish “to long for.” “Sensu,” says WACHTER, “a
trahentibus desumpto quia desideria trahunt, et desi-
derantes trahuntur in rem, eamque vicissim attra-
hunt. Utrumque sane habet suos funiculos, et desi-
derium quo trahimus trahimurque, et res concupita
quae trahit.” -
7. The German gelangen to attain to that which we
have longed for, which we have been long in seeking,
and which at length we have got.
8. The German anlangen, and belangen, pertinere,
as in the phrases cited by WACHTER, was mich belangt,
was mich aniangt, quod ad me spectat. &
From a similar source were probably derived our
lank, lag, linger, &c.
tions.
G R A M M A. R. 129
Grammar.
\-y-
That lagen and langeå, or lengan, should have been
merely different Inodes of pronouncing the same word,
will surprise no one who has observed the frequent
instances, in which the letter n was used by some
Gothic tribes, and omitted by others, in words of
precisely the same origin and import: thus we have
the Gothic munds, and English mouth : the Latin dens,
and English tooth, &c. &c. The Anglo-Saxon verb
langan or lengan had therefore two senses; one being
to make long, the other to be laid on to, connected
with, or dependent on ; and the diversity of its appli-
cation has produced a corresponding difference in the
use of the more modern words which are traceable to
its origin. - - -
1. One class presents either the literal or meta-
phorical conception of that which is stretched out in
length ; and to this class belong the old English pre-
position endelong and Scottish endlang, signifying
extension in length from end to end; the modern
preposition along, as used in the same sense; the
same word along formerly used as we now use long in
the phrase “all night long ;” and, lastly, the verb, to
long for; that is, to stretch out the mind after an object.
2. The other class signifies connection, or depend-
ence. Hence, to belong to (the German antangen or
belangen abovenoticed) is to be holden, as a house
metaphorically is by its owner, or to be bound, as a
son is in the figurative bonds of relationship to his
family. Hence also the now obsolete phrase along of,
or long of, implying to be caused by the person or
thing specified. -
A few examples will illustrate what we have here
said. - - -
Thus DUNBAR, in his Goldin Terge, uses endlang.
Ladys to daunce full Sobirly assayit,
Endlang the trotting river so they may it.
And so GAwiN Doug LAs, in many places, e. gr.
Bot than the women al for drede & affray,
Fled here and there endlang the coist away.
In the romance of Richard Coer de Lion the word
endelang is used adverbially in describing an engine
employed by that monarch at the siege of Acre :
Ovyrtwart & endelang,
With strenges of wyr the stones hang.
The same word occurs in the Scottish Acts of Parlia-
ment, v. ii. p. 19. A. D. 1430 ; “strekande endlang the
coste.”
GoweR and CHAUCER use endelonge.
She slough them in a sodeine rage
Endelonge the borde, as thei ben set.
GOWER,
This lady rometh by the clyffe to play •
With her meyné endlonge the stronde.
CHAUCER.
Tooke justly derives our modern along from on
long, or on length ; which last expression is used by
CHAUCER, in the Testament of Love.
stretched herself along) and rested awhile.” But
Tooke erroneously supposes that our most ancient
English writers only used the word along in the sense
of the Anglo-Saxon gelang ; i.e. “opera, causã, impulsu,
culpâ cujus vis;” and he therefore improperly accuses
Gower of using alonge for endlonge in the following
line,—
I tary forth the night alonge.
WOL. I.
‘‘ And these
wordes said, she streyght her on length, (i. e. she
The force of the word alonge is here the same as that Chap. I.
of long, in MILTON's beautiful lines—
See there the olive grove of Academe,
Plato's retirement, where the attic bird
Trills her thick warbled notes, the summer long.
* - Paradise Regained.
And Gower is not singular in using along, to signify
length of time; for we meet with the following pas-
sage in Robert DE BRUNNE :—
Here I salle the gyue alle myn heritage,
And als along as I lyue to be in thin Ostage.
To along was in like manner used, where we now use
to long ; as in GoweR, -
This worthy Jason sore alongeth
To see the strange regions.
The meaning of this verb, to long, is well illustrated
by Tooke from the Anglo-Saxon “Langath the awuht
Adam up to Gode,” i.e. longeth you, lengtheneth you,
stretcheth you, up to God. *
The preposition along, in the sense of on length, is
now commonly used, as in the following passage from
MILTON's Lycidas
So Lycidas sank low, but mounted high,
Where other groves, and other streams along,
With nectar pure, his oozy locks he laves.
The modern Scottish dialect, for along, in this
sense, uses alongst ; as we say amongst, amidst, whilst,
for among, amid, while ; and So we find alongest and
alongst in old English.
Phormyo was constrayned to cause his people to be
soubdenly embarqued, and to Sayle alongest by the laride.
Nicoll’s Z'huciditles.
They toke their waye towards the sea, alongest the sayd ryuer.
Ibid.
The Turks did keep strait watch and ward in all their ports
thereabout alongst the sea-coast.
ISNOLLES. Hist. Turks.
It is somewhat remarkable, that JoHNson, in citing
this last sentence, should call alongst an adverb ; since
it is manifestly a preposition governing (as the com-
mon grammarians say) the noun “ sea-coast ;” and
the sense is, “ the Turks watched the coast on its
length,” or “ the Turks watched throughout the length
of the coast.”
:4gidst, which is a word exactly of the same nature
as along or alongst, Johnson properly calls a prepo-
sition ; and with the same propriety explains to sig-
nify “in the midst"—
Of the fruit
Of each tree in the garden we may eat;
But of the fruit of this fair tree amidst
The garden, God hath said, ye shall not eat.
MILTON.
Here it is equally manifest that the preposition,
amidst, is nothing more than the noun mid or middle
(from the Latin medius) with the Superlative termina-
tion est, and the corrupted prefix a ; and that the
whole sense would be “in the middest (or middle-
most) part of the garden.”
To return to the preposition, along, in the sense of
“ on length,” we may observe, that it is identical
with the adverb along in the common exclamations
“Go along !”—“Get along !"—that is, “ &loignez
vous ;”—“ abi in longinquum ;”—“ remove yourself
to some distance from this spot.” In like manner
g S
+
130
G R A M M A. R.
Grammar, must we explain the adverb along in the phrase ºf to
Near.
Opposite,
&c.
wka
Mesne.
go along with.” Thus ShakspeaRE—
I your commission will forthwith dispatch,
And he to England shall along with you.
*. Hamlet.
The verb, to belong, must be differently explained :
it is obvious, that this verb implies length or distance,
if at all, only in a very indirect, and indistinct manº tive salvis had in reality the force and effect of a
ner; but refers more distinctly to the notions of
connection and dependence already mentioned : and the
same must be said of the word along prepositionally
used by old writers to signify the relation of an effect
to its cause. In this sense it was followed by upon, on,
and latterly of, as the Anglo-Saxon gelang was by aet.
Thus, in the instance cited by Lys, “ aet the ys
ure lyfe gelang ;” “ on thee does our life depend.
But thus this maiden had wronge
Which was upon the kinge alonge. Gow'ER,
— your loue al fully graunted is
To Troylus, & therto trouth yplight,
That but it were on him alonge, ye molde
Him neuer falsen. CHAUceR.
It is long of yourself; for you were the party that commended
him to me. ARCHB. ABBOT'S Narrative.
The adjectives, near and nigh, are commonly used
in English as prepositions, as is the corresponding
Italian adjective vicino. The Latin propé, which
answers to our preposition, near, is the adverbial form
of the old adjective propus, before alluded to. The
French près is the Italian presso from the Latin parti-
ciple pressus. These words scarcely need illustration :
we may however observe, with Mr. TYR whit, that
next is the superlative of nigh, as the Saxon heat was
of high. This critic was also right in his remark,
(which Tooke unnecessarily censures,) that in modern
use we more commonly employ newt, to signify the
“ nighest following” than the “ nighest preceding ;”
though, in fact, it means simply the nighest. These,
however, are matters of mere idiom.
Although the word opposite be in its Latin original
(oppositus) a participle, yet it was first adopted into
the English language as an adjective, and then em-
ployed colloquially as a preposition. Thus we say,
“ opposite Somerset House”—“ opposite the Horse
Guards.” In like manner, many other adjectives are
used prepositionally, as “to walk round London,” &c.
Tooke therefore properly enumerates among prepo-
sitions, round and around, “whose place” he says,
“ is supplied in the Anglo-Saxon by hueil and on-
hweil ; in the Danish and Swedish, by om-kring; in
Dutch, by om-ring ; and in Latin, by circum ; a Gr.
kepicos, of which circulus is the diminutive.” Hweil,
it will be observed, is our substantive wheel, and is
probably connected with our verb to whirl; and it is
remarkable, that this same hweil forms our adverb
while, and substantive “ a while,” a time ; for the
continued motion of time has been often typified by a
wheel; and by a similar analogy, the year was called
in Latin, annus, from annulus, a ring ; as the Greeks
termed it eviavtos, from its revolving into itself.
The use of the adjective mesne, though not strictly
prepositional in the following passage, may yet serve
in some measure to illustrate the subject of which we
are speaking. -
Contrarie lawe it is, if after the exigent awarded, the appeale
doe abate for insufficiencie, or for that, that he that is outlawed
was imprisoned mesne betweene the awarding of the exigent and
the outlawrie pronounced. STAUNFord, on Prerogative,
3. Participles being merely adjectives involving the
notion of action as in existence, it is naturally to be
inferred, that they may be used as we have seen the
pure adjectives used, to perform the function of a
preposition. We have already had occasion to notice
the Latin ablative case absolute, in the instance of
“ salvis auspiciis,” where we showed that the adjec-
preposition; and this became still more obvious in
considering the old word saufe, which is only the
same adjective transmitted from the Latin language
through the French into English. The case is not
altered, when we find the participle saving, or the old
Scottish sawfande, employed in the same manner.
Thus, in the Act of 1455, we find “ sauſande the
poynts quhilks ar neidful for the conservacion of the
treaty.” So we say in colloquial language “ barring
accidents.” In the Scottish Act of 1456, the participle
belangande occurs with the same prepositional con-
struction. “As to the thirde artikill, belangande, the
sending to France.” In the Act of 1524, we meet
with the expression “enduring the time of his office;”
where, in modern English, we should use during.
legal phraseology the ablative absolute durante vità,
is rendered “for and during the term of his natural
life;” where, as the word during and the word for are
used with exactly the same force in the sentence, it is
plain, that if for be a preposition, during is one also
It happens, however, that our lexicographers have
only acknowledged those participles to be preposi-
tions which are most frequently so employed ; such
as touching and concerning, which are thus noticed by
Dr. JoHNson :- --
“Touching, prep. [This word is originally a par-
ticiple of touch..] With respect, regard, or relation to.”
Touching things which belong to discipline, the church hath
authority to make canons and decrees, even as we read in the
apostles times it did. HookER, book iii.
“ CoNCERNING, prep. [from concern : this word,
originally a participle, has before a noun the force of
a preposition.] Relating to, with relation to.”
There is not anything more subject to errour, than the true
judgment concerning the power and forces of an estate.
BACON.
Many other participles, however, might be pointed
out in various languages, which are plainly used as
prepositions, and some of them so recognised by
grammarians. Thus Cour DE GEBELIN ranks among
prepositions the present participles pendant, durant,
touchant, moyennant, nonobstant, suivant, and the past
participles, attendu, vit, and hormis. So we use pend-
ing, during, hanging, living, failing, considering, omitting,
regarding, respecting, and anciently moiening.
5 & ſp
At whose instigacion and stiring, I have me applied, moiening
the helpe of God, to reduce and translate it. R. COPLAND.
The participle hanging is used in one of our earliest
English statutes, as we now use pending, and the
French pendant : and corresponding to the ablative
absolute pendente lite. “The said accompt to be ij or
iij yere hanging,” Stat. 1. Rich. III. c. 14.
4. Verbs, either singly, or in combination with
In
Chap. I.
rv-, - Qº e &
Participial
phrases.
Verbal
other words, supply the place of prepositions, and Phrases-.
sometimes come to be considered as such. Thus, as
we have seen the adjective sauf and the participle
sawfande, used prepositionally, so we find the impera-
tive of the verb save employed for the same purpose.
G R A M M A. R. 131
Grammar. Dr. Johnson, by oversight, as it should seem, calls
^-y-' this word an adverb . TookE, in his Chapter on pre-
positions, more correctly mentions it thus—
“ SAVE. The imperative of the verb. This prepo-
sitive manner of using the imperative of the verb to
save afforded Chaucer's Sompnour no bad equivoque
against his adversary the Friar.
God save you all, SAVE, this cursed Frere.”
Here the construction is “Save (set aside or except)
this Friar; and then I hope that God will save (deliver
from evil) all the rest of you.”
So in the Squire's Tale.
This strange Knight that came thus sodenly
All armed, saue his hedde—
That is, the Knight was entirely armed, but when
you say entirely, you must save (or except) his head.
The words “ save and except” are often used sy-
nonymously in many of our legal instruments: we
shall not therefore be surprised to find except reckoned
by Dr. Johnson among prepositions—
“ExcEPT. preposit. [from the verb.] This word,
long taken as a preposition or conjunction, is origi-
nally the participle passive of the verb, which, like
most others, had for its participle two terminations,
earcept or excepted. All except one, is all, one excepted.
Except may be, according to the Teutonick idiom, the
imperative mood : all, except one ; that is, all but one,
which you must except.”
“ 1. Exclusively of; without inclusion of.
Richard earcept, those, whom we fight against,
Had rather have us win than him they follow.”
SHAKSPEARE, Rich. III.
For except were anciently used out-take outtak and
outtaken.
Which euery kynde made die
That upon middle erthe stoode
Outtake Noe and his bloode.
But yron was there none, ne stele
For all was golde men myght se
Out-take the fethers and the tre,
And schortly euery thing that doith repare
In firth or feild, flude, forest, erth or are,
Out-tak the mery Nychtyngale Philomene.
G. Douglas,
Gower.
CHAUCER.
But none of them it might beare
Upon his worde to yeue answere
Outtaken one, whiche was a knight.
Tooke has quoted from BEN Jonson the preposition
outcept, which he says is “ the imperative of a mis-
coined verb, whimsically composed of out and capere,
instead of ex and capere.” But this is probably no
more than a miscoinage of Ben Jonson's coarse and
pedantic wit, putting in the mouth of one of his charac-
ters such language as never was spoken. The passage
is from his Tale of a Tub:
“I’ld play him 'gaine a Knight or a good Squire, or Gentleman
of any other countie i'the kingdome—outcept Kent; for there they
landed all Gentlemen.”
Gow ER,
Very similar to the use of the imperatives ercept and
save, as prepositions, is the colloquial expression, “let
alone,” in use among the Irish Peasantry. Thus in
Miss EDGEworth's tale of Ormond, Moriarty Carroll
says: “It might happen to any man, let alone gentle-
man:”—The sense of which expression nearly answers
to the Latin medicam ; but in the construction it is
“let alone gentleman, speak not of that class of
society; for it is not only to them, but to any man Chap. I.
that such an accident might happen.”
Mr. Tooke says with some plausibility that the
French preposition avec is only a contraction of avez
que, have that ; but we must observe that in old
French we find it written, oveke, ove &c.; as in the
letter of Sir Perres DE MonTFoRT (A. D. 1256) before
quoted; and therefore it may possibly be of a different
Origin.
Most of the verbs and participles, which we have
noticed ; together with many others of a like nature,
are acknowledged by grammarians in general to be Abbreviat-
f ed forms.
prepositions, without any change of form or even o
accentuation; but there are other prepositional phrases
which, occurring frequently in conversation, lead to
abbreviations and ellipses, and thus ultimately leave a
single word which performs the function of a preposi-
tion. In order to illustrate what is here meant, we
shall begin with those words which retain the same
sense both in the form of prepositions and in that of
nouns or verbs: and afterwards we shall notice those
prepositions in which the original meaning of the
noun or verb from which they are derived, has become
obsolete, or is to be traced only by analogy.
The substantive Term has been already noticed as
employed prepositionally in our old Statutes : nor was
this a mere legal technicality: in an old poem, en- Where not
entitled Tytus and Gesyppus, published in the beginning obsolete.
of the sixteenth century, we find the following lines:
Tytus his wedynge rynge forthe than dyd take,
And put it on the fynger of his wyfe,
Grauntynge to be her husbonde terme of lyfe.
Here the full construction in modern language
would be “granting to be her husband, during the
term of her life;” but the noun being used absolutely
becomes a sort of preposition ; and if this mode of
speaking had obtained in general use, the word term
would no doubt have been reckoned by modern gram-
marians among our prepositions.
We have already said that the same might have
happened with the word stead, in English ; as it has
with the same word, pronounced statt in German. The
Germans too use our noun craft, (which with them
means strength) as a preposition; as kraft seines Amtes
“ by the power of his office.” So they say “Laut des
briefes,” the word laut (our loud) being the substan-
tive “ sound.” Laut des briefes, then, is originally
“ according to the sound of the letter,” and in its
modern sense “according to the purport of the letter;”
as we say an act “sounds to folly :” and so CHAUCER
Sowning in moral vertue was his speche,
And gladly wolde he lerne and gladly teche.
* Prol. viii.
The Germans likewise use the prepositions diesseits,
and jenseits, literally “this side,” and “yon side.” A
similar use is colloquially made, (particularly in the
West of England) of our common nouns outside and
inside; and the former is used by Col.BRIDGE in his
Christabel.
Outside of her kennel, the mastiff old
Lay fast asleep in the moonlight cold.
No difficulty whatever can occur in the explanation
of words, beginning with the prefix a, or be, most of
which we have already noticed in their adverbial use;
such as along, amidst, around, across, astride, aboard,
below, beside. In all these instances, the nouns or verbs
S 2
132
G R A M M A. R.
Grammar, are in common use : and it is clear that in employing
S-N-' any one of these words to express the relation in which
Less
obvious,
Athwart.
another substantive is placed, we give the name of
that relation, considered as a separate conception of the
mind; in other words we employ a noun in a secondary
use, as a preposition. -
It is unnecessary to explain the mouns round, cross,
stride, board, low, and side; only let it be observed that
these nouns become prepositions, not by the addition
of the prefix a or be (for that is merely an accident of
idiom, and applies equally to the same words when
used as adverbs) but their prepositional force depends
on their expressing a relation of some substantive, as
‘‘ around the tree,” “ across the street,” “ astride the
horse,” “aboard the ship,” “below the hill,” “beside
the church.” -
There is a little, and but a very little more difficulty
in such prepositions as athwart, against, among, about,
behind, between, betwixt, beyond, beneath. At first sight,
it is clear, that they are of the same family with the
preceding ; and a very short investigation suffices to
trace them to the nouns or verbs, of which they are
only a slightly varied form. It must be remembered
that throughout this part of our disquisitions, when we
speak of nouns and verbs, we only conform to the es-
tablished usage ; for, as we have already shown, the
essence of the verb being to assert, it is really the noun
involved in the verb, which furnishes the name of the
particular conception to be found in the adverb, pre-
position, or any other secondary part of speech.
Thus the preposition athwart is derived says Johnson
from a and thwart. - -
“Themistocles made Xerxes post out of Grecia, by giving out a
purpose to break his bridge athwart the Hellespont.”
BACON. Essays.
Here the bridge is not in fact asserted to have borne
the relation of thwartness to the Hellespont, or to have
been thwart with regard to the Hellespont, but the as-
Sertion is supposed and the name of the conception
only is expressed. However, it seems very immaterial,
whether we derive the preposition athwart from the
adjective thwart or from the verb thwart; both which
we happen to have in our language: -
Mov'd contrary with thwart obliquities. MILTON.
Swift as a shooting star -
In autumn thwarts the night IDEM.
Equally immaterial would it have been, whether
our preposition had happened to be written thwart or
athwart; for as we have frequently observed, it is not
the sound of the word, but its manner of signification,
which determines what part of speech it is to be deemed.
There is a conception of obliquity, and thence of harsh-
ness, perversity, &c. &c., which in the various northern
dialects is expressed by this word thwart and similar
articulations, as thwur, twer, dwar, zwar, swer, of which
it may be worth while to notice some instances:
1. Thwar, thwur, thwear, thver. -
Anglo-Saxon, thwur, oblique : thwur, thweor, thwyr,
thwurh, thweorlice, perverse; thweorscipe, thwyrmisse,
perverseness; thweorian, thwyrian, to thwart or op-
pose. w
Gothic, thwains, angry, thwarting.
Runic, thver, contrary, rebellious.
Islandic, thverskytningr, a contrary wind.
2. Twer, tver.
Islandic, tuer, transverse.
Swedish, twert.
Danish, tverer, tvert, tver.
Old German, twerch, the dwarfs supposed to be
a perverse race of beings. -
Armoric, gitwerch, the pigmies.
3. Dwar, dwer.
Armoric duerh, athwart, duerahen oblique
Swedish, dwerg, the dwarfs. -
Dutch, dwars, transverse, dwarsdryven, to thwart,
dwarsdryven, a cross-grained fellow, dwarslyk,
cross-wise, dwarsstraat, a cross street, dwerg,
a dwarf, &c. - - -
Islandic, dwergur, the dwarfs, dwergmal, the echo
or voice of the dwarfs.
Anglo-Saxon, dwerg, dweorh, the dwarfs.
4. Zwer, swer. • - -
German, zwerch, oblique. Zwerg, a dwarf.
Gothic, tuzweryan, to faulter. - º
Dutch, zwerven, to Swerve, zwerver, a wanderer.
English, swerve.
We may observe that the same analogy which
applies to the word thwart applies also to the word
across; for in English we use it adjectively to signify
perverse and peevish, and our old writers also employ
it as a preposition :
Betwixt the midst and these, the Gods assign'd
Two habitable seats to human kind:
And cross their limits cut a sloaping way.
- DRYDEN's Virgil.
Chap. F.
Against seems to be derived merely from the verb go. Against.
In the Anglo-Saxon it is ongeon and ongegen in the
German entgegen ; in the Dutch jegens and tegens.
TookE observes that instead of this preposition the
Danes use the analogous words mod and imod from
their verb moder to meet ; and the Swedes emot from
the verb mota of the same meaning : both which verbs
agree with the Gothic motyan, the Dutch moellen and
the English meet ; to which might be added our mote
in the old word folc-mote, the modern ward-mote, &c.
Among is accurately explained by MINSHEU; being Among,
from the word meng, the root of many terms in the
northern dialects signifying to mingle or mix.
Dutch, mengen, mengelen, to mix.
Anglo-Saxon, maengan, the same.
German, mengen, to mix, menge, a mixed quantity.
English, monger, as in Cheesemonger, Ironmonger, &c.
Mr. Tooke says among is always pronounced amung :
we do not happen to recollect any instance of this in
rhyme, which would be one mode of testing his accu-
racy of observation : and we apprehend that such a
pronunciation is by no means universal, nor even
COII] IIl Oil.
Amonges is used adjectivally by CHAUCER.
“Yf thou castest thy seedes in the feldes, thou shuldst haue in
mynde, that the yeres bene amonges, otherwhyle plentuous, and
otherwhyle bareyn.” - -
Gower uses amonge adverbially—
And tho she toke hir childe in honde
And yafe it souke; and euer amonge
She wepte— - -
He also uses amongest and emonge prepositionally—
I stonde as one amongest all
Which am oute of hir grace fall,
# # * * *
The Kyng with all his hole entent
Then at laste hem axeth this,
What kynge men tellen that he is
Emonge the folke—
G R A M M A R. 133
Grammar. .
\—y—’
About.
|Behind.
Between,
Betwixt,
In the ballad on the Battle of Bruges (A. D. 1301,)
we find both amonges and among used as prepositions.
The kyng of Fraunce made statuz newe, -
. In the lond of Flaundres among false ant trewe,
That the comun of Bruges ful sore con arewe,
Ant seiden amonges hem.
In the Seuyn Sages, it is written omang;-
Lene he was and also lang, -
And most gentil man than omang.
In the Scottish Acts of Parliament, we find amangis.-
That thai ressaue and admitt amangis thame Maister Williame
Lundy. Sc. Acts. A. D. 1567.
In an old English Letter of the year 1258 it is
amanges.
To halden amanges yew ine hord. 1 Faed. 378.
The word meynt appears identical with this prepo-
sition, being merely the participle of the same Anglo-
Saxon verb, mangan, to mingle.
Warme milke she put also therto
With honey meynt. Gower.
For euer of loue the sickenesse
Is meynt with swete & bitternesse. CHAUCER.
Tooke observes that the Danes use, instead of among,
the prepositions mellem and iblandt. Mellem is from
the Danish melerer, French méler, Italian mescolare,
from which source also came the old English ymell.
Herdest thou ever slike a song er now 2
Lo! what a complin is ymell hem alle. -
- - CHAUCER.
The Danish iblandt and Swedish ibland are from the
verbs iblander, and blanda, to blend.
This word is evidently of similar origin with the
French bout, the butt, limit, or end, of any thing;
which MENAGE supposes to be derived from an old
Celtic word bod; and which occurs again in the Ger-
man boden, and in the English bottom, bottomless, &c.
About, is directly from the Anglo-Saxon onboda,
onbuta; and it means on the extremities or limits of
any thing, round about it. -
On the hind-part. In the Gothic Gospels we read
gang hindar mik Satana, “ Get thee behind me, Satan "
Matth. c. viii. v. 33. In the Armoric, hinter is behind.
In the Anglo-Saxon, hindan is the same. In the
modern German, hinten and hinter are behind. In the
Gothic, hindar, that which is left behind. Hence also
the English, to hinder, hind-most, the hind-wheel, hind-
quarter, hinderling, &c.
By twain, by twice.
By twene the waiwe of wode and wroth,
In to his doughter chambre he goth. Gower.
Bytuene Mersh and Aueril, when spray beginneth to springe,
The lutel foul hath hyre wyl on hyre Iud to synge.
Harl. MSS. No. 2253. fol. 63.
This was the forward pleinly for t'endite,
Bitwiazen Theseus and lim Arcite. CHAUceR.
This latter word, it will be observed, very closely
resembles the German zwischen, between, from zwey,
two; as zwischen fünfund sechs, “between five and six.”
Euery man to other will seyne,
That bytwyv you is somme synne,
Romance of the Lyfe of Ipomydon.
Thy wife and thou mote hange fer atwynne,
For that bytwyt you shall be no synne. CHAUCER.
In the year 1420 we find it written betwyz and
betwene. ... (9 Rymer, 916.) -
Sir PHILIP SIDNEY uses betweene as an adjective :
His authoritie hauing bin abused by those great lords, who in . Chap. I.
those betweene times of raigning had brought in the worst kinde
of oligarchie. Arcadia.
In the old English we find from this same source
the adverbs a twayne and otuynne,
with his axe he smote it atwayne. -
- See WHART.on, v. i. p. 156.
He fondred the Sarazyns otuynne.
- ROBERT DE BRUNNE.
This word seems to be of the same origin as the Beyond.
preposition, against; being from the verb gan, gan-
gan, or gongan, to go. Hence says Mr. Tooke,
‘‘ beyond any place” means “ be passed that place,”
or “be that place passed.” It might perhaps be more
correctly explained, “ that place being passed ;” for
as we have before observed, the preposition does not
assert, which is the function of the verb ; but merely
names a conception, which is the function of a noun.
Beneath is by the mether, that is, lower part. In Beneath.
the old English it is written binethen.
Here kirtel , here pilche of ermine,
Here keuerchefs of silk, here smoko line,
Al togidere, with both fest,
Sche to rent binethen here brest.
Rom. of the Sewyn Sages.
Niden and nider, with their derivatives, are found in
many northern dialects, signifying that which is
below, or inferior.
German, mieder, below.
Swedish, nedre, neder.
Danish, ned.
Dutch, meder, down ; Nederland, the Low Coun-
tries, or Netherlands ; beneden, beneath; beneden-
waards, downwards, &c.
Anglo-Saxon, nither, below
Armoric, nithane, under.
Frankish, midana, beneath.
To this same source Mr. Tooke traces the preposi-
tion under, as being originally on neder.
Hitherto we have spoken of words used as prepo- Where ob-
sitions, and also as nouns or verbs in the same, or solete,
nearly the same signification; and in these we have
proceeded from the more to the less obvious. There
is no absolute line to be drawn in matters of this kind
between that which is discoverable at first sight, or
on a short reflection, and that which it requires some
study to make out; because the different capacities,
and the different experience, of different men, must
influence the degrees in this scale But we may pro-
ceed by almost imperceptible degrees from that which
almost all men think clear and self evident, to that
which almost all will admit to be involved in obscu-
rity, and yet the analogical principle, discreetly used,
will give us scarcely less confidence in the latter than
in the earlier stages of this progress.
Following this clue, we come to the preposition With.
with, which will probably be found rather more obscure
in its derivation than any of the words hitherto
examined. There are no less than three etymologies,
to which it has been thought necessary to resort, in
order to account for the different uses of this one
preposition :- - r -
1. The Gothic verb withan, to bind, or join toge-
ther.
2. The Gothic preposition withra, toward, or
against
134 G R A M M A. R.
Grammar. 3. The Anglo-Saxon verb wyrthan, (or rather the pistrini atque horrei,” says Junius, “ with exponitur Chap. L
S-N-7 Gothic wisan) to be. torta.” - -
. . We are inclined to regard the first and second of
these etymologies, though at first sight so widely
different in signification, as originally the same.
When any two visible objects are nearly connected, in
local situation, they must appear to be placed in
apposition to each other, if both be viewed from a
distant point ; but if one be viewed from the other, it
will appear to be placed in opposition. Now, the
preposition with, both in Anglo-Saxon and in English,
expresses these different relations of apposition and
opposition : it is therefore probable, that the original
radix of the word, (so far as these two significations
are concerned,) expressed the idea common to both,
namely, the idea of connection. To exemplify this
observation, let us suppose that John and Andrew
are seen at the distance of half a mile by Peter; they
appear to be close together, to be joined with, or bound
to each other ; but on approaching them he finds
that there is a considerable interval between them,
and the one either stands opposite to the other, or
comes toward him, or stands against him resisting, or
draws back from him. Now all these conceptions of
being joined with, standing opposite to, coming to-
ward, resisting, and drawing back from, with others
of a like kind, will be found to be expressed in different
Teutonic dialects by words obviously related to our
preposition with. This will appear more at large as
we separately examine the above stated etymologies.
1. The idea of connection, or joining together, was
expressed by the Maeso-Gothic verb, withan, of which
the past tense, gawath, occurs in the following passage
of the Codex Argenteus. Thata Goth gawath, Manna
ni skaidai, “What God hath joined together, let not
man put asunder.” (St. Mark, x. 9.) Hence, as a
particular kind of weed is called bindweed, because
it twists round and binds together other plants; so a
particular kind of tree (the willow) was called the
with-tree, or withy-tree, (in old German, weide-baum,
or wette-baum); because its tender twigs were used
to with, (that is, to bind together,) many objects in
rustic economy. The twigs so used for binding were
also called withs, or wythes : and a with or wythe was
a term given to any thing that bound either the body
or the mind.
MoRTIMER, in his Husbandry, speaks of the tree :—
Birch is of use for ox-youks, hoops, screws; wythes for faggots.
Lord BAcon uses this word to signify the twig :-
An Irish rebel put up a petition, that he might be hanged in a
with, and not a halter; because it had been so used with former
rebels.
The two words with and halter are simply binder and
holder; but use, it appears, had appropriated the
former, to a binder made of willow twigs; and the
latter, to a holder made of hemp.
King CHARLEs employs the same word metapho-
rically :—
These cords and wythes will hold men's consciences, when force
attends and twists them.
In different Anglo-Saxon glossaries, we find withig,
the willow; withthe, a hoop, or band ; cynewiththe,
the diadem, the king's band, or “golden round,” as
Shakspeare calls it.
In an Alamannic glossary, “ ubi recensentur res
“Danis quoque,” says the same author, “widde est
copula viminea ; potissimum tamen, ut videtur, copula
ex salignis viminibus contexta, contortave.”
In Dutch, the willow is called wiede, wide, weyde,
70226.
Our word willow itself originally conveyed the same
notion of binding; it being derived from the Anglo-
Saxon wilig, which came from the verb wilan, as withig
did from the verb withan ; and both withan and wilan
signified to bind.
It is little to be doubted, but that our verb wed, to
marry, is radically the same as with ; and means
simply, to join, or bind together.
Wed seems to have been opposed to shed ; the former
signifying to join together, the latter to separate.
Shed is still used in the Scottish dialect, in the parti-
cular sense of separating or dividing the hair on the
forehead ; as in the old ballad—
Janet hath kilted her green kirtle,
A wee abune her knee;
And she hath shed her yellow hair,
A wee abune her bree.
It is obviously derived from the Gothic skaidan, re-
ferred to in the above quotation from the Codex Argen-
teus; and skaidan is identical with the Anglo-Saxon
sceadan, the German scheiden, and Dutch scheyden, to
separate, or put asunder.
Moreover, as there were two verbs, withan and
wilan, signifying to join, so there were two analogous
verbs, skaidan and schalen, signifying to separate. Of
skaidan we have already spoken : schalen is still extant
in German, in the sense of separating the outer coat,
rind, peel, or shell, of a thing from that which is
within : and the substantive schale in that language
is the shell of an egg or nut ; it also signifies a small
cup or saucer for drinking out of, probably because
shells were originally used for that purpose. Con-
nected with this word schale is our substantive shell,
which was written in old English shale, and is the same
word with the scales of a fish, meaning that which is
scaled off, or separated from the main body. Hence,
in Scotland, the kirk-scaling is the departure of the
congregation from church, when they separate in all
directions. Our word shell is also the Danish,
Swedish, and Icelandic, skal or skel, the Anglo-Saxon
scyll, the Dutch schell, schille, the Italian scaglia, and
the French escaille, or écaille. In Dutch also, the
tiles of houses are called schalien, and in Gothic
skalyos. -
To return to the preposition with. WACHTER derives
the German weide, and Frankish wida, a willow, from
the old verb wetten, to bind ; “ab usu, quem arbor
officiosa praebet colonis et hortulanis in jungendis et
alligandis rebus;” and he suggests, that the Latin
vitis, a vine, is so named from its binding round other
trees. Weiden also he explains, to bind, and weid,
wied, wette, a bond. The Frankish languuid, is a
waggon-rope. Wette also signifies, metaphorically,
the law, which binds; and this in butch is wet,
whence wet-boek is a law-book ; wetsteller, wetmaaker,
wet-geever, a legislator; wethouders, magistrates; wet-
geleerde, a lawyer; wetbreker, a lawbreaker; wetloos,
an outlaw ; wettig, legitimate, &c. The verb wetten
is not only to bind, but to bind in wedlock. “Oritur,”
says WACHTER, “ a wette, vineulum, copula, ligamen,
G R A M M A. R.
135
Grammar, unde reliqua, tam verba, quam substantiva, tanquam
\-N-me' ex matrice prodierunt.”
custom for scholars to offer public challenges for disº Chap. I.
putation on any given subject, As the party who \-N-
From all these authorities we may safely conclude,
that we have ascertained the proper origin of our
common preposition with, in the sense of association,
e. gr.
In all thy humours, whether grave or mellow,
Thou'rt such a touchy, testy, pleasing fellow;
Hast so much wit and mirth, and spleen about thee,
There is no living with thee, nor without thee.
Tatler.
2. It is obvious, that in several of the other uses of
this preposition, which Dr. Johnson, points out, it
really expresses no more than the same conception of
joining or binding together, modified by the nature of
the objects spoken of. Such are the following:—
“ in company,”—“in partnership,”—“in appendage,”
—“ in mutual dealing,"—for I am joined with those
with whom I am in company;-I am bound to one
with whom I am in partnership ;-a thing is joined to
that of which it is an appendage;—two persons, who
mutually deal together, are bound by the laws of
honesty to each other;-and so of similar cases. It
is remarkable that. Johnson himself gives the two
following senses of this preposition, in immediate
SUICCéSSIOIl.
4. On the side of ; for—
— O madness of discourse :
That cause sets up with, and against thyself.
- SHAKSPEARE.
5. In opposition to ; in competition, or contest—
— I do contest
As hotly and as nobly with thy love,
As ever against thy valour.
SHAKSPEARE,
This illustrates the transition before mentioned, from
apposition to opposition ; and hence Johnson says,
“With, in composition, signifies opposition or priva-
tion.” Instances of this use of the word in modern
English are, withdraw, withhold, and withstand.
BARB.our uses withsay— -
With right or wrang it have would thay;
And if anie would thane withsay,
Thay would sa do, that thay suld tine,
Eyther land or lyfe, or live in pine.
This is in German widersagen : as in the old bap-
tismal formula “widersagestu dem Teufel und allen seinen
werken 2" “ Dost thou renounce the devil and all his
works 2" And in this sense absagen is also used.
It is observable, that the modern German, which
does not use with, or any similar preposition in the
sense of association, has wider to signify opposition,
both in the simple form, and in a great number of
compounds,--as wiederhalten, to resist ; widerlegen, to
refute ; wider-reden, to reply; wider-sprechen, to con-
tradict, &c.; and so widerschein, a reflected light;
widersinn, an absurdity; widerschall, an echo, &c.
In the Anglo-Saxon, with and wither, are both used
in the sense of opposition, or reflecting back from ; as
withstandian, to resist; with-coren, wither-coren, cursed,
with-secyian, withersecyian, to contradict; with-ladan,
to lead back; with-scuftan, to repel. In the laws of
Canute we find witersacan, apostates. In the old
English laws we have withernam, in the barbarous
Latin of that day, withernamium, a reseizure. This
last word is said to have given an easy victory to an
English lawyer in Italy, at an epoch when it was the
accepted the challenge had the choice of a subject,
our lawyer proposed, as his question, An averia capta
in withermamio replegiari possint ; to which his anta-
gonist, as he did not understand what withernamium
meant, was unable to give any reply.
In the Islandic, we find both vid and vidur signify-
ing against. In the Frankish, wid and with are
“ against,” as with thenne Divvel, “ against the
Devil.” But in most of the other Teutonic dialects,
when the sense is contra, or retro, the letter r is found
in the word. In the Gothic of Ulfila withra signifies
both toward and against, as alla so baurgs usiddya withra
Jaisw, “all the city went out toward Jesus.” Saei nist
withra izwis, faurizwis ist; “He that is not against us,
is for us :” and so in the compounds withrawairthan,
“ opposite,” “over against ;" with raidya, “ he met;"
withragamotyan, “to meet.” In the Frankish, uwidrun-
piotun, is to write in reply. In the Alamannic,
uuidartragan, is to carry back. In the old Salic laws,
widredo is a repeater of his oath, from eid, an oath.
In the Lombard laws, widerboran is a manumitted
slave. This last word is also written guiderbora, as in
the laws of Luitprand, (circ. A. D. 720.) “ Si quis
aldiam alienam aut suam ad uxorem tollere voluerit,
faciat eam guiderboram.” Another remarkable instance
of the use of wider in composition, is in widrigildum,
which some writers confound with wergeldum; but
EccARDUs accurately distinguishes these words,
observing that the latter properly signifies the price,
ransom, or value of a man ; the former, any composi-
tion by which a loss is paid back, or compensated.
Weregelt is well known to the old English and Scottish
law ; (see Fleta, and the Regiam Majestatem.) Were-
geltthef aceording to Fleta is “Latro, qui redimi
potest.” Hence SoMNER derives wer-geld from wer, a
man, and gelt, price. On the other hand, GRotiUs,
(in the preface to the Gothic writers,) defines wedri-
geldium “ quod pro talione datur ;” and this word is
properly derived by WENDELINUs from the Teutonic
weder contra, vicissim, and gelt, aestimatio. It is
differently written, widrigilth, widrigildum, guidrigild,
wedrigildum, wedrigeldum, widrigild.
Widrigilth secundum quod appretiatus fuerit.
Decr. CHILDEB. II. (A.D. 711.)
Suum widrigildun omnino componat.
Decr. LUDov. II. (A. D. 879.)
Si stupri crimine detectae fuerit componat guidrigild suum.
Capitul. ARECH. PRINC. BENEVENT.
Juxta quod widrigild illius est.
- Capit. Lothariſ, IMP. (A. D. 824.)
Perhaps the most remarkable derivation from the
word wither, or wider, now remaining in our language,
is guerdon; and the more so, as the English etymo-
logists in general have entirely mistaken its origin.
The English word guerdon is a mere adoption of
the French guerdon, of which MENAGE thus speaks :-
“Je croy qu'il vient de werdung qui signifie pretii aesti-
matio, et dont les escrivains de la basse Latinité ont
fait aussi werdunia pour dire la mesme chose. De
guerdon les Espagnols ont fait galardon, et les Italiens
guiderdone." SKINNER cites this ; but prefers the
derivation of guerdon by My LIUs from the Dutch weer-
deren, waerderen, aestimare, censere; and this from
weerd, waerd dignus, et weerde, valor, pretium.
136
G R A M M A. R.
*
Grammar. Junius cites the French guerdon, Italian guiderdone,
\-y-' Spanish galardon, and Welsh wherth; “ quae omnia,”
*
widergild and widerbora.
piet.
says he, “valde affinia sunt Teutonico weerde, weerdiie.”
What is meant by galardon being “ valde affine,” to
weerde, we cannot guess; any more than we can tell
how the Italians formed guiderdone out of guerdon :
and as to the base Latin werdunia, we never hap-
pened to meet with that word. The real history of
the word guerdon, however, may, we apprehend, be
very satisfactorily traced, as follows:– -
1. Widerdonum. This word is correctly explained
by DU CANGE, “Voxibrida, a widar Teutonico, contra
sº 5 y 2 J
g y
et donum Latino, munus.
particles with Latin substantives or verbs, is a fact,
which, properly considered, may cast some light on
the true principles of etymology; for those principles
have been hitherto but little studied. Thus we find
our word miscreant to be compounded of the Teutonic
mis (our verb, to miss,) and the Latin credere : and the
French have many such compounds, e. gr. mécompte,
méconnoitre, mécontent, mésaventure, mesoffrir, mésestimer,
médire, méfaire, &c. Widerdonum occurs in the Tabula-
rium Casauriense, (A. p. 876.)
Quia tu dedisti mihi, pro memorată convenientiá, widerdonum,
caballum unum, et argentum solidos centum.
2. Guiderdone, or guidardone. This is merely the
word widerdonum with the Italian articulation gui for
wi, as in guidrigild and guiderbora above noticed, for
The Latin termination um
was universally changed into o by the elder Italians;
and in modern writing it is softened still more into e ;
whence we find in the Pocabolario della Crusca, guider-
done and guidardone, with their derivatives, guiderdo-
mare, guidardonare, guiderdonato, guiderdonatrice, gui-
dardonatrice, guiderdonamento, guidardonamento, &c.
E come i falli meritan punizione, così i beneficii meritan
guiderdone. BoccAcio, (circ. A. D. 1350.) .
E per guidardone del vincitore apparecchio Ghirlande.
- Idem.
3. Guizardonum. This is evidently a mere pro-
vincial corruption of pronunciation from guidardonum.
Item quod nullum mumus, guizardomum, vel xenia aliqua reci-
Statut. MASSIL. (circ. A. D. 1220.)
4. Guiardomum.
Non currant aliquae pacnae, usurae, guiardona, vel expensae.
- * . Statut. VERCEL.
This word is thus explained in an old glossary :-
“ Guiardonum, remuneratio; Ital. guidardone, nostris
guerredon.”
Paris, No. 7657.) -
5. Guiardon, in the Provençal dialect, praemium.
6. Guerredon. The old French word above alluded
to, which is also found in the verbal form, guerre-
donner. - -
Se Dieu sauve le baron,
Ils en auront bon guerredom.
- Roman D'Athis.
Voulons, pour ce, yeeulx guerredomner, et poursuir de faveur
especial. Chart. PHIL. VI. (A. D. 1330.)
7. Guerdon. In the old French translation of the
passage above cited from the Statutes of Marseilles,
the words “ Guizardonum vel xenia,” are rendered
“guerdon, ou estrenne.” .
In English, guerdon is used to signify a just recom-
pense either for good or bad deeds. -
This mixture of Teutonic
(Wide Glossar, Provinc. Lat ex cod. Reg.
He shall by thy revenging hand at once receive the just guerdon
of all his former villainies, KNoLLES. Hist, Turk.
Fame is the spur that the clear spirit doth raise,
(That last infirmity of noble mind)
To scorn delights, and live laborious days;
But the fair guerdon when we hope to find,
Comes the blind fury, with th' abhorred sheers,
And slits the thin spun life. MILTON.
3. Having examined two derivations of our modern
preposition with, we come to the third, which is thus
stated by Mr. HoRNE TookE. - -
“WITH is also sometimes the imperative of wyrthan,
to be. Mr. Tyrwhit, in his glossary, (art. BUT,) has
observed truly, that ‘By and with are often synoni-
mous.' They are always so, when witH is the impe-
Chap. I.
S-N-7
rative of wyrthan : for BY is the imperative of beon, to
be. He has also in his glossary, (art. WITH,) said
truly, that “witH meschance, witH misadventure, witH
sorwe ; 5316, 7797, 69.16, 4410, 5890, 5922, are to
be considered as parenthetical curses.' For the literal
meaning of those phrases is (not God yeve, but) BE
inischance, BE misadventure, BE sorrow, to him or them,
concerning whom these words are spoken. But Mr.
Tyrwhit is mistaken when he supposes— witH evil
prefe, 5829; witH harde grace, 7810; witH sory grace,
12810;’ to have the same meaning; for in those three
instances, witH is the imperative of withan ; nor is
any parenthetical curse or wish contained in either of
those instances.”
There is something ingenious in connecting with
and wyrthan ; and it was probably suggested to Mr.
Tooke by the analogy which with-out and with-in bear
to the Scottish “ but and ben ;" i.e. be-out and be-in.
The Anglo-Saxons also used with-yeondan, for be-
yond; and indeed they employed the separate prepo-
sition with so loosely, as to afford room for supposing
that it was only equivalent to the general expression
of existence, be: for HICKEs, in his Grammatica Anglo-, .
Savonica, explains with by the Latin words juxta, cum,
contra, adversus, pro, circa, circiter, erga, a, ab; and
one of his examples is remarkable, as using with for
by in the sense of near to. “With tha sa, Adriaticum,
juata mare Adriaticum.” .
Still we are in some doubt whether the Anglo-Saxon
wyrthan affords the proper solution of this question;
since we do not find the rever introduced before the th
into either the Anglo-Saxon or English preposition :
in other words, we do not find wyrth used as a prepo-
sition in Saxon, or worth in English : and though
worth is certainly used for be in the parenthetical curse
wo worth ! and in the parenthetical blessing well worth !
it is not quite so clear that with is thus used in the
expressions “ with meschance, with misaventure, or
with sorwe.”
In the vision of Piers Plowman we have the verb
worth, to be. In CIIAUCER we have wo worth, and in
Piers Plowman, well worth, and much wo worth.
And said, mercy madam, your man shall I worth.
Piers Plowman.
Wo worth the faire gemme vertulesse !
Wo worth that hearbe also that doth no bote!
Wo worth the beauty that is routhlesse ! .
Wo worth that might that trede ech under fote!
- - CHAUCER.
Much wo worth the man that misruleth his inwitte ;
And well worth Piers Plowman that pursueth God in his going!
- - Piers Plowman.
The Anglo-Saxon wyrthan, wurthan, or weorthan,
G R A M M A R.
137
Grammar, and the English worth, are from the Gothic wairthan ;
\-N- but perhaps the Anglo-Saxon and English with, used
By.
synonymously with be, are rather from the other
Gothic verb Substantive, wisan : for the different Teu-
tonic tribes used three verbs substantive, (as they are
called,) viz. bean, wisan, and wurthan ; of which we
retain traces in the different tenses of our verb, to be ;
namely, be, was, and were.
From the last-mentioned signification ºf the pre-
position with, the transition is easy to the perposition
by, which in many of its uses is manifestly nothing
more than the imperative be. Dr. Johnson, in his
usual manner, gives no less than twenty-five senses
of this preposition, as denoting the agent, the instru-
ment, the cause, &c., all which is very proper in
lexicography, but will assist us little in grammar,
without some further analysis. The dictionary maker,
moreover, has in general little or nothing to do with
those uses of words which have become entirely
obsolete ; but these may often assist the researches of
the grammarian. Perhaps we may trace all the uses
of this preposition, and of the analogous words in
other Teutonic dialects, to two different origins,
namely, the verbs to be, and to big. When derived
from the former, it is a sort of elliptical expression,
the word agent, instrument, cause, &c. being under-
stood from the context ; when derived from the latter,
it signifies proximity. Thus, in the following examples,
(quoted by Johnson)—“ The grammar of a language
is sometimes to be carefully studied by a grown man;”
—“When Hector fell by Pelides' arms ;”—“If we
give you any thing, we hope to gain by you :"—The
meaning is, “ the grammar is to be studied, there
being a grown man as the student;”—“Hector fell,
there being the arms of Pelides, which caused his
fall;”—“ We hope to gain, there being you, to pro-
amote our gains.” But in the following examples, by
signifies proximity, either stationary or in passage.
A spacious plain, whereon
Were tents of various hue ; by some, were herds
Of cattle grazing. MILTON,
Many beautiful places, standing along the sea-shore, make the
town appear much longer than it is, to those that sail by it.
ADDISON.
SHAKSPEARE puns on the two different uses of the
word by in the following passage:–
So thou may'st say the church stands by thy tabour, if thy tabour
stands by the church.
That is, “if you use the word by improperly, you
may be understood to mean that the church is sup-
ported by means of your tabour; whereas, the fact
merely is, that your tabour happens to be placed near
the church.”
It is in this latter sense of proximity, that we find
the word by used adverbially and as a substantive, and
also (when in composition) adjectivally.
1. Adverbially—
And in it lies the God of Sleep,
. And snorting by,
We may descry
The monsters of the deep. DRYDEN.
- - I did hear -
The galloping of horse. Who wast’t came by ?
- SHAKSPEARE.
2. As a substantive, in the phrase “by the by ;”
anciently written, “upon the by.” - -
YOL. I
This wolf was forced to make bold, ever and anon, with a
sheep, in private, by the by. L'ESTRANGE.
In this instance there is, upon the by, to be noted, the perco-
lation of the verjuice through the wood. BA.cox.
3. “ By, in composition,” says Johnson, “implies
something out of the direct way, and consequently
some obscurity, something irregular, something col-
lateral, or private.” These are instances of the
natural transitions of the mind in the use of words,
and the enumeration is only defective in not specifying
the first link of the chain; thus, a by-slander, one
who stands near. A by-road, a road, which, branch-
ing off from the main road, is of course less frequented
and comparatively obscure ; a by-end, an object
obscurely connected with the known and ostensible
end in view : the by-play of an actor, those actions
and gestures which are carried on apart from the main
business of the scene: a by-law, a law apart from the
public laws of the state, and affecting only a private
body of men : a by-word, a word of reproach, used
aside as it were, and separately from honest and
honourable conversation : a by-name, a surname, or
nickname, added to or substituted for the original and
proper name of the individual : by past times, are those
times which once were passing by us, (as the mari-
ners sailed by the town above spoken of by Addison,)
but which have now passed by, and are gone. Dr.
Johnson says that “this” composition is used at plea-
sure ; but in fact it is very much regulated by cus-
tom ; for several even of the instances which he quotes
would now be considered as antiquated expressions;
such are a by-concernment, by-interest, a by-name, by-
respects, by-views, &c.
To this we may add the use of the word by in the
phrase “ by and by :” and perhaps in the phrase
“ Good by I" *-
By and by seems to signify a time near to the pre-
sent, but not immediately following it, and usually
refers to some action out of the course of that on
which the individual is at the present moment em-
ployed. Thus, when the friar wishes to conceal
Romeo before he opens the door to the nurse, who is
knocking, he says,
Stand up.
Run to my study—(by and by)—God's will.
What wilfulness is this 1 (I come, I come.)
where the passages in parenthesis are addressed to
the nurse. -
So, Othello, speaking alternately to himself and to
Emilia, who is calling for admittance, says,
(Yes!)—'tis Emilia—(by and by) She's dead.
SPENSER uses this expression narratively, to signify
proximity of time.
The noble knight alighted by and by
From lofty steed.
CHAUCER uses it to signify proximity of place.
And so befel that in the tas they found
Two yonge knightes ligging by and by.
The phrase Good by 1 is commonly supposed to be
a mere contraction of the words “Good be with you ;”
but it seems as probably to have been an elliptical
phrase for “ may good be by :” that is, “ may good
be near you, wherever you are ſ”
There is a singular difference in the use of the
T
Chap. I.
S-N-'
138
G R A M M A R.
Grammar preposition by in the sense of proximity, between the
\—V-' English and Scottish dialects.
In the former, by him-
self means “ alone,” “no one else being by:” in the
latter it signifies “ insane,” “beside himself.”
Sitting in some place by himself, let him translate into English
Iris former lesson. ASCHAM.
g And monie a day was by himsel",
He was sae sairly frighted. |BURNS.
In old English be and by are often used indifferently :
e. gr. “Damville be right ought to have the leading
of the army ; but bycause thei be cosen germans to
the admirall, thei be mistrusted.” (See Lodge's
Illustrations, vol. ii. p. 9.) So in the ballad—“ How a
Merchande,” &c.
Bothe be daye and be nyght
So in Montgomir RIE’s “ Cherrie and the Slae.”
I saw nae way qullairby to cum,
Be ony craft to get it clum.
In the description of Cokaigne—
Ther beth briddes mani and fail,
That stinteth neuer bi har migt,
Miri to sing dai and nigt.
In like manner we find byfore and before, bylove and
belove, bycause and because, &c.
By any other cause or matter hadde or made by fore the seid
fyne levied. Stat. 1. RIC. III. c. 7. MS.
He was newly fallen to his fader's herytage, who was so well
by loved in his royalme. BERNERS’s Froissart, fol. lxxxi.
His men murmured and spake of hym otherwyse then they
sholde do bycause of them of the garyson of Dulcen.
- Ibid.
Hn the letter of HENRY III. (A. D. 1958.) which is
one of the most ancient specimens of English extant,
we find biforen— -
Alswo alse hit is bifuren iseid.
- Faedera, vol. i. part i. p. 378.
There are several uses of the preposition by in old
English, which have now become obsolete; as ‘‘ by
daies olde,” which Gower uses for the modern phrase
‘‘ in old times.” . . .
In the romance of Sir Guy we find “ by twenty mile
round about,” instead of “for twenty mile round
about.” King JAMEs I. of Scotland uses by for of
As Tantalus I travail, ay butcles, -
That ever ylike hailith at the well,
Water to draw with buket bottemles—
So by myself this tale I may well telle.
- King's Quair, canto ii. St. 51.
These and many other uses of the analogous prepo-
sitions occur in the Maeso-Gothic, Anglo-Saxon, &c.
In Maeso-Gothic the verb bean or bion is not found,
but the preposition bi exists both separately and in
composition. In its separate use it answers to the
Latin in, pro, cum, contra, Secundum, post, de, and circa.
As a component particle, it appears in bigitan, to find,
(or, as we say, to get at ;) bifotuns, fetters; bihlahyan,
to laugh at ; bimaitan, to cut around; bigaurdans,
girded round about, &c. In Anglo-Saxon, “ be
Petres maessan,” is “ at Peter-mass ;” i. e. “ on the
festival of St. Peter.” “Tha he gehyrde be tham
Haelende,”—“When he heard of the Healer,” i.e. of
Jesus. “Be Wihtgares daege,”—“ In the days of
Whitgar.” “Be hyra wartmum ye hig onenawath,”
—“ By their fruits shall ye know them.” Be also
enters into the composition of several Saxon preposi-
tions, as beforan, before ; betwux, betwix; beheonan,
on this side ; beaftan, or baftan, after; binnan, within;
butan, without ; bufan, upon. In these and in many
other compound words, be is evidently the mere root.
of the verb bean, to be. It is, however, sometimes
written bi, or big, as “se bisceop the him big saet,”—
“The bishop who sate by him," i. e. near him : and
in this sense it may be reasonably supposed to have
some affinity to the verb byan, to inhabit; or bicyian,
to build ; which latter is still retained in the Scottish
verb, to big 5 as in the song of Bessy Bell and Mary
Gray.— * .
They bigg’d a bour on yon burn brae.
From the verb bicyian or by an comes our local ter-
mination in by, so frequent in Yorkshire and Lincoln-
shire, as in Danby, Manby, Ranby, Belby, Kelby,
Welby, Holtby, Boltby, Kirkby, Birkby, Harmby, Barmby,
Hazby, Sarby, Romanby, Normanby, Salmonby, Hareby,
&c. The German preposition bey, which is rendered
by the French chez, and the Latin apud, may perhaps
be in like manner derived, as ADELUNG suggests, from
the old verb bio, bo, bawen, in the sense of dwelling at,
or occupying a certain spot. In the old Prussian lan-
guage bo or po was used prepositionally in this sense;
and hence the Borussi or Porussi, the ancient name of
the Prussians signified those who dwelt near the
Russians, as Pomeranii, the Pomeranians, signified
those who lived (Po-Meere) near the sea. In the
Frankish we find pi, as pi hantun, at hand. In the
Alamannic it is written pii, as pii inkange, near the
entraln Ce. - -
Chap. I.
Among the prepositions compounded with be, or Before,
Ol'.
by, we have already noticed behind, before, between,
beyond, beneath, &c. and we have shown that in the
compound word behind, the simple word hind is a
moun, that is to say, the name of a conception formed
by the mind. There can be little doubt, we appre-
hend, but that before is a preposition of the same
nature as behind; that is to say, that the words hind
and fore were equally in their origin, nouns. We
still use them both adjectivally, even in their separate
State. -
Resistance in fluids arises from their greater pressing on the
fore than hind part of the bodies moving in them.
CHEYNE.
And so they occur in various compound words, as
forewheel, hindwheel, foreman, hinderling, &c.
As we have said that the preposition athwart might
have been thwart, that the preposition across has been
actually written cross, and that the Germans indif-
ferently use anstatt and statt, so it is obvious that the
preposition before would be equally intelligible, and
would convey exactly the same meaning if it were
written fore ; for the prefixes a and be are mere mat-
ter of idiom, and do not alter the meaning of the
words thwart, cross, fore, &c. with which they are
united in common use. Accordingly afore and tofore
were formerly used for before.
Whosoever should make light of any thing afore spoken or
written, out of his own house a tree should be taken, and he
thereon be hanged. ESDRAs, ch. vi. v. 22.
Darie in a vergerys, •
Tofore him mony knyghtis ywis.
Ayng Alisaunder.
And so we still use these expressions in the compound
words aforesaid, aforementioned, heretofore, &c.
A G R A M M A R.
139
\-v- the name of a conception.
truth “a pure idea of intellect,” which sense alone Chap. I.
Fore, therefore, must be considered as a noun, or
Now of what conception
is it the name 2 This question will be best answered
by comparing together several instances of its use.
We have, in English, the words forecastle, foredeck,
fore-end, forefinger, forefoot, forehead, foreland, forelock,
foremast, &c. relating to place; and the words fore-
advise, forebode, forecast, forefather, forenoon, &c. relat-
ing to time. It is plain that there is an analogy
between these two classes of words; for they both
agree in expressing, by the particle fore, one common
conception, namely, that the thing spoken of is before
some other thing, either in place, or in time. A fore-
castle, for instance, is the elevated part of a ship ;
which, as she moves through the water, goes before
the main body of the vessel. A foreland is a part of
the land which projects before the rest into the sea.
To foreadvise is to give advice before the emergency
to which it is applicable, actually occurs. The fore-
noon is that part of the day, which elapses before the
sun reaches the meridian. -
Now, this conception, so expressed by the particle
fore, is not the conception of a real object, but it is the
conception of a relation existing between two objects.
We may give it the name of foreness or of precession,
or any other name that may be thought more suitable;
but the conception itself must unavoidably be formed
by all men. A savage, when in presence of his enemy,
not only apprehends that such enemy exists, but that
he is before him. The same savage, when he perceives
the sun rising, not only knows that a certain portion
of the day is elapsing, but that such portion is before
the noon. In order to know these two facts, however,
he must necessarily be able to form a conception, in
the one case, of a relation of place, and in the other
case, of a relation of time. But the relations of place
and those of time, are, in many instances, if not iden-
tical, at least so closely analogous, as to be expressed
in most languages by the same term ; and thus, in
most languages, we find that the word, which implies
priority of time, expresses also precession in space;
which is the case with the word in question, fore.
Other analogies again coincide with these. The
person that is chief in dignity, rank, or order, is usally
said to precede or go before his inferiors; and the
final cause, motive, or end is placed before the mind
when deliberating on an act to be done.
Lord MonBooDo justly says, “every kind of relation
is a pure idea of intellect, which can never be appre-
hended by sense :” and when Mr. HoRNE Tooke denies
this proposition, he shows strange ignorance of the
human mind. Sense, taking that term in its widest
acceptation, can only apprehend an external object, it
can apprehend the thing, which is before another, or
the thing, before which another is ; but the relation of
place, time, order, causation, or the like, which we
express by the word before, is discerned not by a
simple operation of sense, but by means of an exercise
of our comparing and judging faculties. It is most
extraordinary that Tooke, who asserts universally
that “ prepositions are the names of real objects,”
should say of the preposition, for, “I believe it to be
no other than the Gothic substantive fairina, CAUSE.”
What real object is CAUSE 2 How is causation to be
apprehended by sense That we have a conception
of cause is certain ; but it is equally certain that we
come at it by means of our intellect ; and that it is in
never did nor ever can give.
That the Gothic substantive fairina may have some
etymological affinity to our preposition and conjunc-
tion, for, we do not mean to dispute; nor do we deny
that for often expresses the relation of a final cause
to its effect ; but the reason of this is, that the words
For and Fon E are the same. - -
The identity of these words, both in their simple
and compound states, may be shown in a variety of
instances. ... - -
In “ Christ's descent into hell,”
we have fore used as
we now use for.— -
Fore Adame's sunne fol y wis,
Ich haue tholed al this.
Our common words wherefore and therefore are “for
which,” and “for this;” and the latter is often writ-
ten forthi or forthy, in ancient authors; as the former
is written for why by some of more modern date.
- Forthi myn wonges waxeth won.
f MSS. Harl. No. 2253, fol. 63.
And forthy if it happe in any wise,
That here be any louer in this place.
- CHAUCER's Troilus.
Solyman had three hundred field pieces, that a camel might
well carry one of them, being taken from the carriage; for why,
Solyman purposing to draw the emperor unto battle, had brought
no greater pieces of battery with him.
RNOLLEs’ Hist. Turk.
Forsaid is used as foresaid, or aforesaid, in a docu-
ment of the year 1420. (Rymer, v. ix. p. 916.)
Forlok, for forelook, i. e. foresight, occurs in the
romance of Sir Amadas.
Ther Y had an hondorthe marke of rent,
Y spentte hit all in lyghtte atent,
Of suche forlok was Y.
In the same romance we have fordryvön for foredriven.
Folke fordryvon in the schores
Wrekkyd with the water lay.
So, forward for foreword; i.e. promise made before-
hand.
• Thenke what forward that thou made,
When thou full greyt myster hade.
In the romance of Sir Tristem, edited by Sir WALTER
Scott, the preposition before is written bi for.
The folk stode unfain
Bi for that leuedi fre:
Rouland mi lord is slain,
He speketh no more with me.
Mr. TookE has, with great parade of comment, in
above twenty quarto pages, reviewed the seventeen
significations of the word for, which are given by
GREENwood, and the forty-six by Johnson; besides
reprehending LowTH and TYRwhit, for their remarks
on the same word. The result is, that in Mr. Tooke's
opinion, for always signifies cause. Now, this is an
error. For signifies merely before; and as the final cause
is before the mind of the agent, for may, in some in-
stances, be rendered cause ; but there are other cases
in which the notion of a final cause does not seem to
be involved in the signification of the word for. When
we say “ Christ died for us,” we mean that our salva-
tion was before the contemplation of Christ as the
final cause of his death. When we speak of “ fighting
for the public good,” we mean that the public good
is before the mind of the combatant, as the final cause
T 2
140 G R A M M A R.
a spectator; for this arises from the prevalent dispo- Chap. I.
Grammar. of his fighting. But in the following instances, the
sition to explain intellectual phenomena by material \-y-'
S—y—’ notion of cause is very indistinctly, if at all alluded
to. “ He is big enough for his age;” i.e. having
before your mind his age, considering his age, he is
big enough. “He speaks one word for another,”
i. e. he speaks one word before, or in the place of ano-
ther, he takes the wrong word first. “We sailed
directly for Genoa;” i. e. Genoa was before us as the
object toward which we sailed. In like manner, when
for is used in the sense of equivalence, it is to be ex-
plained by the same word, before. “An eye for an
eye;" i.e. having before you the consideration that an
eye has been destroyed, another eye must be put out.
“To translate line for line;” i.e. laying before you
one line of the original, one line of the translation
must follow. And this notion of equivalence is taken
negatively in the common phrase,_For all. Thus,
“ for all his exact plot, down was he cast from his
greatness;" i. e. the casting him down was effected
before, in presence of, or, as we say, in the teeth of,
all his plot. -
For, when used as a conjunction, has manifestly the
same force.
Heav'n doth with us, as we with torches deal,
Not light them for themselves; for if our virtues
Did not go forth of us, 'twere all alike,
As if we had them not. SHAKSPEARE.
Izet us paraphrase this sentence :-" If our virtues
did not proceed from us to others, we might as well
not have them : and therefore I say, that Heaven uses
us as we do our torches.” The words for and therefore
correspond ; and the word “ therefore,” we know, is
the same as “ that fore,” or “ that being before me.”
Now that which is before the mind of the speaker, and
which leads him to the conclusion that Heaven deals
with us as we with torches, is the reflection—that if
our virtues did not go forth of us, they would be use-
less; and this reflection is marked as being before the
mind of the speaker, by the word for in the original
sentence, as it is, in the paraphrase, by therefore,
Again, in the 9th chapter of St. Mark’s Gospel, verses
5 and 6:— • *
Peter answered and said to Jesus, Master, it is good for us to
be here ; and let us make three tabernacles; one for thee, and
one for Moses, and one for Elias. For he wist not what to say;
jor they were sore afraid. -
Here we see, the construction is, “ They were sore
afraid; and therefore he wist not what to say; and
therefore he talked (as Bishop TAYLoR says,) he knew
not what, but nothing amiss, something prophetical.”
The fear and wonder of the apostles was before the
ecstasy which deprived them of the power of connected
reasoning ; and the ecstasy was before the words which
St. Peter uttered. - -
This double use of the conjunction, for, serves to
explain the double use of the preposition for noticed
by Tooke. “A writ was moved for, for Old Sarum.”
The representation of Old Sarum was before any writ,
as an object in the mind of those who first devised
such an instrument; and the desire of obtaining this
particular writ was before the motion, as an object in
the mind of the mover. -
Nor let it be thought strange, that our wishes and
intentions should be expressed in language which
seems to indicate that they stand before the mind
Tocally, as a tree or a house stands before the eye of
analogies, just as we call a certain faculty of the mind
imagination, which word properly signifies the power
of making visible or tangible representations of sen-
sible objects. So we call another faculty acuteness,
adopting our metaphor from the sharpness of a sword
or knife. So we speak of impressions on the memory
—of reflections on our acquired knowledge, &c. This
disposition to speak of the faculties and operations of
the mind in language originally applied to the powers
and exercises of the body, may be said to have been at
first and in the early stages of human society, a neces-
sity; and if we confined ourselves to the etymological
signification of words, it would be so still. In the
rude ages of Gothic and Grecian barbarism, the action
of taking was expressed by the radical tak or dek,
whence our present verb, to take, and the Ionic ēekw.
From its superior use in taking, the right hand was
called dextra ; he who was expert at any manual em-
ployment, was said by analogy to be dextrous; and by
a further analogy, a superior readiness of contrivance,
or quickness of expedient, (though a mental faculty,)
was called deaterity. This sort of analogy must be
resorted to, not merely by the untutored savage, in
explaining the acts of his mind, but even by the most
profound philosophers in investigating the same
abstruse subject. Even the sublime speculations of
PLATO are necessarily conveyed in metaphors derived
from external and material nature; a circumstance
which has occasioned some modern writers of emi-
nence to form very erroneous notions of the doctrines
of the Grecian sage. Plato never dreamt of “intelli-
gible species,” as actually distinct from the intellect
itself, but merely as distinguishable for purposes of
reasoning. He never thought that the voeſteva were
separate from the ves, as a picture is from the eye of
the painter; but rather held that they were the very
intellectual life and being itself. In like manner,
when we say that motives, or objects of desire or
aversion are before the mind, we do not suppose any
local position*, either of the mind or of the mental
conception; but we adopt an analogy on which are
founded the various uses of the words for, fore, before,
therefore, &c. - -
In the use of the conjunction, for, already noticed,
the words “I say,” or some such phrase, must be
supplied, to make up the full construction of the
sentence : but there is another and now obsolete use
of the same conjunction, in which the sense is perfect
without such addition.
Thus, in the old satire on Grooms and Stable-boys,
MSS. Harl. 2253, fol. 125.
* Totrow 5s at yévos év to ris xdºpas, ael péopâvoi trporéexöuevov,
&öpav 6& trapéxov 30 c. *xet yéveriv traoru, adrb hê pºet’ &vatorêmortas
ãºrrby, Aoylapºº rive yuá00 p.6%) is trisov, trpès $ 5e kai Övelpotoxaplew
BAérovres, kal papºv avaykalov ćivái tre, to by Štrav, u Twi réma, kal
karéxov x&pay rivá. To be uſire év yi, piñré tra kat' otoavöv, 36ew
s
(ELF/Cºl. - -
The third kind of thing (having mentioned intellect and sense)
is space, which is not subject to decay, but affords a place to all
created things ; this is touched without affording the sense of
touch, and is obscurely understood to exist, by a spurious kind
of reasoning. Looking on this, we dream, as it were, and say,
it is somehow necessary that every existing thing should exist in
some certain place, having some definite situation, and that what
is neither on earth nor in heaven, is nothing. - * -
PLATO, in Timaeo,
G R A M M A. R. 141
“ Chelsea Hospital was built for disabled soldiers;” Chap. I.
Whil God wes on erthe and wandrede wyde,
i.e. the future accommodation of disabled soldiers was S-N-7
What was the reson why he molde ryde 2
For he molde no grome to go by ys syde.
Grammar.
*-V-'
And so in “ Christ's descent into Hell,”
For y thyn heste huelde noht,
Duere ich habbe hit her abolit.
The same use of the word for occurs occasionally
in SHAKSPEARE.
Heaven defend your good souls that you think
I will your serious and great bus'ness scant,
For she is with me, Othello.
In these passages, as well as in those before cited,
for may, by transposition, be rendered therefore, as
follows:– “He would not have a groom to go by his
side, and therefore he would not ride;”—“I have not
kept thy commandments, and therefore I have paid
dearly for my conduct;”—“She is with me, but I
will not therefore neglect your business.” Or, to vary
the phrases still more, with the same sense—“ He
would not have a groom to go by his side, and that
determination being before his mind, he would not
ride;”—“I have not kept thy commandments, and
that misconduct having occurred before my present
sufferings, has been their cause ;”—“ She is with me,
but though her society be before my mind as a motive
to idleness, it will not induce me to neglect your
business.” .
We may sum up the different uses of the word, for,
as follows. It is employed either as a preposition, as
a conjunction, as an adverb, as an adjective, or as a
component particle of a word. As a preposition,
when properly used, and without ellipsis, it signifies a
relation, 1st. of place ; 2dly. of time ; 3dly. of rank,
or order; and 4thly. of cause, motive, or object. By
an ellipsis, it may express the negative of its proper
signification; and there are some uses of it in writers
of repute, which are altogether improper. In the
signification of rank, causation, &c. it expresses the
future, co-existent, or previous cause of an action, the
iimitation of a quality, or the equivalence, substitution,
similitude, or opposition of a substance. As a conjunc-
tion, adverb, adjective, or particle, its significations
coincide with some of those which it has as a preposition.
Upon the whole, it denotes that a person or thing is
before another thing in place, time, or order; or that
it is before the mind as a cause or object positively or
relatively: and as similar relations are denoted by the
terms fore, afore, before, tofore, therefore, wherefore,
&c. the inference seems clear that for and fore were
originally the same word.
When for is applied to place, it signifies that which
is before us in intention, as “we sailed for Genoa.”
That which is before us, and becomes in fact the end
of our journey, is expressed by to ; as “we sailed to
Malta.”
When for is applied to time, it signifies, that the
time in question is before the mind of the agent, as
that which either continues, or is intended to conti-
nue, during the whole period of the action. “ He is
chosen for life;” i.e he is chosen to serve for life, life
being before the mind of the elector, as that which is
to form the duration of the service. “He studied for
a year; i. e. placing before your mind a year, that will
be found to equal the time that he studied.
When for is applied to causation, or motive, the
object is future in such sentences as the following 5–
the object before the minds of those who directed the
building. In like manner, when the poet exclaims—
“O ! for a muse of fire " which is equivalent to “I
wish for a muse of fire ;” the muse is before his mind
as the object of his wish. - -
The cause is co-existent in such sentences as these :
—“ Objects depend for their visibility, upon the
light ;” i.e. visibility being before the mind, when we
consider objects, we find that in this respect they de-
pend upon the light. “He does all things for the
love of virtue;” i. e. in every action of his life, the
love of virtue is before his mind as a motive.
The object or cause is past, in such phrases as
these ;--‘‘ to punish a man for his crimes;”—“ to re-
ward him for his valour.” Here the crimes and the
valorous deeds respectively, though they may have
long gone by, are still before the mind of the person
punishing and rewarding.
We find in Robert DE BRUNNE, ther for, employed
to denote a cause, precedent. In speaking of the
murder of Sir John CoMYN, because he refused to rebel
against King Edward, he says—
Sir Jon wild not so, ther for was he dede.
where, according to modern usage, we should say,
“ therefore was he killed.”
For, used after an adjective or adverb, serves to
limit and restrain the quality by reference to some
certain object; as, “big for his age ;" i.e. having
before your mind his age, speaking with reference to
that, you may call him big. “Situated commodiously
for trade;” i. e. trade being before the mind when we
speak of the situation, we may call it commodious.
When for is used after a substantive, it is generally
with reference to some verb, expressed or understood,
and then its use is similar to what we have already
observed in speaking of verbs : e. gr. “ an eye for an
an eye;” “ he takes Richard for Robert;” “ he shot
Peter for a deserter.” Here an eye is before the mind
as being equivalent to an eye : Robert is before the mind
as being the person for whom Richard is substituted.
The character of a deserter is before the mind as that
to which the character of the person shot bore a real
or supposed similitude ; and the context will show
whether it is meant to suggest identity or diversity;
whether the individual was really a deserter, or whe-
ther his being alleged to be so, was merely a pretence
to justify the execution.
Among the uses of the preposition, for, which may
be regarded as improper, or at least have become
obsolete, we may reckon the following, in which
nevertheless, for always retains the sense of before.
1. Mr. TYRwhit, in his Glossary, says, –“ FoR,
prep. Sax. sometimes signifies against,” and among
other instances cites—
Some shall sow the sacke,
For shedding of the wheat. CHAUCER.
Mr. Tooke says, that “this construction is auk-
ward and faulty;” but that “the meaning of for is
equally conspicuous;” “ the cause of sowing the sack
being that the wheat may not be shed.” The shedding
of the wheat is before the mind, but it is not before the
mind as the proper object of the sowing ; that is to
Say, as an end to be attained by sowing the sack; but
142 G R A M M A. R.
speare indeed makes the Duchess of York, when inter- Chap. I.
Grammar. on the contrary, as an end to be prevented ; and as
ceding for the life of her son Aumerle, say— S-N-2
\—y-' this distinction may not immediately appear from the
context, an obscurity is introduced into the sense,
which renders the construction faulty, and has justly
brought it into disuse. -
2. The redundant use of for, preceding to, with an
infinitive, is very ancient in English. It occurs fre-
quently in RoberT DE BRUNNE.
The yere next on hand yede the kyng of France
To the holy land, with his purveiance,
Upon Gode's emmys forto tak vengeance.
So in the song, on the Battle of Lewes, A. D. 1264.
The kyng of Alemaigne, bi mi leaute,
Thritti thousent pound askede he,
Forte make the pees in the countre.
It was probably adopted in imitation of the French
idiom, “ pour prendre,” “pour faire,” &c.; inasmuch
as pour and for equally indicate objects before the mind
as causes of an action past or future; but the cases
differ, because in French the termination er alone does
not sufficiently denote motive, or cause ; whereas, the
preposition to, in English, has that force; and conse-
quently it renders for redundant. This idiom there-
fore is at present confined to the vulgar.
3. The following use of for is elliptical.
For tusks, with Indian elephants lie stroye.
Tusks were not before the monster as the object
which he strove to attain ; but he strove to attain
celebrity, and tusks are before the mind of the narra-
tor in speaking of that celebrity. The full construc-
tion therefore would be, “ he strove with Indian
elephants to attain celebrity for tusks;” but as the
ellipsis introduces an obscurity into the sentence, this
construction is also properly reprobated.
4, Dr. Lowth censures Swift for saying “he
accused the ministers for betraying the Dutch ;” and
DRYDEN, for saying, “you accuse Ovid for luxuriancy
of verse;” both which expressions Mr. Tooke de-
fends. This, however, is a matter of idiom, and it
turns on the force given in English to the verb accuse.
We say, to accuse of a crime or fault, but not to
accuse for a crime or fault; because the crime or fault
is not regarded as the motive directly before the mind
in such an act as accusation. We may reproach a
minister for betraying an ally; or we may censure a
poet for the luxuriancy of his verses; because it is
the nature of censure and reproach to assume the fact
as certain ; whereas, in accusation, properly speak-
ing, the , fact remains in doubt. However this may
be, it is certain that the passages above quoted from
Swift and Dryden are not consistent with modern
idiom ; and they probably were the result of haste in
their composition.
5. A somewhat similar observation may be made
on the expressions “ sick for disgust,” and “sick for
love,” which also come recommended by the appro-
bation of Mr. Tooke. The Lady, in WycHERLEY's
play, says she is “ sick for her gallant;” and Falstaff,
in the 2d part of King Henry IV, says, “I know the
young king is sick for me.” There may be an object
before the mind, occasioning sickness; as in these
cases: but the feeling which constitutes the sickness,
be it disgust, love, or any other, is not in modern use
separated from it, and made a distinct object. Shak-
Yet am I sick for fear,
But here, it would seem, is meant an actual bodily
sickness occasioned by fear : and even in this sense,
the construction, however allowable in poetry, would
appear harsh in common composition, or discourse.
The conjunctional use of the word for has already
been noticed, at some length. The adverbial use is
colloquial, and is generally considered inelegant in
composition. Thus, instead of saying, “ a writ was
moved for,” where for performs the function of an
adverb, it would be advisable to say “a motion was
made for a writ;” but on either construction, for im-
plies that the writ was the object before the motion,
as its cause, in the mind of the mover.
For is used adjectivally in such sentences as the
following :—“It is for the general good of human
society;”—“It were not for your quiet;”—“ Moral
considerations could not move us, were it not for the
will.” Here the general good of human life, and our
own quiet, are laid before us as proper motives to
action ; and the will is stated to be before our capa-
bility of being moved by moral considerations, as the
cause of such capability. In the colloquial phrase of
vulgar combatants, “I am for you ;” the meaning
is, “I am before you, in opposition.”
Lastly, for, when used as a component particle,
agrees with fore when used in the same manner. Thus
we have forbear, forbid, forget, forlorn, forsake, forswear,
and foreclose, forego, foreslack, forespent, forestall, fore-
waste. Some words, too, are written indifferently
either way,+as forward and foreward, forfend and fore-
fend. Dr. Johnson says, “ for has, in composition,
the power of privation, as forbear ; or depravation, as
forswear; and other powers not easily explained.”
The explanation is easy enough, when we consider
the various analogies of that which is before ; inas-
much as it signifies going forth, going out of the ordi-
nary limits, being opposed to, and the like.
To the same original, fore, we may trace many
other English words, as forth, further, first, &c.
The word forth occurs in a charter of King Edward
the Confessor, preserved in the very valuable work of
HICKEs, (Thes. Ling. Sept. v. i. 161.) It there appears
to signify “freely” or “readily ;” and is spelt vorth,
as for is spelt in the same instrument vor; which is
the more remarkable, because the charter relates to
the county of Somerset, where that pronunciation is
still preferved :-
“Ich qucthe eou that ich wille that Gyse Bissop bed thisses
bisSopriches;–swo uol, & swo worth swo hit eni bissop him to
uoren formest haueth on ealle thing.”—“Significamus vobis nos
velle quod episcopus Giso episcopatum possideat—adet, plenè et
liberê per omnia sicut ullus episcoporum praedecessorum suorum
unquam habebat.”
In RobºFT DE BRUNNE we find forthely used for
“ readily;” e. gr. “als forthely as he'—as readily
as he.
Further (sometimes erroneously spelt farther,) was
anciently in English forther; and in High German
forder : e. gr. “Das volk zog nicht forder bis Mirjam
aufgenommen wird;’—“ The people journeyed not
(went no further) till Miriam was brought in again.”
(Numb. c. xii. v. 15.) OTTFRID, in the Frankish Gos-
pels, instead of this word, uses furder; in Anglo-
G R A M M A R.
143
Gramrºar.
or state.”
Saxon it is written forthor : in Low Saxon, vorder,
vurder, vudder; in modern German, vorder is the
foremost part, as “vorderseite des gebaudes,” the front
of a house ;-‘‘ vordertheil des schiffs,” the prow of a
ship. In old German, this word is written furter and
furder. ADELUNG says forder is the comparative of
fort; which, in some modern dialects, is pronounced
furt and furd.
First, in English, was originally fore-est, i. e. fore-
most ; and of the same origin is the German first,
which properly signifies, according to ADELUNG, “the
first and most eminent person of his nation, province,
It is commonly rendered “prince.” In
the German Bible, Abraham and Job are called fürsten,
princes. Furst is written by WILLERAMUs, worst; by
OTTFRID, furista; in Low Saxon, fürste and forste ; in
Swedish, forste ; in Danish, fyrste ; and is the super-
lative (says Adelung) of für. “ Fur and vor (pro-
nounced fore) are sometimes distinguished, (says
WACHTER,) as if vor applied only to time, and not to
place, or to cause ; fur to place and cause, and not to
time; but this distinction is not steadily observed
among us, nor is there any trace of it in the ancient
writers; for the Goths say faur, faura ; the Anglo-
Saxons, for, fore, fyr, fyre; the Franks and Alamans,
fora, furi; the Belgians, vor; the English, for ; the
Swedes, för, &c.” This author adds, that the Greek
Tpo, and the Latin pro, differ not from für and vor,
except in a slight change of the labial articulation,
and in transposing the canine letter, r.
The simple Greek preposition Tpo signifies before,
both in place and time ; and the compounds in which
it has that meaning are innumerable. The adverb
7pwi denotes the early morning, the foremost part of
the day. The adjective Tpwtos, first, is evidently the
superlative of Tpo, as our first is of fore. IIpupa is the
prow, the forepart of the ship.
In Latin, the prepositions pro and praº are both
connected with the Greek Tpo. The ancients also
used pri for pra: ; whence prior and primus, as also
pridem, pridie, princeps, priscus, pristinus ; all relating
to that which is before, in time, or order. Prae sig-
nifies before, in place; e. gr. “I pra; sequar;”—“Go
before, I will follow.” “ Praefert manus;”—“ He
stretches out his hands before him ; he feels his way,
like a person walking in the dark.” “ Praecalvus :"—
“bald before, bald on the fore part of the head;” or
before, in time ; e. gr. “praºcanus,”—“ greyheaded
before his time.” “ Praecocia poma ;”—“ apples,
which grow ripe before the usual time ;” or before as
a cause, e. gr. “ misera pra, amore ;” “ wretched for
love,” love being that which was before her wretched-
ness, as its cause ; or before, as denoting superiority
or excess, as “pra-altus,” excessively high, before all
others in height. In like manner, pro refers to place;
e. gr. “ hasta posita pro aede Jovis Statoris;”—“ a
spear placed before the temple of Jupiter Stator;” or to
time, as “proavus,” “a great grandfather; one who
lived before the grandfather;” or to cause, e. gr.
“ poenam promerui.;”—“I have deserved punishment
for my offences;” my evil deserts are before my punish-
ment, as its cause.
Nor is the Latin language without closer traces of
the Teutonic for, in foris, foras, forum, forceps; for
these words signify respectively, foris, “the door;”
which is in the forepart of the house; foras, “ out of
º
“ the market-place,” or scene of public debates an
trials, which were anciently carried on in an open
space before the houses of the citizens; forceps, “ the
tongs ;” the instrument with which a smith drew
forth hot iron from the fire.
Again, in the base Latin of a subsequent age, we
find such words as foraneus, forensis, forasticus, foresta,
forgeldum, forisfacere, forisbannitus, &c. which appear
to be of a similar origin. Foraneus, forensis, and forås-
ticus, signify that which is forth of the house, or
country; a thing or person that is external, strange,
or foreign. Hence the Italian “grazie foranee,”
“ external advantages;” “forese schiatta,” “a rustic
race,” (che stafuor della città, as it is explained in the
Pocabolario Della Crusca.) So “forastica pugnac,” are
foreign wars, (Epist. 3. S. Bonifac. Archiepisc. Mogunt.)
and “forastici homines,” are strangers, foreigners ;
(Tabul. S. Remigii. Rhemensis.) . Foresta did not
originally signify “a wild, uncultivated tract of ground
with wood, as Dr. JoHNsoN defines our word, forest ;
but rather as GIovaNNI VILLANI defines the Italian
word foresta ; “ luogo di fuora, separato dalla congre-
gazione e coabitazione degli uomini;” “a place that is
forth from cities, and separated from the congregation
and co-habitation of men.” Whether these places did or
did not abound in trees was accidental; but as it gene-
rally happened that they did so, the word forest came to
be considered as indicating a woody tract of country.
It is remarkable, however, that our word wood, itself,
does not seem to have originally had a necessary con-
nection with the notion of a tree, or its substance;
but to have been of the same meaning as wild, weald,
wald, wold, wod, wud, &c. denoting any thing unculti-
vated, savage, fierce, or mad. Hence, the weald of
Kent was the wild, uncultivated part of that county.
“ St. Swithin footed thrice the wold;" i. e. the desart.
OTTFRID, in the Frankish Gospels, translates “the voice
of one crying in the wilderness,”—“ Stimma rua-
fentes in wastinnu waldes.” TATIAN translates uéAt
diptov (wild honey) “wildi honug ;” but the Anglo-
Saxon version renders it “wudu hunig.” This word
wud often occurs in Anglo-Saxon, signifying wild,—as
“ wudu bucca, a wild goat; “wud-culfer,” a wild
pigeon; “ wudu-coc,” a wild cock; which two last
we still call a wood-pigeon, and a woodcock. Wods,
in Gothic, is used for a demoniac madman; e. gr.
“ saei was wods,” “ he that had been possessed of the
devil.” (St. Mark, c. v. v. 18.) In Anglo-Saxon, wod
is used for mad; hence “wode-thistle,” i. e. mad-
thistle, was the name of hellebore, a remedy against
madness. In Frankish, wotnissa was madness. In
Dutch, woede is fury ; in Scottish, wud is mad. The
English wood, in the same sense, has become obsolete;
but is found in SPENSER.
Coal black steeds yborn of hellish brood,
That on their rusty bits did champ as they were wood,
To return to the derivatives of for and forth. For-
geldum was an impost probably on foreign goods :—
Omnibus geldis, tengeldis, horngeldis, forgeldis, penigeldis, &c.
Monast. Anglican. vol. i. p. 372.
Forisfacere, is explained by Ducange, “offendere,
nocere, q. facere foris, i. e. extra rationem.” Here
the Latin foris is unnecessarily substituted for the
Teutonic for. Forfare was the Italian word of which
doors,” abroad, forth, “ from the house ;” forum, forisfacere was the barbarous Latin translation ; and
d Chap. I.
144
G R A M M A. R.
Grammar, for in forfare, was employed exactly as for is in the
\-y-. English forlorn ; and ver (pronounced fer) in the Ger-
man verloren. In a secondary sense, forfare signified
to forfeit lands or goods for one's misdeeds. So forban
was one who acted against the ban or commandment
of the law ; (for Ottfrid translates “ my command-
ment,” “ban minnan,”) and in a secondary sense,
one who was banished, or exiled by command, forth
from the state. In the former signification, the
French still use forban for “a pirate :” and in the
latter, MATTIIEw, of Paris, uses forisbannitus, in his
history, (ad ann. 1245.)
Expulsus a Scotiá, forisbannitus ab Anglià.
Forda is our word ford, which is manifestly from
the Gothic faran.
Non liceat alicui facere dammas, aut fordas, aut alia impedi-
menta in watergangiis. -
Ordinatio Marisci Ramesiensis.
Fordale appears to be of the same origin.
Tendit usque ad magnam aquam de Ayr, et fordales ejusden
prati. Monast. Anglic. vol. i. p. 657.
It is scarcely necessary to trace minutely the
connection of for and fore with the German für,
ver, and vor; the Dutch voor, the French pour, &c.
One or two instances, however may be noted. The
German vorbey is the old English forbi, and Scottish
forbye ; but with some variation in the use. Forbey
sometimes denotes the passing along before a place ;
e. gr. “Die flotte segelte die insel vorbey ;” the fleet
sailed along before the island. Sometimes it denotes
a time that is past, and consequently a time before the
present ; e. gr. “Das jahr ist vorbey;”—the year is
at an end. Forbi is used by RoRERT DE BRUNNE in
the following senses 3–" before,” “ notwithstanding,”
“ away,” “therefrom ;” “forbi euer ilkone,” before
every one. BURNs uses forbye for “besides,” “ over
and above.” The Dutch voor is used in the senses of
before and for, as “voor de deur,” “ before the door ;"
—“’s daages te voore,” “the day before ;”—“ voor alle
dingen,” “ before all things;”—“ dat is voor hem,”
“ that is for him ;”—“ jets voor verlooren houden,’
“ to give up a thing for lost. Voorburg is a fenced
suburb, built before a city. In old French, this was
forsbourg, since corrupted into fauxbourg, faubourg, and
faubour. -
Et pour ladite requeste, le sergent, en la ville et forsbourgs,
n'aura que cinq Sols. Comst. de Tourraine.
. The French hors was anciently written fors ; and
was probably derived, as MENAGE suggests, from the
Latin foris.
BRANToME uses fors in the sense of “except.”
Ne furent à l'offrande, fors Monsieur D'Angoulesme.
And so LA Font'AINE–
Toute la troupe étoit lors endormie,
Fors le galant.
In like manner, hormis has been formed from foris,
missus; and dehors from de foris.’ -
There cannot be any doubt, but that the French
pour is the Teutonic for or fur. In English compound
words adopted from the French, it is spelt and pro-
nounced pur; as purchase, purport, &c. Purfle, which
Johnson defines “a border of embroidery,” is simply
foreworked, or fore-edged, pour-filé.
I saw his sleues purfiled at the hand -
With gris, and that the finest in the land.
- CHAUCER.
Gris is a better sort of fur.
Griseum.) -
Thus have we seen that our words for and fore are
alike connected with words of analogous sound and
sense, both of Gothic and Grecian origin: and it
seems not improbable, that they also agree with the
verb, to fare, which is the Anglo-Saxon and Gothic
faran, to go, or move forward. From fare doubtless
comes the adverb far: and we find in old English, tha
the past tense of fare was fore. -
Thorghe mountayn & more, the Bascles gether weie,
Our nesche and hard thei fore, & did the Walsch men deie.
Robert DE BRUNNE.
Bot he mot quitely go, in world where he fore,
And frely passe him fro, fro whom that he to suore.
IDEM.
(v. DUCANGE, ad voc.
Chap. L.
As before is compounded of be and fore; so but is But,
compounded of be and out. “But,” says SKINNER,
“ ab A. S. bute, butan, praeter, nisi,” &c.—“bute autem
and butan tandem deflecti possint a praep. be, circa,
vel beon esse, and ute, vel utan, foris.” Mr. Tooke,
however, has observed, that this word has in English
two derivations; viz. that just quoted from Skinner,
which is indisputably right ; and another suggested
by Tooke himself, which will require some observa-
tion hereafter.
1. We proceed, however, first, with but in the
obvious sense of be-out; and for the present we assume,
that the meaning of the word out is sufficiently under-
stood, as denoting the opposite to in. By old English
and Scottish writers we find it often written bot, or
bote, possibly from some confusion with respect to its
derivation : however, as there is no regularity in th’s
respect, the orthography may merely have varied
according to the accidental habits of the different
writers. -
But, answering to without is applied to place in the
Scottish dialect, and opposed to ben, i. e. within ; e. gr.
“ blithe was she but and ben,” i.e. she was sprightly
both within the house and without. We find binman
in Anglo-Saxon for bi-innan, or be-innan, in the same
sense as the Scottish ben. The Dutch also use buyten
and binnen, with these significations,—as ‘‘ buyten
deur,” without doors; “ binnens huys,” within doors.
In the old ballad of the Gaberlunzie Man, ascribed to
King JAMES I. of Scotland, in the 15th century, we
find both expressions.
Gae butt the house lass, and waken my bairn,
And bid her come quickly ben.
But, answering to without in the same sense of pri-
vation, is of very ancient use, both in the English and
Scottish dialects. t
Alsua that all the landis of the kinrik be taxt eftar as thai ar
of vale now, and that but fraude or gile.
Scot. Act. Parl. 1424.
The sowking wolf furth strekyng breist and udyr,
About his palpis but fere, as thare modyr
The twa twynnyis. GAWIN DOUGLAS.
But mete or drinke, she dressed her to lie
In a darke corner of the hous alone. CHAUCER.
IBut, in the sense of privation, answering to except,
occurs in our common expression “all but one ;" i.e.
all be-out one, all, if one be-out. In this sense also it
occurs frequently in old English and Scottish,
G R A M M A. R. * 145
Grammar.
-—
What is ther in paradis,
Bot grasse and flure, and greneris?
. . Descr. of Cokaygme.
Quhich has my hert for ever set abuſe,
In perfyte joye that never may remufe
Bot onely deth. £ing's Quair.
In this sense it is sometimes preceded by a negative,
as in the Description of Cokaygne.
Ther nis met bote frute,
Beth ther no men bot two.
So in the Anglo-Saxon Gospels, (Luke c. viii. v.
51.) “ne let he nanne mid hym ingan buton Petrum
et Johannem et Jacobum ;” “ he suffered no man to
go in save Peter and James and John.” And again,
(Luke c. ix. v. 13.) “We nabbath buton fif hlafas and
twegen fixas, buton we gan, and us mete bicyon ;”
“we have no more but five loaves and two fishes,
except we should go and buy meat.”
In Chaucer we find (according to the idiom of that
day,) no less than three negatives preceding but.
Ne neuer y nas but of my body trewe.
That is, “I never was otherwise than true.”
In the present day, we omit the negative; which,
as Mr. Tooke observes, often forms a very blameable
and corrupt abbreviation of construction. Thus we
say, “I saw but two plants;” which, in old English,
would have been “I ne saw but two plants;” I saw
no plants be-out two. So CHILLINGworTH says, “If but
wise men have the ordering of the building;” i. e. if
none have the ordering of the building but wise men
Hence arises the conjunctional force of but, bote, or
bot, answering to unless.
Thus, in the ballad of the Mon in the Mone.—
Nis nowyht in the world, that wot when he syt,
Ne, bote hit bue the hegge, whet wedes he wereth.
So Robert DE BRUNNE,-
For slayn is Kyng Harald, & in lond may non be,
Bot of William he hald for homage & feaute,
So in Kyng Alisaunder,-
Al that we havith wonne and wrought,
Y no holde hit for nought,
Bote we mowe heom wynne,
So in Richard Coer de Lion.—
They tolde hym the hard caas,
Off the Sawdoun's hoost hou it was,
And but he come to hem anon,
They wer forlorne everilkon.
So GAwiN Douglas-
Blyn not, blyn not, thou grete Troian Enee,
Of thy bedis nor prayeris quod sche; i
For bot thou do, thir grete durris, but dred,
And grislie yettis sall neuer warp on bred.
So CHAUcER.—
But he wil hym repente.
But, or bot, in this sense, was often followed by
give or if
Thus, in the Scottish Act of 1424, before quoted,—
Thaisalbe chalangit be the kyng as fautours of sik rebellyng, bot
gif thai haif for thame resonable excusacion.
So in the romance of Sir Tristrem.—
The maiden of heighe kinne
She cald hir maisters thre;
Bot give it be thurch ginne,
A selly man is he.
So in Richard Coer de Lion.—
VOL. I.
Wher thorwgh they myghten not withstonde, Chap. I.
Put yiff Saladyn the Sawdan S-N-"
Come to help with many a man. -
The last sense of this word, but, which we shall
notice, is that of our common conjunction, answering
to the French mais and the Anglo-Saxon, ac.
RoBERT DE BRUNNE commonly spells it bot.
Roberd thouht no gile,
Bot come on gode manere till his brother Henry.
* :* 3. * : ; :*
Roberd bi his letter his brother gan diffie,
Bot gode Anselme, that kept of Canterbirie the see,
Before the barons' lept, kried, pes per charitie.
GAwiN Douglas sometimes spells it bot, and some-
times, though in this sense more rarely, but.
Sic wourdis vane & unsemelie of sound,
Furth warpis wyde this liger fulichelie;
Bot the Troiane baroun unabasitlie,
Na wourdis preisis to render him agane.
Booke x. p. 338.
Quhare some forgadderit all the Troyane army,
And thyck about him flok and cam, but baid,
But nowthir scheild nor wappinis doun they laid.
Booke xii. p. 430.
So King JAMEs I. of Scotland, in his poem of The
King's Quair, uses bot.
Bot for alsmoche as sum micht think or seyne,
Quhat nedis me apoun so lytill evyn
To writt all this 2 I ansuere thus ageyne.
In the poem of Christis Kirk of the Grene, by the
same royal author, however, it is sometimes written
but.
Twa, that wer herdsmen of the herd,
Ran upon udderis, lyk rammis;
But quhair thair gobbis werungeird,
Thay gat upon the gammis.
In the schedule of accusation against King HENRY
VI. presented in Parliament, A. D. 1461, it is written
but. &
Not oonlie in the north parties, but also oute of Scotlond.
So in the English statute of 1483, before referred to.
That such exaccions, called benevolences, afore this tyme takyn
be take for no example, to make suche or any lyke charge here-
after, but it be dampned and annulled for ever.
DUNBAR, in his Goldin Terge, uses but.
All thir bure genyies to do me grivans;
But resoun bure the terge.
# * #. # #
Thick was the schot of grundin arrows keme ;
But Resoun, with the goldin schield sae schene,
Weirly defendit quhosoeir assayit.
And so Montgomery, in the “ Cherrie and the Slae.”
My agony was sae extreme,
I swelt and swound for feir;
But or I walkynt of my dreme,
He spulyied me of my geir.
2. We have seen that in the different uses of bote,
bot or but, these words appear to be used almost in-
differently: and perhaps they may all be referred to
the same derivation, be-out; for that which is out, is
excepted from that which is in ; and it is likewise over
and above that which is in. In this last acceptation,
therefore, it may well answer to the French mais,
which is a corruption of the Latin magis, more ; and
to the Anglo-Saxon ac, or eac, which is from eacan, to
add to ; as in Gothic, there are the conjunction auk,
and the verb aukan, with the same significations, so
U
146 G R A M M A. R.
Grammar, in Greek, av and āvāw ; in Latin, aut and augeo; in Thus have we seen the two connecting links; viz. Chap. I.
\-N-'Alamannic, auh and auhhon ; in Danish, og and Öge; be-out and bet, one or both of which connect what Mr. S-N-2
in Dutch, ook, and the old verb oecken ; and in Eng-
lish, eke, and to eke out.
Mr. Tooke, however, thinks that in this significa-
tion of over and above, the word bot was the imperative
of the Anglo-Saxon and Gothic verb botan, which (he
says) “means to boot, i. e. to superadd, to supply, to
substitute, to atone for, to compensate with, to
remedy with, to make amends with, to add some-
thing more, in order to make up a deficiency in
something else.” We do not mean positively to
reject one of the very few original etymologies in the
first volume of the Diversions of Purley ; but we must
observe, that botan rather means to add something
better than something more. The Gothic botan is our
verb, to boot; and is explained by JUNIUs, proficere,
prodesse, juvare. There can be little doubt but that
it was of the same origin with the Anglo-Saxon, bet,
hetera, best, which two last we retain in our better and
best. In Anglo-Saxon, betan was emendare, and bot,
emendatio. Hence, perhaps, when the conjunction
but implies preference, its original meaning was “bet-
ter.” Thus, “I will not do this, but I will do that,”
means “I will not do this, better I will do that ;” i. e.
I can do it better in fact, or better to my own satis-
faction and pleasure. BEDE says, (i. 26.) “hi lefnysse
onfengon cyrican to timbrianne, and to betanne;”—
“They received permission to repair and amend the
churches.” LUPUs says, (Serm. i. 3.) “ to myclan
bryce sceal micel bot nyde;” — To a great breach
shall need great amends. . -
Hence CHAUGER says, -
God send euery gode man bote of his bale !
Hence, also “ nets to bete,” in old English, was to
mend nets. -
Bot was used in a secondary sense for repentance,
which was supposed to amend men; as “bot theawas
awent ;”—repentance changes manners ; and also for
compensation, as we say, to make amends for any
thing; hence in the Anglo-Saxon feohbote, was a pecu-
niary mulct; and in the old English theftbote, was a fine
for theft. Hence also fire-bote, foldbote, and ploughbote,
were three rights anciently reserved to tenants of
taking what boots, (i. e. profits, or is requisite) for
fuel, for the fold, and for the plough. Straw and hay
being among some of these botes : and the peasantry
making covering for the legs of such materials, those
coverings came to be called boots; and what is now
called a bottle of hay, was the botal, or quantity, usually
led home from the field for bote. The man in the
moon is described in the verses often quoted above, as
bearing his burthen on a bot-fork ; that is, an instru-
ment used to bring home thorns and other materials
used for bote.
Mon in the mone stont and stryt,
On is bot-forke is burthen he bereth.
From this source evidently are our noun, booty; the
Italian, bottino; French, butin ; Spanish, botin ; Dutch,
buyt ; and Danish bytte. Our verb, to boot, also is the
Dutch baeten. Our better is the Dutch, beter; German,
besser ; Gothic, batizo ; Frankish, bezzer; Alamannic,
pezzira ; Danish, bedre; Swedish, baettre; Islandic,
bettri. The oldest form of the positive of these words,
(says ADELUNG,) was in German, bas, and in Lower
Saxon, bat; which brings us back again to the Anglo-
Saxon bet and bot.
LocKE calls “ the several views, postures, stands,
turns, limitations, and exceptions, and several other
thoughts of the mind intimated by this particle but.”
The meaning of out we have hitherto explained Out, In,
only by its opposition to in; but what are these
words 2 To these questions we have little more to
answer, than that inasmuch as they name distinct
conceptions of relation, they must have been origi-
nally nouns. Mr. Tooke observes, “that in the Gothic
and Anglo-Saxon, inna means uterus, viscera, venter,
interior pars corporis;” and that “there are some
etymological reasons, which make it not improbable
that out derives from a word, originally meaning skin.”
If these facts could be well established, they would
prove but little. They would only prove that man,
in the early progress of thought, applied (as it was
most natural he should do) his conceptions of the
relations of place to his own body, and distinguished
the inside of his frame from the outside. “Inna, inne
is also used in a secondary sense,” says Mr. Tooke,
“ for cave, cell, cavern;” that is to say, it is used for
the place in which a man or other animal dwelt.
Thus, in the song on The Battle of Lewes, (A. D.
1264.) we have yn for a place of abode.
Sire Simond de Mounfort hath suore biys chyn,
Heuede he nou here the Erl of Waryn,
Shulde he neuer more com to is yn.
Hence our common noun, an inn, now used for a
house of entertainment for travellers; the place
where, after having been out all day on their journey,
they are in at night. That this word was anciently
applied to a more private and permanent residence,
however, is evident, both from the passage just
quoted, and also from a similar one in the ballad on
The Battle of Bruges, (A. p. 1301.)
Sir Jakes ascapede, by a coynte gym.
Out at one posterne, ther men solde wyn,
Out of the fyhte, hom to ys yn.
The Anglo-Saxon verb innan is our verb, to inn, as
in BUTLER.— -
I'm certain 'tis not in the scrow]
Of all those beasts, and fish, and fowl,
With which, like Indian plantations
The learned stock the constellations;
Nor those that drawn for signs have been
To th’ houses where the planets inn.
From the signification of place, the transition to
signify time, is natural and easy.
Danger before, and in, and after th' act,
You needs must own is great.
DANIEL. Civil War.
The signification of circumstance is still more com-
prehensive.
In all things approving ourselves as the ministers of God; in
much patience, in afflictions, in necessities, in distresses, in stripes,
in imprisonments, in tumults, in labours, in watchings, in fast-
ings. ST. PAUL, 2 Cor.c. vi. v. 4, 5.
Now, if we suppose any given space, or time, or
circumstance, to be represented by a circle, whoever
or whatever is between the periphery and the centre,
bears to the thing given the relation, which we express
by the word in, and whoever or whatever is farther
from the centre than the periphery is, bears to the
whole the relation which we express by the word out;
G R A M M A. R.
147
Eitel at present signifies in German, vain. “Signi-, *P. "
ficatus,” adds WACHER, “ex priori desumptus, quod S-V-
Grammar, and this may be considered, either simply, or with re-
S-N-'ference to some other thing or person. Thus, a per-
son may be said simply to be out of doors, or to play
out of time, or out of tune, or to be out of his senses ;
or with reference to others, he may be said to outdo
them, or to outshine them, or the like. In modern
times, out is not used alone, as a preposition ; but we
find it so used in CHAUCER.—
Thou shuld neuer out this groue pace.
And the correspondent aus and ausser, in German,
have the same force, as “aus dem hause gehen,” to
go out of the house ; “ausser landes,” out of the
country. Most of the Teutonic dialects have this
word, as the Gothic us, uzuh, ut, uta; the Anglo-
Saxon, ut; the Alamannic and Frankish, wº ; the
Dutch, uyt. “Even the Persian ez, and Latin ex,”
says ADELUNG, “belong to this root;” and if so, we
may, of course, add the Greek eş and ek, and the
Latin e. -
We have noticed within and without : but instead of
these, many old writers use inwith and outwith.
Thus, in the Seuyn Sages occurs inwith.
I sal him teche, with hert fre,
So that, inwith yeres thre,
Sal he be so wise of lare,
That ye sal thank me euermare.
BARB.our has outwith-
As he auised nou haue thay done,
And to them outwith sent he soon,
And bad thane harbre thame that night,
And on the morn cum to the fight.
This word occurs in a curious passage of the Scottish
Statute of A. D. 1427.
Item, at na lipirous folk sit to thig, nothir in kirk, nor in kirk
yarde, na in name vthir place within the borowis, bot at thare awin
hospitale, ande at the porte of the toune, ant vthir places oute with
the borowis.
We find in BARB.our the words withouten and
joroutten.—
For he would in his chalmer be,
A wel grete while in priuite
With hym a clerk withoutten mo.
# # sº * * : ;
I ask you respite for to see
This letter, and therewith aduised be
Till to morn that ye be set,
And then, foroutten longer let,
This letter sal I enter here,
As out signifies privation in without, foroutten, and
the like, so it has a like force in the word outlaw ;
which is, in Anglo-Saxon, utlaga. In the charter of
Edward the Confessor, before quoted, we find unlage.
—“And gif what sy mid unlage out of than bissop-
riche geydon;"—and if anything be taken from that
bishoprick with un-law ; i. e. with injustice. Our
word idle is derived from ut, or out. The German
word eitel had for its first signification, empty; in
Frankish, ital. “Sinan stual liaz er italan;" — he
left his seat empty. “ Thaz itala grab ;”—the empty
sepulchre (from which Christ had risen.) “Inti
otage forliaz itale;”—the rich he sent away empty.
“Origo vocis,” says WACHTER, “est a particula
privativa ut, ex;” and we have already seen that ut is
our word out. This etymology may cast some light
on Shakspeare's well known passage—
Of antres vast and desarts idle.
vano nihil sit inertius nec magis vacuum.” Hence, in
the Alamannic, “ ital-ruam,” is vain-glory : and in
the Anglo-Saxon, “ yael-yyip,” is vain-boasting. In
this sense HookER uses idle, “They are not in our
estimation idle reproofs.” •
In is a word of still more general use among the
European nations than out. We find it in the Greek
ev, the Gothic, Italian, and Latin, in ; the French
and Spanish, en ; the Swedish and Islandic, inni; the
Frankish and Alamannic, inna; the Anglo-Saxon,
innan ; and many compound forms, as the Gothic
innathro, within ; and inngaggan, to enter; the Latin
intra, infra, &c.; the Italian and Spanish dentro, the
French dans, and dedans, &c.
The Anglo-Saxon innan sometimes signifies into, as
“heo beseah innan tha byrgenne;” she looked into
the sepulchre : sometimes within, as “innan huse,”
within the house. We find it also further compounded,
as in oninnan and beinnan, e. gr. ‘‘ oninnan me selfum,”
within myself; “beinnan than carcerne,” in the
prison. -
In like manner we find that the Latin in signifies not
only within, but into, toward, and consequently against;
agreeing in this respect with out, which we have seen
not only signifies without, but beyond, and not only
privation, but excess. So, “ in domo,” signifies
within the house ; “Piso in aedem Vestae pervasit,”
Piso came into the temple of Vesta ; “ in meridiem
spectat,” it looks toward the south ; “ haec cum audio
in te dici, excrucior,” when I hear these things said
against thee, I am afflicted. From this last sense it
would seem that the privative force which the Latin
in has in composition is derived; as infelic, inops ;
and so in our English words infamous, inactive, impro-
bable, &c. MILTON, however, seems to have some-
what exceeded the limits of grammatical analogy,
when he invented the word inabstinence,
That thou may'st know
What misery th’ inabstinence of Eve
Shall bring on man.
Mr. Tooke says, “I imagine that of, in the Gothic of off.
and Anglo-Saxon af, is a fragment of the Gothic and
Anglo-Saxon afara posteritas, afora proles, &c.; that
it is a noun Substantive, and means always consequence,
offspring, successor, follower.” That of or af was a noun,
that is, the name of a conception, is not to be doubted;
but to say that it is a fragment of afara, is probably
as correct as to say, that the word iron is a fragment
of the ancient noun substantive, ironmonger: If it be
a fragment of any thing, it is more probably of aft,
which we shall consider under the word after. How-
ever, the nouns, which by long use, for many cen-
turies, and in various dialects, have come to serve as
the most common prepositions, are in general so far
removed from their source, that we cannot trace them
back to it with certainty, as we can the more recently
adopted prepositions, touching, concerning, during, and
others already mentioned. It is very possible that
the term of, af, or ap, may, in certain early dialects
have signified a son ; and indeed some traces of this
seem observable in the Sclavonic of, as Peterhof, the
Welsh ap, as ap-Rice, and the Irish o, as O'Hanlon;
but this fact, if it could be established, would be very
far from proving, that the term might not have been
U 2
148
G R A M M A. R.
Grammar, so applied with reference to a more general idea,
such as that of proceeding from, depending on, or be-
longing to, the parents. -
The preposition of, and the preposition and adverb
off, were anciently the same word, and the subsequent
variation of orthography was merely accidental.
I schall you telle of a kynge,
A doughty man withoute lesynge,
Off body he was styffe and stronge. . .
Lyfe of Ipomydon.
Godwyn, an Erle of Kent, met with Alfred,
Him and alle his feres vntille prison tham led :
Of som smote of ther hedes, of som put out ther iyene.
ROBERT DE BRUNNE.
And at the last, with gret payne,
Kyng Richard wan the Erl off Champagne;
The Erl off Leycetre, Sere Robard,
The Erl off Rychemond and Kyng Richard.
Richard Coer de Lion.
And in the castle off Tyntagill.
Legend of King Arthur.
Quhare sodeynly a turture quhite as calk,
So evinly vpon my hand gan lyt
And wnto me sche turnyt hir full ryt
Off quham the chere, in hir birdis assort,
Gave me in hert kalendis of confort.
KING JAMEs. King's Quair.
Off signifies dissociation, or distance of place; and
this both adverbially and prepositionally.
See
The lurking gold upon the fatal tree;
Then rend it off.
About thirty paces off were placed harquebusiers.
KNoll.E.S.
Cicero's Tusculum was at a place called Grotto ferrate, about
two miles off this town. ADDISON.
DRYDEN.
“Proceeding from" may probably have been the
original sense of the words of and off, both which in
Dutch are written af; as “ Ik weet'er niet af,” I
know nothing of it. “Zyn hoed is af,” his hat is off.
This word af was used in old French ; as “ hostel qf
brebez,” a sheepfold ; hotel au), brebus. It is the
Gothic af, as “wairp af thus,” cast from thee; “af
missilbin tauya niwaiht,” I do nothing of myself. It
is the Lower Saxon, and Swedish af. It is without
doubt the modern German ab, as “die farbe geht ab,”
the colour goes off, or fades; “ das feuer geht ab,” the
fire goes out ; “abhangen,” to depend on ; “ablassen,”
to leave off. And it is probably connected with the
Latin ab, and Greek ato. “Af pro ab scribere anti-
qui solebant,” says PRIscIAN : and we find on an
ancient brazen tablet, ‘‘ qf vobºis” for “a vobis.”
GELLIUs, speaking of the verbs aufugio and aufero,
says, “illud inspici quaerique dignum est, versane sit
et mutata AB praepositio in Av syllabam propter levita-
tem vocis; an potius AV particula Suá sit propria ori-
gine, et proinde, ut pleraeque aliae prepositiones a
Graecis, ita haec quoque inde accepta sit; sicuti est, in
illo versu Homeri :"—
- Ağ purav učv rpóra, kal éo pašav, kal éðelpav.
If the word af was part of aft, it may possibly have
signified “ the back,” and, consequently, “that which
we leave behind ;” that “ before which we are placed;”
or, that “ from which we proceed.” Hence of and
fore may be regarded as expressing different stages of
removal from an object; and thus we may see how
to be fond of an object, to wish for it, and to long
after it, come to be nearly synonymous. w
In many old writers we find of employed as we no
use out of, or from.
I sal the brynge of helpyne.
M.S. Harl, No. 2253, fol. 55.
Mote ye neuer of world wend. . . .
e Idem. No. 913.
Chargit he lous of this ilk mannis handis.
- GAwiN Douglas, book ii. p. 43.
Quhilk that he sayis of Frensche he did translait.
IDEM. Preface,
. There are several other uses of this preposition now
obsolete, among which we may notice the following:
Even like some empty creek, that long hath lain
Left and neglected of the river by. -
- DANIEL’s Musophilus.
How many thousands never heard the name
Of Sydney, or of Spenser, or their bookes,
And yet brave fellowes, and presume of fame.
Ibid.
Lucifer of the brightest and most glorious angel, is become the
blackest and the foulest fiend.
Homily against Disobedicnce, &c.
Bot yif I may with my brother go, 'ſ
Mine hert it breketh of thre. -
Amis and 24 milown.
Then I, whiche had not slept of the hole nyght,
By Morpheus sodaynly had lost my sight.
Goodwyn's Mayden's Dreme,
Sir, said Regnawde, I thank you much of your good will.
Foure Sounes of Aimon.
Holi Chirche was foundid of the apostlis on Crist the stoon.
WICLIF,
Soche an other for to ymake,
That might of beaute be his make. CHAUCER.
Chap. I.
The adverb off “ is generally opposed,” says John- On.
son, “ to on ; as “to lay on, to take off.” On would
seem to apply adhesion to, as off does separation
from ; as to stand on a table, to fall off a table; to be
fastened on, to be cut off; to flow on, as a river, with
continuity, to fly off as a bird, with separation. But
in the signification of belonging to, on was anciently
used where we now use of ; as in the Letter of
HENRY III., A. D. 1258.-‘‘ Henr. thurg Godes ful-
tume, King on Engleneloande, Lhloaverd on Yrloand,
Duk on Norm. on Aquitain, Eorl on Aniou.” In the
old English it was also used for in ; as, in the same
letter, “ to alle hise halde ilaerde ilewed on Hunten-
don schir.” In the Anglo-Saxon, besides this latter
sense it had many others, as “tha comon fram east-
daele to yebiddenne hi on Ierusalem,” then came they
from the east parts to Jerusalem to pray: “ sum feoll
on tha thornas,” some fell among thorns: ‘‘ sceo
fordaelde on laecas eall that hed ahte," she had spent
all her living upon physicians: “ on thone heofen
beseah,” he looked up to heaven: “ eode on anne
munt,” he went up into a mountain : “thare halyan
rode, the ure drihten on throwode,” the holy cross
that our Lord suffered on : “ Hwi ferde ye on westenne
geseon " What went ye out into the wilderness to
see : “For on,” says Hickes, “ sometimes occurs an,
from the Gothic ana.” In Gothic, the preposition ana
is used separately for on or in, as “ana staina,” on a
rock : “ana mesa,” in a charger. The Goths, Franks,
and Alamans, used also an and ana in many. com-
pounds; as, the Gothic anaaukan, to add, or join on
G R A M M A. R.
149
Grammar.
Up. upon,
above, over,
to ; the Frankish anbeten, to pray to, or, as we say,
“ to call upon the name of the Lord.” The Alaman-
nic “ angeuangen,” to lay hands on, or claim. This is
also the Dutch aan, and German an, of which WACH-
TER, in the 5th section of his Prolegomena, gives
many significations; e. gr. denoting connection, as
anbinden, to bind on to ; denoting the direction of an
act toward a particular object, as anblicken, to look
wpon, or toward; denoting continuity of time, as an-
stehem, to stop, to stand as it were on the same point
of time; thus we say, a ship stands on, in the same
course, using the word on for a continuous adherence,
as in the other case it is used for a stationary adhe-
rence. ADELUNG considers the German an to be con-
nected with the French én, the Latin an in composi-
tion, and the Greek ávā, as āvā uéaw, in the middle ;
àvá X66va, on the earth. It is plain that our on,
though in modern use most frequently applied to that
which is higher in place, did not, in its origin, neces-
sarily imply such a position ; for though it was added
to up, in the word upon, it was also added to nether
or neder in the word on-meder, under. It is difficult to
assign with certainty any substantival form of this
word. It has, however, been observed, that both in
the Breton and Turkish languages ana signifies mo-
ther ; and from this circumstance, the learned PEzRoN
derives Diana, the mother of light, from di, day, or
light, and ana, mother. The scriptural word “abba,”
father, is well known; and perhaps from abba and
ana, some etymologists may be inclined to derive our
prepositions off and on.
We proceed to the word upon just noticed, and
with this are connected above and over. The radix up
implying superior elevation is most commonly em-
ployed, in modern English, as an adverb. As a pre-
position, we now use it, only to denote that an action
is directed from a lower to a higher part, as ,
In going up a hill the knees will be most weary. BAcon.
But by old English writers it was used (as we now
use upon,) to signify the being actually placed above
and resting on an object.
Gyfre he rood all be hynde
Up Blaunchard whyt as flour.
MS. Calig. A. ii. fol. 36.
A wel vayre compayneye,
Pppe vayre wyte stedes, & in vayre armure also.
R. GLou CESTER.
And in Rober T DE BRUNNE we find “ up that” used
for “ upon that,” thereupon, upon that account.
Op, the corresponding word in the Dutch language,
is used in the same manner; e. gr. “op een paerd
ryden,” to ride on horseback. So “op den tafel,” is
“ upon the table;” “ op de vloer,” on the floor. And
in the sense of completion, the Dutch op and our up
also agree ; as opeeten, to ert up ; opdrinken, to drink
tºp; opbouwen, to build up ; opgeschikt, drest up.
Mr. Tooke, in his usual manner, raises a dispute
about that, which properly understood, can admit of
no dispute at all; namely, whether up was originally
an adjective, a substant ve, or a verb. “ Upon, up,
over, above,” he justly says, “ have all one common
origin;” and he is clearly right in connecting them
with the Anglo-Saxon ufan, high. He adds not an
irrational conjecture, that uſa, or up may have anciently
meant the same as top, or head; but when he goes on
WOL. I.
to infer from this, and other conjectures of a like kind, , Chap. I,
“ that the names of all abstract relations (as it is
called,) are taken either from the adjectived common
names of objects, or from the participles of common
verbs,” he either means to advance an historical fact,
or to lay down a necessary principle in the constitution
of the human mind. If he means to speak histori-
cally, he asserts what it is utterly impossible either
to prove or disprove: if he means to speak philoso-
phically, his philosophy is destitute of common sense.
We need only examine our own minds with a very
slight degree of attention, to be satisfied that our
conceptions of quality, positive or relative, are just as
essential to human reason, as our conceptions of sub-
stance or of action. “ The relations of place,” says
Tooke, “are more commonly from the names of some
parts of our body; such as head, toe, breast, side, back,
womb, skin, &c.” It would have been equally correct,
or rather equally incorrect, in a philosophical point of
view, to have said, “ the names of various parts of
our body ; as head, toe, breast, side, back, womb, skin,
&c. are from the relations of place.” As matter of
history, both assertions are equally arbitrary. Mr.
Tooke is very positive that the etymologists who de-
rive head from the Scythian ha, German hoch, Dutch
hoog, Alamannic houch, Gothic hauh, and Anglo-
Saxon heah, high, are all wrong; and that it is the
participle of the Anglo-Saxon verb, heafan, to heave.
The fact, no doubt, is, that the same conception, and
the same radical expression, was the origin of them all,
as well as of the Islandic had, and German höhe,
height ; the Anglo-Saxon heafod, Gothic haubith,
Alamannic haubit, Islandic hoffud, Dutch hoofd, the
head; the Anglo-Saxon heafon, heaven; the Alaman-
nic hebig, and Anglo-Saxon ha-fig, heavy, difficult to
heave ; the Alamannic erhafan, to ferment, to raise
dough ; the Anglo-Saxon, haf, fermenting; the An-
glo-Saxon heap, Alamannic houph, Dutch hoop, a
heap; and numerous other cognate words in many
languages. - -
As the Anglo-Saxon uſa is our up, the Dutch op,
and the German auf; so the Anglo-Saxon ufera, the
comparative of wfa, and ofºr, the preposition, are our
wpper and over, the Dutch opper and over, the German
iiber, Alamannic wbar, Frankish upar, Gothic uſar, &c.
The Anglo-Saxons also used uppan or uppon ; and
as they had binnon for be-innon, and baftan for be-aftan,
so they had bufan for be-ufan ; as “ bufan tham
watere," upon the water. This bufan is, no doubt,
the origin of the Dutch boven, above : and our word
above, written in old Scottish abufe, is on-be-ufa; as
the Scottish abune, is on-be-ufan. In Danish, we find
over, ober, over, overste ; in Swedish, uppe, up, Öfwer,
ôfwerste, ofte, ypperst. WACHTER considers iber to be
connected with the Hebrew eber, Persian avar, Greek
iTep, and Latin super; and he traces its significations
from that which is above, in place, to above, in power ;
above, in eminence; above, in the sense of prevailing
over; above, in excellence ; over and above, in abun-
dance; over, in excess: and, again, from that which
is beyond in place, to that which is beyond in quantity;
hence, to overlook, is to look beyond, and therefore
not to notice; while, on the other hand, to look over,
is to examine carefully, by looking from point to point.
After noticing these and many other meanings of this
word, he concludes—“ Uber plures habet significatus
quorum racemationem aliis relinquo, qui hisce inves-
X
150 G R A M M A. R.
Grammar, tigandis et in ordinem digerendis ad taedium usque
\-N-' defatigatus sum."
The adjectival use of over and upperest is common
to signify the power of taking up, or readily compre- Chap. I.
hending any notion; as in the phrase “dull i'th Q-2-
in CHAUCER.
Her ouer lyp wyped she so clene,
That in her cup was no ferthynge sene. -
Prol. Cant. Tales.
By whiche degres men myght climben from the neytherest letter
to the upperest. Boecius, book i.
So, in Kyng Alisaunder,
Theose seresys as y fynde,
Uppwrest folk buth of ynde.
The adverbial use of over, answering to our adverb
too, is curiously marked in a passage of ALEXANDER
Monroomſ ERY's Cherrie and Slae.
All owres ar repute to be vyce;
Owre hich, owre law, owre rasch, owre myce
Owre het, or yit owre cauld.
Up, in the sense of completion, occurs in our word
upshot.
I cannot pursue this business with any safety to the upshot.
* SHAKSPEARE.
Johnsox derives upshot from up and shot; it would
be more proper to derive it from up and shut ; the
shutting up of a business being formerly used for its
close.
Altho’ he was patiently heard, as he delivered his embassage,
yet in the shutting up of all, he received no more but an insolent
ałłSW6. T. KNOLLES.
The Dutch boven corresponds exactly with the old
English aboven, which occurs in the ballad of the
Battle of Lewes.
By God that is aboven ous, he dude muche symme.
That lette passen oversee the Erl of Warynne.
Among the combinations of up and over, we may
notice over that, over against, out owre, and uptak.
Over that was formerly used, as we now use more-
over to signify, “ in addition to.—
That the same fyne be openly and solemply rad proclaymed in
the same court; and over that, a transcript of the same fyne be
sent by the seid justices unto the justices of assize.
Stat. 1. RIC. III. c. 7. MS.
In the same sense, the Anglo-Saxon writers use
“ofer that,” and the Germans iber das. Our com-
pound preposition, over against, is transposed, in the
German gegeniber.
Over against this church stands a large hospital, erected by a
Shoemaker. ADDISON, on Italy.
This is rendered in the Anglo-Saxon gospel foran-
ongean. “ Tha reowon hiy to Gerasenorum rice, that
is foran-ongean Galileam ;”—“And they arrived at
the country of the Gadarenes, which is over against
Galilee.” Luke, c. viii. v. 26.
In the Scottish dialect, we find the compound pre-
position out-owre, which is used in two senses by
BURNs.
The rising moon began to glow'r,
The distant Cumnock hills out-ow'r,
Death and Doctor Hornbook.
He by his shouther ga’e a keek,
An' tumbled, wi' a wintle,
Out-owre, that night,
The word “uptak” is also used colloquially in Scotland,
uptak,” which signifies slow in comprehending an *
idea ; the mental faculty being in this instance, as in
so many others, expressed by reference to a bodily
action.
Our preposition, at, is the Gothic at, and Anglo-At.
Saxon at. It may probably have been connected with
the Latin ad, which Tooke awkwardly derives from
actum. Ad was more probably the root of the verb
addo; though it may not now be easy to trace it in
a substantival form. For ad we sometimes meet with
ar, as, and at. FULVIUs URSINUs quotes, from the
Laws of the Twelve Tables, arvorsom, for adversum ; as
“ arvorsom hostem aeterna auctoritas estod;” that is,
“ against an alien, the right of property is never barred
by prescription ;” whereas, against a Roman it was
so barred. VELIUS LoNGUs says, that the old Romans
not only used arvorsum for adversum, but asvorsarius
for adversarius ; and Voss IU's observes, that in many
ancient books and inscriptions, ad is written at.
The use of the preposition at, in Anglo-Saxon and
old English, was much more loose and comprehen-
sive than in our modern dialect. We find it used
where we should now use to, from, about, of, by, in
with, &c. -
In the romance of the Seuyn Sages, “ at lere" occurs
for “ to learn,” or to be taught. *
The sext maister rase vp onane.
Sir, he said, if thi will were,
Tak thi son to me, at lere.
In the Anglo-Saxon, “ et him” is used for “from
him ;” e. gr. “ animath that pund aet him,” take the
talent from him.
Bishop LATIMER uses at for about, in the following
passage, -
What ado was there made in London at a certain man, because
he said, Burgesses I may, butterflies
CHAUCER uses the phrase “ to take leave at,” for of
She toke her leaue at hem ful thriftely.
This line is very similar to one in the romance of
Octouian Imperator.
At all the cytë she tok her leue.
So, in the Lyfe of Ipomydon,
He toke hys leve at Jason there,
And went forthe ellys where.
In Richard Coer de Lion, we have “to ask at.”
He askyd at all the route,
Gyff ony durste com, and prove
A cours, for hys lemannes love.
“To ask at a person,” is considered, in the present
day, as a Scoticism.
Similar to this is Bishop LATIMER's phrase, “to
learn at.”—
He must study, and he must pray: and how shall he do both
these ? He maye learn at Salomon.
RoberT DE BRUNNE uses at for by.
3 tº
Sen thou has don amisse, at thin vnconyng,
We may not faile at this, to help the in alle thing.
At is also used for by in an old document of the year
1415. (9 Rymer, 301.) -
Besechyng yow, at the reverence of God.
In the romance of The Lyfe of Ipomydon, at is used
for in. -
G R A M M A R.
151
Grammar.
\-y-'
To, too,
.venit ;”
He wold wend into strange contre,
So that ye take it not at greffe.
Robert DE BRUNNE also uses at, where we should
now use with ; as in the form of Baliol's homage to
King Edward.— n -
I Jon Baliol, the Scottis kyng,
I bicom thi man for Scotlond thing;
The whilk I hold, & salle thorgh right,
Clayme to hold, at all my myght.
This lax mode of using the preposition at is observ-
able in our phrase “at all,” which Johnson explains “in
any manner, in any degree ;” and which corresponds
to the Scottish ava ; i.e. af all, or of all.
An' lows'd his ill-tongu'd, wicked scawl,
Was warst ava’. BURNS.
At is sometimes, though awkwardly, cumulated
with other prepositions, as “ at about six o'clock;”
and so in a statute of the year 1495, “ at after none.”
Divers artificers and laborers reteyned to werke, and serve
waste moch part of the day, and deserve not their wagis; sume
tyme in late comyng unto their werke, erly departing therfro, longe
sitting at ther brekfast, at ther dyner and nonemete, and long tyme
of sleping at after none. Stat. 2. HEN. Vil. c. xxii. MS.
Thus also in BARBour we find the expression “ at to
$ 2.
II]. OFI).
That they may this night, if they will
Gang harbry them, and sleep, and rest;
And at to morn, but longer lest
You shall ish forth to the battail.
The origin of the word to, like that of the word at,
Čan at best be but matter of conjecture. It may,
however, be reasonably conjectured, that at and to are
from the same root, “ per anastrophen,” as WAcHTER
expresses it ; that is to say, that the vowel was
sounded before the radical consonant in the one
instance, and after it in the other. The primary con-
ception, common to both words, seems to have been
that of touch, either in consequence of moving the
bodily organs to, or in consequence of their being at
a specified place. Hence, the Latin ad coincides with
both our to and at ; e. gr. “Verres ad Messanam
Verres came to Messina. “Mihi quoque
etiam est ad portum negotium ;” I too have business
also at the harbour.
la maison,”—“il est allé à la campagne.” In the
Devonshire dialect, to is used for at; as “ he lives to
Exmouth :” and we have seen above, that “ at lere"
was used for “ to learn," ad discendum.
Mr. Tooke says, “ the preposition to, in Dutch,
written toe and tot, a little nearer to the original, is
the Gothic substantive, taui or tauhts, i. e. act, effect,
result, consummation ; which Gothic substantive is in-
deed itself no other than the past participle, tauid, or
tauids, of the verb tauyan, agere. In the Teutonic
this verb is written tuan or tuon ; whence the modern
German thun, and its preposition varying like its
verb, tu. In the Anglo-Saxon, the verb is teogan, and
preposition to.” -
In all this, we see nothing of the “real object”
which, according to Mr. Tooke's general theory, every
preposition should signify; and it is a very circuitous
mode of getting at a short monosyllabic preposition,
to suppose that there first existed a dissyllabic verb,
from which was formed a dissyllabic participle, and
that this participle, a little differently articulated, be-
And so in French, “il reste à
came a dissyllabic substantive, which was shortened, Chap. I.
we know not how, or wherefore, into the monosyl-
lable in question. -
The German zu (not tu, as Mr. Tooke supposes,)
answers, like the Latin ad, to our to and at ; e. gr.
“komm zu mir,” come to me; “ zu Windsor,” at
Windsor. WACHTER mentions, as connected with it,
the Gothic at and du ; Anglo-Saxon, at and to ; Frank-
ish and Alamannic az, zuo, zwa, za, ze, zi; Dutch, toe and
tot ; English, to and at. “Omnia," adds this learned
author, “affinia Latino ad ; nam ad et to se mutuo
producunt per anastrophen.”
The various uses of the German zu, the English, to,
too, and at ; the French d, the Latin ad, &c. will illus-
trate each other: and we may consider them as indi-
cating approach to, or arrival at, a place, time, or cir-
cumstance; and thence, as having an objective force
before a verb or substantive ; moreover, since that to
which a person or thing has attained, or which has
come to it, is an addition to it, therefore to denotes ad-
dition ; and thence excess; and thence, in composition,
it has an intensive force; and, lastly, where the inten-
sive force is very slight, the use of to seems almost
superfluous.
In relation to place, we find zu used emphatically in
German, “ die thur ist zu,” exactly corresponding
with our English colloquial phrase—the door is shut
to. So zugang is the Latin aditus, from ad and ire.
Zukunft is the Latin adventus, from ad and venire, the
coming of Christ to the world. The Frankish zuo-
chumft, from zuo and chommen, is the Latin aggressio,
from ad and gradior. In English the preposition to is
not commonly prefixed, as in German to verbs in
composition, but follows them, as “ to fall to,” “ to
bring a ship to ;” and, so in the interjectional phrase,
“Go to "' The Germans say “reit zu,” for, ride on;
“ geh zu,” go on, &c.
In relation to time, we find the German zu macht,
answering to the English “ at night.” Zukunft, in a
secondary sense, signifies the time to come, the French
l'évenir.
In relation to circumstance, the German zufall is
the Latin accidens, from ad and cado, whatever befalls,
or falls to a person ; zubringen, to bring to an end ;
zusagen, to promise to a person. Zu pferde is the
French a cheval, on horseback.
The objective force of zu, before a verb, is well
explained by Dr. Noe HDEN, in his excellent Grammar
of the German Language, (3d edit. p. 388, et seq.)
whence it appears that the action may be either future,
as “ lust zu Spielen,” an inclination to play; or pre-
sent, as “Das vergnügen sie zu sellen,” answering to
the French “j'ai grand plaisir à vous voir,” I have
great pleasure in seeing you; or past, as “mide zu
stehen,” tired of standing.
The English to had formerly a similar objective
force before a substantive; but this construction is
now obsolete. -
They have gruel to potage,
And lekes kynde to companage. TREVISA,
Thothai were fiften winter old,
He dubbed bothe tho bernes bold,
To knightes, in that tide.
Amis and Amilown.
The English too, also, denoting addition, is the
same word as to; and in the Anglo-Saxon and old
English, is written to.
x 2
152
G R A M M A. R.
Grammar.
The arriving to such a disposition of mind, as shall make a man
take pleasure in other men's sins, is evident from the text, and from
experience too. SOUTH. .
The German zu in composition, possesses this same
force ; e. gr. zuname, a name in addition to another
name, the Latin agnomen, from ad and nomen. *
The word toname occurs in Robert DE BRUNNE,
with the same meaning in speaking of Statin, whose
nose had been cut off by King Isaac Comnenus.
For Isaac did him schame, his lord suld be,
Thei called liim this toname, Satin the Nasee.
The German word zugemus (in Frankish, zuomuse,)
signifies, in like manner, vegetables, or garnish of any
kind added to the meat; from the German mus, Frank-
ish muos, Alamannic muas, Gothic mats, French mets,
Anglo-Saxon mete, and English meat. So the Ger-
man verb zugeben is to give something in addition to
the stipulated price.
The secondary sense of our too is excess, as “ too
great ;” that is something added to the proper degree
of greatness ; and in this sense, zu is used in the
German compounds zuhoch, too high, or overhigh ;
zulang, too long, or overlong ; zuwarm, too warm, or
OVēI'WäI’IYl. -
Zw is used with an intensive force in such words as
zubereiten, to make quite ready ; zulassen, to grant to,
&c.; and this may probably have some analogy to
the Greek Ça, which has an intensive force ; as in
&#7Aeros, very rich; {diffeos, very divine; £dkotos,
very furious, &c. The old English to before a verb
or participle, appears to have had nearly this force.
He schal therfore ben islawe,
And afterward al to-drawe. Seuyn Sages.
Th’ emperour saide, I fond hire to rent,
Hire her and hire face isllent. Ibid.
So, in the translation of the Bible, in the time of
Ring HENRY VIII. “ confregit cerebrum ejus,” is
rendered “ all to brake hys brayne panne.” (Judges
c. ix. v. 53.) In the modern editions this is impro-
perly printed “to break.” -
The intensive force of zu is scarcely, if at all per-
ceivable in such words, as zuvor, before; zuwider,
against; zusamen, along with. In English we still
use to, thus in together, and in heretofore, as we formerly
did in tofore and toforme. -
There entered into the place, there I was lodged, a ladie, the
moste semelich & moste goodly to my sight, that euer toforne
appeared to any creature. CHAUCER. Test. of Love.
Tofore the kyng com an harpour,
And made a lay of gret favour.
- Ryng Alisaunder.
To appears to be superfluously used by BARB.our in
the preposition into.
That he would travel owne the sea
And a while into Paris be.
On the other hand, in the preposition unto, the syl-
lable un, which seems to have been originally on,
augments the force of to, and gives it the force of the
Latin usque, ad, and French jusqu'à. -
We have seen that for is unnecessarily prefixed to
to before an infinitive; as “for to go,” which is now
reckoned a vulgarism. “ From to” seems still more
alien to the general idiom of our language; yet it
occurs in poetry.—
For not to have been dipp'd in Lethe's lake,
Could keep the son of Thetis from to die.
And there is something analogous to this in the Ger-
man ohne, zu; e. gr. “ ohne zu wissen,” which we con-
strue, with the participle, “ without knowing,” and
the French, with the infinitive, sans savoir.
As the origin of the word to is matter of con-
jecture, of course we could only indicate conjec-
turally those words with which it may very anciently
have been connected in sound and signification : and
among these, it may be sufficient to notice the
numeral two. This is in Gothic two or twa, in Anglo-
Saxon tu or twa, in Greek évu, in Latin duo, in Welsh
dan, dwy, in Breton dou, in Tartarian tua, in Danish
tu, in Frankish and Alamannic zwei, zwo, in German
zwey, in Dutch twee, in Scottish twae and twa. STA-
DENIUS endeavours to show that it is a word corn-
pounded of the Gothic du, to, and a, or o, one ; so
that it properly signifies “one added to one.”
Chap. I.
Till is used prepositionally and conjunctionally ; Till.
but always, in modern English, with reference to time
alone; e. gr.
Unhappy till the last, the kind, releasing knell.
CowLEY.
Meditate so long, till you make some act of prayer to God, or
glorification of him. J. Taylor.
Dr. Jofi Nson is mistaken in explaining the latter of
these examples, as not signifying “ to the time that,”
but “to the degree that ;” for it palpably refers to
the continuance of the meditation, which must occupy
time. Mr. TookE, on the other hand, is right in say-
ing (with reference to modern usage,) that “we
apply to indifferently either to place or time; but till
to time only and never to place. Thus we may say,
From morn to night th' eternal larum rang ;
Or, from morn till night, &c.
But we cannot say, “From Turkey till England.”
He is, however, entirely mistaken in supposing “ that
till is a word compounded of to and while, i. e. time;”
and that “ the coalescence of these two words to-
hwile took place in the language long before the pre-
sent wanton and superfluous use of the article the,
which by the prevailing custom of modern speech is
now interposed.” For, on the contrary, the custom
of confining the signification of the preposition and
conjunction till to time, is comparatively of very
modern date, and is confined solely to the English
dialect. t •
“ Til,” says HIcKEs, “ is a Cimbric word, signify-
ing ad, usque, and it often occurs (in Anglo-Saxon,) as
“ yearwian tiletanne,” to make ready to eat ; “ cwacth
til him Haelend,” the Saviour said to him. “ Til, in the
old Norwegian and modern Islandic languages, governed
the genitive.” So we find in the Islandic History of
Hialmar, “til borgarinar," ad propugnaculum.
In a marginal note to the letter of Henry III. (A. D.
1258,) we find this word written tel.
And al on tho ilche worden is isend in to aurihce othre slicire,
ouer althare kuneriche on Engleneloand, & ek in tel Irelonde.
RoBERT DE BRUNNE writes it tille.—
A knyght was than among, Sir Richard Seward,
Tille our faith was he long, & with Kyng Edward,
Tille our men he com tite. - -
CHAUCER uses til—
A doly season til a carefull dite
Should correspond. Test of Creseide.
G R A M M A. R. 153
terminum ad quem, vocant; hinc manifestum, sensum vocis a Chap. I.
termino terminante ad terminum intentionalem translatum ‘esse.” ~'
The preposition from is the word fro which we still From.
Grammar. GAwiN Douglas, tyll.
\-y-'
Ane young bullock of cullour quhite as snaw
With hede equale tyll his moder on hicht.
In Octouian Imperator it is written tylle.
Her pauyloun whan they com tylle,
Ther that sche was,
Her maydenys gonne to crye schylle,
Treson, alas !
In like manner we have until, and thertil, for unto
and thereto.
Then strake the dagger untill his heart.
L. Z'homas and Fair Annet.
His owin lady he toke by liue,
And gaf the knyght until his witue.
- Seuyn Sages.
Wntil his toure thus wendes he right,
For to speke with his lady bright. Ibid.
They found the gates shut them untyll.
Adam Bell, &c.
(That is, shut against them.)
I bicom thi man for Scotland thing
With alle the purtenance thertille.
Rob. DE BRUNNE.
Thoffe thei haue not als tyte her wyll,
Yette shall they cura sumtyme thertyll.
Sir Amadas.
And the knight and his lady
Went tham forth with grete solas
To the ship whare his godes in was.
The Erl went with tham thartill.
The word while is used in several of our provin-
cial dialects, and by many old writers for till.
Thus in the Scottish Statute of 1430.
It is statute and ordanit, that the act of the fisching of Sal-
monde, maid be the King that now is and the thre estatis, be
fermly kepit ay furth quhil it be reuokit be the King and the thre
estatis.
So in an historian quoted by Mr. Tooke, vol. i.
p. 363.
“He commaunded her to be bounden to a wylde horse tayle, by
the here of her hedde, and so to be drawen whyle she were
dede.” -
In like manner the word to is used for till in the
romance of Sir Amadas.
And owtte of cuntre wille y wende,
To y haue gold and syluer to spende.
But we have never met with the compound to-
hwile, or to-while in any English or Saxon writer.
The German zuweilen, which is the only compound
resembling it, signifies “ sometimes,” “ now and then,”
and nearly answers to the Scottish adverb whiles, as
in BURNs's inimitable description of Tam O'Shanter.
Whiles haddin' fast his gude blue bonnet,
Whiles crooning o'er some auld Scots sonnet,
Whiles glow’ring round, wi' prudent cares,
Lest bogles catch him unawares.
Seuyn Sages.
SKINNER says that til was used, in his time, in Lin-
colnshire for to : GRose includes it in his Provincial
Glossary as signifying to in the north of England: and
it is to this day very generally so used in the country
parts of Scotland.
Gae farer up the burn til Habbie's how.
ALLAN RAMSAY.
The substantival form of til is to be found in the
word Zil, which WACHTER thus explains :
1. “ Finis, limes, terminus temporis et loci. Anglosax: tell
apud BENson. Graecis réAos, a rexelv finire, terminare ‘’
2. “ Meta jaculantis, scopus agentis, terminus oculi et mentis.
Cum scopus sit terminus agentis, quem Latini finem, et Scholas
use adverbially.
As when a heap of gathered thorns is cast
Now to, now fro, before th' autumnal blast.
In the Anglo-Saxon, and old Scottish dialect it is
often written fra : and there can be little doubt but
that fro and fra are, in fact, the same word with the
English adjective free, the Gothic friya, Anglo-
Saxon friy, freo, Frankish frio, German frey, Swedish
fry, and Dutch vry, all of the same meaning. These
words too, were no doubt, connected with the Ger-
man freude joy, and froh joyful, free from care, which
last is in Frankish fro, and in Dutch wro; and also
with the German fremde, and Anglo-Saxon fremd, a
stranger, one who dwells far from us.
“ From,” says Mr. Tooke, “ means merely begin-
ning, and nothing else. It is simply the Anglo-Saxon
and Gothic noun frum, beginning, origin, source,
foundation, author.” But beginning is not “a real
object,” and, therefore, this etymology, if it prove any
thing, proves that Mr. Tooke's theory of prepositions
is false. The word frum, was no doubt the same as
from, and may have been used to signify that from
which any thing proceeded; but this was probably with
reference to a still more general conception involved
in all the terms that we have above mentioned.
In the Gothic Gospel of St. John, (c. xv. v. 27,)
we have fram and frum in immediate connection.
“ Fram fruma mith missiyuth.” From the beginning
ye are with me.
The Anglo-Saxons used both fram and fra.
The old Scottish writers use fra, and frequently in
the sense of “ from the time of.” Thus GAwiN
Doug LAs, (b. ii. p. 63,) “fra she was loist,” i.e. from
the time that she was lost.
So BARBou R,
And fra he wist what charge they had
He busked him, but mare abad.
Pope.
RoBERT DE BRUNNE uses fro.
Andrew is wroth, the wax him loth, for ther pride
He is than fro, now salle thei go, Schame to betide.
Mr. TookE says, that the preposition through is the Through.
name of a real object, namely, door. This notion he
probably took from the following passage in VERstE-
GAN's Restitution of Decayed Intelligence. “ Dure or
durh, now a door; it is as much to say as through ;
and not improper; because it is a durh-fare or thorow
passage.” Verstegan certainly reasons more cor-
rectly in deriving door from through, than Tooke in
deriving through from door; the more general idea
must have preceded the more particular; men must
have passed through many places before doors were
invented. Nevertheless there may have been a con-
nection between the words through and door, as there
probably was between the words per and porta.
Through is the Gothic thairh; the Anglo-Saxon
thurh ; the old English thurg, thourh, thorgh, thorth,
thorou, &c.; the Alamannic duruh, durich, dhurah ;
the Frankish thuruh, thuruhe, thurah, durh, the Ger-
man durch, the Dutch door, &c.
The following are old English and Scottish ex-
amples :
Henr; thurg Godes Fultume King on Euglene loande.
r Letter Hen. III. I.258.
154
G R A M M A R,
Grammar.
For alle this thraldam, that now on Inglond es,
Throgh Normanz it cam, bondage and destres.
ROBERT DE BRUNNE.
The appel, where thorou al the wold was forlore.”
MS. Homily, temp. RICHARD II.
Sixtene hundred of horsemen hede ther her fyn
Thourh huere oune prude. -
Ballad on Battle of Bruges.
The lady rod thorth Cardeule. Syr. Launfal.
In like manner are the compounds, therthrough,
quhairthroughe, thorghout, and out through.
But whatsoe'er made the debate
Therthrough he died, well I wat. BARBou R.
Sik as has sufficiently of thar awin, quhar throughe thai mai be
punyst gif thai trespass. Scot. Stat. A. D. 1424.
IDiuers ar yit absent, quhairthrow large tyme is spent and
mathing as yit done. Scot. Stat. A. D. 1567.
The kyng thorghout the lond, he did crie his pes,
And with the law than bond, als skille wild he ches,
ROBERT DE BRUNNE.
— an aizle brunt
Her braw new worset apron
Out through, that night.
BURNS, Hallow E'en.
It is probable that one of the most ancient substan-
tival forms of the word through is to be found in the
Anglo-Saxon throt, or English throat.
ADELUNG considers that durch, &c. are connected
with the Greek repew, Latin tero, and Swedish taera,
After,
mination thro in the Gothic uzathro,
to pierce through.
To these we may add the Anglo-Saxon thirliam,
which is our verb to drill a hole, whence naºsethyrl,
was the nostril. -
As that which has been gone through with, or which
is thoroughly effected, is complete, so duruh, durch,
door, &c. in composition signify completeness, or ex-
cellence; as in the Frankish duruhtuan “ to accomp-
lish, " or do thoroughly ; the German durchlawchtig,
and Dutch doorluchtig., “ most illustrious,” or
thoroughly illustrious. -
In this sense we may explain the force of the ter-
extra, com-
pletely, or thoroughly out of. And perhaps to this
source is to be traced the Latin tra, in the preposi-
tions intra, extra, ultra, citra, &c.
Mr. TookE is undoubtedly right in saying that this
word is merely the comparative of aft; and he has
acted with more prudence than usual, in not pretend-
ing to specify any particular object of which aft was
originally the name. It may probably have been a
term applied to the back ; and, as we have before
suggested, the radic of aft, may have been af; but
these are all mere conjectures. It is certain, however,
that our English words aft and after are related to the
Gothic aftara, Anglo-Saxon after, Danish and Swedish
efter, Dutch and Swedish achter, all of the same signi-
fication. In German after is not found in its sepa-
rate state, but enters into many compounds, all with
analogous significations, e. gr. afterdarm, the intes-
tinum rectum; after-geburt, the after-birth ; afterkind,
a posthumous child, &c. What we express by “fore
and aft,” the Danes express by “for og bag ;” and
the Danish bag is no doubt our word back. They
have also bagdeel, the breech, the stern of a ship; and
tilbage, behind, analogous in construction to our old
word to fore. -
The nautical expression abaft is from the Anglo- Chap. I.
Saxon be-aftan, or baftan, as “ gang baftan me Sa- \-~~/
tanas.”—Get thee behind me Satan.
After is poetically used as an adjective in the beau-
tiful ballad of Gil Morice.
To me nae after days, mornichts
Will eir be saft or kind.
It is probable that the Greek ávtåp may have been
the Gothie after, with little, if any change in the pro-
nunciation. Indeed a modern Greek would pronounce
avtap, avtar.
From the signification of that which is behind, in
place, naturally follows the signification of that which
is subsequent in time, as “ the afternoon.” Hence
our modern adverb qfterwards, and the obsolete adverb
eftsoons, signifying shortly afterwards. In this sense
of aft it may have given rise to the Greek áv6ts.
As the effect comes after the cause, in order,
and the copy ºfter the model, we have the expres-
sions “ after our unrighteousness,” “ after Rem-
brandt,” &c. which are expressed according to a
similar analogy in Latin, by the word secundum. In
this manner the Franks used the word after, as “ qfter
kewrahti,” after what we have wrought. A singular
instance of this use of the word after occurs in Kyng
Alisaunder, where the poet is describing certain
‘‘ bestes ferlich,” called “ Deutyrauns—”
More hyben than Olyfaunz;
Blake heueded after a palfray;
Ac in the forehede, parmafay,
Hy have thre hornes.
Having thus examined at length the chief English Obsolete
prepositions now in use, it may not be necessary to andforeign.
consider so minutely the obsolete prepositions of our
own language, or those which are only to be found
in other languages or dialects. Some of these, how-
ever, we will briefly notice.
Mid, used in Anglo-Saxon and old English for with,
is the Gothic mith, Frankish, Alamannic, and German
mit, Dutch met, Danish mód, and probably the Greek
pºetà. It is evidently connected with the verb meet.
Emb, of which we retain a trace in the modern
word embassy, was an Anglo-Saxon preposition signi-
fying about. It seems to have had some analogy to
the Anglo-Saxon substantive wanb, the belly; in the
Scottish dialect wane : and was no doubt connected
with the German um, and the old Latin am. “ The
particle um,” says Dr. NoFHDEN, “is frequently joined
with zu, which expresses the design still more dis-
tinctly. Iiebet die Tugend um glücklich zu Seyn,
love virtue (in order) to be happy.” Festus says,
“Am praepositio loquelaris significat circum :” and
R. STEPHANUs says “ Verisimile est Latinos ambi
suum, unde contractè am, Graecorum au%t, debere.”
The old English whilom seems to be compounded of
while and om, or em. -
The Scottish participles anent and forement are of
doubtful origin ; they may probably be derived from
ent, for end. Rob ERT De BRUNNE uses ent for ended.
Be that the werre was ent, wynter was ther yare.
To Dounſermelyn he went, for rest wild he thare,
The German ohne, without, seems to have some
affinity with our negative prefix un, where that particle
is derived (as it seems to be in some instances) from
wan or want. We have in Burns's poems, wanchancie,
G R A M M A. R.
155
Grammar, wanzestfu,' &c.
S-N-" to ohne is ano, as “ano zwifal,” without doubt.
The Frankish preposition answering
In
the Swabian dialect this is aun ; in Alamannic anoh,
which nearly approaches the Greek avev.
The German preposition wegen, concerning, touch-
ing, &c. is evidently from weg motus, which is our
verb wag, and substantive way.
. The German preposition sonder, and Dutch zonder,
without, or separated from, are doubtless connected
with our words sundry and asunder, and these perhaps
with sand.
The French preposition chez is correctly referred by
Mr. Tooke to the Italian casa, so that “ chez moi" is
literally “ house me," i. e. at my house. J
. The Dutch preposition van, of, or from, is retained
in English as a substantive ; but it does not, as Mr.
TooKE seems to suppose, indicate a real object, but the
relation which that object bears to some other ; for
when we speak of the van of an army, we do not
mean merely to indicate a certain number of soldiers,
but to signify that those soldiers are placed in a certain
relation to the rest of their comrades.
Thus have we considered two of the three methods
by which the relation of a substantive to a verb or to
another substantive, may be expressed in language.
The remaining mode of expressing such relation is by
those changes or inflections of the word itself which
are called cases. Of these we have considered the
general use in treating of nouns and their incidents.
The particular means employed to form such inflec-
tions will be most conveniently considered when we
come to treat of the particles which enter into the
composition of the great majority of words.
III. Having stated first the necessary complexity
of every sentence in which a preposition is employed,
and secondly the origin and use of many known prepo-
sitions, in expressing the relations of substantives, we
have only, in the third place, to subjoin a few remarks
on the relations ordinarily so expressed.
Now relation, which is the fourth of the logical pre-
dicaments, supposes three things, the subject, or thing
related, the object or correlative, and the relation
itself, or circumstance existing in the subject by
means of which it is related to the object, and which
logicians call the foundation. When we say “John
is before Peter,” “John” is the subject, “ Peter” is
the correlative, and ‘‘ before” is the foundation, or,
as we have been accustomed to speak, the conception
of relation, expressed prepositionally.
It is manifest, that the circumstance, whatever it be,
that forms the foundation of a logical relation, or
(which is the same thing) that constitutes (when ex-
pressed in language together with its subject and
object) a preposition, may either be common to the
two terms (as they are called) of the relation, or it may
belong to one of them exclusively. If I say “John
is with Peter,” the relation expressed by the preposi-
tion with belongs equally to Peter and to John; but if
I say John is before Peter, the relation expressed by
the preposition before belongs exclusively to John. In
the first case it is perfectly indifferent whether I say
“John is with Peter,” or “ Peter is with John ;” it is
perfectly indifferent which I make the subject and
which the object of the relation ; but in the other
case, if I were to say “Peter is before John,” I should
not only vary the assertion, but I should directly con-
tradict it. - *
Still the foundation of the relation would Le the
same : and we may illustrate this with the trivial Chap. I.
comparison of two children playing at see-saw. If
John and Peter be equally balanced at the opposite
ends of a plank, John is level with Peter, and Peter is
level with John, and the plank is the measure or
standard of the level; but if John be lighter than
Peter, John at once rises above Peter, and Peter sinks
below John, and the same plank measures the elevation
of one and the depression of the other. What the
supposed plank is to the boys, the preposition is to
the substantives related ; and hence we may easily ex-
plain not only certain diversities in the idioms of
different languages, but some apparent contradictions
in the same idiom. Thus Mr. TookE makes the fol-
lowing just observation on the Dutch preposition van :
“The Dutch,” says he, “ are supposed to use VAN
in two meanings, because it supplies indifferently the
places both of our of and from. Notwithstanding
which, VAN has always one and the same single mean-
ing. And its use, both for of and from, is to be ex-
plained by its different apposition. When it supplies
the place of from, VAN is put in apposition to the same
term to which from is put in apposition. But when it
Supplies the place of of, it is not put in apposition to
the same term to which of is put in apposition, but to
its correlative.” The difference of idiom between the
Dutch and English languages might have been still
more strongly stated ; for “Van Amsterdam gekomen”
signifies “ come from Amsterdam;” whereas “ Pan
Amsterdam geboortig,” is “ born at Amsterdam :”
and our prepositions at and from are commonly used
in senses very opposite to each other.
But it is not only the different use of prepositions in
different languages, but the apparent contradictions
in the same language, which are thus to be explained.
The prepositions for and after are of directly contrary
origin and signification, being (as has been fully
shown) the same as the words fore and aft. Never-
theless we say, “to seek for that which is lost,” and
“ to seek after that which is lost.” The thing sought
is considered as before the mind of the seeker ; and
consequently the seeker is considered as after, or be-
hind the thing sought ; when, therefore, we use the
word before, we specify the relation of which the
thing sought is the subject ; but when we use the
word after, we specify a relation of which the subject
is the seeker: or to use Mr. Tooke's phraseology, we
put before in apposition with the thing sought ; and
after in apposition with the seeker.
From this statement it appears that the subject of the
relation specified may or may not be the logical subject
of the proposition enunciated in the sentence. In the
sentences, “ John seeks for Peter,” and “John seeks
after Peter,” John is the logical subject; but the
former sentence involves the expression of a relation
of which Peter is the subject ; the latter of one the
subject of which is John. The relation of foreness
exists in Peter ; the relation of afterness exists in
John.
How a particular preposition may be employed, in
this respect, is mere matter of idiom, and depends
solely on custom—
Quem penes arbitrium est, et jus, et norma loquendi.
But it will generally be found that the prepositions of
most frequent use are employed with the greatest
latitude, in the earlier stages of a language, and so
continue, until their equivocal signification gives rise
156
G R A M M A. R.
Grammar. to inconveniences which are only to be remedied by
\–V-' confining them to certain forms of construction.
less equivocalness than is found in instituted lan- Chap. I.
guages, suffice to express those various respects, \-y-
Various prepositions may sometimes be used indif-
ferently in a sentence ; and sometimes a particular
preposition is absolutely essential to the sense. This
circumstance depends on the nature of the relation
intended to be expressed. In general, the external
and physical relations of objects must be expressed by
their own proper and peculiar words. Thus we can-
not substitute in for out, or after for before, in speaking
of visible objects, and bodily actions ; but the case is
different when we come to speak of the mind ; for as
the analogy of its states and operations to those of the
material world are very loose and general, so we may
adopt almost any external relation of things as a sym-
bol whereby to explain mental relations. Thus we
may say that a person did a certain act in envy, or out
of envy, or through envy, or from envy, or for envy,
or with envy ; but we cannot say of the same man,
under the same circumstances, that he was in his house
and out of his house, passing through the town, and
distant from the town, walking with another person,
or a mile before him. Still there are limits, fixed by
custon, to the use of each. preposition ; but these
limits vary much in diſferent languages; and hence a
translation, correct in substance, often appears literally
inaccurate. Thus the French “sous peine,” answers
to our “ on pain,” and to the old English “ up peine.”
No more up peine of lesing of your hed. CHAUCER.
Custom also varies in the course of time, as we
have seen in many of the examples already cited, and
which have now become obsolete, as “ to learn at,”
“ to accuse for,” &c. But it must not always be
supposed that the force of a preposition is varied, be-
cause the application is different ; for the difference
may arise from the other words in the sentence; thus
the French 6ter ā and donner a, are our “take from,”
and “give to ;” but in both cases à retains its pri-
mary force, and the apparent opposition depends on
the contrariety between 6ter and donner.
To suppose that the prepositions necessary to any
language could be enumerated a priori would certainly
be absurd. TookE has ridiculed the grammarians who
have attempted to enumerate them, as matter of fact
and history. It has been said, that the Greeks had
eighteen prepositions, the Latins, forty-nine, and the
French, (according to different authors,) thirty-two,
forty-eight, and seventy-five. It is certainly a pos-
sible, but a very useless labour, to ascertain what
words have been used as prepositions in a dead lan-
guage. In a living language it is quite impracticable,
for every day may enhance their number, by new
combinations of thought and expression. A preposi-
tion is not like a piece of money stamped to pass for
a certain value, and which cannot change its denomi-
nation or value. It is a word to which a transient
function is assigned, and which, as soon as it has dis-
charged that office, becomes available again for its
former purposes, as a noun, verb, or other part of
speech.
But although it be not possible to enumerate pre-
positions, yet they may be subjected to a general clas-
sification, according to the great distinctions of rela-
tion in human conceptions. M. Cours DE GEBE LIN
has attempted something of this kind, and Bishop
WILKINs has also given an arrangement of thirty-six
prepositions, “ which,” he says, “ may, with much
which are to be signified by this kind of particle.” It
may be doubted whether either of these schemes be
sufficiently comprehensive, or perfectly philosophical.
Prepositions must be classed, if at all, by their signi-
cation only, as expressing relations of parity or of
disparity, of place, time, motion, order, causation,
&c.; and in forming such an arrangement, the same
word will frequently occur, with different powers,
according as its force is primary, or figurative.
Although the proper function of a preposition be to
modify a substantive, yet in several of the instances
already quoted, we have seen prepositions accumulated
on each other, either as separate words, or as com-
pounds, and, of course, modifying each other. .
In the earlier and less cultivated periods of a lan-
guage, such cumulations of words may be expected
to be more common ; but as grammatical accuracy
and elegance of style prevail, the prepositions (consi-
dered as distinct words,) are usually confined more
strictly to their separate use. We find even in MIL-
Ton, the combination at under, as “ some trifles com-
posed at under twenty ;” but in the present day, such
a construction would hardly be tolerated by the critics.
In more ancient times this sort of construction was
still more prevalent; and we find numberless such
expressions as “ of beyond,” “for against,” and the
like. .
Artifycers and other straungiers, from the parties of beyonde the
See, Stat. l. Ric. III. c. ix.
The shiref of the seid countie of Northumbreland, or wardeyn
of the est and middell marchees for ayenst Scotlond. -
- Stat. 11. HEN. VII, c. ix.
Where the combination has been such as to present
to the mind the ready conception of a new relation, it
has generally been received in language as a new pre-
position, as throughout, into, overthwart; and so perhaps
the Latin intra, extra, &c. Custom too has sometimes
given a distinct force to compounds, which appear
originally to have had no signification different from
that of the simple preposition which formed their
basis. Thus we have in English distinguished within
from in, without from out ; and more slightly unto from
to, untill from till, &c. So in French we find en and
dans, avant and devant, vers and devers, près and auprés,
with more or less of distinction in their modern use
and application ; and, in like manner, the Italians,
from the Latin ante, have formed innarizi, formerly
imanti, and diamzi ; as from pressus they have formed
appresso and d'appresso. -
L'alma Ciprignia imanti i primi albori
Ridendo empia d'amor la terra e'l mare.
ANNIBAL CARO.
Torna amore a l'aratro, e i sette colli,
Ou 'era dianzi il seggio tuo maggiore.
F. M. MoLZA.
Io pur doueua il mio bel sole, io stesso
Seguir col piè, come segu'hor col core;
E le fredde Alpi, e'l Rhen, ch'aspro rigore,
Mai Sempre agghiaccia rimir d'appresso.
: IDEM.
Where the prepositions, as they are called, have
entered into composition with nouns and verbs, they
are in fact no more than adjectival and adverbial par-
ticles, and remain to be considered as such, in a future
part of this essay. It is, however, to be observed, that
when such a composition takes place, the adding of
G R A M M A R. !
f
157
Grammar, the same preposition to the sentence, in a separate
S-N-" form, is a redundancy, to be justified only by the energy
of feeling which sanctions the repetition of words. .
Dr. Johnson, citing the exquisite lines of Hamlet—
O! that this too, too solid flesh would melt,
Thaw, and resolve itself into a dew! *
has frigidly observed, that too “ is doubled to in-
crease its emphasis ;” but that “this reduplication
seems harsh " It is clear, that to repeat and
dwell upon a conception often gives energy and
weight to discourse. In the Andria of TERENCE we
find—“ Quid tibi videtur 2 adeon' ad eum ?” So CIcERo
says “ Nihil non consideratum eribat er ore.” So
VIRGIL–“ Retro sublapsa referri;” in all which in-
stances it is impossible not to see that the repetition
of the preposition is a great beauty. Nor is this ob-
servation to be confined to the repetition of the same
preposition; for it applies substantially to all preposi-
tions, and even adverbs, of similar meaning; as in
TERENCE—“Nonne oportuit praescisse me ante 9"—
“Multa concurrunt simul.” Grammarians of repute,
it must be allowed, have censured these redundancies
of expression, which, doubtless, are to be regarded as
exceptions from a general rule, and ought not to enter
into the ordinary construction of a sentence. But
the censure, when directed against such passages as
we have cited, rather shows an acquaintance with
technicalities, than a nice feeling of the higher powers
of language. . .
In like manner, the omission of prepositions,
, though sometimes 9 wing to a defective construction,
has been in other instances unnecessarily blamed. The
omission of the preposition of is undoubtedly awkward
In the following instances:–
That every person cómyng to suche feires shulde have lawefull
remed of all maner contractes. Stat. 1. Ric. III. c. vi. MS.
But God that is of maist poustè
Reserued to his majestie ;
For to knaw in his prescience, -
BARBOUR.
Of all kind time the first movence.
The kyng Robert wist he was there g
And what kind chiftains with him were. IDEM.
Then should they full enforcedly
Right in mids the kirk assail
The Englishmen. IDEM.
So, in old French, the preposition de is often awk-
wardly omitted. - t -
Wrepoch ab Edenauct, &c. oveke totle orgoyl de Gales—des-
cendirent a la terre nostre seigneurs le rei.
- Zet. P. De Mounfort, A. D. 1256.
Qui la maison son voisin ardoir voit,
De la sienne douter se doit. -
Faut noter— la maison son voisin estre dict à la façon an-
cienne; au lieu de dire “la maison de son voisin.”
H. ESTIENNE.
So, also in Italian, the authors of the Vocabolario della
Crusca observe, on the word casa : “ Nome, dopo di
cui vien lasciato talvolta dagli autori, per proprieta di
linguaggio, l'articolo, o il segnacaso.”
Esi sen' andaron di concordia a casa i prestatori.
- - . BoccACIo.
Cominciano a chiedere il Gonfalone che stava in casa Germa-
nico,-4° Vexillum in domo Germanici situm flagitare occipiunt.”
DAVANZATI, Tacit. Ann.
In the construction of the Latin language, some
grammarians contend, that where a noun is com-
YOL. I. - - - * * : -
Ҽ.
monly said to be governed by another noun, or by a chap. I. .
verb, it is preper to consider that a preposition has
been suppressed ; as, “ Cicero fuit eloquentior (prae)
fratre.” But this seems an unnecessary refinement in
grammar ; for the particle or in eloquentior, and the
termination e in fratre, sufficiently show the relation
between eloquence and frater, which is all the effect
that a preposition could produce. - . - 1.
The same observation may be made on the expres-
sions ire rus, domum, Romam, Hierosolymam, where
Vossius seems to suppose an omission of ad or in ; but
he adds, “ Latinis tam usitata est ha-cellipsis, in ex-
emplis allatis, ut vulgo naturalis sermo existimetur.”
. It may, however, be doubted, whether such con-
structions as alias res improbus, cactera laetus, and the
like, are not to be ranked among the negligences of
composition, though sanctioned by names of high
repute in Roman literature. -
— Ille eam rem adeb sobriè et frugaliter
Accuravit, ut alias res est impensè improbus.
PLAUT. Epid, iv. l.
Excepto quod non simul esses, cattera laetus. -
HoRAT. Ep. i. 10.
Similar observations may be made on the Greek
writers, who are often censured for the omission of
prepositions; and the remark is sometimes just,
though in general the relation is sufficiently expressed,
and the preposition would therefore be superfluous.
The learned LAMBERTUs Bos says, “Praepositionum
ellipsin tantopere amant scriptores Graeci ut interdum
dual praepositiones in unā orationis parte omittantur.
Aristoph. Nub. v. 1083. *Hv to 9To vulcm07s éaoſ : Si
(in) hoc (a) me victus fueris. Plene: ºvels toū-o vuk"Oñs
it’ &uoo.” In this instance it would perhaps have
been better, had the rhythm allowed it, to express
the first of the two prepositions; but the relation of
éu.00 to vukmójs is sufficiently denoted by their respec-
tive terminations. - * *
From all that has here been said of prepositions, the
necessity, and even beauty, of such a part of speech is
sufficiently manifest. “Most, if not all prepositions,”
says HARRIs, “ seem originally formed to denote the
relations of PLACE.” “ Omne corpus,” says ScALIGER,
aut movetur aut quiescit: quare opus fuit aliquà notá,
quae to To 0 significaret, sive esset inter duo extrema,
inter quae motus fit, sive esset in altero extremorum,
in quibus fit quies. Hinc eliciemus praepositionis
essentialem definitionem.” “But though the origi-
nal use of prepositions,” continues Harris, “ was to
denote the relations of place, they could not be con-
fined to this office only. They, by degrees, extended
themselves to subjects incorporeal, and came to denote
relations, as well intellectual as local.” “But how,”
says Cour DE GEBELIN, “ can such words introduce
into the pictures of speech so much harmony and
clearness, and become so necessary, that without
them, language would present but an imperfect deli-
neation ? How can these words produce such power-
ful effects, and diffuse throughout discourse so much
warmth and delicacy 2." The reason, he adds, is
simple: “ There is no object which does not sup-
pose the existence of some other object to which
it is bound, with which, it is connected, to which it in
some way or other bears relation. A valley supposes
the existence of a mountain, a mountain that of less
elevated lands: smoke implies fire, and there is “no
Y
158
G R A M M.A. R.
Grammar: rose without a thorn.” It is of necessity, then, that
*
different objects should be bound together in speech
as they are in nature; and that we should have words
to express the relations which exist among things.”
After this, it may be unnecessary to remark on Mr.
Tooke's sweeping censure of the philosophers, that
* though they have pretended to teach others, they
have none of them known themselves what the nature
of a preposition is.” -
$ 8. Of conjunctions. *
We have seen that a perfect sentence is formed by
a noun and a verb, as, “ John walks ;” that it is com-
plicated by the addition of an adverb, which modifies
the verb, as, “John walks foremost ;” and that it is
rendered still more complex by a preposition which
shows the relation of the noun or verb to another
noun, as, “John walks before Peter ;” but it may be
requisite to connect one sentence either simple or
complex, with another; as “John walks, and Peter
rides.” Now the word which thus conjoins sentences
is called a conjunction.
In the very commencement of our inquiry into this
class of words, we are met by the broad, unqualified
assertion of Mr. TookE, “I deny them to be a sepa-
rate sort of words, or part of speech by themselves.”
Such are the bold, but absurd or unmeaning propo-
sitions which have obtained for this etymologist the
reputation not merely of a grammarian, but of an
absolute inventor of the science of grammar ! He
himself tells us, “ he means to discard all mystery.”
Why, what greater mystery can there possibly be,
what greater confusion in the mind of a student of
grammar than to be told that there is no order, no
classification, among words,--that if is derived from
give, and therefore if and give are words of the same
sort, nay identically the same in all their uses—that
they do not indicate by their use, any different “turns,
stands, postures, &c. of the mind.” The mystery
here discarded is the mystery of learning. The student
is stopped on the very threshold of his studies, by being
assured that there is nothing for him to learn. And
the sage who gives him this precious information,
sets up for the great illuminator of mankind. “I be-
Hieve I differ from all the accounts which have hitherto
been given of language,” says Mr. TookE. Very true:
and every patient in Bedlam differs from all other
persons who give any account of his state of
mind. It is somewhat strange, that in support of
his title to absolute originality and exclusive know-
ledge of grammar, this writer should quote the fol-
lowing (among other) expressions of LoRD BAcon :-
** Quae in naturd fundata sunt, crescunt et augentur;
qua, autem in opinione VARIANTUR, non augentur.” The
science of grammar, which is founded in nature, was
taught, as we have shown above, by PLATO and ARIs-
ToTLE. Since their time it has grown and been in-
creased by the labours of grammarians in all ages,
and in a great variety of languages down to the pre-
sent time ; and now we see it illustrated by appli-
cation to languages dead and living, polished and
barbarous, to the Sanskrit, Hebrew, Latin, and Gothic,
as well as to the English and French, the Soosoo, and
the Chinese: and we find the same principles running
throughout them all, because language is the ex-
pression of thought, and human thought runs in the
same channels, among all mankind, But when at
N.
the close of the eighteenth century of the Christian . Chap. I.
era, an individual professes to set aside every trace
and vestige of the knowledge which preceded him,
his doctrine is not an augmentation, but a variation,
and we may be well assured that, it is founded in the
mere opinion of its pretended inventor. Now what is
opinion ? Mr. Tooke presumes to ridicule Lord
Monboddo's account of it, derived from the Platonic
philosophy, simply because Mr. Tooke could not or
would not understand that philosophy. Plato says
that the subject of opinion is neither to 6v nor to
an év. But this, however paradoxical it may appear
to any person who will not take the trouble to reflect
upon it, will be found extremely clear, with the help
of a very slight degree of attention. By to Öv he
means that which is, in the absolute sense of the word—
that which is, always, and certainly, and without any
variation. By to pun öv he means that which is not at
any time, or in any manner, and cannot be conceived
to be. Thus it is always and certainly true that in
our idea of a circle all the radii are equal; and it is
not at any time or in any manner true that we can
form an idea of a circle with unequal radii. But there
is a third case which is continually occurring to us,
namely, that an object is presented to our observation
which may correspond more or less accurately with
a given idea. We may see for instance a coach-wheel,
or the dome of St. Paul's church, but we can only
form an opinion how nearly either of these approaches
to our idea of a perfect circle; for the life of man
would not suffice to prove such coincidence beyond
the possibility of a doubt. Now, Plato distinguished
this class of objects by the expression to quyvouevov,
which he opposed to to 6v, as in the following cele-
brated passage of the Timaeus—Estiv obv 8) kat' éumv
86tav rpūtov 8tauperéov táče tć to *ON Aév 'ael, Yéveriv
8é 'ex' 3xov kai Ti Tô TITNO'MENON Aév, 3, 84
’88érore to gév 87, NOH'XEI, Aéra Adye replAnn Töv,
'ael kāra ravità èv. To 3’aé AO'EH, per' dugºaews
ãAdya, bočaatöv, Yuyvduevov kai &roXXiaevov ć'vtws
ôé '88érote &v–which passage Cicero has thus freely
rendered :—“ Quid est, quod semper sit, neque ullum
habet ortum ? et quod gignatur, nec unquam sit *
Quorum alterum intelligentid et ratione comprehen-
ditur, quod unum semper atque idem est: alterum
quod affert opinionem per sensus rationis expertes,
quod totum opinabile est; id gignitur et interit, nec
unquam esse vere potest.”—And the general sense of
both these great writers is, that science is founded on
that which is; opinion on that which seems : science
relates to that which is distinctly apprehended, because
it is permanent, immutable, and consonant to the
necessary laws of human existence; opinion to that
which is vague and indistinct, arising from sensible
impressions, and the casual accidents of time and
place. What Mr. Tooke calls his “general doctrine,”
is of this latter kind ; it is an opinion derived from
comparing the sound of words, not only without re-
garding, but often in direct opposition to their sense.
Should any one for a moment conceive that we are
speaking without due respect to the literary repu-
tation of Mr. Tooke, we beg to remind him that we
speak of a passage in which Mr. Tooke himself has
treated the profound wisdom of a PLATO and a CIcERo
with the most sovereign contempt, and has even
represented Lord Monboddo as an idiot, for quoting
their very words. As to Lord Monboddo himself,
&º .
§ * .
G R A M M A R.
159
Grammar.
Defined
from use.
of a word *
TXefinition,
Mr. Tooke -elsewhere says that his Lordship WaS
“incapable of writing a sentence of common English;”.
but this is nothing to his abuse of one of his critics,
the late Mr. WINDHAM, an accomplished scholar, and
as honourable a man as ever existed, but whom Mr.
Tooke calls in his chapter on conjunctions, a “can-
nibal,” and “a cowardly assassin.”
We call a word which conjoins, sentences a con-
junction. But to this also Mr. Tooke objects. “Con-
junctions,” says he, “it seems, are to have their
denomination and definition from the use to which
they are applied : per accidens, essentiam.” This leads
ws to ask what Mr. Tooke understands by the essence
Its sense, or its sound 2 Evidently the
latter, which is in truth, an accident, The words
“ cat,” “ et,” “ and,” are all essentially the same.
The Greek, the Roman, the Englishman, who may
have used each respectively, must have meant and
intended the same thing ; but by the thousand acci-
dents which led to the formation of each separate
language, the expression became varied in sound.
JBesides, this objection involves Mr. Tooke in a gross
inconsistency. He admits that a noun differs from a
verb ; but how does it differ, if not in use How
does the noun love differ from the verb love, or the
noun whip from the verb whip, but in use 2 And if a
noun differs from a verb in its use alone, why should
not a conjunction differ from both, in the same man-
ner This is an essential difference ; because the
essence of a word is the thought which it conveys ;
but there is no more reason for calling the sound
of a word its essence, than for giving that appel-
lation to the colour of the ink with which it is
printed.
“There is not such a thing,” says Mr. TookE, “as
a conjunction in any language, which may not, by a
skilful herald, be traced home to its own family and
origin.” This may, or may not, be the case ; but it
has nothing to do with the science of grammar. Mr.
Tooke has accurately “traced home” some conjunc-
tions: in regard to others, he has been mistaken; but
whether right or wrong in the particular instances, his
general doctrine can derive no benefit from them. To
prove that a word performs one function at one time,
does not disprove its performing another function at
another time. In fact, most of Mr. Tooke's deriva-
tions in this part of his work are borrowed from
former writers; but those writers never conceived
any thing so absurd, as that derivation was the whole
of grammar. - -
The early grammarians included what we call con-
junctions and prepositions, under the name of the con-
nective Xvv8éopos : and the definition given of the
>vvěeguos by Aristotle, though commonly cited as
that of a conjunction, is, in fact, equally applicable to
a preposition. It is in part doubtful, owing to the
diversity of the manuscripts, but, upon the whole,
the following may be regarded as tolerably correct :
>ivěeguos éat, ºwu) do muos, éc r\etdvov prev juvöv utas,
onſlautukùv 6é, Tote?v Teºvkvia uéav anuavtuki), @ww.jv.—
“A connective is a non-significant word, formed to
make one significant expression, out of words (or ex-
pressions) more than one, but (separately) significant.”
According to this definition, of is a connective in the
phrase “the Son of man;" for both son and man are
separately significant; but by the connective they are
so united as to produce a third significant expression,
racter of a conjunction.
According to the same definition, but is a connectiy
“John danced" is one significant expression; and
“ Peter sang" is another significant expression; and
they are both united together, so as to form one con-
tinued sense, by the word “but.”
Subsequent writers, however, perceived that it
would be useful to separate these two classes of con-
nectives ; they therefore gave to that which showed
the relation of word to word the name of preposition;
and to that which showed the relation of sentence to
sentence, the name of conjunction. Hence ScALIGER
says “Conjunctio est quae conjungit orationes plures;”
and SANCTIUs, more briefly, “Conjunctio orationes inter
se conjungit.” HARRIs says, “ The conjunction con-
nects not words but sentences;” and he gives the
definition of a conjunction fully, thus:– “A part of
speech, void of signification itself, but so formed as to
help signification, by making two or more significant
sentences to be one significant sentence.” Wossius says
“Conjunctio est quae sententiam sententiae conjungit;”
and he more formally defines it, “ Dictio invariabilis,
quae conjungit verba, et sententias, actu vel potestate.”
We should be inclined to prefer the following defini-
tion—“A conjunction is a word used to show the
relation of sentence to sentence.” We designedly omit
stating it as a characteristic of the conjunction to be
“void of signification,” or to be “invariable.” Pos-
sibly these expressions may be understood in such
senses, as to agree with the proper idea of a conjunc-
tion ; but they may also serve to give a false idea of it:
and, at all events, they are not essential to the cha-
Neither do we think it neces-
sary to say, that the conjunction unites “verbs and
sentences;” for, according to the definition which we
have heretofore given of a sentence, it is clear that
the uniting of verbs must be the uniting of sentences
Thus “he danced and sang” combine in reality the
two sentences “he danced” and “he sang.” Lastly,
it seems scarcely necessary to add, as Vossius does,
the words “actually or potentially;” for this seems
merely to have relation to those cases which are to be
explained by the figure, ellipsis, so common in all the
constructions of speech.
The main point, however, is, that the conjunction
receives its distinguishing characteristic from showing
the relation of sentences, and not simply of words.
Mons. Cour DE GEBELIN expresses this in his figurative
way, by saying ‘‘ une conjunction est un mot, qui, de
plusieurs tableaux de la parole fait un tout;” for, by
tableaux he does not mean a single object, a single
assertion, or a single sensation, but such a combina-
nation of these as we have called a sentence. -
Mr. Tooke, however, objects that there are cases in
which the words, commonly called conjunctions, do
not connect sentences, or show any relation between
them. “You, and I, and Peter, rode to London, is
one sentence made up of three. Well! So far, mat-
ters seem to go on very smoothly. It is, You rode, I
rode, Peter rode. But let us now change the instance,
and try some others, which are full as common,
though not altogether so convenient. Two AND two
make four ; AB AND BC AND CA form a triangle; John
and Jane are a handsome couple. Does AB form a
triangle, BC form a triangle 2 &c. Is John a couple?
Is Jane a couple 2 Are two, four " To all this we
answer, that if it could be shown that and, or any other
e Chap. I.
in the expression “John danced, but Peter sang;" for -
Y 2
160
G R A M M A. R.
Grammar. word generally used as a conjunction, was occasionally formed on an elliptical construction, and resolve the
Sentences
connected.
Connection
of nouns.
used with a different force and effect, that circumstance
would not make it less a conjunction, when used con-
junctionally. In the instances cited, the word and
serves merely to distribute the whole into its parts, all
which bear relation to the verb : and it is observable,
that though the verb be not twice expressed, yet it is
expressed differently from what it would have been,
had there been only a single nominative. We say,
** John is handsome,' —“ Jane is handsome ;” but we
say John and Jane are a handsome couple. In this
particular, the use of the conjunction differs from that
of the preposition: it varies the assertion, and thus does
potestate, if not actu, (to use the phrase of Vossius,)
combine different sentences ; for though AB does not
form a triangle, yet AB forms one part of a triangle,
and BC forms another part, and CA the remaining
part; and these three parts are the whole. So, when
PERIzoNIUs says “Emi librum x drachmis et iv obolis,”
although the buying was not wholly effected by the
ten drachmas, nor by the four oboli; yet the pur-
chaser did employ ten drachmas in buying, and he did
also employ four oboli in buying. The meaning,
therefore, if fully developed, would exhibit two sen-
tences connected by the conjunction and. Neverthe-
less, if any one contend that the word and, in the
above sentences, does simply and solely connect
together the nouns, then we say it must in such
instances be called a preposition ; but this will in no
degree alter its property or character as a conjunction,
when it is really employed to connect sentences.
* In pursuance of the view exhibited by our definition
of this part of speech, we proceed to consider the
three following subjects : first, the sentences con-
nected; secondly, the different relations between
them, intimated by different conjunctions, or con-
junctional forms; and thirdly, the words or phrases
which are used to imply these relations.
We have, in a former part of this treatise, distin-
guished sentences into enunciative and passionate ; and
in each, the verb, or the interjection, which stands in
the place of a verb, is to be taken as the hinge on
which all the rest of the sentence turns. By means
of this we form an unity of thought, a distinct per-
ception of some fact, or a feeling of some sentiment,
connected with a distinct object. But thoughts and
sentiments do not always succeed each other in the
mind as detached, and perfectly separate things, but
more commonly with associations of similarity or
contrast, with relations of cause and effect, and with
a thousand other modifications and mutual depen-
dencies. Hence these first and elementary unities
become parts of larger unities; the simple sentence
forms only a phrase or paragraph in a more compre-
hensive sentence ; and the longest sentence is more or
less closely connected with what precedes or follows
it, in a long discourse or poem. & - -
When this compression (so to speak) of thoughts
is the closest, it unites mere words, in the manner
we have already described ; thus, in the expressions,
“I paid six shillings and twopence”—“I gave six
shillings want twopence”—“ II est dix heures moins
un quart”—“ XY plus Z” — “AB minus C"—the
words and, want, moins, plus, minus, all serve to con-
nect words, and may be called prepositions if we regard
only what is expressed in their respective sentences;
but if we consider the sentences themselves to be
assertion applying to all the objects as a whole, into
separate assertions applying to the separate objects,
Chap. I.
as parts of that whole, then these same words may
be properly called conjunctions. So, when Hamlet,
addressing the ghost of his father, says,
If thou hast any sound, or use of voice,
- Speak to me!— - -
The word “ or,” if considered as merely pointing out.
a relation between the nouns, “ sound,” and “ use,”
may be called a connective preposition ; but if the sen-
tence be supposed equivalent (as we think it is) to
this, “ if thou hast any sound, or if thou hast any
use of voice,” then or is certainly to be called a
conjunction. - . . . . . . -
Whatever difficulty there may be when the verb is Connection
suppressed, there can be none when it is expressed— of verbs.
e, gr.
Fairy elves,
Whose midnight revels, by a forest side,
Or fountain, some belated peasant sees,
- Or dreams he sees. • - -
Here the sense is clearly, “ the peasant sees revels,
or the peasant dreams that he sees revels,” and the
latter or is therefore clearly a conjunction uniting
those two short sentences, in one longer sentence.
*.
How far these connections may go on, that is to say, Length of
how many conjunctions may be admitted into one "
comprehensive sentence, is a matter not to be deter-
mined by any grammatical rule, but must depend on
the taste and judgment of the writer; and great
writers, more particularly great poets and orators
often seem to indulge in a more than common degree
of continuity. Thus MILTON.— -
Now, Morn, her rosy steps in th’ eastern clime
Advancing, sow'd the earth with orient pearl,
When Adam wak'd, so custom'd ; for his sleep
Was airy, light, from pure digestion bred,
And temp'rate vapours bland, which th' only sound
Of leaves and fuming rills, Aurora's fan,
Lightly dispers'd, and the shrill, matin song
Of birds on ev'ry bough. r
Thus, too, CIcERo—
Potestne tibi hujus vitae lux, Catilina, authujus coeli spiritus
esse jucundus, cum scias, horum esse neminem qui nesciat, te
pridie Kalendas Januarias, Lepido et Tullo Consulibus, stetisse in
comitio cum telo ; manum consulum et principum, Civitatis
interficiendorum causã paravisse; sceleri ac furori tuo non
mentem aliquam aut timorem tuum, sed fortunam Populi Ro-
mani obstitisse 2 * 3
And it is to be observed, that in both these instances,
the following sentence begins with a distinct ex-
pression of relation to that which preceded it. Mil-
ton, having described Adam's sleep as light, goes on
to say, “ so much the more his wonder was” to find
that the rest of Eve had been unquiet : and Cicero
having briefly alluded to the former atrocities of
Catiline, proceeds, “ac jam illa omitto.” Indeed
there are some writers whose sentences, for whole
pages together, are connected, and it is difficult to
detach a short passage so as to show its whole force
and effect, without referring to the previous and sub-
sequent parts of the discourse. For instances of this
continuous style, we may particularly refer to the
Sermons on the Creed by the celebrated Dr. Isaac
BARRow ; who, it must, be confessed, carries this
method to an excess; for even in a continued argu-
ment the mind seems to require some short pauses,
assages.
G R A M M A R. d 161
labour of attempting this proof, he would have found . Chap. I.
p that some, at least, of the terms which he has specified,
Different A very slight degree of reflection must teach any serve to mark useful distinctions; and that that utility
relations of one, that the relations of sentences to each other must has been very well.marked out by Mr. HARRIs, an
* be very various, and consequently that the modes of author whom Mr. Tooke affects to hold in so much, but
Grammar: and resting places, as it were, to enable it to pursue
its steps with regularity and firmness.
marking these different relations ought to be classed
under several different heads. Those persons, how-
ever, whose vanity or ignorance prompts them to
overturn the whole fabric of that wisdom which has
preceded them, uniformly begin by decrying it as mere
rubbish. Thus Mr. Tooke, speaking of conjunctions,
says, “At the same time we shall get rid of that
farrago of useless distinctions into conjunctive, ad-
junctive, disjunctive, subdisjunctive, copulative, negative-
copulative, continuative, subcontinuative, positive, sup-
positive, causal, collective, effective, approbative, discretive,
ablative, presumptive, abnegative, completive, augmenta-
tive, alternative, hypothetical, extensive, periodical,
7motival, conclusive, explicative, transitive, interrogative,
comparative, diminutive, preventive, adequate-preventive,
adversative, conditional, suspensive, illative, conductive,
declarative, &c. &c. which explain nothing ; and (as
most other technical terms are abused) serve only to
throw a veil over the ignorance of those who employ
them.” As this mode of treating a scientific subject
* is extremely flattering to the indolence of mankind in
general, the above passage may not improbably have
produced an injurious effect, in deterring the gram-
matical student from investigations which it falsely
describes as unprofitable : and we therefore think it
proper to examine a declamation, which in any other
point of view would be totally beneath notice.
In the first place, there is a manifest want of good
faith in heaping together a number of words, “ con-
junctive, adjunctive,” &c. &c. &c. which are not to be
found in any one grammatical writer, and presenting
the whole as a “farrago” common to such writers.
This is a mere trick, and a trick extremely unworthy
of any man with the least pretension to literary repu-
tation. The thirty-nine terms above cited are indeed
a “farrago;” they have no meaning as they stand,
they are placed in no order, and they have no relation
to each other ; but whose fault is that 2 Undoubtedly
Mr. Tooke's, for he was the sole author and inventor of
the “farrago” which he pretends to ridicule.
“Most other technical terms,” says he, “ serve only
to throw a veil over the ignorance of those who em-
ploy them.” A profound remark
a surgeon must not speak of the metacarpal bone, or of
the arterial tube; nor an engineer of a counterscarp, or
a ravelin, because these are all technical terms ; and
technical terms are a mere veil for ignorance Mr.
Tooke, however is not original, in applying this sort
of reasoning to grammar. That philosophic statesman,
JAcK CADE, thus reproaches his prisoner Lord SAY,
“It will be proved to thy face, that thou hast men
about thee, that usually talk of a noun and a verb, and
such abominable words, as no Christian ear can endure
to hear.” Admitting however that some technical
terms may be properly employed, Mr. Tooke asserts
that the terms applied to classify conjunctions form
only a “farrago of useless distinctions.” Now, this it
would have been better for him to prove than to assert:
only assertien was the easier process of the two, and
presented the shorter road to celebrity as a grammati-
cal reformer If Mr. Tooke had submitted to the
So, the geometri-
'cian must not tell us of a parallelogram, or of a rhomboid;
such very undeserved, contempt; for whatever may have
been the errors of Harris, they are not a thousandth
part so gross, or so injurious to the science of grammar,
as those into which Tooke himself has falleń. • 4
Mr. Harris exhibits the following scheme of the .
SC116 IIlêe
different species, into which conjunctions may be
divided. “Conjunctions while they connect sentences,
either connect also their meanings or not.” The
first division of them therefore is into connexive and
disjunctive. “Aut sensum conjungunt ac verba,” says
SCALIGER, “aut verba tantum conjungunt, sensum
vero disjungunt.” So says Vossius, “ Aliae sunt
copulativa, ut, et, que, ac; aliae sunt disjunctivae ut vel,
aut.” The former of these terms adds he, is used in a
strict sense, “nam omnis quidem conjunctio copulat;
sed hae simpliciter id praestant citra disjunctionem.
sententiae, aut caussalitatem, vel ratiocinationem.” On
the other hand he defends the expression of disjunctive
conjunctions because by them “ conjunguntur voces
And BoETHIUs
materialiter, disjunguntur formaliter.” A
gives the same reason in different words, where he
says, “ conjunctionem ea quae conjungit inter se, dis-
jungere in tertio.” We do not cite these expressions of
Vossius and Boethius as most happily chosen to illus-
trate the distinction in question ; yet that distinction
is no less obvious than fundamental. Every one must
perceive at first sight, the marked difference between
these two passages, “ Caesar was ambitious and Rome
was enslaved”—“Caesar was ambitious, or Rome was
enslaved.” It is clear that the words and and or alike
join the same sentences ; but it is equally clear that
they join them differently. . In the one case, they inti-
mate, that the propositions stand on the same basis,
and are both meant to be asserted with the same de-
gree of confidence : in the other, that the ground, on
which the one assertion is made, excludes the other ;
and that if not contradictory they are at least meant
to be contradistinguished. Both and and or are con-
junctions ; both mark that a relation exists between
the sentences; but the particular relations, which they
mark, are different : one serves to cumulate, the other
to separate. -
GELLIUs uses the word connexiva for that sort of
conjunction, which Voss IU's calls copulativa ; and the
former term is better suited than the latter to the
scheme adopted by Harris; for he divides “ the con-
junctions, which conjoin both sentences and their
meanings,” i. e. those which may be called connevives,
into copulatives and continuatives. The copulative
(which perhaps might be called the cumulative con-
junction) “ does no more” according to him, “ than
barely couple sentences; and is therefore applicable
to all subjects whose natures are not incompatible.
Continuatives on the contrary, by a more intimate con-
nection, consolidate sentences into one continuous
whole; and are therefore applicable only to subjects
which have an essential coincidence. To explain by
examples,—'Tis no way improper to say Lysippus was
a statuary, AND Priscian was a grammarian—The sun
shineth, AND the sky is clear. But 'twould be absurd to
say Lysippus was a statuary BECAUSE Priscian was a
grammarian—though not to say the sun shineth BECAUSE
.162 G R A M M A. R.
may be admitted into use. Thus we may say, Troy Chap. I.
Grammar, the sky is clear. The reason is, with respect to the -
will be taken UNLEss the Palladium be preserved ; where S
first, the coincidence is merely accidental; with
respect to the last, 'tis essential and founded in nature.”
The Greek name for the copulative (in this sense)
was 24věeapºos avur) getticós ; for the continuative
ovvartik.ds, or rapaavvarrucos.
The continuatives are subdivided by Harris into
suppositive and positive. The suppositives are such as
if; the positives, such as because, therefore, as, &c.
The former denote (necessary) connection, but do
not assert existence; the latter imply both the one
and the other. The Greek term avva"Trucos and the
Latin continuativa was applied to the suppositive con-
junctions, which extend not only to possible but even
to impossible suppositions, as, “ if the sky fall, we
shall catch larks;" the positives were ealled Tapa-
ovvarruco, or subcontinuatival, and assumed the actual
existence of the primary fact ; and this either where
the connection is strictly and logically necessary or
where it is mere matter of analogy, the former case
being expressed by because, &c. the latter by as, &c.
Qf the suppositives, GAZA says, Örapčev aev 8", āco-
Mov6éav 8é Tuva, kai Tâțw öm Movow : PRIscIAN says they
signify to us “ qualis est ordinatio et natura rerum,
cum dubitatione aliquà essentiae rerum.” And ScALIGER
says they conjoin “sine subsistentia necessarid ; potest
enim subsistere, et non subsistere, ºutrumque enim
admittunt.”
The positives are either causal or collective. The
causals are such as because, since, &c. which subjoin
causes to effects ; e. gr. the sun is in eclipse, BECAUSE
the moon intervenes. The collectives are such as sub-
join effects to causes ; e. gr. the moon intervenes, riſ ERE-
FoRE the sun is in eclipse. The causals were called in
Greek’Avtuo Moyukov, and in Latin causales or causativa: ;
the collectives were called in Greek >vXNoºyistukot, and
in Latin collectivae or illativa.
The disjunctive conjunctions are in like manner
divisible into various classes. Their first distinction
is into simple and adversative. A simple disjunctive
conjunction, disjoins and opposes indefinitely as either
it is day, or it is night. An adversative disjoins with a
positive and definite opposition, asserting the one
alternative and denying the other; as it is not day,
BUT it is might. . .
The adversatives admit of two distinctions, first as
they are either absolute or comparative, and secondly
as they are either adequate or inadequate. The abso-
lute adversative is where there is a simple opposition
of the same attribute in different subjects, or of
different attributes in the same subject, or of dif-
ferent attributes in different subjects; as 1. Achilles
was brave, BUT Thersites was not ; 2. Gorgias was a
sophist BUT not a philosopher ; 3. Plato was a philoso-
pher BUT Hippias was a sophist. The comparative
adversative marks the equality or excess of the same
attribute in different subjects, as Nireus was more
beautiful THAN Achilles—Wirgil was As great a poet, As
Cicero was an orator. These relate to substances and
their qualities, but the other sort of adversatives
relate to events, and their causes or consequences.
Mr. Harris applies to these latter the terms adequate
and inadequate; he however confesses that this is a
distinction referring only to common opinion, and the
form of language consonant thereto ; for in strict
metaphysical truth no cause that is not adequate is
any cause at all. With this explanation the terms
the word unless implies that the preservation of the
Palladium will be an adequate preventive of the cap-
ture of Troy. On the other hand, when we say, Troy
will be taken ALTHough Hector defend it, we intimate
that Hector's defending it, though employed to pre-
vent the capture, will be an inadequate preventive.
The following, then, is a comprehensive view of
Mr. Harris's scheme for an arrangement of the
conjunctions. - •
1. copulative vº
1. connexive l, suppositive
- 2. continuative , suppositly
2. positive {
U2. collective
1. causal
1. simple -
2. disjunctive { {. absolute, or comparative
l2. adversative
2. adequate, or inadequate
Priscian distinguishes the subdisjunctive from the
disjunctive : and he gives the former appellation to the
Latin sive, as Alexander SIVE Paris ; where sive has
nearly a similar force with the Greek étt’ obv. In
English we use the conjunction or indifferently as a
disjunctive or subdisjunctive; that is, we say, “Alex-
ander or Paris,” whether Alexander and Paris be two
different persons, or only two different names for the
same person. SCALIGER and VossIUs both approve of
the distinction between the disjunctive and the sub-
disjunctive : and though, in our own language, we
employ the same word for both purposes, yet it may
not be amiss to distinguish its two functions by
appropriate designations.
It remains to be seen what are the conjunctional Conjunc-
Now it is manifest that one tional,
forms in language.
sentence may, and generally speaking, in a long dis-
course, the majority of sentences must serve to lead
the mind from what precedes to what follows. It
would, however, be endless to attempt to point out
the means by which this is effected ; nor would such
an explanation, if practicable, properly fall within the
scope of grammar. The remark nevertheless is
important ; for a sentence is in this respect only the
developement of an operation more briefly effected by
a word or a phrase. In treating of prepositions, we
first considered prepositional phrases, and then showed
how those phrases were gradually compressed into
words constituting that class to which the name of
preposition is usually assigned. It may not be
inecessary to follow exactly the same order of dis-
cussion in this part of our treatise ; but we will begin
with some of the more common conjunctions, and
afterwards advert to phrases, and to certain other
modes by which a connection of thought is kept up
between sentence and sentence. -
“The principal copulative,” says HARRIs, “is and," AND.
which answers to the Greek kāt and the Latin et, and
is found we apprehend substantially in all cultivated
languages. VossIUs considers the Latin et to be
derived per apocopen from the Greek étu, praeterea,
insuper, or more properly speaking to be the very
word &rt only pronounced more briefly by the Latins.
It is remarkable that in the most ancient remains that
we have of the Latin language, the fragments of the
laws of the Twelve Tables, et rarely if ever occurs, but
its place is supplied by the enclitic que, which is pro-
bably of the same origin as the Greek cit. The force
and effect of all these words, as simply coupling toge-
G R A M M A. R. 163
Grammar, ther sentences, will be fully understood from what
\—y—’ has been already said of the copulative conjunctions.
doubted, inasmuch as the former use depends on a Chap. F.
more simple operation of thought than the latter. Eac \–y-
Ac, eke,
Saxon ook, the Dutch oock, Swedish ok, Danish and
than eac from eacan.
Mr. TookE derives our common word and from An-ad,
which he says in Anglo-Saxon signifies dare congeriem.
This etymology is altogether obscure. It has even
been doubted whether Anan which he expounds dare,
to give or grant, had any such meaning ; and what
to make of the syllable ad which he translates con-
geriem we do not know. However, with his usual
confidence in his own judgment, he elsewhere says, “I
have already given the derivation which I believe will
alone stand examination.” SKINNER more modestly, but
with quite as much plausibility, says, “AND–nescio
an a Lat. addere, q. d. add, interjectà per epenthesin n,
ut in render, a reddendo.” A word of this very ancient
use can only be guessed at with much doubt, and may
probably be itself one of the original roots of language.
We find terms of some analogy to it in the early
Gothic dialects. In the Frankish and Alamannic it is
written indi, inti, enti, unte, unde ; in the modern
German und; in Icelandic end, in Lower Saxon wr.
ADELUNG considering (like Skinner) that the letter
m is often inserted in one dialect, while it is omitted
in another, is of opinion that the Latin et, and Greek
&rt are identical in origin with the Teutonic enti, unte,
&c. It is possible too, that our word and may have
a connection with the Moeso-Gothic and, which is
used as a preposition answering to the Greek év, eis,
éri, kará; or with the word andar, which in the same
language means “other." Upon the whole, Skin-
ner's suggestion is probably not remote from the
truth ; for the meaning of and is clearly add ; nay, in
separate sentences we may always substitute the
imperative add for the conjunction and, with little if
any difference in the force or intelligibility of the
sentence. Thus, ‘‘ I rode, add Peter walked, add
James sailed,” will not only convey the same notions,
but will connect them nearly in the same manner, as
if it had been more elegantly written, “I rode, and
Peter walked, and James sailed.”
The Latin ac, which seems to be identical with our
eke, is a copulative of nearly the same force as our
and. The Latin language does not afford any obvious
etymology for the conjunction ac; but of the etymo-
logy of eke there can be no doubt; and TookE wisely
adopts that of JUNIUs. Eke as a conjunction, has
become nearly obsolete in modern English, with the
exception of a few colloquial phrases in which it is
still employed ; but it is clearly the same as the verb
to eke out; and they are both from the Anglo-Saxon
eac, also, again, and eacan to add to, or augment. In
the Gothic, Frankish, and Alamannic we find it
written auk, auh, ouh. The Gothic verb aukan is
manifestly identical with the Greek ávčew and the
Latin augere. In Alamannic and Frankish the verb is
written auchon, auhhon, ouhhon, in Danish oge, in
Islandic auka. ADELUNG says that some of the most
ancient German writers use auch for und (our con-
junction, and). Of similar origin too are the Lower
Icelandic og : and it is observable that in old Frankish
ioh was similarly used for a conjunction. Tooke
reprehends Skinner for deriving eacan from eac, rather
- n. There is no doubt that eac is
the root, and eacan the derivative ; and so far Skinner
is doubtless, right; but that eac itself was used as a
verb before it was used as a conjunction is not to be
might be a verb in a single and simple sentence : it
could not be a conjunction except in a complex sen-
tence, that is, in the union of several sentences. Mr.
Tooke has made an observation which holds true in
several instances, but which like all philosophy that
is founded on mere observation would be calculated to
mislead, if adopted as an universal truth. He remarks
that “ in each language where this imperative is used
conjunctively, the conjunction varies just as the verb
does."—Thus, says he,
“In Danish the conjunction is og and the verb oger.
“In Swedish the conjunction is och and the verb.oka.
“In Dutch the conjunction is ook from the verb oecken.
“In German the conjunction is auch from the verb
auchon. . .
“In Gothic the conjunction is auk and the verb aukan,
“As in English the conjunction is eke or eak from the
verb eacan.” - - -
So far he is right; but on the other hand, the Latin
conjunction ac varies from the verb augeo : the Greek
av wants the characteristic É of ilvãeuv, and the Ice-
landic og differs from the verb auka. - -
As eke varies in a slight degree from the simple
copulative, and, so also is a copulative with a still
more specific meaning ; inasmuch as it implies some-
thing of similitude with what went before. We have
already seen that so when used as a pronoun, was
originally equivalent to “ this,” and when used as
an adverb, to “thus.” Also, therefore, though by long
use it has become a conjunction, may properly be
regarded as an elliptical phrase, meaning “wholly
thus,” or “ in like manner.”
Also
We come now to the continuative conjunctions, If.
that is to say, those which not only connect sentences
and their meanings by coupling them together, but
mark a dependence of one on the other; and this,
first as suppositives—IF is called by Mr. Harris a
suppositive conjunction : some other grammarians
term it a conditional ; but however it may be desig-
nated, the general force and effect of such a con-
junction is obvious in most languages. It serves to
mark the certain dependence of one event on another,
without asserting the absolute existence of either.
We therefore intimate, that if the one be the other must
be its necessary result, that when we are sure of the
one, then we may reckon upon the other also ; or
that the former being given as a datum, the latter
follows by the power of reasoning. Hence the Greek
et, and the Latin si merely expressed being ; for et is
part of the verb ew or eipt, and si is part of siet or sit.
The power of the conjunction et is thus elegantly
illustrated by Plutarch, according to the free trans-
lation of the old English folio : “In logike this con-
junction EI (that is to say if, which is so apt to
continue a speech and proposition) hath a great force,
as being that which giveth forme unto that propo-
sition, which is most agreeable to discourse of reason
and argumentation. And who can deny it 2 consi-
dering that the very brute beasts themselves have in
some sort a certeine knowledge and true intelligence
of the subsistence of things; but nature hath given
to man alone the notice of consequence, and the judge-
ment for to know how to discerne that which fol-
loweth upon every thing. For that it is day, and
that it is light, the very woolves, dogs, and cocks
164 - \
G R A M M A. R.
Grammar, perceive ; but that if it be day, of necessitie it must
make the aire light, there is no creature, save onely
man that knoweth.” The Greek or Latin construction
therefore is “be it that there is day there must be
light.” Again, the German conjunction answering
to our if is wenn, which also signifies when. Hence
the expression, “Wenn man dich fragt, so antworte,"
which signifies “ if any one asks you, answer thus,”
may be rendered with little difference of meaning,
“ when any one asks you, answer thus.” Lastly, the
English if is plainly in signification give ; and hence
Skinner's etymology of it has never been disputed.
He says, “ IF (in agro Linc. gif) ab A.S. gif, si. Hoc
a verbo gifan, dare, q, d. dato." Tooke justly adds
that gif for if is to be found not only in Lincolnshire,
but in all our old writers. It must be observed that
the same letter was variously pronounced g and y in
different dialects, as gate and yate, give and yewe. It is
also to be observed that the participle given (approach-
ing still more nearly to Skinner's dato) was used as
well as the imperative give ; and from these two
Sources we have for the conjunction geve, gef, giffe,
gif, yive, yef, yif. If, if, and gin ; which may be still
further illustrated by tracing the verb, participle, and
substantives, gyffe, yive, yeve, yave, gaff, yew, yth, yeft,
yifte, gytys, yewer, yewoun, yewen, yevyn, &c.; as in the
following examples : -
Hartely myght thei warry me,
That of ther gud had ben so fre,
To gºffe me and to sende.
- Sir Amadas.
Sir Amis answerd tho
Sir, therof yive Y nought a slo
Do al that thou may.
Amis and 4miloun.
Not Avarice the foule caytyfe
Was halfe to grype so ententyfe,
As Largesse is to yeue & spende.
x - - CHAUCER.
And with hys hevy mase of stele
There he gaff ſhe kyng hys dele.
Richard Coer de Lion.
And truely in the blustring of her looke, shee yaue gladnes &
comforte sodainly to all my wittes. CHAUCER. Test. Lov.
The remedy by the seid estatutes is not verray perfite nor
3yevyth certeyn ne hasty remedy.
* Stat. 11. Hem, P.I.I. c. 22. MS.
He gaf gyftys largelyche
Gold & syluer & clodes ryche.
Launfal Miles.
For gret yeftys that she gan bede,
To londe the schypmen gonne her lede.
. Octouian Imperator.
Every astate, feoffenent, yeft, relesse, graunte, lesis and con-
firmacions of landys. - Stat. 1. Rich. III. c. 1. MS.
Provided that this acte—extend not—to any graunte or grauntes,
yeft or yiftis had or made by the kinges letres patentes to the
same Anthony. - AStat. H. i. Hen. VII. c. 31. MS.
Ayenst the sellers, feffours, yewouns or grauntours and his or
their heires. Stat. 1, Rich. III. c. l. MS.
That no artificer ne laborer herafter named take no more ne
gretter wagis then in this estatute is lymytted, upon the payne.
assessed as well unto the taker as to the yever.
Stat. 1 1. Hen. WII. c. 22. MS.
Which lawe by negligence ys disused, and therby grete boldnes
ys goven to sleers and murdrers. Stat. 3. Hen. VII. c. 2. MS.
. Yeoven under our signet. Q. Elizabeth, Let. to Sir W. Cecil.
If the seid lessee or lesses within viii daies warnyng to theym
weven by any of the seid justices of the peas. -
- Stat. 11. Hen. WII. c. 9. MS.
Or yitgeve Virgil stude well before. GAwiN Douglas. .
Eorthliche knyght, or eorthliche kyng
Nis so swete in no thyng ;
Gef he is God, he is mylde. - ... •
Ryng Alisaunder.
He askyd at all the route, - * ,
Gºff ony durste com and prove
A cours for hys lemannes love. g
Richard Coer de Lion.
For giff he be of so grete excellence,
That he of every wight hath cure & charge,
Quhat have I gilt to him, or doon offense 2
K. JAMES I. The King's Quair.
The domes and law pronouncis sche to thaym then,
The feis of thare laubouris equalye *
Gart distribute. Gif dout fallis thareby
Be cut or cavill that plede sone pārtid was.
- GAwiN Douglas.
Ich am comen hider to day,
For to sauen hem, yiue Y may. -
Amis and 24 miloun.
Yef thou me louest ase mon says,
Lemmon as y wene; .
Ant yef hit thi wille be
Thou loke that hit be sene.
MS. Harl. No. 2253, fol. 80.
Wurthe we never for men telde, • ' -
Sith he hath don us thys despyte,
Yiffe he agayn passe quyte.
y - Richard Coer de Lion.
He thought yifich com hir to,
More than ichaue ydo,
The abbesse wil souchy gile.
Æay Le Freine.
The lawe of the landys that yf eny man be slayne in the day,
and the felon not taken, the townshipp wher the deth or murder
is done shal be amerced. Stat. 3. Hen. VII. c. 2. MS.
Gin living worth cou’d win my heart,
You wou'd na speak in vain.
Scots Song.
These words geve, gef, guff, giff, gif, yive, yef, yiffe,
yiff, yif, yf, which in the last eleven examples are
conjunctions, are doubtless the same in origin with
the preceding verbs geve, yewe, guffe, gaff, yave, yewyth,
and the nouns giftys, yeftys, yeft, yiftis, yewouns, yewers;
and in like manner the conjunction gi'n is clearly
nothing more than a new application of the participle
goven, yeoven, or yeven, which is the modern given.
But this new application causes the words if, gif,
gi'n, &c. to express a new “posture, stand, turn, or
thought of the mind,” (as Mr. Locke speaks) and
thus to perform a different function in language, or
become a different “ part of speech,” namely, a con-
junction. Mr. Tooke therefore is right so far as he
follows SKINNER, who first showed the connection
between if and give : but he is wrong, when, trusting
to his own theory, he says “our corrupted if has
always the signification of the English imperative give,
In short he is right where he is not
original, and original only where he is not right. Nor
is his “ additional proof” of much relevancy. “As
an additional proof,” says he, “we may observe, that
whenever the datum upon which any conclusion
and no other.”
depends, is a sentence, the article that if not ex-
pressed is always understood, and may be inserted
after if; as in the instance,— .
- — ‘My largesse .
Hath lotted her to be your brother's mistresse,
Gif shee can be reclam'd; gif not, his prey.’
. Sad Shepherd, act. 2. Sc. 1.
the poet might have said,
* Gif that she can be reclaimed, &c."
Chap. I.
G R A M M A. R.
165
*; But the article that is not understood and cannot be
inserted after if, where the datum is not a sentence but
some noun governed by the verb if or give. Exam.
* How will the weather dispose of you to-morrow :
If fair, it will send me abroad, &c.’” - ,--
Now the whole of this observation turns on the
peculiar idiom of the English language, which admits
one form of ellipsis and not another; for all these
constructions are elliptical; and the word that, which
is a conjunction as well as if, has not the least pre-
tension in such sentences to be called an article. We
shall, have occasion hereafter to notice some other
uses of this conjunction, when we speak of the
phrases O ! si—O ! gi'n, an if, as if, &c. •
The conjunction an, is not mentioned by SKINNER,
JUNIUS, LYE, or any writer of note, before Dr. John-
son, whose account of it is perfectly unintelligible.
He says it is “ sometimes a contraction of and if ;”
sometimes a contraction of “ and before if;” some-
times a contraction of “ as if ;” and to complete this
jumble of inconsistencies, he elsewhere says, “ and
sometimes signifies though, and seems a contraction
of and if.”—And again, “ in and if, the and is redun-
dant.” -
TookE, who has justly reprehended the errors of
Johnson, thus speaks of the word an himself “We
have in English another word, which, though now
rather obsolete, is used frequently to supply the place
of if ; as, “ an you had any eye behind you, you
might see more detraction at your heels, than fortune
before you. Twelfth Night, act ii. sc. 8.” Again, “An
is also a verb, and may very well supply the place of
if ; it being nothing else but the imperative of the
Anglo-Saxon verb anan, which likewise means to give
or grant.” - - * *
This conjectural etymology of Mr. Tooke's is
plausible, though not perfectly satisfactory. The
verb anan, to grant, is of dubious authority. The
supposed instances of its occurrence are rare, and may
possibly be accounted for from casual errors in manu-
scripts. Few words are brought into use as secondary
parts of speech, which have not also a very general
use as primary parts, and that in different dialects ;
but we have in vain sought to trace this verb anan as
a verb or noun in any dialect ancient or modern,
beyond the two or three doubtful instances cited by
Mr. Tooke. We do not positively reject his etymology,
but we must own it appears to us quite as probable
that 'an is only a further corruption than gi'n from
given or yeven ; and this is the more probable because
'an seems never to have been used but in the collo-
quial dialect of homely life, or of distant provinces.
Thus, in Much Ado about Nothing, Beatrice, who
An.
affects a homely and somewhat coarse kind of wit,
replies to the messenger as follows:—
MESS. I see, lady, the gentleman is not in your books.
BEAT. No ; 'an he were, I would burn my study.
So we find, in an old Scotch song—
*An thou wert mine ain thing,
O ! I wou'd lo'e thee, I wou’d lo’e thee .
IBut no serious and polished writer at any period of
our literature uses an for if ; and at present it is not
only “ rather obsolete,” but has long been obsolete
altogether. af - :
The circumstance which tends to give the most
plausibility to Mr. Tooke's etymology, is, that this
YOL, I, .
word is often spelt by old writers and, which may Chap. I.
seem to be a contraction of anned, i. e. granted, if \-y—’
there be such a verb as to an. - . .
Hereafter, litel in a stounde,
Comen vp, out of the grounde,
Amonge the folk sodeynlich,
Grete foxes, and griselich— § . -
Her bytt envenymed was, -
Man ne beest non there nas,
And he were of hem ybite,
That he nas ded, God it wyte. -
- Kyng Alisaunder.
So in an old MS. in the public library at Cambridge— ,
Ther is Leythe, Reythe, and Meythe . . . .
Meythe ouerset Reythe for the defawte of Leythe ;
Bot and Reythe methe com to Leythe, -
Scholden neuer Meythe ouerset Reythe.
In Gammer Gurton's Needle, Diccon says,
It is a murrion crafty drab and froward to be pleased,
And ye take not the better way, your medle yet ye lese it.
Lord BAcon, also, thus writes—
It is the nature of extreme self-lovers, as they will set an house
on fire, and it were but to roast their eggs. •
Still, in the very unsettled state of our ancient
orthography, much stress cannot be laid on this cir-
cumstance : and it seems hardly sufficient to out-
weigh the presumption against the derivation from
anan, arising from the want of correspondent nouns
and verbs in all the Teutonic dialects. -
Whatever be the true etymology of an its gram-
matical force and effect are exactly the same as those
of if . -
Because, since, and as are enumerated by HARRIs as Because,
causal conjunctions. We have already noticed the since, as:
word because as a preposition. It was originally a
phrase or combination of the words by and cause, and
we sometimes find by cause that used in old writers;
e. £r. - -
On me no fatnesse wilbe seene,
By cause that pasture I fynde none.
Ballad of Chichevache, MS. Harl. 2251.
In modern use it commonly signifies a cause prece-
dent ; but formerly it appears to have been applied
to denote the final cause, or object of an action.
The word since will afford scope for more particular
observation. Dr. Johnson, though he calls since an
adverb, has given the following instances of its use
evidently as a conjunction.
1. “ From the time that"— -
He is the most improved mind, since you saw him, that ever
was, without shifting into a new body. POPE.
2. “ Because that"— g
Since the clearest discoveries we have of other spirits, besides
God and our own souls, are imparted by revelation; the information
of them should be taken from thence. LOCKE.
Mr. TookE says “ since is a very corrupt abbre-
viation confounding together different words, and
different combinations of words ;” and he afterwards
classes the different uses of this word under four
heads, viz.- - - -
1. (As a preposition) for siththan, sithence ; or seen
and thenceforward.
2. (As a preposition) for seand, seeing as, or seeing
that. - `-- .
3. (As a conjunction) for seand, seeing, seeing as,
or seeing that.
4. (As a conjunction) for siththe, sith, seen as, or
Seen that,
- Z.
I66
G R A M M A. R.
Grammar. And he adds in a note, “it is likewise used adver-
Sºv-2 bially; as when we say—it is a year since ; i. e. a
year seen.” In short, Mr. Tooke contends that it “ is
the participle of seon, to see.” \
We conceive, that a little investigation will show
this etymology to be entirely erroneous. There are
in English two causal conjunctions, which, as such,
have nearly the same force and effect, viz. since and
seeing ; the latter speaks for itself; the former
requires to be traced to its source.
We say then, that since is a contraction of sith
thence, or sithens, the root of which latter is the word
sith or sithe and we have before shown that sithe is
identical with tide, which in German is pronounced
zeit, in Frankish zit, and citi, and was probably the
origin of the Latin cito, and in all these words the
common idea expressed is time.
Now, as the noun while, which also signified time,
came to be used adverbially in the forms of while,
whiles, whilst, to signify the time during which an
action continued, so the noun sith, time, in the forms
of sith, sithen, sythyn, seththen, was used adverbially
to signify the time from which an event was to be
reckoned.
. This adverb, like most others of a similar con-
struction, came next to be employed prepositionally
and conjunctionaily, with the same reference to time
past. - r
. Finally, as the effect commonly succeeds the cause
in time, sith came to be used as a causal conjunction,
either distinctly referring to time, or without such
distinct reference. * .
The different stages in this progress we shall pro-
ceed to illustrate, by adducing examples of the use of
sith and its derivatives.
1. As a moun, signifying time. -
Whan he him seghth, than was he blithe,
And kest him wel mani a sithe.
- - Seuyn Sages.
And such he was iproued ofte sithes. CHAUCER.
For thi was Tristrem oft
To court cleped fele sithe. Sir Trist remz.
For nede now wo is me,
Said 'I'ristrem that sithe. Ibid.
Thai underfengen him with cher blithe,
And thonged him a thousand sithe.
Sevyn Sages.
2. As an adverb, signifying afterwards, i. e. at a time
subsequent. -
And is sith some dele changed.
The letter told him all the deed,
And he unto his men gart read;
And sithin said them sickerly,
I hope Thomas his prophecy
Of Ersiltoun verifyd be.
Ac Alisaunder, his owen honde,
Biheueded the prince of the londe,
And sithen, withouten any pyté,
Sette on fyre that cyté. Kyng Alisaunder.
He tok that blod that was so bright,
And alied that gentil knight,
That euer was hende in hałe,
And seththen in a bed him dight.
Amis and Amilown.
TREVISA.
BARBour.
He gaffe ther ryche gyftes,
Both to sqwyars and to knyghtes,
Stedes, hawkes, and howndes :
And sythym apon a day
He buskyd hym on hys jorday. Sir Amadas.
3. As a conjunction, simply signifying “from the
time that.” -
To his ostage she went right,
There she nyver come by fore,
Sithe his stedis harborowed thore. *
*. Zyfe of Ipomydon.
Sethe Normans came first into Engelonde. TREVISA,
Nas non so holy prophete
Seththe Adam and Eue the appel ete.
Christ's Descent to Hell.
Siththe that I was born to man, -
Swylke sorwe hadde I never nan. - *
Richard Coer de Lion.
4. As a conjunction, signifying “from the time
that,” with the farther idea of causation. -
Sith so is that sinne was first cause of thraldome, than (i. e.
then) is it thus; that at the time that all this world was in sinne,
than was all this world was in thraldome. CHAUCER, P.T.
For sith the daie is come that I shal die,
I make plainly my confession. CHAUCER, Kn. T.
5. As a conjunction, in relation to cause only.
The wise eke saieth wo him that is alone,
For and he fall he hath none help to rise,
And sith thou hast a felow, tell thy mone.
CHAUCER.
Sithe in thi support myn hope abidith al. LIDGATE-
And therefore madame, if your wil be,
Sithe we have so grete plenté,
Sende hym somme, while we may.
- Lyfe of Ipomydon.
In the Scottish dialect, we find sithin, syne, and sen.
Sithin we have already cited from Barbour. Syme
appears to be a contraction from sithin, or sithen,
used adverbially, and in contradistinction to a time
preceding. -
He busked him, bot mair abade,
And left purpois that he had tane,
And to England again is game,
And syne to Scotland word sent he. BARBOUR.
By processe and by menys favourable,
First of the blisful goddis purveyance, -
And syme throu long and trewe contynance
Of veray faith. The King's Quair.
Till first ae caper, syne anither,
Tam tint his reason a’ thegither. BURNS.
Lang syne, long since, a time long past, is an ex-
pression well known from the admirable song of
Auld lang syne. Sen may possibly have been the
past participle seen, used as a causal conjunction, in
the same manner as we employ the active participle
seeing. - - -
Giff ye be warldly wight that dooth me sike,
Quhy lest God mak you so, my derest hert,
To do a sely prisoner thus smert,
That lufis you all, and wote of nought but wo,
And therefore merci sucte sen it is so.
The King's Quair.
Sensyne, a compound of sen and syne, is used adver-
bially, as in the Scottish translation of the Romance
of Alexander, A. D. 1438.
Sensyne is past anc thousand yeir,
Four hundred and threttie thairto neir,
And auch, and some dele mair I wis.
So in the Act of the Scottish Parliament, A. D. 1540–
All his gudis movable and vnmovable pertening to him, the
tyme of the committing of the said cryme, and sensyne, to be
decernit to pertene to His Grace. -
We now come to the word as, which HARRIS reckons
among the causal conjunctions, ex. gr.
Chap. I
S-N-
- * G R A M M A. R.
167
Grammar.
As when the moon hath comforted the night,
And set the world in silver of her light—
So, when the glories of our lives, &c.
- . . . . - CHAPMAN.
Here we see that as marks an analogical connection
between one set of incidents and another. The first
set are assumed to be well known and certain, the
latter to be equally true but less obvious. Whether
the term causal be strictly applicable to this sort of
analogical connection may perhaps be doubted; but
inasmuch as the certainty in both instances is first
stated, because and as may properly enough be dis-
tinguished by a common appellation from therefore
and so, which mark the less obvious or certain of the
two facts.
Mr. TookE however seems to deny that as is a
conjunction. His words are, “ the truth is that as
is also an article ; and (however and whenever used
in English) means the same as it, or that, or which.”
Why he calls it an article we know not ; for in
another part he says, “I should be sorry if any of
my readers were—to believe—that articles and pro-
nouns are neither nouns nor verbs—for I hope here-
after to satisfy the reader that they are nothing else,
and can be nothing else.” He afterwards published
another volume on grammar; but though it contains
a long chapter on “ the Rights of Man,” it has none
on either article or pronoun. We are therefore left
in the dark, as to Mr. Tooke's opinion of the word
as ; and know not whether he thought it a noun or a
verb ; why, being either, he called it an article ; and
why, if it could at once be either a noun or a verb
and an article, it could not also be a conjunction.
In its etymology indeed Mr. Tooke is certainly
right; as is the German es, it ; and as we have else-
where had occasion to observe, the same word which
signified identity, by an easy transition came to sig-
nify likeness; and hence we often find in our ancient
style the word like, either prefixed pleonastically to
as, or else used with a corresponding force. Of the
former we have an instance in Psalm ciii. 13.
Like as a father pitieth his children; so the Lord pitieth them
that fear him.”
The poet S. DANIEL furnishes an example of the
latter kind.—
O ! thou and I have heard, and read, and known,
Of like-proud states, as woefully incumberd, "
And framed by them examples for our own,
Which now among examples must be numberd.
We use so as a relative to the antecedent as, or as
an antecedent to the relative that ; and so (as Mr.
Tooke justly observes) is the Gothic sa, or so, it or
that ; but so by some of our old writers was used
where we now use as.
Bulsifal neied so loude
That it schrillith into the cloude—
Ac Alisaundre leop on his rugge,
So a goldfynch doth on the hegge ;
Hit monteth, and he let him gon,
- So of bowe doth the flon, Ayng Alisaunder,
In the German translation of the Bible, so is some-
times used as the relative pronoun who, in the same
manner as we employ the pronoun that.
Alle Juden so in AEgyptenland wohneten.
All the Jews which dwell in the land of Egypt,
JEREMIAH, c. 44. v. 1.
Als is also used in the Anglo-Saxon and old English
Latin words quod and ut or uti.
for as : and this word likewise is correctly explained Chap. 1.
by Mr. Tooke, as “a contraction of al and es, or as.” se-
—“ This al,” adds he, “ which in comparisons used
to be very properly employed before the first es or as,
but was not employed before the second, we now in
modern English suppress.” It would not be quite
correct to say that als was never employed before the
second es or as ; for examples of it sometimes occur.
- Vnto the toure he takes the way
Als hastily als euer he may.
Wntil the kirk than went he sone
And herd his mes, als he was wone.
Sewyn Sages. '
Ibid.
From as we naturally pass to the word that, which That.
is also a pronoun conjunctionally used. It is rather
singular that any difficulty should ever have occurred,
respecting either this word, or the corresponding
Mr. Tooke says,
“ that is the article or pronoun that ;” in which he
seems to have copied VossIUs, who says, “ quod
pronomen est, etiam cum dico, gaudeo QUOD veneris;
vel illo Horatii, lib. l. sat. 4.” -
Incolumis laetor quod vivit in urbe.
Nam integrè sit, gaudeo eo nomine, vel lactor ob id,
sive propter id negotium, quod est te venisse.
That quod may be used as a pronoun is no reason
why it should not also be used as a conjunction : and
its use is what determines its grammatical character.
Ut seems to have been an abbreviation of the later
Romans from uti, and is manifestly the Greek con-
junction 67t, which Hoog Eve N justly remarks is
formed by uniting the pronouns 6 and Ti.
Mr. HARRIs calls therefore a collective conjunction, Therefore,
meaning that it subjoins an effect to a cause, e. gr. Wherefore,
& g gº Il.
“The moor, intervenes; therefore the Sun is in then.
eclipse.” And he observes, “we use causals (such
as because) in those instances where the effect being
conspicuous we seek its cause ; and collectives in
demonstrations, and science properly so called, where
the cause being known first, by its help we discern
consequences. Our English word therefore is mani-
festly a phrase, or combination of words reduced by
custom into one ; like the Latin propterea, which for
this reason Vossius excludes from the class of con-
junctions.—“ Quamobrem, quasobres, propterea, quare,
et similia,” says he, “non videntur hujus esse classis;
quia non tam vox unica Sunt, eague composita, quam
plures : cui rei argumento nobis est, quod structura,
quae in simplici voce locum non habet, in earum
singulis observatur. Et vix caussa apparet cur
quamobrem magis sit vox unica, quam eam ob rem; vel
quare quam ed re.” The latter part of this reasoning
does not strictly apply to the English therefore, and
even admitting it to be correct we may still call that
word a conjunction. Its meaning, as we have else-
where had occasion to show, is simply for this (sub-
auditur cause or reason :) and it has two conjunctional
meanings; first when we state the effect as a matter
of fact ; and secondly when we state it as a matter
of reasoning. - .
1. This is the latest parley we will admit,
Therefore to our best mercy give yourselves. -
r SHAKSPEARE.
2. He blushes, therefore he is guilty. -
- Spectator.
The blush is not the cause of the guilt in fact; but it
is the cause of our asserting the person to be guilty.
Z 2 -
168
G R A M M A. R.
Grammar. The statement would be the very reverse, if the fact
alone were considered; for we should then say, “he
English else and the Latin alius; but certainly WAcHTER Chap. I.
was neither idle nor ignorant : and yet he has traced -->
* ~
is guilty, therefore he blushes;” but the full con-
struction in the other sense is, “ he blushes, therefore
I conclude that he is guilty.”
Wherefore is so similar in construction and effect to
therefore, that it needs no further explanation.
Then, used as an adverb, signifies at that time, but
used as a conjunction it not only has that meaning,
but in a secondary sense it means “ in consequence.”
1. My brother's servants
Were then my fellows, now they are my men.
alius quispiam, niemand el, nemo alius.
et primitiva, quae Graecis effertur dºxos. Lat. alius.
this radix, with a similar signification, through a
great variety of languages. The passage is a very
curious one, and well deserves attention. -
“EL, ell, alius, alienus, peregrinus. HENIscHIUS
in Thes. L. Germ. el, alius, jemand el, alter quispiam,
Vox Celtica
Inde composita et derivata in omnibus dialectis, et
praecipue.”
“ CAMBRICA, aliwn, alienus, alon alieni, inimici,
. . SHAKSPEARE, alltud alienigena, advena, alltudo in exilium pellere,
2. If all this be so, then man has a natural freedom. altudaeth exilium, allwlad alienigena, arallu alterare,
- LockB. ellmyn Alamanni, et usurpatur pro peregrino quovis.
We call either and neither, or and nor simple dis-
neither, or, junctives, in conformity with the scheme of HARRIs
above particularised; but they might perhaps be
more appropriately styled alternatives ; either and or
being set in opposition to each other affirmatively;
neither and nor negatively. Either is clearly in origin
a pronoun ; and or is a contraction of other, which is
also a pronoun. In old English other frequently
occurs at length, in the sense of the modern or.
Ful feole and fille
Beoth yfounde, in hearte and wille
That hadde levere a ribandye
Than to here of God, other of seynte Marie.
Kyng Alisaunder.
In a charter of king Edward the Confessor we
have oth for or.
Swo ful and swo forth, swo Duduc Bissop oth any Bissop hit
firmest him toforen havede.
The conjunction or is frequently followed by else."
As nor is by yet. The word else, Mr. Tooke says, is
“ the imperative ales of the (Anglo-Saxon) verb
alesan to dismiss.” The learned HIcKEs, however,
thinks it is contracted from the Latin aliaş : and of
this opinion, which appears to us the more probable,
are SKINNER and MINs HEw. It occurs both in the
Scottish and English idioms, and is written els, elles,
aliorsum,
Box HoRN in Leg. Ant. Brit.”
“GoTHICA, aljath alio, aliorsum, peregre, aljathro
aliunde, aljakunja alienigena, apud JUNIUM in Gloss.
Goth. p. 49” -
** ANGLo-SAxon ICA, elles alias, alioquin, elles—hwar
eltheodig, altheodig, exterus, extraneus,
peregrinus, elreordig, barbarus, apud SoMN. et BENSON.
Quibus addi potest eltheodiscemen peregrini, ex Matth.
xxvii. 7.” -
“ FRANCICA, allasuuara, alio, in Gloss. Pez. eliporo,
alienigena, in Gloss. Boxh. elirarter barbarus, in Gloss.
R. Mauri.”
“ ALAMANNICA, allesuuanan, aliunde, in Gloss.
Keron.” * -
“Is LANDICA, ella, alias, apud VEREL. in Ind.” *-
‘‘ ANGLICA, else, alius, alias, aliter, alioqui, else-
where, alibi.”
“ GERMANICA hodierna, alfanz aliena loquens,
elgötze, Idolum peregrinum, elend terra aliena, buffel
bos peregrinus, &c.”
Wachter goes on to cite the proper names derived
from this root, as Allobroges, Alamanmi and Aliso.
Neither and nor are merely either and or with a
negative particle prefixed to them. To these two
distinct words the Latin nec, or neque, answers when
repeated.
natare neque literas novit,” says, “ neque magis
ellis, ellys, &c.
- To take where a man hath leue negandi adverbium est quam conjunctio.” In this
Good is : and elles he mote leue. GOWER. position we cannot acquiesce, and indeed his subse-
What man that in special - quent argument shows that he had some doubt on the
Hath not him selfe he hath not els point himself. “Certè,” says he, “ in ea particulā
No more the perles than the shels. Inºw duo sunt, Tô ne, quod negandi adverbium est, et to que
Withouten noyse or clatteryng of belles quod copulativa est conjunctio. Utrumque munus
Te Deum was our songe, and nothyng "ºvern praestat neque : ac quatenus negat, adverbium est :
Him behoueth serve himselfe that has no swayn,
Or els he is a fole, as clerkes sayn. IDEM.
Traist not all talis that wanton wowaris tellis
You to defloure purposyng, and not ellis.
GAWIN DOUGLAs.
Frehold withyn the same shirez to the yerely value of xxs at the
leste, or ellys londes and tenº holdyn by custume of manere.
Stat. 1. Ric. III. c. 4. MS.
As though they lacked wysedome and learnyng to be able for
such offices, or elles were no men of conscience, or els were not
meete to be trusted. LATIMER’s Sermons, Ed. 1562.
Than may ye haue baith quaiffis and kellis -
Hich candie ruffes and barlet bellis
All for your weiring and not ellis.
Philotus, Edinburg ed.
Mr. Tooke very angrily accuses his critics of “igno-
rance and idleness,” because they venture to suggest
that el or al (signifying other) is the radix of the
length.
quia verö et disjunctas connectit sententias, quodam-
modo conjunctio est.” We cannot but think that a
little reflection would have shown this very acute and
judicious grammarian, that, under such circumstances
as he describes, a word becomes not merely quodam-
modo, but plainly and altogether a conjunction.
To the simple disjunctives either, or, and neither,
mor, are opposed the simple connectives both, and.
It is sufficient to observe that as either and or are
pronouns used conjunctionally, so is both a pronoun
employed in the same manner, and consequently
converted into a real conjunction.
Of the etymology of but we have already spoken at
It belongs to that class of conjunctions
which Harris calls the adversative absolute. In these
a positive and a negative are both asserted. We have
a remarkable instance of this in MILTON, who
reduplicates the conjunction but in application to
Vossius speaking of the passage “neºue.
Both.
But, ac-
G R A M M A R.
169
Grammar, two different kinds of opposition in the same sen-
\-y—’ tence. . . "
- Than.
. . Virtue may be assailed but never hurt;
But evil on itself shall back recoil,
And mix no more with goodness.
: Whether this reduplicated construction be a beauty,
or a blemish, in style, we shall not here inquire : we
only cite the passage to show the effect of the con-
junction but, which in both cases is as above stated.—
1. It is positively asserted that virtue may be assailed,
and negatively asserted that virtue cannot be hurt.
2. It is negatively asserted that virtue cannot be
hurt, and positively asserted that evil shall recoil on
itself; i. e. shall be hurt. In the one case, the sub-
ject remains the same, but the predicates vary; in the
other, the subjects are opposed to each other, but the
predicates are, if not identical, at least equivalent.
Ac, which was probably identical with eac, eke, and
originally signified also, is found in old English
writers, for but ; e. gr. -
With wraththe to Alisaundre he saide,
“Quik tak thy wed for thy deth.”
Alisaundre, “Nay” onswerith
Wed no schalt thou have of me,
Ac Y wol have wed of the.
- Kyng Alisaunder.
Nor is this surprising, since the French mais and
Italian ma, but, are merely the Latin magis, more ;
and therefore originally signified mere addition, with-
out opposition. -
Than and as (which latter we have already con-
sidered as a causal) are reckoned by Harris among
adversatives of comparison, the former implying
superiority, the latter equality, as, “ Nireus was
fairer than Achilles.”—“ Virgil was not so great as
Cicero.” It is clear, therefore, that these words
having a relative force, must be preceded either by
some separate word, as an antecedent, or by some
inflection which has the force of an antecedent. In
the first of the examples just quoted, the comparative
termination er renders the word fairer the antecedent
of the relative than : -in the second examples so acts
as an antecedent to the relative as. The antecedents,
when consisting of separate words, are commonly
called adverbs, and properly so, inasmuch as they
modify an adjective or another adverb. The relatives
are also called adverbs by many grammarians, but
improperly since they obviously connect sentences.
It is of course matter of mere idiom whether the com-
parison be effected by an inflection in the antecedent,
in the relative, or in both ; or whether it be effected
by a separate word, but in the latter case we call the
relative a conjunction. -
It is also matter of idiom whether the same con-
junction answer one or several purposes. Thus the
Latin ac and atque, which in their first sense are mere
copulatives, become adversatives of comparison in
such phrases as aque ac, aque atque, aliter ac, aliter
atgue. - - - - -
Somnia dormienti, non aquë ac vigilanti probantur. CICERo.
Quae beneficia acqué magna non Sunt habenda atque ea quae
judicio, considerate, constanterque delata sunt. IDEM.
Ego isti nihilo sum aliter ac fui. TERENTIUs.
Nunquam tealiter atque es in animum induxi meum. IdEM. “
So we use as, with the force of a causal conjunction,
or of a relative conjunction, or of the antecedent to
brave, as Alexander.”
such relative; as in the sentence, “Caesar was as
w
So, in Greek, “ the simple disjunctive #, or vel,”-
as HARRIs observes, “ is mostly used indefinitely, so
as to leave an alternative. But when it is used
definitely, so as to leave no alternative, it is then a per-
fect disjunctive of the subsequent from the previous ;
and has the same force with kai e", or et non. It is
thus GAZA explains that verse of Homer"— cº
BáAop” &y& Azov ordov čupeval, 3) &roxérôat
“That is to say, I desire the people should be saved
AND NOT be destroyed; the conjunction ) being
avaipetticos or sublative. It must however be confest,
that this verse is otherwise explained by an ellipsis,
either of uſix\ov, or āvtés, concerning which, see the
commentators.” - p
The grammarians seem to have doubted to what
class they should assign most of these words. Thus
PRISCIAN in one place calls quâm an elective conjunction,
but in another an adverb of comparison. PLINY, ac-
cording to Charisius and Diomedes designated these
words generally as conjunctiones relativa ad aliquid.
Vossius says “ multa esse fatemur, quibus et in
adverbiis et hic (Sc. in conjunctionibus) recte tribua-
tur locus—Ac et atque conjunctiones sunt, cum dico
Brutus Ac vel ATOUE Cassius : adverbia sunt in isto
aliter facit Ac tu, vel ATQUE tu ; nam idem valent ac
adverbium comparandi quâm.”
It is remarkable that all these words, than, as,
quàm, à were originally pronouns; for than and then,
as has been observed, are the same word.—
Than hadde the douke ich vnderstond
A clief steward of alle his lond.
Amis and Amilown.
Hire swyre is whittore then the swon. -
Ballad on Alisown, MS. Harl. No. 2253.
The French que is used both for our than and as,
e.gr. plus que, “more than,” and autant que, “ as much
as.” In some provincial dialects of England they say
“ greater as” for “greater than ;" and in the old
Scottish dialect ma or nor is used for than.
Item it is statut that na man, of quhat estate degre or condicioun
he be of, rydande or gangande in the cuntre, leide nor haif ma
personis with him na may suffice him, or till his estate.
- s : Scot. Act. Parl. A. D. 1424.
I levir haif evir
A foul in hand or tway
Nor sieand ten flieand
About me all the day.
The Cherrie and the Slae.
SKINNER has given two etymologies of the conjunc-
tion unless. He says, “unless, nisi, praeter, praeterquam,
q. d. one less, uno dempto seu excepto : vel potius ab
onlesan dimittere, liberare, q, d. hoc dimisso.” TookE
adopts the latter etymology, only suggesting that it is
from the imperative onles dimitte. It does not appear
to us that there is any reason to believe that the
Anglo-Saxon verb onlesan was ever used in this con-
junctional manner ; and we rather incline to think
that the present word was originally on less that, a
phrase adopted as a literal translation of the French
phrase à moins que.
But alway sister remembre that Charitie is not perfect onles that
it be burninge. Z’reatise of Charitie.
The Dictionnaire de l'Academie says—
A MoINS QUE, Sorte de conjonction qui regit le subjonctif, et
qui signifie, si ce n'est que. Il n'en fera rien A Moins QUE vous
7te Wuy parlier, - -
Unless.
170
G R A M M A R. *-
Grammar.
\-y-'
Though,
This explanation is confirmed by observing, that in
the old Scottish dialect the phrase les than was used
instead of our modern unless. -
That na notaris maid nor to be maid be the imperouris autorite
haue faith in contractis ciuile within the realme, les than he be
examinyt be the ordinare and approuit be the kingis hienes.
Scot. Act. Parl. A. D. 1469.
I shall distroye hyr landis alle
Hyr men sle bothe grete and smalle
Hyr castelle breke and hyr toure
With strenghe take hyr in hyr boure
£esse than she may fynde a knyght -
That for hyr loue with me darre fight. -
The Lyfe of Ipomydon.
Mr. Tooke however is not only very positive in the
etymology which he has borrowed from Skinner, but
is extremely angry at the critics who presume to
question it. What he says further of this word and of
less, lest, and least, we shall have occasion to consider
hereafter. - *
Unless is called by Harris an adversative adequate,
with reference to , the prevention of an event. Mr.
Tooke says this is “a gross mistake ;” but as Mr.
Harris had explained the terms adequate and inadequate
preventive by analogy to adequate and inadequate cause,
and had expressly added that “this distinction has
reference to common opinion and the form of language
consonant thereto,” there was little ground for Mr.
Tooke's objection. When we say, “Troy will be
taken unless the Palladium be preserved,” we mean to
express an opinion that if the Palladium be preserved
Troy will not be taken. That opinion however we
do not assert as a fact : and the fact may eventually
happen to differ from it, without any great impeach-
ment of our judgment in calling unless an adversative
adequate.
Except, which is manifestly the imperative mood of
a verb used conjunctionally, agrees in effect with
wnless. Thus we might say, “Troy will be taken,
eaccept the Palladium be preserved.”
But if is a conjunctional phrase used formerly in a
like señse—
That noon of thoo merchaunts of Venice convey into this said
realme any merchandisez, but yf the same merchaunt and mer-
chaunts bryng with every butte of Malvesy x bowstaves.
Stat. I. Ric. III. c. I l. MS.
Without is also conjunctionally used in the same
Sense.
This realme is like to lacke bothe stuff of artillary, and cºf
artificers of the same, without a provision of due remedy in this
behalf be the more spedely founde. - Ibid.
We find in another statute of the same date, (A. D.
1483) the phrase but if the rather employed, with the
same signification.—
Wheruppon but if the rather a remedy be purveid by youre
most noble grace, of werry likelyhode consequently shall ensue
the destruccion of drapery of all this your seid realme.
Stat. 1. Ric. III, c. 8. MS.
HARRIs calls though, or although, an inadequate
adversative, that is to say a conjunction uniting two
sentences, one of which states an event or circum-
stance, and the other states another event or cir-
cumstance as inadequate to prevent the former ; e. gr.
“ Troy will be taken ALTHough Hector defend it,”
where the conjunction although serves to shew that
Hector defends Troy with a view to prevent its being
taken ; but that this preventive is inadequate to pro-
duce the intended effect. We may, however, observe
that the same conjunction is used, and by a just
analogy to mark an apparent incongruity of qualities,
where the possession of the one does not in fact pre-
clude the existence of the other, as, “ though brave,
yet pious,” “ though learned, yet polite.”
Mr. TookE says “ tho' or though is the imperative
thqf or thafig, from the verb thqftan or thafigan to
allow.” This is one of the few original etymologies
of Mr. Tooke ; and we must confess we think it more
ingenious than sound. In a charter of William the
Conqueror we find, “ic nelle gethaftan that aenig man
this abrecan,” which in the ancient Latin version is
thus rendered, “ ego nolo consentire ut aliquis istud
frangat :” and the same clause occurs in two other
charters, one of HENRY I. the other of HENRY II. in
the latter of which the verb is spelt gethauian, i. e.
gethavian. These examples seem to show that in the
Anglo-Saxon language thaftan or thawian signified to
consent or permit, neither of which ideas has much
in common with the meaning of the conjunction
though. If this however had been the origin of the
conjunction, we might expect to find it in Anglo-
Saxon thqf or thav ; but it is theah. We might also
expect to find the for v in the numerous other
Teutonic dialects in which a similar conjunction or
adverb occurs ; but there is no such thing. ADELUNG,
under the German word doch, says, “ In Low Saxon
this particle is sounded doch and dog, by Ulphila thau, by
Ottfried thoh, by Willeram doh, in Anglo-Saxon theah,
in Dutch doch, in English though, in Danish dog, in
Swedish dock.”
In old English and Scottish we find it written very
variously, thah, thau, thaugh, thoffe, thof, thocht, and
thought. - -
Richard thah thou be euer trichard
Tricchen shalt thou neuer more.
Song on battle of Lewes.
Ant for ir feirnesse, thaw ho be comen of threlle,
Hire wedlac ne scal honout lesen all. -
Vita Sanctæ Margaretat.
Thaugh me slowe feole of heam,
They slowe mo of the kyngis men.
Kyng Alisaunder.
Z'hoffe Y owe syche too. Sir Amadas.
7"hof men wolde alle the londe seche.
MS. Harl. 7333, fol. 125.
Bot thocht I failyeit of rhyming,
Forgif me for my will was gude.
Scottish Rom, of Alerander,
Thocht be na reson persaue I mycht but fale
Quhat than the force of armis coud auale.
GAWiN DOUGLAS.
Thocht he remission
Haif for prodission,
Schame and suspission -
Ay with him dwells. DUNBAR.
The king—woll—that suche possession—veste and be—holy in
the other persone—in like wise as thought he had never be
enfeoffed. Stat. l. Ric. III. c. 5. MS.
It is to be observed that Gawin Douglas and other
Scottish writers spell thocht, the past tense of the
verb, to think, exactly as they do this conjunction.
So that we thocht maist semelye, in ane field,
To de fechtand ennarmed vnder Schield. . * -
GAwiN DougLAS.
But said they sould sound thair retreit
Because they thocht them nae ways meit
Conducters unto me.
ALEX. MontgomeRY.
Chap. J.
G R A M M A. R.
171
Grammar. Add to this that the Anglo-Saxon athoht, the Dutch
~~~ gadacht, and the German gedacht all answer to our
Yet, still,
substantive thought; and upon the whole it may not
be deemed improbable, that the words though and
thought are of the same origin. -
Thus the example, “Troy will be taken though
Hector defend it,” may be paraphrased, “even if it
be thought or supposed that Hector defends Troy, even
if this supposition be admitted, or be true in fact,
still Troy will be taken.”
In confirmation of this etymology we may observe,
that the word suppose is often used in the Scotch
dialect for though.
Yone slae, suppose thou think it sour,
May satisfie to slokkin
Thy drouth now. ALEX. MoRTGoMERY.
Stories to rede ar delectabil
Suppois that thay be nocht but fabil.
For though were also used albe, albeit, howbeit, all had,
all should, all were, all give, &c.
Yet and still are conjunctions used in English as
relatives to the antecedent adversatives though,
suppose, &c.; e. gr
Though Birnam wood be come to Dunsiname,
Yet will I try the last. SHAKSPEARE,
— Though I do contemn report myself,
As a mere sound, I still will be so tender
Of what concerns you in all points of honour,
That the immaculate whiteness of your fame
Shall ne'er be sullied,
The use of yet as a conjunction is directly taken
from its use as an adverb ; e. gr. r x
- —Tarry, Jew ;
The law hath yet another hold of you.
BARBOUR.
MASSINGER.
SHAKSPEARE.
Here yet means, “ at this time, after you have come,
as you suppose, to the end of the legal proceedings,
against you, in addition to these there remains
another.” So, in the above example where it is
employed as a conjunction, yet means—“ at this time,
after Birnam wood has come to Dunsinane, and when
no hope seems to lie in resistance, I will nevertheless
resist.” - -
The etymology of this word has already been con-
sidered in our chapter on adverbs. -
As yet refers primarily to time, so still refers prima-
rily to place, but secondarily to time. Stelle in German
is place, and it answers to our stead, as an meiner stelle,
“ in my stead.” ADELUNG, speaking of this word,
says, “By Notker it is written stal ; in Swedish it is
stálle; in the Anglo-Saxon stealle, steale, in Low
Saxon stede, in the Swiss dialect stahl.” Hence the
Anglo-Saxon, English, Frankish, and German adjective
still, means primarily remaining in the same place,
motionless; and consequently quiet ; secondarily it
means that which remains unchanged by the lapse of
time, or which moves on equally with it.
1. Thy stone, O Sysiphus, stands still.
Such silence waits on Philomela's strain
On some still evening.
2. It hath been anciently reported and is still received.
BACON
Pope.
IDEM,
A generation of still-breeding thoughts.
SHAKSPEARE.
Still as a conjunction is manifestly the adverb so
employed, and the adverb is taken from the adjective :
it is not easy to conceive a direct transition from the
:
.
Anglo-Saxon imperative stell, to our modern con-
junction. The analysis of the above example is, “I
contemn report as far as it merely affects myself; but
at the same time (and indeed at all times alike) I will be
tender of your reputation.” .
In treating of yet, Mr. Tooke has very erroneously
Chap. I.
explained algate as “ meaning no other than all get.”
The very example which he adduces might have
taught him a different origin of this word.
“For albeit tarieng be noyful, algate it is not to be reproued in
yeuynge of iugement, me in vengeaunce takyng.” CHAUCER.
French having long been the fashionable language,
previously to the time of Chaucer, the construction of
his sentences is generally to be explained by reference
to French idioms; and algate, which is literally all
way, was undoubtedly a translation of the French
conjunction toutefois. -
Et se ele eschaoit à autre—tote vois qeele eust este ainsi donée
—il domroit la fie au Roi de Angleterre.
r Treaty, England and France, A. D. 1259.
Si ne pooms a vostre priere entendre quant à ores; totevoies,
pur ceo que nous ne voliens mie q’il fuissent en nostre terre
surpris de leur cors ne de leur biens, si lor avoms suffert de
marchander. Letter K. Edw. I. A. D. 1304.
ToUTEFois. Conjonction adversat ; meanmoins, mais, pourtant.
Tous les hommes recherchent les richesses, and toutesfois on voit
peu d'hommes riches heureux. JDict, de l'Academie.
Kyng Alisaunder leoseth many men
Ac allegate the kynges -
Losen ten ageyns on in werrynges.
Kyng Alisaunder.
Gate, as has already been explained, is the same as
gait from the verb go or gae. -
HARRIs notices another class of conjunctions which
he says “may be properly called adverbial conjunctions,
because they participate the nature both of adverbs
and conjunctions—of conjunctions as they conjoin
sentences; of adverbs as they denote the attributes
of time and place.” Such are when, where, whence,
whither, whenever, wherever, &c. Upon the principles
which we have adopted, these are to be called con-
junctions when they conjoin sentences ; but the name
adverbial, is not at all distinctive, because many other
conjunctions have occasionally an adverbial use ; and
many prepositions when used conjunctionally serve to
mark time or place. The scheme of arrangement
which Harris has followed, is principally directed to
the logical connection of sentences; but the connec-
tions of time and place are merely physical, and should
therefore form a class apart. . The term ordinative
which Voss IUs applies to deinde, postea, &c. may not
improperly designate this class.
Thus, among ordinatives of time we should reckon
whiles, till, o that, or, be. -
- His Lord nold he neuer forsake
Whiles he ware oliue.
Amis and Amiloun.
Full ofte drinkes shee, - - -
Till ye may see
The teares run down her cheeke.
Gammer Gurton’s Needle.
Al the day and al the nyht
O that sprong the day lyht.
Geste of Kyng Horn.
Sathanas Y bynde the her shalt thou lay,
O that come Domesday.
Christ's Descent to Helſ.
He it is my dedly foo;
He schal abeyen it or he goo.
Richard Coer de Zion.
172
G R A M M A. R.
Grammar.
Conjunc-
tional
phrases.
Your madynis than sall haue your geir
Put in gude ordour and effeir
Ilk morning or yowryse. Philotus.
The supper done than vp ye ryse,
To gang ane quhyle as is the gyse;
Be ye haue rowmit ane alley thryse
It is ane myle almaist.” Ibid.
So, where is an ordinative of place in the following
passage. • .
He rails
Even there, where merchants most do congregate.
SHAKSPEARE.
We have seen that several of the conjunctions, now
considered as single words, were formerly phrases;
such are because, therefore, wherefore, quamobrem, and
toutefois; but there are many other conjunctional
phrases, which have been more or less generally
appropriated to the connection of sentences, such are
the following in old English, Scottish, and French,
Howe be it—for als moche—at least waye—not forcing
whether — contrariwise — insafer as —pur ceo qe—cest
asavoir— and over that — coment que—how often, so
often — no the less—neuerthelas—not for thi—nought
gaynstandand—forfered that—set in cais—put the cais—
jorseing that, &c. &c. • -
Howe be it, the kynge held styll his siege. BERNER's Froissart.
Bot for als moche as sum micht think or seyne tº
Quhat nedis me apoun so lytill evyn
To writt all this; I ansuere thus ageyne. -
* - Z'he King's Quair.
. This geare lacketh wethering; at least waye it is not for me to
plough. . . - Rishop LATIMER.
These words goe generally to all the king's tenants—not forcing
whether he haue the reuersion by dyscent.
Sir W. StAUNFord, A. D. 1599.
Contrariwise, certain Laodiceans and lukewarm persons think
they may accommodate points of religion by middle ways.
- - BACON. Essays.
And decernis the saidis actis and euery ame of thame to be
abolishit and extinct for euer, insafer as ony of the saidis actis ar
repugnant and contrarie to the confessioun and word of God
foirsaidis. Scot. Act. Parl, A. D. 1567.,
E purcéo qe aucunes gentz de nre Roiaume se doutent qe les
aides &c. pussent turner en servage a eus e a leurs heirs avoms
graunte pur nous et pur nos heires qe mes tieles aides &c. ne
treroms a custume. Stat. 25. Edw. I. c. 1. A. D. 1297.
Meismes les chartres en toutz leur pointz en ples devaunt eus
e en jugementz les facent alower, cest asavoir la grand chartre
des franchises come ley commune, e la chartre de la forest Solom
l'assise de la forest. - Ibid.
That—the same fyne be openly and solemply rad and pro-
claymed in the same court—And in the same tyme that it is so
redd and proclaymed all plees cesse ; and over that a transcript
of the same fyne be sent by the seid justices unto the justices of
assisez. Stat. 1. Ric. III. c. 7. MS.
It de common droit poit distreiner pur le rent aderere, coment
que tiel done fuit fait sauns fait. LITTLETON, Sect. 214.
How often his eye turned to his attractiue adamant, so often
did an vnspeakeable horrour strike his noble heart.
. - SIR. P. S1DNEY’s Arcadia.
what ansuere thei bare the sothe can I not say
No the les of fele this was the comon sawe.
* R. DE BRUNNE,
Youe knowe, Lordes Syracusans, that we haue hytherto done
in thys warre, as men of honestie : newerthelas, leste there be
anny that wnderstandeth not fullye the affayre, I wolle well declare
yt vnto hym. NicoLLs's Thucydides, fo. 191.
Was mad another statute, that non erle no baroun
No other lorde stoute ne fraunkeleyn of toun.
Tille holy kirke Salle gyue tenement rent no lond,
Not for thi he wille that alle religioun
Haf and hold in skille that gyuen is at resoun.
R. DE BRUNNE.
Item it is ordanyt that all craftis &c be distroyit &c not
gaynstandand ony priuilegis or fredome geifyn in the contrare, , ,
- . e * * * Scot. Act. Parl, A. D. 1424.
He slogh him some that ilk day - * ,
Forfered that he sold oght say. - -
The Seuyn Sages.
With stout curage agane him wend I will
Thocht he in proues pas the grete Achill,
Or set in cais sic armour he weris as he,
Wrocht be the handis of God Vulcanus sle. -
- - GAwiN Douglas.
And put the cais that I may not optene
From Latyne land thaim to expell all clene,
Yit at leist thare may fall stop or delay. IDEM.
It may be ordered that ii or iii of our owne shippes do see the
sayde Frenche soldiers wafted to the coast of France; forseing
that our sayd shippes entre no hauen there.
* t - Q. ELIZABETH to Sir W. Cecil.
It is plain, that these phrases operate, with relation
to the sentences between which they show a relation,
exactly in the same manner as the words do which we
call conjunctions. A phrase is first abbreviated into:
its principal words, and these are again contracted
into one short word. Thus the French c'est asavoir
above quoted was probably first translated into English,
“ it is to know,” or “ it is to wit,” whence we now
have in our legal documents the abbreviated phrase,
“ to wit,” as from the Latin videre licet comes videlicet,
which we have adopted into the English language.
These abbreviations and contractions are very arbitrary
in their use; thus our ancestors in the fifteenth century
used to say where, for that conjunction which we now
express by whereas, i.e. where that.
wher in a statute made in the xvi; yere of the reign of King
Edward the iiijth hit was ordeigned &c &c. Please it theyore
youre highnesse &c to ordeign. . -
Stat. 1. Ric, III. c. 6. MS.
The ordinals, which we have included in the class
of pronouns, such as first, second, &c. necessarily imply
connection, and consequently the adverbs formed
from them, are easily employed with a conjunctional
force, as primo, secundo, tertio, when placed at the
beginnings of sentences. The same also is to be
observed of the adverbs used as relatives to these
antecedents, such as deinde, item, puis, next, syne,
lastly, &c. “ Deinde,” says VossIUs, “ cum verbo
jungitur, ad circumstantiam temporis indicandum,
adverbium est: conjunctio autem, cum tantium ad
orationis juncturum pertinet.”
Accepit conditionem ; dein quaestum occipit. TERENTIUS.
Pergratum mihi feceris; spero item Scaevolae, &c. CICERo.
Ils font estat d’aller a Orleans, a Blois, puis a Tours.
- Dict. de l'Academie.
First ae caper, syne anither. BURNS.
The adverbs where, when, &c. which we have
heretofore shown to be pronouns in origin, have often
the same conjunctional force, and in such case are
properly to be reckoned conjunctions. * -
It remains to be observed that some conjunctions
are used singly, and others in a succession of two or
more. Thus we may say, “John and William came,”
or, “ both John and William came.”—“ It is ordained
that proclamation be made, and that the judgment be
recorded, and furthermore that the record be trans-
mitted.” Where two or more succeed each other,
there is a certain order in the succession ; ex. gr.
“ as—so ;”, “ so—that :” “ when—then,” &c. On
this subject VossIUS thus speaks – “Conjunctioni
chap. I
G R A M M A. R.
173.
Grammar; etiam accidit ordo; secundum quem aliae sunt
praepositiva, ut et, nam; alia postpositivat, ut quoque,
autem; alia communes, ºut equidem, itaque. Igitur
saepiùs postponitur. Enim etiam est particula prae-
positiva, Terent. Phor, act v. sc. viii. Enim nequeo
solus.” Ad postpositivas etiam pertinent encliticae.
Ex his, que interdum alteri verbo jungitur quam
nativus verborum ordo exigebat: ut apud HoRAT.
lib. ii. od. 19. -
Ore pedes tetigitaue crura.
Pro cruraque tetigit. These however are matters
depending on the particular idiom of each lan-
guage, and not governed by the philosophy of general
grammar.
The case is different with the pleonasms and cumu-
lations of conjunctions. These occur in all languages,
and they therefore clearly arise out of principles
common to the human mind in different countries.
Hence VossIUs speaks of expletive conjunctions—
‘‘ Expletiva sunt, quae nullā necessitate sententiae, sed
explendi tantùm gratiâ usurpantur. Ut quae metri
vel ornatſis caussà inseruntur. SALLUST in Catil.
Werúm enimverb is demum mihi vivere, et frui animd
videtur; ubi verilm redundat.” VIRGIL in xii.
& &
Equidem merui mec deprecor inquit. '
Tºlena fuerit sententia, licet equidem tollas.” To this
head are to be referred such expressions as “ an if.”
- Well I know
The clerk will ne'er wear hair on's face that had it.
He will an’ if he live to be a man.
Where either an or if is redundant ; for they both
signify the same, and Johnson is wrong in supposing
that an' in this instance is a contraction of and.
Voss IU's refers these redundancies to the custom of
ancient writers, “ nempe is veterum mos fuit, ut inter-
dum conjungerent voces idem significantes.” But
they are not peculiar to any age or nation : they
are the result of hasty and inconsiderate habits of
speech, which, it is true, are more common in the
first formation of a language, than in more cultivated
and civilized periods of history.
Cumulation, however, is not always redundancy.
Thus when we find a sentence beginning thus—“ but
nevertheless if,” the conjunction but connects it with
what goes before, and if with some subsequent
sentence, and the word nevertheless alone may be
called redundant, and yet not strictly so, since it adds
a great force and emphasis to the word but.
In the Greek language, this cumulation of con-
junctions is frequent; and is sometimes explained b
an ellipsis. Thus HoogeveN says—“ hoc modo àNA&
vövºye redditur nunc maxime, suppressà per ellipsin
voculá čitrote.” Ita SoPHoci, in Electr. v. 413.−
"O Oeol trarpåot avy Yévé0.0é y &AA& vöv |
O Dii patrii, adeste nunc marimè, vel nunc saltem 1
Plenior structura est’Q 9eo: Tarptºot, €tzrote avºyéved 6é
pot, d\\& vövºye avy Yévéoée l—0 Dii patrii, si unquam
alias mihi adfuistis, at nunc adeste saltem 1
And so much for the conjunction, which may be
considered as the completion of the parts of speech
necessary in any language to discourse, so far as it
consists merely of enunciative sentences !
§ 9. Of interjections.
“The brutish, inarticulate interjection," says Mr.
HoRNE TookE, “which has nothing to do with speech,
VOL., I,
to a proposition which
and is only the miserable refuge of the speechless, Chap. I.
has been permitted, because beautiful and gaudy, to
usurp a place among words.” This is what we learn
from Mr. Tooke, on the subject of interjections : and
surely this is sufficiently inconsistent with itself, and
with common experience. How can a class of words
be at once beautiful, gaudy, brutish, and inarticulate 2
And what is meant by saying that the interjection,
which somehow or other has been enabled to occupy
a place among words, has nothing to do with speech,
and is only the miserable refuge of the speechless 2
If some grammarians have reckoned inarticulate
sounds among interjections, it is certain that far the
greater part of the sounds so designated are not only
articulate, but like adverbs, conjunctions, &c. may
generally be traced to a distinct connection with nouns
and verbs. VossIUs, speaking of CHARISIUs, says—
“ Male idem huc refert trit quae murum vox est,
apud Naevium Corollarià, Par ratio erit Aristophani
Bpekekéâ, quae vox est ranarum. Idem censendum de
rei inanimae sono, vel humano quidem, Sed nec ex
instituto aliquid significante, nec animi affectum
testante. Uti bat qui sonus est ex ore cornicinis
lituum eximentis, quemadmodum, ex CESELLIo
VINDICE, observat Charisius. (Utitur Plautus, Pseudo :)
Item bat tatti fluctus quidam, et Sonus vocis effemi-
nation, ut esse in sacris anagmenorum, vocum veterum
interpres scribit ; et ex eo idem Charisius extremo,
lib. ii.” Upon this principle we may admit that
sounds, whether articulate or inarticulate, which are
merely intended as imitations of other sounds, not
proceeding from the human mind nor expressing
human passion or affection, are neither interjections
nor parts of speech.
But excluding these, there are many sounds, more
or less perfectly articulated, which occur constantly
in human speech, but which yet are not to be reduced
to any of the classes which we have hitherto discussed.
These, generally speaking, we reckon among inter-
jections : they do not form part of an enunciative
sentence; but they are commonly thrown in between
such sentences, or the parts of them, according as the
impulse of a strong or sudden feeling dictates.
Now, as a botanist would but imperfectly teach his
science, if he were to tell his scholars that certain
large portions of the vegetable world were beneath
their notice, as weeds ; or as he would be a poor
mineralogist who should disdain to cast an eye on
pebbles; so he is a miserable grammarian who affects
to disregard the numerous interjections and interjectional
phrases which give such force, tenderness, variety,
and truth to the works of the rhetorician and poet,
and contribute so much toward rendering language
an exact picture of the human mind. -
SANCTIUs, like Tooke, denied that the interjection Definition.
was a part of speech; but he did this, with at least
a show of argument: his conclusion was fairly derived
from his premises : only those premises were built
on too narrow and limited a view of his subject.
“Interjectionem non esse partem orationis,” says he,
“sic ostendo : quod naturale est idem est apud
omnes : sed gemitus et signa laetitiae idem sunt apud
omnes : Sunt igitur naturales. Sivero naturales non
sunt partes orationis. Nam eae partes, secundum
Aristotelem, ex instituto, non natură debent constare.”
The error here arises from giving too great a latitude
within certain limits is true ;
2 A
174 ...G. R. A M M A. R.
Grammar...viz. that words are significant ea instituto; for in truth
- distinguish them grammatically into classes, having Chap. I.
this proposition applies only to nouns (i.e. names of * - -
more or less distinctness of conception attäched to \-N-
distinct conception) and to words derived from them.
But in the nature of the human mind, intellect is
mixed up with feeling, the will is often confounded
with the reason ; and our desires, or fears, uncon-
sciously Inodify our conceptions or assertions. We
express in speech the transitions and mixt states of
the mind, as well as its clear, fixt, and determinate
distinctions ; and hence the interjection rises from a
scarcely articulate sound to a passionate, and almost
to an enunciative-sentence. -
According to CHARISIUs, CoMMINIANUs briefly defines
the interjection thus, “pars orationis significans ad-
..fectum animi.”—CAIUS Julius RoMANUs thus, “pars
orationis motum animi significans;” and PALABMon
thus, “ interjectiones sunt quae nihil docibile habent,
significant tamen adfectum animi.” DIoMEDEs gives
the following definition—“ pars orationis adfectum
mentis adsignificans voce incondità.” Vossius how-
ever, observes that apage 1 euge 1 and many others,
are not voces incondita; nor is the signifying an
affection of the mind peculiar to the interjection, for
even adverbs do this, as iracunde, irridenter, timide, &c.
He also censures the following definition, dictio inva-
. riabilis quae interjicitur orationi ad declarandum animi
affectum; for says he, “...interjections are not always
thrown in between the parts of a sentence; since we may
properly begin a sentence with an interjection.” His
own definition is, “ vox affectum mentis significans,
ac citra verbi opem sententiam complems.” This
definition agrees in the main with that which is to
be gathered from the works of that excellent old
grammarian, PRIscIAN ;
passionis animipulsu, per exclamationem, interjicitur:”
and finally from all these authorities it is clear, that
an interjection is a word showing an actual emotion of
the mind, without assertion ; which, therefore, we may
adopt as the definition of this part of speech.
To illustrate this definition, it may be necessary to
explain, first, what we here mean by a word; secondly,
why we say the interjection does not assert any thing,
and, thirdly, what we understand by an emotion of
the mind. -
First then, we take the term word in a large and
comprehensive sense, including not only what HARRIs
calls “voices of art,” but also what he terms “ voices
of nature, expressing those passions and natural
emotions of the mind which spontaneously arise in
the human soul, upon the view, or narrative of
interesting events.” Now, the expressions of mere
passion or emotion, as such, are éither effected with
some degree of volition, or they are extorted by a
physical necessity; but on the one hand, it may be
doubted whether pure physical necessity can operate
so as to produce speech properly so called, that is,
with any the slightest degree of articulation. To take
a striking instance, that of the Philoctetes of Sopho-
CLEs : we find him at one time exclaiming "A a, i, º,
at another A7, af, ai, af, and again:TIatra, Tara, Tatra';
but it is manifest that some power, beyond that of
mere mechanical impulse, must intervene to give even
the slightest of these articulations its difference from
the rest. On the other hand, if we admit that some
degree of thought enters into all those “voices,”
which express the emotions of the human mind, then
it becomes difficult, if not impossible, for us to .
\,
follows.
common, they constitute interjectional phrases, expletive
obscure.
the conjunction, preposition, or adverb, may often be
traced home to its origin in the verb, or noun,
We have said that the interjection does not assert. Do not
It manifests the existence of an emotion, to the sym-
viz. “ vox quae :alicujus
we say it shows actual emotion.
...them—to distinguish, for instance, in this respect,
between O ! is euge evax papae fie harrow !
pax hush hurrah'ſ alas ! bravo! &c. &c.; for such
words may form an ascending gradation from that
which is but just above mechanical impulse to that
which is but just below the assertion of a proposition.
Where indeed such an assertion takes place, that is
(speaking as a grammarian) where a verb is con-
nected with a noun, there is formed a sentence, which
may be resolved grammatically into its separate parts
of speech. But this is not all—the same difficulty
which is found in the ascending scale of expression,
occurs in the descending scale. A whole sentence is
sometimes suddenly interposed in a discourse, by the
mere effect of passion or strong feeling, without any
direct connection with what goes before, or with what
Some such sentences become popular and
parts of the daily conversation of particular sects,
parties, or classes of men ; they become habitual;
they are abbreviated, contracted, corrupted ; they
remain in language as words, sometimes with little
more articulation, or distinct meaning than those
other sounds which are ascribed to the effect of mere
natural impulse. Here then is a wide field for inter-
jectional forms in speech, comprehending the almost
involuntary exclamation, the word more or less signi-
fieant, and the phrase more or less imperfect and
And thus we see, that the interjection, like
pathies of mankind, but it does not declare that
existence as a fact addressed to their judgment. In
this respect therefore it differs from the verb. Again,
It does not merely
name the conception of an emotion, but gives to that
conception a vital energy as it were ; it shows the
speaker to be affected by its impulse, and is thus dis-
tinguished from a noun. It is true, that the limits
between an interjection and a noun or verb are not
always very easy to be observed. The imperative
mood, and the interrogative form of a verb have so
much of animation about them, that they easily pass
into mere interjections, and the same may be said of
the vocative case of nouns. In practice, we should
be inclined to say, that so long as a noun or verb
(distinguishable as such) enters into construction
with other parts of a sentence, or admits of gram-
matical inflection, according to its particular appli-
cation, it is to be considered as not having assumed
the character of a mere interjection ; whilst on the
other hand, the simply articulated exclamation, or
the noun or verb which has lost somewhat of its
original form and signification, while it expresses
emotion, is to be called an interjection. -
Though the interjection itself does not assert, it
may be coupled with an assertion, as one subordinate
sentence is coupled with another in a larger sentence.
This we have already exemplified in the passages—
“O ! that I had wings like a dove "—“O ! that
this too solid flesh would melt "--in both which, the
verbs (had, and would melt) are put in the subjunc-
ºtive mood, as dependent on the interjection, O !—just
G. R. A. M. M. A. R. 175.
if it were only to be met with in the “plays” of Chap. I.
SoPHocLES, PLAUTUs, Molier E, SHAKSPEARE ; or in S-
Grammar, as they would have been had the place of O ! been
* supplied by a verb, such as, “I wish,” “I desire,”
*—-
Emotions
expressed,
or the like. , r -
In an union of this kind the interjection precedes
the sentence with which it is connected ; for it has
Deen observed by Vossius, that though the name
interjection is given on account of its being thrown in
between the parts of a sentence, yet this is not essential
to the character of an interjection. It is , so named,
not because it is always, but because it is generally
so placed. “ Interjectiones dictae sunt quia sæpe
interseramtur orationi, non quod id perpetuum sit.”—
“Interjectio non Semper interjicitur ; quia ab ea
quoque rectè auspicamur.” — “Nec tamen de gada
ejus est, ut interjiciatur; cum per se compleat sen-
tentiam, nec rarö ab ea incipiat oratio.”
It is scarcely necessary to add, that the interjection
may stand quite alone. The mind may be satisfied
with giving utterance to its feelings, without entering
into any train of reasoning whatever; or those feel-
ings may be too intense and overpowering to admit
of any exercise of the discursive faculty. In either
case the interjection, to use the phrase of Vossius,
“ sententiam per se, complet.”
We come now to the most interesting part of this
discussion, namely the consideration of the emotions
expressed by interjections, or interjectional phrases.
And it is to be observed, that we here use the term
emotions, as we before used the term, word, in its most
comprehensive sense, including not only the gentler
movements of the mind which are sometimes so
called, but all kinds and degrees of passion, feeling,
or sentiment, which for the moment exclusively
govern and direct expression in speech. In this view,
so far is the interjection from being a “brutish”
thing, that the nice and philosophical examination of
it, as it has been practised in the different languages
and ages of the world, would furnish matter for a
better treatise than was ever yet written on the sen-
sibilities and sympathies of the human mind. Mr.
Tooke declares that “ the dominion of speech is
erected upon the downfal of interjections.”— If so,
the dominion of speech never was erected, nor ever
will be, till the minds of all men are “ a standing
pool” — incapable of being moved or incited to
action even by the naked calculations of a cold,
exclusive, hateful selfishness. Mr. Tooke himself
uses interjections, especially in those passages which
relate to matters affecting his own personal feelings
and interests. Yet he says, “ where speech can be
the “ romances and novels” of SIDNEY, CERVANTEs,
LE SAGE, FIELDING, how lamentable must be the
taste, how blind the philosophy, which would decline
the examination of this interesting part of speech
The emotions expressed by interjections and inter-
jectional forms of speech may be considered as of
three kinds, each running into the others by scarcely
distinguishable shades. The impulse of the mind,
which leads to the expression, arises either from strong
passion, from milder affections, (that is, emotions in
the narrower sense of the word,) or else from certain
feelings intimately connected with particular objects
or events.
Let us first consider the interjectional expression .
of the stronger passions, such as terror, fear, pain,
sorrow, hatred, eager desire, warm affection, and enthu-
siastic joy.
CHAUCER uses harrow ! as a common exclamation Harrow!
of the vulgar in situations of danger and terror.
Or I will cry out harrowe ? and alas !
And again,
That down he goeth, and crieth harrowe I die.
So in the Proces of the Seuyn Sages.
With both honden here yaulew here
Out of the tresses sche hit tere :
And sche to-cragged hire visage
And gradde harow, with gret rage.
It is probable that this exclamation was brought by
our Norman ancestors from France. In the old
coustumier of Normandy haro or harou is the cry of
the country, for pursuit of felons, or other demand of
justice.
DENYALDUs in his Rollo Normanicus interprets it
ha " Raoul / as a cry addressed to Rollo Duke of Nor-
mandy, whose name was formidable to all evil-doers.
This is what we now call the hue and cry from the
French huer, to hiss or hoot ; in the Statute of West-
minster, the First, A. D. 1272, it is termed crie de pays,
(see the ingenious remarks of the Hon. DAINEs BAR-
RINGTON on the statutes) and in the Statute of Win-
chester, A. D. 1285, heu e cri. -
Other etymologists may perhaps prefer the deri-
vation of this word from the adjective horowe, used in
old English for filthy, odious, in Anglo-Saxon horu,
horuwe, from the Icelandic hor, mucor, probably not
unconnected with the Latin horreo.
And thei wer noughtie, foule, and horowe. tºº
employed, they are totally useless; and are always - CHAUCER.
insufficient for the purpose of communicating our Sometime envious folke with tonges horowe.
thoughts.” “And indeed,” adds he, “where will IDEM.
you look for the interjection ? Will you find it
amongst laws, or in books of civil institutions, in
history, or in any treatise of useful arts or sciences 3
No : you must seek for it in rhetoric and poetry, in
novels, plays, and romances.” Mr. Tooke has for-
gotten one book, in which interjections abound from
the beginning to the end, and fill the mind with
impressions of the highest sublimity and pathos—
That book is the BIBLE. But if the interjection had
only to do with “ rhetoric and poetry,” surely its
sphere would not be narrow. If a knowledge of it
only led us properly to appreciate the lofty mind of
DEMost HENEs or CICERo, to read with true relish the
immortal verses of HomeR, VIRGIL, TAsso, MILTON.—
Be this as it may, the interjection harrow, although
its origin is involved in some obscurity, was evidently
used, either to denote a strong feeling of horror, or a
want of help, in which latter sense it would nearly
resemble the invocations for help, common in old
poetry. -
God help Tristrem, the knight !
He faught for Yngland.
O empti saile ! quhare is the wynd suld blowe
Me to the port quhare gyueth all my game 2
Help Calyope and wynd in Marye name.
The King's Quair.
Sir Tristrem.
It is obvious, that the simpler any articulations are, Ah! Oh!
the more easily they may be adapted by the flexibility
2 A 2
176
G R A M M A R.
Grammar. of the voice to express different states of the mind: a
>-v- slight degree of elevation or depression, of length or
Alas!
shortness, of weakness or force, serves to mark a very
sensible difference in the emotion meant to be ex-
pressed. Hence CINoNIo thus speaks of the Italian
ah and ahi :-‘‘ I varj affetti cui serve questa interiez-
zione ahed ahi sono più di venti; ma v'abbisogna
d'un avvertimento; che nell'esprimerli sempre diver-
sificano il suono, e vagliono quel tanto che, presso i
Latini, ah I proh 1 oh 1 vac 1 hei 1 papa. 1 &c. Ma questa
ë parte spettante a chi pronunzia, che sappia dar loro
l'accento di quell' affetto cui servono ; e sono—
d'esclamazione—di dolersi—di svillaveggiare—di pre-
gare — di gridare minacciando— di minacciare—di
sospirare—disgarare—di maravigliarsi–d'incitare—
—di dsegno—di desiderare—di reprendere—di ven-
dicarsi–di raccomandazione—di commovimento per
allegrezza—di lamentarsi—di beffare—ed altri varj.”
VossIU's observes of the Latin ah, that in ancient books
it is often written a without the aspiration ; as pro
is also written for proh ; and indeed the Greeks write
à without the breathing. Thus the 739th, and the
746th lines of the Philoctetes are both written’A, 3, 4, &.
PRISCIAN, too, says that a is the name of a letter, and
a preposition, and also an interjection. We need
scarcely observe that both ah 1 and oh 1 are used by
English writers as interjections of pain and sorrow.
In youth alone unhappy mortals live -
But ah / the mighty bliss is fugitive. IDRY DEN.
Oh I this will make my mother die with grief.
SHAKSPEARE.
Dr. Johnson says “ah, interjection—a word noting
sometimes dislike and censure—sometimes contempt
and exultation — sometimes, and most frequently,
compassion and complaint.” He also says “ oh,
interjection—an exclamation denoting pain, sorrow, or
surprise.” The Greek 'Id, and Latin Io, varying but
little in sound from O, were also sometimes used to
denote pain or sorrow. Thus Philoctetes in the agony
of his bodily torture cries tº, tº ; and Polymestor in
the Hecuba of Euripides uses the same exclamation.
Thus TIBULLUs says,
Uror, io / remove, sava Puella faces !
Lib. ii. Eleg. 4.
And in CLAUDIAN, Io seems to express the agony of
grief—
Mater io ! seu te Phrygiis in vallibus Ida
Mygdonio buxus circumsonat horrida cantu ;
Seu tu sanguineis ululantia Dindyma, Gallis
Incolis De rapt. Proserp.
The word alas ! was manifestly adopted into the En-
glish language from the French helds ! which is only a
corruption of the Italian ahi lasso, “ah ! weary 1” It
does not appear to have been known in England much
before the time of Chaucer, who frequently uses it.
How shall I doen 2 whan shall she come againe 2
I note alas ! why let I her go. Troilus, book v.
So in the early romances—
Thurch the bodi him pight,
With gile :
To deth he him dight
Allas that ich while !
Allas that he no hadde ywite,
Er the forward were ysmite,
That hye and his Ieman also
Sostren were and twinnes to.
Sir Tristrem.
Iay le Fraine,
Quhat sall I think? Allace quhat reverence
Sall I mester to your excellence 2
The King's Quair.
Evir allace 1. than said scho, -
Am I nocht cleirlie tynt? .
Peblis to the play.
‘Chap. I.
The sensation of weariness, expressed in ahi lasso, is .
also to be found in the Scottish interjectional phrase
“ weary fa’ you.”
Weary fa’ you Duncan Gray ! Old Scottish Song.
The latin vac, which is used only as an interjection,
in that language, is no doubt identical with the
Wae .
Anglo-Saxon wa and Scottish wae, which is our sub-
stantive woe : and it is to be observed that the Latin v
was in all probability pronounced like our w.
Was miscro milli 'TERENTIUS.
HIcKEs reckons wa is me / and, want me ! among the
Anglo-Saxon interjections of grief. In old English
we find “wo the be ſ”—“ woe worth !” &c.; and in .
Scottish “wae's me !” and “wae's my heart."—
Wales wo the be / the fende the confound !
R. DE BRUNNE.
Where ar those worldlyngs, now :
they were about any kyng!
Ah, wae be to you Gregory,
An ill death may you dee'
Ballad of Lord Gregory.
Wae's my heart that we shou'd sunder
LATIMER.
Scottish Song.
From wae it is probable came the verb wail, and
from waile wae came waileway, welaway, and corruptly
welladay. • . .
HIcKEs expounds the Anglo-Saxon wala wa 1 heu !
proh dolor and he adds in a note, “ haec interjectio
frequenter tropice ponitur pro dolore, praecipue in
scriptis Satyrographi, ut,
“Wote no wyght what war is ther that peace reineth
Newhat is witerly weale till welaweye him teache.”
We find it written variously, weylaway, wayleway,
waileway, wel awaie.
Betere hem were at home in huere londe,
Than forte seche Flemmyssh by the see stronde,
Whare routh moni Frensh wyf wryngeth hire honde.
Ant singeth weylaway.
Battle of Bruges.
Sche Seyd wayleway,
When hye herd it was so :
To her maistresse sche gan say,
That hye was boun to go. Sir Tristºrem.
Biclept him in his armes twain,
And oft allas he gan sain,
His song was waileway.
Amis and Afrnilown.
I set hem so a worke, by my faie,
That many a night they songen wel awaie,
CHAUCER.
Connected with wae and wail is the verb waiment,
which Chaucer uses for lament.
The swalow Proigne with a sorowful lay
Whan morow come gan make her waimenting.
Troilus, book ii.
Lastly, the Anglo-Saxon wala (in wala wa) seems
to be still retained in the Scottish interjection waly.—
O waly' waly up the bank,
And waly I waly down the brae :
Scottish Song.
Wo worth them, that euer
Welaway-
Och hone or 0 home 1 and O hone-a-chree appear och hone!
to be exclamations of grief used in the Gaelic language:
G R A M M A. R.
177
Grammar, and this word hone is evidently connected with the
\-y- verb to hone, of which LYE gives this account.—
Pape!
Fie
“ Hone after a thing, anxie rem aliquam appetere, agi
desiderio alicujus rei. Vox agro Devoniensi perfami-
liaris ; ab AS honyian, hogian quae occurrunt apud
Boethium, p. 31, 32. Unde haec nisi a Goth. hunyan 2.
—hwaiva aglu ist thaim hunyandam afar faihu ! quam.
difficile est iis qui soliciti sunt de pecuniis " To hoe
for any thing, according to GRoSE, is in the Berkshire
dialect to long for it,
DANTE commences his seventh
with the exclamation pape l—
Pape ' Satan, pape / Satan, alepe
Comincio Pluto con la voce chioccia.
It is curious to trace this exclamation into the Italian
from the Latin, and into the Latin from the Greek.
In Latin it seems to have chiefly expressed wonder.
Ecquid beo te 2 Mene 2 Papae 'TERENTIUS.
DoNATUs says “papa interjectio mira subito accipi-
3 y
entis:” and R. STEPHANUs says, “ admirantis inter-
jectio, habet enim in se affectum verbi miror.” It is
however admitted to be the Greek raſrat, which is
Imanifestly used by SoPHocLEs as an exclamation of
pain.- -
'Airóxcºxa, Tékuovº Bpúxopal, Tékvov traital.
IIatra, ratrā’ tramrā’ Tratra trairai.
- - - Philoct. v. 752.
It is said to be synonymous with Bagal, called by
ScAPULA “admirantis adverbium.” Perhaps however
it may have had some connection with ºrdnot, which
is used by Homſ.BR, and generally rendered O Dii !
or papa; '
*O trówou, º, uéya Trévôos Axatića ya'av icóvel.
O Dii, certè magnus luctus Achivam terram invadit.
Iliad. a. v. 254.
*o trówou, oiov 6% vu 6ešs Bporo) &ltudavlat'
Papae / ut Scilicet Deos mortales culpant
Odyss. a. V. 32.
In both which instances it is clear, that a strong
feeling of dissatisfaction or reprobation is intended to
be expressed. Plutarch says that this word Törot
signified in the language of the Dryopans the same
as baiuoves ; so that it was originally an invocation
of the minor deities.
Few words in any language more obviously deserve
the title of interjection than fie does in English ; yet
Mr. TookE ranks it among adverbs It is certainly
connected with the Gothic verb fiyan, Anglo-Saxon
feogan, fean, fian, Frankish and Alamannic fien, figen,
all which signify to hate ; but that it is to be re-
garded as the imperative of any of these verbs, may
be doubted. More probably the verb was formed
from the exclamation, of which WACHTER gives the
following account.
Saxones inferiores et Gallos hodiernos, sicut apud
Latinos fu. Germani superiores dicunt phui et pfui,
Graeci ºped. A flatu contra putidum.” R. STEPHANUs
explains the Latin fue “ interjectio ructum expri-
mentis.” (See Plautus, Most. 1. 37.) The Greek peo
sometimes expresses sorrow, and in this sense pro-
bably was the same as the Latin heu l—Thus XENOPHoN
says, ºped & Gºuë) \ºvz)—ºeſ too duépds—both relating
to persons dead : and Sophocles says ºped ráAas, heu,
me miserum ! The same interjection is also used to
express admiration ; as by ARISTOPHANES, ºfteå, pso,
# us?' duopiº Boöweva èv épytóww Yévet—where the
canto of the Inferno,
“ Fi, interjectio aversantis apud
scholiast observes, that ºped commonly expresses com-
plaint or indignation, but here admiration. -
Latin phy is an interjection of admiration, (see Terence,
Adelph. 3. 3. 59.) With the verb fian are connected
feide odium and feind hostis. Feide or fede is ex-
plained by WACHTER “ inimicitia aperta, persecutio,
vindicta. Anglo-Saxon fathth, Island, fad, Latino
Barbaris faida and feida, Belgis veede, Anglis feud."
Thus in the Lombard Laws (lib. 1. tit. 7. art. I and 15.)
we find “faida, id est inimicitia.” From faida was
formed the Barbarous Latin diffidare, which is the
origin of the French defter, and of our verb to defy.
The modern German fehde, the Low-Saxon veide, and
the Danish feide, all signify enmity. Feind, hostis, an
enemy, is properly, says ADELUNG, the participle of
the old verb fian to hate. This word is written by
ULFILA fijand, by KERo and OTTFRIED fiant, by WIL-
LERAM vient, in Anglo-Saxon feond, fynd, in Lower-
Saxon fijnd, in Danish fiende, in Swedish fiende, in
Icelandic fiande ; and in many of those dialects it
receives like the English fiend, the particular signifi-
cation of an enemy to the soul, an evil spirit. So in
old English- - .
The small fendes that bueth nout stronge
He shulen among men gonge.
Christ's Descent to Hell.
In the Scottish dialect the word fient, the Devil, is jocu-
larly employed as a sort of adverb, answering to our
colloquial use of such phrases as “the devil a bit,” &c.
When I looked to my dart,
It was sae blunt,
Fient haet o't wad hae pierced the heart
Of a kail-runt. BURNs,
They loiter, lounging, lank, and lazy,
Tho de'il haet ails them, yet uneasy. IDEM. .
Fie is also related to the interjections fol. 1 and faugh
and they all three express various modifications of
dislike. Thus the French fi, donc 1 is a slight, and often
a sportive reproof, while the English foh T is, as Dr.
Johnson says, “ an interjection of abhorrence.”
Foh one may smell, in such, a will most rank,
Foul disproportions, thoughts unnatural.
SHAKSPEARE.
Both foh ( and faugh are connected with the Anglo-
Saxon fah, and English foe, an enemy; but this
circumstance has led Dr. Johnson into an error in
grammatical reasoning. He says foh ' is from the
Saxon word fah an enemy, “as if one should
at sight of any thing hated cry out a foe!” This
supposes the conception of an enemy to be prior to
the more general emotion of dislike, or at least to
have received a name before the other had been
expressed by a sound. Now the contrary is so obvi-
ously probable, we had almost said so necessarily
true, that it must be taken as a first principle in all
rational etymology.
Chap. I.
So in \-y—’
Most authors reckon such expressions as ah me ! All me! &c.
Otuou ! hei mihi / me miserum, &c. as real interjections.
We may at least rank them among interjectional
phrases.
Cry but ah me / couple but love and dove.
* SHAKSPEARE.
VIGER calls Of an adverb of grieving : and he adds,
* ex of et dativo pºol conflatur novum dolentis adver-
bium oiuot, unde factum est verbum oiutógetv, h. e.
oiuot Aéyetv.” WossIUs, however, contends that a
word like miserum is not to be called an interjection.
1.78
G R A M M A. R.
Grammar. His expressions are these, “sunt alia, quae etsi animi
- adfectum testentur, ad hanc tamen classem non per-
Leeze me !
&c,
O si!
tinent. Ut malum ! quod rectè interjectionum numero
eximit CAELIUS CALCAGNINUs, lib. ii. epist. 8. Sed est
étrºpºvnua per interpositionem, ut Donatus quoque
annotavit super Terentii non uno loco. Similis ratio
in istis, miserum ! infandum ! nefas 1 atque aliis.” How
far an epiphonema per interpositionem, differs from
an interjection, it may not be easy to say. We think,
upon the whole, that when any word expressing
emotion, be it noun, verb, or other part of speech, is
prefixed or thrown into a sentence without connection,
and does not enter into grammatical construction
with the other members of the sentence, it may not
improperly be called an interjection. Thus the furious
clamor of the Jewish populace against our blessed
Saviour—“Apov, "Apov — which is rendered in our
translation by the interjectional phrase, “Away with
him'ſ away with him 1" might perhaps be called an
interjection, though it is in origin an imperative mood.
The same may be said of the expression of Philoctetes,
*OAw\a, and 'AyrdAw)\a (). v. 749 and 752) which
differ but little from the vulgar Irish exclamation,
“I’m kil’t.”—’Apov, "Apov, may be compared, in
point of grammatical form, to the expressions so
common in popular meetings off l off l—down / down /
&c. And “ away with him” may in like manner be
compared to the phrase “ out upon it !” -
Shy. My own flesh and blood to rebel !
SAL. Out upon it, old Carrion
SHAKSPEARE.
From interjections expressing painful emotions we
turn to those which express pleasurable emotions.
In the Scottish dialect we find leeze me ! signifying
“ it is dear to me.” g -
Leeze me on drink! it gie's us mair
* Than either school or college. |BURNS,
Leeze me on your curly pow,
Bonie Davie, daintie Davie || -
Old Scottish Song.
The explanation of this phrase is “ lief is me ;” and
of the meaning of the word lief, carus, dear, we have
already spoken under the head of adverbs.
The construction of the phrase is similar to the
German wohl uns ! and the English well is him 1
Well is him that dwelleth with a wife of understanding, and
that hath not slipped with his tongue ! Beclesiast. xxv. 8.
The Latin amabo, the future tense of the verb amo
I love, is often introduced interjectionally as an ex-
clamation of fondness.—
Vide, amabo, si non cum adspicias os impudens
Widefur. r TERENTIUS.
ENGRAPHIUS, an old commentator, on this passage,
says that amabo is used by the poets without any
meaning ; but on this Vossius justly remarks—“ si
blandientis est, otiosum esse nequit, eum multium
blanditiae et preces valeant.”
Eager desire is shown by such expressions as O si I
—oh, gi'n 1–0 utinam 1–e19e, E: Yáp, &c. r
Quamquam O si solita quicquam virtutisinesset.
- - - VIRGIL.
O 1 gi'n my luve were yon red rose,
That grows upon the castle wa'! - r
- - Old Scottish Song.
outinam tune, cum Lacedæmona classe petebat,
Obrutus insanis esset adulter aquis? OvIDIUS,
*o yépov, etë", as 6vuòs évi stă0soot pixotoriv, s
“as to y&vg6 toilo. - HoNIERUS.
E? yöp Aiyêo 6’ &ng. SoPHOCLES.
Chap. I.
Our common huzza and hurrah! seem to be old Huzza!
German shouts of exultation. It does not appear bravo! &c.
that they were ever used by the more southern
nations. . The Latin words io !' evac 1 and evaw 1 ex-
press nearly similar emotions. We have adopted
into the English language the Italian exclámation of
applause bravo 1 but chiefly at our theatrical enter-
tainments, Some persons affect to distinguish the
persons applauded by the terminations bravo, brava,
bravi; but this seems to be carrying the adoption of a
foreign idiom further than is suited to the character of
a mere interjection. A strong proof that interjections.
are not, as SANCTIUS contends, naturales, et idem apud
omnes, is, that even in such apparently unpremeditated
effusions of joy, people of different countries testify
their feelings differently, and each in the mode of his
own country. Thus where the Englishman exclaims
hurrah / the Frenchman would cry ha ( bon 1 and the
Italian bravo 1 º
Among milder emotions, we may reckon the various
degrees of surprise, sometimes mixt with a certain
dissatisfaction, which are indicated by the Scottish
hech the French comment 1 the English good now !—
good lack l—la, la, la 1–dear me ! sure / &c. &c.
Hech man dear sirs is that the gate
They waste sae mony a braw estate BURNS.
Comment donc 1 qu’est que c'est que ceci On dit que vous
voulez donner votre fille en mariage à un Carême-prenant 1.
- MoLIERE.
Good now / good now how your devotions jump with mine !
• - DRY DEN.
FLAM. My lord, having great and instant occasion to use fifty
talents, hath sent to your lordship to furnish him : nothing
doubting your present assistance therein.
LUcul. La, la, la 1–nothing doubting, says he A noble
gentleman 'tis, if he would not keep so good a house.
- g SHAKSPEARE.
Interjections expressing haste, slowness, and the
like, are common in all languages, and are usually
verbs or adjectives applied to this purpose. Such are
the old English yare the Scottish swith ! and hoolie 1
the German nur frisch I the French allons ! the Italian
piano 1 the modern English make haste come, come !
stay ! stop 1 hold ! &c. &c. n
Yare is from the Anglo-Saxon verb yearwian, to
prepare, as yearwian tiletanne, to prepare for eating.
Heigh, my hearts cheerly, cheerly, my hearts, yare ſ yare I
Take in the top-sail. SHAKSPEARE.
We have noticed swithe among the adverbs. It is
used interjectionally by BURNs. - - * .
Then swith an get a wife to hug.
The same poet uses hoolie for gently.—
But still the mair I'm that way bent,
Something cries hoolie J
I red you, honest man, tak tent
Ye'll shaw your folly.
. There are many interjections expressing slight con-
tempt—some by way of expostulation, as prithee l—
Some indicating the trivial nature of the object, as
pish l—some showing a degree of vexation in the
speaker, as pshaw ſ—some denoting the absurdity of
the thing in question, as the English tut 1 and tush I
the Latin vah / the French bah 1 and the Scottish
hout ! whilst others mark in the speaker a certain.
Hech' &c.
YarC
swith ! &c.
Prithee I
pish 1. &c.
G R A M M A. R.
179
Grammar.
Y
feeling of disgust or weariness, as the English humph
the French ouf / &c. -
TookE ranks prithee! among adverbs. Johnson
does not decide what part of speech it is, but merely
calls it “a familiar corruption of pray thee.” This
corruption, however, becomes in use a real inter-
jection. In the following instance the request is
, merely contemptuous.- -
w Pohl prithee / ne'er trouble thy head with such fancies;
. But rely on the aid thou shalt have from St. Francis.
- Old Song.
In the next, the request is more serious, but still the
abbreviation of the phrase marks a degree of fami-
liarity.— 2 -
Alas! why comest thou, at this dreadful moment,
To shock the peace of my departing soul ?
Away / I prithee leave me ! Row E.
Of the interjection pish ' Dr. Johnson thus speaks—
“ PISH ! interj. a contemptuous exclamation. This
is sometimes spoken and written pshaw 1 I know
not their etymology, and imagine them formed by
chance.” -
she frowned, and cried pish when I said a thing that I stole.
- - Spectator, No. 268.
A peevish fellow has some reason for being out of humour, or
has a natural incapacity for delight, and therefore disturbs all with
pishes and pshaws. Ibid, No. 438.
Pshaw would certainly be an odd way either of speak-
ing or writing pish : and an interjection is no more
formed by chance than a chronometer.
pshaw both appear to be natural exclamations ; but
they express different shades of contempt, the latter
showing more of ill humour and vexation than the
former. t . " •
Dr. Jon Nson says of tutº “ this seems to be the
same with tush '''' and of the latter he says—“TUs H !
interj.—of this word I can find no credible etymology
—an expression of contempt.”
Tut, man! one fire burns out another's burning.
- - * - SHAKSPEARE.
, Tush ’ say they, how should God perceive it; is there know-
ledge in the Most High 2 - Psalm ixxiii.
Among the few interjections, which, WALLIs says,
the English language possesses, he reckons “ tush
contemmentis.” It is probably connected with the
French verb tousser, to cough. Wallis renders it by
sense ! they're aye at sic trash as that, said the brother.
the Latin vah, which sometimes has a similar force.—
Wah 2 leno iniqua me non vult loqui. TERENTIUS.
With this latter the French ball ! is probably con-
‘nected, and it may also have some relation to the
French verb baailler, to yawn.
The interjection hout 1 is very common in the novels
-of the author of Waverley—
Weel, but Tronda kens this lad weel ; and she has often spoke
-to me about him, They call his father the silent man of Sum-
burgh ; and they say he's uncanny—Hout 1 hout ! Nonsense ! non-
The Pirate.
, Hout 1 seems to be an onomatopoeia of the same nature
as the English verb, to hoot, or the Scottish, to hoast,
to cough. - *
Humph 1 appears to be a mere imitation of the
natural expression of contemptuous discontent in the
following passage. -
SEMP, Must he needs trouble me on t—Humph : &
. . . . Boye all others? SHAKSPEARE,
dessus le dos du Bourgeois, qui crie oaf, parce
ºf • * p y e e f
TG/Cly Ou dºwvitat O'O L (tº apºap Teat OſOl).
Pish and
Ouf! a similar expression of the pain arising from Chap. I.
weariness, as in the Bourgeois Gentilhomme of Mo-
LIERE.-- -
Après que l'Invocation est finie les Derviches Otent l'Alcovan de
qu'il est las d’avoir
été long temps en cette posture. - -
Among interjections of soothing and encouraging, Euge! . .
of satisfaction, acquiescence, and the like, we may Gramercie,
reckon euge (well done); 0ápoet (be of good cheer); *
sodes, (I pray you); paramour (for love's sake); gra-
mercy J &c. -
The Latin euge ' was, in its origin, a compound of
the Greek gé and ye. * *
The Greek imperative 6dpoet is rendered by the in-
terjectional phrase “be of good cheer 1" in our trans-
lation of St. Matthew's Gospel, (c. ix. v. 2.)—0ápoet
- In the Latin vul-
gate, the correspondent word is “ confide.” In the
Anglo-Saxon translation, the passage runs thus—
“La 1 bearn, gelufe 1 the bedth thine synna forgifene.”
In the Gothic it is “ thrafstei thur barnilo l afletanda
thus frawaurhteis theinos;” where the verb thrafstyan
appears to have some affinity both with the Greek
6dparew, and the Teutonic trost, whence came the Bar-
barous-Latin trustis and antrustio, the Icelandic traust,
the Dutch troost, the German trost and getrost, the
Scottish tryst, and the English trust and trusty; all which
have the analogous significations offiducia, solatium,&c.
The French courage 1 answers not to the imperative
6dpoet, but to the substantive 0dporos. Of this word,
courage, the Dictionnaire de l'Academie says, “il se dit
quelquefois absolument, par maniere de particule ex-
hortative, courage mes amis' courage, Soldats" Thus
we use the words courage 2 comfort 1 patience 1 &c.
PAND... Courage 1 and comfort 1 all shall yet go well.
K. PHIL. Patience, good lady | Comfort gentle Constance :
- SHAKSPEARE.
The Latin sodes 1
is rendered by R. Stephanus
&eopat ; and he calls it “blandientis vel exhortantis
adverbium, seu mavis interjectio; quasi tu dicas quaso,
rogo, obsecro.” It is a contraction of the phrase si
audes.
Quintilio si quid recitares, corrige, sodes /
Hoc aiebat, et hoc. HoRATIUS.
Paramour ! par charité 1 and such like words and
phrases occur often in our old writers.
He spak vnto the emperoure,
Tak me thi sun, sir, paramoure ?
And I sal teche him ful trewly.
Seuyº Sages.
Ynough that hadde of warldes wele,
Togeder thai leved yeres fele,
Thai ferd miri, and so mot we.
Amen, amen, par charité,
- How a Merchant, &c.
Gramercy, or gramercies / which occurs often, in our
old writers as a mode sometimes serious, sometimes
ironical, of returning thanks, is a contraction of the
JFrench grand merci, great thanks.
When the king understood this word, he was right glad of it,
and said to Regnawde, I right gladly grant this to you : and with
the same ye shall have of me x thousand mark every year for to
maintain your estate. Sir, Said Regnawde, gramercie / and cast
himself at his feet. The Foure Sonnes of Aimon.
GoBBo. God bless your worship ! • '
BASS.. Gramercy / Wouldst thou aught with me? “
. . . . - SHAKSPEARE.
180
G R A M M A R.
Grammar.
Hush
whisht !
&c.
Fool. How do you, gentlemen?
SERV. Gramercies / good fool; how does your mistress 2
SHAKSPEARE.
Grand merci fagon de parler dont onse sert dans le style familier,
pour dire, je vous rends graces. Vows me donnez cela 2 Grand
merci J monsieur. * - IDict. de l’.4cademie.
Merci, peutestre de miseresce, par contraction.
MENAGE.
The signification of thanks is so different from that of
mercy, as to render it probable that there were two
derivations of this word in French, the one pointed
out by. Menage applying to the substantive merci,
mercy; the other from merces, recompense ; in which
latter sense grand merci would be in Latin (cupio tibi)
grandem mercedem. In the other sense, the French
have an interjectional phrase, merci de ma vie 1 an-
swering to our familiar exclamation, mercy upon us !
expressing astonishment. -
Eh 1 is used very ludicrously in the soothing expos-
tulations of the Bourgeois' Gentilhomme, with his danc-
ing and fencing masters, when they quarrel. -
M. D'ARMEs. Comment Petit impertinent :
Journ. Eh 1 mon Màitre d’armes :
M. A DANS. Comment Grand cheval de carosse'
JourD. Eh 'mon Măitre a danser , MOLIERE.
There are many words of admonition, such as
hush land whisht ! to keep silence; gare! to beware, &c.
Hush / seems to be the Gothic imperative hausei !
hear ! from the verb hausyan, which occurs frequently
in Ulfila's translation of the Gospels, e. gr. “Hausei 1
Israel, fan Goth unsar fan ains ist ;” “Hear, O Israel!
the Lord our God is one Lord." (Mark, c. xii. v. 29.)
The verb hausyan is manifestly from auso, the ear.
The denomination of this part of the body has a
similarity in many dialects, which may be divided
into two classes distinguished by the letters s and r.
Of the former class are the Gothic auso, the old Latin
ausis, and the Greek offs ; of the latter are the more
modern Latin auris, the Frankish and Alamannic ora,
ore, or, the Low-Saxon and Dutch oor, the modern
German ohr, the Danish ùre, the Swedish oera, the
Icelandic eyra, the Anglo-Saxon eare, and the English
ear. The Italian orecchio, and Spanish oreja, are cor-
ruptions of the Latin diminutive auriculus, and from
orecchio comes the French oreille. * -
Hark 1 is of the same family. From ohr, the ear,
the Germans have formed hören l to hear, and horchen /
to listen to ; as the Latins, from auris or ausis, had
audire and auscultare; and so the Anglo-Saxons had
hyran and hearchian, which are our hear and harken,
or hark, and of this last the imperative easily becomes
an interjection.
The Scottish exclamation whisht ! may not impro-
bably be of the same origin as hush I We pronounce
this word whist l and use it, as Johnson observes, 1st.
as an interjection commanding silence, 2dly. as an
adverb, 3dly, as a verb, and 4thly, as a noun, the
name of a well known game, requiring silent atten-
tion. BURNs uses whisht as a noun implying silence.
A tight outlandish hizzie, braw,
Came full in sight.
Ye need na doubt I held my whisht.
Nearly similar to this is our word hist 1 of which
JoHNson thus speaks –“ HIST, interj. of this word I
know not the original: probably, it may be a corrup-
tion of hush, hush it, husht, hist.”
the slightest possible.”
with the variation of gender and number, e. gr.
Hist! Romeo, hist / O for a falc’mer's voice,
To lure this tassel-gentle back again!
* - r SHAKSPEARE.
It is, however, to be observed, that the Romans
used the imperfect articulation 'st for the same pur-
pose. R. STEPHANUs says “ST [s] vox est silentium
indicentis. Ter. Phorm. v. l. 16. Quid 2 Non is,
obsecro, es, quem semper te esse dictitasti ?—C. 'st—S.
Quid has metuis fores " The Italians use the word
zitto 1 and the French say chut ! VARCHI, in his Erco-
lano, or Dialogo sopra le lingue, printed at Florence in
1570, says of this word, “Il quale zitto, credo che
sia tolto da Latini, i quali, quando volevano, che
alcuno stesse cheto, usavano profferire verso quel tale,
queste due consonanti 'st, quasi come diciano noi
zitto " It is used substantively for the slightest sound
possible. Thus BoccAccIo says “ senza far motto, o
zitto alcuno ;” “ without uttering a word, or sound,
It is also used adjectively,
E i buon Soldati, in campo, o in citadella,
Sistamno citti in far la sentinella.
Of the French chut 1 the Dictionnaire de l'Academie
merely says, “ Chut, particule dont on Se Sert pour
imposer silence.” * g
Gare 1 is a French interjection, the imperative of
the old verb garer. “On se sert,” says the Diction-
naire de l'Academie, “ pour avertir que l'on se range,
ALLEGRI.
que 1'on se detourne pour laisser passer quelqu'un, ou
quelque chose, gare! gare l gare de la 1 gare l'eau ! il
se dit aussi par maniere d'avertissement et de menace:
ainsi on dit à un jeune escolier gare le fouet !” Hence
the French garenne, a warren, or place for preserving
rabbits : guerir, anciently garir, to cure, to preserve
from disease. Guerite, anciently garite, a watch
tower, or centry box, which is the origin of our word
garret. It is said to have been formerly a custom in
the northern parts of Great Britain, in throwing water
from a high window to cry to the passengers below
gardyloo ! a corruption of the above cited French
phrase gare l'eau ! or gare 1 de l'eau ! The verb garer or
garir, is only another pronunciation of the old Teutonic
waren, which appears in so many forms and dialects.
Hence WACHTER says 1. “Waren oculis usurpare,
spectare, intueri. Hic primus verbi latissime patentis
significatus, quem MYLIUs quoque apud veteres ani-
madvertit in Archaeologo Teutone—Francis wwara saepe
est adspectus, et uwara tuon adspicere—ex eddem
fonte haud dubie est adjectivum war videns, in for-
mulà vetustissimä gewar werden videre, videntem fieri
visu cognoscere. 2. Waren ab oculo corporis trans-
fertur ad oculum animi, et tune significat, quantum
potest, considerare, curare, observare, servare, custodire,
cavere.” Of all these significations he gives exam-
ples, as follows:– -
Considerare, hence the Frankish wuara, considera-
tion, and wuara tuon, to consider ; e. gr. “ Ne duont
thes niet uuara thaz in so salo bim ;” “ nolite atten-
dere quod tam fusca sim;” (Willeram, cap. i. 6.)
hence also in modern German warnehmen is used for,
to observe. - -
Curare, hence the Frankish ungiuueri, carelessness;
wngiuuariu, inconsiderate : in modern German they
sometimes say warlos, for careless. *
Observare, hence the Frankish uwara, observation,
used by WILLERAM, cap. ii. 15. “ Ir doctores ecclesia,
tuot wwara,” “The doctors of the church observe.”
Chap. I,
G R A M M A. R. 181
Grammar.
*
Servare, the Frankish uuar a has also this sense,
e. gr. nim sin mihila uuara, “take much care of him,”
“keep your eye on him.”
Custodire, hence the Dutch waarande, Barbarous-
Latin warenna, French garenne, and English warren.
The Germans also use gewarsame for custodia. -
Cavere, as in the Frankish geuueri cautela. From
the participle warend in the sense of caution, came the
Barbarous-Latin warendia, and warandia, bail ; also
warendator, warandare, and warentizare; the French
garant, garantir, garantie ; the English warrant, gua-
rantee, &c. From waren in this sense came warnen,
to premonish, to provide against danger, to fortify.
In the Icelandic, varman is cautio ; in the German,
andern zu warnung is, “as a warning to others.” In
the Frankish, we have giuuarnot werdet, “ be prepared
for defence,” “arm yourselves.” Ingegin uuidaruuinon
so sculun uuir unsih uwarnon ; “ against the enemy so
should we arm ourselves.” (Ottfried, l. ii. c. 3. 3.)
Warninga is used by Willeram for munitio. Warnitus
signifies, in the Barbarous-Latin of the Frankish Capi-
tulars, “armed.” Hence the Italian guarnire, to for-
tify, and guarnigione, which is the French garnison, and
English garrison. -
Other analogous meanings might be added, as the
German gewarten, to expect or look for ; and gewartig
einer sache seyn, “to be aware that a thing will happen.”
In the Anglo-Saxon we find waer cautus, waere
cautio, werian, to defend, warnian, to beware, &c.
In Dutch waarande, a park for keeping deer or
other animals; waarborg, a surety; waaren, to gua-
rantee ; waarnemen, to regard; waarschouwen, to pre-
monish, &c. -
From these sources lastly come our English words
aware, beware, wary, and others already mentioned.
Ware pronounced by the lower classes of people
war ! is often used as an interjection of premonition,
as in war hawk | a notice to smugglers to avoid the
excisemen. Hence LYE, in his edition of Junius's
Dictionary, says— “WAR, cavere, prospicere. War
heads ! prospicite capitibus, ab A. S. warian ejusdem
significationis.
False witnesse is in word, and also in dede : in word, as for to
bireue thy neighbour's good name.—Ware 2 ye questmongers and
notaries : CHAUCER.
Among words of this kind we may reckon mum !
paa pair peace 1 silence 1
JoHNson says, “ Mumſ, interject. Of this word I
know not the original : it may be observed, that when
it is pronounced, it leaves the lips closed; a word
denoting prohibition to speak, or resolution not to
speak ; silence; hush.” It seems to be connected
with the Latin murmur, the German mummeln, the
Dutch mompelen, and the English to mumble.
Paw 1 is called by R. STEPHANUs an interjection ;
as in Plautus, “ par 1 te tribus verbis volo.” “Malè
autem,” says Vossius, “ quidam interpretes posuère
par inter admirationis interjectiones; nam, ut pluribus
ostendit Jos. Scaliger in Ausonianis lectionibus est
silentium sibi aut alteri imponentis.” It is manifestly
the noun par, used interjectionally, in the sense of
peace, quietness, silence, as we say, “hold your
peace 1" for “be silent "retain your peacefulness and
quietness. So the French use the exclamation pair
MADAME Jourd. Hélas ! mon Mariest devenu fou.
Mons. Journ. Paix º insolente ; portez respect à Monsieur.
le Mamamouchi . MoLIERE,
WOL. I.
. So Johnson says “Peace, interjection. A word . Chap. J.,
commanding silence.” - S-N-2
Hark! peace r
It was the owl that shriek'd, the fatal bellman
Which gives the stern'st good night.
SHAKSPEARf.
In pointing out an object, we say lo l in inviting Lo! Troth!
attention list 1 in giving assurance troth ! and there, &c.
are many other such interjections which occur in
books and conversation, as forsooth l indeed / well /
why I hum ! a'tweel I poz' ' &c. &c. -*
Lo is ranked by Mr. Tooke among adverbs :
why, it would be in vain to ask, since the only thing
he tells us of adverbs is, that they are not a separate
part of speech. He is, however, right in his etymo-
logy. Lo / is the imperative of the verb look, used
interjectionally. The old imperative, loketh, was
used in the same manner. - - - -
Loketh " Attyla the greate conquerour,
Dyed in his slepe, with shame and dishonour.
CHAUCER.
List 1 is in like manner the imperative of the verb
to list or listen. - -
—List list / Oh list /
If thou didst ever thy dear father love.
- SHAKSPEARE.
Troth ! is the old noun troth, truth, trust, fidelity.
D. PED. Now, Signor where's the Count 2 Did you see him 2.
BEN. Troth / my lord, I have played the part of lady Fame.
I found him here, &c. SHAKSPEARE.
This exclamation is still common among the lower
ranks of people in Scotland and Ireland : and it is the
abbreviation of a sentence such as “I say the truth".
—“ the truth is ;” or the like. .
Forsooth is little different, in its original meaning,
from troth ; “the sooth” being “ the truth.”
The pauyloun was wrouth, for sothe ywis,
All of werk of sarsynys. Syr Launfal.
The wer lasted so long,
Till Morgan asked pes,
Thurch pine !
For sothe, withouten les,
His liif he wende to tine. Sir Zºristrent.
Indeed / This word, which Johnson notes as an
adverb, and which is in fact made of the two words
in and deed, serves interjectionally to denote surprise,
with some degree of doubt. -
Well 1 and why I are elliptical interjections com-
monly used at the beginning of a sentence. When
any matter has been stated which is known to, or
admitted by the other party in conversation, the
speaker introduces his next position with the inter-
jection well –or else the person addressed exclaims
well I meaning to deny or dispute what follows. The
meaning is “ so far is spoken well;” but as to what
follows a further consideration is necessary. . When
a certain degree of impatience is meant to be ex-
pressed by the hearer, he exclaims well, well I mean-
ing, so far it is well, but you must proceed.
Why I used in a similar manner, expresses a transient
feeling of hesitation, or surprise. -
You have not been a-bed then 2
Why / no. The day had broke before we parted.
- - - - SHAKSPEARE.
Whence is this 2–Why? From that essential suitableness,
which obedience has to the relation which is between a rational
creature and his Creator, SOUTH,
2 B
18%
G R A M M A R.
speak that yet.
SHAKSPEARE.
Grammar; Ninus’ tomb, man! — Why? you must not
That to P Parbleu ! is an exclamation of the same kind but Chap. I.
~~~ at you answer yramus.
not quite so intelligible: It seems to be connected Q-2-
In the two first of these instances, the speaker
seems to have an indistinct intention of asking why
the question is put : in the third why the fact hap-
- F. It is as if he had said, why do you ask whether
have been a-bed the circumstance is trivial—why
do you ask whence this happens the reason is
obvious—why do you speak of Ninus' tomb you are
not yet come to that part of the play. But in all these
cases the emotion is transient, and satisfies itself, as
it were, with a brief interjection, instead of proceed-
ing to develope itself in a formal interrogatory. -
Johnson calls hum ! an interjection : and says it is
** a sound implying doubt or deliberation.”
See Sir Robert : Hum !
• And never laugh for all my life to come.
POPE.
Poz' is a vulgar colloquial expression introduced
into some of our comedies—a mere abbreviation of
positively; to express that a thing is certain. . . .
... We shall not dwell on the interjections of com-
pellation or inquiry—hollo / eheu ! heus tu ! hark ye
..friend ? &c.—nor on those of vexation, plague on’t 1
peste 1 dear heart / O dear ! &c. &c.
The religious opinions and feelings of mankind have
furnished a great variety of exclamations, impreca-
tions, and asseverations, all constituting interjections
or interjectional phrases. Hence the Greek Nat A&
Ată, the Latin a depol! The old English and French
parde 1 perfay 1 parmafey ! parbleu morbleu ! Christes
Tode 1 ouh for Saint Davy 1 for Seynt Martin by
Seint Eloy 1 with sorwe 1 Godamercy J God's face 1 gadso 1
egad 1 foregad I Gog's soul J by cock's bones 1 odsbodikins 1.
zounds ! besides a number of whimsical and arbitrary
exclamations of a similar nature, as gemini 1 snigs I
cadedis 1 tete ventre by my pan by my top 1 by bread
and salt 1 &c. &c. * * -
EUSTATHIUs contends that in the phrase val uá. Ată
the particle vai has the power of adjuration; but
HoogeveeN more accurately says that val has only the
power of confirming the adjuratory particles uá and
7tpos ; as in Nai uá 706e a chºrtpov (Iliad. a.) Nai Tpos
Tāv 6eſºv. (Aristoph. Nub.) -
Ædepol " is commonly written with an ae, and is
explained to be a contraction of per aldem Pollucis.
Vossius however contends that it should be written
edepol 1 and that it is made up of three words e or me,
a particle of adjuration, deus, and Pollur. But
MEURs1Us suggests that it was originally epol, as we
find Ecastor / Equirine I Ejuno I and that the d was
inserted merely as it is in medecum for mecum. At all
events this is a contracted exclamation relating to
Pollur.
- AEdepo! / mortalem paree parcum prædicas.
• * PIAUTUS.
Parde 1 perfay 1 and parmafey are the French par
Dieu, par foi, and par ma foi. - .
Ah! good-sir host! I haue wedded be,
These moneths two, and nat more, parde .
. - - - CHAUCER, .
When Alexander the king was dead, -
Thát Scotland had to steer and lead,
The land six years and more, perfay f : . .
Lay desolate aftir his day. BARBOUR,
SATHAN. Parmafey rich holde myne
Alle tho that bueth her yne. -
, Christ's Descent to Heſt. '
with morbleu 1 or mort bleu ! which was originally an
imprecation of death with putrefaction, either on the
speaker, if his words should not prove true, or on the
person addressed. They have, however, both dwindled.
into mere ejaculations of surprise. Ventrebleu and
teteblew 1 also occur in a similar sense.
M. DE P. Que me woulez vous -
MEDEcIN. Votis guerir, selon l'ordre qui nous a €té donné.
M. DE P. Parblew ' je ne suis pas malade. MOLIERE.
Comment, Marauts vous avez la hardiesse de vous attaquer
à moi! allons ! morbleu Y tue ! point de quartier : IDEM.
Le MARg. Vous, trève de colere !
* Ou je me facherai. - -
Fächez vous, ventreblew 1
DESTOUCHES.
IE CoM. Moi, je ments tétebleu º mon pere permettez. *
LE MARQ. Tout doux, il n'a pastort, et c'est vous qui mentez.
IDEM.
LE BAR.
For Christes rode 1 for Seynt Martin l are solemn
adjurations signifying before the cross of Christ 1 before
Saint Martin By Saint Loy, or Saint Eloy, is an
asseveration of similar import. *
*
Scho crid merci anough,
And seyd for Cristes rode /
What have Y don wough 2
Whi wille ye spille mi blode 2
- Sir Trist?'em.
Bi God, quoth Erl Florentin,
Who may that be, for Seynt Martin "
That ich here in mi forest blow 2
Guy of Warwick.
The Walsch without the toun euerilkon thei lay,
Whan thei the trumpes herd; that he to bataile blew,
And saw the yates sperd, than gamened than no glewe.
Ouh ! far Saynt Davy the Flemmyng wille him gile.
- R. DE BRUNNE,
There was also a nonne, a prioresse
That of her smiling was simple and coy
Her greatest oth was but by Saint Loy / Chaucer.
The wife of Bath however swears much more roundly
than the prioress. -
But now Sir, let me se, what shal I sain?
Aha! by God! I haue my tale againe.
And we find in old writers some singular adjurations.
of the divinity, as be Goddis face 1 bi Godes ore 1 &c.
Evyn in the Peth was Erle Dawy, • *
And til a gret stane that lay by, -
He sayd “ be Goddis jace, we twa,
The fleycht on us sall samyn ta.” , t
WINToN's Chronicle.
IDEM.
Brengwain the coupe bore
Hem rewe that ſerly fode
He swore bi Godes ore
. In her hond fast it stode.
Godamercy J is the more obvious exclamation (how-
ever improperly introduced into trivial discourse) God
have mercy J - - . . . *
ASir Tristrem,
CHAT. Go to then what is your rede 2 say on your mind.
- . Ye shall me rule herein. - • '
Dic. Godamercy 1 dame chat, in faith,
Thou must the gere begin. Gammer Gurton,
This irreverent and irreligious use of the sacred name
of God being felt to be very reprehensible, the vulgar
resorted to various modes of avoiding its sinfulness,
and yet giving way to their emotions in such excla-
mations as Gog's soul J. Gog's sides 1 cock's bones 1 gadso 1
foregad! egad odsbodikins ! bodikins I Godzounds !
zounds 1 odd so I odd's life 1 slife 1 &c. &c. “
G R A M M A R.
183
Grammar.
Nevertheless it appears to have long prevailed; for the Chap. I.
words pan and top in the following examples both \-N-
-signify head. - * - -
HoDG. Daintrels, Dicéon: Gog's soul / many save
This pece of dry horsbred, .
Chav byt no bit this live long day, .
‘. . . . No crome come in my hed. Ganimer Gurton.
Hope. Gog's sides 1 Diccon, me think ich hear him,
- Antarry, chall mar all. -
See, how he nappeth. See, for cock's bones 1
- How he woil fall from his hors atones. CHAUCER.
TMr. 'Tooke has thought proper to call gadso an
adverb, and to explain it by cazzo, an obscene Italian
word. He is wrong on both points. As egad I was
an evasion of by God; and fore gad 1 of before God;
so gadso 1 was an evasive contraction of by God it is
so, or by God, is it so 2 • T -
Od'sbodikins ! is a diminutive of by God's body; and
this is further corrupted to bodikins 1—So Godzounds !
and zounds ! are by God's wounds—odd's life 1 and
Ibid.
'slife 1 are by God's life. -
SHAL. Bodikins 1 Master Page, though I now be old, and of
the peace, if I see a sword out, my finger itches to be one.
e \ - SHAKSP. Merry Wives. &c.
/ He swore by the wounds in Jesu's side
He would proclaim it far and wide. CoLERIDGE.
Zounds'ſ sirrah, the lady shall be as ugly as I choose : she shall
have a hump on each shoulder, &c. -
- SHERIDAN, The Rivals.
Odd's life / when I ran away with your mother, I would not
have touched any thing old or ugly, to gain an empire. -
- Jbid.
O gemini I was probably an evasive imitation of
O Jesu ! What ad's miggs and 'snigs, odzooks ' and
zooks 1 were meant to resemble, it would perhaps be
difficult to ascertain. All these exclamations oceur
in ludicrous writings about the end of the seventeenth.
and beginning of the eighteenth century.
But the man of Clare Hall that proffer refuses:
'Snigs he'll be beholden to none but the muses.
- GEO. STEPNEY.
Ad’s niggs, crys Sir Domine,
Gemini 3 gomini :
Shall a rogue stay 2 T. ix’URFEY.
Come strike hands I'll take your offer:
Farther on I may fare worse.
Zooks / I can no longer suffer. Midas.
So in French we find, among the vulgar, numerous
exclamations of this kind, which it is not easy to
explain. Such are cadedis 1 pcrguenne / testigué !
morguene ! palsanguenne /
Cadedis 1 is a Gascon expression, perhaps signifying
originally chef de Dieu ! “ by the head of God " for
cap in the Gascon dialect was used for “ head.”
Thus cadet the younger son of a family, anciently
capdet, is derived from the Barbarous-Latin capitetum,
or little chief, the elder being the great chief.
Vindreut deuant une place nommée Malaunoy, dedans laquelle
estoit vn Capitaine Gascom, nommé le Capdet REMONNENT.
- Chronique de Louys XI.
pris là, tous deux, une gueble de com-
Parguenzie 2 J’avons
Medecin malgré lui.
mission
Ibid.
battre,
Ibid.
Palsanguenne / vela un medecin qui me plait. Ibid. -
The custom of swearing by the head of the person
making oath was very ancient : and is forbidden by
our Saviour in the well known text, “Neither shalt
thou swear by thy head; because thou canst not
make one hair white or black.” (S. Matt. c. vi. v. 36.)
Testigué vela justement l'homme qu'il nous faut.
Eh! Morguene laissez nous faire. S'il ne tient qu'a
la vache est à nous. - -
Loue is a gretter lawe, by my pan ' *-
Than may be yeuen to any erthly man. -
CHAUCER.
Sire Simond de Mountfort hath swore bi ys top,
Hevede he nou here Sir Hue de Bigot
Al he shulde graunte hen twelfemonth scot
Shulde he neuer more with his fot pot
To help Wyndesore. -
Battle of Lewes,
The military cries, halt 1 or sus 1 az armes God and
St. George / Bourbon, nostre Dame ! Montjoie, St.
Denys 1 &c. &c. are interjectional forms; as are the
naval exclamations yo, ho l avast ! 'vast heaving ! &c.
Mr. Tooke reckons halt 1 among adverbs, and says
it is the imperative of the Anglo-Saxon verb healdan,
to hold. It was probably borrowed by us as a tech-
nical expression from the French, who use the excla-
mation halte, la 1 derived from the German still halten,
to halt or stop. .
Richard aros, and toke hys wede,
And lept on Favel hys gode stede,
And sayde, Lordynges or sus / or sus!
That hath us warned swete Jesus.
Richard Coer de Lion.
A2 armes / he let crye there,
Ayenst the Sarazyns for to fare.
God and St. George / Talbot and England's right !
Prosper our colours in this dangerous fight.
- SHAKSPEARE.
Ibid.
Instead of the tumult and din of their anarchy, the human voice
divine may yet be heard. The antient spirit may yet revive. The
cry of Bourbon, nostre Dame / and Montjoie Str Denys / may
again resound through France. WILDE. 1793.
We need not dwell on the modern popular cries,
such as England for ever ! vive le roi vivat 1 & basſ
off I encore bis But the old Scottish and English
“ heve and how,” and “ rumbelow !" is singular enough
to be cited. -
With hey and how ! rohumbelow !
The young folk werfull bald.
Pellis to the Play.
They rowede hard, and sungge ther too
With heuelow / and rumbeloo ! -
Richard Coe?" de Lioz.
Your maryners shall synge arowe,
Hey how / and rumby lowe 1
º Squyre of lowe degree.
Salutations and valedictions afford several inter-
jections and interjectional forms, as hail I allhail I
welcome 1 benedicite greeting ! farewell, adieu !
Farewell happy fields
Where joy for ever dwells. Hail horrors ; hail
imfernal world. MILTON.
And while I stode, this derke and pale
Reason began to me her tale :
She saied alhaile / my swete frend.
CHAUCER.
Of hail 1 JUNIUs thus speaks, “ haec salutandi
formula expervetustà Gothorum, Anglo-Saxonumque,
|Francorumque, consuetudine.” Hence in St. Mark's
Gospel, (c. xv. v. 18.) the Greek Xalps Baaixed tâv
'Isèadwu ; and Latin “ave 1 rex Judaeorum,” are ren-
dered in Gothic “ hails thiudan Judaie,”—in Anglo-
Saxon “ halbeo, thu Judea cyning,” and in Frankish
“ heil coning Judeono.” From hail or heil come
the Alamannic heilizen salutare, heilizung salutation.
2 B 2
Grammar.
184
eleganti migratione ab omni pervenit ad totum, a toto
ad sanum et salvum,” and he might have added “a
salvo ad sanctum.” - -
I. In the sense of totus, we find the Greek 6Aos,
the old English hole, and modern English whole.
2. In the sense of sanus are the Gothic hailai sani,
wnhailans agroti, unhaili infirmitates, hailyan sanare—
the Frankish and Alamannic heilon sani, wuanaheilem
aegroti, heilaz sanitatis expers, heil, Sanatus, heilen,
sanare, heillihoor salubrius — the Anglo-Saxon hal
Sanus, unhal aegrotus, unharlo invaletudo, hailan Sanare
—in modern German heil werden to be cured, heilen
to cure, heilbar curable, heilkraut a sanatory herb,
Heilsam salutary, heilung a cure, unheilbar incurable, &c.
—in English hale, heal, health, healthy, healthful, &c.
3. In the sense of salvus we have the Anglo-Saxon
hal salvus, e. gr. “hwa maeg halbeon 7" “Who can
be saved 2'' (Mark x. 26.) hal salus, e. gr. “ the hal
ys of Judeum,” “ salvation is of the Jews,” (John iv.
22.) and halend the Saviour—the Frankish and Ala-
mannic heil and heiler salvus, e. gr. “ ther giloubit
inti gitoufit uuirdit ther uuirdit heil,” “ he who shall
believe and be baptized shall be saved :'' heili salus,
heilen salvare, heilant salvator—in modern German heil
well being, unheil misfortune, Heiland the Saviour, &c.
4. In the sense of sanctus, are the Frankish heilag
and heilig, the Dutch and German heilig, the Swedish
helig, the Anglo-Saxon halig, and the English holy,
sanctus. Hence the German verb heiligen and English
to hallow, sanctificare ; the old English hallows, Saints,
Allhallows all saints, &c.
The Saxons and old English used the expression
was hall 1 salvus sis' in drinking to each other :
whence the wassail or wassel-cup, and wasselling for
carousing. In Mr. DIBDIN's Typographical Antiquities
we find a collation of a MS. English Chronicle, with
Caxton's printed Cronicles of Englond, ed. 1480. The
MS. contains this passage,_
The Inonke toke a cup, and fillede hit with gode ale, and
broughte before the king and sette him on his knees, and saide Sir,
ºil. ! for nevere dayes of yhoure lyf me dronkeyhe suche
&lle, -
In the printed copy, the word is more accurately spelt
wassayl, being derived from the Anglo-Saxon wesan to
be, and hal, well. Wes hal! “ be well,” is therefore
the same, in substantial meaning, as the modern
English compliment, on a similar occasion, your
health ! the French a votre Santé ! the Italian salute / &c.
Welcome l'is a literal translation of the two French
words bien and venu, which when used together as a
substantive, are thus explained in the Dictionnaire de
l'Academie, “BIEN-VENUE, S. f. L'heureuse arrivée de
quelqu'un. Il ne se dit proprement que de la pre-
mière fois qu'on arrive en quelque endroit, ou qu'on
est requ en quelque corps: et parceque la coustume
est de payer quelque droit en y entrant, ou de faire
quelque regale a ceux qui en Sont, on dit payer sa
bienvenue ; donner un repas pour sa bienvenue.” -
Benedicite This Latin verb was used by our ances-
tors, in its proper sense, as an interjection of saluta-
tion, and more loosely as a mere expression of surprise;
as the common people still say bless me ! bless my
soul &c.
ROM.
FRIAR. —
Good morrow, Father .
Benedicite /
SHAKSPEARE,
Wachter thinks that the root of hal was all “quod
- - - A benedicite J
What aileth swiche an old man for to chide 2
- - , CHAUCER.
Greeting ! is a word which has travelled very far
from its origin. In Greek we find ºp'ºw, and kpāga,
clamo, kpavy) and kptyń, clamor. The Gothic greitan,
Cimbric grada, Icelandic graata, Swedish grata, Danish
grade, Spanish gridar, Italian gridare, French crier,
Scottish to greet, Dutch kryten, old English grede,
and modern English to cry, all signify to weep, cry,
call aloud, &c. WACHTER says the old German kreide,
clamor, is from krathen, clamare; and krathen appears
to be connected with our verb to crow, and to give
name to krathe, in Frankish chraio, Dutch kraai, Anglo-
Saxon and Scottish crawe, English a crow, corvus.
From kreide came the Barbarous-Latin crida, and
Italian grida, a proclamation. Gridare in Italian is
explained “ mandar fuori la Voce, con alto Suono—
manifestare, pubblicare—mostrare, far comprendere—
garrire, riprendere.”—Graetig is applied in Suabia to
signify a squalling child.
very early used in Anglo-Saxon and old English, for
to salute or wish joy to a person : and greeting was
consequently used as a noun, signifying salutation or
well being. Thus a charter of King Edward the
Confessor begins, “ Eadweard Kyng gret Rodberd
biscop.” The letter of king Henry III. A. D. 1258,
begins “Henr. thurg Godes fultume King (&c.) send
igretinge to alle hise halde," i. e. Henry by God's
grace King of England, &c. sends salutation to all
his subjects. Afterwards the verb “ send” was
omitted in English, as the correspondent verb had
before been in Latin and French ; for the French
copy of the last mentioned letter has “Henri, par le
grace Dieu, Rey de Engleterre (&c.) a tuz ses feaus,
saluz :” and another letter of the same year begins,
“ Domino Papae, Rex Angliae salutem.” Thus greeting
having lost its use, in regular construction, as a noun;
and its original signification as such, being almost
forgotten, it remains in modern official documents
merely as a sort of interjection.
Farewell ! is absurdly called by Johnson an adverb.
He says: “ FAREwBLL, adv. This word is originally
the imperative of the verb fare well, or fare you well ;
sis felic, abi in bonam rem ; or bene sit tibi ; but in time,
use familiarised it to an adverb ; and it is used both
by those who go, and by those who are left.” So R.
STEPHANUs says of the Latin vale : “imperativus, quo
utimur quum recedimus, vel quum remanentes respon-
demus abeunti.” -
The long day thus gan I prye and poure,
Till Phebus endit had his bemes bryt,
And bad go farewele every lef and floure.
The Ring's Quair.
To fare well is in modern usage applied chiefly to
the food and other enjoyments of life ; and the noun
fare has this among other significations; but they all
come no doubt from the Gothic faran, Anglo-Saxon
faran, Alamannic faren, Cimbric fara, Danish fare,
and Dutch vaeren, to go ; which are connected with
our for, fore, forth, further, &c.
Well to fare 1 is used as an interjectional phrase in
Gammer Gurton's Needle.
Hail fellow Hodg ! and well to fare,
With thy meat, if thou have any
The Italian and French valedictions addio ! and
adieu ! are manifest interjections, being abbreviations
of the phrase “I commit you to God.”
...”
It seems that to greet was .
Chap. I.
* G R A M M A. R.
185
Grammar.
* - -
A more unmeaning exclamation cannot well be
S-V-' conceived, than that appears at first sight to be, which
O yes'
Imitative
sounds.
is used by our public criers to call attention in courts
of justice, &c. viz. O ! yes 1–0 ! yes 1–0 ! yes / It
is however the imperative of the old French oyer,
the modern ouir, a corruption of the Latin audire, to
hear; so that it exactly coincides with the exclamation
hear ! hear ! so much used in our senate. Both O yes .
and hear ! may properly be styled interjections. The
same we may say of many miscellaneous exclamations
applied to incidental circumstances, as “Anon anon,
Sir 1" used by the waiter, Francis, in K. Henry IV—
coming 1 the more modern exclamation of a waiter—
going ! that of an auctioneer—lullaby 1 and hushaby |
those of a nurse lulling and hushing an infant, &c. &c.
Finally, we may revert to the imitative sounds, of
which we before spoke. Although considered as
mere imitations they can hardly be called words, or
reckoned among the parts of speech ; yet it often
happens that a certain degree of mental emotion is
mixed with their utterance, and they may then perhaps
be not improperly denominated interjections. Thus
the lively Scottish poet BURNs gives great animation
to his description by the sounds click 1 jee 1 fuffl &c.
When click 1 the string the sneck did draw ;
And jee 1 the door gaed to the wa'. The Vision.
He bleez'd owne her, and she owne him,
As they wad never mair part;
Till fuſſ 1 he started up the lum,
And Jean had e'en a sair heart.
The German poet BURGER uses similar onomatopoeias
with equal effect,
- Und jedes heer mit sing und sang,
Mit paukenschlag, und kling ! und klang !
Geschmückt Init grünen reisen,
Zog heim zu Seinem hāusen.
The sounds kling and klang are connected with the
German verb klingen to sound (like a bell) with our
words clink and clang, the Greek k\dºw and k\ayº),
and the Latin clungor and clango. The German com-
posite wohlklang signifies harmony. -
Very similar to this is our ding, dong bell ! used
interjectionally.—
Sea-nymphs hourly ring his bell.
Hark! now I hear them—ding, dong, bell /
SHAKSPEARE.
The same may be said of tantara ! tantara 1 imitating
the trumpet; row-de-dow-dow ! the drum; rat-a-tat-tat 1
the knocking at a door, &c.
The German interjection schnapp ! or schnapps 1
(quickly l) may perhaps be ranked among these imi-
tative sounds. It is however connected with the
German verb schnappen, the Swedish snappa, and our
snap. The Dutch say met een snap, “ in a trice.”
The French habber is to snap, and the Italian chiappare
appears to be of the same family. Our word slap 1 is
used like the German schnapp ! in the following lines
of a ludicrous poet, L -
Up comes a man, on a sudden, slap / dash /
Snuffs the candles, and carries away all the cash.
ANSTEY.
The French glou ! glour ! is used to imitate the
gurgling sound of liquor from a bottle, as by Sganarelle
in the Medecin malgré lui.-
Qu'ils sont doux, bouteille jolie :
Qu'ils sont doux, vos petits glou glour 1
MoLIERE.
Hallow e'en,
Ienore.
The songs and cries of birds are imitated by such Chap. I.
sounds as pepe ſ—jug, jug 1–tirra-lirra 1–too-hoo !-
cuckoo : &c.
Now, Swete bird, say ones to me pepe 1
- The King's Quair.
And murmurs musical, and swift jug. 1 jug. 1
CoLERIDGE.
Then nightly sings the staring owl,
Tw-whit! to—who a merry note
SHAKSPEARE.
The lark that tirra-lirra ! chaunts. IDEM.
The cuckoo then on ev'ry tree,
Mocks married men, for thus sings he—
Cuckoo ! cuckoo !—O word of fear !
In like manner many loose syllables and imperfect
articulations are used to imitate human laughter,
coughing, whistling, singing, &c. such as ha 1 ha 1 haſ
—te 1 he ſ—ugh !—whew l—tol de rol lol / &c. which
require no particular notice.
We have not pretended to reduce the great variety
of interjections to a complete and systematic arrange-
ment. The only attempt of the kind which deserves
attention is that of the very ingenious Bishop WILKINs;
but it is a mere outline, and is meant to include only
“ rude, incondite sounds,” the “natural signs of our
mental notions or passions,” and “ several of which
are common with us to brute creatures.” It is as
follows :-
1. Solitary, the result of a surprised
1. judgment, denoting
1. admiration, heigh -
2. doubt or consideration, hem hm hy!
3. contempt, pish ! shy | tysh
2. affection moved by apprehension of good or evil
1. past {. ha I ha / he
g sorrow, hoi ! oh oh ah
2. present { love and pity, ah alack alas !
g hate and anger, vauh ! hau !
desire, O ! O that
aversion, phy :
IDEM.
3. future
2. Social
I. preceding discourse
I. exclaiming, oh Soho !
| 2. silencing, 'st hush |
2. beginning discourse
1. to dispose the senses of the hearer
1. bespeaking attention, ho oh
2. expressing attention, ha!
2. to dispose the affections of the hearer
I. by way of insinuation, eja! now !
- 2. by way of threatening, vap ! wo! *
Even this short scheme shows the error of the learned
WALLIS in supposing that there were but few inter-
jections in the English language : and it furnishes
ground for two or three other observations of some
importance in grammatical science. The first is,
that no precise line can be drawn between inter-
jections consisting of “incondite sounds” the “natural
signs” of mental emotion, and exclamations derived
from a partial exercise of the reasoning faculty; for
among the sounds enumerated by Wilkins we find
alas ! derived from the regular Latin adjective lassus—
alack 1 from the English verb to lack, and Dutch
laecken—hush / from the Gothic verb hausyan—and
va. 1 identical with the English noun woe. That the
noun and the mere incondite sound are used as equi-
valents, and with the same sort of grammatical con-
struction, we see in the following lines of BUTLER,-
186
**
G R A M M A. R.
Grammar. "
Intrust it under solemn vows
Of mum ! and silence, and the rose Hudibras.
We may next observe, that the same interjection
expresses very different emotions. Thus we find Wilkins
describing oh 1 as an expression of sorrow, as an
exclamation preceding discourse, and as bespeaking
attention in discourse. These variations then depend
not on the articulation, but on the intonation; that is,
not on the letters which go to form the word, but on
the elevation or depression of voice in pronouncing it :
but this is not peculiar to the interjection oh ( or to
the “incondite” interjections generally; for the same
may be observed of any nouns or verbs used inter-
jectionally. Thus we say impatiently, “ well 1 and
what of that "–or with patient acquiescence, “well !
never mind: it can't be helped.” So there is great
difference between the affected gravity of Falstaff's
imprecation, plague 1 and the same imprecation
seriously uttered against Apemantus. •
- FALST. A plague of sighing and grief! It blows a man up,
like a bladder. First part Henry IV.
CAPH. Stay, stay, here comes the fool, with Apemantus.
SERV. Hang him . . He'll abuse us.
, ISID. A plague upon him Dog | 7'imon.
The scheme of Wilkins too, short as it is, helps to
illustrate the connection which we have already pointed
out between the interjection on the one hand, and the
vocative case, imperative mood, and interrogative form
of the verb on the other. Wol which he properly
ranks among interjections, is the vocative case of a
noun, so used.—Hush ſ (like hark lo oyez &c.)
is the imperative mood of a verb. The interrogative
is in some degree implied by hem 1 or hm which he
considers as interjections of doubt. It is more dis-
tinctly marked in French by the word puis, as ex-
plained in the Dictionnaire de l'Academie. “On dit,
par ellipse, et par interrogation, et puis 1 pour dire,
eh bien qu’en arrivera-t-il 2 que s'ensuivra-t-il
que fera-t-on apres?, Ou bien, qu’en arriva-t-il 2
que s'ensuivit-il 2" .
Thus have we shown the propriety of ranking the
interjection as a separate part of speech, determinable
as all the other parts of speech are, not by its sound
or derivation, but by its use in the particular passage
which may be under consideration. We have shown
that it evinces actual emotion of the mind, but does
not assert the existence of such emotion. Lastly, we
have endeavoured to illustrate the nice shades and
gradations by which as emotion passes into conception
or assertion, in the human mind, and vice versd; so
the interjection rises to a noun, a verb, or a phrase,
and the phrase, verb, or noun sinks into an interjec-
tion. And thus have we concluded our survey of
words as distributed into those classes which gram-
marians call the parts of speech. :
§ 10. Of particles.
Having treated of sentences and words, it only
remains to inquire whether we cannot carry our
grammatical analysis still further, and examine the
constituent parts of words. Now, words may be
resolved into syllables; and syllables may be resolved
into the articulations, which are marked in writing by
fetters: and this part of grammar is called orthography;
but as it relates wholly to the sound, and not at all to
the signification of words, it has nothing to do with
our present inquiry. It is part of the art of grammar;
but no part of the science. - -
Chap. 3.
Nevertheless, though we have called words “ the
primary integers of significant language,” and have
denominated the classes into which they are divided
the parts of speech; yet, even with reference to signi-
fication, there are cértain fractions, if we may so
speak, which go to make up these integers. Thus if
we say, “Johnson was learned”—“ Friendship is delight-
ful;” each of these sentences, as a sentence, contains
three, and only three, significant parts; viz, a subject,
a predicate, and a copula; and each of these parts is
a word. But if we take one of these words, and
inquire how it comes to possess its actual signification,
we may find that this is owing to the peculiar force
and effect of its separate portions. Thus, in the word
Johnson, there are two portions, John and son, which,
taken separately, would be significant; and which,
when put together, form a third signification relating
to the two former. Again, in the word friendship,
there are two portions, friend and ship, each signifi-
cant, when taken separately; and the relation of the
word friend to friendship is very obvious, but the re-
lation of ship to friendship is not equally so, at first
sight, though it may be discovered by study and
reflection, as will hereafter be shown. Lastly, the
word learned, may, in like manner, be divided into
two portions, learn and ed, of which the former has a
clear meaning of its own ; but the latter, if it ever
had a distinct and separate meaning, has long since
lost it, and serves only to mark that learned is a parti-
ciple of the verb to learn. The three words, Johnson,
friendship, and learned, therefore, are manifest com-
pounds, each consisting of a primary part, which is
modified by a secondary part. John is modified by
son, friend by ship, and learn by ed. The primary parts
in such compounds are commonly words, that is, when
used separately, they have a plain and distinct signi-
fication of their own. The secondary parts may or may
not have such separate signification; and their signi-
fication, if any, may be more or less obvious. These
secondary parts, we call particles, when so used in .
composition. Thus, we say, that in the word Johnson,
son is a particle; in the word friendship, ship is a
particle; and in the word learned, ed is a particle.
Particles modify words in three different ways, and
with three different effects.
1. In the ordinary compounds, such as Johnson,
"toon mete, overtake, forewarn, erewhile, elsewhere, there
is no alteration of the principal word, either by chang-
ing the grammatical class to which it belongs, or by
varying the grammatical construction of the sentence
in which it is used. -
2. In such compounds as friendship, bisyhed, procu-
rour, gadelyng, avette, masterless, delightful, blaunchard,
lovely, lolich, sweetly, &c. the grammatical class of the
word is more or less altered ; thus, from the personal
substantive, friend, we form the abstract substantive,
friendship ; from the common substantive apis, we
form the diminutive substantive avette ; from the
common adjective blanche, we form the diminutive
adjective blaunchard; from the adjective busy, we form
the substantive bisyhed; from the substantive master,
we form the adjective masterless; from the adjective
sweet, we form the adverb 'sweetly, and so forth. -
3. In such compounds as growen, bean, makede,
walked, monethes, children, &c. the principal word is
6 R A M M A. R.
187
Grammar, varied in its construction, by the particles en, on, ede,
' ed, es, &c.; and thus are formed those inflections,
Son, &c,
respectively, as particles.
-*
which grammarians call declensions and conjugations.
Weshall trace the first sort of compounds, beginning
with the more obvious, and proceeding to the more
obscure. - - * --
The word Johnson was manifestly in its origin
nothing more than John's son. Thus in all languages
have been formed patronymics, the most ancient of all
family names. The Greeks did this in several instances,
whence such names as AEacides, Pelides, Atrides, &c.;
but the Romans adopted it generally at a very
early period of their history. “Remarquons sur
les noms propres des familles Romaines, (says M.
DE BRossEs,) qu'il n'y en a pas un seul chez eux, qui
ne soit terminé en ius, desinence fort semblable a
1' ºtos des Grecs, c'est-a-dire filius—par oil on pour-
rait conjecturer que les noms des familles, du moins
ceux des anciennes maisons, seraient du genre patro-
nimique.” Thus Caecilius was Caeculae duos, Julius,
Juli tºos, AEmilius, AEmili ºtos, &c. Mr. TookE says,
“I think it not unworthy of remark, that whilst the
old patronymical termination of our northern ancestors
was son, the Sclavonic and Russian patronymic was
of. Thus whom the English and Swedes named Peter-
son, the Russians called Peterhof. And as a polite
foreign affectation afterwards induced some of our
ancestors to assume Fitz (i. e. fils or filius,) instead of
son ; so the Russian affectation, in more modern times
changed of to vitch, (i. e. fitz, fils, or filius) and Peter-
hof, became Petrovitch, or Petrowitz.” The Irish
patronymic O' may possibly be of the same origin as
the Russian of The Welsh 'P is well known to be
ap, an abbreviation of mab, a son, as Price for Ap
Rhys, Powell for Ap Hoël, &c. : the Scottish high-
landers used the cognate word mac, a son, for their
patronymical prefix, as in Mac Donald, i. e. the son
of Donald, Mac Kenzie, (i. e. the son of Kenneth,)
&c.; while the lowland Scotch used still a different
mode of expressing the same thing, by prefixing to
the son's name the genitive case of the father's, as
Watt's Robin, for Robert the son of Walter; Sim's Will,
for William, the son of Simon, whence arose such
family names, as Watts, Sims, and the like : and so
much for the partieles son, ius, fitz, of, vich, mac, O', 'P,
and 'S. - . . .
. The proper name, Johnson, is no less obviously a
compound, than watchman, spearman, boat-hook, and
thousands of similar words in common use. There
are also many that have fallen into disuse, though still
perfectly intelligible; e. gr. nonemete, a meal formerly
eaten by artificers at noon, but which seems to be
distinguished from dinner.
Divers artificers and laborers reteyned to werke and serve,
waste moch part of the day, and deserve not their wagis, summe
tyme in late commyng, unto their werke, erly departing therfro,
Honge sitting at ther brekfast at ther dyner and nonemete, and
long tyme of sleping at after none.
g ty - § Stat. 2 HEN. VII. c. xxii. MS.
And as we have the word noon meat, so we have the
words, noontide, noonday, mid-day, mid-night, forenoon,
afternoon, &c. all nouns compounded on similar prin-
ciples; for as noon modifies meat, so mid modifies night,
and fore modifies noon : and thus noon, mid, and fore
are equally to be considered in these three instances
So, in the compound verb
overtake,...over is a particle modifying take; and this
particle, over, is sometimes corrupted into or, as in the
word orlop, which is a platform of planks laid over S
the beams in the hold of a ship of war; so named
from the Dutch overloopen, to run over, and anciently
written in English overlopps. . .
Somuche as they shall put greater nomber of people in the
castelies and ouerlopps of their shypps they shalbe the more
oppressed. - Nicolls's Thucydides, fol. 191. a.
In Danish, this same preposition over, written ober, is
used as a particle in compound nouns, as oberdommer
the chief-justice. - t
We have already noticed the particle fore which
occurs in forewarn, and in many other compound
verbs: e. gr. -
- Forwakit and forwallouit thus musing
Wery forlyin, Ilestnyt sodaynlye.
- - - The King's Quair.
Erewhile and elsewhere are compound adverbs, of
which we have already noticed the constituent parts
ere, else, while, and where. In addition to what we
have said of else, we may observe that the particle el
occurs with a similar effect in the Danish eller, “ or,"
and ellers, “ else.”
In proceeding to compounds, which, by course of
time, and change of pronunciation, have become less
obvious, we will begin as before, with some proper
names. M. DE BRoss Essays with great truth, “ tous
les mots formant les noms propres, ou appellatifs des
personnes, ont, en quelque langage que ce soit, un
origine certaine, une signification déterminée, une
étymologie veritable.” VERSTEGAN has preserved a rude
distich not unworthy of notice, in this respect.
In foord, in ham, in ley, in tun
The most of English surnames run.
Thus, says he, “ the sirname of Rainford, now
Rainsford, seemeth to have risen by reason that the
first of this name had his dwelling at a passage or
foord caused through raine.”—“ Ham originally sig-
nifieth a coverture or place of shelter, and is thence
grown to signifie one's home, as now uncomposed
we pronounce it — it is one of our greatest ter-
minations of sirnames, as of Denham, for having his
home or residence downe in a valley; of Higham for
the situation of his ham or home upon high ground ;
and accordingly of many others.”—“ Legh, ley, or lea,
howsoever wee do now distinguish these terminations,
I take them to have been anciently all one, and to
signifie ground that lieth unmanured and wildly over-
growne;”—hence Berkley, “ of birch trees, anciently
called berk,” Bromley, “ of the store of broom,” and
Bramley, “ of lee or legh ground bearing brambles,”
Of the name Lesley, he relates this story, “A combat
being once fought in Scotland betweene a gentleman
of the family of the Lesleyes and a knight of Hungary,
wherein the Scottish gentleman was victor: in memory
thereof, and of the place where it happened, these
ensuing verses doe in Scotland yet remaine. -
* Betweene the lesse ley, and the mare
- He slew the Knight, and left him there.”
“Though the name of hedge doe anciently appertaine
to our language, yet we also used sometimes for the
same thing the name of tun. In the Netherlands they
yet call it a tuyn ; and in some parts of England they
will say “ hedging and tining.” Our ancestors, in time
of war, to prevent themselves from being spoyled,
would, in stead of a palizado as is now used, cast a
ditch, and make a strong hedge about their houses,
Ford, &c.


188
G R A M M A R.
Grammar, and the houses, so environed about with tunes or
\-v- hedges, gat the name of tunes annexed unto them. As
Worth,
Cote-tun, now Cotton, for that his cote, or house, was
fenced or tuned about ; North-tun, now Norton in
regard of the opposite situation thereof from South-
tun now Sutton. Moreover, when necessity, by reason
of warres and troubles, caused whole thorpes to be with
such tunes environed about, those enclosed places did
thereby take the name of tunes afterward pronounced
towns.” To the same effect JUNIUs says “Town, villa,
vicus, pagus, et in genere, quilibet locus conclusus et
circumseptus. A. S. tun, Al. zun, B. tuyn, sunt ab A.S.
tynan, betyman, claudere, circumsepire.” And LYE says
“ time the door, fores claudere, ab A.S. tyman claudere;"
which expression “tine the door” is also noticed by
GRose in his Provincial Glossary. In Dutch, tuyn is in
its first sense the hedge of a garden, and then the
garden itself: it is also used for some other inclosed
places, as een hout-tuyn, a wood-yard. So in Scotland,
the toon means the inclosure round about a farm-house,
and in Cornwall the town-place means the farm-yard.
The last mentioned particle ton, has much affinity in
point of signification to the particle worth, also very
common in English names of places and thence of
persons. “ Anciently,” says VERSTEGAN, “ it was
wearth, and weard, whereof yet the name of werd
remaineth to divers places in Germany, as Thonawerd,
(Donawert, Danubii Insula,) Keyserwerd, Bomelswerd
and the like ; and in England, to the same sense
and signification, the names of Tamwoorth, Kenelme-
woorth, and the like. A wearth or werd is a place situate
betweene two rivers, or the nooke of land where two
waters, passing by the two sides thereof, doe enter
into the other ; such nooks of ground having of old
time beene chosen out for places of safety, where
people might bewarded or defended in.” Verstegan has
here described only one kind of worth or wearth; for
this word, (which is the same with garth or yard,)
signifies any inclosure whatever. Indeed its first sig-
nification is the act of girding or surrounding, then the
thing which girds or surrounds, then the thing girded
or surrounded, then, the purpose for which it is sur-
rounded, namely to guard it, then the thing guarded,
the person guarding it, and so forth. -
1. The act of girding or surrounding is expressed by
the Gothic verb gaurdan, the Anglo-Saxon gyrdan, the
Frankish and Alamannic gurten and curten, the Danish
gyrde, the Icelandic gyrda, the Swedish giorda, the
Dutch gorten, the German gurten, and the English to
gird: and all these have an evident affinity to the
Greek kipkos, and the Latin circus, circulus, circum, &c.
2. Various things used for girding or surrounding
were hence named ; e. gr. -
A belt, which is tied round the body of a man, horse,
&c.; in Gothic gaird, in modern German gurt, in
English girth and girdle, in Anglo-Saxon and Danish
gyrdel, in Alamannic gurdel, in Dutch gortel.
A curtain, which is drawn round a bed ; in Dutch
gordyn, in later Latin, Italian and Spanish cortina, in
old French courtine, in English curtain.
The bark, which surrounds the body of a tree, in
Latin cortex. -
A hedge, which surrounds a garden or other inclosed
place ; in Anglo-Saxon geard, in Swedish girde, in
Danish gierde; and so the act of bedging round about
a place, is in Cimbric gertha, in Swedish gaerda, and
in Danish at gierde. -
C iſ
Lastly any hoop or band which surrounds things, is Chap. I.
called in the north of England a garth.
3. Among things surrounded, which derive their
names from this source, may be particularised the
following. The old Latin cors, cortis signified a farm-
yard, or inclosed space before a country house; whence
the Barbarous-Latin curtis, Italian corte, old French
court and English court often applied formerly as the
name of a country house. The Gothic gards, Danish
gaard, Icelandic gard, Cimbric garthur, signified a
house or farm ; the Anglo-Saxon geard, or yeard
an inclosed space, as win-gaard a vineyard, ortgeard,
an orchard or garden (in Gothic aurtigards, from
the Gothic waurts, and Anglo-Saxon wurt or ort,
a root) the Frankish and Alamannic gardo and karto,
Welsh gardd, Danish gaard, Dutch gaerde, Italian
giardino, Spanish gardin, French jardin, German
garten and English garden, hortus. The modern
English yard, the provincial English garth, and the
old English wearth or worth, are only variations in
pronunciation from the Anglo-Saxon geard or yeard.
In the north of England garth is still used generally
for a yard or inclosed place ; so churchgarth is a
churchyard, stockgarth a rickyard, &c. and in Scotland
ward is used in the same sense. -
His braw calf-ward, whare gowans grew,
Sae white and bonie,
Nae doubt they'll rive it wi' the plew. BURNs.
Hence originated many English names of places, and
consequently of persons; as Kenilworth, i. e. Kenelm's
wearth, or Kenelm's inclosure; Wordsworth, i. e. the
Wurts' wearth, or garden of roots (as before explained
under the word orchard ;) Holdsworth, i. e. the Holts'
wearth, or inclosure of trees; Applegarth, the inclosure
of apple-trees; Haygarth the hay-yard; Hoggart the
hog-yard (or sheep inclosure, some sheep being pro-
vincially called hogs,) Garth, the inclosure, &c.
Moreover, as places were often inclosed for defence,
gard, and its cognate sounds came to signify a fortified
place, or city. Hence the Cimbric gard and garthur,
a fortification; the Icelandic gard, a city; the Sclavo-
nic terminations grod and gradz, as in Novogrod cas-
trum novum, and Belgrade castrum album ; the Ger-
man termination gard, as in Stutgard, (from stut a
horse,) civitas equaria : hence also the French boule-
vard, corrupted from burgward, in Barbarous-Latin
burgwardium, munitio oppidi. :
4. The English verbs to guard and to ward, which
are the same word differently pronounced, agree with
the Gothic wardyan, Anglo-Saxon weardian, Alemannic
uuarten, Icelandic varda, Italian guardare, Spanish
guardār, and French garder, to protect, and keep.
Hence the Anglo-Saxon weard, which is both custos
and custodia. So in English we have guard, guardian,
and warden, the person who defends, protects, or
keeps ; ward the act of safe custody, the place where
prisoners or others are safely kept, and the person who
is under the protection of a guardian. The Anglo-
Saxon weard custos appears frequently in composition,
as dureweard a porter, in old English a gateward—
Werys nou this gate ward 2 -
Me thuncheth he is a coward.
- Christ's Descent to Hell.
Many other employments were designated by this
particle ward, which have since become proper names
of families, whence Howard, Hayward, Woodward,
G R A M M A. R.
189."
Grammar. Stewart, Stoddart, &c. Howard, says VERsTEGAN,
~~’ “came of Holdward, which signifieth the governour
or keeper of a castle, fort, or hold of warre.” Hayward
was the person who had the care of the hedges.
He hath hewe sumwher a burthen of brere,
Thare fore sum hayward hath taken ys wed.
Hallad of the Man in the Moon.
Woodward is explained by LYE, “sylvae custos,
saltuarius.”—Stewart is from the Anglo-Saxon and old
English stiward or styward, and modern English steward.
The kyng com in to halle,
Among his knylites alle,
Forth he clepeth Athebrus,
His stiward, and him seide thus;
Stivard tac thou here -
My fundling forto lere. -
Geste of Kyng Horn,
The styward walkyd there withall
Among the lordes in the hall.
The styward tolde Rychard the Kyng
Some anon off that tydng
Sir Cleges.
Richard Coer de Lion.
Stywarde, as thou art me lefe,
Let no monwytte of my myschefe.
Sir Amadas.
That every styward, understyward, baillif, commissarie or
other mynystre holdyng and rulyng any of the seid courtes that
doth the contrary of this ordinance shall forfeit an C. s.
Stat. 1. RIC. III. c. 6. MS.
In the Icelandic, this word is stivardur, from stia
opus and vardur custos : and the word stia seems to
be connected with the Italian stivare, to stow goods or
ballast in a ship. - *.
Stoddart is from the old English Stodward, equorum
custos. A family of this name was anciently settled near
Stodmarsh, in Kent, and the name of the place as well
as that of the person was derived from the Anglo-Saxon
stod and steda, (in Swedish stod, in Alamannic stud, in Ice-
landic stedda,) a horse, whence come our modern steed
and stud. In the Anglo-Saxon also are found stod-hors,
a stallion, (connected with the Danish stod-hest,) stod-
myra, a stud-mare, and stod-fold an inclosure where
horses are kept. DUCANGE explains stuot, equus
admissarius; and WACHTER gives the same explanation
of the old German stut. The modern German stute
is a brood mare. In old English we also find stot,
used for a horse. -
This Reue sate upon a right good stot,
That was all pomell gray, and hight Scot,
CHAUCER.
Hence are derived many other old English names of
places and persons, as Stodham, Stodinton, Stodelegh,
Studlay, Stoteville, Stuteville, Stovel, Stotfald, Stoutes-
feld, Stutfeld, Stottesden, Stottesdon, Stuton, Stoteng-
hem, Stoteney, Stotesbrok, Stotholme, &c.
Reverting to the particle worth, we must observe
that it has sometimes a very different origin from that
which we have above noticed ; for the substantive
worth, value, derived from wirthan, to be, is often
used as a particle. Hence the substantive worthship,
or worship, is estimation, and the verb to worship, to
hold in esteem or reverence.
The profit and the worship of the same roialme. - .
- Treaty HEN. V. A. D. 1420.
Thaime, as our fadir and modir, we shall have and worship
Ibid.
WOL, I
Chap. I.
\-,-
Thowe haste onowryd all my fest,
And worschepyd me also. Sir Cleges.
In this sense, magistrates are called “Your Worship;”
and designated “ Worshipful.” We find in old
English the adjective derworth, signifying precious.
*
Now Jesu for thi derworth blode. -
MS. Harl. No. 913. fol. 29. b.
So, in Danish, elksværdig is “worthy of love or
esteem,” from varer to be, vard, worth.
Thus have we examined the particles, ford, ham,
ley, ton, worth, garth, ward, &c. which enter into
the formation of so many proper names. Nor should
the grammarians disregard this class of words; for in
them are often preserved many traces of connection
between different dialects, contributing much to the
illustration of the whole. Thus the English name
Fairfax, i. e. fair-haired, retains the Anglo-Saxon feaw
crimis. The Spanish Ferdinand, shows the connection
of Spain with the Goths, being derived from pferd
dienend equo serviens. The Scottish Telfair, anciently
written tailufeir, talliefeir, and tailefeir, is the French.
taille-fer, cut-iron; as Playfair is plifer, or bend-
II'OI! -
The particles stead, rick, and dom, are often applied, Stead, &c.
in modern use, to express locality. Stead, which we
have before had occasion to notice, is the Anglo-Saxon
noun sted, Gothic stads, Alamannic stat, Dutch stad,
and old English stede, a place. This word is used as a
particle in gyrdylstede, hache-styd, &c.
To ech a stede the Kyng hym sente
He wan the fyght. Octouian Imperator.
Some he hytte on the bacyn, -
That he cleff him to the chyn;
And some to the gyrdy! stede.
- Richard Coer de Lion.
Thei myghtt not passe the dure threscwold,
Nor lope ouer the hache-styd. .
The Huntyng of the Hare.
And so in the modern words bedstead, roadstead, home-
stead, with which agrees the Danish fyr-sted, a fire-place.
Rick is the obsolete English noun riche, and
modern German reich, a kingdom.
- He that made heaven and erthe
And sun and mone for to shine
Bring ous into his riche, -
And scheld ous fram helle pine.
- Legend of Seynt Katerine.
It is used as a particle in the modern English word
bishoprick, as in old English it was usual to say kingriche.
Over londes he gan fare,
With sorwe and reweful chere,
Seven king riche and mare
Tristrem to finde there. Sir Tristren.
Thar salbe rasyt a general gelde or magif it misteris throu
the haile kinryk. Scottish Act. Parl. A. D. 1424.
Dom is a particle of obscure origin, but of very
extensive use in the different northen dialects. In the
Anglo-Saxon, dom is judgment, from deman to judge,
whence our words doom and deem, and the proper
name demster, a judge. In Frankish duam is power,
which WACHTER and ADELUNG seem to consider as the
primary signification, from whence the special power
of jurisdiction was derived as a secondary signification
was derived. Hence the Anglo-Saxon cynedom,
Dutch koningdom, and English kingdom, first for the
power, and then for the territories of a king. So, in
Frankish rihthuom is empire, and hertuom government,
2 C
196)
G. R. A. M. M. A. R.
Grammar, and in modern German kaiserthum is the empire, herzog-
•-2- thum a dukedom, bisthum a bishoprick, &c. and in these
senses dom is probably connected with the Patin domo
and dominus. Where down merely signifies a general.
state, it may perhaps be connected with the verb: do, as
the correspondent German particle thum, in the same
sense, may with the verb thun. Thus we have
Gate, gang, freedom and thraldom, the Germans alterthum, &c.
are. Gate, gang, and fare, atl originally signify going, as
in Ludgate, gangway, thoroughfare. Hence the did
English algate, the old Scottish howgate, the Danish
ºmellemgang, an intercession or going between ; the
Dutchgangbaar, passable; the Scottish auldfarran, &c.
&c As we have seen the word worth corrupted into the
Leman, &c. particle wor, in worship, so lief-man was corrupted
into le-man, wyfman into wo-man, god-sib into gos-sip-
sib is a relation ; whence Rob ERT DE BRUNNE uses
the word sibred for kindred. In our modern word
harbour, the particle bour has undergone such a
change as not to be easily recognised.
was, in old English, herbarewen.
Herkneth hideward horsmen,
A tidyng ichou telle;
That ye shulen hongen
And herbarewen in helle.
Satire on Horsemen.
It is the same verb as the modern German herber-
gen, and comes from the Alamannic hereberga, com-
pounded of her an army and berg a fortification.
Hereberga therefore first meant the safe quarters of
an army; thence any place of safe resort, and thence
a place of safe resort for travellers, or for ships.
Hence the Dutch herberg, Italian albergo, Spanish
alvergue, and French auberge. Hence also the old
English herbergeour, a person sent before to announce
the approaching arrival of an army at its quarters, or
of a traveller at his inn; which word we have cor-
rupted to harbinger, used generally as a forerunner or
precursor. w
And now of love they treat, tiſt th' ev'ning star,
Love's harbinger appear'd. MILTON.
The particle bour, in our word neighbour, is of a
different origin. It seems to agree with the German
bar in machbar, which some derive from nach nigh
and bauer an inhabitant. Bar, however, is a particle
of extensive use in German, and may, in its various
applications, come from baren to bear, or faren to do ;
as in lastbar, brauchbar, dienstbar, &c. &c.
Certain particles are frequently confounded with
words which they resemble only in sound. Thus the
Nightin- particle gale, in nightingale, has no relation to the noun
gale, gale, a breeze ; but like the German nachtigal is de-
roundelay, rived from the Icelandic gala, and Anglo-Saxon galan,
&c. to sing: and these seem to have some relation to
wail, whence the name of the bird called the wodewale.
To gale was metaphorically used, in old English, for
“ to jest.”
And whan the Sompnour herd the Frere gale.
-- CHAUCER.
So round in roundelay, has no relation to a circle;
but is derived from the verb to roun, to sing, or hum
over a song, whence a song was called a roun.
Lenten ys come, with loue, to touhe,
With blosmen, and with briddes roune.
MS. Harl. 2253, fol. 71, b,
Geynest under gore, herkne to my roun.
* Ibid, fol. 63, b,
To harbour, -
Lace did not anciently mean the elegant manufac- Chap. 1,
ture. So termed in modern days, but any thing which \-y
served for the purpose of a girdle or strap : whenee
the anias or anlace, was a kind of knife or dagger,
so called because it hung on a lace or strap at the
girdle, as described by CHAUCER.
A dagger hanging by a las had he.
The modern particle lass in cutlass seems to have been
ignorantly taken from the old word anias. *
The numbers one, two, and ten are not at first sight One, two,
obvious as particles, when entering into the compound &c.
words eleven, twelve, forty, &c.; but we easily see
that the particle on, in the old English onlevene, is the
numeral word one.
onlevene thousand off our meyné
Ther were slayn withouten pyté.
Richard Coer de Lion.
So the particles twa and zuue in the Gothic twalif, and
Frankish zuwelf, twelve, are easily recognised as the
numerals twa and zuwei two, in those languages
respectively: hence the Gothic twalif, Swedish and
Icelandic tolf, Dutch twaalf, Anglo-Saxon twelf,
Frankish zuwelif, and German zwólf, all evidently
mean two left, as onlevene means one left, over and
above the perfect number ten. -
In like manner the particle tig, which Junius Sup-
poses to have been the old Gothic numeral ten, is
seen in twaintig, thrinstig, fidwortig, which are twenty,
thirty, and forty, in that language. And this same
particle tig was also retained in old English ; as in
the letter of HENRY III. before quoted.
Witnesse usseluen aet Lunden, thane egtetenthe day on the
monthe of Octobr’ in the two and fowertigthe yeare of ure
cruninge. . .
There are numberless other compounds of the kind
which we have hitherto considered ; namely those,
which merely unite two conceptions, without chang-
ing the grammatical class, to which the principal
portion of the word belongs. e
We now come to words in which, by a slight
inflection, the class that the word belongs to is altered.
Friendship is such a compound, and the word friend, Ship, head,
which forms the primary part of it, is sufficiently &c.
obvious ; but what is ship 2 In order to answer this
we must look through the other dialects, in which it
occurs. The Germans use the termination schaft,
the Dutch schap, and the Swedes skap ; and these are
manifestly from the Gothic skapan, Anglo-Saxon
scaapan, or scyppan, Frankish and Alamannic scaffen,
Dutch scheppen, Icelandic skapa and skipa, Danish
skaber, and old English to shup, i.e. to shape, make,
or do. - w
The shuppare that huem shupte
To shome he huem shadde. -
- - Satire on Horsemen.
Wymmen were the beste thing,
. That shup our heye heuene kyng. - -
* MS. Harl. 2253, fol, 71. b.
Friendship therefore is the action, the work, of a
friend.: CHAUCER uses gladshipe.
That gladshipe he hathal forsake,
In Danish we find selkskab, a fellowship; in Anglo-
Saxon ealdor-scipe, cynescipe, sib-scipe, &c. In Ger-
man herrschaft, eigenschaft, gesellschaft, &c. &c.
The particle scape, in landscape, is the same as ship;
for we find in Anglo-Saxon landscipe, in Dutch land-
Schap, and in German landschafft,
G R A M M A. R.
R9].
Grammar. The particle head or hood has nothing to do with:
*-v- the common noun head, from which some ignorant
The diminutive et is from the French, ette and Chap. T. "
Italian etto. . Thus we find the word baronette long Q-->
grammaarians have supposed it to be derived. It is
the Saxon had, status, and is probably connected
with the pronoun hyt, it. In Danish the particle is
hed; in German heit, and keit. Heit is used in Frankish
as a word signifying “person”—e.gr. der ander heit
Gotes “the second person of the Divinity.” We find
in Frankish magadheit virginity, uuipheit woman-
hood; in Anglo-Saxon cniht-hade childhood, preost-
hade priesthood ; in German freyheit freedom,
onenschheit human nature, einsamkeit solitariness,
seligkeit happiness. In old English the particle head
occurs in many compounds now disused ; as yunghead,
wighthede, fairehéd, brotherhede, boldehed, bisyhed, &c.
The particle mess has been still more absurdly
derived from the French nez, the nose. How any human
being could ever have dreamt that greatness, in the
abstract, was named from a great nose, redness from a
red nose, or sweetness from a sweet nose, it is diffi-
cult to conceive. Ness appears to be nothing more
than the French termination esse, preceded by the
Saxon infinitive termination en. Thus, from great
would be formed the verb greaten, which would be
converted into the abstract greaten-esse ; so, sweet,
sweeten, sweeten-esse ; red, redden, redden-esse, &c. It
must not however be omitted that the learned Hickes,
with some doubt, suggests this termination to be
taken from the Gothic eins.
Ess or esse is a particle common to the Anglo-
Saxons, and the French; and it is probably a mere
corruption of the Latin termination etia or itia.
Gower uses tristesse. Some old English words ending
in ess have, by modern corruption, been used as
plurals; such are riches and alms, anciently richesse,
and almesse or elinesse.
Dame richesse on her honáe gan lede
A yong man full of semelyhede. CHAUCER.
Sende god biforen him man,
The while he mai, to heuene;
For betere is on elmesse biforem,
Thanne ben after seuene.
Digby MS. (circa 1066.)
Besides these terminations of abstract nouns, we
have, from the Latin and French, ance, ence, dige, ery,
our, ty, as in finance, credence, courage, drapery, honour,
piety, all which are manifestly framed, in the Latin
original, by combining pronouns, and participial termi-
nations with the radical word.
Other terminations of abstract nouns we have from
Teutonic sources; such are ledge, red, er, th, t, &c.
as in knowledge, kindred, sibred, hunger, murder, death,
sloth, drift, thrift. Ledge seems to be formed from
lagen, and red from raeden.
Henry the Third's letter,are counsellors, and in Scotland
they still say “I read you not to do such a thing,”
Er, th, and t are
the two-
for “I advise you not to do it.”
probably remains of Teutonic pronouns ;
latter are still used in the conjugation of our verbs.
Hence death is that which dieth or maketh to die ;
drift is that which hath driven; thrift that which hath
thriven, &c.
Ling is a diminutive, which Wachter thinks to be
derived from langen, in the sense of tangere or of perti-
mere ; thus the Anglo-Saxons used the word eorth-ling
for a husbandman, as we use worldling for a man of
this world.
Thus radesmen, in King
prior to its institution as a separate dignity by King
James E. --- - -
But he wer. prelat, other baronette. -
- - - - MS. Cotton, Calig, A2, fol. 33. .
Mr. Tyrwhit thinks that doucet was the name of a
particular kind of musical instrument; it was pro-
bably no more than ouradjective dulcet. -
Ther were trumpes and trumpetes
Lowde shalmys and doucetes. f_IDGATE.
i.e. “ there were large and small instruments of the
trump kind ; and there were loud and soft instruments
of the shawm kind.' -
Full, less, and some are particles which give an
adjectival force to a compound. The particles full
and some are obviously identical with the words full
and some. The particle less, in such words as hopeless,
restless, deathless, motionless, &c. Mr. Tooke explains
to be the imperative les 1 which (he says) is dismiss.
It does not appear that les means dismiss : and if it .
did, how are we to explain by dismiss, the word (less)
the comparative of little. It is well known that many
adjectives are used as comparatives which have little
or no affinity with the pesitives. Thus àpečva v is
used as the comparative of &Ya60s, melior of bonus, and
better of good. So less seems to have been an adjective
originally implying want. When compared with little,
therefore, it would signify that quality in a stronger
degree : but when compounded with such words as
those above quoted, it might denote a total want or
privation of the ideas they express. In the following .
instance, it appears to be used in the sense of wanting
honour, evil, worthless, as we now say a loose, bad man.
Pysshopes ant barouns come to the kynges pes,
Ase men that weren fals, fykel, ant les.
MS. Harl, 2253, fol. 59. b.
Ish is a particle of very ancient and general use,
as in reddish, Turkish, &c. It signifies “ of the
nature or substance of a thing;” and seems to have
an affinity to the Greek verb etcw and termination
wkos. It is undoubtedly the German ische, the Dutch .
sche, the Frankish isc, the Italian esco, and the French
esque. In the Edda of SAEMUND, the first man (or .
perhaps the first substance) is called ask.
Unst thriar comw ar thui tide.
Until three came out of that company.
Auffigir og astgier aser ad huse.
Powerful and lovely asans to the house.
Fundu a lande lyte meigande.
They found on the land powerless.
AsK og emblo oerloeg lausa.
Ask and emblo strengthless.
** Ab hoe asko vel asco, primo condito homine,”
says Hickes, “ venit proprium nomen aſse apud
Anglo-Saxones;” and he cites various instances in
which asc appears, to have the general signification
of man.
Ard is an adjectival particle somewhat similar in
effect to ish, but appearing to have been derived into
English immediately from the French. We find in
old English lyard, bayard, blaunchard, trichard, caynard,
&c. now obsolete; but we still retain drunkard, coward,
braggart, and Some others. It seems to exist, as a
word, in the Scottish airt, a quarter of the heavens or
portion of the earth. -
2 c 2
192
G R A M M A R.
Grammar. The adjectival particle wise we have already shown
W.
2 to be the same as the word guise.
Rightwise has been
corrupted into righteous.
It is scarcely necessary to dwell on such particles
as mis from the verb to miss—wan (as in the Scottish
wanchancy) from the noun want—fold as in twofold,
from the verb to fold—the Latin pler, as in dupler
from the verb plico. We have specified enough to
show, that the generality of particles which serve the
purpose of changing the grammatical class to which
a word belongs, originally existed in a separate shape,
as significant words.
Declension It is certainly not so easy to prove that the particles
and conju- used in the declension of nouns and conjugation of
gation.
the words to which they have been traced.
verbs were originally significant words ; yet we
cannot but agree with Mr. TookE that there is good
reason to believe that they were.
One forcible reason for this opinion is, that what
is done in some languages by terminations, is done in
other languages by separate words, by prepositions, by
adverbs, by auxiliary verbs, &c.; but we have already
shown, that not only the auxiliary verbs, but the
adverbs and prepositions, were significant; and hence
it is reasonable to infer that what stands in their place
is significant also. -
The noun substantive, for instance, in some lan-
guages may be varied in gender, number and case,
by its terminations. Thus the Latins expressed the
children of the two sexes by the words puer and
puella. Puer signifies what we mean by a man-child.
We have therefore reason to believe, that as man is a
word significant of a male of the human kind, so er
when standing alone had a similar signification : and
in fact we find that er is to this day the German mas-
culine pronoun he. Puella signifies a girl : if we call
pu-er a he-child, we may call pu-ella a she-child : and
in fact illa is the Latin feminine pronoun she. In like
manner our feminine particle ess, as in shepherdess, is
found in the Italian pronoun essa, she.
The terminations of number and case, are not
very clearly to be traced to their origin; but they
seem in general to be pronouns. Thus the nomina-
tive case lapis a stone, is evidently made up of two
parts, lap, which conveys the conception of stone
through all its inflections, and is, which distinguishes
this particular case. Now is is a Latin pronoun. So
we think the final o in homo, is the Greek article 6,
and the final a in musa the Greek article j.
In nouns adjective, we have already said that the .
termination ly is the Gothic substantive leik, body; and
if the ly in greatly have a separate meaning, it is pro-
bable that the er and est in greater and greatest, have
also separate meanings,
Various explanations have been attempted of the ter-
minations of the Latin and Greek verbs: and though
they may none of them be perfectly satisfactory
throughout, yet it can hardly fail to be admitted,
that some of the particles have a connection with
Thus in
capiam, capies, capiet, the termination am has certainly
some analogy to the Latin me, or the Greek eue ; the
termination es to ov, and the termination et to Tus.
M. de Brosses, after following the radical sound cap
through all its developements in the verb capio, con-
cludes with a just observation. “ Toute cette com-
position est l'ouvrage non d'une combinaison reflechie,
ni d'une philosophie raisonnée mais d'une metaphy-
sique d'instinct.” Now instinct could never have led
men to form a complicated, and beautiful system out
of sounds altogether unmeaning; but it might easily
lead to the gradual combination of known elements,
until they formed at length the complete structure of
a language.
As the effect of a particle in declension or conjuga-
Chap. I.
\-y-
tion is sometimes supplied by a word ; so on the other
hand it is sometimes produced by a mere change in
articulation ; and this seems to be natural to mankind,
because we find it in different languages, and in very
various languages, and in very various ways, as TvTTw,
tv7rw—capio, cepi—sing, sang—man, men, &c.
Lastly we must observe, that there are numerous
causes of anomaly in language, which render it more
particularly difficult to systematise and explain the
minor portions of speech, such as the prepositions,
auxiliary verbs, and particles. One of these causes is
a mistaken notion of analogies between particular
words, where no such analogy exists. Thus our word
further, which was the comparative of forth, has
been supposed by many persons to be the compa-
rative of far, and has therefore been erroneously
written farther. A still more striking instance is that
of the word coud, which we always pronounce pro-
perly, but spell could, inserting the l, without any
reason whatever, but that there is an l in would and
should. The two latter words are from the Anglo-
Saxon wille and sceal, the former is from the Anglo-
Saxon cwethan ; and was always written in old English
couthe, cowthe, or coude. - ... '
That though he had me bete on every bone,
He couthe winne agen my love anone. CHAUCER.
He thowght to taste if he cowthe,
And on he put in his mowth. Sir Cleges.
Sir, quod this knyght myld of speche,
Wold God I cow the your sonne teche : -
- Lyfe of Ipomydon.
Ac he no couthe neuer mo
Chese the better of hem to. Amis and Amiloun.
Whiche was right displesant to the kyng, but he coude nat
amende it. BERNERs's Froissart, fol. 43.
Another and a more effective cause of anomaly is
the love of euphony, or easy pronunciation, which
leads the ignorant especially to corrupt words by
abbreviations and changes, as Godild ! for God yelde,
i.e. reward him. Gossip for god-sib, &c.
Allowing for the obscurities which these and other
causes spread over the minor portions of speech, it
may fairly be said, that in regard to particles, as well
as to words, we have established the great principle of
transition, by which significant sounds pass from one
class and description of signs into another. The noun
or verb becoming a particle, and the particle coalescing
with another verb or noun, serve to modify their signi-
fication, and determine their grammatical use. And,
finally, we may conclude, that language is, throughout,
a combination of significant sounds, fitted to express
thoughts and emotions, as they exist interchangeably
in the human mind, -- -
T-
Logic.
L O G I C.
INTRODUCTORY SECTION.
LoGIC in the most extensive sense which it can with
*—S.-) propriety be made to bear, may be considered as the
Science and also as the Art of Reasoning. It investigates
the principles on which argumentation is conducted,
and furnishes rules to secure the mind from error in
its deductions. Its most appropriate office, however,
is that of instituting an analysis of the process of the
mind in Reasoning : and in this point of view it is, as
has been stated, strictly a Science : while considered
in reference to the practical rules above mentioned, it
may be called the Art of Reasoning. This distinction,
as will hereafter appear, has been overlooked, or not
clearly pointed out by most writers on the subject, Logic
having been in general regarded as merely an Art;
and its claim to hold a place among the Sciences having
been expressly denied. .
Considering how early Logic attracted the attention
of philosophers, it may appear surprising that so
little progress should have been made, as is confes-
sedly the case, in developing its principles, and per-
fecting the detail of the system : and this circum-
stance has been brought forward as a proof of the
barrenness and futility of the study. But a similar
argument might have been urged with no less plausi-
bility, in past ages, against the study of Natural Phi-
losophy, and very recently against that of Chemistry.
No Science can be expected to make any considerable
progress, which is not cultivated on right principles.
Whatever may be the inherent vigour of the plant, it
will neither be flourishing nor fruitful till it meet with
a suitable soil and culture : and in no case is the remark
more applicable than in the present ; the greatest
mistakes having always prevailed respecting the nature
of Logic, and its province having in consequence been
extended by many writers to subjects with which it
has no proper connection. Indeed, with the exception
of Aristotle, (who is himself not entirely exempt from
the errors in question,) hardly a writer on Logic can
be mentioned who has clearly perceived, and steadily
kept in view throughout, its real nature and object.
Before his time, no distinction was drawn between
the Science of which we are speaking, and that which is
now usually called Metaphysics : a circumstance which
alone shews how, small was the progress made in
earlier times. Indeed those who first turned their
attention to the subject, hardly thought of inquiring
into the process of Reason itself, but confined them-
selves almost entirely to certain preliminary points,
the discussion of which is (if logically considered)
subordinate to that of the main inquiry.
VOL. I.
disputation
Zeno the Eleatic, whom most accounts represent Introduc-
as the earliest systematic writer on the subject of
Logic, or as it was then called, Dialectics, divided his
work into three parts; the first of which (upon Con-
sequences) is censured by Socrates [Plato, Parmen.]
for obscurity and confusion. In his second part,
however, he furnished that interrogatory method of
[éptbrmats] which Socrates adopted,
and which has since borne his name. The third part
of his work was devoted to what may not improperly
be termed the art of wrangling, [ćptstucjj which Sup-
plied the disputant with a collection of sophistical
questions, so contrived that the concession of some
point which seemed unavoidable, immediately involved
some glaring absurdity. This, if it is to be esteemed
as at all falling within the province of Logic, is cer-
tainly not to be regarded (as some have ignorantly or
heedlessly represented it) as its principal or proper
business. - The Greek philosophers generally have
unfortunately devoted too much attention to it : but
we must beware of falling into the vulgar error of
supposing the ancients to have regarded as a serious
and intrinsically important study, that which in fact
they considered as an ingenious recreation. The dispu-
tants diverted themselves in their leisure hours by
making trial of their own and their adversary's acute-
ness, in the endeavour mutually to perplex each other
with subtle fallacies; much in the same way as men
amuse themselves with propounding and guessing
riddles, or with the game of chess ; to each of which
diversions the sportive disputations of the ancients
bore much resemblance. They were closely analogous
to the wrestling and other exercises of the gymnasium,
these last being reckoned conducive to bodily vigour
and activity, as the former were to habits of intellec-
tual acuteness; but the immediate object in each was
a sportive, not a serious contest; though doubtless
fashion and emulation often occasioned an undue
importance to be attached to success in each. -
Zeno then is hardly to be regarded as any further a
logician than as to what respects his erotetic method
of disputation ; a course of argument constructed on
this principle being properly an hypothetical sorites,
which may easily be reduced into a series of syllo-
.gisms.
To Zeno succeeded Euclid of Megara, and Antis-
thenes, both pupils of Socrates. The former of these
prosecuted the subject of the third part of his prede-
cessor's treatise, and is said to have been the author
of many of the fallacies attributed to the Stoical school.
2 D
tory
Section.
194 L O G. I. C.
Logic. Of the writings of the latter nothing certain is known: lost te the world for about two centuries, but seem to Introduc-
S-N- if, however, we suppose the above mentioned sect to have been but little studied for a long time after their sº
be his disciples in this study, and to have retained his
principles, he certainly took a more correct view of
the subject than Euclid. The Stoics divided all Nekrø,
every thing that could be said, into three classes:
1st, the simple term ; 2d, the proposition ; 3d, the
syllogism ; viz. the hypothetical ; for they seem to
have had little notion of a more rigorous analysis of
argument than into that familiar form.
We must not here omit to notice the merits of
Archytus, to whom we are indebted for the doctrine
of the categories. He, however, (as well as the other
writers on the subject,) appears to have had no dis-
tinct view of the proper object and just limits of the
science of Logic ; but to have blended with it Meta-
physical discussions not strictly connected with it,
and to have dwelt on the investigation of the nature
of terms and propositions, without maintaining a con-
stant reference to the principles of Reasoning, to
which all the rest should be made subservient.
The state then in which Aristotle found the Science,
(if indeed it can properly be said to have existed at all
before his time,) appears to have been nearly this :
the division into simple terms, propositions and syllo-
gisms, had been slightly sketched out ; the doctrine
of the categories, and perhaps that of the opposition
of propositions, had been laid down; and, as some
believe, the analysis of species into genus and diffe-
rentia, had been introduced by Socrates. These, at
best, were rather the materials of the system than the
system itself; the foundation of which indeed he dis-
tinctly claims the merit of having laid ; and which
remains fundamentally the same as he left it.
It has been remarked, that the Logical system is one
of those few theories which have been begun and per-
fected by the same individual. The history of its dis-
covery, as far as the main principles of the science are
concerned, properly commences and ends with Aris-
totle. And this may perhaps in part account for the
subsequent perversions of it. The brevity and sim-
plicity of its fundamental truths, (to which indeed all
real science is perpetually tending,) has probably led
many to suppose that something much more com-
plex, abtruse, and mysterious, remained to be disco-
vered. The vanity by which all men are prompted
unduly to magnify their own pursuits, has led unphi-
losophical minds, not in this case alone, but in many
others, to extend the boundaries of their respective
; Sciences, not by the patient developement and just
application of the principles of those Sciences, but by
wandering into irrelevant subjects. The mystical em-
ployment of numbers by Pythagoras, in matters
utterly foreign to Arithmetic, is perhaps the earliest
instance of the kind. A more curious and important
one is the degeneracy of Astronomy into judicial Astro-
logy; but none is more striking than the misapplica-
tion of Logic, by those who have treated of it as
“ the art of rightly employing the rational faculties,”
or who have intruded it into the province of Natural
Philosophy, and regarded the syllogism as an engine
for the investigation of nature: overlooking the bound-
less field that was before them within the legitimate
limits of the Science; and not perceiving the import-
ance and difficulty of the task of completing and
properly filling up the masterly sketch before them.
The writings of Aristotle were not only absolutely
plications and perversions of it in later years.
recovery. An Art, however, of Logic, derived from
the principles traditionally preserved by his disciples,
seems to have been generally known, and was em-
ployed by Cicero in his philosophical works; but the
pursuit of the science seems to have been abandoned
for a long time. Early in the Christian era, the Peripa-
tetic doctrines experienced a considerable revival; and
we meet with the names of Galen and Porphyry as
Logicians: but it is not till the fifth century that Aris-
totle's Logical works were translated into Latin by the
celebrated Boethius. Not one of these seems to have
made any considerable advances in developing the
Theory of Reasoning. Of Galen's labours little is
known ; and Porphyry's principal work is merely on
the predicables. We have little of the Science till the
revival of learning among the Arabians, by whom
Aristotle's treatises on this as well as on other subjects
were eagerly studied.
Passing by the names of some Byzantine writers of
no great importance, we come to the times of the
Schoolmen, whose waste of ingenuity and frivolous
subtilty of disputation need not be enlarged upon. It
may be sufficient to observe, that their fault did not
lie in their diligent study of Logic, and the high
value they set upon it, but in their utterly mistaking
the true nature and object of the science ; and by
attempting to employ it for the purpose of physical
discoveries, involving every subject in a mist of
words, to the exclusion of sound philosophical inves-
tigation. Their errors may serve to account for the
strong terms in which-Bacon sometimes appears to
censure Logical pursuits ; but that this censure was
intended to bear against the extravagant perversions,
not the legitimate cultivation of the Science, may be
proved from his own observations on the subject, in
his Advancement of Learning.
His moderation, however, was not imitated in other
quarters. Even Locke confounds in one sweeping
censure the Aristotelic theory, with the absurd misap-
His
objection to the Science, as unserviceable in the disco-
very of truth, (which has of late been often repeated)
while it holds good in reference to many (misnamed)
Logicians, indicates that with regard to the true nature
of the Science itself he had no clearer notions than they
have, of the proper province of Logic, viz. Reasoning ;
and of the distinct character of that operation from the
observations and experiments which are essential to
the study of nature. -
An error apparently different, but substantially the
same, pervades the treatises of Watts and other
modern writers on the subject. Perceiving the inade-
quacy of the syllogistic theory to the vast purposes to
which others had attempted to apply it, he still craved
after the attainment of some equally comprehensive
and all-powerful system ; which he accordingly
attempted to construct, under the title of The Right
Use of Reason ; which was to be a method of invigo-
rating and properly directing all the powers of the
mind : a most magnificent object indeed, but one
which not only does not fall under the province of
Logic, but cannot be accomplished by any one Science
or system that can even be conceived to exist. The
attempt to comprehend so wide a field is no extension
of Science, but a mere verbal generalization, which
ection.
- L. O. G. I. C. 195
Logic. leads only to vague and barren declamation. In every of whom Warburton tells a story in his Div. Leg.) one Introduc-
^-y-' pursuit, the more precise and definite our object, the should complain of a reading glass for being of no sºy
- more likely we are to attain some valuable result ; if, service to a person who had never learned to read. Section.
like the Platonists, who sought after the aird yabov, In fact, the difficulties and errors above alluded to S-'
the abstract idea of good, we pursue some specious are not in the process of Reasoning itself, (which alone
but ill-defined scheme of universal knowledge, we is the appropriate province of Logic,) but in the sub-
shall lose the substance while grasping at a shadow, ject matter about which it is employed. This process
and bewilder ourselves in empty generalities. will have been correctly conducted if it have con-
It is not perhaps much to be wondered at, that in formed to the Logical rules which preclude the possi-
still later times several ingenious writers, forming bility of any error creeping in between the principles
their notions of the Science itself from professed from which we are arguing, and the conclusions we
masters in it, such as have just been alluded to, and deduce from them. But still that conclusion may be
judging of its value from their failures, should have false, if the principles we start from are so. In like
treated the Aristotelic system with so much reproba- manner, no Arithmetical skill will secure a correct
tion and scorn. Too much prejudiced to bestow on it result to a calculation, unless the data are correct
the requisite attention for enabling them clearly to from which we calculate ; nor does any one on that
understand its real character and object, or even to account undervalue Arithmetic ; and yet the objection
judge correctly from the little they did understand, against Logic rests on no better foundation.
they have assailed the study with a host of objections, There is in fact a striking analogy in this respect
so totally irrelevant, and consequently impotent, that, between the two Sciences. All numbers (which are
considering the talents and general information of the subject of Arithmetic) must be numbers of some
those from whom they proceed, they might excite things, whether coins, persons, measures, or anything
astonishment in any one who did not fully estimate else; but to introduce into the Science any notice of
the force of very early prejudice. the things respecting which calculations are made,
Logic has usually been considered by these objectors would be evidently irrelevant, and would destroy its .
as professing to furnish a peculiar method of Reason- scientific characters: we proceed therefore with arbi-
ing, instead of a method of analyzing that mental pro- trary signs representing numbers in the abstract. So
cess which must invariably take place in all correct also does Logic pronounce on the validity of a regu-
Reasoning; and accordingly they have contrasted the larly-constructed argument equally well, though
ordinary mode of reasoning with the syllogistic ; and arbitrary symbols may have been substituted for the
have brought forward with an air of triumph the argu- terms, and consequently without any regard to the
mentative skill of many who never learned the system: things signified by those terms. And the probability
a mistake no less gross than if any one should regard of doing this (though the employment of such arbi-
Grammar as a peculiar language, and contend against trary symbols has been absurdly objected to, even by
its utility on the ground that many speak correctly writers who understood not only Arithmetic but Alge-
who never studied the principles of Grammar ; bra) is a proof of the strictly scientific character of the
whereas Logic, which is, as it were, the Grammar of system. But many professed Logical writers, not
Reasoning, does not bring forward the regular syllo- attending to the circumstances which have been just
gism as a distinct mode of argumentation, designed to mentioned, have wandered into disquisitions on various
be substituted for any other mode; but as the form to branches of knowledge ; disquisitions which must
which all correct Reasoning may be ultimately reduced, evidently be as boundless as human knowledge itself,
and which consequently serves the purpose (when we since there is no subject on which Reasoning is not
are employing Logic as an Art) of a test to try the employed, and to which consequently Logic may not
validity of any argument, in the same manner as by be applied. The error lies in regarding every thing as
chemical analysis we develope and submit to a distinct the proper province of Elogic, to which it is applicable.
examination the elements of which any compound A similar error is complained of by Aristotle, as having
body is composed, and are thus enabled to detect any taken place with respect to Rhetoric ; of which indeed
latent sophistication and impurity. - we find specimens in the arguments of several of the
Complaints have also been made that Logic leaves interlocutors in Cic. de Oratore.
untouched the greatest difficulties, and those which From what has been said, it will be evident that
are the sources of the chief errors in Reasoning; viz. the there is hardly any subject to which it is so difficult
ambiguity or indistinctness of terms, and the doubts to introduce the student in a clear and satisfactory
respecting the degrees of evidence in various proposi- manner, as the one we are now engaged in. In any
tions : an objection which is not to be removed by other branch of knowledge, the reader, if he have
any such attempt as that of Watts to lay down “ rules any previous acquaintance with the subject, will
for forming clear ideas, and for guiding the judgment;” usually be so far the better prepared for comprehend-
but by replying that no Art is to be censured for not ing the exposition of the principles; or if he be
teaching more than falls within its province, and entirely a stranger to it, will at least come to the
indeed more than can be taught by any conceivable study with a mind unbiassed, and free from prejudices
art. Such a system of universal knowledge as should and misconceptions; whereas in the present case it
instruct us in the full meaning of every term, and the cannot but happen that many who have given some
truth or falsity, certainty or uncertainty, of every attention to Logical pursuits, (or what are usually con-
proposition, thus superseding all other studies, it is sidered as such) will frequently have rather been
most unphilosophical to expect or even to imagine. bewildered by fundamentally erroneous views, than
And to find fault with Logic for not performing this prepared by the acquisition of just principles for ulte-
is as if one should object to the Science of Optics for, rior progress; and that not a few who pretend not to
not giving sight to the blind ; or as if (like the man' any aequaintance whatever with the Science, will yet
2 D 2
196
L O G. I. C.
Logic.
have imbibed either such prejudices against it, or such
false notions respecting its nature, as cannot but prove
obstacles in their study of it.
There is, however, a difficulty which exists more
or less in all abstract pursuits, though it is perhaps
more felt in this, and often occasions it to be rejected
by beginners as dry and tedious; viz. the difficulty of
perceiving to what ultimate end,--to what practical
or interesting application the abstract principles lead
which are first laid before the student ; so that he
will often have to work his way patiently through the
most laborious part of the system before he can gain
any clear idea of the drift and intention of it.
This complaint has often been made by chemical
students, who are wearied with descriptions of oxygen,
hydrogen, and other invisible elements, before they
have any knowledge respecting such bodies as com-
monly present themselves to the senses. And accord-
ingly some teachers of Chemistry obviate in a great
degree this objection, by adopting the analytical
instead of the synthetical mode of procedure, when
they are first introducing the subject to beginners;
i. e. instead of synthetically enumerating the elemen-
tary substances, proceeding next to the simplest com-
binations of these, and concluding with those more
complex substances which are of the most com-
mon occurrence, they begin by analyzing these last,
and resolving them step by step into their simple
elements; thus presenting the subject at once in an
interesting point of view, and clearly setting forth the
object of it. The synthetical form of teaching is
indeed sufficiently interesting to one who has made
considerable progress in any study; and being more
concise, regular, and systematic, is the form in which
our knowledge naturally arranges itself in the mind,
and is retained by the memory : but the analytical is
the more interesting, easy, and natural kind of intro-
duction, as being the form in which the first invention
or discovery of any kind of system must originally have
taken place. - -
It may be advisable, therefore, to begin by giving
a slight sketch, in this form, of the Logical system,
before we enter regularly upon the details of it. The
reader will thus be presented with a kind of imaginary
history of the course of inquiry by which the Logical
system may be conceived to have occurred to a
philosophical mind.
In every instance in which we reason, in the strict
sense of the word, i. e. make use of arguments, whe-
ther for the sake of refuting an adversary, or of con-
veying instruction, or of satisfying our own minds on
any point, whatever may be the subject we are engaged
on, a certain process takes place in the mind, which
is one and the same in all cases, provided it be
correctly conducted.
Of course it cannot be supposed that every one is
even conscious of this process in his own mind, much
less is competent to explain the principles on which it
proceeds ; which indeed is, and cannot but be, the
case with every other process respecting which any
system has been formed ; the practice not only may
exist independently of the theory, but must have pre-
ceded the theory; there must have been language
before a system of Grammar could be devised; and
musical compositions previous to the science of Music.
This by the way will serve to expose the futility of
the popular objection against Logic, that men may
reason very well who know nothing of it. The Introduc-
parallel instance adduced, shews that such an objec-
tion might be applied in many other cases, where its
absurdity would be obvious ; and that there is no rea-
son for deciding thence, either that the system has no
tendency to improve practice, or that even if it had
not, it might not still be a dignified and interesting
pursuit.
One of the chief impediments to the attainment of
a just view of the nature and object of Logic, is the
not fully understanding, or not sufficiently keeping in
mind, the SAMENEss of the Reasoning process in all
cases; if, as the ordinary mode of speaking would
seem to indicate, Mathematical Reasoning, and Theo-
logical, and Metaphysical, and Political, &c. were
essentially different from each other, i. e. different
kinds of reasoning, it would follow, that supposing
there could be at all any such Science as we have
described Logic, there must be so many different
species, or at least different branches of Logic. And
such is perhaps the most prevailing notion. Nor is
this much to be wondered at ; since it is evident to
all that some men converse and write in an argumen-
tative way, very justly on one subject, and very
erroneously on another, in which again others excel,
who fail in the former. This error may be at once
illustrated and removed, by considering the parallel
instance of Arithmetic, in which every one is aware
that the process of a calculation is not affected by the
nature of the objects whose numbers are before us :
but that (e. g.) the multiplication of a number is
the very same operation, whether it be a number of
men, of miles, or of pounds; though nevertheless
men may perhaps be found who are accurate in calcu-
lations relative to Natural Philosophy, and incorrect
in those of Political Economy, from their different
degrees of skill in the subjects of these two Sciences ;
not surely because there are different arts of Arithme-
tic applicable to each of these respectively.
Others again, who are aware that the simple system
of Logic may be applied to all subjects whatever, are
yet disposed to view it as a peculiar method of Reason-
ing, and not as it is, a method of unfolding and
analyzing our Reasoning: whence many have been led
(e. g. the author of the Philosophy of Rhetoric) to
talk of comparing syllogistic Reasoning with moral
Reasoning, and to take it for granted that it is possible
to reason correctly without reasoning Logically; which
is in fact as great a blunder as if any one were to mis-
take Grammar for a peculiar language, and to suppose
it possible to speak correctly without speaking Gram-
matically. They have in short considered Logic as
an Art of Reasoning; whereas, so far as it is an Art, it
is the Art of Reasoning : the Logician's object being,
not to lay down principles by which one may reason,
but by which all must reason, even though they are not
distinctly aware of them : to lay down rules, not
which may be followed with advantage, but which
cannot possibly be departed from in sound reasoning.
These misapprehensions and objections being such as
lie on the very threshold of the subject, it would have
been hardly possible, without noticing them, to con-
vey any just notion of the nature and design of the
Logical system. .
Supposing it then to have been perceived that the
operation of Reasoning is in all cases the same, the
analysis of that operation could not fail to strike the
tory
Section.
L O G. I. C.
197
employed to designate a premiss, whether it came first Introduc-
Logic, mind as an interesting matter of inquiry; and
either from error or from design; and even those who
are not misled by these fallacies, are so often at a loss
to detect and expose them in a manner satisfactory to
others, or even to themselves, it could not but appear
desirable to lay down some general rules of Reasoning,
applicable to all cases, by which a person might be
enabled the more readily and clearly to state the
grounds of his own conviction, or of his objection to
the arguments of an opponent, instead of arguing at
random without any fixed and acknowledged princi-
ples to guide his procedure. Such rules would be
analogous to those of Arithmetic, which obviate the
tediousness and uncertainty of calculations in the
head, wherein, after much labour, different persons
might arrive at different results, without any of them
being able distinctly to point out the error of the rest.
A system of such rules, it is obvious, must, instead of
deserving to be called the Art of wrangling, be more
justly characterised as “ the Art of cutting short
wrangling,” by bringing the parties to issue at once,
if not to agreement, and thus saving a waste of
ingenuity.
In pursuing the supposed investigation, it will be
found that every conclusion is deduced, in reality,
from two other propositions, (thence called pre-
mises ;) for though one of these may be, and com-
monly is, suppressed, it must nevertheless be under-
stood as admitted ; as may easily be made evident by
supposing the DENIAL of the suppressed premiss,
which will at once invalidate the argument : e.g. if
any one from perceiving that the world exhibits
marks of design, infers that “ it must have had an
intelligent author," though he may not be aware in
his own mind of the existence of any other premiss,
he will readily understand, if it be denied that
“whatever exhibits marks of design must have had an
intelligent author,” that the affirmative of that propo-
sition is necessary to the validity of the argument.
An argument thus stated regularly and at full length
is called a Syllogism ; which therefore is evidently not
a peculiar kind of argument, but only a peculiar form
of expression, in which every argument may be
stated. When one of the premises is suppressed,
(which for brevity's sake it usually is) the argument
is called an Enthymeme. And it may be worth while
to remark, that when the argument is in this state,
the objections of an opponent are (or rather appear
to be) of two kinds; viz. either objections to the
assertion itself, or objections to its force as an argu-
ment; e. g. in the above instance, an atheist may be
conceived either denying that the world does exhibit
marks of design, or denying that it follows from thence
that it had an intelligent author. The only difference
in the two cases is, that in the one the expressed pre-
miss is denied, in the other the suppressed; for the
force as an argument of either premiss depends on the
other premiss : if both be admitted, the conclusion
legitimately connected with them cannot be denied.
It is evidently immaterial to the argument whether
the conclusion be placed first or last ; but it may be
proper to remark, that a premiss placed after its con-
clusion is called the reason of it, and is introduced by
one of those conjunctions which are called causal;
viz. “ since,” “ because,” &c. which may indeed be
often occasions error and perplexity, that both these
classes of conjunctions have also another signification,
being employed to denote, respectively, cause and
effect, as well as premiss and conclusion : e. g. if I
say, (to use an instance employed by Aristotle) “ yon-
der is a fixed star, because it twinkles,” or, “it
twinkles, and therefore is a fixed star,” I employ
these conjunctions to denote the connection of pre-
miss and conclusion ; for it is plain that the twink-
ling of the star is not the cause of its being fixed, but
only the cause of my knowing that it is so : but if I say,
“ it twinkles because it is a fixed star,” or it is a fixed
star, and therefore twinkles,” I am using the same
conjunctions to denote the connection of cause and
effect; for in this case the twinkling of the star, being
evident to the eye, would hardly need to be proved,
but might need to be accounted for. There are,
however, many cases in which the cause is employed
to prove the existence of its effect; especially in argu-
ments relating to future events : the cause and the
reason, in that case, coincide ; and this contributes to
their being so often confounded together in other
cases. In an argument, such as the example above
given, it is, as has been said, impossible for any one,
who admits both premises, to avoid admitting the
conclusion ; but there will be frequently an apparent
connection of premises with a conclusion which does
not in reality follow from them, though to the inat-
tentive or unskilful the argument may appear to be
valid : and there are many other cases in which a
doubt may exist whether the argument be valid or
not ; i. e. whether it be possible or not to admit the
premises, and yet deny the conclusion. It is of the
highest importance, therefore, to lay down some
regular form to which every valid argument may be
reduced, and to devise a rule which shall prove the
validity of every argument in that form, and conse-
quently the unsoundness of any apparent argument
which cannot be reduced to it —e.g. if such an
argument as this be proposed, “every rational agent
is accountable ; brutes are not rational agents; there-
fore they are not accountable :” or again, “ all
wise legislators suit their laws to the genius of their
nation; Solon did this ; therefore he was a wise
legislator :" there are some, perhaps, who would not
perceive any fallacy in such arguments, especially if
enveloped in a cloud of words; and still more when
the conclusion is true, or, which comes to the same
point, if they are disposed to believe it; and others
might perceive indeed, but might be at a loss to
explain the fallacy. Now these (apparent) arguments
exactly correspond respectively with the following,
the absurdity of the conclusions from which is manifest:
“ every horse is an animal; sheep are not horses;
therefore they are not animals:” and, “all vegetables
grow ; an animal grows; therefore it is a vegetable.”
These last examples, it has been said, correspond
exactly (considered as arguments) with the former;
the question respecting the validity of an argument
being, not whether the conclusion be true, but whe-
ther it follows from the premises adduced. This mode
of exposing a fallacy, by bringing forward a similar
one whose conclusion is obviously absurd, is often,
and very advantageously, resorted to in addressing
\–2– moreover, since (apparent) arguments which are or last ; the illative conjunctions, “therefore,” &c. sº.
unsound and inconclusive, are so often employed designate the conclusion. It is a circumstance which f
198
L. O. G. I. C.
Logic.
those who are ignorant of Logical rules; but to lay
down such rules, and employ them as a test, is evi-
dently a safer and more compendious, as well as a
more philosophical mode of proceeding. To attain
these, it would plainly be necessary to analyze
some clear and valid arguments, and to observe in what
their conclusiveness consists. Let us suppose, then,
such an examination to be made of the syllogism
above mentioned: “whatever exhibits marks of design
had an intelligent author.”
The world exhibits marks of design ; therefore the
world had an intelligent author. In the first of these
premises we find it assumed universally of the class of
“, things which exhibit marks of design,” that they
had an intelligent author ; and in the other premiss,
“ the world" is referred to that class as comprehended
in it': now it is evident, that whatever is said of the
whole of a class, may be said of any thing compre-
hended in that class ; so that we are thus authorized
to say of the world, that it had an intelligent author.
Again, if we examine a syllogism with a negative con-
clusion, as, e.g. “ nothing which exhibits marks of
design could have been produced by chance : the world
exhibits, &c.; therefore the world could not have
been produced by chance.” The process of Reasoning
will be found to be the same ; since it is evident,
that whatever is denied universally of any class, may
be denied of any thing that is comprehended in that
class.
On further examination it will be found, that all
valid arguments whatever may be easily reduced to
such a form as that of the foregoing syllogisms ; and
that consequently the principle on which they are
constructed is the universal principle of Reasoning.
So elliptical indeed is the ordinary mode of expression,
even of those who are considered as prolix writers,
i.e. so much is implied and left to be understood in
the course of argument, in comparison of what is
actually stated, (most men being impatient, even to
excess, of any appearance of unnecessary and tedious
formality of statement,) that a single sentence will
often be found, though perhaps considered as a single
argument, to contain, compressed into a short com-
pass, a chain of several distinct arguments; but if
each of these be fully developed, and the whole of
what the author intended to imply be stated expressly,
it wiłl be found that all the steps even of the longest
and most complex train of Reasoning, may be reduced
into the above form. -
It is a mistake (which might appear scarcely worthy
of notice had not so many, even esteemed writers,
fallen into it) to imagine that Aristotle and other
Logicians meant to propose that this prolix form of
unfolding arguments should universally supersede, in
argumentative discourses, the common forms of ex-
pression; and that to reason Logically, means, to state
all arguments at full length in the syllogistic form :
and Aristotle has even been charged with inconsistency
for not doing so; it has been said, that “ in his
Treatises of Ethics, Politics, &c. he argues like a
rational creature, and never attempts to bring his own
system into practice :” as well might a Chemist be
charged with inconsistency for making use of any of
the compound substances that are commonly em-
ployed, without previously analyzing and resolving
them into their simple elements; as well might it be
imagined that, to speak grammatically, means, to
parse every sentence we utter.
sue the illustration) keeps by him his tests and his
method of analysis, to be employed when any sub-
stance is offered to his notice, the composition of which
has not been ascertained, or in which adulteration is
suspected. Now a fallacy may aptly be compared to
some adulterated compound ; it consists of an inge-
nious mixture of truth and falsehood, so entangled,
so intimately blended, that the falsehood is (in the
chemical phrase) held in solution : one drop of sound
Logic is that test which immediately disunites them,
makes the foreign substance visible, and precipitates
it to the bottom.
But to resume the investigation of the principles of
Reasoning : the maxim resulting from the examination
of a syllogism in the foregoing form, and of the
application of which every valid argument is in reality
an instance, is, “ that whatever is predicated (i.e.
affirmed or denied) universally, of any class of things,
may be predicated, in like manner, (viz. affirmed or
denied) of any thing comprehended in that class.”
This is the principle, commonly called the dictum de
omni et nullo, for the establishment of which we are
indebted to Aristotle, and which is the keystone of his
whole Logical system. It is not a little remarkable
that some, otherwise judicious writers, should have
been so carried away by their zeal against that philo-
sopher, as to speak with scorn and ridicule of this
principle, on account of its obviousness and simplicity;
though they would probably perceive at once, in any
other case, that it is the greatest triumph of philoso-
phy to refer many, and seemingly very various, phe-
nomena to one, or a very few, simple principles ; and
that the more simple and evident such a principle is,
provided it be truly applicable to all the cases in
question, the greater is its value and scientific beauty.
If, indeed, any principle be regarded as not thus appli-
cable, that is an objection to it of a different kind.
Such an objection against Aristotle's dictum, no one
has ever attempted to establish by any kind of proof ;
but it has often been taken for granted ; it being (as
has been stated) very commonly supposed, without
examination, that the syllogism is a distinct kind of
argument, and that the rules of it do not apply, nor
were intended to apply, to all Reasoning whatever.
Under this misapprehension, Campbell (Philosophy of
Rhetoric) labours, with some ingenuity, and not
without an air of plausibility, to shew that every
syllogism must be futile and worthless, because the
premises virtually assert the conclusion : little dream-
ing, of course, that his objections, however specious,
lie against the process of Reasoning itself universally;
and will therefore, of course, apply to those very
arguments which he is himself adducing. -
It is much more extraordinary to find another
author (Dugald Stewart) adopting, expressly, the very
same objections, and yet distinctly admitting within
a few pages, the possibility of reducing every course
of argument to a series of syllogisms.
The same writer brings an objection against the
dictum of Aristotle; which it may be worth while
to notice briefly, for the sake of setting in a clearer
light the real character and object of that principle.
Its application being, as has been seen, to a regular
and conclusive syllogism, he supposes it intended to
prove and make evident the conclusiveness of such a
syllogism; and remarks how unphilosophical it is to
The Chemist (to pur- Introduc-
tory
Section.
L O G. H. C.
199
Hlogic.
attempt giving a demonstration of a demonstration.
And certainly the charge would be just, if we could
imagine the Logician's object to be, to increase the
certainty of a conclusion which we are supposed to
have already arrived at by the clearest possible mode
of proof. But it is very strange that such an idea
should ever have occurred to one who had even the
slightest tincture of Natural Philosophy : for it might
as well be imagined that a Natural Philosopher or a
Chemist's design to strengthen the testimony of our
senses by a priori reasoning, and to convince us that
a stone when thrown will fall to the ground, and that
gunpowder will explode when fixed, because they
shew that according to their principles those pheno-
mena must take place as they do. But it would be
reckoned a mark of the grossest ignorance and stu-
pidity, not to be aware that their object is not to
prove the existence of an individual phenomenon,
which our eyes have witnessed, but (as the phrase is)
to account for it : i. e. to shew according to what
principle it takes place ;-to refer, in short, the
individual case to a general law of nature. The object
of Aristotle's dictum is precisely analogous : he
had, doubtless, no thought of adding to the force of
any individual syllogism ; his design was to point
out the general principle on which that process is con-
ducted which takes place in each syllogism. And as
the laws of nature (as they are called) are in reality
merely generalized facts, of which all the phenomena
coming under them are particular instances ; so the
proof drawn from Aristotle's dictum is not a distinct
demonstration brought to confirm another demonstra-
tion, but is merely a generalized and abstract state-
ment of all demonstration whatever; and is therefore
in fact, the very demonstration which (mutatis mutandis)
accommodated to the various subject matters, is
actually'employed in each particular case. -
In order to trace more distinctly the different steps
of the abstracting process, by which any particular
argument may be brought into the most general form,
we may first take a syllogism stated accurately and at
full length, such as the example formerly given,
** whatever exhibits marks of design, &c.,” and then
somewhat generalize the expression, by substituting
(as in Algebra) arbitrary unmeaning symbols for the
significant terms that were originally used ; the syllo-
gism will then stand thus ; “every B is A.; C is B ;
therefore C is A.” The Reasoning is no less evidently
valid when thus stated, whatever terms A, B, and C,
respectively may be supposed to stand for : Such
terms may indeed be inserted as to make all, or any
of, the assertions false; but it will still be no less im-
possible for any one who admits the truth of the
premises, in an argument thus constructed, to deny the
conclusion; and this it is that constitutes the conclu-
siveness of an argument. .
Viewing then the syllogism thus expressed, it ap-
pears clearly, that “A stands for any thing whatever
that is predicated of a whole class,” (viz. of every B)
“ which comprehends or contains in it something else,”
viz. C., of which B is, in the second premiss affirmed;
and that consequently the first term (A) is, in the con-
clusion, predicated of the third C.
Now to assert the validity of this process, now
before us, is to state the very dictum we are treating
of with hardly even a verbal alteration, viz.:
1. Any thing whatever, predicated of a whole class,
2. Under which class something else is contained,
3. May be predicated
tained.
• The three members into which the maxim is here
distributed, correspond to the three propositions of
the syllogism to which they are intended respectively
to apply. -
The advantage of substituting for the terms, in a
regular syllogism, arbitrary unmeaning symbols such
as letters of the alphabet, is much the same as in
Mathematics : the Reasoning itself is then considered,
by itself, clearly, and without any risk of our being
misled by the truth or falsity of the conclusion, which
are, in fact, accidental and variable ; the essential
point, being, as far as the argument is concerned, the
connection between the premises and the conclusions.
We are thus enabled to embrace the general principle
of all Reasoning, and to perceive its applicability to
an indefinite number of individual cases. That
Aristotle, therefore, should have been accused of
making use of these symbols for the purpose of
darkening his demonstrations, and that too, by per-
sons not unacquainted with Geometry and Algebra,
is truly astonishing. If a Geometer, instead of desig-
nating the four angles of a square, by four letters,
were to call them north, south, east, and west, he
would not render the demonstration of a theorem the
easier ; and the learner would be much more likely
to be perplexed in the application of it.
It belongs then exclusively to a syllogism, properly
so called (i. e. a valid argument, so stated that its
conclusiveness is evident from the mere form of the
expression) that if letters or any other unmeaning
symbols be substituted for the several terms, the
validity of the argument shall still be evident.
Whenever this is not the case, the supposed argu-
ment is either unsound and sophistical, or else may
be reduced, (without any alteration of its meaning)
into the syllogistic form ; in which form, the test
just mentioned may be applied to it.
What is called an unsound or fallacious argument,
i. e. an apparent argument which is, in reality, none,
cannot, of course, be reduced into this form ; but
when stated in the form most nearly approaching to
this that is possible, its fallaciousness becomes more
evident, from its nonconformity to the foregoing
rule : e. g. “whoever is capable of deliberate crime
is responsible ; an infant is not capable of deliberate
crime; therefore, an infant is not responsible :'' here,
the term “responsible" is affirmed universally of
“ those capable of deliberate crime;” it might, there-
fore, according to Aristotle's dictum, have been
affirmed of any thing contained under that class; but
in the instance before us nothing is mentioned as con-
tained under that class, only the term infant is
excluded from that class ; and though what is affirmed
of a whole class may be affirmed of any thing that is
contained under it, there is no ground for supposing
that it may be denied of whatever is not so contained ;
for it is evidently possible that it may be applicable to
a whole class and to something else besides: to say,
e.g. that all trees are vegetables, does not imply that
nothing else is a vegetable. It is evident, therefore,
that such an apparent argument as the above does not
comply with the rule laid down, and is consequently
invalid.
Again, in this instance, “food is necessary to life;
of that which is so con-
S-N-2
Introduc-
tory
Section.
200
L O G. I. C.
Logic.
corn is food; therefore corn is mecessary to life :” the
term “ necessary to life” is affirmed of food, but not
wniversally ; for it is not said of every kind of food; the
meaning of the assertion being manifestly that some
food is necessary to life: here again therefore the rule
has not been complied with, since that which is pre-
dicated, (i.e. affirmed or denied,) not of the whole, but
of a part only of a certain class, cannot be predicated
of any thing, whatever is contained under that class.
The fallacy in this last case is, what is usually
described in Logical language as consisting in the
“ non-distribution of the middle term.” In order to
understand this phrase, it is necessary to observe,
that a proposition being an expression in which one
thing is affirmed or denied of another ; e.g. “A is B,”
both that of which something is said, and that which
is said of it, (i.e. both A and B,) are called “Terms,”
from their being (in their nature) the extremes or
boundaries of the proposition; and there are, of course,
two, and but two, terms in a proposition, (though it
may so happen that either of them may consist either
of one word, or of several ;) and a term is said to be
“ distributed,” when it is taken universally, so as to
stand for every thing it is capable of being applied to;
and consequently “ undistributed,” when it stands
for a part only of the things signified by it; thus, “all
food,” or every kind of food, are expressions which
imply the distribution of the term “ food;” “ some
food " would imply its non-distribution : and it is also
to be observed, that the term of which, in one pre-
miss, something is affirmed or denied, and to which
in the other premiss something else is referred as con-
tained in it, is called the “middle '' term in the syl-
logism, as standing between the other two, (viz. the
two terms of the conclusion,) and being the medium
of proof. Now it is plain, that if in each premiss a
part only of this middle term is employed, i. e. if it
be not at all distributed, no conclusion can be drawn.
Hence, if in the example formerly adduced, it had
been merely stated that “ something” (not “whatever,”
or “every thing”) “ which exhibits marks of design,
is the work of an intelligent author,” it would not
have followed, from the world's exhibiting marks of
design, that that is the work of an intelligent author.
It is to be observed, also, that the words “all,”
and “every,” which mark the distribution of a term,
and “ some,” which marks its non-distribution, are
not always. introduced : they are frequently under-
stood, and left to be supplied by the context; e.g.
“ food is necessary :” viz. “ some food;” “ man is
mortal;” viz. “ every man.” Propositions thus ex-
pressed are called by Logicians “ indefinite,” because
it is left undetermined by the form of the expression
whether the “subject,” (the term of which some-
thing is affirmed or denied being called the “subject”
of the proposition, and that which is said of it, the
“ predicate”) be distributed or not. Nevertheless it
is plain that in every proposition the subject either is,
or is not, distributed, though it be not declared whe-
ther it is or not ; consequently every proposition,
whether expressed indefinitely or not, must be either
“ universal” or “particular;” those being called
universal, in which the predicate is said of the whole
of the subject, (or in other words, where the subject
is distributed ;) and those, particular, in which it is
said only of a part of the subject : e.g. “ All men
are sinful,” is universal ; “ some men are sinful,”
particular : and this division of propositions is in Introduc-
Logical language said to be according to their
“ quantity.” -
IBut the distribution or non-distribution of the
predicate is entirely independent of the quality of the
proposition ; nor are the signs “all” and “ some ''
ever affixed to the predicate ; because its distribution
depends upon, and is indicated by the “ quality” of
the proposition ; i.e. its being affirmative or negative;
it being a universal rule, that the predicate of a nega-
tive proposition is distributed, and, of an affirmative,
undistributed. The reason of this may easily be under-
stood, by considering that a term which stands for a
whole class may be applied to (i.e. affirmed of) any
thing that is comprehended under that class, though
the term of which it is thus affirmed may be of much
narrower extent than that other, and may, therefore,
be far from coinciding with the whole of it : thus it
may be said with truth, that “ the Negroes are un-
civilized,” though the term uncivilized be of much
wider extent than “Negroes,” comprehending, be-
sides them, Hottentots, &c. : so that it would not be
allowable to assert, that “ all who are uncivilized are
Negroes;" it is evident, therefore, that it is a part
only of the term “ uncivilized '' that has been affirmed
of “Negroes:” and the same reasoning applies to
every affirmative proposition ; for though it may so
happen that the subject and predicate coincide, i.e. are
of equal extent, as, e.g. “ all men are rational ani-
mals,” (it being equally true, that “ all rational
animals are men,) yet this is not implied by the form of
the expression ; since it would be no less true, that
“ all men are rational animals,” even if there were
other rational animals besides man. -
It is plain, therefore, that if any part of the predi-
cate is applicable to the subject, it may be affirmed,
and, of course, cannot be denied of that subject; and
consequently, when the predicate is denied of the sub-
ject, it is implied that no part of that predicate is
applicable to that subject ; i. e. that the whole of the
predicate is denied of the subject: for to say, e.g. that
“ no beasts of prey ruminate,” implies that beasts of
prey are excluded from the whole class of ruminant
animals, and consequently that “no ruminant animals
are beasts of prey.
tioned rule, that the distribution of the predicate
is implied in negative propositions, and its non-
distribution in affirmatives.
It is to be remembered, therefore, .that it is not
sufficient for the middle term to occur in a universal
proposition, since if that proposition be an affirmative,
and the middle term be the predicate of it, it will not
be distributed : e.g. if in the example formerly given
it had been merely asserted, that “all the works of an
intelligent author shew marks of design,” and that
“ the universe shows marks of design,” nothing could
have been proved ; since, though both these proposi-
tions are universal, the middle term is made the pre-
dicate in each, and both are affirmative; and accord-
ingly the rule of Aristotle is not here complied with,
since the term, “ work of an intelligent author,”
which is to be proved applicable to “the universe,”
is not affirmed of the middle term, (“what shows
marks of design,”) under which “universe" is con-
tained ; but the middle term on the contrary is
affirmed of it. -
If, however, one of the premises be negative,
And hence results the above men-
tory
Section.
Lo G I c.
201
Logic.
the middle term may then be made the predicate
of it, and will thus, according to the above remark,
be distributed : e.g. “ no ruminant animals are
predacious; the lion is predacious ; therefore the
lion is not ruminant :” this is a valid syllogism ; and
the middle term (predacious) is distributed by being
made the predicate of a negative proposition. The
form, indeed, of the syllogism, is not that prescribed
by the dictum of Aristotle, but it may easily be
reduced to that form, by stating the first proposition
thus ; no predacious animals are ruminant ; which is
manifestly implied (as was above remarked) in the
assertion, that “no ruminant animals are predacious.”
The syllogism will thus appear in the form to which
the dictum applies.
It is not every argument, indeed, that can be reduced
to this form by so short and simple an alteration as in
the case before us : a longer and more complex pro-
cess will often be required ; and rules will hereafter
be laid down to facilitate this process in certain cases:
but there is no sound argument but what can be
reduced into this form, without at all departing from
the real meaning and drift of it: and the form will be
found (though more prolix than is needed for ordi-
nary use) the most perspicuous in which an argument
can be exhibited. .
All reasoning whatever, then, rests on the one sim-
ple principle laid down by Aristotle; that, “what is
predicated, either affirmatively or negatively, of a term
distributed, may be predicated, in like manner, (i. e.
affirmatively or negatively) of any thing contained
under that term.” So that when our object is to prove
any proposition, i. e. to shew that one term may
rightly be affirmed or denied of another, the process
which really takes place in our minds is, that we
refer that term (of which the other is to be thus pre-
dicated,) to some class, (i.e. middle term) of which
that other may be affirmed, or denied, as the case may
loe. Whatever the subject matter of an argument
may be, the Reasoning itself, considered by itself, is
in every case the same process; and if the writers
against Logic had kept this in mind, they would have
been cautious of expressing their contempt of what
they call “ syllogistic Reasoning,” which is in truth
all Reasoning; and instead of ridiculing Aristotle's
principle for its obviousness and simplicity, would
have perceived that these are in fact its highest
praise : the easiest, shortest, and most evident theory,
provided it answer the purpose of explanation, being
ever the best.
If we conceive an inquirer to have reached, in his
investigation of the theory of Reasoning, the point
to which we have now arrived, a question which
would be likely next to engage his attention, is, that
of predication; i.e. since in Reasoning we are to find
a middle term, which may be predicated affirmatively
of the subject in question, we are led to inquire what
terms may be affirmed, and what denied, of what others.
It is evident that proper names, or any other terms,
which denote each but a single individual, as “Caesar,”
“ the Thames,” “ the Conqueror of Pompey,”
“ this river,” (hence called in Logic, “ singular
terms”) cannot be affirmed of any thing besides them-
selves, and are therefore to be denied of any thing
else; we may say, “ this river is the Thames,” or
“ Caesar was the conqueror of Pompey;” but we
cannot say of any thing else that it is the Thames.
WOL. I.
On the other hand, those terms which are called Introduc-
“ common,” as denoting any one individual of a
whole class, as “ river,” “ conqueror,” may o
course be affirmed of any, or all that belong to that
class ; as, “ the Thames is a river;” “ the Rhine
and the Danube are rivers.”
Common terms, therefore, are called “ predica-
bles,” (viz. affirmatively predicable,) from their
capability of being affirmed of others: a singular term
on the contrary may be subject of a proposition,
but never the predicate, unless it be of a negative
proposition; (as, e.g. the first-born of Isaac was not
Jacob ;) or, unless the subject and predicate be only
two expressions for the same individual object, as in
some of the above instances.
The process by which the mind arrives at the
notions expressed by these “ common” (or in popular
language, “ general”) terms, is properly called gene-
ralization; though it is usually (and truly) said to be
the business of abstraction ; for generalization is one
of the purposes to which abstraction is applied: when
we draw off, and contemplate separately, any part of an
object presented to the mind, disregarding the rest
of it, we are said to abstract that part. Thus, a per-
son might, when a rose was before his eyes or mind,
make the scent a distinct object of attention, laying
aside all thought of the colour, form, &c.; and thus,
though it were the only rose he had ever met with,
he would be employing the faculty of abstraction;
but if, in contemplating several objects, and finding
that they agree in certain points, we abstract the cir-
cumstances of agreement, disregarding the differences,
and give to all and each of these objects a name appli-
cable to them in respect of this agreement, i.e. a
common name, (as: “ rose,”) we are then said to
generalize. Abstraction, therefore, does not neces-
sarily imply generalization, though generalization
implies abstraction. -
Much meedless difficulty has been raised respecting
the results of this process; many having contended, and
perhaps more having taken for granted, that there must
be some really existing thing, corresponding to each
of these general or common terms, and of which such
term is the name, standing for and representing it :
e.g. that as there is a really existing being cor-
responding to the proper name AEtna, and signifying
it, so the common term “mountain,” must have some
one really existing thing corresponding to it, and of
course distinct from each individual mountain, (since
the term is not singular, but common,) yet existing
in each, since the term is applicable to each of them.
“When many different men,” it is said, “are at the
same time thinking or speaking about a mountain,
i. e. not any particular one, but a mountain generally,
their minds must be all employed on something ; which
must also be one thing, and not several, and yet can-
not be any one individual :” and hence a vast train of
mystical disquisitions about ideas, &c. has arisen,
which are at best nugatory, and tend to obscure our
view of the process which actually takes place in the
mind. - *
The fact is, the notion expressed by a common
term is merely an inadequate (or incomplete) notion
of an individual; and from the very circumstance of
its inadequacy, it will apply equally well to any one
of several individuals : e. g. if I omit the mention
and the consideration of every circumstance which
2 E
tory
f (A Section.
202
L O G. H. C.
Logic.
distinguishes AEtna from any other mountain, I then
form a notion (expressed by the common term
mountain) which inadequately designates AEtna, and
is equally applicable to any one of several other
individuals.
Generalization, it is plain, may be indefinitely
extended by a further abstraction applied to common
terms : e. g. as by abstraction from the term Socrates
we obtain the common term philosopher; so from
“ philosopher,” by a similar process, we arrive at the
more general term “ man ; ” from “man” to
“ animal,” &c.
The employment of this faculty at pleasure has been
regarded, and perhaps with good reason, as the cha-
racteristic distinction of the human mind from that of
the brutes. We are thus enabled, not only to sepa-
rate, and consider singly, one part of an object pre-
sented to the mind, but also to fix arbitrarily upon
whatever part we please, according as may suit the
purpose we happen to have in view : e.g. any indivi-
dual person to whom we may direct our attention,
may be considered either in a political point of view,
and accordingly referred to the class of merchant,
farmer, lawyer, &c. as the case may be ; or physio-
logically, as negro, or white man ; or theologi-
cally, as Pagan or Christian, Papist or Protestant ;
or geographically, as European, American, &c. &c.
And so, in respect of anything else that may be the
subject of our Reasoning : we arbitrarily fix upon and
abstract that point which is essential to the purpose in
hand ; so that the same object may be referred to
various different classes, according to the occasion.
Not, of course, that we are allowed to refer anything
to a class to which it does not really belong; which
would be pretending to abstract from it something
that was no part of it ; but that we arbitrarily fix on
any part of it which we choose to abstract from the
rest. It is important to notice this, because men are
often disposed to consider each object as really and
properly belonging to some one class alone, from their
having been accustomed, in the course of their own
pursuits, to consider in one point of view only things
which may with equal propriety be considered in other
points of view also ; i.e. referred to various classes,
(or predicates.) And this is that which chiefly consti-
tutes what is called narrowness of mind: e.g. a mere Introduc-
Botanist might be astonished at hearing such plants as
clover and lucerne included, in the language of a
farrner, under the term “grasses,” which he has been
accustomed to limit to a tribe of plants widely different
in all Botanical characteristics; and the mere farmer
might be no less surprised to find the troublesome
“ weed,” (as he has been accustomed to call it,)
known by the name of couch grass, and which he has
been used to class with nettles and thistles, to which
it has no Botanical affinity, ranked by the Botanist as a
species of wheat, (Triticum Repens.) And yet neither
of these classifications is in itself erroneous or irra-
tional ; though it would be absurd in a Botanical
treatise to class plants according to their Agricultural
use ; or in an Agricultural treatise, according to the
structure of their flowers. - -
The utility of these considerations, with a view to
the present subject, will be readily estimated, by
recurring to the account which has been already given
of the process of Reasoning; the analysis of which
shews, that it consists in referring the term we are
speaking of to some class, viz. a middle term; which
term again is referred to or excluded from (as the case
may be) another class, viz. the term which we wish to
affirm or deny of the subject of the conclusion. So
that the quality of our Reasoning in any case must
depend on our being able, correctly, clearly, and
promptly, to abstract from the subject in question that
which may furnish a middle term suitable to the
OCCàS10 Il-
The imperfect and irregular sketch which has here
been attempted, of the Logical System, may suffice
(even though some parts of it should not be at once
fully understood by those who are entirely strangers
to the study) to point out the general drift and pur-
pose of the Science, and to render the details of it both
more interesting and more intelligible. The analytical
form, which has here been adopted, is, generally'
speaking, the best suited for introducing any science in
the plainest and most interesting form ; though the
synthetical, which will henceforth be employed, is the
most regular and the most compendious form for
storing it up in the memory.
CHAPTER I.
OF THE opBRATIONS OF THE MIND AND of TERMrs.
THERE are three operations of the mind which are
concerned in argument : 1st. Simple Apprehension;
2d. Judgment; 3d. Discourse or Reasoning. 1st. Sim-
ple apprehension is the notion (or conception) of any
object in the mind, analogous to the perception of the
senses. It is either incomplex or complex : incom-
plex apprehension is of one object, or of several with-
out any relation being perceived between them, as of
“a man,” “a horse,” “ cards: ” complex is of
several with such a relation, as of “a man on horse-
back,” “a pack of cards.”
2d. Judgment is the comparing together in the
mind two of the notions, (or ideas) whether complex
or incomplex, which are the objects of apprehension,
and pronouncing that they agree or disagree with each
other; (or that one of them belongs or does not belong
to the other.) Judgment therefore is either affirmative
or negative. e -
3d. Reasoning (or discourse) is the act of proceed-
ing from one judgment, to another founded upon it,
(or the result of it.) *
§ 2. Language affords the signs by which these
operations of the mind are expressed and communi-
cated. An act of Apprehension expressed in language,
is called a Term ; an act of Judgment, a Proposition ;
an act of Reasoning, an Argument or Syllogism ; as
€. 9". . -
“Every dispensation of Providence is beneficial;
Afflictions are dispensations of Providence,
Therefore they are beneficial :” is a Syllogism ;
tory
Section.
*= *
Chap. I.
L' O G. I. C. 203
that under “verb,” we do not include the infinitive, Chap. ;
Logic. (the act of Reasoning being indicated by the word
which is properly a noun substantive, nor the parti-S-N-7
*—y–’ “ therefore,”) it consists of three Propositions, each of
which has (necessarily) two Terms, as “beneficial,”
“ dispensations of Providence,” &c.
Language is employed for various purposes, e.g.
the province of an historian is to convey information ;
of an orator, to persuade, &c. Logic is concerned
with it only when employed for the purpose of
Reasoning, (i. e. in order to convince ;) and whereas, in
reasoning, Terms are liable to be indistinct, (i. e. with-
out any clear determinate meaning,) Propositions, to
be false, and Arguments, inconclusive, Logic undertakes
directly and completely to guard against this last
defect, and incidentally and in a certain degree
against the others, as far as can be done by the proper
use of language : it is, therefore, (when regarded as an
art”) “ the art of employing language properly for the
purpose of Reasoning.” Its importance no one can
rightly estimate who has not long and attentively con-
sidered how much our thoughts are influenced by
words, and how much error, perplexity, and labour,
are occasioned by a faulty use of language.
A Syllogism being, as aforesaid, resolvable into three
Propositions, and each Proposition containing two
Terms ; of these Terms, that which is spoken of, is
called the Subject; that which is said of it, the Predi-
cate; and these two together are called the Terms, (or
extremes,) because, logically, the subject is placed
first, and the predicate last : and, in the middle, the
Copula, which indicates the act of Judgment, as by it,
the Predicate is affirmed or denied of the Subject. It
must be either is or Is NoT ; the substantive verb
being the only verb recognised by Logic : all others
are resolvable, by means of the verb, “ to be,” and
a participle or adjective ; e.g. “ the Romans con-
quered:” the word “conquered" is both Copula and
Predicate, being equivalent to “ were (Cop.) victorious”
(Pred.)t -
§ 3. It is evident that a Term may consist
either of one word or of several ; and that it is not
every word that is capable of being employed by
itself as a Term ; e.g. adverbs, prepositions, &c. and
also nouns in any other case besides the nominative.
A noun may be by itself a Term ; a verb (all except
the substantive verb used as the Copula,) is resolvable
into the Copula and Predicate, to which it is equiva-
lent, and indeed is often so resolved in the mere ren-
dering out of one language into another; as “ ipse
adest,” he is present. It is to be observed, however,
* It is to be observed, however, that as a science is conversant
about knowledge only, an art is the application of knowledge to
practice ; hence Logic (as well as any other system of know-
ledge) becomes, when applied to practice, an art ; while con-
fined to the theory of Reasoning, it is strictly a science: and it is
as such that it occupies the higher place in point of dignity, since
it professes to develope some of the most interesting and curious
intellectual phenomena. *
# It is proper to observe, that the Copula, as such, has no
relation to time; but expresses merely the agreement or disagree-
ment of two given terms : hence, if any other tense of the sub-
stantive verb, besides the present, is used, it is either to be under-
stood as the same in sense, (the difference of tense being regarded
as a matter of grammatical convenience only ;) or else, if the cir-
cumstance of time really do modify the sense of the whole propo-
sition, so as to make the use of that tense an essential, then this
circumstance is to be regarded as a part of one of the terms :
** at that time,” or some such expression, being understood.
Sometimes the substantive verb is both Copula and Predicate ;
i. e. where existence only is predicated : e. g. Deus est.
ciple, which is a noun adjective. They are verbals,
being related to their respective verbs in respect of
the things they signify , but not verbs, inasmuch as they
differ entirely in their mode of signification. It is worth
observing, that an infinitive (though it often comes
last in the sentence) is never the Predicate, except
when another infinitive is the Subject. It is to be
observed, also, that in English there are two infinitives,
one, in “ing,” the same in sound and spelling as the
participle present, from which, however, it should be
carefully distinguished; e.g. “ rising early is health-
ful,” and “ it is healthful to rise early,” are
equivalent. *
An adjective (including participles) cannot, by
itself, be made the Subject of a Proposition ; but is
often employed as a Predicate ; as “Crassus was
rich ;” though some choose to consider some substan-
tives as understood in every such case, (e. g. rich
man) and consequently do not reckon adjectives
among simple Terms ; i. e. words which are capable,
simply, of being employed as Terms. This, however,
is a question of no practical consequence.
Of simple Terms, then, (which are what the first
part of Logic treats of) there are many divisions;* of
which, however, one will be sufficient for the present
purpose ; viz. into singular and common ; because,
though any Term whatever may be a Subject, none but
a common Term can be affirmatively predicated of
several others. A singular Term stands for one indivi-
dual, as ‘‘ Caesar,” “ the Thames;" (these, it is
plain, cannot be said [or predicated] affirmatively, of
any thing but themselves.) A common Term stands
for several individuals : i.e. can be applied to any of
them, as comprehending them in its single signification ;
as “man,” “river,” “great.” The notions expressed
by these common Terms, we are enabled to form, by
the faculty of abstraction : for by it, in contemplating
any object (or objects,) we can attend exclusively to
some particular circumstances belonging to it, [some
certain parts of its nature as it were] and quite with-
hold our attention from the rest. When, therefore,
we are thus contemplating several individuals which
resemble each other in some part of their nature, we
can (by attending to that part alone, and not to those
points in which they differ) assign them one common
name, which will express or stand for them merely as
far as they all agree ; and which of course will be
applicable to all or any of them ; (which process is
called generalization,) and each of these names is
called a common Term, from its belonging to them all
alike ; or a Predicable, because it may be predicated
affirmatively of them, or of any one of them.
Generalization (as has been remarked) implies
abstraction, but it is not the same thing ; for there
may be abstraction without generalization : when we
are speaking of an individual, it is usually an abstract
notion that we form ; e. g. suppose we are speaking
of the present King of France; he must actually be
* The usual divisions of words into univocal, equivocal, and
analogous, and into words of the first and second intention,
however, are not, strictly speaking, divisions of words, but divi-
sions of the manner of employing them . the same word may be
employed either univocally, equivocally, or analogously; either
in the first intention or in the second.
2 E 2
204
L. O. G. I. C. .
regarded as containing under it only individuals, is Chap. I.
called infima (the lowest) Species. \- ~/.
Mogic. either at Paris or elsewhere; sitting, standing, or in
\-v- some other posture; and in such and such a dress, &c.
Yet many of these circumstances, (which are separable
accidents, (vide $ 7.) and consequently) which are
regarded as non-essential to the individual, are quite
disregarded by us; and we abstract from them what
we consider as essential ; thus forming an abstract
notion of the individual. Yet there is here no
generalization.
§ 4. Whatever Term can be affirmed of several
things, must express either their whole essence, which
is called the Species ; or a part of their essence, (viz.
either the material part, which is called the Genus, or
the formal and distinguishing part, which is called
Differentia,) or in common discourse, characteristic, or
something joined to the essence, whether necessarily,
which is called a property, or contingently, which is
an accident.
Every Predicable expresses either
ſº Y
The whole essence or part of its or something
of its subject: eSSen Ce joined to its
viz.: Species— | €SSCIn Ce
tº- o Y.
Genus—Difference
£r k- - | gº –Y
Property Accident
ſ g g Y
universal peculiar universal
but not but not and pe-
peculiar universal culiar || -Y
inseparable—separable.
lt is evident from what has been said, that the Genus
and Difference put together make up the Species :
e.g. “ rational” and “animal" constitute “man ;”
so that, in reality, the Species contains the Genus
(i.e. implies it ;) and when the Genus is called a
whole, and is said to contain the Species, this is only a
metaphorical expression, signifying that it comprehends
the Species, in its own more extensive signification :
e.g. if I predicate of Caesar that he is an animal, I say
the truth indeed, but not the whole truth ; for he is not
only an animal, but a man ; so that “man” is a more
full and complete expression than “animal;”, which
for the same reason is more extensive, as it contains,
(or rather comprehends) and may be predicated of,
several other Species, i.e. “ beast,” “ bird,” &c. In
the same manner the name of a Species is a more exten-
sive, but less full and complete term than that of an
individual, (viz. a singular term;) since the Species
may be predicated of each of these. [Note, that
Genus and Species are commonly said to be predicated
in quid, (ti) (i.e. to answer to the question “what?”
as, “ what is Caesar : " Answer, “a man :” “ what
is a man : " Answer, “ an animal.”) Difference, in
“ quale quid;'' (rotov ti) Property and Accident in
quale (Totov).] -
§ 5. A Genus, which is also a Species, is called a
subaltern Genus or Species; as “bird,” which is the
Genus of “pigeon, (i.e. of which “pigeon '' is a
Species) is itself a Species of “ animal.” A Genus
which is not considered as a Species of anything, is
called summum (the highest) Genus ; a Species which
is not considered as a Genus of any thing, i.e., is
When I say of a magnet, that it is “a kind of iron
ore,” that is called its proximum Genus, because it is
the closest (or lowest) Genus that can be predicated
of it : “mineral ” is its more remote Genus.
When I say that the Differentia of a magnet is its
“ attracting iron,” and that its Property is “polarity,”
these are called respectively a specific Difference and
Property; because magnet is an infima Species, (i.e.
only a Species.) - -
When I say that the Differentia of iron ore is its
“ containing iron,” and its Property “ being attracted
by the magnet,” these are called respectively, a generie
Difference and Property, because iron ore is a subaltern
Species or Genus, being both the Genus of magnet, and
a Species of nineral.
That is the most strictly called a Property, which
belongs to the whole of a Species, and to that Species
alone ; as polarity to the magnet. . [And such a pro-
perty, it is often hard to distinguish from the
Differentia ; but whatever you consider as the most
essential to the nature of a Species with respect to the
matter you are engaged in, you must call the Diffe-
rentia ; as “rationality” to “man;" and whatever you
consider as rather an accompaniment (or result) of that
Difference, you must call the Property; as the “ use
of speech” seems to be a result of rationality.] But
very many Properties which belong to the whole of a
Species are not peculiar to it ; as, “ to breathe air”
belongs to every man, but not to man alone ; and it
is, therefore, strictly speaking, not so much a Pro-
perty of the Species “man,” as of the higher, i. e.
more comprehensive, Species, which is the Genus of
that, viz. of “ land animal.”. Other Properties, as
some Logicians call them, are peculiar to a Species, but
do not belong to the whole of it: e. g. man alone can
be a poet, but it is not every man that is so. These,
however, are more commonly and more properly
reckoned as Accidents.
For that is most properly called an Accident, which
may be absent or present, the essence of the Species.
continuing the same ; as, for a man to be “walking,”
or a “ native of Paris :” of these two examples, the
former is what Logicians call a separable Accident,
because it may be separated from the individual :
(e.g. he may sit down;) the latter is an inseparable
Accident, being not separable from the individual,
(i.e. he who is an individual of Paris can never be
otherwise ;) “from the individual,” I say, because
every Accident must be separable from the Species, else
it would be a Property.
Let it here be observed, that both the general name.
“Predicable,” and each of the classes of Predicables,
(viz. Genus, Species, &c.) are relative ; i.e. we cannot
say what Predicable any Term is, or whether it is any at
all, unless it be specified of what it is to be predicated:
e.g. the Term “red " would be considered a Genus,
in relation to the Terms “ pink,” “scarlet,” &c. it
might be regarded as the Differentia, in relation to
“ red rose;”—as a property of “blood ;”—as an
Accident of “a house,” &c. -
And universally, it is to be steadily kept in mind,
that no “ common Terms” have, as the names of
individuals have, any real thing existing in nature cor-
responding to them ; (Tööe ru, as Aristotle expresses it,
though he has been represented as the champion of
L. O. G. I. C.
205
Logic.
the opposite opinion : vide Categ. c. 3.) but is
merely a name denoting a certain inadequate notion
which, our minds have formed of an individual, and
which, consequently, not including any thing wherein
that individual differs from certain others, is applicable
equally well to all or any of them : thus “man”
denotes no real thing (as the sect of the Realists
maintained,) distinct from each individual, but merely,
any man, viewed inadequately, i.e. so as to omit and
abstract from all that is peculiar to each individual;
by which means the Term becomes applicable alike to
any one of several individuals, or (in the plural) to
several together ; and we arbitrarily fix on the circum-
stance which we thus choose to abstract and consider
separately, disregarding all the rest; so that the same
individual may thus be referred to any of several
different Species, and the same Species to several
Genera, as suits our purpose. Thus it suits
the farmer's purpose to class his cattle with his
ploughs, earts, and other possessions, under the
name of “stock :” the naturalist, suitably to his pur-
pose, classes them as “ quadrupeds,” which Term
would include wolves, deer, &c., which to the farmer
would be a most improper classification : the
commissary, again, would class them with corn,
cheese, fish, &c. as “ provision.” That which is
most essential in one view, being subordinate in
another. e
§ 6. An individual is so called because it is inca-
pable of logical Division ; which is a metaphorical
expression to signify “ the distinct (i.e. separate)
enumeration of several things signified by one common
name.” This operation is directly opposite to genera-
lization, (which is performed by means of abstrac-
tion ;) for as in that, you lay aside the difference by
which several things are distinguished, so as to call
them all by one common name, so, in Division, you
add on the differences, so as to enumerate them by
their several particular names. Thus, “ mineral" is
said to be divided into “stones, metals,” &c.; and
metals again into “gold, iron,” &c. and these are
called the parts (or members) of the Division.
The rules for Division are three : 1st, each of the
parts, or any of them short of all, must contain less
(i.e. have a narrower signification) than the thing
divided. 2d. All the parts together must be exactly
equal to the thing divided ; (therefore we must be
careful to ascertain that the summum Genus may be
predicated of every Term placed under it, and of
nothing else.) 3d. The parts or members must be
opposed ; i. e. must not be contained in one another :
e.g. if you were to divide “book” into “poetical,
historical, folio, quarto, French, Latin,” &c. the mem-
bers would be contained in each other ; for a French
book may be a quarto, and a quarto, French, &c.
You must be careful, therefore, to keep in mind the
principle of Division with which you set out : e.g. whe-
ther you begin dividing books according to their mat-
ter, their language, or their size, &c. these being also
so many cross Divisions. And when any thing is capable
(as in the above instance) of being divided in several
different ways, we are not to reckon one of these as
the true, or real, or right one, without specifying
what the object is which we have in view : for one
mode of dividing may be the most suitable for one
purpose, and another, for another; as e.g. one of
the above modes of dividing books would be the most
suitable to a bookbinder; another in a philosophical, Chºp. [.
S-N-"
and the other in a philological view.
. It must be carefully remembered, that the word
“ Division,” as employed in Logic, is, as has been
observed already, metaphorical ; for to divide, means
originally and properly to separate the component
parts of any thing, each of which is of course abso-
lutely less than the whole : e.g. a tree (i.e. any indi-
vidual tree) might be divided “ physically,” as it is
called, into root, trunk, branches, leaves, &c. Now
it cannot be said that a root or a leaf is a tree :
whereas in a logical Division each of the members is,
in reality, more than the whole : e.g. if you divide
tree (i. e. the Genus, tree) into oak, ash, elm, &c. we
may say of the oak, or of any individual oak, that
“it is a tree;” for by the very word “ oak,” we
express not only the general notion of a tree, but
more, viz. the peculiar characteristic (i.e. difference)
of that kind of tree.
It is plain, then, that it is logically only, i. e. in our
mode of speaking, that a Genus is said to contain (or
rather, comprehend) its Species ; while metaphysi-
cally, i.e. in our conceptions, a Species contains,
i. e. implies, its Genus.
Care must be taken not to confound a physical Divi-
sion with a Logical, against which a caution is given
under R. l. -
§ 7. Definition is another metaphorical word, which
literally signifies, “ laying down a boundary ;” and is
used in Logic to signify an expression which explains
any term, so as to separate it from every thing else, as
a boundary separates fields. A nominal Definition
(such as are those usually found in a dictionary of one's
own language) explains only the meaning of the term,
by giving some equivalent expression, which may
happen to be better known. Thus you might define
a “ Term,” that which forms one of the eatremes or
boundaries of a “ Proposition;” and a “ Predicable,”
that which may be -predicated ; “ decalogue,” “ten
commandments;” “ telescope,” an instrument for
viewing distant objects, &c. A real Definition is one
which explains and unfolds the nature of the thing ;
and each of these kinds of Definition is either accidental
or essential. An essential Definition assigns (or lays
down) the constituent parts of the essence, (or nature.)
An accidental Definition (which is commonly called a
Description) assigns the circumstances belonging to the
essence, viz. Properties and Accidents, (e. g. causes,
effects, &c.) thus, “man” may be described as “an
animal that uses fire to dress his food,” &c. [And
here note, that in describing a Species, you cannot men-
tion any thing which is strictly an Accident, because if
it does not belong to the whole of the Species, it can-
not define it: in describing an individual, on the
contrary, you enumerate the Accidents, because by
them it is that one individual differs from another,
and in this case you add the Species: e.g. “Philip was
a man of Macedon, who subdued Greece,” &c. Indi-
viduals, it is evident, can be defined in this way
alone.]
Lastly, the essential Definition is divided into
physical (i. e. natural) and Logical or Metaphysical :
the physical Definition lays down the real parts of the
essence which are actually separable; the logical, lays
down the ideal parts of it, which cannot be separated
except in the mind: thus, a plant would be defined
physically, by enumerating the leaves, stalks, roots,
206
L. O. G. I. C.
Logic.
&c. of which it is composed : logically, it would be
defined an organized being, destitute of sensation ;
the former of these expressions expressing the Genus,
the latter, the Difference : for a logical Definition must
always consist of the Genus and Differentia, which are
the parts of which Logic considers everything as con-
sisting, and which evidently are separable in the
mind alone. Thus “man” is defined “ a rational
animal,” &c. So also a “Proposition " might be
defined, physically, a Subject and Predicate combined
by a Copula: the parts here enumerated being actually
separable ; but logically it would be defined “a sen-
tence which affirms or denies;” and these two parts
of the essence of a Proposition (which are the Genus
and Differentia of it) can be separated in the mind
only. And note, that the difference is not always
one quality, but is frequently compounded of several
together, no one of which would alone suffice.
Definitions are divided into nominal and real,
according to the object accomplished by them ; whether
to explain, merely, the meaning of the word, or the
nature of the thing : they were divided into accidental,
physical, and logical, according to the means employed
by each for accomplishing their respective objects,
whether it be the enumeration of attributes, or of the
physical or the metaphysical parts of the essence.
These, therefore, are evidently two cross divisions.
In this place we are concerned with nominal Definitions
only, (except, indeed, of logical Terms,) because all
that is requisite for the purposes of Reasoning (which
is the proper province of Logic,) is, that a Term shall
not be used in different senses : a real Definition of any C
thing belongs to the science or system which is em-
*
hip
loyed about that thing. It is to be noted, that in ... Chap.
ploy 8 \
Mathematics the nominal and real Definition exactly
coincide ; the meaning of the word, and the nature of
the thing, being exactly the same. This holds good also
with respect to logical Terms, most legal, and many
ethical terms.
It is scarcely credible how much confusion has
arisen from the ignorance of these distinctions which
has prevailed among logical writers.
The principal rules for Definition are three ; viz.
1st. The Definition must be adequate ; i.e. neither too
extensive nor too narrow for the thing defined :
e.g. to define “fish,” “ an animal that lives in the
water,” would be too extensive, because many insects,
&c. live in the water; to define it, “ an animal that
has an air-bladder,” would be too narrow ; because
many fish are without any. .
2d. The Definition must be in itself plainer than the
thing defined, else it would not explain it : I say, “in.
itself,” (i. e. generally) because, to some particular
person, the term defined may happen to be even more
familiar and better understood, than the terms of the
definition. :
3d. It must be couched in a convenient number of
appropriate words, (if such can be found suitable for
the purpose :) for figurative words (which are opposed
to appropriate) are apt to produce ambiguity or indis-
tinctness : too great brevity may occasion obscurity;
and too great proliarity, confusion.
CHAPTER II.
OF PROPOSITIONS,
§ 1. THE second part of Logic treats of the Proposi-
tion; which is, “Judgment expressed in words.”
A proposition is defined logically “a sentence indica-
tive,” i.e. affirming or denying ; (this excludes com-
mands and questions.) “Sentence” being the Genus,
and “indicative” the Difference, this definition expresses
the whole essence ; and it relates entirely to the words
of a Proposition. With regard to the matter, its Pro-
perty is to be true or false, and therefore it must not
be ambiguous, (for that which has more than one
meaning, is in reality several Propositions;) nor imper-
ºfect, nor ungrammatical, for such an expression has no
‘meaning at all. -
Since the Substance (i. e.
sentence (whether it be a Proposition or not) may be
expressed either absolutely,
death;” “ did Caesar deserve death?”) or under an
hypothesis, (as, “if Caesar was a tyrant, what did he
deserve º’’ ‘‘ Was Caesar a hero or a villain 2’’ ‘‘ If
Caesar was a tyrant, he deserved death;” “ he was
either a hero or a villain,”) on this we found the
division of Propositions according to their substance;
viz. into categorical and hypothetical. And as Genus
is said to be predicated in quid (what,) it is by the
members of this division that we answer the question,
what is this Proposition? (quae est propositio.) Answer,
categorical or hypothetical.
Genus, or material part) of
a Proposition is, that it is a sentence; and since every
(as “Caesar deserved
Categorical Propositions are subdivided into pure,
which asserts simply or purely, that the Subject does
or does not agree with the predicate, and modal, which
expresses in what mode (or manner) it agrees ; e. g.
“an intemperate man will be sickly;” “Brutus killed
Caesar;” are pure. “An intennperate man will pro-
bably be sickly;” “ Brutus killed Caesar justly s” are
modal At present we speak only of pure categorical
Propositions, -
It being the Differentia of a Proposition, that it affirms
or denies, and its Property to be true or false ; and Dif-
ferentia being predicated in quale quid ; Property in
quale, we hence form another division of Propositions,
viz. according to their quality, into affirmative, and
negative, (which is the quality of the expression, and
therefore (in Logic) essential;) and into true and false,
(which is the quality of the matter, and therefore acci-
dental.) An affirmative Proposition is one whose Copula
is affirmative, as ‘‘ birds fly;” “ not to advance is to
go back;" a negative proposition is one whose Copula
is negative, as “man is not perfect;” no “miser is
happy.” * - -
Another division of Propositions is according to
their quantity, (or extent;) if the Predicate is said of
the whole of the Subject, the Proposition is universal : if
of a part of it only, the Proposition is particular, (or
partial;) e.g. “England is an island;” “all tyrants are
miserable;” “no miser is rich;” are universal Propo-
I, O G. I. C. 207
\
Logic. , sitions, and their Subjects are therefore said to be
distributed, being understood to stand, each, for the
whole of its significates : but, “ some islands are fertile;”
“ all tyrants are not assassinated ;” are particular, and
their Subjects, consequently not distributed, being taken
to stand for a part only of their significates.
As every Proposition must be either affirmative or
negative, and must also be either universal or parti-
cular, we reckon in all, four kinds of pure categorical
Propositions, (i.e. considered as to their quantity and
quality both;) viz. universal affirmative, whose symbol
(used for brevity,) is 4; universal negative, E5 par-
ticular affirmative, I; particular negative, O.
§ 2. When the subject of a Proposition is a common
Term, the universal signs (“all, no, every,”) are used
to indicate that it is distributed, (and the Proposition
consequently is universal ;) the particular signs,
(“ some, &c.") the contrary; should there be no sign
at all to the common Term, the quantity of the Pro-
position (which is called an indefinite Proposition) is
ascertained by the matter ; i. e. the nature of the
connection between the extremes; which is either
necessary, impossible, or contingent. In necessary
and in impossible matter, an indefinite is understood
as a universal : e.g. “birds have wings;" i. e. all :
“ birds are not quadrupeds;" i. e. none : in contingent
matter, (i. e. where the terms partly (i.e. sometimes)
agree, and partly not,) an indefinite is understood as a
particular; e.g. “food is necessary to life;” “birds
sing ;” i.e. some do; “ birds are not carnivorous;”
i. e. “ some are not,” or, ‘‘ all are not.”
As for singular Propositions, (viz. those whose Sub-
ject is either a proper name, or a common Term with a
singular sign,) they are reckoned as universals, (see
ch. iv. § 2.) because in them we speak of the whole of
the subject; e.g. when we say, “Brutus was a
Roman,” we mean, the whole of Brutus : this is the
general rule; but some singular Propositions may
fairly be reckoned particular; i.e. when some qualify-
ing word is inserted, which indicates that you are not
speaking of the whole of the subject; e.g. “Caesar
was not wholly a tyrant;” “this man is occasionally
intemperate ;” “ non omnis moriar.” It is evident
that the Subject is distributed in every universal Propo-
sition, and never in a particular ; (that being the very
difference between universal and particular Proposi-
tions ;) but the distribution or non-distribution of the
Predicate, depends (not on the quantity, but) on the
quality, of the Proposition; for, if any part of the Pre-
dicate agrees with the Subject, it must be affirmed and
not denied of the Subject ; therefore, for an affirmative
Proposition to be true, it is sufficient that some part of
the Predicate agree with the Subject; and (for the
same reason) for a negative to be true, it is necessary
that the whole of the Predicate should disagree with
the Subject: e.g. it is true that “ learning is useful,”
though the whole of the Term “useful” does not
agree with the Term “learning,” (for many things are
useful besides learning,) but “no vice is useful,”
would be false, if any part of the Term “useful” agreed
with the Term “vice;” (i.e. if you could find any one
useful thing which was a vice.) The two practical
rules then to be observed respecting distribution, are,
lst. All universal Propositions (and no particular)
distribute the Subject.
2d. All negative, (and no affirmative) the Predi-
cate. : . . . • - -
It may happen indeed, that the whole of the Predi- Chap. i.
cate in an affirmative may agree with the Subject;
e.g. it is equally true, that “all men are rational
animals;” and ‘‘ all rational animals are men :” but
this is merely accidental, and is not at all implied in the
form of expression, which alone is regarded in Logic.
Of Opposition.
§ 3. Two Propositions are said to be opposed to
each other, when having the same Subject and Predi-
cate; they differ in quantity, or quality, or both. It
is evident, that with any given Subject and Predicate,
you may state four distinct Propositions, viz. A., E., I,
and O ; and any two of these are said to be opposed ;
hence there are four different kinds of opposition, viz.
1st, the two universals (A and E) are called contraries
to each other; 2d. the two particular, (I and O,)
subcontraries ; 3d. A and I, or E and O, subalterns;
4th. A and O, or E and I, contradictories. As it is
evident that the truth or falsity of any Proposition (its
quantity and quality being known,) must depend on
the matter of it, we must bear in mind that, “ in neces-
sary matter all affirmatives are true and negatives false ;
in impossible matter, vice versd; in contingent matter, all
universals false, and particulars true;” (e. g. “ all
islands, (or, some islands,) are surrounded by water,"
must be true, because the matter is necessary : to say,
“ no islands, or some — not, &c.” would have been
false; again, “ some islands are fertile ; “ some are not
fertile,” are both true, because it is contingent matter:
put “all” or “no” instead of “some,” and the propositions
will be false.) Hence it will be evident, that contraries
will be both false in contingent matter, but never both
true : Subcontraries, both true in contingent matter,
but never both false : contradictories, always one true
and the other false, &c. with other observations, which
will be immediately made on viewing the scheme 2 in
which the four Propositions are denoted by their
symbols; the different kinds of matter, by the initials
n, i, c, and the truth or falsity of each Proposition in
each matter, by the letter v. for (verum) true, f. for
(falsum) false.

| v. A.—contraries E. f. n
i. f. V. 1
c. f. | f. c
, * Q
UD G Q Uſ)
§: O *S C
cº- %. & §
§2. %2. ' -->
cº- *. º
CD & Gº , gº
5 & Qa. c
CD Sº Qº (ſ)
º |
n. V / f. n
i. f V. I
c. v. I. subcontraries— O. V. C.
By a careful study of this scheme, bearing in mind,
and applying the above rule concerning matter,
the learner will easily elicit all the maxims relating
208
I, O G. I. C.
Logic.
to Opposition; as that, in the subalterns, the truth of
the particular (which is called the subalternate) follows
from the truth of the universal (subalternans) and the
falsity of the universal from the falsity of the particular:
that subalterns differ in quantity alone; contraries, and
also subcontraries in quality alone; contradictories, in
both : and hence, that if any Proposition is known to
be true, we infer that its contradictory is false ; if false,
its contradictory true, &c.
Of Conversion.
§ 4. A Proposition is said to be converted when its
Terms are transposed: when nothing more is done, this
is called simple Conversion. No Conversion is of any
use, unless it be illative; i.e. when the truth of the
converse follows from the truth of the exposita, (or pro-
position given;) e. g.
“No virtuous man is a rebel, therefore
No rebel is a virtuous man.”
“Some boasters are cowards, therefore
Some cowards are boasters.”
Conversion can then only be illative when no Term is
distributed in the converse, which was not distributed in
the exposita : (for if that be done, you will employ a
Term universally in the converse, which was only used
partially in the exposita.) Hence, as E distributes both
Terms, and I neither, these Propositions may be illa-
tively converted in the simple manner; (vid. Rule 2.)
But as A does not distribute the Predicate, its simple
Conversion would not be illative ; (e.g. from “all
birds are animals,” you cannot infer that “all animals
are birds,”) as there would be a Term distributed in
the converse, which was not before. We must there-
fore limit its quantity from universal to particular, and
the Conversion will be illative : (e.g. “ some animals
are birds;”) this might be fairly named Conversion by
limitation ; but is commonly called “ Conversion per
accidens.” E may thus be converted also. But in
O, whether the quantity be changed or not, there will
still be a Term (the Predicate of the converse) distri-
buted, which was not before : you can therefore only
convert it by changing the quality; i, e, considering
the negative as attached to the Predicate instead of to the chap. II.
Copula, and thus regarding it as I. One of the Terms Chap. III.
will then not be the same as before ; but the Proposi-
tion will be aequipollent; (i.e. convey the same mean-
ing,) e.g. “ some members of the University are not
learned :” you may consider “ not learned” as the Pre-
dicate, instead of “ learned;" the Proposition will then
be I, and of course may be simply converted, “ some
who are not learned are members of the University.”
This may be named Conversion by negation ; or as it
is commonly called, by contra-position. A may also
be fairly converted in this way, e. g.
“Every poet is a man of genius; therefore
He who is not a man of genius, is not a poet:”
(or, “None but a man of genius can be a poet.")
For (since it is the same thing, to affirm some Attri-
bute of the Subject, or to deny the absence of that Attri-
bute,) the original Proposition is precisely aequipollent
to this, - -
subj. pred.
r—) . f Ye
“No poet is not a man of genius;”
which, being E, may of course be simply converted.
Thus, in one of these three ways, every Proposition
may be illatively converted : viz. “ E, I, simply ; 4,
O, by negation ; A, E, limitation.” Note, that as it
was remarked, that in some affirmatives, the whole of
the Predicate does actually agree with the Subject;
so, when this is the case, A may be illatively converted,
simply ; but this is an aecidental circumstance. In a
just definition, this is always the case ; for there the
Terms being exactly equivalent, (or, as they are called,
convertible Terms) it is no matter which is made the
Subject, and which the Predicate, e.g. “ a good
government is that which has the happiness of the
governed for its object;” if this be a right definition, it
will follow that “a government which has the happi-
ness of the governed for its object, is a good one.”
Most Propositions in Mathematics are of this descrip-
tion : e. g.
“All equilateral triangles are equiangular;” and
“All equiangular triangles are equilateral."
C H A P T E R III.
OF ARGUMENTS.
§ 1. THE third operation of the mind, viz. Reason-
ing (or discourse) expressed in words, is Argument ;
and an Argument stated at full length, and in its regu-
lar form is called a Syllogism : the third part of Logic
therefore treats of the Syllogism. Every Argument
consists of two parts; that which is to be proved ; and
that by means of which it is proved: the former is called
before it is proved the Question; when proved, the Con-
clusion, (or inference ;) that which is used to prove it,
if stated last, (as is often done in common discourse,) is
called the Reason, and is introduced by “ because,” or
some other casual conjunction ; (e. g. “Caesar
deserved death, because he was a tyrant, and all
tyrants deserve death.") If the Conclusion be stated
last, (which is the strict logical form, to which all
Reasoning may be reduced,) then that which is
employed to prove it is called the Premises ; and the
Conclusion is then introduced by some illative con-
junction, as “therefore” e. g.
“All tyrants deserve death;
Caesar was a tyrant;
therefore he deserved death.”
Since then an Argument is an expression in which
“ from something laid down and granted as true, (i. e.
the Premises) something else, (i. e. the Conclusion) beyond
this, must be admitted to be true, as following necessarily,
(or resulting) from the other;” and since Logic is
L O G. I. C. 209
Or a Term not distributed ; for as it is then used to Chap. III.
Logic. wholly concerned in the use of language, it follows
stand for a part only of its signification, it may happen s—y—’
S-N-' that a Syllogism (which is an Argument stated in a
regular logical form,) must be “an Argument, so
expressed, - that the conclusiveness of it is manifest
from the mere force of the expression,” i. e. without con-
sidering the meaning of the Terms: e. g. in this syllo-
gism, “B is A, C is B, therefore C is A :” the Con-
clusion is unevitable, whatever Terms A, B, and C,
respectively, are understood to stand for. And to
this form, all legitimate Arguments may ultimately be
brought. -
§ 2. The rule or axiom, (commonly called “ dictum
de omni et nullo,”) by which Aristotle proves the
validity of this Argument is this : “ whatever is pre-
dicated of a Term distributed, whether affirmatively or
negatively, may be predicated in like manner, of every
thing contained under it.” Thus, in the examples above,
A is predicated of B distributed, and C is contained
wnder B, (i. e. is its Subject;) therefore A is pre-
dicated of C : so “all tyrants, &c.” (p. 208.) This
rule may be ultimately applied to all Arguments; (and
their validity ultimately rests on their conformity
thereto ;) but it cannot be directly and immediately
applied to all, even of pure categorical Syllogisms ;
for the sake of brevity therefore some other axioms
are commonly applied in practice, to avoid the occa-
sional tediousness of reducing all Syllogisms to that
form in which Aristotle's dictum is applicable.
We will speak first of pure categorical Syllogisms;
and the axioms or canons by which their validity is to
be proved: viz. first, if two Terms agree with one and the
same third, they agree with each other: second, if one
Term agrees and another disagrees with one and the same
third, these two disagree with each other. On the former of
these canons rests the validity of affirmative conclu-
sions; on the latter, of negative: for no Syllogism can
be faulty which does not violate these canons; none
correct which does : hence on these two canons are
built the rules or cautions whicn are to be observed
with respect to Syllogisms, for the purpose of ascer-
taining whether those canons have been strictly
observed or not.
1st. Every Syllogism has three, and only three Terms;
viz. the two Terms (or extremes, as they are commonly
called) of the Conclusion, (or question;) (whereof first,
the Subject is called the minor Term; second, the Predi-
cate, the major;) and third, the middle Term, with which
each of them is separately compared, in order to judge
of their agreement or disagreement with each other.
If therefore there were two middle Terms, the extremes,
(or Terms of the Conclusion) not being both compared to
the same, could not be compared to each cther.
2d. Every syllogism has three, and only three Pro-
positions; viz. first, the major Premiss, (in which the
major Term is compared with the middle;) second, the
minor Premiss, (in which the minor Term is compared
with the middle ;) and third, the Conclusion, in which
the minor Term is compared with the major.
3d. Note, that if the middle Term is ambiguous, there
are in reality two middle Terms, in sense, though but one
in sound. An ambiguous middle Term is either an
equivocal Term, used in different senses in the two
Premises; (e.g. -
“ Light is contrary to darkness;
Feathers are light; therefore
Feathers are contrary to darkness.")
WOL. I.
that one of the extremes may have been compared
with one part of it, and the other, with another part
of it ; e. g.
** White is a colour,
Black is a colour; therefore
Black is white.”—Again,
“Some animals are beasts,
Some animals are birds; therefore
Some birds are beasts.”
The middle Term therefore must be distributed once, at
least, in the Premises; (i.e. by being the Subject of
an universal, or Predicate of a negative, Ch. II. § 2.
p. 207.) and once is sufficient ; since if one extreme
has been compared to a part of the middle Term, and
another to the whole of it, they must have been both
compared to the same. -
4th. No Term must be distributed in the Conclusion which
was not distributed in one of the Premises; for that (it
is called an illicit process, either of the major or the
minor Term) would be to employ the whole of a Term
in the Conclusion, when you had employed only a .
part of it in the Premiss; and thus, in reality, to
introduce a fourth Term ; e. g.
“ All quadrupeds are animals,
A bird is not a quadruped ; therefore
It is not an animal.”—Illicit process of the major.
5th. From negative Premises you can infer nothing. For
in them the middle is pronounced to disagree with
both extremes; not to agree with both; or to agree with
one, and disagree with the other; therefore they cannot
be compared together; e. g.
“A fish is not a quadruped,”
“A bird is not a quadruped,” proves nothing.
6th. If one Premiss be negative, the conclusion must be
negative; for in that Premiss the middle Term is pro-
nounced to disagree with one of the extremes, and in
the other Premiss, (which of course is affirmative, by
the preceding rule) to agree with the other extreme ;
therefore the extremes disagreeing with each other,
the conclusion is negative. In the same manner it
may be shewn, that to prove a negative conclusion one of
the Premises must be a negative.
By these six rules, all Syllogisms are to be tried;
and from them it will be evident; first, that nothing can
be proved from two particular Premises; (for you will
then have either the middle Term undistributed, or an
illicit process; e. g.
“Some animals are sagacious ;
Some beasts are not sagacious ;
& 9 y
Some beasts are not animals.")
And for the same reason secondly, that if one of the
Premises be particular, the Conclusion must be par-
ticular; e.g. from
“ All who fight bravely deserve reward; -
Some soldiers fight bravely;” you can only infe
that some soldiers deserve reward.
For to infer a universal Conclusion, would be an
illicit process of the minor. But from two universal
2 F
*
210
L O G. I. C.
Logic.
Premises you cannot always infer a universal Con-
clusion; e. g. - . -
“All gold is precious,
All gold is a mineral ; therefore
Some mineral is precious.”
And even when we can infer a universal, we are
always at liberty to infer a particular ; since what is
predicated of all may of course be predicated of some,
Of Moods.
§ 3. When we designate the three Propositions of a
Syllogism in their order, according to their respective
quantity and quality, (i. e. their symbols) we are said to
determine the Mood of the Syllogism ; e. g. the
example just above, “all gold, &c.” is in the Mood
A, A, I. As there are four kinds of Propositions, and
three Propositions in each Syllogism, all the possible
ways of combining these four, (A, E, I, O,) by threes, are
sixty-four. For any one of these four may be the major
Premiss; each of these four majors may have four dif-
ferent minors, and of these sixteen pairs of Premises,
each may have four different Conclusions. 4 × 4
(=16) × 4 = 64. This is a mere arithmetical calcu-
lation of the Moods, without any regard to the Logical
rules : for many of these Moods are inadmissible in
practice, from violating some of those rules ; e. g.
the Mood E, E, E, must be rejected, as having nega-
tive Premises; I, O, O, for particular Premises; and
many others for the same faults. By examination then
of all, it will be found that of the sixty-four, there
remain but twelve Moods, which can be used in a
legitimate Syllogism, viz. A, A, A, A, A, I, A, E, E,
A, E, O, A, I, I, A, O, O, E, A, E, E, A, O, E., I, O,
I, A., I, I, E, O, O, A, O.
Of Figure.
§ 4. The Figure of a Syllogism consists in the situa-
tion of the middle Term with respect to the extremes
of the conclusion, (i. e. the major and minor term.)
When the middle Term is made the subject of the major
Premiss, and the Predicate of the minor, that is called the
first Figure ; (which is far the most natural and clear
of all, as to this alone, Aristotle's dictum may be at
once applied.) In the second Figure the middle Term
is the Predicate of both Premises: in the third, the
Subject of both : in the fourth, the Predicate of the
major Premiss, and the Subject of the minor. (This
is the most awkward and unnatural of all, being the
very reverse of the first.) Note, that the proper order
is to place the major Premiss first, and the minor
second; but this does not constitute the major and
minor Premises ; for that Premiss (wherever placed)
is the major which contains the major Term, and the
minor, the minor, (v. R. 2. p. 209.) Each of the
allowable Moods mentioned above, will not be allow-
able in every Figure ; since it may violate some of the
foregoing rules, in one Figure, though not in another :
e.g. I, A., I, is an allowable Mood in the third Figure;
but in the first, it would have an undistributed middle. So
A, E, E, would in the first Figure have an illicit process
of the major, but is allowable in the second ; and
A, A, A, which in the first Figure is allowable, would
in the third have an illicit process of the minor: all
which may be ascertained by trying the different
Moods in each Figure, as per scheme.
Let A represent the major Term, C the minor, B the Chap. III.
middle. . - \-N-
1st Fig. 2d Fig. 3d Fig. 4th Fig.
JB, A, A, B, B, A, A, B,
C, B, C, B, B, C, B, C,
C, A, C, A, C, A, C, A.
The Terms alone being here stated, the quantity and
quality of each Proposition (and consequently the
Mood of the whole Syllogism) is left to be filled up :
(i.e. between B, and A, I may place either a negative
or affirmative Copula; and I may prefix either a universal
or particular sign to B.) By applying the Moods then
to each Figure, it will be found that each Figure will
admit six Moods only, as not violating the rules
against undistributed middle, and against illicit process:
and of the Moods so admitted, several (though valid)
are useless, as having a particular Conclusion, when
a universal might have been drawn; e. g. A, A, I, in
the first Figure, ***
“All human creatures are entitled to liberty;
All slaves are human creatures ; therefore
Some slaves are entitled to liberty.”
Of the twenty-four Moods then (six in each Figure)
five are for this reason neglected : for the remaining
nineteen, Logicians have devised names to distinguish
both the Mood itself, and the Figure in which it is
found ; since when one Mood (i.e. one in itself, with-
out regard to Figure) occurs in two different Figures,
(as E, A, E, in the first and second) the mere letters
denoting the Mood would not inform us concerning
the Figure. In these names then, the three vowels
denote the Propositions of which the Syllogism is
composed; the consonants (besides their other uses,
of which hereafter) serve to keep in mind the Figure
of the Syllogism.
Fig. 1. baſbAra, cElArEnt, dArLI, fErLOque
- prioris.
Fig. 2. cFs ArE, cAmEstrEs, fEstIno, barOkO,
secundae.
- - tertia, dArAptſ, dIsAmIs, dAtls|I, fElApton,
Fig. sº fEris'O, habet: quarta insuper
addit. -
Fig. 4. bramAntIp, cAmEnEs, dImāris, fElApo,
frEsis0n.
By a careful study of these mnemonic lines (which
must be committed to memory) you will perceive
that A can only be proved in the first Figure, in which
also every other Proposition may be proved ; that the
second proves only negatives; the third only particulars,
&c.; with many other such observations, which will
readily be made, (on trial of several Syllogisms, in
different Moods) and the reasons for which will be
found in the foregoing rules. E. G. to shew why the
second Figure has only negative Conclusions, we have
only to consider, that in it the middle Term being the
Predicate in both Premises, would not be distributed
unless one Premiss were negative; (v. R. 2. p. 205.)
therefore the conclusion must be negative also, by
R. 6. p. 209. One Mood in each Figure may suffice in
this place by way of example; first, Barbara, viz. (bAr.)
Every B is A ; (bA) every C is B ; therefore (rA)
every C is A, e.g. let the major Term (which is
represented by A) be “one who possesses all virtue;”
Lo G I c.
21 1
Logic.
the minor term (C) “every man who possesses one
virtue;” and the middle term (B) “every one who
possesses prudence ;” and you will have the cele-
brated argument of Aristotle, Eth. sixth book, to
prove that the virtues are inseparable; viz.
“ He who possesses prudence, possesses all virtue;
He who possesses one virtue, must possess pru-
dence; therefore -
He who possesses one, possesses all.”
Second, Camestres, (cAm) every A is B; (Es) no C
is B ; (trºS) no C is A. Let the major term (A) be
“true philosophers,” the minor (C) “the Epicu-
reans ;” the middle (B) “reckoning virtue a good in
itself;" and this will be part of the reasoning of
Cicero, Off book first and third, against the Epicu-
reans. Third, Darapti, viz. (dA) every B is A;
(r.Ap) every B is C; therefore (t1.) Some C is A. e. g.
“Prudence has for its object the benefit of indivi-
duals ; -
But prudence is a virtue ; therefore
Some virtue has for its object the benefit of the
individual,” is part of Adam Smith's reasoning,
(Moral Sentiments,) against Hutcheson and others,
who placed all virtue in benevolence. Fourth, Camenes,
viz. (cAm) every A is B; (En,) no B is C; there-
fore (Es,) no C is A, e. g.
“Whatever is expedient, is conformable to nature;
Whatever is conformable to nature, is not hurtful
to society; therefore e
What is hurtful to society is never expedient,”
is part of Cicero's argument in Off third book: but
it is an inverted and clumsy way of stating what
would much more naturally fall into the first Figure;
for if you examine the propositions of a Syllogism in
the fourth Figure, beginning at the Conclusion, you will
see that as the major Term is predicated of the minor,
so is the minor of the middle, and that, again of the
major : so that the major appears to be merely pre-
dicated of itself. Hence the five Moods in this Figure
are seldom or never used ; some one of the fourteen
(Moods with names) in the first three F igures, being the
forms into which all Arguments may most readily be
thrown ; but of these, the four in the first Figure are
the clearest and most natural ; as to them, Aristotle's
dictum will immediately apply. And as it is on this
dictum that all Reasoning ultimately depends, so all
Arguments may be somehow or other brought into
some one of these four Moods; and a Syllogism is, in
that case, said to be reduced : (i. e. to the first Figure.)
These four are called the perfect Moods, and all the
rest, imperfect.
Ostensive Reduction.
§ 5. In reducing a Syllogism, we are not of course
allowed to introduce any new Term or Proposition,
having nothing granted but the truth of the Pre-
mises; but these Premises are allowed to be illu-
tively converted, (because the truth of any Proposition
implies that of its illative converse) or transposed: by
taking advantage of this liberty, where there is need,
we deduce in Figure one, from the Premises origin-
ally given, either the very same Conclusion as the original
one, or another from which the original Conclusion
follows, by illative Conversion; e. g. Darapti.
“ All wits are dreaded ;
All wits are admired ;
Some who are admired are dreaded.”
Into Darii, by converting by limitation (per accidens)
the minor Premiss.
“ All wits are dreaded;
Some who are admired are wits; therefore
Some who are admired are dreaded."
Camestres.
“All true philosophers account virtue a good in
itself; -
The advocates of pleasure do not account, &c.
Therefore they are not true philosophers.”
Reduced to Celarent, by simply converting the minor,
and then transposing the Premises. .
“Those who account virtue a good in itself, are
not advocates of pleasure;
All true philosophers account virtue, &c.; therefore
No true philosophers are advocates of pleasure.”
This Conclusion may be illatively converted into the
original one.
Baroko, e.g.
“Every true patriot is a friend to religion;
Some great statesmen are not friends to religion;
Some great statesmen are not true patriots.”
To Ferio, by converting the major by negation (con-
traposition) vide Ch. II. § 4.
“He who is not a friend to religion, is not a true
patriot;
Some great statesmen, &c.”
and the rest of the Syllogism remains the same ;
only that the minor Premiss must be considered as
affirmative, because you take “ not a friend to reli-
gion" as the middle Term. In the same manner
Bokardo to Darii; e.g.
“Some slaves are not discontented;
All slaves are wronged; therefore
Some who are wronged are not discontented.”
Convert the major by negation (contraposition)
and then transpose them ; the Conclusion will be the
converse by negation of the original one, which therefore
may be inferred from it; e.g.
“All slaves are wronged;
Some who are not discontented are slaves;
Some who are not discontented are wronged.”
In these ways (which are called Ostensive Reduction,
because you prove in the first Figure, either the very
same conclusion as before, or one which implies it) all
the imperfect Moods may be reduced to the four per-
fect ones. But there is also another way, called
reductio ad impossibile, 2
§ 6. By which we prove (in the first Figure) not
directly that the original Conclusion is true, but that
it cannot be false; i. e. that an absurdity would follow
from the supposition of its being false; e. g.
“All true patriots are friends to religion;
Some great statesmen are not friends to religion;
Some great statesmen are not true patriots.
If this conclusion be not true, its contradictory must
be true; viz. - -
Chap. III.
2 F 2.
212
L O G. I. C.
“All great statesmen are true patriots.”
S-N-' Let this then be assumed, in the place of the minor
Premiss of the original Syllogism, and a false con-
clusion will be proved; e. g. bar. -
“All true patriots are friends to religion;
bA, All great statesmen are true patriots;
rA, All great statesmen are friends to religion :”
for as this Conclusion is the contradictory of the
original minor Premiss, it must be false, since the
Premises are always supposed to be granted; there-
fore one of the Premises (by which it has been cor-
rectly proved) must be false also ; but the major Pre-
miss (being one of those originally granted) is true;
therefore the falsity must be in the minor Premiss;
which is the contradictory of the original Conclusion ;
therefore the original Conclusion must be true. This
is the indirect mode of Reasoning.
§ 7. This kind of Reduction is seldom employed but
for Baroko and Bokardo, which are thus reduced by
those who confine themselves to simple Conversion,
and Conversion by limitation, (per accidens ;) and they
framed the names of their Moods with a view to Chap. III.
point out the manner in which each is to be reduced ;
viz. B, C, D, F, which are the initial letters of all the
Moods, indicate to which Mood of the first Figure,
(Barbara, Celarent, Darii, and Ferio,) each of the
others is to be reduced ; m, indicates that the Pre-
mises are to be transposed ; s, and p, that the Propo-
sition denoted by the vowel immediately preceding,
is to be converted ; s, simply, p, per accidens, (by
limitation :) thus, in Cannestres, (see example, p. 211,)
the C, indicates that it must be reduced to Celarent ;
the two ss, that the minor Premiss and Conclusion
must be converted simply ; the m, that the Premises
must be transposed. K, (which indicates the reduction
ad impossibile) is a sign that the Proposition denoted
by the vowel immediately before it, must be left out,
and the contradictory of the Conclusion substituted ;
viz. for the minor premiss in Baroko, and the major in
Bokardo. But it has been already shewn, that the
Conversion by contraposition, (by negation,) will
enable us to reduce these two Moods, ostensively.
CHAPTER IV.
OF MODAL
Of Modals.
§ 1. HITHER.To we have treated of pure categorical
Propositions, and the Syllogisms composed of such : a
Modal Proposition may be stated as a pure one, by attach-
ing the Mode to one of the Terms : and the Proposition
will in all respects fall under the foregoing rules; e. g.
“ John killed Thomas wilfully and maliciously ;” here
the Mode is to be regarded as part of the Predicate.
“It is probable that all knowledge is useful ;” “ pro-
bably useful” is here the Predicate; but when the
Mode is only used to express the necessary, contingent,
or impossible connection of the Terms, it may as well
be attached to the Subject : e.g. “ man is necessarily
mortal;” is the same as, ‘‘ all men are mortal:” and
“ this man is occasionally intemperate,” has the force
of a particular : (vide Part II. § 2. p. 207.) It is
thus that two singular Propositions may be contradic-
tories; e.g. “ this man is never intemperate,” will be
the contradictory of the foregoing. Indeed every sign
(of universality or particularity) may be considered as
a Mode. Since, however, in all Modal Propositions,
you assert that the dictum (i.e. the assertion itself) and
the mode, agree together, or disagree, so, in some
ases, this may be the most convenient Way of stating
subj. cop.
pred. subject.
-)
a Modal, purely: e.g. “It is impossible that all men
subject.
should be virtuous.” Such is a proposition of St.
subj. cop. pred. subject.
--~ 2- ºr T ~ --—-->
Paul's : “ This is a faithful saying, &c. that Jesus
subject.
Christ came into the world to save sinners.” In these
cases, one of your Terms (the Subject) is itself an entire
Proposition. Thus much for Modal Propositions.
SYLLOGISMS, AND OF ALL ARGUMENTS BESIDES REGULAR AND pure CATEGORICAL syllogismis.
Of Hypotheticals.
§ 2. A hypothetical Proposition is defined to be,
two or more categoricals united by a Copula, (or conjunc-
tion ;) and the different kinds of hypothetical Propo-
sitions are named from their respective conjunctions ;-
viz. conditional, disjunctive, causal, &c.
When a hypothetical Conclusion is inferred from
a hypothetical Premiss, so that the force of the
Teasoning does not turn on the hypothesis, then the
hypothesis (as in Modals) must be considered as part
of one of the Terms ; so that the Reasoning will be, in
effect, categorical : e. g. s
predicate.
Ay- –Y
“ Every conqueror is either a hero or a villain :
Caesar was a conqueror; therefore *
predicate.
f Y
He was either a hero or a villain.”
“Whatever comes from God is entitled to reverence ;
subject.
If the Scriptures are not wholly false, they must
come from God ; -
If they are not wholly false, they are entitled to
reverence.”
But when the Reasoning itself rests on the hypothe-
sis, (in which way a categorical Conclusion may be
drawn from a hypothetical Premiss,) this is what is
called a hypothetical Syllogism ; and rules. have been
devised for ascertaining the validity of such Arguments,
at once, without bringing them into the categorical
form. (And note, that in these Syllogisms the hypo-
thetical Premiss is called the major, and the categorical
one, the minor.) They are of two kinds, conditional
and disjunctive.
Chap. IV.
\—y-'
Lo G I c.
213
Logic.
must be cheap :”
structive.
the consequent, or deny the antecedent, you can Infer Chap IV.
nothing ; for the same consequent may follow from \-y-'
Of Conditionals:
§ 3. A Conditional Proposition has in it an illative
jorce ; i.e. it contains two, and only two categorical
Propositions, whereof one results from the other, (or,
follows from it,) e. g. -
antecedent.
“ If the Scriptures are not wholly false,
consequent.
they are entitled to respect.”
That from which the other results, is called the antece-
dent; that which results from it, the consequent, (con-
sequens ;) and the connection between the two, (ex-
pressed by the word “if”) the consequence, (conse-
quentia.) . The natural order is, that the antecedent
should come before the consequent; but this is fre-
quently reversed : e.g. “the husbandman is well off
if he knows his own advantages;” Virg. Geor. And
note, that the truth or falsity of a conditional Propo-
sition depends entirely on the consequence : e.g. “ if
Logic is useless, it deserves to be neglected ;” here
both antecedent and consequent are false : yet the
whole proposition is true ; i.e. it it true that the
consequent follows from the antecedent. “ If Crom-
well was an Englishman, he was an usurper,” is just
the reverse case : for though it is true that “ Crom-
well was an Englishman,” and also that “ he was an
usurper," yet it is not true that the latter of these
Propositions depends on the former ; the whole Propo-
sition, therefore, is false, though both antecedent and
consequent are true. A Conditional Proposition, in
short, may be considered as an assertion of the validity
of a certain Argument; since to assert that an Argu-
ment is valid, is to assert that the Conclusion neces-
sarily results from the Premises, whether those Pre-
mises be true or not. The meaning, then, of a
Conditional Proposition is this ; that, the antecedent
being granted, the consequent is granted : which may be
considered in two points of view : first, if the antece-
dent be true, the consequent must be true ; hence the
first rule ; the antecedent being granted, the consequent
may be inferred : secondly, if the antecedent were true,
the consequent would be true ; hence the second rule ;
the consequent being denied, the antecedent may be denied ;
for the antecedent must in that case be false; since if
it were true, the consequent (which is granted to be
false) would be true also : e.g. “if this man has a
fever, he is sick 5” here, if you grant the antecedent,
the first rule applies, and you infer the truth of the
consequent ; “he has a fever, therefore he is sick:"
if A is B, C is D ; but A is B, therefore C is D, (and
this is called a constructive Conditional Syllogism ;)
but if you deny the consequent (i. e. grant its contradic-
tory,) the second rule applies, and you infer the contra-
dietory of the antecedent : “he is not sick, therefore he
has not a fever:” this is the destructive Conditional
Syllogism : if A is B, C is D ; C is not D, therefore
A is not B. Again, “if the crops are not bad, corn
for a major; then, “ but the crops
are not bad, therefore corn must be cheap," is con-
“Corn is not cheap, therefore the crops
are bad,” is destructive. “If every increase of popu-
lation is desirable, some misery is desirable ; but no
misery is desirable, therefore, some increase of popu-
lation is not desirable,” is destructive, But if you affirm
other antecedents: e. g. in the example above, a man
may be sick from other disorders besides a fever;
therefore it does not follow from his being sick, that
he has a fever; nor (for the same reason) from his not
having a fever, that he is not sick. There are, there-
fore, two, and only two kinds of Conditional Syllo-
gisms ; the constructive, founded on the first rule, and
answering to direct Reasoning; and the destructive on
the second, answering to indirect. And note, that a
conditional Proposition may (like the categorical A,) be
converted by negation; i.e. you may take the contradic-
tory of the consequent, as an antecedent, and the contra-
dictory of the antecedent, as a consequent : e.g. “ if this
man is not sick, he has not a fever.” By this con-
version of the major Premiss, a constructive Syllogism
may be reduced to a destructive, and vice versa. (Se
§ 6. Ch. IV. p. 21.4. ) * -
Of Disjunctives.
§ 4. A disjunctive Proposition may consist of any
number of categoricals; and, of these, some one, at least,
must be tº ue, or the whole Proposition will be false :
if, therefore, one or more of these categoricals be
denied, (i. e. granted to be false,) you may infer that
the remaining one, or (if several) some one of the
remaining ones is true: e.g. “ either the earth is
eternal, or the work of chance, or the work of an
intelligent being; it is not eternal, nor the work of
chance; therefore it is the work of an intelligent
being.” “ It is either spring, summer, autumn, or
winter; but it is neither spring nor summer, there-
fore it is either autumn or winter.” Either A is B,
or C is D : but A is not B, therefore C is D. Note,
that in these two examples (as well as very many
others,) it is implied not only that one of the mem-
bers (the categorical Propositions) must be true, but
that only one can be true; so that, in such cases, if one
or more members be affirmed, the rest may be denied;
[the members may then be called exclusive :] e.g. “it
is summer, therefore it is neither spring, autumn, nor
winter;” “ either A is B, or C is D ; but A is B,
therefore C is not D.” But this is by no means uni-
versally the case ; e.g. “ virtue tends to procure us
either the esteem of mankind or the favour of God :''
here both members are true, and consequently from
one being affirmed, we are not authorized to deny the
other. It is evident that a disjunctive Syllogism may
easily be reduced to a conditional : e. g. if it is not
spring or summer, it is either autumn or winter, &c.
The Dilemma.
§ 5. Is a complex kind of Conditional Syllogism.
1st. If you have in the major Premiss several antece-
dents all with the same consequent, then these antece-
dents, being (in the minor), disjunctively granted, (i. e.
it being granted that some one of them is true,) the
one common consequent may be inferred, (as in the case
of a simple constructive syllogism :) e.g. if A is B,
C is D; and if X is Y, C is D ; but either A is B, or
X is Y; therefore C is D. “If the blest in heaven
have no desires, they will be perfectly content; so
they will, if their desires are fully gratified; but
214
L O G. I. C.
Now an opponent might deny either of the minor Pre- Chap. IV.
mises in the above Syllogisms, but he could not deny S-V-'
Logic. either they will have no desires, or have them fully
gratified; therefore they will be perfectly content.”
Note, in this case, the two conditionals which make
up the major Premiss may be united in one Proposi-
tion by means of the word “whether -" e.g. “ whe-
ther the blest, &c. have no desires, or have their
desires gratified, they will be content.”
2d. But if the several antecedents have each a
different consequent, then the antecedents, being as
before, disjunctively granted, you can only disjunc-
tively infer the consequents: e.g. if A is B, C is D;
and if X is Y, E is F : but either A is B, or X is Y ;
therefore either C is I), or E is F. “If AEschines
joined in the public rejoicings, he is inconsistent ; if
he did not, he is unpatriotic ; but he either joined, or
not, therefore he is either inconsistent or ünpatriotic.”
(Demost. For the Crown) This case, as well as the
foregoing, is evidently constructive. In the destruc-
tive form, whether you have one antecedent with
several consequents, or several antecedents, either
with one, or with several consequents; in all these
cases, if you deny the whole of the consequent or con-
sequents, you may in the conclusion, deny the whole
of the antecedent or antecedents : e.g. “ if this fact
be true, it must be recorded either in Herodotus, Thu-
cydides, or Xenophon : it is not recorded in any of the
three, therefore it is not true.” “If the world existed
from eternity there would be records prior to the
Mosaic ; and if it were produced by chance, it would
not bear marks of design : there are no records prior
to the Mosaic; and the world does bear marks of design;
therefore it neither existed from eternity, nor is the
work of chance.” These are commonly called Dilem-
mas, but hardly differ from simple conditional Syllo-
gisms. Nor is the case different if you have one antece-
dent with several consequents, which consequents you
disjunctively deny ; for that comes to the same thing
as wholly denying them ; since if they be not all true,
the one antecedent must equally fall to the ground ;
and the Syllogism will be equally simple: e.g. “if
we are at peace with France by virtue of the treaty
of Paris, we must acknowledge the sovereignty of
Buonaparte ; and also we must acknowledge that of
Louis ; but we cannot do both of these ; therefore we
are not at peace,” &c.; which is evidently a plain
destructive. The true dilemma is, ‘‘ a conditional
Syllogism with several antecedents in the major, and a
disjunctive minor ;" hence,
3d. That is most properly called a destructive Dilemma,
which has (like the constructive ones) a disjunctive
minor Premiss ; i. e. when you have several antecedents
with each a different consequent ; which consequents,
(instead of wholly denying them, as in the last case,)
you disjunctively deny ; and thence, in the Conclusion,
deny disjunctively the antecedents : e. g. if A is B,
C is D ; and if X is Y, E is F : but either C is not D,
or E is not F ; therefore, either A is not B, or X is not
Y. “ If this man were wise, he would not speak
irreverently of Scripture in jest; and if he were
good he would not do so in earnest ; but he does it,
either in jest or in earnest; therefore he is either
not wise or not good.” Every Dilemma may be
reduced into two or more simple Conditional Syllo-
gisms : e. g. “if Æschines joined, &c. he is incon-
sistent ; he did join, &c. therefore he is inconsistent :
and again, if AEschines did not join, &c. he is unpa-
triotic ; he did not, &c. therefore he is unpatriotic.”
both ; and therefore he must admit one or the other of
the Conclusions: for, when a Dilemma is employed, it
is supposed that some one of the antecedents must be
true, (or, in the destructive kind, some one of the con-
sequents false,) but that we cannot tell which of them
is so ; and this is the reason why the argument is
stated in the form of a Dilemma. From what has
been said, it may easily be seen that all Dilemmas are
in fact conditional syllogisms; and that disjunctive Syl-
logisms may also be reduced to the same form : but as
it has been remarked, that all Reasoning whatever may
ultimately be brought to the one test of Aristotle's
“ dictum,” it remains to shew how a Conditional Syl-
logism may be thrown into such a form that that test
will at once apply to it; and this is called the
Reduction of Hypotheticals.
§ 6. For this purpose we must consider every
Conditional Proposition as a universal affirmative
categorical Proposition, of which the Terms are entire
Propositions, viz. the antecedent answering to the
Subject, and the consequent to the Predicate ; e.g. to
say, “if Louis is a good king, France is likely to
prosper;" is equivalent to saying, “ the case of Louis
being a good king, is a case of France being likely to
prosper :” and if it be granted, as a minor Premiss to
the Conditional Syllogism, that “ Louis is a good
king ;” that is equivalent to saying, “the present
case is the case of Louis being a good king :” from
which you will draw a conclusion in Barbara, (viz.
“ the present case is a case of France being likely to
prosper,”) exactly equivalent to the original Conclusion
of the Conditional Syllogism ; viz. “France is likely
to prosper.” As the constructive condition may thus
be reduced to Barbara, so may the destructive in like
manner, to Celarent, e.g. “if the Stoics are right,
pain is no evil : but pain is an evil; therefore, the
Stoics are not right ;” is equivalent to, “the case of
the Stoics being right, is the case of pain being no
evil; the present case is not the case of pain being no
evil; therefore the present case is not the case of the
Stoics being right.” This is Camestres, which of
course is easily reduced to Celarent. Or, if you will,
all Conditional Syllogisms may be reduced to Barbara,
by considering them all as constructive; which may
be done, as mentioned above, by converting by nega-
tion the major Premiss, (see p. 212. § 3. Ch. IV.) The
reduction of Hypotheticals may always be effected in
the manner above stated ; but as it produces a cir-
cuitous awkwardness of expression, a more convenient
form may in some cases be substituted : e. g. in the
example above, it may be convenient to take, “ trate,”
for one of the Terms: “ that pain is no evil is not
true; that pain is no evil is asserted by the Stoics;
therefore something asserted by the Stoics is not
true.” Sometimes again it may be better to unfold
the argument into two Syllogisms: e. g. in a former
example; first, “Louis is a good king; the governor
of France is Louis; therefore the governor of France
is a good king.” And then, second, “every country
governed by a good king is likely to prosper,” &c.
[A. Dilemma is generally to be reduced into two or
more categorical Syllogisms.] And when the ante-
cedent and consequent have each the same Subject,
you may sometimes reduce the Conditional by merely
L O G. I. C. 215
view, have fallen into the common error of confound- Chap. IV,
Logic. substituting a categorical major Premiss for the con-
ing Logical with Rhetorical distinctions, and have \–V-”
- ditional one : e. g. instead of “...if Caesar was a tyrant,
he deserved death ; he was a tyrant, therefore he
deserved death ;” you may put for a major, ‘‘ all
tyrants deserve death,” &c. But it is of no great
consequence, whether Hypotheticals are reduced in the
most meat and concise manner or not; since it is not
intended that they should be reduced to categorical,
in ordinary practice, as the readiest way of trying their
validity, (their own rules being quite sufficient for
that purpose ;) but only that we should be able, if re-
quired, to subject any argument whatever to the test
of Aristotle's dictum, in order to shew that all Reason-
ing turns upon one simple principle.
Of Enthymeme, Sorites, &c.
§ 7. There are various abridged forms of Argument
which may be easily expanded into regular Syllogisms:
such as, first, the Enthymeme, which is a Syllogism
with one Premiss suppressed. As all the Terms will
be found in the remaining Premiss and Conclusion, it
will be easy to fill up the Syllogism by supplying the
Premiss that is wanting, whether major or minor :
e.g. “Caesar was a tyrant; therefore he deserved
death.” “A free nation must be happy; therefore
the English are happy.”
This is the ordinary form of speaking and writing.
It is evident that Enthymemes may be filled up
hypothetically.
2d. When you have a string of Syllogisms, in which
the Conclusion of each is made the Premiss of the
next, till you arrive at the main and ultimate Conclu-
sion of all, you may sometimes state these briefly, in
a form called Sorites ; in which the Predicate of the
first proposition is made the Subject of the next; and
so on, to any length, till finally the Predicate of the
last of the Premises is predicated (in the Conclusion)
of the Subject of the first : e. g. A is B, B is C, C is D,
D is E.; therefore A is E. “The English are a brave
people; a brave people are free; a free people are
happy; therefore the English are happy.” A Sorites
then has as many middle Terms as there are interme-
diate Propositions between the first and the last; and
consequently it may be drawn out into as many sepa-
rate Syllogisms; of which the first will have, for its
najor Premiss, the second ; and for its minor, the first
of the Propositions of the Sorites; as may be seen by
the example. It is also evident, that in a Sorites you
cannot have more than one negative Proposition, and
one particular; for else, one of the Syllogisms would
have its Premises both negative or both particular,
(vid. p. 209.) A string of Conditional Syllogisms may
in like manner be abridged into a Sorites ; e. g. if A
is B, C is D ; if C is D, E is F; if E is F, G is H.;
but A is B, therefore G is H. “If the Scriptures are
the word of God, it is important that they should be
well explained; if it is important, &c. they deserve
to be diligently studied; if they deserve, &c. an order
of men should be set aside for that purpose : but
the Scriptures are the word, &c.; therefore an order
of men should be set aside for the purpose, &c.”
Hence, it is evident, how injudicious an arrangement
has been adopted by former writers on Logic, who
have treated of the Sorites and Enthymeme before they
entered on the subject of Hypotheticals. -
Those who have spoken of induction or of example,
as a distinct kind of Argument in a Logical point of
wandered from their subject as much as a writer on
the orders of Architecture would do, who should
introduce the distinction between buildings of stone and
of marble. Logic takes no cognizance of induction, for
instance, or of a priori reasoning, &c. as distinct
Forms of argument; for when thrown into the syllogistic
form, and when letters of the alphabet are substituted
for the Terms (and it is thus that Argument is properly
to be brought under the cognizance of Logic,) there
is no distinction between them; e.g. a Property which
belongs to the ox, sheep, deer, goat, and antelope,
belongs to all horned animals; rumination belongs
to these ; therefore, to all. This, which is an induc-
tive argument, is evidently a Syllogism in Barbara.
The essence of an inductive argument (and so of the
other kinds which are distinguished for it,) consists,
not in the form of the Argument, but in the relation
which the Subject matter of the Premises bears to that
of the Conclusion.
3d. There are various other abbreviations commonly
used, which are so obvious as hardly to call for expla-
nation : as, where one of the Premises of a Syllogism
is itself the Conclusion of an Enthymeme which is
expressed at the same time : e. g. “all useful studies
deserve encouragement; Logic is such, (since it helps
us to reason accurately,) therefore it deserves encou-
ragement;” here, the minor Premiss is what is called
an Enthymematic sentence. The antecedent in that minor
Premiss, (i. e. that which makes it Enthymematic,) is
called by Aristotle the Prosyllogism.
It is evident that you may for brevity substitute for
any term an equivalent ; as in the last example, “it’
for “Logic ;” “such” for “a useful study,” &c.
4th. And many Syllogisms, which at first appear
faulty, will often be found, on examination, to contain
correct reasoning, and, consequently, to be reducible
to a regular form ; e.g. when you have, apparently,
negative Premises, it may happen, that by considering
one of them as affirmative, (see Ch. II. § 4. p. 208.) the
Syllogism will be regular : e.g. “no man is happy
who is not secure; no tyrant is secure ; therefore no
tyrant is happy,” is a Syllogism in Celarent.” Some-
times there will appear to be too many terms; and
yet there will be no fault in the Reasoning, only an
irregularity in the expression : e.g. “no irrational
agent could produce a work which manifests design ;
the universe is a work which manifests design; there-
fore no irrational agent could have produced the
universe.” Strictly speaking, this Syllogism has five
Terms; but if you look to the meaning, you will see,
that in the first Premiss (considering it as a part of this
Argument,) it is not, properly, “an irrational agent"
that you are speaking of, and of which you predicate
that it could not produce a work manifesting design;
but rather it is this “work,” &c. of which you are
speaking, and of which it is predicated that it could
* If this experiment be tried on a Syllogism which has really
negative Premises, the only effect will be to change that fault into
another: viz. an excess of Terms, or, (which is substantially the
same) an undistributed middle ; e.g. “ an enslaved people is not
happy; the English are not enslaved; therefore they are happy :”
if “enslaved” be regarded as one of the Terms, and “not en-
slaved” as another, there will manifestly be four. Hence you may
see how very little difference there is in reality between the dif-
ferent faults which are enumerated.
216 L O G. I. C.
3.3
not be produced by an irrational agent; if then you
S-v-' state the Propositions in that form, the Syllogism will
be perfectly regular.
Thus, such a Syllogism as this, “every true
patriot is disinterested; few men are disinterested;
therefore few men are true patriots;” might ap-
pear at first sight to be in the second Figure,
and faulty; whereas it is Barbara, with the Premises
transposed ; for you do not really predicate of “few
men,” that they are “ disinterested,” but of “ disin-
terested persons,” that they are “few.” Again, “none
but candid men are good reasoners; few infidels are
candid; few infidels are good reasoners.” In this it
will be most convenient to consider the major Pre-
miss as being “all good reasoners are candid,” (which
of course is precisely aequipollent to its illative con-
verse by negation ;) and the minor Premiss and Con-
clusion may in like manner be fairly expressed thus
—“ most infidels are not candid; therefore most
infidels are not good reasoners :” which is a regular
Syllogism in Camestres. Or, if you would state it in
the first Figure, thus—those who are not candid (or
uncandid) are not good reasoners; most infidels are
not candid; most infidels are not good reasoners.
§ 8. The foregoing rules enable us to develope the
principles on which all Reasoning is conducted, what-
ever be the Subject matter of it, and to ascertain the
validity or fallaciousness of any apparent argument,
as far as the form of expression is concerned; that being
alone the proper province of Logic. -
But it is evident that we may nevertheless remain
liable to be deceived or perplexed in Argument by the
assumption of false or doubtful Premises, or by the
employment of indistinct or ambiguous terms; and,
accordingly, many Logical writers, wishing to make
their systems appear as perfect as possible, have
undertaken to give rules “for attaining clear ideas,”
and for “guiding the judgment;” and fancying or
professing themselves suecessful in this, have con-
sistently enough denominated Logic, the “Art of
using the Reason ;” which in truth it would be, and
would supersede all other studies, if it could alone
ascertain the meaning of every Term, and the truth or
falsity of every Proposition, in the same manner as it
actually can the validity of every Argument. And Chap. IV.
they have been led into this, partly by the consider- \-y—
ation that Logic is concerned about the three opera-
tions of the mind—simple Apprehension, Judgment,
and Reasoning; not observing that it is not equally
concerned about all; the last operation being alone its
appropriate province ; and the rest being treated of
only in reference to that. -
The contempt justly due to such pretensions has
most unjustly fallen on the Science itself, much in the
same manner as Chemistry was brought into disrepute
among the unthinking by the extravagant pretensions
of the Alchemists. And those Logical writers have
been censured, not (as they should have been) for
making such professions, but for not fulfilling them.
It has been objected, especially, that the rules of Logic
leave us still at a loss as to the most important and
difficult point in Reasoning ; viz. the ascertaining the
sense of the terms employed, and removing their
ambiguity. A complaint resembling that made
(according to a story told by Warburton in his Div.
Leg.) by a man who found fault with all the read-
ing-glasses presented to him by the shopkeeper; the
fact being that he had never learnt to read. In the
present case, the complaint is the more unreasonable,
inasmuch as there neither is, nor ever can possibly be,
any such system devised as will effect the proposed
object of clearing up the ambiguity of Terms. It is,
however, no small advantage, that the rules of Logic,
though they cannot alone, ascertain and clear up
ambiguity in any Term, point out in which Term of an
Argument it is to be song.ht for, directing our attention
to the middle Term, as the one on the ambiguity of
which a fallacy is likely to be built.
It will be useful, however, to class and describe the
different kinds of ambiguity which are to be met with ;
and also the various ways in which the insertion of
false, or, at least, unduly assumed Premises, is most
likely to elude observation. And though the remarks
which will be offered on these points may not be con-
sidered as strictly forming a part of Logic, they cannot
be thought out of place, when it is considered how
essentially they are connected with the application
of it.
Lo G I c. tº : 217
CHAPTER V.
OF FALLACIP. S.
instructing in the principles and the terms of his chap. V.
system, should totally lay these aside when he came to S-2-2
describe plants, and should adopt the language of the
Introduction.
Logic. By a Fallacy is commonly understood, “any unsound
S-N-" mode of arguing, which appears to demand our con-
viction, and to be decisive of the question in hand,
when in fairness it is not so.” As we consider the ready
detection and clear exposure of Fallacies to be both
more extensively important, and also more difficult
than many are aware of, we propose to take a Logical
view of the subject ; referring the different Fallacies
to the most convenient heads, and giving a scientific
analysis of the procedure which takes place in each.
After all, indeed, in the practical detection of each
individual Fallacy, much must depend on natural and
- acquired acuteness; nor can any rules be given, the
mere learning of which will enable us to apply them
with mechanical certainty and readiness : but still we
shall find that to take correct general views of the
subject, and to be familiarized with scientific discus-
sions of it, will tend, above all things, to engender
such a habit of mind as will best fit us for practice.
Indeed the case is the same with respect to Logic in
general ; scarce any one would in ordinary practice,
state to himself either his own or another's reasoning
in Syllogisms in Barbara at full length ; yet a fa-
miliarity with Logical principles, tends very much,
(as all feel, who are really well acquainted with
them,) to beget a habit of clear and sound Reasoning.
The truth is, that in this, as in many other things, there
are processes going on in the mind (when we are
practising any thing quite familiar to us) with such
rapidity as to leave no trace in the memory; and we
often apply principles which did not, as far as we are
conscious, even occur to us at the time. ~
It would be foreign, however, to the present pur-
pose, to investigate fully the manner in which certain
studies operate in remotely producing certain effects
on the mind ; it is sufficient to establish the fact, that
habits of scientific analysis (besides the intrinsic beauty
and dignity of such studies) lead to practical advan-
tage. It is on Logical principles therefore that we
propose to discuss the subject of Fallacies : and it
might, indeed, seem to be unnecessary to make any
apology for so doing, after what has been formerly
said, generally, in defence of Logic : if the majority
of Logical writers had not usually followed a very
opposite plan. Whenever they have to treat of any
thing that is beyond the mere elements of Logic, they
totally lay aside all reference to the principles which
they have been occupied in establishing and explaining,
and have recourse to a loose, vague, and popular kind
of language; such as would be the best suited indeed
to an exoterical discourse, but seems strangely incon-
gruous in a professed Logical treatise. What should
we think of a Geometrical writer, who, after having
gone through the Elements with strict definitions and
demonstrations, should, on preceding to Mechanics,
totally lay aside all reference to scientific principles,
all use of technical terms, and treat of the subject in
undefined terms, and with probable and popular argu-
ments? It would be thought strange, if even a
Botanist, when addressing those whom he had been
YOL. I.
vulgar. Surely it affords but too much plausibility to
the cavils of those who scoff at Logic altogether, that
the very writers who profess to teach it, should never
themselves make any application of, or reference to
its principles, on those very occasions, when, and
when only, such application and reference are to be
expected. If the principles of any system are well
laid down, if its technical language is well framed,—
then, surely those principles and that language will
afford, (for those who have once thoroughly learned
them,) the best, the most clear, simple, and concise
method of treating any subject connected with that
system. Yet even the accurate Aldrich, in treating
of the Dilemma and of the Fallacies, has very much
forgotten the Logician, and assumed a loose and rhe-
torical style of writing, without making any application
of the principles he had formerly laid down, but on
the contrary, sometimes departing widely from them.
The most experienced teachers, when addressing
those who are familiar with the elementary principles
of Logic, think it requisite, not indeed to lead them,
on each occasion, through the whole detail of those
principles, when the process is quite obvious, but
always to put them on the road, as it were, to those
principles, that they may plainly see their own way to
the end, and take a scientific view of the subject: in
the same manner as Mathematical writers, avoid
indeed the occasional tediousness of going all through
a very simple demonstration which the learner, if he
will, may easily supply ; but yet always speak in strict
Mathematical language, and with reference to Mathe-
matical principles, though they do not always state
them at full length. We would not profess, therefore,
any more than they do, to write (on subjects connected
with the science,) in a language intelligible to those
who are ignorant of its first rudiments; to do so,
indeed, would imply that we were not taking a scien-
tific view of the subject, nor availing ourselves of the
principles which had been established, and the accurate
and concise technical language which had been framed.
§ 1. The division of Fallacies into those in the
words, IN DICTIONE, and those in the matter
EXTRA DICTIONEM, has not been, by any
writers hitherto, grounded on any distinct principle;
at least, not on any that they have themselves adhered
to. The confounding together, however, of these
two classes is highly detrimental to all clear notions
concerning Logic ; being obviously allied to the pre-
vailing erroneous views which make Logic the art of
employing the intellectual faculties in general, having the
discovery of truth for its object, and all kinds of know-
ledge for its proper subject matter ; with all that train
of vague and groundless speculations which have led
to such interminable confusion and mistakes, and
afforded a pretext for such clamorous censures.
It is important, therefore, that rules should be
given for a division of Fallacies into Logical, and Non-
logical, on such a principle as shall keep clear of all
this indistinctness and perplexity.
2 G
218
L O G I C.
except its non-distribution: for though in such cases Chap. V.
the Conclusion does not follow, and though the rules S-Y-
Logic. If any one should object that the division we adopt
is in some degree arbitrary, placing under the one head
Pallacies, which many might be disposed to place
under the other, let him consider not only the in-
distinctness of all former divisions, but the utter
impossibility of framing any that shall be completely
secure from the objection urged, in a case where men
have formed such various j vague notions, from the
very want of some clear principle of division. Nay,
from the elliptical form in which all Reasoning is
usually expressed, and the peculiarly involved and
oblique form in which Fallacy is for the most part
conveyed, it must of course be often a matter of
doubt, or rather, of arbitrary choice, not only to
which genus each kind of Fallacy should be referred,
but even to which kind to refer any one individual
Fallacy: for since in any course of argument, one
Premiss is usually suppressed, it frequently happens,
in the case of a Fallacy, that the hearers are left to
the alternative of supplying either a Premiss which is
not true, or else, one which does not prove the conclu-
sion; e. g. if a man expatiates on the distress of the
country, and thence argues that the government is
tyrannical, we must suppose him to assume either that
“ every distressed country is under a tyranny,” which
is a manifest falsehood, or, merely that “every
country under a tyranny is distressed,” which, how-
ever true, proves nothing, the middle term being
undistributed. Now, in the former case, the Fallacy
would be referred to the head of “eatra dictionem ;”
in the latter, to that of “ in dictione :” which are we
to suppose the speaker meant us to understand 2 surely
just whichever each of his hearers might happen to
prefer : some might assent to the false Premiss ;
others, allow the unsound Syllogism : to the Sophist
himself it is indifferent, as long as they can but be
brought to admit the conclusion.
Without pretending then to conform to every one's
mode of speaking on the subject, or to lay down rules
which shall be, in themselves, (without any call for
Iabour or skill in the person who employs them,)
readily applicable to, and decisive on each individual
case; we propose a division which is at least perfectly
clear in its main principle, and coincides, perhaps, as
nearly as possible with the established notions of
Logicians on the subject.
§ 2. In every Fallacy, the conclusion either does, or
does not follow from the Premises : where the conclu-
sion does not follow from the Premises, it is manifest
that the fault is in the Reasoning, and in that alone ;
these, therefore, we call Logical Fallacies,” as being
properly violations of those rules of Reasoning which
it is the province of Logic to lay down. Of these,
however, one kind are more purely Logical, as ex-
hibiting their fallaciousness by the bare form of the
expression, without any regard to the meaning of the
terms: to which class belong : 1st, undistributed
middle; 2d. illicit process ; 3d. negative Premises, or
affirmative conclusion from a negative Premiss, and
vice versd : to which may be added, 4th, those which
have palpably (i. e. expressed) more than three terms.
The other kind may be most properly called semi-
logical ; viz. all the cases of ambiguous middle term
* Just as we call that a criminal Court in which crimes are
judged. -
of Logic shew that it does not, as soon as the ambiguity
of the middle term is ascertained, yet the discovery and
ascertainment of this ambiguity requires attention to
the sense of the term, and knowledge of the subject
matter; so that here, Logic ‘‘ teaches us not how to
find the Fallacy, but only where to search for it,” and
on what principles to condemn it. Accordingly it has
been made a subject of bitter complaint against Logic,
that it presupposes the most difficult point to be already
accomplished, viz. the sense of the terms to be ascer-
tained. A similar objection, might be urged against
every other art in existence; e. g. against Agriculture,
that all the precepts for the cultivation of land presup-
pose the possession of a farm ; or against Perspective,
that its rules are useless to a blind man. The objec-
tion is indeed peculiarly absurd when urged against
Logic, because the object which it is blamed for not
accomplishing, cannot possibly be within the province
of any one art whatever. Is it indeed possible or con-
ceivable that there should be any method, science, or
system, that should enable one to know the full and
exact meaning of every term in existence? The utmost
that can be done is to give some general rules that
may assist us in this work ; which is done in the two
first parts of Logic.
The very author of the objection says, “this (the
comprehension of the meaning of general terms) is a
study which every individual must carry on for himself;
and of which no rules of Logic (how useful soever
they may be in directing our labours) can supersede
the necessity.” D. Stewart, Phil, vol. ii. ch. ii. s. 2.
Nothing perhaps tends more to conceal from men
their imperfect conception of the meaning of a term,
than the circumstance of their being able fully to
comprehend a process of Reasoning in which it is in-
volved, without attaching any distinct meaning, or
perhaps any meaning at all to that term ; as is evident
when A B C, are used to stand for terms, in a regular
Syllogism : thus a man may be familiarized with a
terma, and never find himself at a loss from not com-
prehending it; from which he will be very likely to
infer that he does comprehend it, when perhaps he
does not, but employs it vaguely and incorrectly,
which leads to fallacious reasoning and confusion. It
must be owned, however, that many Logical writers
have, in great measure, brought on themselves the
reproach in question, by calling Logic “ the right
use of Reason,” laying down “ rules for gaining clear
ideas,” and such-like &\agovela, as Aristotle calls it.
Rhet. book i. ch. ii.
§ 3. The remaining class (viz. where the Con-
clusion does follow from the Premises) may be called
the Material, or Non-logical Fallacies: of these there
are two kinds ; 1st. when the Premises are such as
ought not to have been assumed ; 2d. when the
Conclusion is not the one required, but irrelevant;
which Fallacy is called “ ignoratio elemchi,” because
your argument is not the elemchus, (i. e. proof of the
contradictory) of your opponent's assertion, which it
should be ; but proves, instead of that, some other
proposition resembling it. Hence, since Logic defines
what Contradiction is, some may choose rather to
range this with the Logical Fallacies, as it seems, so
far, to come under the jurisdiction of that art. never-
theless, it is perhaps better to adhere to the original
L O G. I. C.
210
Logic. division, both on account of its clearness, and also
*-y-' because few would be inclined to apply to the Fallacy
* in question the accusation of being inconclusive, and
consequently illogical reasoning : besides which, it
seems an artificial and circuitous way of speaking, to
suppose in all cases an opponent and a contradiction ;
the simple statement of the matter being this, I am
required, by the circumstances of the case, (no matter
why) to prove a certain Conclusion ; I prove, not that,
but one which is likely to be mistaken for it ;—in this
lies the Fallacy.
It might be desirable therefore to lay aside the name
of “ignoratio elemchi,” but that is so generally adopted
as absolutely to require some mention to be made of
it. The other kind of Fałlacies in the matter will
comprehend, (as far as the vague and obscure language
of Logical writers will allow us to conjecture,) the
Rallacy of “non causa pro causd,” and that of “petitio
principii :" of these, the former is by them distinguished
into “a non verd pro verd, and “a non tali pro tali ;” this
last would appear to be arguing from a case not
parallel as if it were so ; which, in Logical language, is,
having the suppressed Premiss false; (for it is in that
the parallelism is affirmed) and the “ a non verd pro verá"
will in like manner signify the expressed Premiss being
false ; so that this Fallacy will turn out to be, in plain Chap. V.
terms, neither more nor less than falsity, (or unfair \-y-
assumption) of a Premiss.
The remaining kind, “petitio principii,” (begging
the question) takes place when a Premiss, whether
true or false, is either plainly equivalent to the Con-
clusion, or depends on it for its own reception. It is
to be observed, however, that in all correct Reasoning
the Premises must, virtually, imply the conclusion ; so
that it is not possible to mark precisely the distinction
between the Fallacy in question and fair argument ;
since that may be correct and fair Reasoning to one
person, which would be, to another, begging the
question, since to one the Conclusion might be more
evident than the Premiss, and to the other, the reverse.
The most plausible form of this Fallacy is arguing in
a circle; and the greater the circle, the harder to
detect. - -
§ 4. There is no Fallacy that may not properly be
included under some of the foregoing heads; those
which in the Logical Treatises are separately enu-
merated, and contradistinguished from these, being in
reality instances of them, and therefore more properly
enumerated in the subdivision thereof; as in the
Scheme annexed.
Fallacies.
frº- Logical. Non-logical or Material. —
(i. e. when the fault is, strictly, in the very process (i.e. when the conclusion does follow from the
of Reasoning; the conclusion not following from - Premises.)
the Premises.)
* Purely-logical. (§ 7.) Semi-logical. Tremiss unduly assumed. Conclusion irrevelanº
(i. e. where the fallacious- (the middle term being (ignoratio elemchi.)
ness is apparent from the ambiguous in sense.) º
mere form of expression.) r— (§ 12.) (§ 13.) T |
- (Petitio principii.) Premiss false or #
‘Undistributed Illicit process, &c." Premiss depend- unsupported.
middle. ing on the Con- -
‘in itself, from the context,’ clusion,
r—; t •TTY }. ſ' circle assuming a proposition"
accidentally. from some connection & g a prop
between the different not the very same as i
the question, but un-
Senses, {} ºr tº tº º
| fairly implying it.
'resemblance, analogy. cause and gº
effect, &c. | r
^ (§ 10.) (§ 11.) Y
Fallacy of Division fallacia acci-
and Composition. dentis, &c.
& TY
^ (§ 16.) (§ 15.) ($ 14.) § 14.)
Fallacy of Fallacy of shifting Fallacy of using Fallacy of appeals to the
objections, &c. ground,
passions; ad hominem ;
complex andge- &
ad verecundiam, &c.
neral terms.
‘to something wholly
irrelevant.
§ 5. On each of the Fallacies which have been thus
enumerated and distinguished, we propose to offer
some more particular remarks: but before we proceed to
this, it will be proper to premise two general observa-
tions, ist, on the importance, and 2d. the difficulty,
from Premiss to Pre-
miss alternately.
of detecting and describing Fallacies; both have been
already slightly alluded to, but it is requisite that they
should here be somewhat more fully and distinctly set
forth.
1st. It seems by most persons to be taken for granted
2 G 2
220
L O G I C.
neous nations in Theology; where the most important Chap. W.
terms are analogical; and yet, they are continually \-N-
Logic. that a Fallacy is to be dreaded merely as a weapon
\—- fashioned and wielded by a skilful Sophist : or if they
allow that a man may with honest intentions slide
into one, unconsciously, in the heat of argument, still
they seem to suppose that where there is no dispute,
there is no cause to dread Fallacy ; whereas there is
much danger, even in what may be called solitary
Reasoning, of sliding unawares into some Fallacy, by
which one may be so far deceived as even to act upon
the Conclusion thus obtained. By solitary Reasoning
is meant the case in which we are not seeking for
arguments to prove a given question, but labouring to
elicit from our previous stock of knowledge some
wseful inference. To select one from innumerable exam-
ples which might be cited, and of which some more will
occur in the subsequent part of this Essay; it is not
improbable that many indifferent sermons have been
produced by the ambiguity of the word “plain :” a
young divine perceives the truth of the maxim, that
“ for the lower orders one's language cannot be too
plain;" (i.e. clear and perspicuous, so as to require
no learning nor ingenuity to understand it,) and when
he proceeds to practice, the word “plain” indistinctly
flits before him, as it were, and often checks him in
the use of ornaments of style, such us metaphor, epi-
thet, antithesis, &c. which are opposed to “plainness”
in a totally different sense of the word, being by no
means necessarily adverse to perspicuity, but rather,
in many cases, conducive to it ; as may be seen in
several of the clearest of our Lord's discourses,
which are of all others the most richly adorned with
figurative language. So far indeed is an ornamented
style from being unfit for the vulgar, that they are
pleased with it even in excess. Yet the desire to be
“plain,” combined with that dim and confused notion
which the ambiguity of the word produces in such as
do not separate in their minds, and set distinctly before
themselves, the two meanings, often causes them to
write in a dry and bald style, which has no advantage
in point of perspicuity, and is least of all suited to the
taste of the vulgar. The above instance is not drawn
from mere conjecture, but from actual experience of
the fact.
Another instance of the strong influence of words
on our ideas may be adduced from a widely different
subject: most persons feel a certain degree of surprise
on first hearing of the result of some late experiments
of the agricultural Chemists, by which they have ascer-
tained that universally what are called heavy soils are
specifically the lightest ; and vice versd. Whence this
surprise? for no one ever distinctly believed the esta-
blished names to be used in the literal and primary
sense, in consequence of the respective soils having
been weighed together ; indeed it is obvious on a
moment's reflection that tenacious clay soils (as well as
muddy roads) are figuratively called heavy from the
difficulty of ploughing or passing over them, which
produces an effect like that of bearing or dragging a
heavy weight; yet still the terms, “light” and
“heavy,” though used figuratively, have most un-
doubtedly introduced into men's minds something of
the ideas expressed by them in their primitive sense.
So true is the ingenious observation of Hobbes, that
“words are the counters of wise men, and the money
of fools.”
More especially deserving of attention is the in-
fluence of analogical terms in leading men into erro-
employed in Reasoning without due attention (oftener
through want of caution than by unfair design) to their
analogical nature ; and most of the errors into which
Theologians have fallen may be traced, in part, to this
CałłSe. -
Thus much, as to the extensive practical influence
of Fallacies, and the consequent high importance of
detecting and exposing them.
§ 6. 2dly. The second remark is, that while sound
Reasoning is ever the more readily admitted, the more
clearly it is perceived to be such, Fallacy, on the
contrary, being rejected as soon as perceived, will, of
course be the more likely to obtain reception, the
more it is obscured and disguised by obliquity and com-
plexity of expression: it is thus that it is the most likely
either to slip accidentally from the careless reasoner,
or to be brought forward deliberately by the Sophist.
Not that he ever wishes that obscurity and complexity
to be perceived; on the contrary it is for his purpose
that the expression should appear as clear and simple
as possible, while in reality it is the most tangled net.
he can contrive. Thus, whereas it is usual to express.
our Reasoning elliptically, so that a Premiss, (or even
two or three entire steps in a course of argument)
which may be readily supplied, as being perfectly
obvious, shall be left to be understood, the Sophist in
like manner suppresses what is not obvious, but is in
reality the weakest part of the argument; and uses
every other contrivance to withdraw our attention (his
art closely resembling the juggler's) from the quarter
where the Fallacy lies. Hence the uncertainty before
mentioned, to which class any individual Fallacy is to
be referred : and hence it is that the difficulty of
detecting and exposing Fallacy, is so much greater
than that of comprehending and developing a process
of sound argument. It is like the detection and
apprehension of a criminal in spite of all his arts of
concealment and disguise; when this is accomplished,
and he is brought to trial with all the evidence of his
guilt produced, his conviction and punishment are
easy; and this is precisely the case with those Fallacies
which are given as examples in Logical Treatises; they
are in fact already detected, by being stated in a plain
and regular form, and are, as it were, only brought up
to receive sentence. Or again, fallacious Reasoning may
be compared to a perplexed and entangled mass of
accounts, which it requires much sagacity and close
attention to clear up, and display in a regular and in-
telligible form ; though when this is once accomplished,
the whole appears so perfectly simple, that the un-
thinking are apt to undervalue the skill and pains
which have been employed upon it.
Moreover, it should be remembered that a very long
discussion is one of the most effectual veils of Fallacy.
Sophistry, like poison, is at once detected, and nau-
seated when presented to us in a concentrated form ;
but a Fallacy which when stated barely, in a few
sentences, would not deceive a child, may deceive half
the world if diluted in a quarto volume. To speak
therefore of all the Fallacies that have ever been enu-
merated as too glaring and obvious to need even being
mentioned, because the simple instances given in
books, and there stated in the plainest and conse-
quently most easily detected form, are such as would
(in that form) deceive no one ; this, surely, shews
L O G I. C.
221
Logic.
{ –
either extreme weakness, or else unfairness. It may
readily be allowed, indeed, that to detect individual
Fallacies, and bring them under the general rules, is a
harder task than to lay down those general rules; but
this does not prove that the latter office is trifling or
useless, or that it does not essentially conduce to the
performance of the other : there may be more inge-
nuity shewn in detecting and arresting a malefactor,
and convicting him of the fact, than in laying down a
law for the trial and punishment of such a person ; but
the latter office, i. e. that of a legislator, is surely
neither unnecessary nor trifling.
It should be added that a close observation and
Logical analysis of fallacious arguments, as it tends
(according to what has been already said) to form a
habit of mind well suited for the practical detection
of Fallacies; so, for that very reason, it will make us
the more careful in making allowance for them ; i. e.
bearing in mind how much men in general are liable
to be influenced by them : e. g. a refuted argument ought
to go for nothing ; but in fact it will generally prove
detrimental to the cause, from the Fallacy which will
be presently explained. No one is more likely to be
practically aware of this, and to take precautions
accordingly, than he who is most versed in the whole
theory of Fallacies; for the best Logician is the least
likely to calculate on men in general being such.
Of Fallacies in form.
§ 7. Enough has already been said in the preceding
compendium ; and it has been remarked above, that
it is often left to our choice to refer an individual
Fallacy to this head or to another.
To the present class we may the most conveniently
refer those Fallacies, so common in practice, of sup-
posing the Conclusion false, because the Premiss is
false, or because the argument is unsound ; and
inferring the truth of the Premiss from that of the
Conclusion; e. g. if any one argues for the existence
of a God, from its being universally believed, a man
might perhaps be able to refute the argument by pro-
ducing an instance of some nation destitute of such
belief; the argument ought then (as has been observed
above) to go for nothing : but many would go further,
and think that this refutation had disproved the exist-
ence of a God; in which they would be guilty of an
illicit process of the major term; viz. “ whatever is
universally believed must be true; the existence of a
God is not universally believed ; therefore it is not
true.” Others again from being convinced of the
truth of the Conclusion would infer that of the Pre-
mises; which would amount to the Fallacy of undis-
tributed middle : viz. “ what is universally believed,
is true ; the existence of a God is true ; therefore it
is universally believed.” Or, these Fallacies might
be stated in the hypothetical form ; since the one
evidently proceeds from the denial of the antecedent
to the denial of the consequent ; and the other from
the establishing of the consequent to the inferring of
the antecedent; which two Fallacies correspond re-
spectively with those of illicit process of the major,
and undistributed middle.
Fallacies of this class are very much kept out of
sight, being seldom perceived even by those who
employ them ; but of their practical importance there
can be no doubt, since it is notorious that a weak
argument is always, in practice, detrimental ; and that
there is no absurdity so gross which men will not Chap. V.
readily admit, if it appears to lead to a Conclusion of ~~
what they are already convinced. Even a candid and
sensible writer is not unlikely to be, by this means.
misled, when he is seeking for arguments to support
a Conclusion which he has long been fully convinced
of himself; i.e. he will often use such arguments as
would lever have convinced himself, and are not
likely to convince others, but rather (by the operation
of the converse Fallacy) to confirm in their dissent
those who before disagreed with him.
It is best therefore to endeavour to put yourself in
the place of an opponent to your own arguments, and
consider whether you could not find some objection to
them. The applause of one's own party is a very
unsafe ground for judging of the real force of an ar-
gumentative work, and consequently of its real utility.
To satisfy those who were doubting, and to convince
those who were opposed, is the only sure test; but
these are seldom very loud in their applause, or very
forward in bearing their testimony.
Of Ambiguous middle.
§ 8. That case in which the middle is undistributed,
belongs of course to the preceding head, the fault
being perfectly manifest from the mere form of the
expression : in that case the extremes are compared
with two parts of the same term; but in the Fallacy
which has been called semi-logical, (which we are
now to speak of) the extremes are compared with
two different terms, the middle being used in two
different senses in the two Premises.
And here it may be remarked, that when the argu-
ment is brought into the form of a regular Syllogism, the
contrast between these two senses will usually appear
very striking, from the two Premises being placed
together ; and hence the scorn with which many have
treated the very mention of the Fallacy of equivocation,
deriving their only notion of it from the exposure of it
in Logical Treatises; whereas, in practice it is common
for the two Premises to be placed very far apart, and
discussed in different parts of the discourse; by which
means the inattentive hearer overlooks any ambiguity
that may exist in the middle term. Hence the advan-
tage of Logical habits, to fix our attention strongly and
steadily on the important terms of an argument.
One case which may be regarded as coming under
the head of Ambiguous middle, is, what is called
“Fallacia Figura Dictionis,” the Fallacy built on the
grammatical structure of language, from men's usually
taking for granted that paronymous words, (i. e. those
belonging to each other, as the substantive, adjective,
verb, &c. of the same root) have a precisely corres-
pondent meaning : which is by no means universally
the case. Such a Fallacy could not indeed be even
exhibited in strict Logical form, which would
preclude even the attempt at it, since it has two
middle terms in sound as well as sense ; but nothing
is more common in practice than to vary continually
the terms employed, with a view to grammatical con-
venience; nor is there any thing unfair in such a
practice, as long as the meaning is preserved unaltered:
e.g. “murder should be punished with death; this
man is a murderer ; therefore he deserves to die; "
&c. &c. Here we proceed on the assumption (in this
case just) that to commit murder and to be a mur-
derer, to deserve death and to be one who ought to
*
222
L O G I. C.
Logic. -
die, are, respectively, equivalent expressions; and it
would frequently prove a heavy inconvenience to be
debarred this kind of liberty ; but the abuse of it
gives rise to the Fallacy in question : e.g. projectors
are unfit to be trusted ; this man has formed a project,
therefore he is unfit to be trusted : * here the Sophist
proceeds on the hypothesis that he who forms a project
must be a projector; whereas the bad sense that
commonly attaches to the latter word, is not at all
implied in the former.
This Fallacy may often be considered as lying not in
the middle, but in one of the terms of the Conclusion;
so that the Conclusion drawn shall not be; in reality,
at all warranted by the Premises, though it will
appear to be so, by means of the grammatical affinity
of the words : e.g. “to be acquainted with the guilty
is a presumption of guilt ; this man is so acquainted;
therefore we may presume that he is guilty :” this
argument proceeds on the supposition of an exact
correspondence between “ presume” and “ presump-
tion,” which however does not really exist; for
“ presumption” is commonly used to express a kind
of slight suspicion ; whereas “to presume” amounts to
absolute belief.
The above remark will apply to some other eases
of ambiguity of term ; viz. the Conclusion will often
contain a term, which (though not as here, different in
eapression from the corresponding one in the Premiss,
yet) is liable to be understood in a sense different from
that which it bears to the Premiss; though of course
such a Fallacy is less common, because less likely to
deceive, in those cases, than in this ; where the term
used in the Conclusion, though professing to correspond
with one in the Premiss, is not the very same in
expression, and therefore is more certain to convey a
different sense; which is what the Sophist wishes.
There are innumerable instances of a non-corres-
pondence in paronymous words, similar to that above
instanced ; as between art and artful, design and
designing, faith and faithful, &c.; and the more slight
the variation of meaning, the more likely is the Fallacy
to be successful; for when the words have become so
widely removed in sense as “pity” and “pitiful,”
every one would perceive such a Fallacy, nor could it
be employed but in jest. .
This Fallacy cannot in practice be refuted, by stating
merely the impossibility of reducing such an argument
to the strict Logical form ; (unless indeed you are
addressing regular Logicians,) you must find some way
of pointing out the non-correspondence of the terms
in question; e.g. with respect to the example above,
it may be remarked, that we speak of strong or faint
“ presumption,” but yet we use no such expression
in conjunction with the verb “presume,” because
the word itself implies strength.
No Fallacy is more common in controversy than the
present, since in this way the Sophist will often be
able to misinterpret the propositions which his oppo-
ment admits or maintains, and so employ them against
him : thus in the examples just given, it is natural to
conceive one of the Sophist's Premises to have been
borrowed from his opponent.
Perhaps a dictionary of such paronymous words as
do not regularly correspond in meaning, would be
nearly as useful as one of synonyms ; i. e. properly
w-ºr-
* Wealth of Nations, A. Smith : Usury.
speaking, of pseudo-synonyms. The present Fallacy is Chap, W.
nearly allied to, or rather perhaps may be regarded s-y-
as a branch of that founded on Etymology ; viz. when
a term is used, at one time, in its customary, and at
another, in its Etymological sense. Perhaps no
example of this can be found that is more extensively
and mischievously employed than in the case of the
word representative : assuming that its right meaning
must correspond exactly with the strict and original
sense of the verb represent, the Sophist persuades the
multitude, that a member of the House of Commons
is bound to be guided in all points by the opinion of
his constituents; and, in short, to be merely their
spokesman : whereas law and custom, which in this
case may be considered as fixing the meaning of the
term, require no such thing, but enjoin the represen-
tative to act according to the best of his own judgment,
and on his own responsibility. H. Tooke has furnished
a whole magazine of such weapons for any Sophist
who may need them, and has furnished some speci-
mens of the employment of them. -
§ 9. It is to be observed, that to the head of
Ambiguous middle should be referred what is called
“ Fallacia plurium Interrogationum,” which may very
properly be named, simply, “ the Fallacy of Interro-
gation ;” viz. the Fallacy of asking several questions
which appear to be but one ; so that whatever one
answer is given, being of course applicable to one only
of the implied questions, may be interpreted as applied
to the other; the refutation is, of course, to reply
separately to each question, i.e. to detect the ambiguity.
We have said several “questions which appear to be
but one, for else there is no Fallacy , such an example
therefore, as ‘‘ estne homo animal et lapis 2" which
Aldrich gives, is foreign to the matter in hand ; for
there is nothing unfair in asking two distinct ques-
tions, or asserting two distinct propositions, distinctly
and avowedly. -
This Fallacy may be referred, as has been said, to
the head of Ambiguous middle ; in all Reasoning
it is very common to state one of the Premises in
form of a question, and when that is admitted, or
supposed to be admitted, then to fill up the rest; if
then one of the terms of that question be ambi-
guous, whichever sense the opponent replies to, the
Sophist assumes the other sense of the term in the
remaining Premiss. It is therefore very common to
state an unequivocal argument, in form of a question
so worded, that there shall be little doubt which reply
will be given: but if there be such doubt, the Sophist
must have two Fallacies of equivocation ready : e. g.
the question “whether any thing vicious is expedient,”
discussed in Cic. Off, book iii. (where, by the bye, he
seems not a little perplexed with it himself,) is of the
character in question, from the ambiguity of the word
“ expedient,” which means sometimes, “ conducive to
temporal prosperity,” sometimes, “ conducive to the
greatest good :” whichever answer therefore was
given, the Sophist might have a Fallacy of equivoca-
tion founded on this term ; viz. if the answer be in the
negative, his argument Logically developed, will
stand thus, “ what is vicious is not expedient ;
whatever conduces to wealth and aggrandizement is
expedient, therefore it cannot be vicious :” if, in the
affirmative, then thus, ‘‘ whatever is expedient is
desirable ; something vicious is expedient, therefore
desirable.” -
L O G. I. C.
223
Logic.
This kind of Fallacy is frequently employed in such
a manner, that the uncertainty shall be, not about
the meaning, but the extent of a term, i. e. whether it
is distributed or not: e.g. “ did A B in this case act from
such and such a motive º’’ which may imply either,
** was it his sole motive 2" or “ was it one of his
motives 2'' in the former case the term “ that which
actuated A B" is distributed ; in the latter not : now
if he acted from a mixture of motives, whichever
answer you give, may be misrepresented and thus
disproved. -
§ 10. In some cases of Ambiguous middle, the term
in question may be considered as having in itself, from
its own equivocal nature, two significations; (which
apparently constitutes the “ Fallacia equivocationis of
Logical writers ;) others again have a middle term
which is ambiguous from the context, i. e. from what
is understood in conjunction with it : this division will
be found useful, though it is impossible to draw the
line accurately in it.
There are various ways in which words come to have
two meanings; Ist. by accident ; (i. e. when there is
no perceptible connection between the two meanings)
as “light” signifies both the contrary to “ heavy,”
and the contrary to “dark.” Thus, such proper
names as John or Thomas, &c. which happen to
belong to several different persons, are ambiguous,
because they have a different signification in each case
where they are applied. Words which fall under this
first head are what are the most strictly called equivocal.
2dly. There are several terms in the use of which
it is necessary to notice the distinction between first
and second intention: the “first intention” of a
term, (according to the usual acceptation of this
phrase,) is a certain vague and general signification
of it, as opposed to one more precise and limited,
which it bears in some particular art, science, or
system, and which is called its “ second intention.”
Thus, among farmers in some parts, the word “beast”
is applied particularly and especially to the ox kind;
and ‘‘ bird,” in the language of many sportsmen is
in like manner appropriated to the partridge : the
common and general acceptation (which every one
is well acquainted with) of each of those two words,
is the first intention of each ; the other, its second
intention.
It is evident that a term may have several second
intentions, according to the several systems into
which it is introduced, and of which it is one of the
technical terms: thus line signifies, in the Art Military,
a certain form of drawing up ships or troops ; in
Geography, a certain division of the earth ; to the
fisherman, a string to catch fish, &c. &c.; all which
are so many distinct second intentions, in each of
which there is a certain signification of “ extension
in length” which constitutes the first intention, and
which corresponds pretty nearly with the employ-
ment of the term in Mathematics.
It will sometimes happen, that a term shall be
employed always in some one or other of its second
intentions ; and never, strictly, in the first, though that
first intention is a part of its signification in each case.
It is evident, that the utmost care is requisite to avoid
confounding together, either the first and second
intentions, or the different second intentions with each
other.
3dly. When two or more things are connected by
resemblance or analogy, they will frequently have the Chap. º
same name. Thus a “blade of grass,” and the con-
trivance in building called a “ dove-tail,” are so called
from their resemblance to the blade” of a sword, and
the tail of a real dove : but two things may be con-
nected by analogy, though they have in themselves no
resemblance : for analogy is the resemblance of ratios,
(or relations) thus, as a sweet taste gratifies the palate,
so does a sweet sound gratify the ear; and hence the
same word, “ sweet,” is applied to both, though no
flavour can resemble a sound in itself: so, the leg of
a table, does not resemble that of an animal ; nor the
foot of a mountain that of an animal : but the leg
answers the same purpose to the table, as the leg of an
animal to that animal; the foot of a mountain has the
same situation relatively to the mountain, as the foot
of an animal, to the animal; this analogy therefore
may be expressed like a Mathematical analogy; (or
proportion) leg : animal :: supporting stick : table.—
In all these cases, (of this 3d head) one of the meanings
of the word is called by Logicians proper, i.e. original
or primary; the other improper, secondary or trans-
ferred: thus, sweet, is originally and properly applied
to tastes ; secondarily and improperly (i.e. by analogy,)
to Sounds : thus also, dove-tail is applied secondarily
though not by analogy, but by direct resemblance to
the contrivance in building so called. When the
secondary meaning of a word is founded on some
fanciful analogy, and especially when it is introduced
for ornament sake, we call this a metaphor; as when
we speak of “a ship's ploughing the deep.” The
turning up of the surface being essential indeed to
the plough, but incidental only to the ship ; but if the
analogy be a more important and essential one, and
especially if we have no other word to express our
meaning but this transferred one, we then call it
merely an analogous word, (though the metaphor is
analogous also ;) e.g. one would hardly call it meta-
phorical or figurative language to speak of the leg of a
table, or mouth of a river.
4thly. Several things may be called by the same
name, (though they have no connection of resemblance
or analogy) from being connected by vicinity of time or
place ; under which head will come the connection of
cause and effect, or of part and whole, &c. Thus a
door signifies both an opening in the wall, (more
strictly called the door-way,) and a board which closes
it: which are things neither similar nor analogous.
When I say, “the rose smells sweet ’’ and “ I smell
the rose :” the word “smell” has two meanings in
the latter sentence, I am speaking of a certain sensa-
tion in my own mind ; in the former, of a certain
quality in the flower, which produces that sensation,
but which of course cannot in the least resemble it :
and here the wordsmell, is applied with equal propriety
to both. Thus we speak of Homer, for “ the works
of Homer ;” and this is a secondary or transferred
meaning : and so it is when we say, ‘‘ a good shot, "
for a good marksman : but the word “ shot” has two
other meanings, which are both equally proper; viz.
the thing put into a gun in order to be discharged from
it, and the act of discharging it.
-—4
* Unless indeed the primary application of the term be to the
leaf of grass, and the secondary, to cutting instruments; which
is perhaps more probable ; but the question is unimportant in
the present case, -
224
L O G. I. C.
are equal to two right angles: A B C, is an angle of Chap. V.
a triangle ; therefore A B C, is equal to two right \--
Logic. Thus, “ learning" signifies either the act of ac-
~~ quiring knowledge, or , the knowledge itself; e. g.
“ he neglects his learning.” “ Johnson was a man
of learning.” Possession is ambiguous in the same
manner; and a multitude of others. Much confusion
often arises from ambiguity of this kind, when unper-
ceived ; nor is there any point in which the copious-
ness and consequent precision of the Greek language
is more to be admired than in its distinct terms for
expressing an act, and the result of that act; e. g.
Tpāśis “ the doing of anything ;” ºrpāqua, “ the thing
done ;” so, öda is and 8tºpov, Aſſyrts and Ajulua, &c. It
will very often happen, that two of the meanings of a
word will have no connection with one another, but
will each have some connection with a third. Thus
“ martyr,” originally signified a witness, thence it
was applied to those who suffered in bearing testimony
to Christianity ; and thence again it is often applied to
sufferers in general : the first and third significations
are not the least connected. Thus “post” signifies
originally a pillar, (póstum, from pono;) then a distance
marked out by posts; and then the carriages,
Imessengers, &c. that travelled over this distance.
Innumerable other ambiguities might be brought
under this fourth head, which indeed comprehends
all the cases which do not fall under the three others.
The remedy for ambiguity is a definition of the
term which is suspected of being used in two senses ;
viz. a verbal, not necessarily a real definition ; as was
remarked in the Compendium.
But here it may be proper to remark, that for the
avoiding of Fallacy or of verbal controversy, it is only
requisite that the term should be employed uniformly
in the same sense as far as the existing question is con-
cerned. Thus, two persons might, in discussing the
question, whether Buonaparte was a GREAT man, have
some difference in their acceptation of the epithet
“ great,” which would be non-essential to that ques-
tion; e.g. one of them might understand by it nothing
more than eminent intellectual, and moral qualities ;
while the other might conceive it to imply the per-
formance of splendid actions : this abstract difference
of meaning would not produce any disagreement in
the existing question, because both those circum-
stances are united in the case of Buonaparte ; but if
one of the parties understood the epithet “great” to
imply GENERosity of character, &c. then there would
be a disagreement. Definition, the specific for am-
biguity, is to be employed and demanded with a
view to this principle; it is sufficient on each occasion
to define a term as far as regards the question in hand.
Of those cases in which the ambiguity arises from
the context, there are many species; several of which
Logicians have enumerated, but have neglected to
refer them, in the first place, to one common class,
(viz. the one under which they are here placed ;) and
have even arranged some under the head of Fallacies
“ in dictione,’’ and others, “ extra dictionem.”
We may consider, as the first of these species, the
Fallacy of “ Division” and that of “Composition,”
taken together, since in each of these the middle term
is used in one Premiss collectively, in the other, dis-
tributively : if the former of these is the major Premiss,
and the latter the minor, this is called the “Fallacy of
division;” the term which is first taken collectively
being afterwards divided; and vice versd. The ordinary
examples are such as these ; all the angles of a triangle
angles. Five is one number; three and two are five ;
therefore three and two are one number; or, three
and two are two numbers, five is three and two,
therefore five ds two numbers : it is manifest that the
middle term, three and two, (in this last example) is
ambiguous, signifying, in the major Premiss “ taken
9 3
distinctly,” in the minor, “taken together :” and so
of the rest. -
To this head may be referred the Fallacy by which
men have sometimes been led to admit, or pretend to
admit, the doctrine of necessity ; e. he who neces-
sarily goes or stays (i. e. in reality, “who neces-
sarily goes, or who necessarily stays”) is not a free agent;
you must necessarily go or stay ; (i. e. “ you must
necessarily take the alternative,”) therefore you are not
a free agent. Such also is the Fallacy which probably
operates on most adventurers in lotteries; e. g. the
gaining of a high prize is no uncommon occurrence ;
and what is no uncommon occurrence may reasonably
be expected ; therefore the gaining of a high prize
“ may reasonably be expected :'' the conclusion when
applied to the individual, (as in practice it is) must
be understood in the sense of “ reasonably expected
by a certain individual ;” therefore for the major Premiss
to be true the middle term must be understood to
mean, ‘‘ no uncommon occurrence to Some one
particular person ;” whereas for the minor (which has
been placed first) to be true, you must understand it
of “ no uncommon occurrence to some one or other;”
and thus you will have the Fallacy of Composition.
There is no Fallacy more common, or more likely
to deceive than the one now before us : the form in
which it is most usually employed, is, to establish
some truth, separately, concerning each single member
of a certain class, and thence to infer the same of the
whole collectively : thus some infidels have laboured to
prove concerning some one of our Lord's miracles, that
it might have been the result of an accidental conjunc-
ture of natural circumstances; next, they endeavour
to prove the same concerning another ; and so on ;
and thence infer that all of them might have been so.
They might argue in like manner, that because it is
not very improbable one may throw sixes in any one
out of a hundred throws, therefore it is no more impro-
bable that one may throw sixes a hundred times
running.
This Fallacy may often be considered as turning on
the ambiguity of the word “ all;" which may easily
be dispelled by substituting for it the word “each”
or “every,” where that is its signification ; e.g. “all
these trees make a thick shade’’ is ambiguous, mean-
ing, either “every one of them,” or “all together.”
This is a Fallacy with which men are extremely apt
to deceive themselves : for when a multitude of par-
ticulars are presented to the mind, many are too weak
or too indolent to take a comprehensive view of them;
but confine their attention to each single point, by
turns; and then decide, infer, and act, accordingly ;
e.g. the imprudent spendthrift, finding that he is able
to afford this, or that, or the other expense, forgets
that all of them together will ruin him.
To the same head may be reduced that fallacious
reasoning by which men vindicate themselves to their
own conscience and to others, for the neglect of those
undefined duties, which though indispensable, and
L O G. I. C.
225
very difficult to ascertain wherein they conceived them Chap. V.
to differ, and what, according to them, is the nature S-V-2
*
Logic, therefore not left to our choice whether we will practise
-V-' them or not, are left to our discretion as to the mode,
and the particular occasions of practising them ; e. g.
“I am not bound to contribute to this charity in par-
ticular; nor to that ; nor to the other :” the practical
conclusion which they draw, is, that all charity may
be dispensed with. -
As men are apt to forget that any two circum-
stances (not naturally connected) are more rarely to
be met with combined than separate, though they be
not at all incompatible ; so also they are apt to imagine
from finding that they are rarely combined, that there
is an incompatibility; e. g. if the chances are ten to one
against a man's possessing strong reasoning powers, and
ten to one against exquisite taste, the chances against
the combination of the two (supposing them neither
connected nor opposed) will be a hundred to one. Miany
therefore, from finding them so rarely united, will
infer that they are in some measure incompatible;
which Fallacy may easily be exposed in the form of
|Undistributed middle : “ qualities unfriendly to each
other are rarely combined; excellence in the reasoning
powers and in taste are rarely combined; therefore
they are qualities unfriendly to each other.”
§ 11. The other kind of ambiguity arising from the
context, and which is the last case of Ambiguous
middle that we shall notice, is the “fallacia accidentis,”
together with its converse “fallacia a dicto secundum
quidad dictum simpliciter s” in each of which the middle
is used in one Premiss to signify something considered
simply, in itself, and as to its essence ; and in the
other Premiss, so as to imply that its accidents are taken
into account with it : as in the well-known example,
“what is bought in the market is eaten ; raw meat is
bought in the market; therefore raw meat is eaten.”
Here the middle has understood in conjunction with
it, in the major Premiss “ as to its substance merely :”
in the minor, “ as to its condition and circumstances.”
To this head perhaps, as well as to any, may be
referred the Fallacies which are frequently founded on
the occasional, partial, and temporary variations in
the acceptation of some term, arising from circum-
stances of person, time, and place, which will occasion
something to be understood in conjunction with it
beyond its strict literal signification; e. g. the phrase
“Protestant ascendancy,” having become a kind of
watch-word or gathering-cry of a party, the expression
of good wishes for it would commonly imply an ad-
herence to certain measures not literally expressed by
the words; to assume therefore that one is unfriendly
to “ Protestant ascendancy” in the literal sense,
because he has declared himself unfriendly to it when
implying and connected with such and such other
sentiments, is a gross Fallacy; and such an one as
perhaps the authors of the above would much object
to, if it was assumed of them that they were adverse
to “ the cause of liberty throughout the world,” and
to “a fair representation of the people,” from their
objecting to join with the members of a factious party
in the expression of such sentiments.
Such Fallacies may fairly be referred to the present
head.
§ 12. Of the Non-logical (or material) Fallacies,
and first of begging the question.
The indistinct and unphilosophical account which
has been given by Logical writers of the Fallacy of
“ non-causd,” and that of “petitio principii," makes it
WOL. I.
of each ; without therefore professing to conform
exactly to their meaning, and with a view to distinct-
ness only, which is the main point, let us confine the
name “petitio principii” to those cases in which the
Premiss either appears manifestly to be the same as
the Conclusion, or is actually proved from the Conclu-
sion, or is such as would naturally and properly so be
proved ; (as if one should attempt to prove the being
of a God from the authority of holy writ;) and to the
other class be referred all other cases, in which the
Premiss (whether the expressed or the suppressed one)
is either proved false, or has no sufficient claim to be
received as true. Let it however be observed, that in
such cases (apparently) as this, we must not too
hastily pronounce the argument fallacious ; for it
may be perfectly fair at the commencement of an argu-
ment to assume a Premiss that is not more evident than
the Conclusion, or is even ever so paradoxical, provided
you proceed to prove fairly that Premiss : and in like
manner it is both usual and fair to begin by deducing
your Conclusion from a Premiss exactly equivalent to
it ; which is merely throwing the proposition in ques-
tion into the form in which it will be most conveniently
proved. Arguing in a circle however must necessarily
be unfair; though it frequently is practised unde-
signedly ; e. g. some Mechanicians, attempt to prove,
(what they ought to lay down as a probable but
doubtful hypothesis,) that every particle of matter
gravitates equally; “ why?" because those bodies
which contain more particles ever gravitate more
strongly, i. e. are heavier : ‘‘ but (it may be urged)
those which are heaviest are not always more bulky;”
“ no, but still they contain more particles, though
more closely condensed ; “ how do you know that 2"
“because they are heavier;” “how does that prove it?”
“ because all particles of matter gravitating equally,
that mass which is specifically the heavier, must needs
have the more of them in the same space.”
Obliquity and disguise being of course most im-
portant to the success of the petitio principii, as well
as of other Fallacies, the Sophist will in general either
have recourse to the circle, or else not venture to state
distinctly his assumption of the point in question, but
will rather assert some other proposition which implies
it; thus keeping out of sight (as a dexterous thief does
stolen goods) the point in question, at the very
moment when he is taking it for granted: hence the
frequent union of this Fallacy with “ignoratio elemchi:”
vide § 14. The English language is perhaps the more
suitable for the Fallacy of petitio principii, from its
being formed from two distinct languages, and thus
abounding in synonymous expressions which have no
resemblance in sound, and no connection in etymology;
so that a Sophist may bring forward a proposition
expressed in words of Saxon origin, and give as a
reason for it, the very same proposition stated in
words of Norman origin; e.g. “ to allow every man
an unbounded freedom of speech, must always be,
on the whole, advantageous to the State ; for it is
highly conducive to the interest of the community,
that each individual should enjoy a liberty perfectly
unlimited of expressing his sentiments.”
§ 13. The next head is, the falsity, or at least,
undue assumption of a Premiss when it is not equiva-
lent to, or dependent on the Conclusion; which, as has
2 H
226
L. O. G. I. C.
general, however, the more skilful, Sophist will avoid Chapi,V.
a direct assertion of what he means unduly to assume; .
Logic. been before said, seems to correspond nearly with the
S-v-" meaning of Logicians, when they speak of “non causa
pro causd :" this name indeed would seem to apply a
much narrower class, there. being one species of
arguments, which are from cause to effect; in which of
course two things are necessary; 1st, the sufficiency of
the cause; 2d. its estabłishment; these are the two
Premises; if therefore the former be unduly assumed,
we are arguing from that which is not a sufficient
eause as if it were so; e.g. as if one should contend from
such a man's having been unjust or cruel, that he will
certainly be visited with some heavy temporal judg-
ment, and come to an untimely end. In this instance
the Sophist, from: having assumed in the Premiss,
the (granted) existence of a pretended, cause, infers in
the conclusion the existenee of the pretended effect,
which, we have supposed to be the Question: or vice
versd; the pretended, effect may be employed to esta-
blish the cause ; e. g. inferring sinfulness from tem-
poral calamity ... but when both the pretended cause,
and effect are granted, i. e. granted to exist, then the
Sophist will infer something from their pretended:
connection ; i.e. he will assume as a Premiss, that “of
these two admitted facts, the one is the cause of the
other;"as the opponents of the Reformation assumed
that it was the cause of the troubles which took place
at that period, and thence inferred that it was an evil.
Such an argument as either of these might strictly be
Galled “ non causa pro-causd ;” but it is not probable,
that the Logical writers intended any such limitation,
(which indeed would be wholly unnecessary and im-
pertinent,) but rather that they were confounding
together cause and reason ; the sequence of Conclusion
from Premises being perpetually mistaken for that of
effect from physical cause. It is indeed a very necessary
caution in philosophical investigation not to assume too
hastily that one thing is the cause of another, when
perhaps it is only an accidental concomitant ; (as was
the case in the assumption of the Premises of the last
mentioned examples :) but investigation is a perfectly
distinct business from argumentation ; and to mingle
together the rules of the two, (as Logical writers have
generally done, especially in the present case,) tends
only to produce confusion in both. It may be better
therefore to drop the name which tends to perpetuate
this confusion, and simply state (when such is the case)
that the Premiss is unduly assumed ; i. e. without
being either self-evident, or satisfactorily proved.
The contrivances by which men may deceive them-
selves or others, in assuming Premises unduly, so that
that undue assumption shall not be perceived, (for it is in
this the Fallacy consists) are of course infinite. Some-
times (as was before observed) the doubtful Premiss is
suppressed, as if it were too evident to need being
proved, or even stated, and as if the whole question
turned on the establishment of the other Premiss.
Thus H. Tooke proves, by an immense induction,
that all particles were originally nouns or verbs; and
thence concludes, that in reality they are so still, and
that the ordinary division of the parts of speech is
absurd; keeping out of sight, as self-evident, the
other Premiss, which is absolutely false; viz. that the
meaning and force of a term; now and for ever, must
be that, which it, or its root originally bore.
Sometimes men are shamed into admitting an un-
founded assertion, by being assured, that it is so
evident it would argue great weakness to doubt it. In
since that might direct. the reader's attention to the
consideration of the question whether, it be true ormot,
since that which is indisputable does not sonoften need
to be asserted: it succeeds better; therefore, if you
allude to the proposition as something curiors, and re-
markable; just as the Reyal Society were imposed on
by being asked to account for the faet that a vesselſ of
water received; no addition to its weight by a live fish
put into it; while they were seeking for the cause,
they forgot to ascertain the fact, and thus admitted
without Suspicion: a mere fiction. Thus an eminent
Scotch writer, instead of asserting that, “ the advocates
of Logic have been worsted and driven from the field
in every controversy,” (an assertion, which if made;
would have been the more readily ascertained to be
perfectly groundless,) merely observes; that, “it is a
circumstancer not a little remarkable.”
Brequently the Fallacy of ignoratio: elemchi is cafled
in to the aid of this ; ii. e. the Premiss is assumed on
the ground of another proposition, somewhat like it,
having been proved; thus in arguing by example, &c.
the parallelism of two cases is often assumed from
their being in some respects alike, though perhaps they
differ in the very point which is essential to the argui-
ment ; e.g. from the circumstance that some men of
humble station, who have been well educated, are apt
to think themselves above low drudgery, it is argued
that universal, education of the lower order; would
beget general idleness-: this argument rests of course
on the assumption of parallelism in the two cases, viz.
the past and the future ; whereas there is a circum-
stance that is absolutely essential, in which they differ;
for when education is universal it must cease to be a
distinction; which is probably the very circumstance
that renders men too proud for their work.
This very same Fallacy is often resorted to on the
opposite side ; an attempt is made to invalidate some
argument from example, by pointing out a difference
between the two cases, though they agree in every
thing that is essential to the question. Eastly, it may
be here remarked, conformably with what has been
formerly said, that it will often be left to your choice
whether to refer this or that fallacious argument to
the present head, or that of Ambiguous middle; “if
the middle term is here used in this sense, there is an
ambiguity; if in that sense, the proposition is false.”
§ 14. The last kind of Fallacy to be discussed is that
of Irrelevant Conclusion; commonly called ignoratio
elenchi, Various kinds of propositions are, according
to the occasion, substituted for the one of which proof
is required. -
Sometimes the particular for the universal; some-
times a proposition with different terms: and various
are the contrivances employed to effect and to conceal
this substitution, and to make the Conclusion which the
Sophist has drawn, answer, practically, the same pur-
pose as the one he ought to have established. We say;
“ practically the same purpose,” because it will very
often happen that some emotion will be excited—some
sentiment impressed on the mind—(by a dexterous
employment of this Fallacy), such as shall bring men
into the disposition requisite for your purpose, though
they may not have assented to, or even siated dis-
tinctly in their own minds the proposition which it was
your business to establish. Thus if a Sophist has to
L O G. I. C. 227
of ignoratio elemchi, which is a very common and suc-. Chºp.”
*Login, defend one who has been guilty ofsome serious offence,
- eessful practice; viz. the Sophist proves, or disproves,
*~~' which he wishes to extenuate, though he is unable dis-
#inctly to prove that it is not such, yet if he can:suc-
ceedin-making the audience laugh at some casual matter,
he has gained practically the same point. So also if
any one has pointed out the extenuating:circumstances
in some particular-case of offence, so as to shew that it
differs widely from the generality of the same class,
the Sophist, if herfind himself unable to disprove these
oircumstances, may do away the force of them, by
simply referring the action to that:very class, which no
one can deny that it belongs to, and the very name of
which will excite a feeling of disgust sufficient to
eounteract ſhe extenuation; e.g. let it be a case of
peculation, and that many mitigating circumstances
have been brought forward which cannot be denied;
the sophistical opponent will reply, “well, but after
all, the man is a rogue, and there is an end of it ;”
now in reality this was (by hypothesis) never the
question; and the mere assertion of what was never
denied, ought not, in fairness, to be regarded as deci-
sive ; but, practically, the odiousness of the word,
arising in great measure from the association of those
very circumstances which belong to most of the class,
but which we have supposed to be absent in this par-
ticular instance, excites precisely that feeling of disgust,
which in effect destroys the force of the defence. In
like manner we may refer to this head all cases of
improper appeals to the passions, and every thing else
which is mentioned by Aristotle as extraneous to the
matter in hand, (éºw 78 7pdf//watos.) * *
In all these cases, as has been before observed, if
the Fallacy we are now treating of be employed for
the apparent establishment, not of the ultimate Con-
clusion, but (as it very commonly happens) of a
Premiss, (i.e. if the Premiss required be assumed on
the ground that some proposition resembling it has
been proved,) then there will be a combination of this
Fallacy with the last mentioned. A good instance
of the employment and exposure of this Fallacy
occurs in Thucydides, in the speeches of Cleon and
Diodotus concerning the Mitylenaeans : the former
(over and above his appeal to the angry passions of his
audience,) urges the justice of putting the revolters to
death ; which, as the latter remarked, was nothing to
the purpose, since the Athenians were not sitting in
judgment, but in deliberation, of which the proper
end is expediency. 2
It is evident that ignoratio elemchi may be employed
as well for the apparent refutation of your opponent's
proposition, as for the apparent establishment of your
own; for it is substantially the same thing to prove what
was not denied, or to disprove what was not asserted :
the latter practice is not less common, and it is more
offensive, because it frequently amounts to a personal
affront, in attributing to a person opinions, &c. which
he perhaps holds in abhorrence. Thus, when in a
discussion one party vindicates, on the ground of gene-
ral expediency, a particular instance of resistance to
Government in a case of intolerable oppression, the
opponent may gravely maintain that “we ought not
to do evil that good may come:” a proposition which
of course had never been denied, the point in dispute
Joeing “ whether resistance in this particular case were
doing revil or not.” In this example it is to be re-
marked, (and the remark will apply very generally,)
that the Fallacy of petitio principii is combined with that
not thesproposition which is really in question, but one
which so implies.it. as to proceed on the supposition
that it is already decided, and can admit of no doubt;
by this means his “assumption of the point in
question” is so indirect and oblique, that it may easily
escape notice; and he thus establishes, practically,
his Conclusion, at the very moment when he is with-
drawing your attention from it to another question.
There are certain kinds of argument recounted and
named by Logical writers, which we should by no means
universally call Fallacies; but which when unfairly
used, and so far as they are fallacious, may very well be
referred to the present head; such as the “ argumentum
ad hominem,” or personal argument, “ argumentum ad
verecundiam,” “argumentum ad populum,” &c. all of
them regarded as contradistinguished from “argu-
mentum ad rem,” or according to others (meaning
probably the very same thing) “ adjudicium.” These
have all been described in the lax and popular language
before alluded to, but not scientifically : the “ argu-
mentum ad hominem” they say, “is addressed to the
peculiar circumstances, character, avowed opinions,
or past conduct of the individual, and therefore has a
reference to him only, and does not bear directly
and absolutely on the real question, as the argumen-
tum ad rem' does :” in like manner the ‘‘ argumentum
ad verecundiam” is described as an appeal to our reve-
rence for some respected authority, some venerable
institution, &c. and the “argumentum ad populum,” as
an appeal to the prejudices, passions, &c. of the mul-
titude, and so of the rest. Along with these is
usually enumerated “ argumentum ad ignorantiam,”
which is here omitted as being evidently nothing more
than the employment of some kind of Fallacy, in the
widest sense of that word, towards such as are likely
to be deceived by it. It appears then, (to speak rather
more technically,) that in the “argumentum ad homi-
mem” the Conclusion which actually is established, is
not the absolute and general one in question, but
relative and particular ; viz. not that “such and such
is the fact,” but that “ this man is bound to admit it, in
conformity to his principles of Reasoning, or in consis-
tency with his own conduct, situation, &c.” Such a
Conclusion it is often both fair and necessary to esta-
blish, in order to silence those who will not yield to
fair general argument; or to convince those whose
weakness and prejudices would not allow them to
assign to it its due weight : it is thus that our Lord
on many occasions silences the cavils of the Jews ; as
in the vindication of healing on the Sabbath, which is
paralleled by the authorized practice of drawing out a
beast that has fallen into a pit. All this, as we have
said, is perfectly fair, provided it be done plainly,
knowingly, and avowedly ; but if you attempt to substi-
tute this partial and relative Conclusion for a more
general one—if you triumph as having established
your proposition absolutely and universally, from
having established it, in reality, only as far as it relates
to your opponent, then you are guilty of a Fallacy of
the kind which we are now treating of : your Conclu-
sion is not in reality that which was, by your own
account, proposed to be proved : the fallaciousness
depends upon the deceit or attempt to deceive. The
same observations will apply to “argumentum ad vere-
cundiam,” and the rest.
- 2 H 2
228
L O G. I. C. **
and objections against a vacuum ; but one of them Chap. V.
must be true.” :* -
Logic. It is very common to employ an ambiguous term
S-V-' for the purpose of introducing the Fallacy of Irrelevant
Conclusion; i.e. when you cannot prove your propo-
sition in the sense in which it was maintained, to
prove it in some other sense; e.g. those who contend
against the efficacy of faith, usually employ that word
in their arguments in the sense of mere belief, unac-
companied with any moral or practical result, but
considered as a mere intellectual process ; and when
they have thus proved their Conclusion, they oppose
it to one in which the word is used in a widely
different sense.
§ 15. The Fallacy of ignoratio elemchi is no where
more common than in protracted controversy, when
one of the parties, after having attempted in vain to
maintain his position, shifts his ground as covertly as
possible to another, instead of honestly giving up the
point. An instance occurs in an attack made on the
system pursued at one of our Universities. The ob-
jectors finding themselves unable to maintain their
charge of the present neglect of Mathematics in that
place, (to which neglect they had attributed the late
general decline in those studies,) they shifted their
ground, and contended that that University was never
famous for Mathematicians ; which not only does not
establish, but absolutely overthrows their own origi-
nal assertion; for if it never succeeded in those pur-
suits, it could not have caused their late decline.
A practice of this nature is common in oral contro-
versy especially; viz. that of combating both your
opponent's Premises alternately, and shifting the
attack from the one to the other, without waiting to
have either of them decided upon before you quit it.
It has been remarked above, that one class of the
propositions that may be, in this Fallacy, substituted
for the one required, is the particular for the universal :
nearly akin to this is the very common case of proving
something to be possible when it ought to have been
proved highly probable; or probable, when it ought to
have been proved necessary; or, which comes to the very
same, proving it to be not necessary, when it should
havé been proved not probable; or improbable, when
it should have been proved impossible. Aristotle, (in
Rhet. book ii.) complains of this last branch of the
Fallacy, as giving an undue advantage to the respon-
dent : many a guilty person owes his acquittal to this;
the jury considering that the evidence brought does
not demonstrate the absolute impossibility of his being
innocent, though perhaps the chances are innumerable
against it.
§ 16. Similar to this case is that which may be
called the Fallacy of objections; i.e. shewing that there
are objections against some plan, theory or system,
and thence inferring that it should be rejected ; when
that which ought to have been proved, is, that there
are more, or stronger objections against the receiving
than the rejecting of it. This is the main, and almost
universal Fallacy of infidels, and is that of which
men should be first and principally warned. This is
also the stronghold of bigoted anti-innovators, who
oppose all reforms and alterations indiscriminately;
for there never was, nor will be, any plan executed
or proposed, against which strong and even unan-
swerable objections may not be urged ; so that unless
the opposite objections be set in the balance on the
other side, we can never advance a step. “There
are objections," said Dr. Johnson, “ against a plenum,
The very same Fallacy indeed is employed on the
other side, by those who are for overthrowing what-
ever is established as soon as they can prove an
objection against it, without considering whether more
and weightier objections may not lie against their own
schemes: but their opponents have this decided advan-
tage over them, that they can urge with great plausi-
bility, “we do not call upon you to reject at once
whatever is objected to, but merely to suspend your
judgment and not come to a decision as long as there
are reasons on both sides:” now since there always will
be reasons on both sides, this non-decision is practically
the very same thing as a decision in favour of the
existing state of things ; the delay of trial becomes
equivalent to an acquittal.”
§ 17. Another form of ignoratio elemchi, which is
also rather the most serviceable on the side of the
respondent, is, to prove or disprove some part of that
which is required, and dwell on that, suppressing all
the rest. * > :
Thus, if a University is charged with cultivating
only the mere elements of Mathematics, and in reply
a list of the books studied there is produced, should
even any one of those books be not elementary, the
charge is in fairness refuted; but the Sophist may
then earnestly contend that some of those books are
elementary ; and thus keep out of sight the real
question, viz. whether they are all so.
Hence the danger of ever advancing more than can
be well maintained ; since the refutation of that will
often quash the whole : a guilty person may often
escape by having too much laid to his charge : so he
may also by having too much evidence against him,
i.e. some that is not in itself satisfactory: thus, a
prisoner may sometimes obtain acquittal by shewing
that one of the witnesses against him is an infamous
informer and spy; though perhaps if that part of the
evidence had been omitted, the rest would have been
sufficient for conviction.
Cases of this nature might very well be referred
also to the Fallacy formerly mentioned, of inferring
the Falsity of the Conclusion from the Falsity of a
Premiss, which indeed is very closely allied to the
present Fallacy: the real question is “whether or
not this Conclusion ought to be admitted ;” the Sophist
confines himself to the question, “whether or not
it is established by this particular argument ;” leaving it
to be inferred by the audience, if he has carried his
point as to the latter question, that the former is
thereby decided. - -
§ 18. It will readily be perceived that nothing is
less conducive to the success of the Fallacy in question
than to state clearly, in the outset, either the propo-
sition you are about to prove, or that which you ought
to prove ; it answers best to begin with the Premises,
and to introduce a pretty long chain of argument before
you arrive at the Conclusion. The careless hearer
takes for granted, at the beginning, that this chain
* “ Not to resolve, is to resolve.” Bacon.
How happy it is for mankind that in the most momentous con-
cerns of life their decision is generally formed for them by
external circumstances; which thus saves them not only from
the perplexity of doubt and the danger of delay, but also from
the pain of regret, since we acquiesce much more cheerfully in
that which is unavoidable. -
1, O G. I. C.
229
are intended for serious conviction, when they are Chap. V.
Logic, will lead to the Conclusion required; and by the time
thoroughly exposed. There are several different kinds S-V-
\-v-' you are come to the end, he is ready to take for
granted that the Conclusion which you draw is the one
required ; his idea of the question having gradually
become indistinct. This Fallacy is greatly aided by
the common practice of suppressing the Conclusion and
leaving it to be supplied by the hearer, who is of
course less likely to perceive whether it be really that
“ which was to be proved,” than if it were distinctly
stated. The practice therefore is at best suspicious ;
and it is better in general to avoid it, and to give and
require a distinct statement of the Conclusion intended.
§ 19. Before we dismiss the subject of Fallacies, it
may not be improper to mention the just and inge-
nious remark, that Jests are Fallacies; i. e. Fallacies so
palpable as not to be likely to deceive any one, but
yet bearing just that resemblance of argument which
is calculated to amuse by the contrast; in the same
manner that a parody does, by the contrast of its levity
with the serious productiou which it imitates. There
is indeed something laughable even in Fallacies which
of joke and raillery, which will be found to corres-
pond with the different kinds of Fallacy: the pun (to
take the simplest and most obvious case) is evidently
a mock argument founded on a palpable equivocation
of the middle term ; and the rest in like manner will
be found to correspond to the respective Fallacies, and
to be imitations of serious argument. It is probable
indeed that all jests, sports, or games, (Tatētaº) pro-
perly so called, will be found, on examination, to be
imitative of serious transactions : but to enter fully
into this subject would be unsuitable to the present
occasion.
We shall conclude the consideration of this subject
with some general remarks on the legitimate province
of Reasoning, and on its connection with Inductive
philosophy, and with Rhetoric : on which points much
misapprehension has prevailed, tending to throw
obscurity over the design and use of the Science under
consideration. *
230 L iO G. I. C.
IE S S A Y •,
ON THE
~s PROVINCE OF REASONING.
their inaccurate expressions. This inaccuracy seems Essay on
chiefly to have arisen from a vagueness in the use of .
the word induction, which is sometimes emploved to ºf
3. ploy Reasoning."
designate the process of investigation and of collect- f
Logic. Logic being concerned with the theory of Reasoning
<-N-" it is evidently necessary, in order to take a correct
view of this Science, that all misapprehensions should
be removed, relative to the occasions on which the
Reasoning process is employed, the purposes it has
in view, and the limits within which it is confined.
Simple and obvious as such questions may appear
to those who have not thought much on the subject,
they will appear on further consideration to be in-
volved in much perplexity and obscurity, from the
vague and inaccurate language of many popular
writers. To the confused and incorrect notions that
prevail respecting the Reasoning process, may be
traced most of the common mistakes respecting the
Science of Logic, and much of the unsound and un-
philosophical argumentation which is so often to be
met with in the works of ingenious writers.
These errors have been incidentally adverted to in
the foregoing part of this article; but it may be
desirable, before we dismiss the subject, to offer on
these points some further remarks, which could not
have been there introduced without too great an in-
terruption to the developement of the system. Little
or nothing indeed remains to be said that is not
implied in the principles which have been already laid
down; but the results and applications of those prin-
ciples are liable in many instances to be overlooked if
not distinctly pointed out. These supplementary
observations will neither require, nor admit of, so
systematic an arrangement as has hitherto been
arrived at, as they will be such as are suggested
principally by the objections and mistakes of those
who have misunderstood, partially, or entirely, the
nature of the Logical system. -
Of Induction.
§ 1. Much has been said by some writers of the
superiority of the Inductive to the Syllogistic method
of seeking truth, as if the two stood opposed to each
other; and of the advantage of substituting the Orga-
non of Bacon for that of Aristotle, &c. &c. which indi-
cates a total misconception of the nature of both.
There is, however, the more excuse for the confusion
of thought which prevails on this subject, because
eminent Logical writers have treated or at least
have appeared to treat of Induction as a distinct
kind of argument from the Syllogism ; which if it
were, it certainly might be contrasted with the Syllo-
gism : or rather the whole Syllogistic theory would
fall to the ground, since one of the very first prin-
ciples it establishes, is that all Reasoning, on whatever
subject, is one and the same process, which may be
clearly exhibited in the form of Syllogisms. It is
hardly to be supposed, therefore, that this was the
meaning of those writers; though it must be admitted
that they have countenanced the error in question, by
ing facts; sometimes the deducing of an inference from
those facts. The former of these processes (i. e. that
of observation and experiment) is undoubtedly distinct
from that which takes place in the Syllogism; but
then it is not a process of argument; the latter again is
an argumentative process; but then it is, like all
other arguments, capable of being Syllogistically
expressed. And hence Induction has come to be
regarded as a distinct kind of argument from the
Syllogism. This Fallacy cannot be more concisely
or clearly stated, than in the technical form with
which we may now presume our readers to be familiar.
Induction is distinct from Syllogism :
Induction is a process of Reasoning ; therefore
There is a process of Reasoning distinct from Syl-
logism.
Here, “Induction” which is the middle term, is
used in different senses in the two Premises.
In the process of Reasoning by which we deduce,
from our observation of certain known cases, an in-
ference with respect to unknown ones, we are employ-
ing a Syllogism in Barbara with the major” Premiss
suppressed ; that being always substantially the same,
as it asserts that “what belongs to the individual or
individuals we have examined, belongs to the whole
class under which they come :” e. g. from an exami-
nation of the history of several tyrannies, and finding
that each of them was of short duration, we con-
clude that “the same is likely to be the case with all
tyrannies;” the suppressed major Premiss being easily
supplied by the hearer; viz. “ that what belongs to
the tyrannies in question is likely to belong to all.”
Induction, therefore, so far forth as it is an argu-
ment, may of course he stated Syllogistically ; but so
far forth as it is a process of inquiry with a view to
obtain the Premises of that argument, it is of course
out of the province of Logic. Whether the Induction
(in this last sense) has been sufficiently ample, i. e.
takes in a sufficient number of individual cases,—
whether the character of those cases has been correctly
ascertained—and how far the individuals we have
examined are likely to resemble, in this or that circum-
stance, the rest of the class, &c. &c. are points that
require indeed great judgment and caution ; but this
judgment and caution are not to be aided by Logic,
because they are, in reality, employed in deciding
whether or not it is fair and allowable to lay down your
Premises ; i. e. whether you are authorized or not,
to assert that “what is true of the individuals you
----sº-
* Not the minor, as Aldrich represents it.
L () G. I. C. 23.1
instance above, if the conclusion had been drawn, not Essay on
respecting tyrannies in general, but respecting this or ºf
that tyranny, that it was not likely to be lasting, each Reasoning.
Logic. have examined, is true of the whole class:” and that
S-N-1 this, or that is true of those individuals. Now, the
rules of Iłogic: have nothing to do with the truth; or
falsity, of the Premises; but merely teach us to decide
(not, whether the Premises are fairly laid down, but)
whether the Conclusior-fºllows fairly from the Premises
Orº not: . ..
- *.
Whether the Premises, may fairly be assumed; or
not, is a point, which cannot be decided without a
competent knowledge of the nature of the subject, e.g.
in Natural Philosophy; in which the circumstances
which in any case affect the result, are usually far
more chearly ascertained, a single instance is often
accounted a sufficient Induction: e. g. having once
ascertained that an individual magnet-will attractiron,
we are authorized to conclude that this property is
universal : in the affairs of human life, a much fuller
Induction is required; as in the former example. In
short the degree of evidence for any proposition, we
originally assume as a Premiss, (whether the expressed,
or the suppressed one) is not to be learned from Logic,
nor indeed from any one distinct Science; but is the
province of whatever Science furnishes the subject
matter of your argument. None but a Politician can
judge rightly of the degree of evidence of a proposi-
tion, in Politics; a Naturalist, in Natural History,
&c. &c. e. g. from examination of many horned
animals; as sheep, cows, &c. a Naturalist finds that
they have cloven feet; now his skill as a Naturalist
is to be shewn in judging whether these animals
are likely to resemble in the form of their feet all
other horned animals; and it is the exercise of this
judgment, together with the examination of indivi-
duals, that constitutes what is usually meant by the
Inductive process ; which is that by which we gain new
truths, and which is not connected with Logic ; being
not what is strictly called Reasoning, but Investigation.
But when this major Premiss is granted him, and is
combined with the minor, viz. that the animals he has
examined have cloven feet, then he draws the conclusion
Logically : viz. that “ the feet of all horned animals
are cloven.” Again, if from several times meeting
with ill-luck on a Friday, any one concluded that
Friday, universally; is an unlucky day, one would
object to his Induction ; and yet it would not be, as an
argument; illogical; since the conclusion follows fairly,
if you grant his implied Premiss, that the events which
happened on those particular Fridays are such as must
happen on all Fridays;” but we should objeet to his
laying down this Premiss; and therefore should justly
say that his Induction was faulty, though his argument
Was: Correot.
And here it may be remarked that the ordinary rule
for fair argument, viz. that in an Enthymeme the
suppressed Premiss should be always the one of whose
truth least doubt can exist, is not observed in Induc-
tion; for the Premiss which is usually the more doubt-
ful of the two, is, in that, the major ; it being in few
cases quite certain that the individuals respecting
which some point has been ascertained are to be fairly
regarded as a sample of the whole class ; the major
Premiss, nevertheless is seldom expressed, for the
reason just given, that it is easily understood, as being
mutatis mutandis, the same in every Induction.
What has been said of Induction will equally apply
to Example, which differs from it only in having a
singular instead of a general conclusion : e. g. in the
of the eases addueed to prove this, would have been
called: an Example:
On the Discovery of Truth.
§ 2. Whether, it is by a process of Reasoning that
New Truths are brought to light, is a question, which
seems to be decided in the negative by what has beer,
already said, though many eminent writers, seem to
have taken for granted the affirmative. It is perhaps,
in a great measure, a dispute concerning the use of
words ; , but it is not for that reason either uninterest-
ing or unimportant, since an inaccurate use of lam-
guage may often, in matters of Science, lead, to con-
fusion of thought, and to erroneous conclusions. And in
the present instance much of the undeserved contempt
which has been bestowed on the Logical system may
be traced to this source ; for when any one has laid
down that “Reasoning is important in the discovery
of Truth,” and that “Logic is of no service in the dis-
covery of Truth,” each of which propositions is true
in a certain sense of the terms employed, but not in
the same sense; he is naturally led to conclude that
there are processes of Reasoning to which the Syllo-
gistic theory does not apply, and of course to mis-
conceive altogether the nature of the Science.
In maintaining the negative side of the above ques-
tion, three things, are to be premised: first, that it is
not contended that Discoveries of any kind of Truth
can be made (or at least are usually made) without
Reasoning ; only that Reasoning is not the whole of
the process, nor the whole of that which is important
therein : secondly, that Reasoning shall be taken in
the sense, not of every exercise of the Reason, but of
Argumentation, in which we have all along used it,
and in which it has been defined by all the Logical
writers, viz. “from certain granted propositions to
infer another proposition as the consequence of them:”
thirdly, that by a “ New Truth,” be understood
Something neither expressly nor virtually asserted
before, not implied and involved in any thing already
known.
To prove then this point demonstratively becomes
in this manner perfectly easy; for since all Reasoning
(in the sense above defined) may be resolved into
Syllogisms; and since even the objectors to Logic
make it a subject of complaint, that in a Syllogism the
Premises do virtually assert the Conclusion, it follows:
at once that no New Truth (as above defined) can be
elicited by any process of Reasoning.
It is on this ground indeed, that the justly celebrated
author of the Philosophy of Rhetoric objects to the
Syllogism altogether, as necessarily involving a petitio
principii; an objection which, of course, he would not
have been disposed to bring forward, had he perceived
that, whether well or ill founded, it lies against all
arguments whatever.
Had he been aware that a Syllogism is no distinct
kind of argument otherwise than in form, but is, in
fact, any argument whatever stated regularly and at
full length, he would have obtained a more correct
view of the object of all Reasoning, which is merely to
expand and unfold: the assertions wrapt up, as it were,
and implied in those with which we set out, and to
232
L O G I. C.
Logic.
bring a person to perceive and acknowledge the full
force of that which he has admitted,—to contemplate
it in various points of view,--to admit in one shape
what he has already admitted in another, and to give
up and disallow whatever is inconsistent with it.
Nor is it always a very easy task even to bring
before the mind the several bearings, the various
applications,—of any one proposition. A common
term comprehends several, often numberless individu-
als, and these often, in some respects, widely differing
from each other; and no one can be, on each occasion
of his employing such a term, attending to and fixing
his mind on each of the individuals, or even of the
species so comprehended. It is to be remembered
too, that both Division and Generalization are in a
great degree arbitrary; i. e. that we may both divide
the same genus on several different principles, and
may refer the same species to several different classes,
according to the nature of the discourse and drift of
the argument; each of which classes will furnish a
distinct middle term for an argument, according to
the question : e. g. if we wished to prove that “a
horse feels,” (to adopt an ill-chosen example from
the above writer,) we might refer it to the genus
“ animal ;” to prove that “ it has only a single
stomach,” to the genus of “ non-ruminants ;” to
prove that it is “likely to degenerate in a very cold
climate,” we should class it with “ original produc-
tions of a hot climate, &c. &c.” Now each of these,
and numberless others to which the same thing might
be referred, are implied by the very term “ horse ;”
yet it cannot be expected that they all be at once
present to the mind whenever that term is uttered.
Much less, when instead of such a term as that, we
are employing terms of a very abstract, and perhaps
complex signification,” as “government, justice, &c.”
The ten Categoriest or Predicaments which Aris-
totle and other Logical writers have treated of, being
certain general heads or summa-genera, to one or
more of which every term may be referred, serve the
purpose of marking out certain tracks, as it were,
which are to be pursued in searching for middle terms
in each argument respectively; it being essential that
we should generalize on a right principle, with a view
to the question before us; or, in other words, that we
should abstract that portion of any object presented
to the mind, which is important to the argument in
hand. There are expressions in common use which
have a reference to this caution ; such as “ this is a
question, not as to the nature of the object, but the
magnitude of it :” “this is a question of time, or of
place, &c." i.e. “ the subject must be referred to this
or to that Category.”
With respect to the meaning of the terms in ques-
tion, “Discovery,” and “New Truth;” it matters not
whether we confine ourselves to the narrowest sense,
* On this point there are some valuable remarks in the Philo-
sophy of Rhetoric itself, book iv. ch. vii.
t The Categories enumerated by Aristotle, are obota, Trégov,
trotov, trgbati, trów, tróte, kefor6a, #xeiv, troićiv, trgazelv; which are
usually rendered, as adequately as perhaps they can be in our
language, Substance, Quantity, Quality, Relation, Place, Time,
Situation, Possession, Action, Suffering. The catalogue has been
by some writers enlarged, as it is evident may easily be done by
subdividing some of the heads ; and by others curtailed, as it is
no less evident that all may ultimately be referred to the two
heads of Substance and Attribute, or in the language of some
Logicians, Accident.
or admit the widest, provided we do but distinguish ;
there certainly are two kinds of “New Truth, and of
“Discovery,” if we take those words in the widest
sense in which they are ever used. First, such Truths
as were, before they were discovered, absolutely
unknown, being not implied by any thing we previ-
ously knew, though we might perhaps suspect them
as probable; such are all matters of fact strictly so
called, when first made known to one who had not
any such previous knowledge, as would enable him to
ascertain them a priori ; i.e. by Reasoning ; as if we
inform a man that we have a colony at Botany Bay;
or that the earth is at such a distance from the sun ;
or that platina is heavier than gold. The communi-
cation of this kind of knowledge is most usually and
most strictly called information: we gain it from obser-
vation; and from testimony ; no mere internal workings
of our own minds, (except when the mind itself is the
very object to be observed,) or mere discussions in
words, will make these known to us; though there is
great room for sagacity in judging what testimony to
admit, and forming conjectures that may lead to profit-
able observation, and to experiments with a view to it.
The other class of Discoveries is of a very different
nature ; that which may be elicited by Reasoning, and
consequently is implied in that which we already know,
we assent to on that ground, and not from observa-
tion or testimony : to take a Geometrical truth upon
trust, or to attempt to ascertain it by observation, would
betray a total ignorance of the nature of the Science.
In the longest demonstration the Mathematical teacher
seems only to lead us to make use of our own stores,
and point out to us how much we had already admit-
ted ; and in the case of many Ethical propositions,
we assent at first hearing, though perhaps we had
never heard or thought of the proposition before ; so
also do we readily assent to the testimony of a respect-
able man who tells us that our troops have gained a
victory; but how different is the nature of the assent
in the two cases. In the latter, we are ready to thank
the person for his information, as being such as no
wisdom or learning would have enabled us to ascer-
tain ; in the former we usually exclaim “ very true !”
“ that is a valuable and just remark; that never struck
me before ''' implying at once our practical ignorance
of it, and also our consciousness that we possess, in
what we already know, the means to ascertain the
truth of it.
To all practical purposes, indeed, a Truth of this
description may be as completely unknown to a man
as the other; but as soon as it is set before him, and
the argument by which it is connected with his pre-
vious notions is made clear to him, he recognises it as
something conformable to, and contained in his former
belief. l
It is not improbable that Plato's doctrine of Remi-
niscence arose from a hasty extension of what he had
observed in this class, to all acquisition of knowledge
whatever. - -
His Theory of ideas served to confound together
matters of fact respecting the nature of things, (which
may be perfectly new to us,) with propositions relating
to our own motions, and modes of thought ; (or to speak
perhaps more correctly, our own arbitrary signs) which
propositions must be contained and implied in those
very complex notions themselves ; and whose truth
is a conformity, not to the nature of things, but to
Essay on
the Pro-
vince of
Reasoning,
L O G. I. C. 233
(i.e. cannot recollect) the name of some person or ; O ſº .
place ; perhaps we even try to think of it, but in vain; ...;
at last some one reminds us, and we instantly recog- Reasoning.
mise it as the one we wanted to recollect ; it may be S-N-2
Logic. our own hypothesis. Such are all propositions in pure
Q-y—’ Mathematics, and many in Ethics, viz. those which
involve no assertion as to real matters of fact. It has
been rightly remarked, that Mathematical propositions
are not properly true or false in the same sense as any
proposition respecting real fact is so called ; and
hence the truth (such as it is) of such propositions is
necessary and eternal ; since it amounts only to this,
that any complex notion which you have arbitrarily
framed, must be exactly conformable to itself. The
proposition that “the belief in a future state, com-
bined with a complete devotion to the present life, is
not consistent with the character of prudence,” would
be not at all the less true if a future state were
a chimera, and prudence a quality which was
nowhere met with ; nor would the truth of the
Mathematician's conclusion be shaken, that “ circles
are to each other as the squares of their diameters,”
should it be found that there never had been a circle
or a square, conformable to the definition, in rerum
*aturd.
The Ethical proposition just instanced, is one of
those which Locke calls “trifling,”
dicate is merely a part of the complex idea implied by
the subject; and he is right, if by “trifling " he means
that it gives not, strictly speaking, any information ;
but he should consider that to remind a man of what
he had not, and what he would have thought of, may
be, practically, as valuable as giving him information ;
and that most propositions in the best sermons, and
all in pure Mathematics, are of the description which
he censures.
It is indeed rather remarkable that he should speak so
often of building Morals into a demonstrative Science,
and yet speak so slightingly of those very propositions
to which we must absolutely confine ourselves, in order
to give to Ethics even the appearance of such a
Science; for the instant you come to an assertion
respecting a matter of fact, as that “men (i.e. actually
existing men) are bound to practise virtue,” or “ are
liable to many temptations,” you have stepped off the
ground of strict demonstration, just as when you pro-
ceed to practical Geometry.
But to return : it is of the utmost importance to
distinguish these two kinds of Discovery of Truth;
to the former, as we have said, the word “infor-
mation” is most strictly applied ; the communication
of the latter is more properly called “ instruction.”
We speak of the usual practice ; for it would be
going too far to pretend that writers are uniform and
consistent in the use of these, or of any other term.
We say that the Historian gives us information res-
pecting past times; the Traveller, respecting foreign
countries : on the other hand, the Mathematician gives
‘instruction in the principles of his Science; the Moralist
instructs us in our duties ; and we generally use the
expressions “a well-informed man,” and “a well-
instructed man,” in a sense conformable to that which
has been here laid down. However, let the words be
used as they may, the things are evidently different,
and ought to be distinguished. It is a question com-
paratively unimportant, whether the term “Discovery”
shall or shall not be extended to the eliciting of those
Truths, which, being implied in our previous know-
ledge, may be established by mere strict Reasoning.
Similar verbal questions indeed might be raised res-
pecting many other cases: e.g. one has forgotten
WOL. I.
because the Pre-
asked, was this in our mind or not? The answer is, that
in one sense it was, and in another sense, it was not.
Or, again, suppose there is a vein of metal on a man's
estate which he does not know of ; is it part of his
possessions or not? and when he finds it out and works
it, does he then acquire a new possession or not Cer-
tainly not, in the same sense as if he has a fresh estate
bequeathed to him, which he had formerly no right to ;
but to all practical purposes, it is a new possession.
This case indeed may serve as an illustration of the one
we have been considering ; and in all these cases, if
the real distinction be understood, the verbal question
will not be of much consequence. To use one more
illustration; Reasoning has been aptly compared
to the piling together blocks of stone ; on each of
which, as on a pedestal, a man can raise himself a
small, and but a small, height above the plain ; but
which, when skilfully built up, will form a flight of
steps, which will raise him to a great elevation. Now .
(to pursue this analogy) when the materials are all
ready to the builder's hand, the blocks ready dug and
brought, his work resembles one of the two kinds of
Discovery just mentioned, viz. that to which we have
assigned the name of instruction : but if his materials
are to be entirely, or in part, provided by himself-if he
himself is forced to dig fresh blocks from the quarry,
this corresponds to the other kind of Discovery.
We have hitherto spoken of the employment of
argument in the establishment of those hypothetical
Truths (as they may be called) which relate only to
our own abstract notions; it is not, however, meant
to be insinuated that there is no room for Reasoning
in the establishment of a matter of fact ; but the other
class of Truths have first been treated of, because in
discussing subjects of that kind the process of Rea-
soning is always the principal, and often the only thing
to be attended to, if we are but certain and clear as to
the meaning of the terms ; whereas, when assertions
respecting real existence are introduced, we have the
additional and more important business of ascertain-
ing and keeping in mind the degree of evidence for
those facts, since, otherwise, our Conclusions could not
be relied on, however accurate our Reasoning ; but,
undoubtedly, we may by Reasoning arrive at matters
of fact, if we have matters of fact to set out with as data;
only that it will very often happen that “from certain
facts,” as Campbell remarks, “we draw only probable
Conclusions;” because the other Premiss introduced
(which he overlooked) is only probable. He observed
that in such an instance, for example, as the one lately.
given, we infer from the certainty that such and such
tyrannies have been short-lived, the probability that
others will be so ; and he did not consider that there
is an understood Premiss which is essential to the
argument; (viz. that all tyrannies will resemble those
we have already observed) which being only of a pro-
bable character, must attach the same degree of un-
certainty to the Conclusion. An individual fact is not
unfrequently elicited by skilfully combining, and
Reasoning from, those already known ; of which
many curious cases occur in the detection of crimi-
nals by officers of justice and Barristers, who acquire
by practice such dexterity in that particular depart-
2 I
234
L O G I C.
what has been said, that in Mathematics, and in such Essay on
Ethical propositions as we were lately speaking of, the Pro:
we do not allow the possibility of any but a Logical ...
g º . . e. , º, Reasoning.
Discovery ; i. e. no proposition, of that class, can be
Logic, ment, as sometimes to draw the right Conclusion from
S-N-" data, which might be in the possession of others, with-
out being applied to the same use. In all cases of
the establishment of a general fact from Induction,
S-N-1
that general fact (as has been formerly remarked) is
ultimately established by Reasoning ; e. g. Bakewell,
the celebrated cattle-breeder, observed, in a great
number of individual beasts, a tendency to fatten
readily, and in a great number of others the absence
of this constitution; in every individual of the former
description, he observed a certain peculiar make, though
they differed widely in size, colour, &c. Those of
the latter description differed no less in various points,
but agreed in being of a different make from the
others : these facts were his data ; from which, com-
bining them with the general principle that Nature
is steady and uniform in her proceedings, he Logically
drew the conclusion that beasts of the specified make
have universally a peculiar tendency to fattening : but
then his principal merit consisted in making the ob-
servations, and in so combining them as to abstract
from each a multitude of cases, differing widely in
many respects, the circumstances in which they all
agreed; and also in conjecturing skilfully how
far those circumstances were likely to be found
in the whole class; the making such observa-
tions, and still more the combination, abstraction,
and judgment employed, are what men commonly
mean (as was above observed) when they speak of
Induction ; and these operations are certainly distinct
from Reasoning. The same observations will apply
to numberless other cases, as, for instance, to the
Discovery of the law of “vis inertial,” and the other
principles of Natural Philosophy. -
But to what class, it may be asked, should be re-
ferred the Discoveries thus made 3 All would agree
in calling them, when first ascertained, “New Truths,”
in the strictest sense of the word ; which would seem
to imply their belonging to the class which may be
called, by way of distinction, “Physical Discoveries:”
and yet their being ultimately established by Reason-
ing, would seem, according to the foregoing rule, to
refer them to the other class, viz. what may be called
“ Logical Discoveries ;” since whatever is established
by Reasoning, must have been contained and virtually
asserted in the Premises. In answer to this, it is to
be observed, that they certainly do belong to the
latter class, relatively, to a person who is in possession
of the data; but to him who is not, they are New Truths
of the other class; for it is to be remembered, that
the words “ Discovery" and “ New Truths” are ne-
cessarily relative : there may be a proposition which
is to one person absolutely known ; to another, (viz.
one to whom it has never occurred, though he is in
possession of all the data from which it may be proved)
it will be, when he comes to perceive it, by a process
of instruction, what we have called a Logical Discovery;
to a third, (viz. one who is ignorant of these data,) it
will be absolutely unknown, and will have been, when
made known to him, a perfectly and properly New
Truth, a piece of information,--a Physical Discovery
as we have called it. To the Philosopher, therefore, who
arrives at the Discovery by Reasoning from his obser-
vations, and from established principles combined with
them, the Discovery is of the former class; to the mul-
titude, probably of the latter, as they will have been most
likely not possessed of all his data. It follows from
true, which was not implied in the definitions we set
out with, which are the first principles: for since
these propositions do not profess to state any matter
of fact, the only Truth they can possess, consists in
conformity to the original principles; to one, there-
fore, who knows these principles, such propositions
are Truths already implied, since they may be de-
veloped to him by Reasoning, if he is not defective
in the discursive faculty; to one who does not under-
stand those principles, (i. e. is not master of the defini-
tions) such propositions are absolutely unmeaning. On
the other hand, propositions relating to matters of fact,
may be, indeed, implied in what he already knew ;
(as he who knows the climate of the Alps, the Andes,
&c. &c. has virtually admitted the general fact, that
“ the tops of mountains are comparatively cold;")
but as these possess an absolute and physical Truth,
they may also be absolutely “ new,” their Truth not
being implied by the mere terms of the propositions.
The truth or falsity of any proposition concerning a
triangle, is implied by the meaning of that and of the
other Geometrical terms; whereas, though one may
understand (in the ordinary sense of that word) the
full meaning of the terms, “ moon” and “inhabited,”
and of all the other terms in the language, he cannot
thence be certain that the moon is, or is not, inhabited.
It has probably been the source of much perplexity
that the term “true” has been applied indiscriminately
to two such different classes of propositions. The
term definition is used with the same laxity; and
much confusion has thence resulted.
Such Definitions as the Mathematical, must imply
every attribute that belongs to the thing defined;
because that thing is merely our meaning, which
meaning the Definition lays down ; whereas, real
Substances, having an independent existence, may
possess innumerable qualities (as Locke observes)
not implied by the meaning we attach to their names,
or, as Locke expresses it, by our ideas of them.
“Their nominal essence (to use his language) is not
the same as their real essence :” whereas the nominal
essence, and the real essence, of a circle, &c. are the
same. A Mathematical Definition, therefore, cannot
properly be called true, since it is not properly a
proposition, (any more than an article in a Dictionary,)
but merely an explanation of the meaning of a term.
Perhaps in Definitions of this class, it might be better
to substitute (as Aristotle usually does) the imperative
mood for the indicative ; thus bringing them into the
form of postulates; for the Definitions and the pos-
tulates in Mathematics differ in little or nothing but
the form of expression : e.g. “let a four-sided figure,
of equal sides and right angles, be called a square,”
would clearly imply that such a figure is conceivable,
and that the writer intended to employ that term to
signify such a figure ; which is precisely all that is
intended to be asserted. If, indeed, a Mathematical
writer mean to assert that the ordinary meaning of
the term is that which he has given, that, certainly,
is a proposition, which must be either true or false ;
but in defining a new term, the term indeed may be
ill-chosen and improper, or the Definition may be
self-contradictory, and consequently unintelligible;
L O G. I. C.
235
Logic: , but the words, “true,” and “false,” do not apply.
*-Y" The same may be said of what are called nominal
Definitions of other things, i. e. those which merely
explain the meaning of the word ; viz. they can be
true or false only when they profess (and so far as they
profess) to give the ordinary and established meaning
of the term. But those which are called real Defini-
tions, viz. which unfold the nature of the thing,
(which they may do in various degrees,) to these the
epithet “true " may be applied; and to make out
such a Definition will often be the very end (not as in
Mathematics the beginning) of our study.
In Mathematics there is no such distinction between
nominal and real Definition ; the meaning of the term,
and the nature of the thing, being one and the same :
so that no correct Definition whatever of any Mathema-
tical term can be devised, which shall not imply
every thing which belongs to the term.
When it is asked, then, whether such great Dis-
coveries, as have been made in Natural Philosophy,
were accomplished, or can be accomplished by Rea-
soning 2 the inquirer should be reminded, that the
question is ambiguous ; it may be answered in the
affirmative, if by “Reasoning" is meant to be in-
cluded the assumption of Premises ; to the right
performance of that work, is requisite, not only in
many cases, the ascertainment of facts, and of the
degree of evidence for doubtful propositions, (in which
observation and experiment will often be indispensa-
ble,) but also a skilful selection and combination of
known facts and principles; such as implies, amongst
other things, the exercise of that powerful abstraction
which seizes the common circumstances—the point
of agreement—in a number of, otherwise dissimilar,
individuals : it is in this that the greatest genius is
shewn. But if “Reasoning ” be understood in the
limited sense in which it is usually defined, then we
must answer in the negative; and reply that such
Discoveries are made by means of Reasoning combined
with other operations. -
In the process we have been speaking of, there is
much Reasoning throughout ; and thence the whole
has been carelessly called a “Process of Reasoning.”
It is not, indeed, any just ground of complaint that
the word Reasoning is used in two senses; but that
the two senses are perpetually confounded together:
and hence it is that some Logical writers fancied that
Reasoning (viz. that which Logic treats of) was the
method of discovering Truth ; and that so many other
writers have accordingly complained of Logic for not
accomplishing that end, urging that “ Syllogism
(i. e. Reasoning; though they overlooked the co-
incidence) never established any thing that is, strictly
speaking, unknown to him who has granted the
Premises : and proposing the introduction of a certain
** rational Logic " to accomplish this purpose; i. e.
to direct the mind in the progress of investigation.
Supposing that some such system could be devised—
that it could even be brought into a Scientific form,
(which he must be more sanguine than Scientific who
expects,) that it were of the greatest conceivable
utility, and that it should be allowed to bear the
name of “Logic,” since it would not be worth while
to contend about a word, still it would not, as these
writers seem to suppose, have the same object proposed
with the Aristotelian Logic ; nor be in any respect a
rival to that system. A plough may be a much more
ingenious and valuable instrument than a flail, but it tº. Oºl
never can be substituted for it.
e Pro-
vince of
Those Discoveries of general laws of Nature, &c. Reasoning.
of which we have been speaking, being of that cha-
racter which we have described by the name of
“Logical Discoveries,” to him who is in possession of
all the Premises from which they are deduced; but being,
to the multitude (who are unacquainted with many of
those Premises) strictly “ New Truths;" hence it is,
that men in general give to the general facts, and to
them, most peculiarly, the name of Discoveries; for to
themselves they are such, in the strictest sense ; the
Premises from which they were inferred being not
only originally unknown to them, but frequently
remaining unknown to the very last : e. g. the general
conclusion concerning cattle, which Bakewell made
known, is what most Agriculturists (and many others
also) are acquainted with ; but the Premises he set
out with, viz. the facts respecting this, that, and the
other, individual ox, (the ascertainment of which
facts was his first Discovery) these are what few know,
or care to know, with any exact particularity.
And it may be added, that these discoveries of parti-
cular facts, which are the immediate result of observation,
are, in themselves, uninteresting and insignificant,
till they are combined so as to lead to a grand general
result ; those who on each occasion watched the
motions, and registered the date of a comet, little
thought, perhaps, themselves, what magnificent results
they were preparing the way for. So that there is an
additional cause which has confined the term Discovery
to these grand general conclusions ; and, as was just
observed, they are, to the generality of men, per-
fectly New Truths in the strictest sense of the word,
not being implied in any previous knowledge they
possessed. Very often it will happen, indeed, that
the conclusion thus drawn will amount only to a
probable conjecture ; which conjecture will dictate to
the inquirer such an experiment, or course of experi-
ments, as will fully establish the fact ; thus Sir H.
Davy, from finding that the flame of hydrogen gas
was not communicated through a long slender tube,
conjectured that a shorter, but still slenderer tube,
would answer the same purpose; this led him to try
the experiments, in which, by continually shortening
the tube, and at the same time lessening its bore,
he arrived at last at the wire-gauze of his safety-
lamp.
It is to be observed also, that whatever credit is con-
veyed by the word “Discovery,” to him who is regarded
as the author of it, is well deserved by those who
skilfully select and combine known Truths, (especially
such as have been long and generally known,) so as to
elicit important, and hitherto unthought-of, conclu-
sions; theirs is the master mind; apxitektovik) @povnats'
whereas men of very inferior powers may sometimes,
by immediate observation, discover perfectly new
facts, empirically, and thus be of service in furnishing
materials to the others; to whom they stand in the
same relation (to recur to a former illustration) as the
brickmaker or stonequarrier, to the architect. It is
peculiarly creditable to A. Smith, and to Mr. Malthus,
that the data from which they drew such important
Conclusions had been in every ones hands for cen-
turies.
As for Mathematical Discoveries, they (as we have
before said) must always be of the description to which
2 I 2
236
L O G. I. C.
Logic. we have given the name of “Logical Discoveries;”
\-N-7
since to him who properly comprehends the meaning
of the Mathematical terms, (and to no other are the
Truths themselves, properly speaking, intelligible,)
those results are implied in his previous knowledge,
since they are Logically deducible therefrom. It is
not, however, meant to be implied that Mathematical
Discoveries are effected by pure Reasoning, and by
that singly. For though there is not here, as in Phy-
sics, any exercise of judgment as to the degree of evi-
dence of the Premises, nor any experiments and obser-
vations, yet there is the same call for skill in the
selection and combination of the Premises in such a
manner as shall be best calculated to lead to a new,
that is, unperceived and unthought-of Conclusion.
In following, indeed, and taking in a demonstration,
nothing is called for but pure Reasoning; but the
assumption of Premises is not a part of Reasoning, in
the strict and technical sense of that term. Accord-
ingly, there are many who can follow a demonstration,
or any other train of argument, who would not suc-
ceed well in framing one of their own.*
For both kinds of Discovery then, the Logical, as
well as the Physical, certain operations are requisite,
beyond those which can fairly be comprehended under
the strict sense of the word “Reasoning;” in the Logical,
is required a skilful selection and combination of known
Truths ; in the Physical we must employ, in addition
(generally speaking) to that process, observation and
experiment. It will generally happen, that in the
study of Nature, and, universally, in all that relates to
matters of fact, both kinds of investigation will be
united ; i.e. some of the facts or principles you
reason from as Premises, must be ascertained by
observation ; or, as in the case of the safety-lamp,
the ultimate Conclusion will need confirmation from
experience; so that both Physical and Logical Dis-
covery will take place in the course of the same
process: we need not, therefore, wonder, that the
two are so perpetually confounded. In Mathematics,
on the other hand, and in great part of the discussions
relating to Ethics and Jurisprudence, there being no
room for any Physical Discovery whatever, we have
only to make a skilful use of the propositions in our
possession, to arrive at every attainable result.
The investigation, however, of the latter class of
subjects differs in other points also from that of the
former ; for setting aside the circumstance of our
having, in these, no question as to facts, no room for
observation,-there is also a considerable difference in
what may be called the process of Logical investigation;
the Premises on which we proceed being of so different
a nature in the two cases.
To take the example of Mathematics, the defini-
tions, which are the principles of our Reasoning, are
very few, and the axioms still fewer; and both are,
for the most part, laid down, and placed before the
student in the outset; the introduction of a new defi-
nition or axiom, being of comparatively rare occur-
rence, at wide intervals, and with a formal statement;
besides which, there is no room for doubt concerning
either. On the other hand, in all Reasonings which
regard matters of fact, we introduce, almost at every
step, fresh and fresh propositions (to a very great
* Hence the Student must not confine himself to this passive
kind of employment, if he would become truly a Mathematician,
number) which had not been elicited in the course of
our Reasoning, but are taken for granted; viz. facts
and laws of Nature which are here the principles of
our Reasoning, and maxims, or “ elements of belief,”
which answer to the axioms in Mathematics. If, at
the opening of a Treatise, for example, on Chemistry,
on Agriculture, on Political Economy, &c. the author
should make, as in Mathematics, a formal statement
of all the propositions he intended to assume, as
granted throughout the whole work, both he and his
readers would be astonished at the number : and, of
these, many would be only probable, and there would
be much room for doubt as to the degree of proba-
bility, and for judgment, in ascertaining that degree.
Moreover, Mathematical axioms are always em-
ployed precisely in the same simple form ; e. g. the
axiom that “things equal to the same, are equal to
one another,” is cited, whenever there is need, in those
very words; whereas the maxims employed in the other
class of subjects, admit of, and require, continual mo-
difications in the application of them : e.g. “the sta-
bility of the laws of Nature,” which is our constant
assumption in inquiries relating to Natural Philosophy,
assumes many different shapes, and in some of them,
does not possess the same absolute certainty as in
others: e.g. when from having always observed a cer-
tain sheep ruminating, we infer, that this individual
sheep will continue to ruminate, we assume that “the
property which has hitherto belonged to this sheep,
will remain unchanged;” when we infer the same pro-
perty of all sheep, we assume that “the property which
belongs to this individual, belongs to the whole
species:" if, on comparing sheep with some other
kinds of horned animals, and finding that all agree in
ruminating, we infer that, “ all horned animals rumi-
nate,” we assume that “ the whole of a genus or class
are likely to agree in any point wherein many species
of that genus agree ;” or in other words, “ that if
one of two properties, &c. has often been found ac-
companied by another, and never without it, the
former will be universally accompanied by the latter;”
now all these are merely different forms of the
maxim, that “ nature is uniform in her operations;”
which, it is evident, varies in expression in almost
every different case where it is applied, and admits of
every degree of evidence, from absolute moral cer-
tainty, to mere conjecture.
The same may be said of an infinite number of
principles and maxims appropriated to, and employed
in each particular branch of study. Hence, all such
Reasonings are, in comparison of Mathematics, very
complex; requiring so much more than that does,
beyond the process of merely deducing the Conclusion
Logically, from the Premises ; so that it is no wonder
that the longest Mathematical demonstration should be
so much more easily constructed and understood, than
a much shorter train of just Reasoning concerning
real facts. The former has been aptly compared to a
long and steep, but even and regular, flight of steps,
which tries the breath, and the strength, and the
perseverance, only; while the latter resembles a short,
but rugged and uneven, ascent up a precipice, which
requires a quick eye, agile limbs, and a firm step ; and
in which we have to tread now on this side, now on
that ; ever considering, as we proceed, whether this
projection will afford room for our foot, or whether
some loose stone may not slide from under us.
Essay on
the Pro-
vince of
Reasoning.
L O G. I. C.
237
Logic. As for those Ethical and Legal Reasonings which
S-a-’ were lately mentioned, as in some respects resembling
those of Mathematics, (viz. such as keep clear of
all assertions respecting facts,) they have this ulf-
subject of which we would predicate something, to a Essay on
class to which that predicate will (affirmatively or the Prº:
negatively) apply; in the other we seek to find com- *:::::.
prehended, in the subject of which we have predicated ſº
ference; that not only men are not so completely
agreed respecting the maxims and principles of Ethics
and Law, but the meaning also of each term cannot
be absolutely, and for ever, fixed by an arbitrary de-
finition ; on the contrary, a great part of our labour
consists in distinguishing accurately the various senses
in which men employ each term, ascertaining which
is the most proper, and taking care to avoid con-
founding them together.
Of Inference and Proof.
§ 3. Since it appears, from what has been said, that
universally a man must possess something else besides
the Reasoning faculty, in order to apply that faculty
properly to his own purpose, whatever that purpose
may be ; it may be inquired whether some theory
could not be made out, respecting those “ other
operations,” and “ intellectual processes distinct from
Reasoning, which it is necessary for us sometimes to
employ in the investigation of truth;”* and whether
rules could not be laid down for conducting them.
Something has, indeed, been done in this way by
more than one writer; and more might probably be
accomplished by one who should fully comprehend
and carefully bear in mind the principles of Logic,
properly so called ; but it would hardly be possible
to build up any thing like a regular Science, respecting
these matters, such as Logic is, with respect to the
theory of Reasoning. It may be useful, however, to
observe, that these “ other operations” of which we have
been speaking, and which are preparatory to the exercise
of Reasoning, are of two kinds, according to the nature
of the end proposed; for Reasoning comprehends In-
ferring and Proving ; which are not two different
things, but the same thing regarded in two different
points of view : (like the road from London to York,
and the road from York to London,) he who
infers, f proves; and he who proves, infers ; but the
word “infer" fixes the mind first on the Premiss,
and then on the Conclusion ; the word “prove,” on
the contrary, leads the mind from the Conclusion to
the Premiss. Hence, the substantives derived from
these words respectively, are often used to express
that which, on each occasion, is last in the mind ;
Inference being often used to signify the Conclusion,
(i. e. Proposition inferred) and Proof, the Premiss.
We say also “How do you prove that " and “What
do you infer from that 2" which sentences would not
be so properly expressed if we were to transpose
those verbs. One might, therefore, define Proving,
“ the assigning of a reason or argument for the sup-
port of a given proposition ;” and “ Inferring,” the
“ deduction of a Conciusion from given Premises.” In
the one case our Conclusion is given, (i.e. set before us)
and we have to seek for arguments; in the other, our
Premises are given, and we have to seek for a Con-
clusion ; i. e. to put together our own propositions,
and try what will follow from them ; or, to speak
more Logically, in the one case, we seek to refer the
* D. Stewart.
+ We mean, of course, when the word is understood to imply
correct Inference.
something, some other term to which that predicate M
had not been before applied. Each of these is a
definition of Reasoning.
To infer, then, is the business of the Philosopher ;
to prove, of the Advocate; the former, from the great
mass of known and admitted truths, wishes to elicit
any valuable additional truth whatever, that has
been hitherto unperceived; and, perhaps, without
knowing, with certainty, what will be the terms of
his Conclusion. Thus the Mathematician, e. g. seeks
to ascertain what is the ratio of circles to each other,
or what is the line whose square will be equal to a
given circle: the Advocate, on the other hand, has a
proposition put before him, which he is to maintain
as well as he can ; his business, therefore, is to find
middle terms, (which is the inventio of Cicero ;) the
Philosopher's, to combine and select known facts, or
principles, suitably for gaining from them conclusions
which, though implied in the Premises, were before
unperceived ; in other words, for making “Logical
Discoveries.” Such are the respective preparatory
processes in these two branches of study. They are
widely different ;-they arise from, and generate, very
different habits of mind; and require a very different
kind of training and precept. The Lawyer, or Con-
troversialist, or, in short, the Rhetorician in general,
who is, in his own province, the most skilful, may be
but ill-fitted for Philosophical investigation, even
where there is no observation wanted ; – when the
facts are all ready ascertained for him. And again,
the ablest Philosopher may make an indifferent dis-
putant; especially, since the arguments which have
led him to the conclusion, and have, with him, the
most weight, may not, perhaps, be the most power-
ful in controversy. The commonest fault, however,
by far, is to forget the Philosopher or Theologian,
and to assume the Advocate, improperly. It is there-
fore of great use to dwell on the distinction between
these two branches : as for the bare process of Rea-
soning, that is the same in both cases; but the pre-
paratory processes which are requisite in order to
employ Reasoning profitably, these we see branch off
into two distinct channels. In each of these undoubt-
edly, useful rules may be laid down ; but they should
not be confounded together. Bacon has chosen
the department of Philosophy, giving rules in his
Organon, (not only for the conduct of experiments to
ascertain new facts, but also for the selection and
combination of known facts and principles,) with a
view of obtaining valuable Inferences; and it is proba-
ble that a system of such rules is what some writers
mean (if they have any distinct meaning) by their
proposed “Logic.” In the other department, precepts
have been given by Aristotle and other Rhetorical
writers, as a part of their plan. How far these pre-
cepts are to be considered as belonging to the present
system,--whether “method" is to be regarded as a
part of Logic,+whether the matter of Logic is to be
included in the system,--whether Bacon's is properly
to be reckoned a kind of Logic ; all these are merely
verbal questions relating to the extension, not of the
Science, but of the name. The bare process of . Rea-
soning, i. e. deducing a Conclusion from Premises,
238
I, O G. I. C.
to shew the application of it to all Reasoning, in Essay on
pointing out the difference between Verbal and Real the Pº;
Questions, and the probable origin of Campbell's Kºng.
mistake ; for to trace any error to its source, will - |
Logic. must ever remain a distinct operation from the assump-
S-N-'tion of Premises, however useful the rules may be
that have been given, or may be given, for conducting
this fatter process, and others connected with it ; and
however properly such rules may be subjoined to the
precepts of that system to which the name of Logic is
applied in the narrowest sense. Such rules as we now
allude to may be of eminent service ; but they must
always be, as we have before observed, comparatively
vague and general, and incapable of being built up into
a regular demonstrative theory like that of the Syllo-
gism ; to which theory they bear much the same
relation as the principles and rules of Poetical and
Rhetorical criticism, to those of Grammar ; or those
of practical Mechanics, to strict Geometry. We find
no fault with the extension of a term ; but we would
suggest a caution against confounding together, by
means of a common name, things essentially different :
and above all we deprecate the sophistry of striving to
depreciate what is called “ the school Logic,” by per-
petually contrasting it with systems with which it has
nothing in common but the name; and whose object
is essentially different.
It is not a little remarkable that writers whose
expressions tend to confound together, by means of
a common name, two branches of study which have
nothing else in common, (as if they were two different
plans for attaining one and the same object,) have them-
selves complained of one of the effects of this confu-
sion, viz. the introduction, early in the career of Aca-
demical Education, of a course of Logic; under which
name, they observe “men now universally comprehend
the works of Locke, Bacon, &c.” which, as is justly
remarked, are unfit for beginners. Now this would not
have happened, if men had always kept in mind the
meaning or meanings of each name they used. And it
may be added, that, however justly the word Logic may
be thus extended, we have no ground for applying to the
Aristotelian Logic, the remarks above quoted respect-
ing the Baconian; which the ambiguity of the word,
if not carefully kept in view, might lead us to do.
Grant that Bacon's work is a part of Logic ; it no more
follows from the unfitness of that for learners, that
the elements of the theory of Reasoning should be
withheld from them, than it follows that the elements
of Euclid, and common Arithmetic, are unfit for boys,
because Newton's Principia, which also bears the title
of Mathematical, is above their grasp. Of two branches
of study which bear the same name, or even of two
parts of the same branch, the one may be suitable to
the commencement, the other to the close, of the
Academical career.
At whatever period of that career it may be proper
to introduce the study of such as are usually called
Metaphysical writers, it may be safely asserted, that
those who have had the most experience in the busi-
ness of giving instruction in Logic, properly so called,
together with other branches of knowledge, prefer
and generally pursue the plan of letting their pupils
enter on that study next in order, after the elements
of Mathematics.
Of Verbal and Real Questions.
§ 4. The ingenious author of the Philosophy of Rhe-
toric having maintained, or rather assumed, that Logic
is applicable to Verbal controversy alone, there may be
an advantage, though it has been our aim throughout
often throw more light on the subject in hand than
can be obtained if we rest satisfied with merely detect-
ing and refuting it.
Every Question that can arise, is in fact a Question
whether a certain Predicate is or is not applicable to
a certain subject; and whatever other account may
be given by any writer of the nature of any matter of
doubt or debate, will be found, ultimately, to resolve
itself into this. But sometimes the Question turns on
the meaning and extent of the terms employed; some-
times on the things signified by them. If it be made
to appear therefore, that the opposite sides of a certain
Question may be held by persons not differing in their
opinion of the matter in hand, then that Question may
be pronounced Verbal, as depending on the different
senses in which they respectively employ the terms.
If on the contrary it appears that they employ the
terms in the same sense, but still differ as to the appli-
cation of one of them to the other, then it may be
pronounced that the Question is Real,—that they differ
as to the opinions they hold of the things in Question.
If, for instance, two persons contend whether
Augustus deserved to be called a great man, then if
it appeared that the one included under the term
“ great,” disinterested patriotism, and on that ground
excluded Augustus from the class, as wanting in that
quality, and that the other also gave him no credit for
that quality, but understood no more by the term
“great,” than high intellectual qualities, energy of
character, and brilliant actions, it would follow that
the parties did not differ in opinion except as to the
use of a term, and that the Question was Verbal. If
again it appeared that the one did give Augustus credit
for such patriotism as the other denied him, both of
them including that idea in the term great, then the
Question would be Real. Either kind of Question, it is
plain, is to be argued according to Logical principles;
but the middle terms employed would be different ;
and for this reason among others it is important to
distinguish Verbal from Real controversy. In the
former case, e.g. it might be urged with truth, that
the common use of the expression “great and good’’
proves that the idea of good is not implied in the
ordinary sense of the word great ; an argument which
could have, of course, no place in deciding the other
Question.
It is by no means to be supposed that all Verbal
Questions are trifling and frivolous; it is often of the
highest importance to settle correctly the meaning of
a word, either according to ordinary use or according
to the meaning of any particular writer, or class
of men ; but when Verbal Questions are mistaken for
Real, much confusion of thought and unprofitable
wrangling will be generally the result. Nor is it
always so easy and simple a task, as might at first sight
appear, to distinguish them from each other : for
several objects to which one common name is applied
will often have many points of difference, and yet that
name may perhaps be applied to them all in the
same sense, and may be fairly regarded as the genus
they come under, if it appear that they all agree in
what is designated by that name, and that the differ-
ences between them are in points not essential to the
L O G. I. C. 239
and species were some real THINGs, existing inde- .# Oſt
Ile PI'O-
pendently of our conceptions and expressions, and that, vince of
as in the case of singular terms, there is some real Reasoning.
Logic. character of the genus. A cow and a horse differ in
S-N-7 many respects, but agree in all that is implied by the
term “ quadruped,” which is therefore applicable to
both in the same sense. So also the houses of the
ancients differed in many respects from ours, and
their ships, still more ; yet no one would contend that
the terms “house” and “ship,” as applied to both, were
ambiguous, or that oikos might not fairly be rendered
house, and va6s, ship : because the essential charac-
teristic of a house is, not its being of this or that form
or materials, but its being a dwelling for men ; these
therefore would be called two different kinds of houses;
and consequently the term “house” would be applied
to each, without any equivocation, in the same sense :
and so in the other instances. On the other hand,
two or more things may bear the same name, and may
also have a resemblance in many points, and may from
that resemblance have come to bear the same name,
and yet if the circumstance which is essential to each
be wanting in the other, the term may be pronounced
ambiguous : e. g. the word “Priest” is applied to
the ministers of the Jewish and of the Pagan religions,
and also to those of the Christian ; and doubtless the
term is so used in consequence of their being both
ministers, (in some sort) of religion. Nor would every
difference that might be found between the Priests of
different religions constitute the term ambiguous,
provided such differences were non-essential to the
idea suggested by the word Priest ; as e. g. the Jewish
Priest served the true God, and the Pagan, false Gods :
this is a most important difference, but does not con-
stitute the term ambiguous, because neither of these
circumstances is implied and suggested by the term
‘Iepels, which accordingly was applied both to Jewish
and Pagan Priests. But the term ‘Iepe is does seem to
have implied the office of offering sacrifice, atoning for
the sins of the people, and acting as mediator between
man and the object of his worship ; and accordingly
that term is never applied to any one under the Chris-
tian system, except to the one great Mediator. The
Christian ministers not having that office which was
implied as essential in the term ‘Iepeys, were never
called by that name, but by that of Tpeogºtcp.os. It
may be concluded therefore, that the term Priest is
ambiguous, as corresponding to the terms ‘Iepe is and
Tpeofºtepos respectively, notwithstanding that there
are points in which these two agree. These therefore
should be reckoned, not two different kinds of Priests,
but Priests in two different senses; since, (to adopt
the phraseology of Aristotle,) the definition of them
so far forth as they are Priests, would be different.
It is evidently of much importance to keep in mind
the above distinctions, in order to avoid, on the one
hand, stigmatizing as Verbal controversies, what in
reality are not such, merely because the Question
turns on the applicability of a certain Predicate to a
certain subject ; or on the other hand, falling into the
opposite error of mistaking words for things, and judg-
ing of men's agreement or disagreement in opinion in
every case, merely from their agreement or disagree-
ment in the terms employed.
Of Realism.
§ 5. Nothing has a greater tendency to lead to the
mistake just noticed, and thus to produce undetected
Verbal Questions and fruitless Logomachy, than the
prevalence of the notion of the Realists, that genus
individual corresponding to each, so in common terms
also there is something corresponding to each, which
is the object of our thoughts when we employ any
such term.* Few, if any indeed, in the present day
avow and maintain this doctrine ; but those who are
not especially on their guard, are perpetually sliding
into it unawares. Nothing so much conduces to this
as the transferred and secondary use of the words
“ same,” “one and the same,” “identical, &c.” when
it is not clearly perceived and carefully borne in mind
that they are employed in a secondary sense, and that
more frequently even than in the primary. Suppose
e.g. a thousand persons are thinking of the sun, it
is evident it is one and the same individual object on
which all these minds are employed; so far all is clear:
but suppose all these persons are thinking of a tri-
angle ; not any individual triangle, but triangle in
general; and considering perhaps the equality of its
angles to two right angles; it would seem as if in this
case also, their minds were all employed on ‘‘ one and
the same” object ; and this object of their thoughts,
it may be said, cannot be the mere word triangle, but
that which is meant by it; nor again, can it be every-
thing that the word will apply to, for they are not
thinking of triangles, but of one thing : those who
do not acknowledge that this “one thing” has an
existence independent of the human mind, are in
general content to tell us by way of explanation, that
the object of their thoughts is the abstract “idea” of
a triangle ; an explanation which satisfies, or at least
silences many, though it may be doubted whether they
very clearly understand what sort of a thing an idea is,
which may thus exist in a thousand different minds at
once, and yet be “one and the same.”
The fact is, that “unity” and “ sameness” are in
such cases employed, not in the primary sense, but to
denote perfect similarity. When we say that ten thou-
sand different persons have all “one and the same "
idea in their minds, or are all of “one and the same'
opinion, we mean no more than that they are all
thinking exactly alike , when we say that they are all
in the “ same” posture, we mean that they are all
placed alike : and so also they are said all to have the
“ same " disease when they are all diseased alike.
The origin of this secondary sense of the words,
“same,” “one,” “identical,” &c. (an attention to which
would clear away an incalculable mass of confused
Reasoning and Logomachy,) is easily to be traced to the
use of language and of other signs, for the purpose of
mutual communication. If any one utters the “ one
single” word “ triangle,” and gives “one single”.
definition of it; each person who hears him forms a
certain notion in his own mind, not differing in any
respect from that of each of the rest ; they are said
therefore to have all ‘‘ one and the same” notion,
because, resulting from, and corresponding with, that
which is in the primary sense “ one and the same”
expression ; and there is said to be “one single” idea
of every triangle, (considered merely as a triangle,)
because one single name or definition is equally appli-
cable to each. In like manner all the coins struck by
9
* A doctrine commonly, but falsely, attributed to Aristotle,
who expressly contradicts it. Categories, repl ovatas.
240
L O G. I. C.
Logic, the same single die, are said to have “one and the
S-N-" same” impression, merely because the one descrip-
tion which suits one of these coins will equally suit
any other that is exactly like it.
It is not intended to recommend the disuse of the
words “ same,” “ identical,” &c. in this transferred
sense ; which, if it were desirable, would be utterly
impracticable; but merely, a steady attention to the
ambiguity thus introduced, and watchfulness against Essay on
the errors thence arising. The difficulties and per- ...;
plexities which have involved the questions respecting Reasoning.
personal identity, among others, may be traced prin- ~’
cipally to the neglect of this caution. But the further
consideration of that question would be unsuitable to
the subject of this article.
R H E TO R. I. C.
IN T R O DU C T O R Y S E CT I O N.
Of Rhetoric various definitions have been given
\–2–2 by different writers; who, however, seem not so
much to have disagreed in their conceptions of the
nature of the same thing, as to have had different
things in view while they employed the same terms.
Not only the word Rhetoric itself, but also those used
in defining it, have been taken in various senses; as
may be observed with respect to the word “Art” in
Cic. de Orat. where a discussion is introduced as to
the applicability of that term to Rhetoric ; manifestly
turning on the different senses in which “Art” may
be understood.
To enter into an examination of all the definitions
that have been given, would lead to much uninterest-
ing and uninstructive verbal controversy. It is suffi-
cient to put the reader on his guard against the
common error of supposing that a general term has
some real object, properly corresponding to it, inde-
pendent of our conceptions ;-that, consequently,
some one definition is to be found which will com-
prehend every thing that is rightly designated by
that term ; –and that all others must be erroneous ;
whereas in fact it will often happen, as in the present
instance, that both the wider, and the more restricted
sense of a term, will be alike sanctioned by use,
(the only competent authority ;) and that the conse-
quence will be a corresponding variation in the defi-
nitions employed, none of which perhaps may be
fairly chargeable with error, though none can be
framed that will apply to every acceptation of the
term.
It is evident that in its primary signification,
Rhetoric had reference to public Speaking alone, as
its etymology implies : but as most of the rules for
speaking are of course applicable equally to writing,
an extension of the term naturally took place ; and
we find even Aristotle, the earliest systematic writer
on the subject whose works have come down to
us, including in his Treatise such compositions as
were not intended to be publicly recited.* And even
as far as relates to Speeches, properly so called, he
takes, in the same Treatise, at one time a wider, and
at another a more restricted view of the subject ;
inclu 'ing under the term Rhetoric, in the opening of
his work, nothing beyond the finding of topics of
Perstasion, as far as regards the matter of what is
* Arist, Rhet, book iii.
WQL. I. -
spoken ; and afterwards embracing the consideration Introduc-
of Style, Arrangement, and Delivery.
tory
The invention of Printing, by extending the sphere Section.
of operation of the writer, has of course contributed
to the extension of those terms which in their primary
signification had reference to Speaking alone. Many
objects are now accomplished through the medium of
the Press, which formerly came under the exclusive
province of the Orator; and the qualifications re-
quisite for success are so much the same in both cases,
that we apply the term “Eloquent” as readily to a
Writer as to a Speaker; though etymologically con-
sidered it could only belong to the latter. Indeed
“Eloquence” is often attributed even to such com-
positions, e. g. Historical works, as have in view an
object entirely different from any that could be pro-
posed by an Orator; because some part of the rules
to be observed in Oratory, or rules analogous to these,
are applicable to such compositions. Conformably
to this view therefore, some writers have spoken of
Rhetoric as the Art of Composition, universally ; or,
with the exclusion of Poetry alone, as embracing all
Prose composition.
A still wider extension of the province of Rhetoric
has been contended for by some of the ancient writers;
who thinking it necessary to include, as belonging to
the Art, every thing that could conduce to the attain-
ment of the object proposed, introduced into their
systems Treatises on Law, Morals, Politics, &c. on
the ground that a knowledge of these subjects was
requisite to enable a man to speak well on them ; and
even insisted on Virtue,” as an essential qualification
of a perfect Orator, because a good character, which
can in no way be so surely established as by deserving
it, has great weight with the audience.
These notions are combated by Aristotle ; who
attributes them either to the ill-cultivated understand-
ing (ārauðevoča) of those who maintained them, or to
their arrogant and pretending disposition, &Nagoveda;
i. e. a desire to extol and magnify the Art they pro-
fessed. In the present day, the extravagance of such
doctrines is so apparent to most readers, that it would
not be worth while to take Imuch pains in refuting
them. It is worthy of remark however, that the very
same erroneous view is, even now, often taken of
Logic, (as was remarked under that article ;) which
* See Quinctilian.
2 K 24 |
\-2-’
242 R H E T O R. I. C.
have occasion for rules of a different kind from those Introduc.
employed in its discovery. Accordingly, when we ... tory
remarked, in the article above alluded to, that it is a Section. j
common fault, for those engaged in Philosophical and STV"
Rhetoric. has been considered by some as a kind of system of
S-N-2 universal knowledge, on the ground that argument
may be employed on all subjects, and that no one
can argue well on a subject which he does not under-
stand ; and which has been complained of by others
as not supplying any such universal instruction as its
unskilful advocates have placed within its province;
such as in fact no one Art or System can possibly
afford. -
The error is precisely the same in respect of
Rhetoric and of Logic; both being instrumental arts;
and, as such, applicable to various kinds of subject-
matter, which do not properly come under them.
So judicious an author as Quinctilian would not
have failed to perceive, had he not been carried away
by an inordinate veneration for his own Art, that as
the possession of building materials is no part of the
art of Architecture, though it is impossible to build
without materials, so, the knowledge of the subjects
on which the Orator is to speak, constitutes no part of
the art of Rhetoric, though it be essential to its suc-
cessful employment; and that though virtue and the
good reputation it procures, add materially to the
Speaker's influence, they are no more to be, for that
reason, considered as belonging to the Orator, as such,
than wealth, rank, or a good person, which manifestly
have a tendency to produce the same effect.
In the present day however, the province of
Rhetoric, in the widest acceptation that would be
reckoned admissible, comprehends all “ Composition
in Prose ;” in the narrowest sense, it would be limited
to “ Persuasive Speaking.”
We propose in the present article to adopt a middle
course between these two extreme points ; and to
treat of Argumentative Composition generally, and ex-
clusively; considering Rhetoric (in conformity with
our original plan, and with the very just and philoso-
phical view of Aristotle) as an off-shoot from Logic.
It was remarked in our article on that Science, that
Reasoning may be considered as applicable to two
purposes, which we ventured to designate respectively
by the terms “ Inferring, and Proving ;” i. e. the
ascertainment of the truth by investigation, and the
establishment of it to the satisfaction of another : and
it was there remarked, that Bacon, in his Organon, had
laid down rules for the conduct of the former of these
processes, and that the latter belonged to the province
of Rhetoric : and it was added, that to infer is to be
regarded as the proper office of the Philosopher ; –to
prove, of the Advocate. It is not however to be un-
derstood that Philosophical works are to be excluded
from the class to which Rhetorical rules are appli-
cable; for the Philosopher who undertakes, by writing
or speaking, to convey his notions to others, assumes
for the time being, the character of Advocate of the
doctrines he maintains ; the process of investigation
must be supposed completed, and certain conclusions
arrived at by that process, before he begins to impart
his ideas to others in a treatise or lecture; the object
of which must of course be to prove the justness of
those conclusions. And in doing this, he will not
always find it expedient to adhere to the same course
of reasoning by which his own discoveries were ori-
ginally made; other arguments may occur to him after-
wards, more clear or more concise, or better adapted to
the understanding of those he addresses. In explain-
ing therefore, and establishing the trath, he may often
Theological inquiries, to forget their own peculiar
office, and assume that of the Advocate, improperly,
this caution is to be understood as applicable to the
process of forming their own opinions; not, as excluding
them from advocating by all fair arguments, the con-
clusions at which they have arrived by candid inves-
tigation. But if this candid investigation do not take
place in the first instance, no pains that they may
bestow in searching for arguments, will have any ten-
dency to ensure their attainment of truth. If a man,
begins (as is too plainly a frequent mode of proceed-"
ing) by hastily adopting or strongly leaning to some
opinion, which suits his inclination, or which is sanc-
tioned by some authority that he blindly venerates,
and then studies with the utmost diligence, not as
an Investigator of Truth, but as an Advocate labour-
ing to prove his point, his talents and his researches,
whatever effect they may produce in making converts
to his notions, will avail nothing in enlightening his
own judgment and securing him from error.
Composition however, of the Argumentative kind,
may be considered (as has been above stated) as
coming under the province of Rhetoric. And this
view of the subject is the less open to objection, in-
asmuch as it is not likely to lead to discussions that
can be deemed superfluous, even by those who may
choose to consider Rhetoric in the most restricted
sense, as relating only to “ Persuasive Speaking ;”
since it is evident that Argument must be, in most cases
at least, the basis of Persuasion.
We propose then, to treat first, and principally, of
the Discovery of Arguments, and of their Arrange-
ment; secondly, to lay down some Rules respecting
the excitement and management of the Passions, with
a view to the attainment of any object proposed,—-
principally, Persuasion in the strict sense, i. e. the
influencing of the Will ; thirdly, to offer some re-
marks on Style ; and fourthly, to treat of Elocution.
It may be expected that before we proceed to treat
of the Art in question, we should present our readers
with a sketch of its history. Little however is re-
quired to be said on this head, because the present is
not one of those branches of study in which we can
trace with interest a progressive improvement from
age to age. It is one, on the contrary, to which more
attention appears to have been paid, and in which
greater proficiency is supposed to have been made, in
the earliest days of Science and Literature, than at any
subsequent period. Among the ancients, Aristotle,
who was the earliest, may safely be pronounced to be
also the best, of the systematic writers on Rhetoric.
Cicero is hardly to be reckoned among the number ;
for he delighted so much more in the practice than
in the theory of his art, that he is perpetually drawn off
from the rigid Philosophical analysis of its principles,
into discursive declamations, always eloquent indeed,
and often highly interesting, but adverse to regularity
of system, and frequently as unsatisfactory to the
practical student as to the Philosopher. He abounds
indeed with excellent practical remarks, though the
best of them are scattered up and down his works
with much irregularity; but his precepts, though of
great weight, as being the result of experience, are
R H E T O R I C. 243
Rhetoric
\-y-
not often traced up by him to first principles ; and
we are frequently left to guess, not only on what basis
his rules are grounded, but in what cases they are
applicable. Of this latter defect a remarkable instance
will be hereafter cited. -
Quinctilian is indeed a systematic writer; but
cannot be considered as having much extended the
Philosophical views of his predecessors in this depart-
ment. He possessed much good sense, but this was
tinctured with pedantry;-with that àNagovela as Aristo-
tle calls it, which extends to an extravagant degree the
province of the Art which he professes. A great part of
his work indeed is a Treatise on education generally, in
the conduct of which he was no mean proficient ; for
such was the importance attached to public Speaking,
even long after the downfall of the Republic had cut
off the Orator from the hopes of attaining, through
the means of this qualification, the highest political
importance, that he who was nominally a Professor
of Rhetoric, had in fact the most important branches
of instruction intrusted to his care.
Many valuable maxims however are to be found in
this author; but he wanted the profundity of thought,
and power of analysis which Aristotle possessed.
The writers on Rhetoric among the ancients whose
works are lost, seem to have been numerous; but
most of them appear to have confined themselves to
a very narrow view of the subject; and to have been
occupied, as Aristotle complains, with the minor
details of style and arrangement, and with the sophis-
tical tricks and petty artifices of the Pleader, instead
of giving a masterly and comprehensive sketch of the
essentials.
Among the moderns, few writers of ability have
turned their thoughts to the subject; and but little
has been added, either in respect of matter, or of
system, to what the ancients have left us. It were
most unjust however to leave unnoticed Dr. Camp-
bell's Philosophy of Rhetoric : a work which does
not enjoy indeed so high a degree of popular
favour as Dr. Blair's, but is incomparably superior
to it, not only in depth of thought and ingenious
original research, but also in practical utility to
the student. The title of Dr. Campbell's work has
perhaps deterred many readers, who had concluded it
to be more abstruse and less popular in its character
than it really is. Amidst much however that is
readily understood by any moderately intelligent
reader, there is much also that calls for some exertion
of thought, which the indolence of most readers re-
fuses to bestow. And it must be owned that he also
in some instances perplexes his readers by being per-
plexed himself, and bewildered in the discussion of
questions through which he does not clearly see his
way. His great defect, which not only leads him
into occasional errors, but leaves many of his best
ideas but imperfectly developed, is his ignorance and
utter misconception of the nature and object of Logic,
on which some remarks were made in our article on
that Science. Rhetoric being in truth an off-shoot of
Logic, that Rhetorician must labour under great
disadvantages who is not only ill-acquainted with that
system, but also utterly unconscious of his deficiency.
From a general view of the history of Rhetoric,
two questions naturally suggest themselves, which on
examination will be found very closely connected
together : 1st, what is the cause of the careful and
extensive cultivation, among the ancients, of an Art Introduc-
which the moderns have comparatively neglected ;
and 2dly, whether the former or the latter are to be
regarded as the wiser in this respect;-in other words,
whether Rhetoric be worth any diligent cultivation.
With regard to the first of these questions, the
answer generally given is that the nature of the Govern-
ment in the ancient democratical States caused a
demand for public speakers, and for such speakers as
should be able to gain influence not only with edu-
cated persons in dispassionate deliberation, but with
a promiscuous multitude ; and accordingly it is re-
marked, that the extinction of liberty brought with it,
or at least brought after it, the decline of Eloquence;
as is justly remarked (though in a courtly form) by
the author of the dialogue on Oratory, which passes
under the name of Tacitus : “Quid enim opus est longis
in Senatu sententiis, cum optimi cito consentiant 2 quid,
nultis apud populum concionibus, cum de Republica non
imperiti et multi deliberent, sed sapientissimus, et unus 2"
This account of the matter is undoubtedly correct
as far as it goes ; but the importance of public speaking
is so great, in our own, and all other countries that are
not under a despotic Government, that the apparent
neglect of the study of Rhetoric seems to require
some further explanation. Part of this explanation
may be supplied by the consideration, that the dif-
ference in this respect between the ancients and our-
selves, is not so great in reality as in appearance.
When the only way of addressing the public was by
orations, and when all political measures were debated
in popular assemblies, the characters of Orator,
Author, and Politician, almost entirely coincided ;
he who would communicate his ideas to the world,
or would gain political power, and carry his legis-
lative schemes into effect, was necessarily a Speaker;
since as Pericles is made to remark by Thucydides,
“one who forms a judgment on any point, but can-
not explain himself clearly to the people, might as
well have never thought at all on the subject.” The
consequence was, that almost all who sought, and all
who professed to give, instruction, in the principles
of Government, and the conduet of judicial proceed-
ings, combined these, in their minds and in their
practice, with the study of Rhetoric, which was
necessary to give effect to all such attainments; and
in time the Rhetorical writers (of whom Aristotle
makes that complaint) came to consider the Science
of Legislation and of Politics in general, as a part of
their own Art. -
Much therefore of what was formerly studied under
the name of Rhetoric is still, under other names, as
generally and as diligently studied as ever.
It cannot be denied however that a great difference,
though less, as we have said, then might at first
sight appear, does exist between the ancients and the
moderns in this point ;--that what is strictly and
properly called Rhetoric, is much less studied, at
least less systematically studied, now, than formerly.
Perhaps this also may be in some measure accounted
for from the circumstances which have been just
noticed. Such is the distrust excited by any suspicion
of Rhetorical artifice, that every speaker or writer
who is anxious to carry his point, endeavours to dis-
own or to keep out of sight, any superiority of
* Thucydides, book ii.
tory
ection,
2 K 2
244 R H E To R I c.
(while that continues to be the case,) prove rather Introduc-
an impediment than a help ; as indeed will be found sºy
Rhetoric, skill; and wishes to be considered as relying rather
S-N-2 on the strength of his cause, and the soundness of his
views, than on his ingenuity and expertness as an
advocate. Hence it is, that even those who have
paid the greatest and the most successful attention to
the study of Composition and of Elocution, are so far
from encouraging others by example or recommenda-
tion to engage in the same pursuit, that they labour
rather to conceal and disavow their own proficiency;
and thus, theoretical rules are decried, even by those
who owe the most to them. Whereas among the
ancients, the same cause, did not, for the reasons
lately mentioned, operate to the same extent; since,
however careful any speaker might be to disown the
artifices of Rhetoric properly so called, he would not be
ashamed to acknowledge himself, generally, a student,
or a proficient in an Art which was understood to
include the elements of Political wisdom.
With regard to the other question proposed, viz.
concerning the utility of Rhetoric, it is to be observed
that it divides itself into two ; 1st, whether Oratorical
skill be, on the whole a public benefit, or evil ; and
2ndly, whether any artificial System of Rules is con-
ducive to the attainment of that skill. The former
of these questions was eagerly debated among the
ancients ; on the latter but little doubt seems to
have existed. With us, on the contrary, the state of
these questions seems nearly reversed. It seems
generally admitted that skill in Composition and in
Speaking, liable as it evidently is, to abuse, is to be
considered, on the whole, as advantageous to the
public ; because that liability to abuse is neither in
this, nor in any other case, to be considered as con-
clusive against the utility of any kind of art, faculty,
or profession ;--because the evil effects of misdirected
power, require that equal powers should be arrayed
on the opposite side ;-and because truth having
an intrinsic superiority over falsehood, may be ex-
pected to prevail when the skill of the contending
parties is equal ; which will be the more likely to
take place, the more widely such skill is diffused.
But many, perhaps most persons, are inclined to the
opinion that Eloquence either in writing or speaking,
is either a natural gift, or at least, is to be acquired
only by practice, and is not to be attained or im-
proved by any system of rules. And this opinion is
favoured not least by those (as has been just ob-
served) whose own experience would enable them
to decide very differently ; and it certainly seems
to be in a great degree practically adopted. Most
persons, if not left entirely to the disposal of chance
in respect of this branch of education, are at least
left to acquire what they can by practice, such as
school or college exercises afford, without much care
being taken to initiate them systematically into the
principles of the Art; and that, frequently, not so
much from negligence in the conductors of education,
as from their doubts of the utility of any such regular
system.
It certainly must be admitted, that rules not con-
structed on broad Philosophical principles, are more
likely to cramp, than to assist the operations of our
faculties;–that a pedantic display of technical skill is
more detrimental in this than in any other pursuit,
since by exciting distrust, it counteracts the very pur-
pose of it;-that a system of rules imperfectly com-
prehended, or not familiarized by practice, will,
in all other Arts likewise;—and that no system can
be expected to equalise men whose natural powers are.
different : but none of these concessions at all inva-
lidate the positions of Aristotle; that some succeed
better than others in explaining their opinions, and
bringing over others to them ; and that, not merely by
superiority of natural gifts, but by acquired habit; and
that consequently if we can discover the causes of this
superior success, the means by which the desired end
is attained by all who do attain it, we shall be in pos-
session of rules capable of general application : Örep
éart, says he, texvils épyov.” Experience so plainly
evinces, what indeed we might naturally be led
antecedently, to conjecture, that a right judgment on
any subject is not necessarily accompanied by skill
in effecting conviction,-nor the ability to diseover
truth, by a facility in explaining it, that it might be
matter of wonder how any doubt should ever have
existed as to the possibility of devising, and the utility
of employing, a System of Rules for “Argumentative
Composition,” generally, distinctfrom any system con-
versant about the subject-matter of each composition,
It is probable that the existing prejudiees on this
subject may be traced in great measure to the im-
perfect or incorrect notions of some writers, who
have either confined their attention to trifling minutiae
of style, or at least have in some respect failed to
take a sufficiently comprehensive view of the principles
of the Art. One distinction especially is to be clearly
laid down and carefully borne in mind by those who
would form a correct idea of those principles; viz.,
the distinction already noticed under the article Logic,
between an Art, and the Art. “An Art of Reasoning"
would imply, “a System of Rules by the observance
of which one may Reason correctly ;” “ the Art of
Reasoning” would imply a System of Rules to which
every one does conform, (whether knowingly, or not)
who reasons correctly : and such is Logic, considered
as an Art.
would imply “ a System of Rules by which a good
Composition may be produced;” “ the Art of Compo-
sition,”—“ such rules as cvery good Composition must
conform to,” whether the author of it had them in
his mind or not. Of the former character appear to
have been (among others) many of the Logical and
Rhetorical Systems of Aristotle's predecessors in those
departments: he himself evidently takes the other and
more Philosophical view of both branches : as appears
(in the case of Rhetoric) both from the plan he sets
out with, that of investigating the causes of the suc-
cess of all who do succeed in effecting conviction, and
from several passages occurring in various parts of
his Treatise, which indicate how sedulously he was on
his guard to conform to that plan. Those who have
not attended to the important distinction just alluded
to, are often disposed to feel wonder, if not weariness,
at his reiterated remarks, that “all men effect per-
suasion either in this way or in that ;” “it is impossible
to attain such and such an object in any other way;”
&c. which doubtless were intended to remind his
readers of the nature of his design ; viz. not, to teach
an Art of Rhetoric but the Art;-not to instruct them
—a
.* Rhet, book i. ch. i.
In like manner “ an Art of Composition”
ection.
- R H E T O R. I. C.
245
Rhetoric. merely how conviction, might be produced but how it
\-N-7 7m tº St.
If this distinction were carefully kept in view by
the teacher and by the learner of Rhetoric, we should
no longer hear complaints of the natural powers
being fettered by the formalities of a System ; since
no such complaint can lie against a System whose
Rules are drawn from the invariable practice of all
who succeed in attaining their proposed object. No
one would expect that the study of Sir Joshua Rey-
nolds's lectures, would cramp the genius of the painter.
No one complains of the Rules of Grammar as fettering
Introduc-
tory
Section.
Chap. I.
Language ; because it is understood that correct use S-V-
is not founded on Grammar, but Grammar upon cor-
rect use. A just system of Logic or of Rhetoric, is
analogous, in this respect, to Grammar.
CHAPTER I.
of THE INVENTION, ARRANGEMENT, AND INTRODUCTION OF ARGUMENTS.
IT has been formerly remarked in our Treatise on
ToGIC, that in the process of Investigation properly so
called, viz. that by which we endeavour to discover
Truth, it must of course be uncertain to him who is en-
tering on that process, what the conclusion will be, to
which his researches will lead ; but that in the pro-
cess of conveying truth to others, by reasoning, (i. e.
that which according to the view we have at present
taken, may be termed the Rhetorical process,) the
conclusion or conclusions which are to be established
must be present to the mind of him who is conduct-
ing the Argument, and whose business is to find
Proofs of a given proposition.
It is evident therefore, that the first step to be
taken by him, is, to lay down distinctly in his own
mind, the proposition or propositions to be proved.
It might indeed at first sight appear superfluous even
to mention so obvious a rule; but experience shows
that it is by no means uncommon for a young or ill-
instructed writer to content himself with such a
vague and indistinct view of the point he is to aim at,
that the whole train of his reasoning is in consequence
affected with a corresponding perplexity, obscurity,
and looseness. It may be worth while therefore to
give some hints for the conduct of this preliminary
process, the choice of propositions. Not, of course
that we are supposing the author to be in doubt what
opinion he shall adopt : the process of Investigation
(which does not fall within the province of Rhetoric)
being supposed to be concluded; but still there will
often be room for deliberation as to the form in which
an opinion shall be stated, and, when several propo-
sitions are to be maintained, in what order they shall
be placed.
On this head therefore we shall proceed to propose
some rules; after having premised (in order to antici-
pate some objections or doubts which might arise)
one remark relative to the object to be effected. This
is of course, what may be called, in the widest sense
of the word, Conviction; but under that term are
comprehended 1st, what is strictly called Instruction ;
and 2ndly, Conviction in the narrower sense; i. e. the
Conviction of those who are either of a contrary
opinion to the one maintained, or who are in doubt
whether to admit or deny it. By Instruction on the
other hand, is commonly meant the Conviction of those
who have neither formed an opinion on the subject,
nor are deliberating whether to adopt or reject the
proposition in question, but are merely desirous of
ascertaining what is the truth in respect of the case
before them. The former are supposed to have before
their minds the terms of the proposition maintained,
and are called upon to consider whether that particular
proposition be true or false; the latter are not sup-
posed to know the terms of the conclusion, but to be
inquiring what proposition is to be received as true.
It is evident that the speaker or writer is, relatively
to these last, (though not to himself,) conducting a
process of Investigation ; as is plain from what has
been said of that subject, in the article Logic.
The distinction between these two objects gives rise
in some points to corresponding differences in the mode
of procedure, which will be noticed hereafter ; these
differences however are not sufficient to require that
Rhetoric should on that account be divided into two
distinct branches, since, generally speaking, though
not universally, the same rules will be serviceable for
attaining each of these objects.
§ 1. The first step is, as we have observed, to lay
down, (in the author's mind,) the proposition or pro-
positions to be maintained, clearly, and in a suitable
form. He who makes a point of observing this rule,
and who is thus brought to view steadily the point he
is aiming at, will be kept clear, in a great degree, of
some common faults of young writers; viz. entering
on too wide a field of discussion, and introducing many
propositions not sufficiently connected; an error which
destroys the unity of the composition. This last error
those are apt to fall into, who place before themselves
a Term instead of a Proposition ; and imagine that be-
cause they are treating of one thing, they are discussing one
question. In an Ethical work, for instance, one may be
treating of virtue, while discussing all or any of these
questions; “Wherein virtue consists 3" “Whence our
notions of it arise "“Whence it derives its obligation ?”
&c., but if these questions were confusedly blended
together, or if all of them were treated of within a short
compass, the most just remarks and forcible argu-
ments would lose their interest and their utility in so
perplexed a composition.
Nearly akin to this fault, is the other just men-
tioned, that of entering on too wide a field for the
length of the work; by which means the writer is
confined to barren and uninteresting generalities ; as
e.g. in general exhortations to virtue, (conveyed, of
course, in very general terms,) in the space of a dis-
course only of sufficient length to give a charac-
teristic description of some one branch of duty, or
246 R H E T O R I. C.
Rhetoric. of some one particular motive to the practice of it.
~~' Unpractised composers are apt to fancy that they
1st, Into Irregular, and Regular, i. e. Syllogisms; Chap. I.
these last into Categorical and Hypothetical ; and the S-V-
shall have the greater abundance of matter, the wider
extent of subject they comprehend ; but experience
shows that the reverse is the fact : the more general
and extensive view will often suggest nothing to the
mind but vague and trite remarks, when upon narrow-
ing the field of discussion, many interesting questions
of detail present themselves. Now a writer who is
accustomed to state to himself precisely, in the first
instance, the conclusions to which he is tending, will
be the less likely to content himself with such as
consist of very general statements; and will often be
led, even where an extensive view is at first proposed,
to distribute it into several branches, and waiving the
discussion of the rest, to limit himself to the full de-
velopement of one or two ; and thus applying, as it
were, a microscope to a small space, will present to
the view much that a wider survey would not have
exhibited.
It may be useful, for one who is about thus to lay
down his propositions, to ask himself these three
guestions : 1st, What is the fact 2ndly, Why (i. e.
from what Cause) is it so ; or, in other words, how
is it accounted for 2 and 3rdly, What Consequence
results from it 2
The last two of these questions, though they will
not in every case suggest such answers, as are strictly
to be called the Cause and the Consequence of the
principal truth to be maintained, may, at least, often
furnish such propositions as bear a somewhat similar
relation to it.
It is to be observed that in recommending the writer
to begin by laying down in his own mind the propo-
sitions to be maintained, it is not meant to be implied
that they are always to be stated first ; that will de-
pend upon the nature of the case, and rules will here-
after be given on that point.
It is to be observed also, that by the words “ Pro-
position” or “Assertion,” throughout this Treatise, is
to be understood some conclusion to be established for
itself; not with a view to an ulterior conclusion: those
propositions which are intended to serve as premises,
being called, in allowable conformity with popular
usage, Arguments ; it being customary to argue in the
enthymematic form, and to call, for brevity's sake,
the expressed premiss of an enthymeme, the argument
by which the conclusion of it is proved.
Of Arguments.
§ 2. Arguments are divided according to several dif-
ferent principles; i.e. logically speaking, there are
several divisions of them. And these cross-divisions have
proved a source of endless perplexity to the Logical and
Rhetorical student, because the writers on those sub-
jects have not been aware of them. Hardly any thing
perhaps has contributed so much to lessen the interest
and the utility of systems of Rhetoric, as the indistinct-
ness hence resulting. When in any subject the mem-
bers of a division are not opposed, but are in fact
members of different divisions crossing each other, it
is manifestly impossible to obtain any clear notion of
the species treated of; nor will any labour or ingenuity
bestowed on the subject be of the least avail, till the
original source of perplexity is removed ;-till, in
short, the cross-division is detected and explained.
Arguments then may be divided,
former into Syllogisms in the first Figure, and in the
other figures, &c. &c.
2ndly, They are frequently divided into “ Moral,”
(or “Probable,") and “Demonstrative,” (or “Ne-
cessary.”)
3rdly, Into “Direct” and “ Indirect,” (or reductio
ad absurdum,) the Deictic and Elenctic of Aristotle.
4thly, Into Arguments from “ Example,” from
“Testimony,” from “Cause to Effect,” from “Ana-
logy,” &c. &c.
It will be perceived on attentive examination, that
several of the different species just mentioned will
occasionally contain each other ; e. g. a probable Argu-
ment may be at the same time a Categorical Argument,
a Direct Argument, and an Argument from Testimony,
&c.; this being the consequence of Arguments having
been divided on several different principles; a circum-
stance so obvious the moment it is distinctly stated,
that we apprehend such of our readers as have not
been conversant in these studies, will hardly be dis-
posed to believe that it could have been (as is the fact)
generally overlooked, and that eminent writers should
in consequence have been involved in inextricable
confusion. We need only remind them however of the
anecdote of Columbus breaking the egg; that which is
perfectly obvious to any man of common sense, as
soon as it is mentioned, may nevertheless fail to occur,
even to men of considerable ingenuity.
It will also be readily perceived, on examining the
principles of these several divisions, that the last of
them alone is properly and strictly a division of Argu-
ments as such. The 1st is evidently a division of the
Forms of stating them ; for every one would allow that
the same Argument may be either stated as an enthy-
meme, or brought into the strict syllogistic form ; and
that either categorically or hypothetically, &c., e.g.
“Whatever has a beginning has a cause ; the earth
had a beginning, therefore it had a cause ;” or, “If
the earth had a beginning it had a cause : it had a be-
ginning,” &c. every one would call the same Argument,
differently stated. This, therefore, evidently is not a
division of Arguments as such.
The 2nd is plainly a division of Arguments according
to their subject-matter, whether Necessary or Probable,
certain or uncertain. In Mathematics, e.g. every pro-
position that can be stated is either an immutable
truth, oran absurdity and contradiction; while in human
affairs the propositions which we assume are only true
for the most part, and as general rules; and in Physics,
though they must be true as long as the laws of nature
remain undisturbed, the contradiction of them does
not imply an absurdity; and the conclusions of course,
in each case, have the same degree and kind of cer-
tainty with the premises. This, therefore, is properly
a division, not of Arguments as such, but of the Pro-
positions of which they consist.
The 3rd is a division of Arguments according to the
purpose for which they are employed;—according to
the intention of the reasoner; whether that be to esta-
blish “directly” (or “ostensively") the conclusion drawn,
or (“indirectly”) by means of an absurd conclusion
to disprove one of the premises: (i. e. to prove its
contradictory) since the alternative proposed in every
valid Argument is, either to admit the conclusion, or
to deny one of the premises. Now it may so happen
R H E T O R. I. C.
.247
It is
probable, e. g. that many have been induced to admit
the doctrine of Transubstantiation, from its clear
connection with the infallibility of the Romish Church;
and many others, by the very same Argument, have sur-
rendered their belief in that infallibility. Again, Berkley
and Reid seem to have alike admitted that the non-
existence of matter was a necessary consequence of
Locke's Theory of Ideas ; but the former was hence
led, bond fide, to admit and advocate that non-exis-
tence, while the latter was led by the very same
Argument to reject the Ideal Theory. Thus, we see
it is possible for the very same Argument to be Direct
to one person, and Indirect to another ; leading them
to different results, according as they judge the origi-
ºnal conclusion, or the contradictory of a premiss, to be
the more probable. This, therefore, is not properly
a division of Arguments as such, but a division of the
purposes for which they are employed.
The 4th, which alone is properly a division of Argu-
ments as such, and accordingly will be principally
treated of, is a division according to the “relation of
the subject-matter of the premises to that of the con-
clusion.” We say, “ of the subject-matter,” because
the logical connection between the premises and con-
clusion is independent of the meaning of the terms
employed, and may be exhibited with letters of the
alphabet substituted for the terms; but the relation
we are now speaking of between the premises and
conclusion, (and the varieties of which form the seve-
ral species of Arguments,) is in respect of their subject-
matter; as e. g. an “Argument from Cause to Effect”
is so called and considered, in reference to the rela-
tion existing between the premiss, which is the
Cause, and the conclusion, which is the Effect; and
an “Argument from Example,” in like manner, from
the relation between a known and an unknown in-
stance, both belonging to the same class. And it is
plain that the present division, though it has a re-
ference to the subject-matter of the premises, is yet
not a division of propositions considered by them-
selves, (as in the case with the division into probable
and demonstrative,) but of Arguments considered as
such ; for when we say, e. g. that the premiss is a
Cause, and the conclusion the Effect, these expres-
sions are evidently relative, and have no meaning,
except in reference to each other ; and so also when
we say that the premiss and the conclusion are two
parallel cases, that very expression denotes their
relation to each other.
In distributing, then, the several kinds of Argu-
ments, according to this division, it will be found
convenient to lay down first two great classes, under
one or other of which all can be brought ; viz.
1st, such Arguments as might have been employed
to account for the fact or principle maintained, sup-
posing its truth granted ; 2nd, such as could not be
so employed. The former class (to which in this
Treatise, the name of “A priori'' Argument will be con-
fined,) is manifestly Argument from Cause to Effect;
since to account for any thing, signifies to assign the
Cause of it. The other class, of course, comprehends
all other Arguments, of which there are several kinds,
which will be mentioned hereafter.
The two sorts of proof which have been just
spoken of, Aristotle seems to have intended to de-
Rhetoric, that in some cases, one person will choose the former,
J– and another the latter, of these alternatives.
signate by the titles of 3rt for the latter, and 8tor, for Chap. I.
the former; but he has not heen so clear as could
be wished, in observing the distinction between
them. The only decisive test by which to distinguish
the Arguments which belong to the one, and to the
other of these classes is, to ask the question, “Sup-
posing the proposition in question to be admitted,
would this Argument serve to account for the truth,
or not 2" It will then be readily referred to the
former or to the latter class, according as the answer
is in the affirmative or the negative, as, e. g. if a
murder were imputed to any one on the grounds of
his “having a hatred to the deceased, and an interest
in his death,” the Argument would belong to the
former class ; because, supposing his guilt to be ad-
mitted, and an inquiry to be made how he came to
commit the murder, the circumstances just mentioned
would serve to account for it ; but not so, with respect
to such an Argument as his “ having blood on his
clothes;
other class.
2 3
which would therefore be referred to the
And here let it be observed, once for all, that when
we speak of arguing from Cause to Effect, it is not
intended to maintain the real and proper efficacy of
what are called Physical Causes to produce their re-
spective Effects, nor to enter into any discussion of the
controversies which have been raised on that point,
which would be foreign from the present purpose.
The word “Cause,” therefore, is to be understood as
employed in the popular sense; as well as the phrase
of “accounting for" any fact.
As far, then, as any Cause, popularly speaking, has
a tendency to produce a certain Effect, so far its
existence is an Argument for that of the Effect. If
the Cause be fully sufficient, and no impediments in-
tervene, the Effect in question follows certainly ; and
the nearer we approach to this, the stronger the
Argument. -
This is the kind of Argument which produces,
(when short of absolute certainty,) that species of
the Probable which is usually called the Plausible.
On this subject Dr. Campbell has some valuable re-
marks in his Philosophy of Rhetoric (book i. § 5. ch. vii.)
though he has been led into a good deal of perplexity,
partly by not having logically analyzed the two
species of probabilities he is treating of, and partly
by departing, unnecessarily, from the ordinary use
of terms, in treating of the Plausible as something
distinct from the Probable, instead of regarding it as a
species of Probability.
This is the only kind of Probability which poets, or
other writers of fiction, aim at ; and in such works
it is often designated by the term “natural.” Writers
of this class, as they aim not at producing belief, are
allowed to take their “Causes” for granted, (i. e. to
assume any hypothesis they please,) provided they
make the Effects follow naturally ;
representing,
that is, the personages of the fiction as acting, and
the events as resulting, in the same manner as might
have been expected, supposing the assumed circum-
stances to have been real. And hence, the great
Father of Criticism establishes his paradoxical imaxim,
that impossibilities which appear probable, are to be
preferred to possibilities which appear improbable.
For, as he justly observes, the impossibility of the
hypothesis, as e. g. in Homer, the familiar intercourse
of God with mortals, is no bar to the kind of Pro-
248
R H E To R I c.
which could not be used to account for the fact in Chap. I.
question, supposing it granted,") may be sub-divided S-V-
Rhetoric. bability 'required, if those mortals are represented as
\-v-'acting in the manner men naturally would have done
under those circumstances. .
The Probability, then, which the writer of fiction
aims at, has, for the reason just mentioned, no tendency
to produce a particular, but only a general belief;
i. e. not that these particular events actually took
place, but that such are likely, generally, to take
place under such circumstances.* In Argumentative
Compositions however, as the object of course is to
produce conviction as to the particular point in ques-
tion, the Causes from which our Arguments are drawn,
must be such as are either admitted, or may be proved,
to be actually existing, or likely to exist.
On the appropriate use of this kind of Argument,
(which is probably the étkös of Aristotle, though un-
'fortunately he has not furnished any example of it,)
some-Rules will be laid down hereafter; our object at
present having been merely to ascertain the nature
of it. And here it may be worth while to remark,
that though we have applied to this mode of Rea-
soning the title of “a priori,” it is not meant to be
maintained that all such Arguments as have been by
other writers so designated, correspond precisely with
what has been just described, f The phrase, “a
priori" Argument, is not, indeed, employed by all in
the same sense; it would however generally be under-
stood to extend to any argument drawn from an
antecedent or forerunner, whether a Cause or not ;
e. g. “the mercury sinks, therefore it will rain.”
Now this Argument being drawn from a circumstance
which though an antecedent, is in no sense a Cause,
would fall not under the former, but the latter, of the
classes laid down, since when rain comes, no one
would account for the phenomenon by the falling of
the mercury; and yet most, perhaps, would class
this among “a priori" Arguments. In like manner
the expression, “a posteriori'' Arguments, would not
in its ordinary use, coincide precisely, though it would,
very nearly, with the second class of Arguments.
The division, however, which has here been adopted,
appears to be both more Philosophical, and also more
precise, and consequently more practically useful than
any other ; since there is so easy and decisive a test by
which an Argument may be at once referred to the
one or to the other of the classes described.
The second, then, of these classes, (viz. “Arguments
s−
* On which ground Aristotle contends that the end of Fiction
is more Philosophical than that of History, since it aims at
general, instead of particular Truth.
+ Some Rhetorical students accordingly, partly with a view
to keep clear of any ambiguity that might hence arise, and
partly for the sake of brevity, have found it useful to adopt, in
drawing up an outline, or analysis of any composition, certain
arbitrary symbols, to denote, respectively, each class of Argu-
ments and of Propositions; viz. A, for the former of the two
classes of Arguments just described, (to denote “A priori,” or
“Antecedent,” probability,) and B, for the latter, which, as
consisting of several different kinds, may be denominated “the
Body of evidence.” Again, they designate the proposition,
which accounts for the principal and original assertion, by a
small “a,” or Greek a, to denote its identity in substance with
the Argument bearing the symbol “A,” though employed for a
different purpose; viz. not to establish a fact that is doubtful,
but to account for one that is admitted. The proposition, again,
which results as a Consequence or Corollary from the principal
one, they designate by the symbol C. There seems to be the
same convenience in the use of these symbols as Logicians have
found in the employment of A, E, 1, 0, to represent the four
kinds of Propositions according to quantity and quality.
into two kinds; which will be designated by the
terms “ Sign” and “Example.”
By “Sign,” (so called from the Xmue?ov of Ari-
‘stotle,) is meant a species of Argument of which the
analysis is as follows: As far as any circumstance is,
what may be called, a Condition of the existence of a
certain effect or phenomenon, so far it may be in-
ferred from the existence of that Effect : if it be a
Condition absolutely essential, the Argument is, of
course, demonstrative ; and the Probability is the
stronger in proportion as we approach to that case.
Of this kind is the Argument in the instance lately
given : a man is suspected as the perpetrator of the
supposed murder, from the circumstance of his clothes
being bloody; the murder being considered as in a
certain degree a probable condition of that appearance;
i. e. it is presumed that his clothes would not otherwise
have been bloody. Again, from the appearance of
ice, we infer, decidedly, the existence of a temperature
below freezing point, that temperature being an essen-
tial Condition of the crystallization of water.
Among the circumstances which are conditional to
any Effect, must evidently come the Cause or Causes;
and if there be only one possible Cause, this being
absolutely essential, may be demonstratively proved
from the Effect : if the same Effect might result from
other Causes,then the Argumentis, at best,but probable.
But it is to be observed, that there are also many
circumstances which have no tendency to produce a
certain Effect, though it cannot exist without them,
and from which Effect, consequently, they may be
inferred, as Conditions, though not Causes ; e. g. a
man’s “being alive one day,” is a circumstance ne-
cessary, as a Condition, to his “dying the next ;"
but has no tendency to produce it : his having been
alive, therefore, on the former day, may be proved
from his subsequent death, but not vice versd. *
It is to be observed therefore, that though it is very
common for the Cause to be proved from its Effect,
it is never so proved, so far forth as [?] it is a Cause,
but so far forth as it is a condition, or necessary
circumstance.
A Cause, again, may be employed to prove an
Effect, (this being the first class of Arguments already
described,) so far as it has a tendency to produce the
Effect, even though it be not at all necessary to it; (i. e.
when other Causes may produce the same Effect,)
and in this case, though the Effect may be inferred
from the Cause, the Cause cannot be inferred from the
Effect; e.g. from a mortal wound you may infer
death, but not vice versd.
Lastly, when a Cause is also a necessary or proba-
ble condition, i. e. when it is the only possible or
likely Cause, then we may argue both ways; e.g. we
may infer a General's success from his known skill,
or, his skill, from his known success : these two
* It is however very common, in the carelessness of common
language, to mention; as the Causes of phenomena, circumstances
which every one would allow, on consideration, to be not Causes,
but only Conditions, of the Effects in question; e. g. it would
be said of a tender plant, that it was destroyed in consequence
of not being covered with a mat; though every one would mean
to imply that the frost destroyed it; this being a Cause too well
known to need being mentioned; and that which is spoken of as
the Cause, viz. the absence of a covering, being only the Condition,
without which the real Cause could not have operated.
R H E T O R. I. C.
249
Rhetoric. Arguments belonging, respectively, to the two classes
^-y-' originally laid down. And it is to be observed that,
sions are as equivocal and uncertain in their significa- Chap. I.
tion as the original one. It is in vain to attempt S-N-2
in such Arguments from Sign as this last, the con-
clusion which follows, logically, from the premiss,
being the Cause from which the premiss follows,
physically, i. e. as a natural Effect, there are in this
case two different kinds of Sequence opposed to each
other. In Arguments of the first class, on the con-
trary, these two kinds of Sequence are combined ;
i. e. the Conclusion which follows logically from the
premiss, is also the Effect following physically from it
as a Cause; a General's skill, e. g. being both the Cause
and the Proof of his being likely to succeed.
It is most important to keep in mind the distinction
between these two kinds of Sequence, which are, in
Argument, sometimes combined, and sometimes op-
posed. There is no more fruitful source of confusion
of thought than that ambiguity of language employed
on these subjects, which tends to confound together
these two things, so entirely distinct in their nature.
There is hardly any argumentative writer on subjects
involving a discussion of the Causes or Effects of any
thing, who has clearly perceived and steadily kept in
view the distinction we have been speaking of, or
who has escaped the errors and perplexities thence
resulting. The wide extent accordingly, and the
importance of the mistakes and difficulties arising out
of the ambiguity complained of, is incalculable. To
dilate upon this point as fully as might be done with
advantage, would lead us beyond our present limits;
but it will not be foreign to the purpose of this article
to offer some remarks on the origin of the ambiguity.
complained of, and on the cautions to be used in
guarding against being misled by it.
The premiss by which any thing is proved, is not
necessarily the Cause of the fact's being such as it is ;
but it is the Cause of our knowing and being con-
vinced that it is so ; e. g. the wetness of the earth is
not the Cause of rain, but it is the Cause of our
knowing that it has rained. These two things, the
premiss which produces our conviction, and the Cause
which produces that of which we are convinced, are
the more likely to be confounded together, in the
looseness of colloquial language, from the circum-
stance that (as has been above remarked) they fre-
quently coincide; as, e.g. when we infer that the
ground will be wet, from the fall of rain which pro-
duces that wetness. And hence it is that the same
words have come to be applied, in common, to each
kind of Sequence; e.g. an Effect is said to “follow”
from a Cause, and a Conclusion to “follow " from
the premises; the words “Cause ’’ and “Réâson,”
are each applied indifferently, both to a Cause, pro-
perly so called, and to the premiss of an Argument;
though “Reason,” in strictness of speaking, should
be confined to the latter. “Therefore,” “ hence,”
** consequently,” &c., and also, “ since,” “ because,”
and “why," have likewise a corresponding ambiguity.
The multitude of the words which bear this double
meaning, (and that, in all languages,) greatly in-
creases our liability to be misled by it ; since thus the
very means men resort to for ascertaining the sense of
any expression, are infected with the very same ambi-
guity; e. g. if we inquire what is meant by a “Cause,”
we shall be told that it is that from which something
“ follows;” or, which is denoted by the words
“ therefore,” “ consequently,” &c. all which expres-
WOL. I.
ascertaining by the balance the true amount of any
commodity, if false weights are placed in the oppo-
site scale. Hence it is that so many writers, in
investigating the Cause to which any fact or phe-
nomenon is to be attributed, have assigned that which
is not a Cause, but only a Proof that the fact is so ;
and have thus been led into an endless train of errors
and perplexities.
Several, however, of the words in question, though
employed indiscriminately in both significations,
seem (as was observed in the case of the word
“Reason,”) in their primary and strict sense, to be
confined to one, “8),” in Greek, and “ ergo,” or
“itaque,” in Latin, seem originally and properly to
denote the Sequence of Effect from Cause; “dpa,”t
and “igitur,” that of conclusion from premises. The
English word “ accordingly,” will generally be found
to correspond with the Latin “itaque.”
The interrogative “why,” is employed to inquire,
either, 1st, the “Reason,” (or “ Proof;”) 2ndly, the
“Cause ;” or 3rdly, the “object proposed,” or final
Cause: e. g. 1st, Why are the angles of a triangle
equal to two right angles 2 2nd, Why are the days
shorter in winter than in summer ? 3rd, Why are the
works of a watch constructed as they are 2 If any
one were to ask “Why the Gospel-revelation is to
be received 2" he might intend by this question any
one of these three inquiries ; which would of course
require very different answers.
It is to be observed that the discovery of Causes
belongs properly to the province of the Philosopher;
that of “Reasons,” strictly so called, (i.e. Arguments)
to that of the Rhetorican ; and that, though each will
have frequent occasion to assume the character of the
other, it is most important that these two objects
should not be confounded together.
Of Signs then one kind are such as from a certain
Effect or phenomenon, infer the “Cause” of it; and
the other, such as, in like manner, infer some “Con-
dition” which is not the Cause. Of these last, one spe-
cies is the Argument from Testimony; the premiss
being the existence of the Testimony, the Conclusion,
the truth of what is attested ; which is considered as
a “Condition” of the Testimony having been given ;
since it is evident that so far only as this is allowed,
(i.e. so far only as it is allowed that the Testimony
would not have been given, had it not been true,) can
this Argument have any force.
Testimony is of various kinds; but the distinction.
between them is so obvious, as well as the various
circumstances which add to, or diminish the weight of
any Testimony, that it is not necessary to enter into
any detailed discussion of the subject. It may be worth
remarking, however, that one of the most important
distinctions is between Testimony to matters of Fact,
and to Doctrines or Opinions: in estimating the weight
of the former, we look chiefly to the honesty of the
witness, and his means of obtaining information ; in
the latter, his ability to judge is equally to be taken
* Most Logical writers seem not to be aware of this, as they
generally, in Latin Treatises, employ “ ergo " in the other sense;
it is from the Greek épyq), i.e. “in fact.”
+ "Apa having a signification of fitness or coincidence; whence
&pa.
2 L
250
R H E T O R. I. C.
regarded as less probable than any other; since the Chap. I.
letters of which the Iliad is composed, if shaken toge- -v-
Rhetoric. into consideration. With respect however to the cre-
\-N- dibility of witnesses, it is evident that when many
sº
coincide in their testimony, (where no previous concert
can have taken place,) the probability resulting from
this concurrence does not rest on the supposed veracity
of each considered separately, but on the improbability
of such an agreement taking place by chance. For
though in such a case each of the witnesses should be
considered as unworthy of credit, and even much more
likely to speak falsehood than truth, still the chances
might be infinite against their all agreeing in the same
falsehood. This remark is applied by Dr. Campbell
to the Argument from Testimony; but he might have
extended it to other Arguments also, in which a simi-
lar calculation of chances will enable us to draw a
Conclusion, sometimes even amounting to moral cer-
tainty, from a combination of data which singly would
have had little or no weight ; e. g. if any one out of
a hundred men throw a stone which strikes a certain
object, there is but a slight probability, from that fact
alone, that he aimed at that object; but if all the
hundred threw stones which struck the same object,
no one would doubt that they aimed at it. . It is from
such a combination of Argument that we infer the
existence of an intelligent Creator from the marks of
contrivance visible in the Universe, though many of
these are such as, taken singly, might well be conceived
undesigned and accidental; but that they should all
be such, is morally impossible. Great care is requisite
in setting forth clearly, especially in any popular dis-
course, Arguments of this nature ; the generality of
men being better qualified for understanding, (to use
Lord Bacon's words,) “ particulars, one by one,” than
for taking a comprehensive view of a whole; and
therefore in a Galaxy of Evidence, as it may be called,
in which the brilliancy of no single star can be pointed
out, the lustre of the combination is often lost on them.
Hence it is, as was remarked in the Treatise on Fal-
lacies, that the sophism of “Composition,” as it is
called, so frequently misleads men: it is not impro-
bable, (in the above example,) that each of the stones,
considered separately, may have been thrown at ran-
dom; and therefore the same is concluded of all, con-
sidered in conjunction. Not that in such an instance as
the above, any one would reason so weakly ; but that
a still greater absurdity of the very same kind is in-
volved in the rejection of the evidences of our religion,
will be plain to any one who considers, not merely the
individual force, but the number and variety of those
evidences. *
And here it may be observed, that though the
easiest and most popular way of practically refuting
the Fallacy just mentioned, (or indeed any Fallacy,)
is, by bringing forward a parallel case, where it leads
to a manifest absurdity, a metaphysical objection
may still be urged against many cases in which we
thus reason from calculation of chances ; an objection
not likely indeed practically to influence any one, but
which may afford the Sophist a triumph over those who
are unable to find a solution. If it were answered then
to those who maintain that the universe, which exhi-
bits so many marks of design, might be the work of
non-intelligent causes, that no one would believe it
possible for such a work as the Iliad, e. g. to be pro-
duced by a fortuitous shaking together of the letters
of the alphabet, the Sophist might challenge us to
explain why even this last Supposition should be
ther at random, must fall in some form or other; and
though the chances are millions of millions to one against
that, or any other determinate order, there are precisely
as many chances against one, as against another : and in
like manner, astonished as we should be, and convinced
of the intervention of artifice, if we saw any one draw
out all the cards in a pack in regular sequences, it is
demonstrable that the chances are not more against
that order, than against any one determinate order we
might choose to fix upon. The multitude of the chances,
therefore, he would say, against any series of events,
does not constitute it improbable; since the like hap-
pens to every one every day; e. g. a man walking
through London streets on his business, meets acci-
dentally hundreds of others passing to and fro on
theirs : and he would not say at the close of the day
that any thing improbable had occurred to him; yet it
would almost baffle calculation to compute the Chances
against his meeting precisely those very persons, in the
order, and at the times and places of his meeting each.
The paradox thus seemingly established, though few
might be practically misled by it, many would be
at a loss to solve. The truth is, that any supposition
is justly called improbable, not from the number of
chances against it, considered independently, but from
the number of chances against it compared with those
which lie against some other supposition: we call the
drawing of a prize in the lottery improbable, though
there be but five to one against it, because there are
more chances of a blank; on the other hand, if any
one was cast on a desert island under circumstances
which warranted his believing that the chances were
a hundred to one against any one's having been there
before him, yet if he found on the sand pebbles so
arranged as to form the letters of a man's name, he
would not only conclude it probable, but absolutely
certain that some human being had been there; be-
cause there would be millions of chances against those
forms having been produced by the fortuitous action
of the waves. So also, in the instance above given,
any unmeaning form into which a number of letters
might fall, would not be called improbable, countless
as the chances are against that particular order, be-
cause there are just as many against each one of all
other unmeaning forms ; but if the letters formed a
coherent poem, it would then be called incalculably
improbable that this form should have been fortuitous,
though the chances against it remain the very same ;
because there must be much fewer chances against the
supposition of its having been the work of design. The
probability in short, of any supposition, is estimated
from a comparison with each of its alternatives.
The foregoing observations however, as was above
remarked, are not confined to Arguments from Testi-
mony, but apply to all cases in which the degree of
probability is estimated from a calculation of chances.
Before we dismiss the consideration of Signs, it may
be worth while to notice another case of combined
Argument different from the one lately mentioned, yet
in some degree resembling it. The combination just
spoken of is where several Testimonies or other Signs,
singly perhaps of little weight, produce jointly, and
by their coincidence, a degree of probability far ex-
ceeding the sum of their several forces, taken sepa-
rately; in the case we are now about to notice, the
R H E T O R. I C.
25 I:
Rhetoric combined force of the series of Arguments resulting
from the order in which they are considered, and from
their progressive tendency to establish a certain con-
clusion. E. g. one part of the law of nature called the
“ vis inertiae,” is established by the Argument we
allude to ; viz. that a body set in motion will eter-
nally continue in motion with uniform velocity in a
right line, so far as it is not acted upon by any causes
which retard or stop, accelerate or divert its course.
Now, as in every case which can come under our ob-
servation, some such causes do intervene, the as-
sumed supposition is practically impossible, and we
have no opportunity of verifying the law by direct
experiment ; but we may gradually approach indefi-
nitely near to the case supposed : and on the result of
such experiments our conclusion is founded. We find
that when a body is projected along a rough surface,
its motion is speedily retarded and soon stopped ; if
along a smoother surface, it continues longer in mo-
tion; if upon ice, longer still, and the like with regard
to wheels, &c. in proportion as we gradually lessen the
friction of the machinery; if we remove the resistance
of the air, by setting a wheel or pendulum in motion
under an air-pump, the motion is still longer continued.
Finding then that the effect of the original impulse is
more and more protracted, in proportion as we more
and more remove the impediments to motion from
friction and resistance of the air, we reasonably conclude
that if this could be completely done, (which is out of
our power,) the motion would never cease, since what
appear to be the only causes of its cessation, would be
absent.
Again, in arguing for the existence and moral at-
tributes of the Deity from the authority of men's
opinions, great use may be made of a like progressive
course of Argument, though it has been often over-
looked. Some have argued for the being of a God from
the universal or at least general consent of mankind ;
and some have appealed to the opinions of the wisest
and most cultivated portion, respecting both the exist-
ence and the moral excellence of the Deity. It cannot
be denied that there is a presumptive force in each of
these Arguments ; but it may be answered that it is
conceivable an opinion common to almost all the spe-
cies, may possibly be an error resulting from a con-
stitutional infirmity of the human intellect ;-that if
we are to acquiesce in the belief of the majority, we
shall be led to Polytheism ; such being the creed of
the greater part : and that though more weight may
reasonably be attached to the opinions of the wisest
and best-instructed, still, as we know that such men
are not exempt from error, we cannot be perfectly safe
in adopting the belief they hold, unless we are con-
vinced that they hold it in consequence of their being
the wisest and best instructed;— so far forth as they
are such. Now this is precisely the point which may
be established by the above-mentioned progressive
Argument. Nations of Atheists, if there are any such,
are confessedly among the rudest and most ignorant
savages : those who represent their God or Gods as
malevolent, capricious, or subject to human passions
and vices, are invariably to be found, (in the present
day at least,) among those who are brutal and unci-
vilized ; and among the most civilized nations of the
ancients, who professed a similar creed, the more en-
lightened members of society seem either to have
rejected altogether, or to have explained away, the
popular belief. The Mahometan nations, again, of the Chap. R-
present day, who are certainly more advanced in civi-
lisation than their Pagan neighbours, maintain the
unity and the moral excellence of the Deity; but the
nations of Christendom, whose notions of the divine
goodness are more exalted, are undeniably the most
civilized part of the world, and possess, generally
speaking, the most cultivated and improved intellec-
tual powers. Now if we would ascertain, and appeal
to, the sentiments of man as a rational being, we
must surely look to those which not only prevail
most among the most rational and cultivated, but to-
wards which also a progressive tendency is found in
men in proportion to their degrees of rationality and
cultivation. It would be most extravagant to suppose
that man's advance towards a more improved and
exalted state of existence should tend to obliterate
true and instil false notions. On the contrary we are
authorized to conclude, that those notions would be
the most correct, which men would entertain, whose
knowledge, intelligence, and intellectual cultivation
should have reached the highest pitch of perfection;
and that those consequently will approach the nearest
to the truth which are entertained, more or less, by
various nations, in proportion as they have advanced
towards this civilized state.
Many other instances might be adduced, in which
truths of the highest importance may be elicited by
this process of Argumentation, which will enable us
to decide with sufficient probability what consequence
would follow from an hypothesis which we have never
experienced ; it might, not improperly, be termed
the Argument from Progressive Approach.
The third kind of Arguments to be considered
being the other branch of the second of the two classes
originally laid down, may be treated of under the
general name of Example, taking that term in its
widest acceptation, so as to comprehend the Argu-
ments designated by the various names of Induction,
Experience, Analogy, Parity of Reasoning, &c. all of
which are essentially the same, as far as regards the
fundamental principles we are here treating of ; for in
all the Arguments designated by these names it will
be found, that we consider one or more, known, indi-
vidual objects or instances, of a certain class, as fair
specimens, in respect of some point or other, of that
class ; and consequently draw an inference from them
respecting either the whole class, or other, less known,
individuals of it. In Arguments of this kind then it
will be found, that universally we assume as a major
premiss that what is true, (in regard to the point in
question,) of the individual or individuals which we
bring forward and appeal to, is true of the whole class
to which they belong; the minor premiss next asserts
something of that individual; and the same is then
inferred respecting the whole class : whether we stop
at that general eonclusion, or descend from thence to
another, unknown, individual; in which last case,
which is the most usually called the Argument from
Example, we generally omit, for the sake of brevity,
the intermediate step, and pass at once in the expres-
sion of the Argument from the known, to the un-
known, individual. This ellipsis however does not,
as some seem to suppose, make any essential difference
in the mode of Reasoning; the reference to a common
class being always, in such a case, understood, though
not expressed; for it is evident that there can be no
2 L 2
252 - R H E T O R I C.
Rhetoric, reasoning from one individual to another, unless they
S-N-' come under some common genus, and are considered
in that point of view ; e. g. *
Astronomy was decried Geology is likely to be
at its first introduction, as decried, &c.
adverse to religion :
e-º-
-* Q}
º $
ſº §
º §
- Ts S.
every Science is likely to be decried at its first in-
troduction, as adverse to religion. gº
This kind of example, therefore, appears to be a
compound Argument, consisting of two enthymemes ;
and when (as often happens) we infer from a known
Effect a certain Cause, and again, from that Cause,
another unknown Effect, we then unite in this ex-
ample, the argument from Effect to Cause, and that
from Cause to Effect, e.g. we may from the marks of
Divine benevolence in this world, argue, that “the
like will be shown in the next ;” through the inter-
mediate conclusion, that “ God is benevolent.” This
is not indeed always the case; but there seems to be
in every example, a reference to some Cause, though
that Cause may frequently be unknown ; e.g. we
suppose, in the instance above given, that there is
some Cause, though we may be at a lost to assign it,
which leads men generally to decry a new Science.
The term “ Induction” is commonly applied to such
Arguments as stop short at the general conclusion ;
and is thus contradistinguished, in common use, from
Example. There is also this additional difference,
that when we draw a general conclusion from several
individual cases, we use the word Induction in the
singular number, while each one of these cases, if the
application were made to another individual, would
be called a distinct Example. This difference, however,
is not essential, since whether the inference be made
from one instance or from several, it is equally called an
Induction, if a general conclusion be legitimately drawn,
and this is to be determined by the nature of the sub-
ject-matter; in the investigation of the laws of nature, a
single experiment, fairly and carefully made, is usually
allowed to be conclusive, because we can then pretty
nearly ascertain all the circumstances operating : a
Chemist who has ascertained, in a single specimen of
gold, its capability of combining with mercury, would
not think it necessary to try the same experiment with
several other specimens, but would draw the conclu-
sion concerning those metals universally, and with
certainty; in human affairs on the contrary, our un-
certainty respecting many of the circumstances that
may affect the result, obliges us to collect many coin-
ciding instances to warrant even a probable conclusion.
From one instance, e.g. of the assassination of an
Usurper, it would not be allowable to infer the cer-
tainty, or even the probability, of a like fate attending
all Usurpers.” &
Experience, in its original and proper sense, is
applicable to the premises from which we argue,
not to the inference we draw. Strictly speaking, we
know by Experience only the past, and what has passed
under our own observation; thus, we know by Erpe-
rience that the tides have daily ebbed and flowed,
during such a time; and from the Testimony of others
as to their own experience, that they have formerly
* See article Logic, “On the Province of Reasoning,”(p. 230.)
done so ; and from this experience, we conclude, by Chap. I.
Induction, that the same phenomenon will continue. S-N-7
The word Analogy again is generally employed in
the case of Arguments in which the instance adduced
is somewhat more remote from that to which it is
applied; e. g. a physician would be said to know by
experience the noxious effects of a certain drug on
the human constitution if he had frequently seen men
poisoned by it; but if he thence conjectured that it
would be noxious to some other species of animal, he
would be said to reason from Analogy; the only dif-
ference being that the resemblance is less, between a
man and a brute, than between one man and another;
and accordingly it is found that many brutes are not
acted upon by some drugs which are pernicious to
man. But more strictly speaking, Analogy ought to
be distinguished from direct resemblance, with which
it is often confounded in the language even of eminent
writers (especially on Chemistry and Natural History)
in the present day. Analogy being a “resemblance
of ratios,” that should strictly be called an Argu-
ment from Analogy, in which the two cases (viz. the
one from which, and the one to which we argue) are
not themselves alike, but stand in a similar relation to
something else; or in other words that the common
genus which they both fall under, consists in a rela-
tion. Thus an egg and a seed are not in themselves
alike, but bear a like relation to the parent bird and
to her future nestling, on the one hand, and to the old
and young plant on the other, respectively; this rela-
tion being the genus which both fall under : and
many Arguments might be drawn from this Analogy.
Again the fact that from birth different persons have
different bodily constitutions, in respect of complec-
tion, stature, strength, shape, liability to particular
disorders, &c. which constitutions, however, are ca-
pable of being, to a certain degree, modified by regi-
men, medicine, &c. affords an Analogy by which we
may form a presumption, that the like takes place in
respect of mental qualities also ; though it is plain
that there can be no direct resemblance either between
body and mind, or their respective attributes.
In this kind of Argument one error, which is very
common, and which is to be sedulously avoided, is
that of concluding the things in question to be alike,
because they are Analogous;–to resemble each other in
themselves, because there is a resemblance in the
relation they bear to certain other things; which is
manifestly a groundless inference. Another caution is
applicable to the whole class of Arguments from Ex-
ample ; viz. not to consider the resemblance or Ana-
logy to extend further (i. e. to more particulars) than
it does. The resemblance of a picture to the object
it represents, is direct; but it extends no further than
the one sense of seeing is concerned. In the parable
of the unjust steward an Argument is drawn from
Analogy, to recommend prudence and foresight to
Christians in spiritual concerns; but it would be ab-
surd to conclude that fraud was recommended to our
imitation; and yet mistakes very similar to such a
perversion of that Argument are by no means rare.f
* Aoyáv Špoićrms, Aristotle. $
+ “Thus, because a just Analogy has been discerned between th
metropolis of a country, and the heart of the animal body, it has
been sometimes contended that its increased size is a disease,
that it may impede some of its most important functions, or even
be the cause of its dissolution.” Copleston's Inquiry into the
R H E TO R. I. C. 253
.* horned, to pronounce the animal a ruminant. Whereas Chap. I.
Rhetoric. The Argument from Contraries, (ºf evavitºv) noticed
on the other hand, the fable of the countryman, who S-2-’
S—— by Aristotle, falls under the class we are now treating
of ; as it is plain that Contraries must have something
in common ; and it is so far forth only as they agree,
that they are thus employed in Argument. Two
things are called “Contrary,” which, coming under the
same class, are the most dissimilar in that class. Thus,
virtue and vice are called Contraries, as being, both,
“ moral habits,” and the most dissimilar of moral habits.
mere dissimilarity, it is evident, would not constitute
Contrariety; for no one would say that virtue was con-
trary to a mathematical problem, the two things
having nothing in common. In this then, as in other
Arguments of the same class, we may infer that the
two Contrary terms have a similar relation to the same
third, or respectively to two corresponding, (i. e. in
this case, Contrary) terms : we may conjecture e. g.
that since virtue may be acquired by education, so
may vice ; or again, that since virtue leads to happi-
ness, so does vice to misery.
The phrase “Parity of Reasoning,” is commonly
employed to denote Analogical Reasoning.
Aristotle, in his Rhetoric, has divided Examples into
Real and Invented: the one being drawn from actual
matter of fact ; the other, from a supposed case. And
he remarks, that though the latter is more easily ad-
duced, the former is more convincing. If however
due care be taken, that the fictitious instance,—the
supposed case, adduced, be not wanting in probability,
it will often be no less convincing than the other. For
it may so happen, that one, or even several historical
facts may be appealed to, which being nevertheless
exceptions to a general rule, will not prove the pro-
bability of the conclusion. Thus, from several known
instances of ferocity in black tribes, we are not autho-
rized to conclude, that blacks are universally, or gene-
rally ferocious; and in fact, many instances may be
brought forward on the other side. Whereas in the
supposed case, (instanced by Aristotle, as employed
by Socrates,) of mariners choosing their steersman by
lot, though we have no reason to suppose such a case
ever occurred, we see so plainly the probability, that if
it did occur, the lot might fall on an unskilful person,
to the loss of the ship, that the argument has con-
siderable weight against the practice, so common in
the ancient republics, of appointing magistrates by
lot. There is, however, this important difference;
that a fictitious case which has not this intrinsic pro-
bability, has absolutely no weight whatever; so that
of course such arguments might be multiplied to any
amount without the smallest effect : whereas any
matter of fact which is well established, however
wnaccountable it may seem, has some degree of weight
in reference to a parallel case ; and a sufficient number
of such arguments may fairly establish a general rthe,
even though we may be unable, after all, to account
for the alleged fact in any of the instances; e.g. no sa-
tisfactory reason has yet been assigned for a connection
between the absencº of upper cutting teeth, or of the
presence of horns, aſſid rumination; but the instances,
are so numerous and constant of this connection, that
no Naturalist would hesitate, if on examination of a
new species he found those teeth absent, and the head
Doctrines of Necessity and Predestination, note to Disc. iii. q.v.
for a very able dissertation on the subject of Analogy, in the
course of an analysis of Dr. King's Discourse on Prettestination.
- ***
**.
obtained from Jupiter the regulation of the weather,
and in consequence found his crops fail, does not go
one step towards proving the intended conclusion ;
because that consequence is a mere gratuitous assump-
tion without any probability to support it. ... There is
an instance of a like error in a tale of Cumberland's,
intended to prove the advantage of a public over a
private education; he represents two brothers edu-
cated, on the two plans respectively; the former
turning out very well, and the latter very ill; and had
the whole been matter of fact, a sufficient number of
such instanees would have had weight as an Argument;
but as it is a fiction, and no reason is shown why the
result should be such as represented, except the sup-
posed superiority of a public education, the Argument
involves a manifest petitio principii; and resembles the
appeal made in the well-known fable, to the picture of
a man conquering a lion ; a result which might just
as easily have been reversed, and which would have
been so, had lions been painters. It is necessary, in
short, to be able to maintain, either that such and
such an event did actually take place, or that, under a
certain hypothesis, it would be likely to take place.
Under the head of Invented Example, a distinction is
drawn by Aristotle, between rapaſ}oxi et Adryos : from
the instances he gives, it is plain that the former cor-
responds (not to Parable, in the sense in which we
use the word, derived from that of Tapapox) in the
Sacred Writers, but) to Illustration ; the latter to
Fable or Tale. In the former, an allusion only is made
to a case easily supposable ; in the latter, a fictitious
story is narrated. Thus, in his instance above cited,
of Illustration, if any one, instead of a mere allusion,
should relate a tale, of mariners choosing a steersman by
lot, and being wrecked in consequence, Aristotle would
evidently have placed that under the head of Logos.
The other method is of course preferable, from its
brevity, whenever the allusion can be readily under-
stood : and accordingly it is common, in the case of
well-known fables, to allude to, instead of narrating,
them. That, e.g. of the horse and the stag, which he
gives, would, in the present day, be rather alluded to
than told, if we wished to dissuade a people from
calling in a too powerful auxiliary. It is evident that
a like distinction might have been made in respect of
historical examples; those cases which are well
known, being often merely alluded to, and not recited.
The word “Fable” is at present generally limited to
those fictions in which the resemblance to the matter
in question is not direct, but analogical ; the other
class being called Novels, Tales, &c. Those resem-
blances are, (as Dr. A. Smith has observed) the most
striking, in which the things compared are of the most
dissimilar nature ; as is the case in what we call Fables;
and such accordingly are generally preferred for Argu-
mentative purposes, both from that circumstance
itself, and also on account of the greater brevity which
is, for that reason not only allowed but required in
them.* For a Fable spun out to a great length becomes
an Allegory, which generally satiates and disgusts ; on
the other hand, a fictitious Tale, having a more direct,
* A Novel or Tale may be compared to a Picture; a Fable to
a Device. -
254 R H E T O R. I. C.
Rhetoric, and therefore less striking, resemblance to reality, re-
\-v- quires that an interest in the events and persons should
mal evidence of Christianity in general, proves the most Chap. I.
satisfactory to a believer's mind, but is not that which S-N-"
be created by a longer detail, without which it would be
insipid. The Fable of the Old Man and the Bundle of
Sticks, compared with the Iliad, may serve to exem-
plify what has been said; the moral conveyed by each
being the same, viz. the strength acquired by union,
and the weakness resulting from division ; the latter
fiction would be perfectly insipid if conveyed in a
few lines; the former, in twenty-four books, insup-
portable.
Of the various uses, and of the real or apparent re-
futation, of Examples, (as well as of other Arguments),
we shall treat hereafter; but it may be worth while
here to observe, that we have been speaking of Ex-
ample as a kind of Argument, and with a view therefore
to that purpose alone; it often happens, that a resem-
blance, either direct, or analogical, is introduced for
other purposes; viz. not to prove anything, but either
to illustrate and explain one's meaning, (which is the
strict etymological use of the word Illustration,) or to
amuse the fancy by ornament of language. It is of
course most important to distinguish, both in our
own compositions and those of others, between these
different purposes.
Of the various use and order of the several kinds of Pro-
position and of Argument, in different cases.
§ 3. The first rule to be observed is, that it should
be considered, whether the principal object of the
discourse be, to give satisfaction to a candid mind, and
convey instruction to those who are ready to receive it,
or to compel the assent, or silence the objections, of an
opponent. The former of these purposes is, in general,
principally to be accomplished by the former of those
two great classes into which arguments were divided;
(viz. by those from Cause to Effect,) the other, by the
latter
To whatever class, however, the Arguments we
resort to may belong, the general tenour of the reason-
ing will, in many respects, be affected by the present
consideration. The distinction in question is never-
theless in general little attended to. It is usual to call
an Argument, simply, strong or weak, without refer-
ence to the purpose for which it is designed ; whereas
the Arguments which afford the most satisfaction to a
candid mind, are often such as would have less weight
in controversy than many others, which again would be
less suitable for the former purpose.* E. g. the inter-
* Our meaning cannot be better illustrated than by an instance
referred to in that incomparable specimen of Reasoning, Dr.
Paley's Horae Paulinae. “ When we take into our hands the
letters,” (viz. St. Paul's Epistles,) “ which the suffrage and
consent of antiquity hath thus transmitted to us, the first thing
that strikes our attention is the air of reality and business, as
well as of seriousness and conviction, which pervades the whole.
Let the sceptic read them. If he be not sensible of these qualities
in them, the argument can have no weight with him. If he be ;
if he perceive in almost every page the language of a mind actu-
ated by real occasions, and operating upon real circumstances, I
would wish it to be observed, that the proof which arises from
this perception is not to be deemed occult or imaginary, because
it is incapable of being drawn out in words, or of being conveyed
to the apprehension of the reader in any other way, than by send-
ing him to the books themselves.” p. 403.
There is also a passage in Dr. A. Smith's Theory of Moral Sen-
timents, which illustrates very happily one of the applications of
the principle in question: “Sometimes we have occasion to
makes the most show in the refutation of infidels; the
Arguments from Analogy on the other hand, which
are the most unanswerable, are not so pleasing and
consolatory. t
Rule second. Matters of Opinion, (as they are called;
i. e. where we are not said properly to know, but to
judge,) are established chiefly by Antecedent-proba-
bility; (Arguments of the first class, viz. from Cause
to Effect,) though the testimony of wise men is also
admissible; past Facts, chiefly by Signs, of various
kinds ; (that term, it must be remembered, including:
Testimony,) and future events by Antecedent-proba-
bilities and Examples. -
Example, however, is not excluded from the proof of
matters of opinion; since a man's judgment in one
case, may be aided or corrected by an appeal to his
judgment in another similar case. It it in this way
that we are directed, by the highest authority, to guide.
our judgment in those questions, in which we are
most liable to deceive ourselves ; viz. what, on each
occasion, ought to be our conduct towards another;
we are directed to frame for ourselves a similar Sup-
posed case, by imagining ourselves to change places
with our neighbour, and then considering how, in that
case, we should wish to be treated.
It happens more frequently, however, that, when in
the discussion of matters of opinion, an Example is in-
troduced, it is designed, not for Argument, but, strictly
speaking, for Illustration ;—not to prove the proposition
in question, but to make it more clearly understood;
e.g. the Proposition maintained by Cicero, (de Off.
book iii.) is what may be accounted a matter of opinion;
viz. that “nothing is expedient which is dishonourable;”
when then he adduces the Example of the supposed
design of Themistocles to burn the allied fleet, which
he maintains, in contradiction to Aristides, would not
have been expedient, because it would have been
unjust, it is manifest, that we must understand the
instance brought forward as no more than an Illustra-
tion of the general principle he intends to establish ;
since it would be a plain begging of the question to
argue from a particular assertion, which could only
defend the propriety of observing the general rules of justice by
the consideration of their necessity to the support of society.
We frequently hear the young and the licentious ridiculing the
most sacred rules of morality, and professing, sometimes from
the corruption, but more frequently from the vanity of their
hearts, the most abominable maxims of conduct. Our indigna-
tion rouses, and we are eager to refute and expose such detest-
able principles. But though it is their intrinsic hatefulness and
detestableness which originally inflames us against them, we are
unwilling to assign this as the sole reason why we condemn them,
or to pretend that it is merely because we ourselves hate and
detest them. The reason, we think, would not appear to be con-
clusive. Yet, why should it not; if we hate and detest them
because they are the natural and proper objects of hatred and
detestation But when we are asked why we should not act in
such or such a manner, the very question seems to suppose that,
to those who ask it, this manner of acting does not appear to be
for its own sake the natural and proper object of those sentiments.
We must show them, therefore, that it ought to be so for the
sake of something else. Upon this account we generally cast
about for other arguments, and the consideration which first
occurs to us, is the disorder and confusion of Society which
would result from the universal prevalence of such practices. We
seldom fail, therefore, to insist upon this topic.” (Part ii. Sec. ii.
p. 151, 152, vol. i. ed. 1812.)
R H E TO R I. C. 255
sion might be allowed) unplausible; and this prejudice Chap. H.
is to be removed by the Argument from Cause to S-V-
Rhetoric. be admitted by those who assented to the general
S-N-" principle. -
It is important to distinguish between these two uses
of Example ; that on the one hand we may not be led
to mistake for an Argument such an one as the fore-
going; and that on the other hand, we may not too
hastily charge with sophistry him who adduces such an
one simply with a view to explanation. .
It is also of the greatest consequence to distinguish
between Examples (of the invented kind,) properly
so called, i. e. which have the force of Arguments, and
Comparisons introduced for the ornament of style, in
the form, either of simile, as it is called, or a Metaphor.
Not only is an ingenious comparison often mistaken
for a proof, though it be such as, when tried by the
rules laid down in the present Article, and under the
head of LoGIC, affords no proof at all; but also on the
other hand, a real and valid argument is not unfre-
quently considered merely as an ornament of style, if
it happen to be such as to produce that effect; though
there is evidently no reason why that should not be
fair Analogical Reasoning, in which the new idea in-
troduced by the Analogy chances to be a sublime or a
pleasing one. E. g. “The efficacy of penitence, and
piety, and prayer, in rendering the Deity propitious,
is not irreconcileable with the immutability of his
nature, and the steadiness of his purposes. It is not
in man's power to alter the course of the sun ; but it
is often in his power to cause the sun to shine or not
to shine upon him ; if he withdraws from its beams,
or spreads a curtain before him, the sun no longer
shines on him ; if he quits the shade, or removes the
curtain, the light is restored to him ; and though no
change is in the mean time effected in the heavenly lu-
minary, but only in himself, the result is the same as if it
were. Nor is the immutability of God any reason why
the returning sinner, who tears away the veil of pre-
judice or of indifference, should not again be blessed
with the sunshine of divine favour.” The image here
introduced is ornamental, but the Argument is not the
less perfect; since the case adduced fairly establishes
the general principle required, that “a change effected
in one of two objects having a certain relation to
each other, may have the same practical result as if it
had taken place in the other.”
The mistake in question, is still more likely to
occur when such an Argument is conveyed in a single
term employed metaphorically; as is generally the
case where the allusion is common and obvious ; e. g.
“we do not receive as the genuine doctrines of the pri-
mitive Church what have passed down the polluted stream
of Romish tradition.” The Argument here is not the less
valid for being conveyed in the form of a Metaphor.
The employment, in questions relating to the future,
both of the Argument from Example, and of that from
Cause to Effect, may be explained from what has been
already said concerning the connection between them;
some cause, whether known or not, being always
supposed, whenever an Example is adduced. -
Rule third. When Arguments of each of the two
formerly-mentioned classes are employed, those from
Cause to Effect (Antecedent-probability) have usually
the precedence. -
Men are apt to listen with prejudice to the Argu-
ments adduced to prove any thing which appears
abstractedly improbable ; i. e. according to what has
been above laid down, unnatural, or (if such an expres-
Effect, which thus prepares the way for the reception
of the other Arguments; e. g. if a man who bore a good
character, were accused of corruption, the strongest
evidence against him might avail little ; but if he were
proved to be of a covetous disposition, this, though it
would not alone be allowed to substantiate the crime,
would have great weight in inducing his judges to lend
an ear to the evidence. And thus, in what relates to
the future also, the a priori Argument and Example
support each other, when thus used in conjunction and
in the order prescribed ; a sufficient cause being esta-
blished, leaves us still at liberty to suppose that there
may be circumstances which will prevent the effect
from taking place ; but Examples subjoined show that
these circumstances do not, at least always, prevent
that effect; and on the other hand, Examples intro-
duced at the first, may be suspected of being excep-
tions to the general rule, (unless they are very
numerous,) instead of being instances of it; which an
adequate cause previously assigned, will show them to
be ; e. g. if any one had argued, from the temptations
and opportunities occurring to a military commander,
that Buonaparte was likely to establish a despotism
on the ruins of the French Republic, this Argument,
by itself, would have left men at liberty to suppose
that such a result would be prevented by a jealous
attachment to liberty in the citizens, and a fellow
feeling of the soldiery with them ; then, the Examples
of Caesar and of O. Cromwell would have proved, that
such preventives are not to be trusted.
Aristotle accordingly has remarked on the expe-
diency of not placing Examples in the foremost rank
of Arguments; in which case, he says, a considerable
number would be requisite; whereas, in confirmation,
even one will have much weight. This observation,
however, he omits to extend, as he might have done,
to Testimony and every other kind of Sign, to which
it is no less applicable.
Another reason for adhering to the order here pre-
scribed is, that if the Argument from Cause to Effect,
were placed after the others, a doubt might often
exist, whether we were engaged in proving the point in
question, or (assuming it as already proved) in seeking
only to account for it; that Argument being, by the
very nature of it, such as would account for the truth
contended for, supposing it were granted. Constant
care, therefore, is requisite to guard against any con-
fusion or indistinctness as to the object in each case
proposed ; whether that be, when a proposition is
admitted, to assign a cause which does account for it,
(which is one of the classes of Propositions formerly
noticed) or, when it is not admitted, to prove it by an
Argument of that kind which would account for it, if it
were granted.
With a view to the Arrangement of Arguments, no
rule is of more importance than the one now under
consideration ; and Arrangement is a more important
point than is generally supposed; indeed it is not per-
haps of less consequence in Rhetoric than in the Mili-
tary Art; in which it is well known, that with an
equality of forces, in numbers, courage, and every
other point, the manner in which they are drawn up,
so as either to afford mutual support, or on the other
hand, even to impede and annoy each other, may make
the difference of victory or defeat.
256
R H E 'I' O R. I. C.
Rhetoric.
E. g. in the statement of the Evidences of our Reli-
S-\r-gion, so as to give them their just weight, much
depends on the Order in which they are placed. The
Antecedent-probability that a Revelation should be
given to man, and that it should be established by
miracles, all would allow to be, considered by itself,
in the absence of strong direct testimony, utterly
insufficient to establish the Conclusion. On the other
hand, miracles considered abstractedly, as represented
to have occurred without any occasion or reason for
them being assigned, carry with them such a strong
intrinsic improbability as could not be wholly sur-
mounted even by such evidence as would fully estab-
lish any other matters of fact. But the evidences of
the former class, however inefficient alone towards
the establishment of the Conclusion, have very great
weight in preparing the mind for receiving the other
Arguments ; which again, though they would be
listened to with prejudice if not so supported, will
then be allowed their just weight. The writers in
defence of Christianity have not always attended to
this principle; and their opponents have often availed
themselves of the knowledge of it, by combating in
detail Arguments the combined force of which would
have been irresistible. They argue respecting the
credibility of the Christian miracles, abstractedly, as if
they were insulated occurrences, without any known
or conceivable purpose ; as e.g. “what testimony is
sufficient to establish the belief that a dead man
was restored to life 2'' and then they proceed to show
that the probability of a Revelation, abstractedly con-
sidered, is not such at least as to establish the fact
that one has been given. Whereas, if it were first
proved (as may easily be done) merely that there is
no such abstract improbability of a Revelation as to
exclude the evidence in favour of it, and that if one
were given, it might be expected to be supported by
miraculous evidence, then, just enough reason would
be assigned for the occurrence of miracles, not indeed
to establish them, but to allow a fair hearing for the
Arguments by which they are proved.
The importance attached to the Arrangement of
Arguments by the two great rival orators of Athens,
may serve to illustrate and enforce what has been
said. Afschines strongly urged the judges (in the cele-
brated contest concerning the crown) to confine his
adversary to the same order in his reply to the charges
brought, which he himself had observed in bringing
them forward. Demosthenes however was far too
skilful to be thus entrapped ; and so much import-
ance does he attach to this point, that he opens his
speech with a most solemn appeal to the Judges for
an impartial hearing; which implies, he says, not only
a rejection of prejudice, but no less also a permission
for each speaker to adopt whatever Arrangement
he should think fit. And accordingly he proceeds
to adopt one very different from that which his anta-
gonist had laid down ; for he was no less sensible
than his rival that the same Arrangement which is
the most favourable to one side, is likely to be the
least favourable to the other.
It is to be remembered however, that the rules
which have been given respecting the Order in
which different kinds of Argument should be arranged,
relate only to the different kinds of Arguments ad-
duced in support of each separate Proposition ; since
of course the refutation of an opposed assertion, effected
by means of signs, may be followed by an a priori Chap. I.
Argument in favour of our own Conclusion; and the S-SA-
like in many other such cases.
Rule fourth. A Proposition that is well known (whe-
ther easy to be established or not) should in general
be stated at once, and the Proofs subjoined ; but if it
be not familiar to the hearers, and especially if it be
likely to be unacceptable, it is usually better to state
the Arguments first, or at least some of them, and
then introduce the Conclusion.
There is no question relating to Arrangement more
important than the present ; and it is therefore the
more unfortunate that Cicero, who possessed so much
practical skill, should have laid down no rule on this
point, (though it is one which evidently had engaged
his attention,) but should content himself with saying
that sometimes he adopted the one mode and some-
times the other,” (which doubtless he did not do at
random,) without distinguishing the cases in which
each is to be preferred, and laying down principles to
guide our decision. Aristotle also, when he lays
down the two great heads into which speech is divi-
sible, the Proposition and the Proof, f is equally
silent as to the order in which they should be placed ;
though he leaves it to be understood, from his
manner of speaking, that the Conclusion (or Ques-
tion) is to be first stated, and then the Premises, as
in Mathematics. This indeed is the usual and natural
way of speaking or writing; viz. to begin by declar-
ing your Opinion, and then to subjoin the Reasons for
it. But there are many occasions on which it will be
of the highest consequence to reverse this plan. It
will sometimes give an offensively dogmatical air to a
Composition to begin by advancing some new and un-
expected assertion; though sometimes again this may
be advisable, when the Arguments are such as can
be well relied on, and the principal object is to excite
attention, and awaken curiosity. And accordingly,
with this view, it is not unusual to present some doc-
trine, by no means really novel, in a new and para-
doxical shape. But when the Conclusion to be
established is one likely to hurt the feelings and
offend the prejudices of the hearers, it is essential to
keep out of sight, as much as possible, the point to
which we are tending, till the principles from which
it is to be deduced shall have been clearly established;
because men listen with prejudice, if at all, to Argu-
ments which are avowedly leading to a Conclusion
which they are indisposed to admit ; whereas if we
thus, as it were, mask the battery, they will not be
able to shelter themselves from the discharge. The
observance accordingly, or neglect, of this rule, will
often make the difference of success or failure.
And it will often be advisable to advance very gra-
dually to the full statement of the Proposition
required, and to prove it, if one may so speak, by
instalments, establishing separately, and in order,
each part of the truth in question. It is thus that
Aristotle establishes many of his doctrines, and among
others his definition of happiness, in the beginning of
the Nicomachean Ethics; he first proves in what it does
not consist, and then establishes, one by one, the
several points which together constitute his notion.
Rule fifth. If the Argument a priori has been
introduced in the proof of the main Proposition in
* De Orat. + Rhet. book iii.
R. H. E. T. O. R. I. C. 257
the conduct of it will not be unsuitable in this place. Chap. I.
Rhetoric question, there will generally be no need of afterwards
In the first place, it is to be observed that there is Q-->
\-y-/adducing Causes to account for the truth established ;
(since that will have been already done in the course
of the Argument,) on the other hand, it will often be
advisable to do this when Arguments of the other
class have alone been employed.
For it is in every case agreeable and satisfactory,
and may often be of great utility, to explain, where it
can be done, the Causes which produce an Effect that
is itself already admitted to exist. But it must be
remembered that it is of great importance to make it
clearly appear which object is, in each case, proposed;
whether to establish the fact, or to account for it ;
since otherwise we may often be supposed to be em-
ploying a feeble Argument; for that which is a satis-
factory explanation of an admitted fact, will frequently
be such as would be very insufficient to prove it, Sup-
posing it were doubted.
Rule sirth. Refutation of Objections should gene-
rally be placed in the midst of the other Arguments,
but nearer the beginning than the end.
If indeed very strong Objections have obtained
much currency, or have been just stated by an oppo-
ment, so that what is asserted is likely to be regarded
as paradoxical, it may be advisable to begin with a
Refutation ; but when this is not the case, the men-
tion of Objections in the opening will be likely to give
a paradoxical air to our assertion, by implying a con-
sciousness that much may be said against it. If
again all mention of Objections be deferred till the
last, the other Arguments will often be listened to
with prejudice by those who may suppose us to be
overlooking what may be urged on the other side.
Sometimes indeed it will be difficult to give a satis-
factory Refutation of the opposed Opinions till we
have gone through the Arguments in support of our
own : even in that case however it will be better to
take some brief notice of them early in the Composi-
tion, with a promise of afterwards considering them
more fully, and refuting them. This is Aristotle's
usual mode of procedure.
A Sophistical use is often made of this last rule,
when the Objections are such as cannot really be satis-
factorily answered. The skilful Sophist will often, by
the promise of a triumphant Refutation hereafter,
gain attention to his own statement, which, if it be made
plausible, will so draw off the hearer's attention from
the Objections, that a very inadequate fulfilment of
that promise will pass unnoticed, and due weight will
not be allowed to the Objections.
It may be worth remarking, that Refutation will
often occasion the introduction of fresh Propositions;
i. e. we may have to disprove Propositions, which,
though incompatible with the principal one to be
maintained, will not be directly contradictory to it;
e.g. Burke, in order to the establishment of his
theory of beauty, refutes the other theories which
have been advanced by those who place it in “fitness”
for a certain end—in “proportion "–in “perfec-
tion,” &c.; and Dr. A. Smith, in his Theory of Moral
Sentiments, combats the opinion of those who make
expediency the test of virtue—of the advocates of a
“Moral sense,” &c. which doctrines respectively are
at variance with those of these authors, and imply,
though they do not express, a contradiction of them.
Though we are at present treating principally of
the proper collocation of Refutation, some remarks on
VOL., I,
(as Aristotle remarks, Rhet. book ii., apparently in oppo-
sition to some former writers) no distinct class of
refutatory Arguments, since they become such merely
by the circumstances under which they are em-
ployed. -
There are two ways in which any Proposition may
be refuted; * first, by proving the contradictory of it;
second, by overthrowing the Arguments by which it
has been supported. The former of these is less
strictly and properly called Refutation, being only
accidentally such, since it might have been employed
equally well had the opposite Argument never existed;
and in fact it will often happen that a Proposition
maintained by one author may be in this way refuted
by another, who had never heard of his Arguments.
Thus Pericles is represented by Thucydides as prov-
ing, in a speech to the Athenians, the probability of
their success against the Peloponnesians, and thus,
virtually, refuting the speech of the Corinthian am-
bassador at Sparta, who had laboured to show the
probability of their speedy downfalt In fact, every
one who argues in favour of any Conclusion is virtu-
ally refuting, in this way, the opposite Conclusion.
But the character of Refutation more strictly be-
longs to the other mode of procedure ; viz. in which
a reference is made, and an answer given, to some
specific Arguments in favour of the opposite Conclu-
sion. This may consist either in the denial of one of
the Premises, or an Objection against the conclusiveness
of the Reasoning. And here it is to be observed that the
Objection is often supposed, from the mode in which
it is expressed, to belong to this last class, when in
truth it does not, but consists in the contradiction of
a Premiss; for it is very common to say, “I admit
your principle, but deny that it leads to such a con-
sequence;” “ the assertion is true, but it has no
force as an Argument to prove that Conclusion;” this
sounds like an objection to the Reasoning itself, but it.
will often be found to amount only to a denial of the
suppressed Premiss of an Enthymeme; the assertion
which is admitted being only the expressed Premiss,
whose force as an Argument must of course depend
on the other Premiss which is understood. Thus
Warburton admits that in the Law of Moses the doc-
trine of a future state was not revealed ; but contends
that this, so far from disproving, as the Deists pre-
tend, his Divine mission, does, on the contrary, estab-
lish it. But the Objection is not to the Deist's Argu-
ment properly so called, but to the other Premiss,
which they so hastily took for granted, and which he
disproves, viz. “ that a divinely-commissioned Law-
giver would have been sure to reveal that doctrine."
The Objection is then only properly said to lie against
the Reasoning itself, when it is shown that granting
all that is assumed on the other side, whether ex-
pressed or understood, still the Conclusion contended
for would not follow from the Premises, either on
account of some ambiguity in the Middle Term, or
* 'AvravXAoylagös and évaraqis of Aristotle, book ii.
† The speeches indeed are avowedly the composition of the
historian ; but he professes to give the substance of what was
either actually said, or likely to be said, on each occasion ; and
the Arguments urged in the speeches now in question are un-
jºy such as the respective speakers would be likely to
employ. . -
2 M.
258
R H E T O B I. C.
Rhetoric, some other fault of that class. (See LoGIC, chapter on
\-y-/ Fallacies.)
It may be proper in this place to remark, that
“Indirect Reasoning ” is sometimes confounded with
“Refutation,” or supposed to be peculiarly connected
with it; which is not the case; either Direct or Indirect
Reasoning being employed indifferently for Refutation
as well as for any other purpose. The application of
the term “elenctic,” (from éAéºxetv to refute or dis-
prove,) to Indirect, Arguments, has probably contri-
buted to this confusion ; which, however, principally
arises from the very circumstance that occasioned such
a use of that term ; viz. that in the Indirect method
the absurdity or falsity of a Proposition (opposed to our
own) is proved; and hence is suggested the idea of
an adversary maintaining that Proposition, and of the
Refutation of that adversary being necessarily accom-
plished in this way. But it should be remembered
that Euclid and other mathematicians, though they
can have no opponent to refute, often employ the
Indirect Demonstration ; and that on the other
hand, if the contradictory of an opponent's Premiss
can be satisfactorily proved in the Direct Method, the
Refutation is sufficient. It is true however that
while in science the Direct Method is considered pre-
ferable, in controversy the Indirect is often adopted
by choice, as it affords an opportunity for holding up
an opponent to scorn and ridicule, by deducing some
very absurd Conclusion from the principles he main-
tains, or according to the mode of arguing he em-
ploys. Nor indeed can a fallacy be so clearly exposed
to the unlearned reader in any other way. For it is
no easy matter to explain, to one ignorant of Logic,
the grounds on which you object to an inconclusive
Argument, though he will be able to perceive its cor-
respondence with another brought forward to illus-
trate it, in which an absurd Conclusion may be
introduced, as drawn from true Premises. *
It is evident that either the Premiss of an opponent
or his Conclusion may be disproved, either in the
Direct or in the Indirect Method; i. e. either by prov-
ing the truth of the Contradictory, or by showing that
an absurd Conclusion may fairly be deduced from the
Proposition in question : when this latter mode of
Refutation is adopted with respect to the Premiss,
the phrase by which this procedure is usually desig-
nated, is, that the “Argument proves too much;” i.e.
that it proves, besides the Conclusion drawn, another,
which is manifestly inadmissible ; e. g. the Argu-
ment by which Dr. Campbell labours to prove that
every correct Syllogism must be nugatory, as involv-
ing a “petitio principii,” proves, if admitted at all,
more than he intended, since it may easily be shown
to be equally applicable to all Reasoning whatever.
It is worth remarking, that that which is in sub-
stance an Indirect Argument, may easily be altered in
form so as to be stated in the Direct Mode. For,
strictly speaking, that is Indirect Reasoning in which
we assume as true-the Proposition whose Contradic-
tory it is our object to prove ; and deducing regularly
from it an absurd Conclusion, infer thence that the
Premiss in question is false; the alternative proposed
in all correct Reasoning being either to admit the
Conclusion, or to deny one of the Premises ; but by
adopting the form of a Destructive Conditional,” the
* See LoG1c.
same Argument as this in substance may be stated , Chap. I.
directly ; e.g. we may say, let it be admitted that no
testimony can satisfactorily establish such a fact as
is not agreeable to our experience ; thence it will
follow, that the Eastern Prince judged wisely and
rightly in at once rejecting, as a manifest falsehood,
the account given him of the phenomenon of ice;
but he was evidently mistaken in so doing; there-
fore the Principle assumed is unsound. Now the
substance of this Argument remaining the same, the
form of it may be so altered as to make the Argu-
ment Direct ; viz. “if it be true that no testimony,
&c. that Eastern Prince must have judged wisely, &c.
but he did not ; therefore that Principle is not
true.”
Universally indeed a Conditional Proposition may
be regarded as an assertion of the validity of a certain
Argument; the Antecedent corresponding to the
Premises, and the Consequent to the Conclusion; and
neither of them being asserted as true, only the
dependence of the one on the other ; the alternative
then is, to admit the Consequent, (which forms the
Constructive Syllogism,) or to deny the Antecedent,
which forms the Destructive; and the former accord-
ingly corresponds to Direct Reasoning, the latter to
Indirect, being, as has been said, a mode of stating
it in the Direct form, as is evident from the examples
adduced.
The difference between these two modes of stating
such an Argument is considerable, when there is a
long chain of Reasoning ; for when we employ the
Categorical form, and assume as true the Premises we
design to disprove, it is evident we must be speaking
ironically, and in the character, assumed for the mo-
ment, of an adversary; when, on the contrary, we
use the hypothetical form, there is no irony. Butler's
Analogy is an instance of the latter procedure;
he contends that if such and such objections are
admissible against Religion, they must be applied
equally to the constitution and course of nature.
Had he, on the other hand, assumed, for the Argu-
ment's sake, that such objections against Religion are
valid, and had thence proved the condition of the
natural world to be totally different from what we see
it to be, his Arguments, which would have been the
same in substance, would have assumed an ironical
form. This form has been adopted by Burke in his
celebrated Defence of Natural Society, by a late noble
Lord;” in which, assuming the person of Bolingbroke,
he proves, according to the principles of that author,
that the Arguments he brought against ecclesiastical,
would equally lie against civil institutions.
It is in some respects a recommendation of this
latter method, and in others an objection to it, that
the sophistry of an adversary will often be exposed by
it in a ludicrous point of view; and this, even where
no such effect is designed; the very essence of jest
being its mimic sophistry.T This will often give addi-
tional force to the Argument, by the vivid impression
which ludicrous images produce : but again, it will
not unfrequently have this disadvantage, that weak
men, perceiving the wit, are apt to conclude that
nothing but wit is designed, and lose sight perhaps of
* This is an Argument from Analogy, as well as Bishop But-
ler's, though not relating to the same point, Butler's being a
defence of the Doctrines of Religion.
f See Logic, Chapter on Fallacies, at the conclusion.
R H E T O R. I. C. 259
Conclusion drawn may nevertheless be true : yet men Chap. J.
Rhetoric, a solid and convincing Argument, which they regard r
are apt to take for granted that the Conclusion itself S-v-/
S-N-' as no more than a good joke. Having been warned
that “ ridicule is not the test of truth,” and that
“ wisdom and wit” are not the same thing, they
distrust every thing that can possibly be regarded as
witty; not having judgment to perceive the combina-
tion, when it occurs, of wit with sound Reasoning.
The ivy-wreath completely conceals from their view
the point of the Thyrsus : and moreover if such a
mode of Argument be employed on serious subjects,
the “weak brethren” are sometimes scandalized by
what appears to them a profanation; not having dis-
cernment to perceive when it is that the ridicule
does, and when it does not, affect the solemn subject
itself. But for the respect paid to Holy Writ, the
taunt of Elijah against the prophets of Baal would
probably appear to such persons irreverent. And the
caution now implied will appear the more important
when it is considered how large a majority they are,
who, in this point, come under the description of
** weak brethren :” he that can laugh at what is
Iudicrous, and at the same time preserve a clear dis-
cernment of sound and unsound Reasoning, is no
ordinary man.
It may be observed generally that too much stress
is often laid, especially by unpractised Reasoners, on
Refutation; (in the strictest and narrowest sense, i. e.
of Objections to the Premises, or to the Reasoning,)
they are apt both to expect a Refutation where none
can fairly be expected, and to attribute to it, when
satisfactorily made out, more than it really accom-
plishes.
For first, not only specious, but real and solid Argu-
ments, such as it would be difficult or impossible to
refute, may be urged against a Proposition which is
nevertheless true, and may be satisfactorily established
by a preponderance of probability. It is in strictly
scientific Reasoning alone that all the Arguments
which lead to a false Conclusion must be fallacious :
in what is called moral or probable Reasoning, there
may be sound Arguments and valid objections on
both sides;* e. g. it may be shown that each of two
contending parties has some reason to hope for suc-
cess; and this, by irrefragable Arguments on both
sides, leading to Conclusions which are not contra-
dictory to each other; for though only one party can
obtain the victory, it may be true that each has some
reason to expect it. The real question in such cases is,
which event is the more probable;—on which side the
evidence preponderates. Now it often happens that
the inexperienced Reasoner, thinking it necessary that
every Objection should be satisfactorily answered,
will have his attention drawn off from the Arguments
of the opposite side, and will be occupied perhaps in
making a weak defence, while victory was in his
hands. The Objection perhaps may be unanswerable,
and yet may safely be allowed, if it can be shown that
more and weightier Objections lie against every other
supposition. This is a most important caution for
those who are studying the Evidences of Religion.
Secondly, the force of a Refutation is often over-
rated : an Argument which is satisfactorily answered
ought to go for nothing; but it is possible that the
* “There are objections against a Plenum, and objections
against a Vacuum ; but one of them must be true,” Johnson.
g
is disproved, when the Arguments brought forward to
establish it have been satisfactorily refuted; assuming,
when perhaps there is no ground for the assumption,
that these are all the Arguments that could be urged.
This may be considered as the fallacy of denying the
Consequent of a Conditional Proposition, from the
Antecedent having been denied : “if such and such
an Argument be admitted, the Assertion in question
is true ; but that Argument is inadmissible; therefore
the Assertion is not true.” Hence the injury done to
any cause by a weak advocate; the cause itself appear.
ing to the vulgar to be overthrown when the Argu.
ments brought forward are answered.
On the same principle is founded a most important
maxim, that it is not only the fairest, but also the
wisest plan, to state Objections in their full force; at
least, wherever there does exist a satisfactory answer
to them ; otherwise, those who hear them stated more
strongly than by the uncandid advocate who had
undertaken to repel them, will maturally enough con-
clude that they are unanswerable. It is but a mo-
mentary and ineffective triumph that can be obtained
by manoeuvres like those of Turnus's charioteer, who
furiously chased the feeble stragglers of the army,
and evaded the main front of the battle.
Rule seventh. The Arguments which should be
placed first in order are, catteris paribus, the most
Obvious, and such as naturally first occur. -
This is evidently the natural order; and the adher-
ence to it gives an easy, natural air to the Composi-
tion. It is seldom therefore worth while to depart
from it for the sake of beginning with the most power-
ful Arguments, (when they happen not to be also the
most Obvious) or on the other hand, for the sake of
reserving these to the last, and beginning with the
weaker : or, again, of imitating, as some recommend,
Nestor's plan of drawing up troops, placing the best
first and last, and the weakest in the middle. It will
be advisable however (and by this means you may
secure this last advantage) when the strongest Argu-
ments naturally occupy the foremost place, to recapitu-
late in a reverse order; which will destroy the appear-
ance of anti-climax, and is also in itself the most easy
and natural mode of recapitulation. Let, e.g. the
Arguments be A, B, C, D, E, &c. each less weighty
than the preceding; then in recapitulating proceed
from E to D, C, B, concluding with A.
Of Introduction.
§ 4. A Proem, Exordium, or Introduction, is, as
Aristotle has justly remarked, not to be accounted one
of the essential parts of a Composition, since it is not
in every case necessary. In most, however, except
such as are extremely short, it is found advisable to
premise something before we enter on the main Argu-
ment, to avoid an appearance of abruptness, and to
facilitate, in some way or other, the Object proposed.
In larger works this assumes the appellation of Pre-
face or Advertisement ; and not unfrequently two are
employed, one under the name of Preface, and ano-
ther, more closely connected with the main work,
under that of Introduction.
The rules which have been laid down already will
2 M 2
260
R H E T O R. I. C.
Rhetoric, apply equally to that preliminary course of Argument
S-y- of which Introductions often consist.
The writers before Aristotle, are censured by him
for inaccuracy, in placing under the head of Introduc-
tions, as properly belonging to them, many things
which are not more appropriate in the beginning than
elsewhere; as, e. g. the contrivances for exciting
the hearer's attention ; which, as he observes, is an
improper Arrangement; since, though such an In-
troduction may sometimes be required, it is, generally
speaking, anywhere else rather than in the beginning,
that the attention is likely to flag.
The rule laid down by Cicero, (De Orat.) not to
compose the Introduction first, but to consider first
the main Argument, and let that suggest the Exordium,
is just and valuable; for otherwise, as he observes,
seldom any thing will suggest itself but vague gene-
ralities; “ common" topics, as he calls them, i. e.
what would equally well suit several different Com-
positions ; whereas the Introduction, which is com-
posed last, will naturally spring out of the main
subject, and appear appropriate to it.
1. One of the Objects most frequently proposed in
an Introduction, is, to show that the subject in question
is important, curious, or otherwise interesting, and
worthy of attention. This may be called an “Intro-
duction inquisitive.”
2. It will frequently happen also, when the point
to be proved or explained is one which may be very
fully established, or on which there is little or no
doubt, that it may nevertheless be strange, and different
from what might have been expected ; in which case
it will often have a good effect in rousing the atten-
tion, to set forth as strongly as possible this para-
dowical character, and dwell on the seeming improba-
bility of that which must, after all, be admitted. This
may be called an “Introduction paradoxical.”
fl-
* “If you should see a flock of pigeons in a field of corn; and
if (instead of each picking where and what it liked, taking just
as much as it wanted, and no more) you should see ninety-nine
of them gathering all they got, into a heap; reserving nothing
for themselves, but the chaff and the refuse ; keeping this heap
for one, and that the weakest, perhaps worst, pigeon of the
flock; sitting round, and looking on, all the winter, whilst this
one was devouring, throwing about, and wasting it; and if a
pigeon, more hardy or hungry than the rest, touched a grain of
the hoard, all the others instantly flying upon it, and tearing it
to pieces; if you should see this, you would see nothing more
than what is every day practised and established among men.
Among men, you see the ninety and nine toiling and scraping
together a heap of superfluities for one, (and this one too, often-
times the feeblest and worst of the whole set, a child, a woman,
a madman, or a fool ;) getting nothing for themselves all the
while, but a little of the coarsest of the provision, which their
own industry produces; looking quietly on, while they see the
fruits of all their labour spent or spoiled; and if one of the
number take or touch a particle of the hoard, the others joining
against him, and hanging him for the theft.
3. What may be called an “Introduction cor-
Chap. },
rective,” is also in frequent use; viz. to show that the S-V-
subject has been neglected, misunderstood, or misrepre-
sented by others. This will, in many cases, remove a
most formidable obstacle in the hearer's mind, the
anticipation of triteness, if the subject be, or may
supposed to be, a hacknied one ; and it may also
serve to remove or loosen such prejudices as might
be adverse to the favourable reception of our Argu-
mentS. -
4. It will often happen also, that there may be need
to explain some peculiarity in the mode of Reasoning
to be adopted; to guard against some possible mistake
as to the Object proposed ; or to apologise for some
deficiency: this may be called the “ Introduction
preparatory.”
5. And lastly, in many cases there will be occasion
for what may be called a “Narrative Introduction,”
to put the reader or hearer in possession of the out-
line of some transaction, or the description of some
state of things, to which references and allusions are
to be made in the course of the Composition. Thus,
in Preaching, it is generally found advisable to detail,
or at least briefly to sum up, a portion of Scripture
history, or a parable, when either of these is made
the subject of a Sermon.
Two or more of the Introductions that have been
mentioned are often combined, especially in the Pre-
face to a work of any length.
And very often the Introduction will contain appeals
to various passions and feelings in the hearers ; espe-
cially a feeling of approbation towards the Speaker,
or of prejudice against an opponent who has pre-
ceded him; but this is, as Aristotle has remarked, by
no means confined to Introductions.”
“ There must be some very important advantages to account
for an institution, which, in the view of it above given, is so
paradoxical and unnatural.
“ The principal of these advantages are the following :” &c.—
Paley's Moral Philosophy, book iii. part i. c. 1 and 2.
* It has not been thought necessary to treat of Conclusion,
Peroration, or Epilogue, as a distinct head : the general rules,
that a Conclusion should be neither sudden and abrupt, (so
as to induce the hearer to say, “I did not know he was going
to leave off,”) nor, again, so long as to excite the hearer's im-
patience after he has been led to expect an end, being so obvious
as hardly to need being mentioned. The matter of which the
concluding part of a Composition consists, will, of course,
vary according to the subject and the occasion ; but that
which is most appropriate, and consequently most frequent (in
Compositions of any considerable length,) is a Recapitulation,
either of a part or the whole of the Arguments that have been
used ; respecting which a remark has been made at the end
of Section 3.
Anything relative to the Feelings and the Will, that may be
especially appropriate to the Conclusion, will be mentioned in
its proper place. -
R H E T O R. I. C.
CHAPTER II.
OF PERSUASION.
PERSUASION, properly so called, i.e., the Art of in-
\-y-' fluencing the Will, is the next point to be considered.
And Rhetoric is often regarded (as was formerly re-
marked) in a more limited sense, as conversant about this
head alone. But even, according to that view, the rules
above laid down will be found not the less relevant ;
since the Conviction of the understanding (of which
we have hitherto been treating) is an essential part of
Persuasion, and will generally need to be effected by
the Arguments of the Writer or Speaker. For in order
that the Will may be influenced, two things are re-
quisite; viz. that the proposed Object should appear
desirable ; and that the Means suggested should be
proved to be conducive to the attainment of that Ob-
ject; and this last, evidently, must depend on a process
of Reasoning. In order, e.g. to induce the Greeks to
unite their efforts against the Persian invader, it was
necessary to prove that cooperation could alone render
their resistance effectual, and also to awaken such
feelings of patriotism, and abhorrence of a foreign yoke,
as might prompt them to make these combined efforts.
For it is evident, that however ardent their love of
liberty, they would make no exertions if they appre-
hended no danger; or if they thought themselves able,
separately, to defend themselves, would be backward
to join the confederacy; and on the other hand, that
if they were willing to submit to the Persian yoke, or
valued their independence less than their present ease,
the fullest conviction that the Means recommended
would secure their independence, would have no
practical effect.
Persuasion, therefore, depends on 1st, Argument,
(to prove the expediency of the Means proposed) and
2ndly, 'What is usually called Exhortation, i. e. the
incitement of men to adopt those Means, by repre-
senting the End as sufficiently desirable. It will hap-
pen indeed, not unfrequently, that the one or the other
of these Objects will have been already, either wholly
or in part, accomplished, so that the other shall be
the only one that it is requisite to insist on ; viz.
sometimes the hearers will be sufficiently intent on the
pursuit of the End, and will be in doubt only as to
the Means of attaining it; and sometimes, again, they
will have no doubt on that point, but will be indif-
ferent, or not sufficiently ardent, with respect to the
proposed End, and will need to be stimulated by
Exhortations. Not sufficiently ardent, we have said,
because it will not so often happen that the Object in
question will be one to which they are totally indif-
ferent, as that they will, practically at least, not
reckon it, or not feel it, to be worth the requisite
pains. No one is absolutely indifferent about the
attainment of a happy immortality; and yet a great
part of the Preacher's business consists of Exhortation,
i. e. endeavouring to induce men to use those exertions
which they themselves know to be necessary for the
attainment of it. -
Aristotle, and many other writers, have spoken of Chap. II.
Appeals to the Passions as an unfair mode of influenc- \-2–
ing the hearers; in answer to which Dr. Campbell
has remarked, that there can be no Persuasion without
an address to the Passions:* and it is evident, from
what has been just said, that he is right, if under the
term Passion is included every active principle of our
nature. This however is a geater latitude of meaning
than belongs even to the Greek word IIá0m, though
the signification of that is wider than, according to
ordinary use, that of our term “ Passions.” But
Aristotle by no means overlooked the necessity for
Persuasion, properly so termed, calling into action
Some motive that may influence the Will; it is plain
that whenever he speaks with reprobation of an
appeal to the Passions, his meaning is, the excitement
of such feekings as ought not to influence the decision
of the question in hand. A desire to do justice may
be called, in Dr. Campbell's wide acceptation of the
term, a Passion : this is what ought to influence a
Judge ; and no one would ever censure a Pleader for
striving to excite and heighten this desire; but if the
decision be influenced by an appeal to Anger, Pity,
&c. the feelings thus excited being such as ought not
to have operated, the Judge must be allowed to have
been unduly biassed ; and that this is Aristotle's mean-
ing is evident from his characterising the introduction
of such topics, as éºw Too Tpdquatos, “ foreign to the
matter in hand.” And it is evident that as the motives
—r
* “ To say, that it is possible to persuade without speaking
to the passions, is but at best a kind of specious nonsense. The
coolest reasoner always in persuading, addresseth himself to the
passions some way or other. This he cannot avoid doing, if he
speak to the purpose. To make me believe, it is enough to show
me that things are so ; to make me act, it is necessary to show
that the action will answer some End. That can never be an End
to me which gratifies no passion or affection in my nature. You
assure me, ‘It is for my honour.” Now you solicit my pride, with-
out which I had never been able to understand the word. You
say, ‘ It is for my interest.' Now you bespeak my self-love.
* It is for the public good.' Now you rouse my patriotism. “It
will relieve the miserable.' Now you touch my pity. So far
therefore it is from being an unfair method of persuasion to
Imove the passions, that there is no persuasion without moving
them. -
R. “ But if so much depend on passion, where is the scope for
argument 2 Before I answer this question, let it be observed,
that, in order to persuade, there are two things which must be
carefully studied by the orator. The first is, to excite some
desire or passion in the hearers; the second is, to satisfy their
judgment that there is a connection between the action to which
he would persuade them, and the gratification of the desire or
passion which he excites. This is the analysis of persuasion.
The former is effected by communicating lively and glowing
ideas of the object; the latter, unless so evident of itself as to
Supersede the necessity, by presenting the best and most forcible
arguments which the nature of the subject admits. In the one
lies the pathetic, in the other the argumentative. These incor-
porated together constitute that vehemence of contention to
which the greatest exploits of Eloquence ought doubtless to be
ascribed,”—Campbell's Philosophy of Rhetoric, book i, c. vii.
Sec. 4, o
262
R H E T O R. I. C.
Rhetoric, which ought to operate will be different in different
S-N-' cases, the same may be objectionable and not fairly
admissible in one case, which in another would be
perfectly allowable.* An instance occurs in Thucy-
dides, in which this is very judiciously and neatly
pointed out : in the debate respecting the Mityleneans,
who had been subdued after a revolt, Cleon is intro-
duced contending for the justice of inflicting on them
capital punishment ; to which Diodotus is made to
reply, that the Athenians are not sitting in judgment
on the offenders, but in deliberation as to their own
interest; and ought therefore to Čonsider, not the
right they may have to put the revolters to death, but
the expediency or inexpediency of such a procedure.
In judicial cases, on the contrary, any appeal to the
personal interests of the Judge, or even to public
expediency, would be irrelevant. In framing laws
indeed, and (which comes to the same thing) giving
those decisions which are to operate as precedents,
the public good is the Object to be pursued; but in
the mere administering of the established laws, it is
inadmissible. -
There are many feelings, again, which it is evident
should in no case be allowed to operate, as Envy,
thirst for Revenge, &c. &c. the excitement of which
by the Orator is to be reprobated as an unfair artifice;
but it is not the less necessary to be well acquainted
with them, in order to allay them when previously
existing in the hearers, or to counteract the efforts
of an adversary in producing or influencing them.
It is evident, indeed, that all the weaknesses, as well
as the powers of the human mind, and all the arts by
which the Sophist takes advantage of these weak-
nesses, must be familiarly known by a perfect Orator;
who, though he may be of such a character as to dis-
dain employing such arts, must not want the ability
to do so, or he would not be prepared to counteract
them. An acquaintance with the nature of poisons is
necessary to him who would administer antidotes.
The active principles of our nature may be classed
in various ways; the arrangement adopted by Mr.
Dugald Stewartf is, perhaps, the most correct and
convenient ; the heads he enumerates are Appetites,
(which have their origin in the body,) Desires, and
Affections; these last being such as imply some kind
of disposition relative to another Person ; to which
must be added, Self-love, or the desire of Happiness
as such, and the Moral faculty, called by some writers
Conscience, by others the Moral sense, and by Dr. A.
Smith, the sense of Propriety.
Under the head of Affections may be included the
sentiments of Esteem, Regard, Admiration, &c. which
it is so important that the audience should feel to-
wards the Speaker. Aristotle has considered this as
a distinct head, separating the consideration of the
speaker's Character ("H60s toſ Méyovros) from that of
the disposition of the hearers ; under which, how-
ever, it might, according to his own views, have been
included; it being plain from his manner of treating
of the speaker's Character, that he means, not his
real character, (according to the fanciful notion of
Quinctilian,) but the impression produced on the minds
of the hearers, by the speaker, respecting himself.
He remarks, justly, that the Character to be established
wrºr- -v-f-sºw-ºrus
* See the Treatise on FALLAcies, sec. 14.
† Outlines of Moral Philosophy.
is that of, 1st, Good Principle, 2ndly, Good Sense,
and 3rdly, Goodwill and friendly disposition towards
the audience addressed ; * and that if the Orator can
completely succeed in this, he will persuade more
powerfully than by the strongest Arguments. He
might have added, (as indeed he does slightly hint at
the conclusion of his Treatise,) that, where there is an
opponent, a like result is produced by exciting the
contrary feelings respecting him ; viz. holding him
up to contempt, or representing him as an object of
reprobation or suspicion.
To treat fully of all the different emotions and
springs of action which an Orator may at any time
find it necessary to call into play, or to contend
against, would be to enter on an almost boundless
field of Metaphysical inquiry, which does not pro-
perly fall within the limits of the subject now before
us : and on the other hand, a brief definition of each
passion, &c. &c. a few general remarks on it, could
hardly fail to be trite and uninteresting. A few mis-
cellaneous Rules therefore may suffice, relative to the
conduct, generally, of those parts of any Composition
which are designed to influence the Will.
§ 1. The first and most important point to be
observed in every address to any Passion, Sentiment,
Feeling, &c. is, that it should not be introduced as
such, and plainly avowed; otherwise the effect will
be, in great measure, if not entirely lost. This cir-
cumstance forms a remarkable distinction between
the head now under consideration, and that of Argu-
mentation. When engaged in Reasoning, properly
so called, our purpose not only need not be concealed,
but may, without prejudice to the effect, be distinctly
declared : on the other hand, even when the feelings
we wish to excite are such as ought to operate, so
that there is no reason to be ashamed of the endea-
vours thus to influence the hearer, still, our purpose
and drift should be, if not absolutely concealed, yet
not openly declared, and made prominent. Whether
the motives which the Orator is endeavouring to call
into action, be suitable or unsuitable to the occasion,
such as it is right, or wrong for the hearer to act
upon, the same rule will hold good. In the latter case
it is plain, that the speaker who is seeking to bias
unfairly the minds of the audience will be the more likely
to succeed by going to work clandestinely, in order that
his hearers may not be on their guard, and prepare and
fortify their minds against the impressions he wishes
to produce; in the other case, where the motives dwelt
on are such as ought to be present and strongly to
operate, men are not likely to be pleased with the
idea that they need to have these motives urged upon
them, and that they are not already sufficiently under
the influence of such sentiments as the occasion calls
for. A man may indeed be convinced that he is in
such a predicament, and may ultimately feel obliged to
the Orator for exciting or strengthening such senti-
ments; but while he confesses this, he cannot but
feel a degree of mortification in making the con-
fession, and a kind of jealousy of the apparent as-
sumption of superiority in a speaker, who seems
to say, “now I will exhort you to feel as you ought
on this occasion;” “I will endeavour to inspire you
with such noble and generous, and amiable senti-
ments as you ought to entertain;" which is, in effect,
* Aperh, ºpóvnois, Eövoia, book ii. c. i.
Chap. II,
*
R. H. E. T. O. R. I. C. 263
emotion. With respect to Argument itself indeed, Chap. II,
Rhetoric, the tone of him who avows the purpose of Exhorta-
different occasions will call for different degrees of S-N-
\-y- tion. The mind is sure to revolt from the humi-
liation of being thus moulded and fashioned, in respect
to its feelings, at the pleasure of another; and is
apt, perversely, to resist the influence of such a dis-
cipline.
Whereas there is no such implied superiority in
avowing the intention of convincing the understand-
ing : men know, and (what is more to the purpose)
feel, that he who presents to their minds a new and
cogent train of Argument, does not necessarily pos-
sess or assume any offensive superiority, but may, by
merely having devoted a particular attention to the
point in question, succeed in setting before them
Arguments and Explanations which had not occurred
to themselves; and even if the Arguments adduced,
and the Conclusions drawn, should be opposite to
those with which they had formerly been satisfied,
still there is nothing in this so humiliating, as in
that which seems to amount to the imputation of a
moral defect.
It is true that Sermons not unfrequently prove •
popular, which consist avowedly and almost exclu-
sively of Exhortation, strictly so called,—in which
the design of influencing the sentiments and feelings
is not only apparent, but prominent throughout ; but
it is to be feared, that those who are the most pleased
with such discourses are more apt to apply these
Exhortations to their neighbours than to themselves ;
and that each bestows his commendation rather from
the consideration that such admonitions are much
needed, and must be generally useful, than from finding
them thus useful to himself.
When indeed the speaker has made some progress
in exciting the feelings required, and has in great
measure gained possession of his audience, a direct
and distinct Exhortation to adopt the conduct recom-
mended will often prove very effectual; but never
can it be needful or advisable to tell them (as some
do) that you are going to exhort them. - - r &
It will, indeed, sometimes happen that the excite-
ment of a certain feeling will depend, in some mea-
sure, on a process of Reasoning; e. g. it may be
requisite to prove, where there is a doubt on the sub-
ject, that the person recommended to the Pity, Gra-
titude, &c. of the hearers, is really an object deserving
of these sentiments: but even then, it will almost
always be the case, that the chief point to be accom-
plished shall be to raise those feelings to the requisite
height, after the understanding is convinced that the
occasion calls for them. And this is to be effected
not by Argument, properly so called, but by present-
ing the circumstances in such a point of view, and so
fixing and detaining the attention upon them, that
corresponding sentiments and emotions shall gra-
dually, and as it were spontaneously, arise.
§ 2. Hence arises another Rule, closely connected
with the foregoing, though it also so far relates to
Style that it might with sufficient propriety have been
placed under that head; viz. that in order effectually
to excite feelings of any kind, it is necessary to em-
ploy some copiousness of detail, and to dwell some-
what at large on the several circumstances of the case
in hand; in which respect there is a wide distinction
between strict Argumentation, with a view to the
conviction of the understanding alone, and the at-
tempt to influence the will by the excitement of any
Copiousness, Repetition, and Expansion; the chain of
Reasoning employed, may, in itself, consist of more
or fewer links; abstruse and complex Arguments
must be unfolded at greater length than such as are
more simple ; and the more uncultivated the au-
dience, the more full must be the explanation and
illustration, and the more frequent the repetition of
the Arguments presented to them ; but still the same
general principle prevails in all these cases; viz. to
aim merely at letting the Arguments be fully under-
stood and admitted ; this will indeed occupy a shorter
or longer space, according to the nature of the case
and the character of the hearers; but all Expansion
and Repetition beyond what is necessary to accomplish
conviction, is in every instance tedious and disgust-
ing. On the contrary, in a description of anything
that is likely to act on the feelings, this effect will by
no means be produced as soon as the understanding
is sufficiently informed ; detail and expansion are
here not only admissible, but absolutely necessary, in
order that the mind may have leisure and opportunity
to form vivid and distinct ideas. For as Quinctilian.
well observes, he who tells us that a city was sacked,
although that one word implies all that occurred,
will produce little, if any, impression on the feelings,
in comparison of one who sets before us a lively
description of the various lamentable circumstances;
to tell the whole, he adds, is by no means the same
as to tell every thing. -
§ 3. It is not however, always advisable to enter
into a direct detail of circumstances, which would often
have the effect of wearying the hearer beforehand,
with the expectation of a long description of some-
thing in which he probably does not as yet feel much
interest; and would also be likely to prepare him too
much, and forewarn him as it were of the object pro-
posed,—the design laid against his feelings. It will
often, therefore, have a better effect to describe
obliquely, (if we may so speak,) by introducing cir-
cumstances connected with the main Object or event,
and affected by it, but not absolutely forming a part
of it. And circumstances of this kind may not unfre-
quently be selected, so as to produce a more strik-
ing impression of anything that is in itself great and
remarkable, than could be produced by a minute and
direct description; because in this way the general
and collective result of a whole, and the effects pro-
duced by it on other objects, may be vividly impressed
on the hearer's mind; the circumstantial detail of
collateral circumstances not drawing off the mind from
the contemplation of the principal matter as one and
complete. Thus the woman's application to the King
of Samaria, to compel her neighbour to fulfil her
agreement of sharing with her the infant's flesh, gives
a more frightful impression of the horrors of the
famine than any more direct description could have
done; since it presents to us the picture of that hard-
ening of the heart to every kind of horror, and that
destruction of the ordinary state of human sentiment,
which is the result of long-continued and extreme
misery. Nor could any detail of the particular vexa-
tions suffered by the exiled Jews for their disobe-
dience, convey so lively an idea of them as that
description of their result contained in the denuncia-
tion of Moses; “in the evening thou shalt say, would
264
R H E To R I c.
compare with another, should be one which ought to Chap. II.
excite the feeling in question in a higher degree than ~~
Rhetoric, God it were morning, and in the morning thou shalt
\—y—’ say, would God it were evening.”
In the poem of Rokeby, a striking exemplification
occurs of what has been said : Bertram in describing
the prowess he had displayed as a Buccaneer, does
not particularize any of his exploits, but alludes to
the terrible impression they had left : -
“Panama's maids shall long look pale,
When Risingham inspires the tale;
Chili's dark matrons long shall tame,
The froward child with Bertram's name,”
The first of Dramatists, who might have been per-
haps the first of Orators, has afforded some excellent
exemplifications of this rule, especially in the speech
of Antony over Caesar's body.
§ 4. Comparison is one powerful means of exciting
or heightening any emotion; viz. by presenting a
parallel between the case in hand and some other that
is calculated to call forth such emotions: taking care,
of course, to represent the present case as stronger
than the one it is compared with, and such as ought
to affect us more powerfully.
When several successive steps of this kind are em-
ployed to raise the feelings gradually to the highest
pitch, (which is the principal employment of what
Rhetoricians call the Climax,”) a far stronger effect
is produced than by the mere presentation of the
most striking object at once. It is observed by all
travellers who have visited the Alps, or other stu-
pendous mountains, that they form a very inadequate
notion of the vastness of the greater ones, till they
ascend some of the less elevated, (which yet are huge
mountains,) and thence view the others still towering
above them. And the mind, no less than the eye,
cannot so well take in and do justice to any vast
object, at a single glance, as by several successive
approaches and repeated comparisons. Thus in the
well-known Climax of Cicero in the Oration against
Verres, shocked as the Romans were likely to be at
the bare mention of the crucifixion of one of their
citizens, the successive steps by which he brings
them to the contemplation of such an event, were
calculated to work up their feelings to a much higher
pitch : “It is an outrage to bind a Roman citizen ;
to scourge him is an atrocious crime; to put him to
death is almost parricide ; but to crucify him—what
shall I call it 2"
It is observed, accordingly, by Aristotle, in speaking
of Panegyric, that the person whom we would hold
up to admiration, should always be compared, and
advantageously compared, if possible, with those that
are already illustrious, but if not, at least with some
person whom he excells : to excel, being in itself,
he says, a ground of admiration. The same rule will
apply, as has been said, to all other feelings as well
as to Admiration : Anger, or Pity, for instance, are
more effectually excited if we produce cases such
as would call forth those passions, and which though
similar to those before us, are not so strong; and so
with respect to the rest. *
When it is said, however, that the Object which we
* An analogous Arrangement of Arguments in order to set forth
the full force of the one we mean to dwell upon, would also
receive the same appellation, and in fact is very often combined
and blended with that which is here spoken of,
that other, it is not meant that this must actually be,
already, the impression of the hearers: the reverse
will more commonly be the case ; that the instances
adduced will be such as actually affect their feelings
more strongly than that to which we are endeavouring
to turn them, till the flame spreads, as it were, from
the one to the other. This will especially hold good in
every case where self is concerned ; e. g. men feel na-
turally more indignant at a slight affront offered to
themselves, or those closely connected with them,
than at the most grievous wrong done to a stranger;
if therefore you would excite their utmost indignation
in such a case, it must be by comparing it with a pa-
rallel case that concerns themselves ; i.e. by leading
them to consider how they would feel were such and
such an injury done to themselves. And, on the other
hand, if you would lead them to a just sense of their
own faults, it must be by leading them to contemplate
like faults in others ; of which the celebrated parable
of Nathan, addressed to David, affords an admirable
instance. *
§ 5. Another Rule, (which also is connected in
some degree with Style) relates to the tone of feeling
to be manifested by the writer or speaker himself, in
order to excite the most effectually the desired emo-
tions in the minds of the hearers. And this is to be
accomplished by two opposite methods: the one,
which is most obvious, is to express openly the feeling
in question ; the other, to seem labouring to suppress
it : in the former method, the most forcible remarks
are introduced,—the most direct as well as impas-
sioned kind of description is employed,—and some-
thing of exaggeration introduced, in order to carry
the hearers as far as possible in the same direction in
which the Orator seems to be himself hurried, and
to infect them to a certain degree with the emotions
and sentiments which he thus manifests : the other
method, which is often no less successful, is to ab-
stain from all remarks, or from all such as come up
to the expression of feeling which the occasion seems
to authorize,_to use a gentler mode of expression
than the case might fairly warrant, to deliver “ an
unvarnished tale,” leaving the hearers to make their
own comments,—and to appear to stifle and studiously
to keep within bounds such emotions as may seem
natural. This produces a kind of reaction in the
hearers' minds; and being struck with the inadequacy
of the expressions, and the laboured calmness of the
speaker's manner of stating things, compared with
what he may naturally be disposed to feel, they will
often rush into the opposite extreme, and become the
more strongly affected by that which is set before
them in so simple and modest a form. And though
this method is in reality more artificial than the other,
the artifice is the more likely (perhaps for that very
reason) to escape detection; men being less on their
guard against a speaker who does not seem so much
labouring to work up their feelings, as to repress or
moderate his own ; provided that this calmness and
coolness of manner be not carried to such an ex-
treme as to bear the appearance of affectation; which
caution is also to be attended to in the other mode of
procedure no less; an excessive hyperbolical exag-
geration being likely to defeat its own object.
Aristotle mentions, (Rhet, bookix.) though very briefly,
R H E T O R. I. C. 265
often have the effect of a train of sound Argument. Chap. II.
Rhetoric, these two modes of rousing the feelings, the latter
This artifice falls under the head of “ Irrelevant Con-S-N-2
S–V-' under the name of Eironeia, which in his time was
commonly employed to signify, not according to the
modern use of “ Irony, saying the contrary to what
is meant,” but, what later writers usually express by
Litotes, i. e. “ saying less than is meant.”
The two methods may often be both used on the
same occasion, beginning with the calm, and pro-
ceeding to the impassioned, afterwards, when the
feelings of the hearers are already wrought up to a
certain pitch : 6tav éxm 767 Tois àkpoatãs, ka? Toémon
évôsatágat.* Universally indeed it is a fault carefully
to be avoided, to express feeling more vehemently than
that the audience can go along with the speaker;
who would, in that case, as Cicero observes, seem
like one raving among the sane, or intoxicated in the
midst of the sober. And accordingly, except where
from extraneous causes the audience are already in an
excited state, we must carry them forward gradually,
and allow time for the fire to kindle. The blast which
would heighten a strong flame, would, if applied too
soon, extinguish the first faint spark. The speech of
Antony over Caesar's corpse, which has been already
mentioned, affords an admirable example of that com-
bination of the two methods which has been just
spoken of.
Generally however, it will be found that the same
Orators do not excel equally in both modes of exciting
the Passions ; and it should be recommended to each
to employ principally that in which he succeeds best,
since either, if judiciously managed, will generally
prove effectual for its object. The well-known tale
of Inkle and Yarico, which is an instance of the
eatenuating method, (as it may be called,) could not,
perhaps, have been rendered more affecting, if equally
so, by the most impassioned vehemence and rhetorical
heightening. *
§ 6. When the occasion or Object in question is not
such as calls for, or as is likely to excite in those par-
ticular readers or hearers, the emotions required, it
is a common Rhetorical artifice to turn their attention
to some Object which will call forth these feelings;
and when they are too much excited to be capable of
judging calmly, it will not be difficult to turn their
Passions, once roused, in the direction required, and
to make them view the case before them in a very
different light. When the metal is heated, it may
easily be moulded into the desired form. Thus vehe-
ment indignation against some crime, may be directed
against a person who has not been proved guilty of
it; and vague declamations against corruption, op-
pression, &c. or against the mischiefs of anarchy; with
high-flown panegyrics on liberty, rights of man, &c.
or on social order, justice, the constitution, law,
religion, &c. will gradually lead the hearers to take
for granted, without proof, that the measure pro-
posed will lead to these evils or these advantages ;
and it will in consequence become the object of
groundless abhorrence or admiration. For the very
utterance of such words as have a multitude of what
may be called stimulating ideas associated with them,
will operate like a charm on the minds, especially of
the ignorant and unthinking, and raise such a tumult
of feeling, as will effectually blind their judgment;
so that a string of vague abuse or panegyric, will
* Aristotle, Rhet, book iii, ch. vii.
WOL. I.
clusion," or ignoratio elemchi, mentioned in the Treatise
on Fallacies. (Art. Logic, ch. v. sec. 14.) Mr. Ben-
tham has treated of the employment of these “pas-
sion-kindling appellatives,” as he calls them, under
the head of “ Fallacies of Confusion,” in his work
entitled the Book of Fallacies. Many other observa-
tions, also occurring in that Treatise, will be found
very nearly to coincide with that which has been said
in the fifth chapter of the Article on Logic just referred
to ; though not to be so strictly tried by Logical rules.
Of many popular Sophisms he has given (though in
a singular manner,) an able exposure; and of many
others, unfortunately, the most striking exemplifica-
tions may be found in his own reasonings; in which
petitio principii in particular, occurs perpetually; as
well as the one now before us, the employment of
wituperative, or as he calls it “Dyslogistic " language:
that also which we there described as the “Fallacy of
Objections,” (which might be called by a lover of
new-coined epithets, in language similar to that often
employed by Mr. Bentham, a reverse-of-wrong-for-
right-mistaking Fallacy) is skilfully described, and
but too often employed; as if, because existing abuses
are maintained by those who have an interest in keeping
what they have, no apprehension were to be enter-
tained of new evils being introduced through the
interested conduct of those who wish to acquire what
they have not ; * and as if, because many are mis-
led by a blind veneration for “Authority” and the
“ Wisdom of our Ancestors,” there did not exist also,
as antagonist muscles, as it were, to these, an equally
blind craving after novelty for its own sake, and a
veneration for the ingenuity of one's own inventions.
It is matter of regret that the powers of such a mind
as that of Mr. Bentham, should be to so great a degree
wasted. Such, however, must always be the case, when
a Scientific work is composed (with whatever sincerity)
for party purposes, or with any object foreign to the
precise End of the Science in question. Many Arguments
accordingly are, in the work alluded to, stigmatised
as Fallacies, which may be, either sophistical, or
sound and fair, according to the circumstances in
which they are employed ; such as that a certain
proposed reformation ought to be effected “gradually;”
that we must “ wait awhile, the present not being
the time for such and such a measure ; ” or that
“ this or that proposal comes from a suspicious
quarter,” &c. which are topics that may be fairly or
unfairly urged. And it is but too plain that the line
is drawn not with a view to the mode in which, but
to the object for which, and the party by which, each
Argument is urged. It is only when certain classes
* Those who will not profit by the lessons of the French
Revolution, are not, perhaps, likely to learn wisdom from the great
historian of Greece, who has so well described the workings of
human passion as manifested in the civil commotions of Corcyra
and other States. But to such as are willing to receive instruc-
tion, that which he affords can never cease to be valuable.—
'Ev 6'oëv tº Kepkipg rà troAA& &utów ºrpoeroNuñ0m, kal &mdaſa
#Spel uév ćpxéuevo to TAéov i aw?pogium, into róv, Thy Tipoptav Tra-
pagxávrov, Št &vtauvváuevo Spdaetav IIFNIA3 AE TH> EIOOTITAX
AIIAAAAEEIONTEX Tives, uſixtata 5’ &v Ště, tra.0ovs étribuproºvtes tú.
rów TréAas #xetv, trapā ātkmv yiyvágicolev * * * * * Evvrapax6évros
Te roß Stov, Šs Tov kapov Toorov, Tſ tróAel, kal Tów vöuav ºpatho-ara.
# &v0pwireia púris, éta,0üta Kai trap& rous vópovs &ölkeſv, &aaévm
é6%Aworev, &kpaths uév Špyſis of ora, kpeta owy Śē toū āticatov,
IIOAEMIA AE TOT IIPOTXO'NTox. Thucyd. book iii, sec. 84.
2 N
266
R H E T O R. I. C.
authority than others for the most part ventured to Chap. II.
assume. It is by the expression of wise, amiable, and S-M-'
Rhetoric. of Propositions are distinctively pointed out as abso-
\--~~' lutely false, inadmissible, or irrelevant, or certain
deductions from true ones shown to be unfair, that
any useful warning can be supplied. The hopes,
therefore, which the author entertains (p. 410,) that by
the general study and adoption of his Principles, de-
bates may be cleared and shortened, (each Fallacy being
detected, exposed by name, and exploded, as soon as
uttered) seem more sanguine than well-founded. If
the general adoption, by the great majority of the
audience, of the same system, means, their being of
the same party, no doubt they would readily and easily
silence by clamour every opposite Argument; but if
they are merely to agree in adopting Logical prin-
ciples as ill-defined as those we are speaking of, the
proposed plan for the ready exposure of each fallacious
Argument, resembles that by which children are de-
luded, of catching a bird by laying salt on its tail ;
the existing doubts and difficulties of debate being no
greater than, on the proposed system, would be found
in determining what Arguments were, and what not,
to be classed with the Fallacies in question.
The work, however may be read safely, and, per-
haps, not without advantage, by those who have
sufficient interest in the subject to encounter the
obscurity of the style, and sufficient patience in in-
vestigation, and power of discrimination, to separate
the particles of gold-dust from the mass of sand and
weed with which they are blended. It has been
thought advisable therefore to make this reference
to an author who is, perhaps, too generally regarded,
except by the very small number of disciples who
idolize him, with that unmixed contempt which is due
to a portion only (though certainly no inconsiderable
portion) of his tenets. Among posterity, the opinions
entertained of him may probably be less violently
contrasted, and, on the whole, more favourable ; at
least it usually happens that those who have manifested
any considerable original powers, and have elicited
valuable truths, however contaminated by the most
extravagant errors, are remembered, even more fa-
vourably than is strictly their due ; their absurdities
are gradually forgotten, like the inscription on plaster
on the light-house of Pharos, which mouldered away
by the action of the weather ; while the value of
their discoveries is durably recorded, and becomes
more and more conspicuous, like the inscription en-
graved on the marble beneath.
§ 7. In raising a favourable impression of the
speaker, or an unfavourable one of his opponent, a
peculiar tact will of course be necessary; especially
in the former, since direct self-commendation will
usually be disgusting to a greater degree, even than
a direct personal attack on another; though, if the
Orator is pleading his own cause, or one in which he
is personally concerned, (as was the case in the speech
of Demosthenes concerning the Crown,) a greater
allowance will be made for him on this point ; espe-
cially if he be a very eminent person, and one who
may safely appeal to public actions performed by him.
Thus Pericles is represented by Thucydides as claim-
ing directly, when speaking in his own vindication,
exactly the qualities (good Sense, good Principle, and
Good-will,) which Aristotle lays down as constituting
the character which we must seek to appear in. But
then it is to be observed, that the historian represents
him as accustomed to address the people with more
generous Sentiments, that Aristotle recommends the
speaker to manifest his own character ; but even this
must generally be done in an oblique” and seemingly
incidental manner, lest the hearers be disgusted with
a pompous and studied display of fine sentiments;
and care must also be taken not to affront them by
seeming to inculcate as something likely to be new to
them, maxims which they regard as almost truisms.
Of course the application of this last caution must
vary according to the character of the persons ad-
dressed ; that might excite admiration and gratitude
in one audience, which another would receive with
indignation and ridicule. Most men, however, are
disposed rather to overrate than to extenuate their own
moral judgment; or at least to be jealous of any one's
appearing to underrate it.
Universally indeed, in the Arguments used, as well
as in the appeals made to the Feelings, a consideration
must be had of the hearers, whether they are learned or
ignorant,--of this or that profession,-nation,-cha-
racter, &c. and the address must be adapted to each ;
so that there can be no excellence of writing or
speaking in the abstract; nor can we any more pro-
nounce on the Eloquence of any Composition, than
upon the wholesomeness of a medicine, without know-
ing for whom it is intended. The less enlightened
the hearers, the harder, of course, it is to make them
comprehend a long and complex train of Reason-
ing ; so that sometimes the Arguments, in themselves
the most cogent, cannot be employed at all with
effect ; and the rest will need an expansion and copious
illustration which would be needless, and therefore
tiresome, (as has been above remarked,) before a
different kind of audience : on the other hand, their
feelings may be excited by much bolder and coarser
expedients; such as those are the most ready to em-
ploy, and the most likely to succeed in, who are them-
selves but a little removed above the vulgar; as may
be seen in the effects produced by fanatical preachers.
But there are none whose feelings do not occasionally
need and admit of excitement by the powers of Elo-
quence ; only there is a more exquisite skill required
in thus affecting the educated classes than the popu-
lace.f
* E. g. “It would be needless to impress upon you the
maxim,” &c. “You cannot be ignorant,” &c. &c. “I am not.
advancing any high pretensions in expressing the sentiments
which such an occasion must call forth in every honest heart,”
&c. g
+ “The less improved in knowledge and discernment the
hearers are, the easier it is for the speaker to work upon their
passions, and by working on their passions, to obtain his end.
This, it must be owned, appears on the other hand, to give a
considerable advantage to the preacher, as in no congregation
can the bulk of the people be regarded as on a footing, in point
of improvement, with either House of Parliament, or with the
Judges in a Court of Judicature. It is certain, that the more
gross the hearers are, the more avowedly may you address your-
self to their passions, and the less occasion there is for argu-
ment; whereas, the more intelligent they are, the more covertly
must you operate on their passions, and the more attentive must
you be in regard to the justness, or at least the speciousness of
your reasoning. Hence some have strangely concluded, that
the only scope for eloquence is in haranguing the multitude;
that in gaining over to your purpose men of knowledge and
breeding, the exertion of Oratorical talents hath no influence.
This is precisely as if one should argue, because a mob is much more
easily subdued than regular troops, there is no occasion for the
R. H. E. T. O. H. I. C. - 267
Rhetoric.
In no point more than in that now under considera-
\-y-'tion, viz. the Conciliation (to adopt the term of the
Latin writers) of the hearers, is it requisite to con-
sider who and what the hearers are ; for when it is
said that good Sense, good Principle, and Good-will,
constitute the character which the speaker ought to
establish of himself, it is to be remembered that
every one of these is to be considered in reference to
the opinions and habits of the audience. To think
very differently from his hearers, may often be a
sign of the Orator's wisdom and worth ; but they
are not likely to consider it so. A witty Satirist,”
has observed, that “it is a short way to obtain the
reputation of a wise and reasonable man, whenever
any one tells you his opinion, to agree with him.
Without going the full length of completely acting
on this maxim, it is absolutely necessary to remem-
ber, that in proportion as the speaker manifests
his dissent from the opinions and principles of his
audience, so far he runs the risk at least, of im–
pairing their estimation of his judgment. But this
it is often necessary to do when any serious object
is proposed; because it will commonly happen that
the very End aimed at shall be one which im-
plies a change of sentiments, or even of principles
and character, in the hearers. Those indeed who aim
only at popularity, are right in conforming their sen-
timents to those of the hearers, rather than the
contrary; but it is plain that though in this way they
obtain the greatest reputation for Eloquence, they de-
serve it the less; it being much easier, according to
the tale related of Mahomet, to go to the mountain,
than to bring the mountain to us.f There is but
little Eloquence in convincing men that they are in
the right, or inducing them to approve a character
which coincides with their own.
art of war, nor is there a proper field for the exertion of military
skill, unless when you are quelling an undisciplined rabble.
Every body sees in this case, not only how absurd such a way of
arguing would be, but that the very reverse ought to be the con-
clusion. The reason why people do not so quickly perceive the
absurdity in the other case, is, that they affix no distinct mean-
ing to the word eloquence, often denoting no more by that term
than simply the power of moving the passions. But even in
this improper acceptation, their notion is far from being just ;
for wherever there are meri, learned or ignorant, civilized or
barbarous, there are passions; and the greater the difficulty is
in affecting these, the more art is requisite.” Campbell's Rhetoric,
book i. ch. x. Sec. 2. p. 224, 225.
* Swift.
+ “Little force is necessary to push down heavy bodies placed
on the verge of a declivity, but much force is requisite, to stop
them in their progress, and push them up. If a man should
say, that because the first is more frequently effected than the
last, it is the best trial of strength, and the only suitable
use to which it can be applied, we should at least not think
him remarkable for distinctness in his ideas. Popularity alone,
therefore, is no test at all of the eloquence of the speaker,
no more than velocity alone would be, of the force of the
external impulse originally given to the body moving. As in
this the direction of the body, and other circumstances, must
be taken into the account; so in that, you must consider the
tendency of the teaching, whether it favours or opposes the
vices of the learers. To head a sect, to infuse party-spirit, to
make men arrogant, uncharitable, and malevolent, is the easiest
task imaginable, and to which almost any blockhead is fully
equal. But to produce the contrary effect, to subdue the spirit
of faction, and that monster spiritual pride, with which it is
invariably accompanied, to inspire equity, moderation, and
charity into men's sentiments, and conduct with regard to others,
is the genuine test of eloquence.” Campbell's Rhetoric, book i.
ch, X, Sec. 5, p. 239.
2 3
The Christian preacher therefore is in this respect Chap. II.
placed in a difficult dilemma, since he may be sure
that the less he complies with the depraved judgments
of man's corrupt nature, the less acceptable is he
likely to be to that depraved judgment.
But he who would claim the highest rank as an
Orator, (to omit all higher considerations) must be
the one who is the most successful, not in gaining
popular applause, but in carrying his point, whatever
it be. The preacher, however, who is intent on this
object, should use all such precautions as are not
inconsistent with it, to avoid raising unfavourable
impressions in his hearers. Much will depend on a
gentle and conciliatory manner; nor is it necessary
that ae should, at once, in an abrupt and offensive
form, set forth all the differences of sentiment between
himself and his congregation, but win them over by
degrees; and in whatever point, and to whatever
extent, he may suppose them to agree with him, it
is allowable, and for that reason advisable, to dwell
on that agreement; as the Apostles began every ad-
dress to the Jews by an appeal to the Prophets,
whose authority they admitted ; and as St. Paul
opens his discourse to the Athenians (though unfor-
tunately the words of our translation are likely to
convey an opposite idea,”) by a commendation of their
respect for religion. And above all, where censure is
called for, the speaker should avoid, on Christian, as
well as on Rhetorical principles, all appearance of
exultation in his own Superiority,+of contempt, or
of uncharitable triumph in the detection of faults ;
‘‘ in meekness, instructing them that oppose them-
selves.” -
Of intellectual qualifications, there is one which
it is evident, should not only not be blazoned forth,
but should in a great measure be concealed, or kept
out of sight; viz. Rhetorical skill; since whatever is
attributed to the Eloquence of the speaker is so much
deducted from the strength of his cause. Hence,
Pericles is represented by Thucydides as artfully claim-
ing, in his vindication of himself, the power of ex-
plaining the measures he proposes, not, Eloquence
in persuading their adoption. And accordingly a
skilful Orator seldom fails to notice and extol the
Eloquence of his opponent, and to warn the hearers
against being misled by it. It is a peculiarity there-
fore in the Rhetorical art, that in it, more than in any
other, vanity has a direct and immediate tendency to
interfere with the proposed object. Excessive vanity
may indeed, in various ways, prove an impediment to
success in other pursuits; but in the endeavour to
Persuade, all wish to appear excellent in that art,
operates as a hindrance. A Poet, a Statesman, or a
General, &c. though extreme covetousness of ap-
plause may mislead them, will, however, attain their
respective Ends, certainly not the less for being ad-
mired as excellent in Poetry, Politics, or War ; but
the Orator attains his End the better the less he is
regarded as an Orator; if he can make the hearers
believe that he is not only a stranger to all unfair
artifice, but even destitute of all Persuasive skill what-
ever, he will Persuade them the more effectually; and
if there ever could be an absolutely perfect Orator,
9 3
* Aetorišauploved Tépous, not “too superstitious,” but (as almost
all commentators are now agreed) “very much disposed to the
worship of Divine beings.” *
_*
2 N 2
268
R H E T O R. I C.
decided vituperation, in the eyes of those imbued Chap. II.
Rhetoric, no one would, at the time at least, discover that he
was so. And this consideration may serve to account
for the fact which Cicero remarks upon (De Oratore,
book i.) as so inexplicable; viz. the small number of
persons who, down to his time, had obtained high
reputation as Orators, compared with those who had
attained excellence in other pursuits. Few men are
destitute of the desire of admiration ; and most are
especially ambitious of it in the pursuit to which they
have chiefly devoted themselves; the Orator therefore
is continually tempted to sacrifice the substance to
the shadow, by aiming rather at the admiration of
the hearers, than their conviction ; and thus to fail
of that excellence in his art which he might other-
wise be well qualified to attain, through the desire of
a reputation for it. And on the other hand, some may
have been really Persuasive speakers, who yet may
not have ranked high in men's opinion, and may not
have been known to possess that art of which they
gave proof by their skilful concealment of it. There
is no point, in short, in which report is so little to
be trusted. - - -
Of the three points which Aristotle directs the
Orator to claim credit for, it might seem at first sight
that one, viz. “Good-will,” is unnecessary to be
mentioned ; since Ability and Integrity would appear
to comprehend, in most cases at least, all that is
needed ; a virtuous man, it may be said, must wish well
to his countrymen, or to any persons whatever, whom
he may be addressing. But on a more attentive con-
sideration, it will be manifest that Aristotle had good
reason for mentioning this head ; if the speaker were
believed to wish well to his Country, and to every
individual of it, yet if he were suspected of being
unfriendly to the political or other Party to which
his hearers belonged, they would listen to him with
prejudice. The abilities and the conscientiousness of
Phocion seem not to have been doubted by any ; but
they were so far from gaining him a favourable hear-
ing among the Democratical party at Athens, (who
knew him to be no friend to Democracy,) that they
probably distrusted him the more; as one whose
public spirit would induce him, and whose talents
would enable him, to subvert the existing Consti-
tution.
One of the most powerful engines, accordingly,
of the Orator, is this kind of appeal to party-spirit.
Party-spirit may, indeed, be considered in another
point of view, as one of the Passions which may be
directly appealed to, when it can be brought to operate
in the direction required ; i. e. when the conduct the
writer or speaker is recommending appears likely
to gratify party-spirit ; but it is the indirect appeal to
it which is now under consideration ; viz. the favour,
credit, and weight which the speaker will derive from
appearing to be of the same party with the hearers,
or at least not opposed to it. And this is a sort of
credit which he may claim more openly and avowedly
than any other ; and likewise may throw discredit on
his opponent in a less offensive, but not less effectual
manner. A man cannot say in direct terms, “I
am a wise and worthy man, and my adversary the
reverse;” but he is allowed to say, “I adhere to
Whig or Tory principles,” (as the case may be,) and
“my opponent the reverse ;” which is not regarded
as an offence against modesty, and yet amounts
virtually to as strong a self-commendation, and as
with party-spirit, as if every kind of merit and of S-N-
demerit had been enumerated : for to zealous party
men, zeal for their party will very often either imply,
or stand as a substitute for, every other kind of
worth.
Hard, indeed, therefore is the task of him whose
object is to counteract party-spirit and to soften the
violence of those prejudices which spring from it.*
His only resource must be to take care that he give
no ground for being supposed imbued with the violent
and unjust prejudices of the opposite party,’—that he
give his audience credit (since it rarely happens but
that each party has some tenets that are reasonable,)
for whatever there may be that deserves praise, that
he proceed gradually and cautiously in removing the
errors with which they are infected,—and above all,
that he studiously disclaim and avoid the appearance
of any thing like a feeling of personal hostility, or
personal contempt.
If the Orator's character can be sufficiently esta-
blished in respect of Ability, and also of Good-will
towards the hearers, it might at first sight appear as
if this would be sufficient ; since the former of these
would imply the Power, and the latter, the Inclination,
to give the best advice, whatever might be his Moral
character; but Aristotle (in his Politics) justly remarks
that this last is also requisite to be insisted on, in
order to produce entire confidence ; for, says he,
though a man cannot be suspected of wanting Good-
will towards himself, yet many very able men act
most absurdly, even in their own affairs, for want of
Moral virtue, being either blinded or overcome by
their Passions, so as to sacrifice their own most im-
portant interests to their present gratification ; and
much more, therefore, may they be expected to be thus
seduced by personal temptations, in the advice they
give to others. Pericles, accordingly, in the speech
which has been already referred to, is represented by
Thucydides as insisting not only on his political
ability and his patriotism, but also on his unim-
peached integrity, as a qualification absolutely neces-
sary to entitle him to their confidence : for “the
man,” says he, “ who possesses every other requisite,
but is overcome by the temptation of a bribe, will be
ready to sell every thing for the gratification of his
avarice.”
From what has been said of the speaker's recom-
mendation of himself to the audience, and establish-
ment of his authority with them, sufficient Rules
may readily be deduced for the analogous process,
the depreciation of an opponent. Both of these, and
especially the latter, under the offensive title of
personality, are by many indiscriminately decried as
unfair Rhetorical artifices; and, doubtless they are,
in the majority of cases, sophistically employed; and
by none more effectually than by those who are per-
petually declaiming against such Fallacies ; the un-
thinking hearers not being prepared to expect them from
* “Of all the prepossessions in the minds of the hearers,
which tend to impede or counteract the design of the speaker,
party-spirit, where it happens to prevail, is the most pernicious,
being at once the most inflexible, and the most unjust. * * * *
Violent party men not only lose all sympathy with those of the
opposite side, but even contract an antipathy to them. This,
on some occasions, even the divinest eloquence will not surmount.”
Campbell's Rhetoric.
R H E T O R. I. C.
269
.Rhetoric, those who represent themselves as holding them in such
\-y-'abhorrence. But surely it is not in itself an unfair topic
of Argument, in cases not admitting of decisive and un-
questionable proof, to urge that the one party deserves
the hearers' confidence, or that the other is justly an
object of their distrust. “If the measure is a good
one,” says Mr. Bentham, “ will it become bad be-
cause it is supported by a bad man : if it is bad, will
it become good, because supported by a good man 2
If the measure be really inexpedient, why not at once
show that it is so Your producing these irrelevant
and inconclusive Arguments, in lieu of direct ones,
though not sufficient to prove that the measure you
thus oppose is a good one, contributes to prove that
you yourself regard it as a good one.” Now there is
no doubt that the generality of men are too much
disposed to consider more, who proposes a measure,
than what it is that is proposed ; and probably would
continue to do so, even under a system of annual
Parliaments and universal suffrage; and if a warning
be given against an excessive tendency to this way of
judging, it is reasonable, and may be useful ; nor
should any one escape censure who confines himself to
these topics, or dwells principally on them, in cases
where “ direct” Arguments are to be expected ; but
they are not to be condemned in toto as “irrelevant
and inconclusive,” because they are only probable, and
not in themselves decisive; it is only in matters of
strict science, and that too, in arguing to scientific
men, that the character of the advisers (as well as all
other probable Arguments,) should be wholly put out
of the question. And it is remarkable that the neces-
sity of allowing some weight to this consideration, in
political matters, increases in proportion as any
country enjoys a free government; if all the power
be in the hands of a few of the higher orders, who
have the opportunity at least, of obtaining education,
it is conceivable, whether probable or not, that they
may be brought to try each proposed measure exclu-
sively on its intrinsic merits, by abstract Arguments;
but can any man, in his senses, really believe that the
great mass of the people, or even any considerable
portion of them, can ever possess so much political
knowledge, patienee in investigation, and sound Logic,
(to say nothing of candour,) as to be able and willing
to judge, and to judge correctly, of every proposed
political measure, in the abstract, without any regard
to their opinion of the person who proposes it
And it is evident that in every case, in which the
hearers are not completely competent judges, they
not only will, but must, take into consideration the
characters of those who propose, support, or dissuade
any measure ;-the persons they are connected with,
the designs they may be supposed to entertain, &c.;
though, undoubtedly, an excessive and exclusive re-
gard to Persons rather than Arguments, is one of the
chief Fallacies against which men ought to be cau-
tioned.
In no way, perhaps, are men, not bigoted to party,
more likely to be misled by their favourable or un-
favourable judgment of their advisers, than in what
relates to the authority derived from Eaperience; not
that Experience ought not to be allowed to have great
weight; but that men are apt not to consider with
sufficient attention, what it is that constitutes Ex-
perience in each point ; so that frequently one man
shall have credit for much Experience, in what relates
to the matter in hand, and another, who, perhaps, Chap. II.
possesses as much, or more, shall be underrated as \–N2–/
wanting it. The vulgar, of all ranks, need to be
warned, 1st, that time alone does not constitute Expe-
rience ; so that many years may have passed over a
man's head, without his even having had the same
opportunities of acquiring it, as another, much
younger : 2nd, that the longest practice in conducting
any business in one way, does not necessarily confer
any Experience in conducting it in a different way;
e.g. an experienced Husbandman, or a Minister of
State, in Persia, would be much at a loss in Europe;
and if they had some things less to learn than an entire
novice, on the other hand they would have much to
unlearn : and, 3rd, that merely being conversant about
a certain class of subjects, does not confer Experience
in a case where the Operations, and the End proposed,
are different. It is said that there was an Amsterdam
merchant, who had dealt largely in corn all his life,
who had never seen a field of wheat growing; this
man had doubtless acquired, by Experience, an accu-
rate judgment of the qualities of each description of
corn,-of the best methods of storing it,-of the arts
of buying and selling it at proper times, &c.; but he
would have been greatly at a loss in its cultivation ;
though he had been, in a certain way, long conversant
about corn. Nearly similar is the Experience of a prac-
tised Lawyer, (supposing him to be nothing more) in a
case of Legislation ; because he has been long conversant
about Law, the unreflecting attribute great weight to
his judgment; whereas his constant habits of fixing
his thoughts on what the law is, and withdrawing it
from the irrelevant question of what the law ought
to be ;-his careful observance of a multitude of
Rules, (which afford the more scope for the dis-
play of his skill, in proportion as they are arbitrary,
unreasonable, and unaccountable,) with a studied in-
difference as to that which is foreign from his business,
the convenience or inconvenience of those Rules, may
be expected to operate unfavourably on his judgment
in questions of Legislation ; and are likely to counter-
balance the advantages of his superior knowledge, even
in such points as do bear on the question. The con-
sideration then of the character of the speaker, and
of his opponent, being of so much importance, both.
as a legitimate source of Persuasion, in many instances,
and also as a topic of Fallacies, it is evidently incum-
bent on the Orator to be well versed in this branch
of the art, with a view both to the justifiable advance-
ment of his own Cause, and to the detection and
exposure of unfair artifice in an opponent. It is
neither possible, nor can it, in justice be expected, that
this Inode of Persuasion should be totally renounced
and exploded, great as are the abuses to which it is
liable ; but the speaker is bound, in conscience, to
abstain from those abuses himself, and, in prudence,
to be on his guard against them in others.
It only remains to observe, on this head, that, as
Aristotle teaches, the place for the disparagement of
an opponent is, for the first speaker, near the close of
his discourse, to weaken the force of what may be
said in reply; and, for the opponent, near the open-
ing, to lessen the influence of what has been already
said.
§ 8. Either a personal prejudice, such as has been
just mentioned, or some other passion unfavourable
to the speaker's Object, may already exist in the
270
R H E T O R I. C.
Rhetoric.
minds of the hearer, which it must be his business
to allay. - -
It is obvious that this will the most effectually be
done, not by endeavouring to produce a state of
perfect calmness and apathy, but by exciting some
contrary emotion. And here it is to be observed that
some passions may be, Rhetorically speaking, oppo-
site to each other, though in strictness they are not
so ; viz. whenever they are incompatible with each
other : e. g. the opposite, strictly speaking, to Anger,
would be a feeling of Good-will and approbation
towards the person in question; but it is not by
the excitement of this, alone, that Anger may be
allayed ; for Fear is, practically, contrary to it also ;
as is remarked by Aristotle; who Philosophically
accounts for this, on the principle that Anger im-
plying a desire to inflict punishment, must imply also
a supposition that it is possible to do so; and accord-
ingly men do not, he says, feel Anger towards one who
is so much superior as to be manifestly out of their
reach ; and the Object of their Anger ceases to be so,
as soon as he becomes an Object of Apprehension. Of
course the converse also of this holds good ; Anger,
when it prevails, in like manner subduing Fear.
Compassion, likewise, may be counteracted either by Chap. II,
Disapprobation, by Jealousy, by Fear, or by Disgust
ºsmº
and Horror; and Envy, either by Good-will, or by Chap. III.
Contempt. -
This is the more necessary to be attended to, in order
that the Orator may be on his guard against inadver-
tently defeating his own Object, by exciting feelings
at variance with those he is endeavouring to pro-
duce, though not strictly contrary to them. Aristotle
accordingly notices, with this view, the difference
between the “ Pitiable,” (éAgeuvov) and the “Horrible
or Shocking,” (8euvov;) which, as he observes, excite
different feelings, destructive of each other; so that the
Orator must be warned, if the former is his Object,
to keep clear of any thing that may excite the latter. .
It will often happen that it will be easier to give anew
direction to the unfavourable passion, than to subdue
it ; e. g. to turn the indignation or the laughter of
the hearers against a different object. Indeed, when-
eyer the case will admit of this, it will generally prove
the more successful expedient, because it does not
imply the accomplishment of so great a change in the
minds of the hearers
CHAPTER III.
OF STYLE,
Though the consideration of Style has been laid
down as holding a place in a Treatise of Rhetoric, it
would be neither necessary nor pertinent, to enter fully
into a general discussion of the subject, which would
evidently embrace much that by no means peculiarly
belongs to our present inquiry. It is requisite for an
Orator, e. g. to observe the rules of Grammar; but
the same may be said of the Poet and the Historian,
&c. nor is there any peculiar kind of grammatical
propriety belonging to Persuasive or Argumentative
compositions; so that it would be a departure from
our subject to treat at large, under the head of Rhe-
toric, of such rules as equally concern every other of
the purposes for which Language is employed.
Conformably to this view we shall, under the pre-
sent head, notice but slightly such principles of com-
position as do not exclusively or peculiarly belong to
the present subject; confining our attention chiefly to
such observations on Style as have an especial reference
to Argumentative and Persuasive works. -
§ 1. It is sufficiently evident (though the maxim is
often practically disregarded) that the first requisite of
Style not only in Rhetorical, but in all compositions,
is Perspicuity; since, as Aristotle observes, language
which is not intelligible, or not clearly and readily in-
telligible, fails, in the same proportion, of the purpose
for which language is employed. And it is equally self-
evident (though this truth is still more frequently
overlooked) that Perspicuity is a relative quality, and
consequently cannot properly be predicated of any
work, without a tacit reference to the class of readers
or hearers for whom it is designed. Nor is it enough
that the Style be such as they are capable of under-
standing, if they bestow their utmost attention : the
J.
degree and the kind of attention, which they have been
accustomed, or are likely to bestow, will be among
the circumstances that are to be taken into the account,
and provided for. The kind, as well as the degree, of
attention, is mentioned, because Some hearers and
readers will be found slow of apprehension indeed,
but capable of taking in what is very copiously and
gradually explained to them , while others on the
contrary, who are much quicker at catching the sense
of what is expressed in a short compass, are incapable
of long attention, and are not only wearied, but abso-
lutely bewildered, by a diffuse Style. r
When a numerous and very mixed audience is to be
addressed, much skill will be required in adapting the
Style, (both in this, and in other respects,) and indeed
the Arguments also, and the whole structure of the
discourse, to the various minds which it is designed
to impress ; nor can the utmost art and diligence
prove after all more than partially successful in such
a case ; especially when the diversities are so many
and so great, as exist in the congregations to which
most Sermons are addressed, and in the readers for
whom popular works of an argumentative, instructive,
and hortatory character, are intended. It is possible,
however, to approach indefinitely to an object which
cannot be completely attained, and to adopt such a
Style and such a mode of Reasoning, as shall be level
to the comprehension of the greater part, at least, even
of a promiscuous audience, without being distasteful
to any.
It is obvious, and sufficiently well known, that ex-
treme conciseness is ill suited to hearers or readers,
whose intellectual powers and cultivation are but
Small: the usual expedient, however, of employing a
R H E T O R I. C. 271
is to prefer terms of Saron origin, which will gener- Chap, III,
ally be more familiar to them, than those derived S-N-2
Rhetoric, prolir Style by way of accommodation to such minds,
! is seldom successful : most of those who could have
comprehended the meaning, if more briefly expressed,
and many of those who could not do so, are likely to be
bewildered by tedious expansion; and being unable
to maintain a steady attention to what is said, they
forget part of what they have heard before the whole is
completed. Add to which, that the feebleness produced
by excessive dilution, (if such an expression may be
allowed,) will occasion the attention to languish ; and
what is imperfectly attended to, however clear in itself,
will usually be but imperfectly understood. Let not an
author, therefore, satisfy himself by finding that he
has expressed his meaning so that, if attended to, he
cannot fail to be understood ; he must consider also
(as was before remarked) what attention is likely to
be paid to it : if on the one hand much matter is ex-
pressed in very few words, to an unreflecting audience,
or if, on the other hand, there is a wearisome prolixity,
the requisite attention may very probably not be be-
Stowed. * W.
The best general rule for avoiding the disadvantages
both of conciseness and of prolixity, is to employ
Repetition : to repeat, that is, the same sentiment and
Argument in many different forms of expression; each
in itself brief, but all, together, affording such an ex-
pansion of the sense to be conveyed, and so detaining
the mind upon it, as the case may require. Cicero
among the ancients, and Burke among the modern
writers, afford, perhaps, the most abundant practi-
cal exemplifications of this rule. The latter some-
times shows a deficiency in correct taste, and lies open
to Horace's censure of an author, “Qui variare cupit
rem prodigaliter unam ;” but it must be admitted that
he seldom fails to make himself thoroughly under-
stood, and does not often weary the attention, even
when he offends the taste of his readers.
Care must of course be taken that the repetition
may not be too glaringly apparent; the variation must
not consist in the mere use of other, synonymous,
words; but what has been expressed in appropriated
terms may be repeated in metaphorical; the antece-
dent and consequent of an Argument, or the parts of
an antithesis may be transposed; or several different
points that have been enumerated, presented in a varied
order, &c.
It is not necessary to dwell on-that obvious rule laid
down by Aristotle, to avoid uncommon, as they are
vulgarly called, hard words, i. e. those which are such
to the persons addressed ; but it may be worth re-
marking, that to those who wish to be understood by
the lower orders, one of the best principles of selection
* It is remarked by Anatomists that the nutritive quality is not
the only requisite in food;—that a certain degree of distention of
the stomach is required, to enable it to act with its full powers;–
and that it is for this reason hay and straw must be given to
horses, as well as corn, in order to supply the necessary bulk.
Something analogous to this takes place with respect to the gene-
rality of minds, which are incapable of thoroughly digesting and
assimilating what is presented to them, however clearly, in a very
small compass. Many a one is capable of deriving that instruc-
tion from a moderate sized volume, which he could not receive
from a very small pamphlet, even more perspicuously written, and
containing every thing that is to the purpose. It is necessary that
the attention should be detained for a certain time on the subject:
and persons of unphilosophical mind, though they can attend tò
what they read or hear, are unapt to dwell upon it in the way of
subsequent meditation. --
le
from the Latin, (either directly or through the me-
dium of the French,) even when the latter are more
in use among persons of education. Our language
being (with very trifling exceptions) made up of these
elements, it is very easy for any one, though unac-
quainted with Saxon, to observe this precept, if he has
but a knowledge of French or of Latin ; and there is a
remarkable scope for such a choice as we are speaking
of, from the multitude of synonymes derived, respec-
tively, from those two sources. The compilers of our
Liturgy being anxious to reach the understandings of
all classes, at a time when our language was in a less
settled state than at present, availed themselves of this
circumstance in employing many synonymous, or nearly
synonymous expressions, most of which are of the des-
cription just alluded to. Take as an instance, the Ex-
hortation: “acknowledge” and “confess;” “dissemble”
and “cloak ;” “humble” and “lowly;” “goodness”
and “mercy;” “assemble” and “ meet together :”
and here it may be observed that, as in this last in-
stance, a word of French origin will very often not
have a single word of Saxon derivation corresponding
to it, but may find an exact equivalent in a phrase of
two or more words: e.g. “ constitute,” “go to make
up;” “arrange,” “put in order;” “substitute,” “put
in the stead,” &c. &c.
It is worthy of notice that a Style, composed chiefly
of the words of French origin, while it is less intelli-
gible to the lowest classes, is characteristic of those
who in cultivation of taste are below the highest. As
in dress, furniture, deportment, &c. So also in language,
the dread of vulgarity constantly besetting those who
are half conscious that they are in danger of it, drives
them into the extreme of affected finery. So that the
precept which has been given with a view to perspi-
cuity, may, to a certain degree, be observed with an
advantage in point of elegance also. *
In adapting the Style to the comprehension of the
illiterate, a caution is to be observed against the am-
biguity of the word “Plain ;" which is opposed some-
times to Obscurity, and sometimes to Ornament; the
vulgar require a perspicuous, but by no means, a dry
and unadorned Style ; on the contrary, they have a
taste rather for the over-florid, tawdry, and bombastic :
nor are the ornaments of style by any means necessarily
inconsistent with perspicuity; Metaphor, which is
among the principal of them, is indeed, in many cases,
the clearest mode of expression that can be adopted;
it being usually much easier for uncultivated minds to
comprehend a similitude or analogy, than an abstract
term. And hence the language of savages, as has
often been remarked, is highly metaphorical ; and
such appears to have been the case with all languages
in their earlier, and consequently ruder and more
savage state ; many terms relating to the mind and
its operations, being, as appears from their etymo-
logy, originally metaphorical, though by long use
they have ceased to be so : e. g. the words “ponder,”
“ deliberate,” “reflect,” and many other such, are evi-
dently drawn by analogy from external sensible bodily
actions.
In respect to the Construction of sentences, it is
an obvious caution to abstain from such as are too
long ; but it is a mistake to suppose that the obscu-
rity of many long sentences depends on their length
272
R H E T O R I. C.
Rhetoric, alone; a well constructed sentence of very consider-
\-N-' able length may be more readily understood, than a
shorter one which is more awkwardly framed. If a
sentence be so constructed that the meaning of each
part can be taken in as we proceed, (though it be
evident that the sense is not brought to a close) its
length will be little or no impediment to perspicuity;
but if the former part of the sentence convey no dis-
tinct meaning till we arrive nearly at the end, how-
ever plain it may then appear, it will be on the whole
deficient in perspicuity; for it will need to be read
over, or thought over, a second time, in order to be
fully comprehended; which is what few readers or
hearers are willing to be burthened with. Take as an
instance such a sentence as this : “It is not without a
degree of patient attention and persevering diligence,
greater than the generality are willing to bestow,
though not greater than the object deserves, that the
habit can be acquired of examining and judging of
our own conduct with the same accuracy and impar-
tiality as that of another:” this labours under the defect
we are speaking of, which may be remedied by some
such alteration as the following: “the habit of examin-
ing our own conduct as accurately as that of another,
and judging of it with the same impartiality, cannot be
acquired without a degree of patient attention and per-
severing diligence, not greater indeed than the object
deserves, but greater than the generality are willing to
bestow.” The two sentences are nearly the same in
length, and in the words employed ; but the alteration
of the arrangement allows.the latter to be understood
clause by clause, as it proceeds. The caution just given
is the more necessary to be insisted on, because an
author is apt to be misled by reading over a sentence
to himself, and being satisfied on finding it perfectly
intelligible, forgetting that he himself has the advan-
tage, which a hearer has not, of knowing at the begin-
ning of the sentence what is coming in the close.
Universally, indeed, an unpractised writer is liable
to be misled by his own knowledge of his own mean-
ing, into supposing those expressions clearly intelli-
gible, which are so to himself; but which may not be
so to the reader, whose thoughts are not in the same
train. And hence it is that some do not write or speak
with so much perspicuity on a subject which has long
been very familiar to them, as on one which they un-
derstand indeed, but with which they are less intimately
acquainted, and in which their knowledge has been
more recently acquired. In the former case it is a matter
of some difficulty to keep in mind the necessity of
carefully and copiously explaining principles which by
long habit have come to assume in our minds the
appearance of self-evident truths. So far is Blair's
notion from being correct, that obscurity of Style
necessarily springs from indistinctness of Conception.
The foregoing rules have all, it is evident, pro-
ceeded on the supposition that it is the writer's inten-
tion to be understood ; and this cannot but be the case
in every legitimate exercise of the Rhetorical art; and
generally speaking, even where the design is Sophisti-
cal. For, as Dr. Campbell has justly remarked, the
Sophist may employ for his purpose what are in them-
selves real and valid Arguments, since probabilities
may lie on opposite sides, though truth can be but on
one ; his fallacious artifice consisting only in keeping
out of sight the stronger probabilities which may be
urged against him, and in attributing an undue weight
to those which he has to allege.
premiss which there is no sufficient ground for ad-
mitting; or he may draw off the attention of the hearers
to the proof of some irrelevant point, &c. according to
the various modes described in the Treatise on FALLA-
cIEs ; but in all this there is no call for any departure
from perspicuity of Style, properly so called ; not
even when he avails himself of an ambiguous term.
“For though,” as Dr. Campbell says, “a Sophism can
be mistaken for an Argument only where it is not rightly
understood,” it is the aim of him who employs it, rather
that the matter should be misunderstood than not under-
stood ; –that his language should be deceitful rather
than obscure or unintelligible. The hearer must not
indeed form a correct, but he must form some, and if
possible, a distinct, though erroneous idea of the Ar-
guments employed, in order to be misled by them.
The obscurity in short, if it is to be so called, must
not be obscurity of Style; that must be, not like a mist
which dims the appearance of objects, but like a
coloured glass which disguises them. -
There are, however, certain spurious kinds, as they
may be called, of writing or speaking, (distinct from
what is strictly termed Sophistry,) in which obscurity
of Style may be apposite. The object which has all
along been supposed, is that of convincing or persuad-
ing ; but there are some kinds of Oratory, if they are
to be so named, in which different ends are pro-
posed. One of these ends is, (when the cause is such
that it cannot be sufficiently supported even by speci-
ous Fallacies,) to appear to say something, when there is
in fact nothing to be said; so as at least to avoid the
ignominy of being silenced. To this end, the more
confused and unintelligible the language, the better,
provided it carry with it the appearance of profound
wisdom, and of being something to the purpose.
“Now though nothing (says Dr. Campbell) would
seem to be easier than this kind of Style where an
Author falls into it naturally ; that is, when he de-
ceives himself as well as his reader, nothing is more
difficult when attempted of design. It is beside requi-
site, if this manner must be continued for any time,
that it be artfully blended with some glimpses of
meaning ; else, to persons of discernment, the charm
will at length be dissolved, and the nothingness of
what has been spoken will be detected ; nay even the
attention of the unsuspecting multitude, when not
relieved by any thing that is level to their compre-
hension, will infallibly flag. The Invocation in the
Dunciad admirably suits the Orator who is unhappily
reduced to the necessity of taking shelter in the unin-
telligible:
“Of darkness visible so much be lent,
As half to show, half veil the deep intent.”
Chap. viii. Sec. 1. p. 119.
This artifice is distinguished from Sophistry, pro-
perly so called, (with which Dr. Campbell seems to
confound it,) by the circumstance that its tendency is
not, as in Sophistry, to convince, but to have the ap-
pearance of arguing, when in fact, nothing is urged;
for in order for men to be convinced, on however in-
sufficient grounds, they must (as was remarked above)
understand something from what is said, though, if it
be fallacious, they must not understand it rightly ; buf,
if this cannot be accomplished, the Sophist's next
Or again he may, Chap.
either directly or indirectly, assume as self-evident a \-N-2
R H E TO RI c.
273
Rhetoric, resort is the unintelligible, which indeed is very often
\-y-' intermixed with the Sophistical, when the latter is of
employ them, is to amuse their audience with specious Chap. III.
emptiness. S-N-"
occupy time.
bringing them forward.
itself too scanty or too weak. Nor does the adoption
of this Style serve merely to save his credit as an
Orator or Author; it frequently does more : ignorant
and unreflecting persons, though they cannot be,
strictly speaking, convinced, by what they do not un-
derstand, yet will very often suppose, each, that the
rest understand it ; and each is ashamed to acknow-
ledge, even to himself, his own darkness and per-
plexity; so that if the speaker with a confident air an-
nounces his conclusion as established, they will often,
according to the maxim “omne ignotum pro mirifico,”
take for granted that he has advanced valid Arguments,
and will be loth to seem behind hand in comprehend-
ing them. It usually requires that a man should have
some confidence in his own understanding, to venture
to say, “what has been spoken is unintelligible to me.”
Another purpose sometimes answered by a discourse
of this kind, is that it serves to furnish an excuse,
flimsy indeed, but not unfrequently sufficient, for men
to vote or act according to their own inclinations;
which they would perhaps have been ashamed to do,
if strong Arguments had been urged on the other side,
and had remained confessedly unanswered ; but they
satisfy themselves if something has been said in favour
of the course they wish to adopt, though that some-
thing be only fair-sounding sentences that convey no
distinct meaning. They are content that an answer
has been made, without troubling themselves to con-
sider what it is.
Another end, which in speaking, is sometimes pro-
posed, and which is, if possible, still more remote
from the legitimate province of Rhetoric, is to
When an unfavourable decision is ap-
prehended, and the protraction of the debate may
afford time for fresh voters to be summoned, or
may lead to an adjournment, which will afford
scope for some other manoeuvre;—when there is a
chance of so wearying out the attention of the
hearers, that they will listen with languor and
impatience to what shall be urged on the other side;—
when an advocate is called upon to plead a cause in
the absence of those whose opinion it is of the utmost
importance to influence, and wishes to reserve all his
Arguments till they arrive, but till then, must appa-
rently proceed in his pleading ; in these and many
similar cases, which it is needless to particularize, it is
a valuable talent to be able to pour forth with fluency
an unlimited quantity of well-sounding language which
has little or no meaning;-which shall not strike the
hearers as unintelligible or nonsensical, though it con-
vey to their minds no distinct idea. Perspicuity of
Style, real, not apparent perspicuity, is in this case
never necessary, and sometimes, studiously avoided.
If any distinct meaning were conveyed, and that which
was said were irrelevant, it would be perceived to be
so, and would produce impatience in the hearers, or
afford an advantage to the opponents; if, on the other
hand, the speech were relevant, and there were no
Arguments of any force to be urged, except such as
either had been already dwelt on, or were required to
be reserved (as in the case last alluded to) for a fuller
audience, the speaker would not further his cause by
So that the usual resource on
these occasions, of such Orators as thoroughly under-
stand the tricks of their art, and do not disdain to
Vol. 1.
Another kind of spurious Oratory, and the last that
will be noticed, is that which has for its object the
hearer's admiration of the Eloquence displayed. This,
indeed, constitutes one of the three kinds of Oratory
enumerated by Aristotle, and is regularly treated of by
him along with the deliberative and judicial branches;
though it hardly deserves the place he has bestowed
on it. -
When this is the end pursued, perspicuity is not
indeed to be avoided, but it may often without detriment
be disregarded.* Men frequently admire as eloquent,
* In Dr. Campbell's ingenious dissertation, (Rhetoric, book ii.
c. vii.) “on the causes that nonsense often escapes being de-
tected, both by the writer and the reader,” he remarks, (sec. 2,)
that “there are particularly three sorts of writing wherein we are
liable to be imposed upon by words without meaning.”
“The first is, where there is an exuberance of metaphor. Nothing
is more certain than that this trope, when temperately and appo-
sitely used, serves to add light to the expression, and energy to
the sentiment. On the contrary, when vaguely and intemperately
used, nothing can serve more effectually to cloud the sense, where
there is sense, and by consequence to conceal the defect, where
there is no sense to show. And this is the case, not only where
there is in the same sentence a mixture of discordant metaphors,
but also where the metaphoric Style is too long continued, and
too far pursued. [Ut modicus autem atque opportunus translationis
wsus illustrat orationem : ita frequens et obscurat et tadio complet;
continuus vero in allegorian et aenigmata erit. Quint, lib. viii. C. vi.]
The reason is obvious. In common speech the words are the im-
mediate signs of the thought. But it is not so here; for when a
person, instead of adopting metaphors that come naturally and
opportunely in his way, rummages the whole world in quest of
them, and piles them one upon another, when he cannot so pro-
perly be said to use metaphor, as to talk in metaphor, or rather
when from metaphor he runs into allegory, and thence into enigma,
his words are not the immediate signs of his thought; they
are at best but the signs of the signs of his thought. His writing
may then be called, what Spenser not unjustly styled his Fairy
Queen, a perpetual allegory or dark conceit, Most readers will
account it much to bestow a transient glance on the literal sense,
which lies nearest; but will never think of that meaning more
remote, which the figures themselves are intended to signify. It
is no wonder then that this sense, for the discovery of which it is
necessary to see through a double veil, should, where it is, more
readily escape our observation, and that where it is wanting we
should not so quickly miss it.
“There is, in respect of the two meanings, considerable variety
to be found in the tropical Style. In just allegory and similitude
there is always a propriety, or, if you choose to call it, congruity,
in the literal sense, as well as a distinct meaning or sentiment
suggested, which is called the figurative sense. Examples of this
are unnecessary. Again, where the figurative sense is unexcep-
tionable, there is sometimes an incongruity in the expression of
the literal sense. This is always the case in mixed metaphor, a
thing not unfrequent even in good writers. Thus, when Addison
remarks that “there is not a single view of human nature, which
is not sufficient to eatinguish the seeds of pride,’ he expresses a
a true sentiment somewhat incongruously; for the terms eatin-
guish and seeds here metaphorically used, do not suit each other.
In like manner, there is something incongruous in the mixture of
tropes employed in the following passage from Lord Bolingbroke :
* Nothing less than the hearts of his people will content a patriot
Prince, nor will he think his throne established, till it is esta-
blished there.” Yet the thought is excellent. But in neither of
these examples does the incongruity of the expression hurt the
perspicuity of the sentence. Sometimes, indeed, the literal
meaning involves a direct absurdity. When this is the case, as
in the quotation from Z'he Principles of Painting given in the pre-
ceding chapter, it is natural for the reader to suppose that there
must be something under it; for it is not easy to say how ab-
surdly even just sentiments will sometimes be expressed. But
when no such hidden sense can be discovered, what, in the first
view conveyed to our minds a glaring absurdity, is rightly on re-
flection denominated nonsense. We are satisfied that De Piles
neither thought, nor wanted his readers to think, that Rubens
was really the original performer, and God the copier. This
O
274 e R H E TO R. I. C.
Thetoric, and sometimes admire the most, what they do not at sounding words be arranged in graceful and Sonorous Chap. III.
-v- all, or do not fully comprehend, if elevated and high periods. Those of uncultivated minds especially, are S-N-
then was not his meaning. But what he actually thought and
wanted them to think, it is impossible to elicit from liis words.
His words then may justly be attributed bold, in respect of their
literal import, but wrimeaning in respect of the author's intention.
“lt may be proper here to observe, that some are apt to con-
found the terms absurdity and nonsense as synonymous, which they
manifestly are not. An absurdity, in the strict acceptation, is a
proposition either intuitively or demonstratively false. Of this
kind are these : ‘Three and two make seven.” “All the angles of a
triangle are greater than two right angles.” That the former is false
we know by intuition ; that the latter is so, we are able to demon-
strate. But the term is further extended to denote a notorious
falsehood. If one should affirm, that “at the vernal equinox the
sun rises in the north and sets in the south,’ we should not hesi-
tate to say, that he advances an absurdity; but still what he
affirms has a meaning ; insomuch, that on hearing the sentence
we pronounce its falsity. Now nonsense is that whereof we cannot
say either that it is true, or that it is false. Thus, when the Teutonic
Theosopher enounces, that ‘ all the voices of the celestial joyful-
ness, qualify, commix, and harmonize in the fire which was from
eternity in the good quality,” I should think it equally imperti-
nent to aver the falsity as the truth of this enunciation. For,
though the words grammatically form a sentence, they exhibit to
the understanding no judgment, and consequently admit neither
assent nor dissent. . In the former instances I say the meaning, or
what they affirm, is absurd ; in the last instance I say there is no
meaning, and therefore properly nothing is affirmed. In popular
language, I own, the terms absurdity and nonsense are not so ac-
curately distinguished. Absurd positions are sometimes called
nonsensical. It is not common, on the other hand, to say of
downright nonsense, that it comprises an absurdity.
“Further, in the literal sense there may be nothing unsuitable,
and yet the reader may be at a loss to find a figurative meaning,
to which his expressions can with justice be applied. Writers
immoderately attached to the florid, or highly figured diction, are
often misled by a desire of flourishing on the several attributes
of a metaphor, which they have pompously ushered into the dis-
course, without taking the trouble to examine whether there be any
qualities in the subject, to which these attributes can with justice
and perspicuity be applied.” This immoderate use of metaphor,
i)r. Campbell observes, “ is the principal source of all the non-
sense of Orators and Poets.
“The second species of writing wherein we are liable to be
imposed on by words without meaning, is that wherein the
terms most frequently occurring, denote things which are of a
complicated nature, and to which the mind is not sufficiently
familiarised. Many of those notions which are called by Philo-
sophers mixed modes, come under this denomination. Of these
the instances are numerous in every tongue; such as government,
church, state, constitution, polity, power, commerce, legislature,
Jurisdiction, proportion, symmetry, elegance. It will considerably
increase the danger of our being deceived by an unmeaning use of
such terms, if they are besides (as very often they are) of so in-
determinate, and consequently equivocal significations, that a
writer, unobserved either by himself or by his reader, may slide
from one sense of the term to another, till by degrees he fall into
such applications of it as will make no sense at all. It deserves
our notice also, that we are in much greater danger of terminating
in this, if the different meanings of the same word have some
affinity to one another, than if they have none. In the latter case,
when there is no affinity, the transition from one meaning to
another, is taking a very wide step, and what few writers are in
any danger of ; it is, besides, what will not so readily escape the
observation of the reader. So much for the second cause of
deception, which is the chief source of all the nonsense of writers
on politics and criticism. 4
“The third and last, and I may add, the principal species of
composition, wherein we are exposed to this illusion by the abuse
of words, is that in which the terms employed are very abstract,
and consequently of very extensive signification. It is an obser-
vation that plainly ariseth from the nature and structure of lan-
guage, and may be deduced as a corollary from what hath been
said of the use of artificial signs, that the more general any name
is, as it comprehends the more individuals under it, and conse-
quently requires the more extensive knowledge in the mind that
would rightly apprehend it, the more it must have of indistinct- .
ness and obscurity. Thus the word lion is more distinctly ap-
prehended by the mind than the word beast, beast than animal,
apt to think meanly of any thing that is brought down
perfectly to the low level of their capacity; though to
do this with respect to valuable Truths which are not
trite, is one of the most admirable feats of genius; they
. admire the profundity of one who is mystical and ob-
scure; mistaking the muddiness of the water for depth;
and magnifying in their imaginations what is viewed
through a fog ; and they conclude that brilliant lan-
guage must represent some brilliant ideas, without
troubling themselves to inquire what those ideas are.
Many an enthusiastic admirer of a “fine discourse,”
or a piece of “fine writing,” would be found on ex-
amination to retain only a few sonorous, but empty
phrases; and not only to have no notion of the general
drift of the Argument, but not even to have ever con-
sidered whether the Author had any such drift or not.
It is not meant to be insinuated that in every such
case the composition is in itself unmeaning, or that
the Author had no other object than the credit of Elo-
quence: he may have had a higher end in view ; and
he may have expressed himself very clearly to some
hearers, though not to all : but it is most important
to be fully aware of the fact, that it is possible to
obtain the highest applause from those who not only
receive no edification from what they hear, but abso-
lutely do not understand it. So far is popularity from
being a safe criterion of the usefulness of a Preacher.
§ 2. The next quality of Style to be noticed is what
may be called Energy ; the term being used in a
wider sense than the 'Evépycta of Aristotle, and nearly
corresponding with what Dr. Campbell calls Vivacity;
so as to comprehend every thing that may conduce to
-stimulate attention,--to impress strongly on the mind
the Arguments adduced,—to excite the Imagination,
and to arouse the Feelings. …”
This Energy then, or Vivacity of Style, must de-
pend (as is likewise the case in respect of Perspicuity.)
on three things; 1st, the Choice of words, 2d, their
Number, and 3d, their Arrangement.
With respect to the Choice of words, it will be most
convenient to consider them under those two classes
which Aristotle has described under the titles of
—-w
animal than being. But there is, in what are called abstract sub-
jects, a still greater fund of obscurity, than that arising from
the frequent mention of the most general terms. Names must
be assigned to those qualities as considered abstractly, which
never subsist independently, or by themselves, but which consti-
tute the generic characters and the specific differences of things.
. And this leads to a manner which is in many instances remote
from the common use of speech, and therefore must be of more
difficult conception.” (Book ii. sec. 2. p. 102, 103.)
It is truly to be regretted that an author who has written so justly
on this subject, should within a few pages so strikingly exemplify
the errors he has been treating of, by indulging in a declamation
against Logic which could not even to himself have conveyed any
distinct meaning. When he says that a man who had learned
Logic was “qualified without any other kind of knowledge, to
defend any position whatever, however contradictory to common
sense ; and that that art observed the most absolute indifference
to truth and error,” he cannot mean that a false conclusion could
be logically proved from true premises ; since, ignorant as he
was of the subject, he was aware, and has in another place dis-
tinctly acknowledged, that this is not the case ; nor could he
mean merely that a false conclusion could be proved from a false
premiss, since that would evidently be a nugatory and ridiculous
objection. He seems to have had, in truth, no meaning at all ;
though like the authors he had been so ably criticising, he was
perfectly unaware of the emptiness of what he was saying. •
R H E T O. R. I. C. 275
either more or less General terms according to the Chap. III.
Rhetoric. Kipta and Eéva, for which our language does not afford,
. objects he is speaking of. There is, however, in S-N-2
\–2–’ precisely corresponding names : “Proper,” “ Appro-
priate,” or “Ordinary,” terms, will the most nearly
designate the former; the latter class including all.
others;–all that are in any way removed from com-
mon use ;-whether uncommon terms, or ordinary.
terms, either transferred to a different meaning from
that which strictly belongs to them, or employed in.
a different manner from that of common discourse.
All the Tropes and Figures, enumerated by Gram-
matical and Rhetorical Writers, will of course fall
under this head.
With respect then to “Proper’ terms, the prin-
cipal rule for guiding our Choice with a view to Energy,
is to prefer, ever, those words which are the least
abstract and general. Individuals alone having a real
existence,” the terms denoting them (called by Logi-
cians “Singular terms,”) will of course make the most.
vivid impression on the mind, and exercise most the
power of Conception; and the less remote any term
is from these, i. e. the more specific, the more Energy
it will possess, in comparison of such as are more
general. The impression produced on the mind by
a Singular term, may be compared to the distinct
view taken in by the eye of any object (suppose a
man) near at hand, in a clear light, which enables us
to distinguish the features of the individual ; in a
fainter light, or rather farther off, we merely perceive
that the object is a man ; this corresponds with the
idea conveyed by the name of the Species; yet further
off, or in a still feebler light, we can distinguish
merely some living object, and at length, merely some
object ; these views corresponding respectively with the
terms denoting the genera, less or more remote : and
as each of these views conveys, as far as it goes, an
equally correct impression to the mind, (for we are
equally certain that the object at a distance is some-
thing, as that the one close to us is such and such an
individual,) though each, successively, is less vivid ;
so, in language, a General term may be as clearly
wnderstood, as a Specific or Singular term, but will
convey a much less forcible impression to the hearer's
mind. “The more General the terms are,” (as Dr.
Campbell justly remarks,) “ the picture is the fainter;
the more Special they are, the brighter. The same
sentiment may be expressed with equal justness, and
even equal perspicuity, in the former way, as in the
latter ; but as the colouring will in that case be more
languid, it cannot give equal pleasure to the fancy,
and by consequence will not contribute so much
either to fix the attention, or to impress the me-
mory.”
It might be supposed at first sight, that an Author
has little or no Choice on this point, but must employ
* Thence called by Aristotle, (Categ. sec. 3,) “primary sub-
stances,” (Trpárat āq (al,) Genus and Species, being denominated
“secondary,” as not properly denoting a “really-existing-thing,”
(rööe Ti,) but rather an attribute. He has, indeed, been con-
sidered as the great advocate of the opposite doctrine; i. e. of
the system of “Realism ;” which was certainly embraced by
many of his professed followers ; but his own language is suf-
ficiently explicit. II&ora ö& 80ſa Šoke? Tööe Ti a muatvely. 'Eml
w&v 3v táv Trparov gotów &vauqto Sātmtov kai äAm6és . Šarruv, Šti Tööe
Ti or muatvei &ropov yap, kai év & puðuº Tb ÖmXéuevov čotiv. 'Firl 6&
Töv Sevrépav Šalčºv, PAINETAI pièv Šuota's tº a Xijuari Tris Trpoon-
Tyoptas rôe ti ornuatveiv, 3rav čitrn, Śvēpatros, ) {6}ov O'r MHN TE
AAHOE3’ &AA& PaxNov trolov Ti a muatver k. T. A. Aristotle,
Categ. Sec. 3. -
almost every case, great room for such a Choice as we
are speaking of ; for, in the first place, it depends on
our Choice whether or not we will employ terms more
General than the subject requires ; which may almost
always be done consistently with Truth and Propriety,
though not with Energy : if it be true that a man
has committed murder, it may be correctly asserted,
that he has committed a crime; if the Jews were
“ exterminated,” and “ Jerusalem demolished ” by
“ Vespasian's army,” it may be said, with truth, that
they were “subdued" by “an Enemy,” and their
“ Capital’’ taken. This substitution then of the
General for the Specific, or of the Specific for the Sin-
gular, is always within our reach ; and many, espe-
cially unpractised Writers, fall into a feeble Style by
resorting to it unnecessarily ; either because they
imagine there is more appearance of refinement or of
profundity, in the employment of such terms as are
in less common use among the vulgar, or, in some
cases, with a view to give greater comprehensiveness
to their Reasonings, and to increase the utility of
what they say by enlarging the field of its application.
Inexperienced Preachers frequently err in this way, by
dwelling on Virtue and Vice, Piety and Irreligion, in
the abstract, without particularizing ; forgetting that
while they include much, they impress little or
nothing.
The only Appropriate occasion for this Generic
language, (as it may be called,) is when we wish to
avoid giving a vivid impression,--when our Object is
to Soften what is offensive, disgusting, or shocking ;
as when we speak of an “execution,” for the inflic-
tion of the sentence of death on a criminal ; of which
kind of expressions, common discourse furnishes
numberless instances. On the other hand, in An-
tony's speech over Caesar's body, his object being to
excite horror, Shakspeare puts into his mouth the most
particular expressions : “ those honourable men (not,
who killed Caesar, but) whose daggers have stabbed
Caesar.” -
But in the second place, not only does a regard for
Energy require that we should not use terms more
General than are exactly adequate to the objects
spoken of, but we are also allowed, in many cases, to
employ less General terms than are exactly Appropriate.
In which case we are employing words not “Appro-
priate,” but belonging to the second of the two classes
just mentioned. The use of this Trope,” (enumerated
by Aristotle among the Metaphors, but since more
commonly called Synecdoche) is very frequent, as it con-
duces much to the Energy of the expression, without
occasioning, in general, any risk of its meaning being
mistaken. The passage cited by Dr. Campbell, f from
one of our Lord's discourses, (which are in general
of this character,) together with the remarks made
upon it, will serve to illustrate what has been just said:
“‘Consider,’ says our Lord, the lilies how they
grow : they toil not, they spin not ; and yet I say
* From Tperò ; any word turned from its primary signi-
fication. -
+ The ingenious Author cites this in the Section treating of
“Proper terms,” which is a trifling oversight; as it is plain that
“ lily” is used for the Genus “flower,”—“Solomon,” for the
Species “King,” &c. g
2 O 2
red
276
R H E T O R. I. C.
Rhetoric. unto you, that Solomon in all his glory, was not arrayed
like one of these. If then God so clothe the grass
which to-day is in the field, and to-morrow is cast
into the oven, how much more will he clothe you?”
Het us here adopt a little of the tasteless manner of
modern paraphrasts, by the substitution of more Gene-
ral terms, one of their many expedients of infrigidat-
ing, and let us observe the effect produced by this
change. “Consider the flowers, how they gradually
increase in their size, they do no manner of work,
and yet I declare to you, that no king whatever, in his
most splendid habit, is dressed up like them. If then
God in his providence doth so adorn the vegetable
productions, which continue but a little time on the
land, and are afterwards devoted to the meanest uses,
how much more will he provide clothing for you ?’
How spiritless is the same sentiment rendered by these
small variations 2 The very particularizing of to-day
and to-morrow, is infinitely more expressive of transi-
toriness, than any description wherein the terms are
General, that can be substituted in its room.” It is a
remarkable circumstance that this characteristic of
Style is perfectly retained in translation, in which every
other excellence of expression is liable to be lost;
so that the prevalence of this kind of language in
the Sacred writers, may be regarded as something
providential. It may be said with truth, that the book
which it is the most necessary to translate into every
language, is chiefly characterised by that kind of
excellence in diction which is least impaired by trans-
lation.
But to proceed with the consideration of Tropes ;
the most employed and most important of all those
kinds of expressions which depart from the plain and
strictly Appropriate Style, all that are called by
Aristotle, Eóva,-is the Metaphor, in the usual and
limited sense ; viz. a word substituted for another,
on account of the Resemblance or Analogy between
their significations. The Simile or Comparison may
be considered as differing in form only from a Meta-
phor; the Resemblance being in that case stated,
which in the Metaphor is implied. Each may be
founded either on Resemblance, strictly so called, i. e.
direct Resemblance between the objects themselves
in question, (as when we speak of “ table-land,” or
compare great waves to mountains,) or on Analogy,
which is the Resemblance of ratios,-a similarity of
the relations they bear to certain other objects ; as
when we speak of the “ light of reason,” or of “ re-
velation ;” or compare a wounded and captive warrior
to a stranded ship.f The Analogical Metaphors and
Comparisons are both the most frequent and the most
striking. They are the most frequent, because almost
every object has such a multitude of relations, of
different kinds, to many other objects ; and they are
the most striking, because (as Dr. A. Smith has well
remarked,) the more remote and unlike in themselves
any two objects are, the more is the mind impressed
and gratified by the perception of some point in which
they agree. *
It has been already observed, under the head of
Example, (chap. 1,) that we are carefully to distin-
guish between an Illustration, i. e. an Argument from
* Luke, ch. xii. ver. 27, 28.
+ Roderic Dhu, in the Lady of the Lake.
Analogy or Resemblance, and what is properly called Chap. III.
a Simile or Comparison, introduced merely to give
force or beauty to the expression. The aptness and
beauty of an Illustration sometimes leads men to over-
rate, and sometimes to underrate, its force as an
Argument. (Vol. i. p. 255.)
With respect to the choice between the Metaphori-
cal form and that of Comparison, it may be laid down
as a general rule, that the former is always to be pre-
ferred,” wherever it is sufficiently simple and plain
to be immediately comprehended ; but that which as
a Metaphor would sound obscure and enigmatical,
may be well received if expressed as a Comparison.
We may say, e.g. with propriety, that “Cromwell.
trampled on the laws :'' it would sound flat to say
that “ he treated the laws with the same contempt.
as a man does any thing which he tramples under his
feet.” On the other hand it would be harsh and
obscure to say, “the stranded vessel lay shaken by
the waves,” meaning the wounded chief tossing on
the bed of sickness ; it is therefore necessary in such
a case to state the Resemblance. But this is never to
be done more fully than is necessary to perspicuity,
because all men are more gratified at catching the
Resemblance for themselves, than at having it pointed
out to them.t And accordingly the greatest masters
of this kind of Style, when the case will not admit of
pure Metaphor, generally prefer a mixture of Meta-
phor with Simile ; first pointing out the similitude,
and afterwards employing metaphorical terms which
imply it ; or, vice versd, explaining a Metaphor by a
statement of the Comparison. To take examples
from an Author who particularly excels in this point;
(speaking of a morbid Fancy,)
“— like the bat of Indian brakes,
Her pinions fan the wound she makes,
And soothing thus the dreamer's pain,
She drinks the life-blood from the vein.”:
The word “ like ’’ makes this a Comparison ; but
the three succeeding lines are Metaphorical. Again,
to take an instance of the other kind,
“ They melted from the field, as snow,
When streams are swoln, and south winds blow,
I)issolves in silent dew :”$
Of the words here put in italics, the former is a
Metaphor, the latter, introduces a Comparison.
Though the instances here adduced are taken from a
Poet, the judicious management of Comparison which
they exemplify, is even more essential to a Prose
writer, to whom less licence is allowed in the em-
ployment of them. It is a remark of Aristotle,
(Rhet. book iii. c. 4,) that the Simile is more suitable
in Poetry, and that Metaphor is the only ornament of
language in which the Orator may freely indulge.
He should therefore be the more careful to bring a
Simile as near as possible to the Metaphorical form.
The following is an example of the same kind of
expression : “These metaphysic rights entering into
common life, like rays of light which pierce into a
dense medium, are, by the laws of nature, refracted
**Early j čuc&v ueta pop&, Suaq,épaara ºrpo6éore, Sib firrov #65,
3rt uakporépaſs k. T. A. Aristotle, Rhet. book iii. c. 10.
+ Tö wav6&vely paštws #55 ºptoet. Aristotle, Rhet. book iii, c. 5.
# Itokeby. § Marmion,
R H E TO RIC.
277
Thetoric, from their straight line. Indeed, in the gross and
\-Nº-' complicated mass of human passions and concerns, the
primitive rights of man undergo such a variety of refrac-
tions and reflections, that it becomes absurd to talk of
them as if they continued in the simplicity of their
original direction.”
Metaphors may be employed, as Aristotle observes,
either to elevate or to degrade the subject, according
to the design of the Speaker; being drawn from
similar or corresponding objects of a higher or lower
character. Thus a loud and vehement Speaker may
be described either as bellowing, or as thundering. And
in both cases, if the Metaphor is apt and suitable to
the purpose designed, it is alike conducive to Energy.
He remarks that the same holds good with respect to
Epithets also, which may be drawn either from the
highest or the lowest attributes of the thing spoken
off Metonymy likewise (in which a part is put
for a whole, a cause for an effect, &c.) admits of a
similar variety in its applications.
Any Trope (as is remarked by Dr. Campbell,) adds
force to the expression, when it tends to fix the mind
on that part, or circumstance, in the object spoken
of, which is most essential to the purpose in hand.
Thus, there is an Energy in Abraham's Periphrasis for
“God,” when he is speaking of the allotment of Di-
vine punishment: “shall not the Judge of all the earth
do right " If again we were alluding to His omniscience,
it would be more suitable to say, “this is known only
to the Searcher of hearts :" if, to his power, we should
speak of Him as “ the Almighty,” &c.
Of Metaphors, those generally conduce most to
that Energy or Vivacity of Style we are speaking of,
which illustrate an intellectual by a sensible object ;
the latter being always the most early familiar to the
mind, and generally giving the most distinct im-
pression to it. Thus we speak of “ umbridled rage,”
“ deep-rooted prejudice,” “ glowing eloquence,’’ a
“ stony heart,” &c. And a similar use may be made
of Metonymy also; as when we speak of the “Throne,”
or the “Crown " for “Royalty,”—the “sword” for
“ military violence,” &c. -
But the highest degree of Energy (and to which
Aristotle chiefly restricts the term) is produced by
such Metaphors as attribute life and action to things
inanimate ; and that, even when by this means the
last nientioned rule is violated, i. e. when sensible
objects are illustrated by intellectual. For the dis-
advantage is overbalanced by the vivid impression
produced by the idea of personality or activity ; ; as
when we speak of the rage of a torrent, a furious
storm, a river disdaining to endure its bridge, &c. §
Many such expressions, indeed, are in such common
F------
* Burke, On the French Revolution.
+ A happier example cannot be found than the one which
Aristotle cites from Simonides, who, when offered a small price
for an Ode to celebrate a victory in a mule-race, expressed his
contempt for half-asses, (hutovot) as they were commonly called ;
but when a larger sum was offered, addressed them in an Ode as
“Daughters of Steeds swift-as-the-storm.” &exxotóðav 65 yarpes
introv.
f The figure called by Rhetoricians Prosopopoeia (literally,
Personification) is, in fact, no other than a Metaphor of this
kind ; thus, in Demosthenes, Greece is represented as addressing
the Athenians. So also in the book of Genesis, (chap. iv. ver, 10,)
“ the voice of thy brother's blood crieth unto me from the
ground.”
§ Pontem indignatus,
use as to have lost all their Metaphorical force, since Chap. III.
they cease to suggest the idea belonging to their
primary signification, and thus are become, practically,
Proper terms. But a new, or at least, unhackneyed,
Metaphor of this kind, if it be not far-fetched and
obscure, adds greatly to the force of the expression.
This was a favourite figure with Homer, from whom
Aristotle has cited several examples of it; as “ the
raging arrow,” “ the darts eager to taste of flesh,”
“the shameless "(or as it might be rendered with more
exactness, though with less dignity, “ the provoking)
stone” (Aaas āvatóñs) which mocks the efforts of Sisy-
phus, &c. Our language possesses one remarkable ad-
vantage, with a view to this kind of Energy, in the
constitution of its genders. All nouns in English,
which express objects that are really neuter, are con-
sidered as strictly of the neuter gender ; the Greek
and Latin, though possessing the advantage, which is
wanting in the languages derived from them, of
having a neuter gender, yet lose the benefit of it by
fixing the masculine or feminine genders upon many
nouns denoting things inanimate ; whereas in Eng-
lish, when we speak of any such object in the mascu-
line or feminine gender, that form of expression at
once confers personality upon it. When “Virtue,”
e.g. or our “Country,” are spoken of as females,
or “ Ocean " as a male, &c. they are, by that very
circumstance, personified ; and a stimulus is thus given
to the imagination, from the very circumstance that
in calm discussion or description, all of these would
be neuter; whereas in Greek or Latin, as in French
or Italian, no such distinction could be made. The
employment of “ Virtus,” and “’Ape1),” in the femi-
nine gender, can contribute, accordingly, no animation
to the Style, when they could not, without a Solecism,
be employed otherwise.
There is, however, very little, comparatively, of
Energy produced by any Metaphor or Simile that is
in common use, and already familiar to the hearer;
indeed, what were originally the boldest Metaphors,
are become, by long use, virtually, Proper terms; as
is the case with the words “ source,” “reflection,”
&c. in their transferred senses ; and frequently are
even nearly obsolete in the literal sense, as in the
words “ardour,” “acuteness,” “ ruminate,” &c. If,
again, a Metaphor or Simile that is not so hackneyed
as to be considered common property, be taken from
any known Author, it strikes every one, as no less a
plagiarism than if an entire argument or description
had been thus transferred. And hence it is, that, as
Aristotle remarks, the skilful employment of these,
more than of any other, ornaments of language, may
be regarded as a mark of genius ; (€vºváis a juetov,)
not that he means to say, as some interpreters sup-
pose, that this power is entirely a gift of nature, and
in no degree to be learnt ; on the contrary, he ex-
pressly affirms, that the “ perception of Resem-
blances,”f on which it depends, is the fruit of
“Philosophy;”; but he means that Metaphors are
not to be, like other words and phrases, selected from
* There is a peculiar aptitude in some of these expressions which
the modern student is very likely to overlook; an arrow or
dart, from its flying with a spinning motion, quivers violently when
it is fixed ; thus suggesting the idea of a person trembling with
eagerness.
+ Tö 8potov Šgåv. Aristotle, Rhet, book ii.
# ‘Pāov čk piñocoptas, Ibid. book ii, and iii.
278
R H E To R I c.
We expect, indeed, and excuse in ancient writers, Chap, IIf:
as a part of the unrefined simplicity of a ruderlanguage, S-V-'
Rhetoric, comon use, and transferred from one composition to
\—y-Z another,” but must be formed for the occasion. Some
care is accordingly requisite, in order that they may
be readily comprehended, and may not have the ap-
pearance of being far-fetched and extravagant ; for
this purpose it is usual to combine with the Metaphor
a Proper term which explains it; viz. either attri-
buting to the term in its transferred sense, something
which does not belong to it in its literal sense ; or,
vice versa, denying of it in its transferred sense, some-
thing which does belong to it in its literal sense.
To call the Sea the “ watery bulwark” of our island,
would be an instance of the former kind ; an example
of the latter is the expression of a writer who speaks
of the dispersion of some hostile fleet by the winds
and waves, “ those ancient and unsubsidized allies of
England.”
It is hardly necessary to mention the obvious and
hackneyed cautions against mixture of Metaphors;t
and against any that are complex and far-pursued, so
as to approach to Allegory. In this last case, the
more apt and striking is the Analogy suggested, the
more will it have of an artificial appearance ; and
will draw off the reader's attention from the subject,
to admire the ingenuity displayed in the Style. Young
writers, of genius, ought especially to be admonished
to ask themselves frequently, not whether this or that
is a striking expression, but whether it makes the
meaning more striking than another phrase would,—
whether it impresses more forcibly the sentiment to be
conveyed. -
It is a common practice with some writers to en-
deavour to add force to their expressions by accumu-
lating high-sounding Epithets, # denoting the greatness,
beauty, or other admirable qualities of the things
spoken of ; but the effect is generally the reverse of
what is intended. Most readers, except those of a
very vulgar or puerile taste, are disgusted at studied
efforts to point out and force upon their attention
whatever is remarkable ; and this, even when the ideas
conveyed are themselves striking. But when an at-
tempt is made to cover poverty of thought with
mock sublimity of language, and to set off trite sen-
timents and feeble arguments by tawdry magnificence,
the only result is, that a kind of indignation is super-
added to contempt ; as when (to use Quinctilian's
comparison) an attempt is made to supply, by paint,
the natural glow of a youthful and healthy com-
plexion. § *
* 'Ovic ēart trap' &AAov Aašeiv. Aristotle, Rhet. book iii.
‘h Dr. Johnson justly, censures Addison for speaking of
“, bridling in his muse, who longs to launch into a nobler strain ;”
“which,” says the Critic, “is an act that was never restrained by
a bridle.” Some, however, are too fastidious on this point.
Words, which by long use in a transferred sense, have lost nearly
all their metaphorical force, may fairly be combined in a man-
ner which, taking them literally, would be incongruous. It
would savour of hypercriticism to object to such an expression as
** fertile source.”
: Epithets, in the Rhetorical sense, denote, not every adjec-
tive, but those only which do not add to the sense, but signify
something already implied in the noun itself; as, if one says,
“ the glorious sun ;” on the other hand, to speak of the “rising.”
or ‘‘ oneridian sun,” would not be considered as, in this sense,
employing an Epithet.
§ “A principal device in the fabrication of this Style,” (the
mock-eloqueñt,) “is to multiply epithets, dry epithets, laid on
the outside, and into which none of the vitality of the sentiment
is found to circulate. You may take a great number of the words
such a redundant use of Epithets as would not be tole-
rated in a modern, even in a translation of their works ;
the “white milk,” and “dark gore,” &c. of Homer,
must not be retained, at least, not so frequently as
they occur in the cricinal. Aristotle, indeed, gives us
to understand that in his time this liberty was still
allowed to Poets ; but later taste is more fastidious.
He censures, however, the adoption by prose writers.
of this, and of every other kind of ornament that
might seem to border on the poetical ; and he bestows
on such a Style, the appellation of “frigid,” (\ºvkpov,)
which, at first sight may appear somewhat remark-
able, (though the same expression, “frigid,” might
very properly be so applied by us,) because “warm,”
“glowing,” and such like Metaphors, seem naturally
applicable to poetry. This very circumstance, how-
ever, does not in reality account for the use of the
other expression. We are, in poetical prose, re-
minded of, and for that reason disposed to miss, the
“ warmth and glow" of poetry : it is on the same
principle that we are disposed to speak of coldness in
the rays of the moon, because they remind us of sunshine,
but want its warmth ; and that (to use an humbler
and more familiar instance) an empty fire-place is
apt to suggest an idea of cold.
The use of Epithets however, in prose composition,
is not to be proscribed ; as the judicious employment
of them is undoubtedly conducive to Energy. It is
extremely difficult to lay down any precise rules on.
such a point. The only safe guide in practice must
be a taste formed from a familiarity with the best
Authors, and from the remarks of a skilful Critic, on
one's own composition. It may, however, be laid.
down as a general caution, more particularly needful
for young writers, that an excessive luxuriance of
Style, and especially a redundancy of Epithets, is the
worse of the two extremes; as it is a positive fault,
and a very offensive one ; while the opposite is but the
absence of an excellence. It is also an important
rule that the boldest and most striking, and almost
poetical, turns of expression, should be reserved (as
Aristotle has remarked, book iii. c. 7,) for the most im-
passioned parts of a discourse; and that an Author should
guard against the vain ambition of expressing every
thing in an equally high-wrought, brilliant, and forcible
Style. The neglect of this caution often occasions.
the imitation of the best models to prove detrimental.
When the admiration of some fine and animated pas-
sages leads a young writer to take those passages for
his general model, and to endeavour to make every
sentence he composes equally fine, he will, on the
contrary, give a flatness to the whole, and destroy the
effect of those portions which would have been forci-
ble if they had been allowed to stand prominent. To
brighten the dark parts of a picture, produces much
the same result as if one had darkened the bright
parts; in either case there is a want of relief and
contrast ; and Composition, as well as Painting, has
its lights and shades, which must be distributed.
out of each page, and find that the sense is neither more nor less
for your having cleared the composition of these Epithets of
chalk of various colours, with which the tame thoughts had
submitted to be rubbed over, in order to be made fine.” Foster,
Essay iv.
R H E T O R. I. C.
279
disgusting. Critics treating on this subject have gone Chap. III.
into opposite extremes; some fancifully attributing S-v-'
Thetoric, with no less skill, if we would produce the desired
\-y-' effect.* -
In no place, however, will it be advisable to introduce
any Epithet which does not fulfil one of these two
purposes; 1st, to Eaplain a Metaphor; a use which has
been noticed under that head, and which will justify,
and even require, the introduction of an Epithet,
which, if it had been joined to the Proper term, would
have been glaringly superfluous; thus, AEschylus,f
speaks of the “winged hound of Jove,” meaning the
Eagle : to have said the “ winged eagle,” would have
had a very different effect : 2dly, when the Epithet,
expresses something which, though implied in the
subject, would not have been likely to occur at once
spontaneously to the hearer's mind, and yet is im-
portant to be noticed with a view to the purpose in
hand. Indeed it will generally happen, that the Epi-
thets employed by a skilful Orator, will be found to
be, in fact, so many abridged arguments, the force of
which is sufficiently conveyed by a mere hint ; e. g.
if any one says, “we ought to take warning from the
bloody revolution of France,” the Epithet suggests
one of the reasons for our being warmed ; and that,
not less clearly, and more forcibly, than if the Argu-
ment had been stated at length. -
With respect to the use of Antiquated, Foreign,
New-coined or New-compounded words,f or words
applied in an unusual sense, it may be sufficient to
observe, that all writers, and prose writers most,
should be very cautious and sparing in the use of
them ; not only because in excess they produce a
barbarous dialect, but because they are so likely
to suggest the idea of artifice; the perception of
which is most especially adverse to Energy. The
occasional apt introduction of such a term, will some-
times produce a powerful effect; but whatever may
seem to savour of affectation, or even of great solici-
tude and study in the Choice of terms, will effectually
destroy the true effect of Eloquence. The language
which betrays art, and carries not an air of simplicity
and sincerity, may, indeed, by some hearers, be thought
not only very fine, but even very Energetic ; this very
circumstance, however, may be taken for a proof that it
is not so; for if it had been, they would not have thought
about it, but would have been occupied, exclusively,
with the subject. An unstudied and natural air, there-
fore, is an excellence to which the true Orator, i. e.
he who is aiming to carry his point, will be ready to
sacrifice any other that may interfere with it.
The principle here laid down will especially apply
to the Choice of words, with a view to their Imitative,
or otherwise, Appropriate sound The attempt to make
the sound an echo to the sense, is indeed more fre-
quently to be met with in poets than in prose writers;
but it may be worth remarking, that an evident effort
after this kind of excellence, as it is offensive in any
kind of Composition, would in prose appear peculiarly
* Omnia vult helle Mažho dicere; die aliquando
Et bene; dic neutrum ; dic aliquando male.
+ Pronetheus. z
: It is a curious instance of whimsical inconsistency, that
many who, with justice, censure as pedantic, the frequent intro-
duction of Greek and Latin words, neither object to, nor refrain
from, a similar pedantry with respect to French and Italian.
This kind of affectation is one of the “ dangers ” of “a little
learning :” those who are really good linguists are seldom so
anxious to display their knowledge.
to words, or combinations of words, an Imitative power
far beyond what they can really possess,” and repre-
senting this kind of Imitation as deserving to be
studiously aimed at ; and others, on the contrary,
considering nearly the whole of this kind of excellence
as no better than imaginary, and regarding the exam-
ples which do occur, and have been cited, of a con-
gruity between the sound and the sense as purely
accidental. The truth probably lies between these
two extremes. In the first place, that words denoting
sounds, or employed in describing them, may be Imi-
tative of those sounds, must be admitted by all; indeed
this kind of Imitation is, to a certain degree, almost
unavoidable, in our language at least, which abounds
perhaps more than any other, in these, as they may
be called, naturally expressive terms; such as “hiss,”
“ rattle,” “clatter,” “splash,” and many others.
In the next place, it is also allowed by most, that
quick or slow motion may, to a certain degree at least,
be imitated or represented by words ; many short
syllables (unincumbered by a clash either of vowels,
or of consonants coming together,) being pronounced
in the same time with a smaller number of long syl-
lables, abounding with these incumbrances, the former
seems to have a natural correspondence to a quick,
and the latter to a slow motion, since in the one a
greater, and in the other a less space, seem to be
passed over in the same time. In the ancient Poets,
their hexameter verses being always considered as of
the same length, i. e. in respect of the time taken to
pronounce them, whatever proportion of dactyls or
spondees they contained, this kind of Imitation of
quick or slow motion, is the more apparent ; and
after making all allowances for fancy, it seems im-
possible to doubt that in many instances it does exist;
as, e. g. in the often-cited line which expresses the
rolling of Sisyphus's stone down the hill:
* * *
A60ts étevta Teóověe kvXévôeto Atlas &vatóñs.
The following passage from the AEmeid can hardly
be denied to exhibit a correspondence with the slow
and quick motions at least, which it describes ; that of
the Trojans laboriously hewing the foundations of a
tower on the top of Priam's palace, and that of its
sudden and violent fall :
+ “ Aggressi ferrö circiim, quâ siimma labantes,
Júnctiºnäs tabulata (labat, divellimus altis
Sedibits, impilimusque, ## lapsii répentā ritinam
Cum sönitu d'édit, et Dâmâum sipër agºnimă late
Comcidit.”
* Pope has accordingly been justly censured for his incon-
sistency in making the Alexandrine represent both a quick and a
slow motion :
1. “ Flies o'er the unbending corn, and skims along the main.”
2. “Which, like a wounded snake, drags its slow length along.”
In the first instance, he forgot that an Alerandrine is long, from
containing more feet than a common verse ; whereas a long
hearameter has but the same number of feet as a short one, and
therefore being pronounced in the same time, seems to move more
rapidly.
+ The slow movement of this line would be much more per-
ceptible, if we pronounced (as doubtless the Latins did,) the
doubled consonants : “ ag-gres-si fer-ro sum-ama :” but in Eng-
lish, and consequently in the English way of reading Latin or
Greek, the doubling of a consonant only serves to fix the place
of the accent ; the latter of the two being never pronounced,
except in a very few compound words; as “innate,” “conna-
tural,” “poor-rate,” “hop-pole.”
280
R H E TO R. I. C.
appearance of strength to what is weak, it adds Chap. III:
weakness to what is strong; and if pleasing to those S-N-
Thetoric. But, lastly, it seems not to require any excessive
~~' exercise of fancy to perceive, if not, properly speaking,
an Imitation, by words, of other things besides sound
and motion, at least, an Analogical aptitude. That
there is at least an apparent Analogy between things
sensible, and things intelligible, is implied by number-
less Metaphors; as when we speak of “ rough, or
harsh, soft, or smooth manners,” “turbulent passions,”
the “stroke, or the storms of adversity,” &c. Now
if there are any words, or combinations of words,
which have in their sound a congruity with certain
sensible objects, there is no reason why they should
not have the same congruity with those emotions,
actions, &c. to which these sensible objects are ana-
logous. Especially, as it is universally allowed that
certain musical combinations are, respectively, appro-
priate to the expression of grief, anger, agitation, &c.
On the whole, the most probable conclusion seems
to be, that many at least of the celebrated passages
that are cited as Imitative in sound, were, on the
one hand, not the result of accident, nor yet, on the
other hand, of study ; but that the idea in the au-
thor's mind spontaneously suggested appropriate
sounds: thus, when Milton's mind was occupied with
the idea of the opening of the infernal gates, it seems
natural that his expression—
“And on their hinges grate harsh thunder,”
should have occurred to him without any distinct
intention of imitating sounds.
It will be the safest rule, therefore, for a prose
writer at least, never to make any distinct effort after
this kind of Energy of expression, but to trust to the
spontaneous occurrence of suitable sounds on every
occasion where the introduction of them is likely to
have a good effect.
It is hardly necessary to give any warning, gene-
rally, against the unnecessary introduction of Technical
language of any kind, when the meaning can be
adequately, or even tolerably, expressed in common,
i.e. unscientific words; the terms and phrases of Art
have an air of pedantic affectation, for which they do
not compensate, by even the smallest appearance of
increased Energy. But there is an apparent exception
to this rule, in the case of what may be called the
“Theological Style;” a peculiar phraseology, adopted
more or less by a large proportion of writers of
Sermons and other religious works; consisting partly
of peculiar terms, but chiefly of common words used
in a peculiar sense or combination, so as to form alto-
gether a kind of diction widely differing from the
classical standard of the language. This phraseology
having been formed partly from the Style of some
of the most eminent Divines, partly, and to a much
greater degree, from that of the Scriptures, i.e. of
our Version, has been supposed to carry with it an
air of appropriate dignity and sanctity, which greatly'
adds to the force of what is said. And this may, per-
haps, be the ease when what is said is of little or no
intrinsic weight, and is only such meagre common-
place as many religious works consist of ; the asso-
ciations which such language will excite in the minds
of those accustomed to it, supplying, in some degree,
the deficiencies of the matter. But this diction,
though it may serve as a veil for poverty of thought,
will be found to produce no less the effect of obscur-
ing the lustre of what is truly valuable: if it adds an
of narrow and ill cultivated mind, it is in a still higher
degree repulsive to persons of taste. It may be said,
indeed, with truth, that the improvement of the ma-
jority is a higher object than the gratification of a re-
fined taste in a few ; but it may be doubted whether any
real Energy, even with respect to any class of hearers,
is gained by the use of such a diction as that of which
we are speaking. For it will often be found that what
is received with great approbation, is yet, even if,
strictly speaking, understood, but very little attended
to or impressed upon the minds of the hearers. Terms
and phrases which have been long familiar to them,
and have certain vague and indistinct notions asso-
ciated with them, men often suppose themselves to
understand much more fully than they do ; and still
oftener give a sort of indolent assent to what is said,
without making any effort of thought. It is justly
observed by Mr. Foster, (Essay iv.) when treating on
this subject, that “ with regard to a considerable
proportion of Christian readers and hearers, a re-
formed language would be excessively strange to
th 2m ;” but that “ its being so strange to them, would
be a proof of the necessity of adopting it, at least, in
part, and by degrees. For the manner in which some
of them would receive this altered diction, would
prove that the customary phraseology had scarcely
given then any clear ideas. It would be found that
the peculiar phrases had been not so much the vehi-
cles of ideas, as the substitutes for them. These
readers and hearers have been accustomed to chime
to the Sound, without apprehending the sense ; inso-
much, that if they hear the very ideas which these
phrases signify, expressed ever so simply in other
language, they do not recognise, them.” He observes
also, with much truth, that the studied incorporation
and imitation of the language of the Scriptures in
the texture of any Discourse, neither indicates reve-
rence for the Divine composition, nor adds to the
dignity of that which is human ; but rather diminishes
that of such passages as might be introduced from the
sacred writings in pure and distinct quotation, standing
contrasted with the general Style of the work.
Of the Technical terms, as they may be called, of
Theology, there are many the place of which might
easily be supplied by corresponding expressions in com-
mon use; there are others, doubtless, which, denoting
ideas exclusively belonging to the subject, could not
be avoided without a tedious circumlocution ; these,
therefore, may be admitted as allowable peculiarities
of diction ; and the others, perhaps, need not be en-
tirely disused : but it is highly desirable that both
should be very frequently exchanged for words or
phrases entirely free from any Technical peculiarity,
even at the expense of some circumlocution. Not that
this should be done so constantly as to render the terms
in question obsolete; but by introducing frequently both
the term and a sentence explanatory of the same idea,
the evil just mentioned,—the habit of not thinking,
or not thinking attentively, on the meaning of what
is said, will be, in great measure, guarded against,-
the Technical words themselves will make a more
forcible impression,-and the danger of sliding into
unmeaning cant will be materially lessened. Such
repetitions, therefore, will more than compensate for,
or rather will be exempt from, any appearance of
R H E T O R. I C
281
*
Rhetoric tediousness, by the addition both of Perspicuity and
*—v- Energy.”
It may be asserted, with but too much
truth, that a very considerable proportion of Christians
have a habit of laying aside, in a great degree, their
common sense, and letting it, as it were, lie dormant,
when points of Religion come before them ;—as if
Reason were utterly at variance with Religion, and
the ordinary principles of sound judgment were to
be completely superseded on that subject; and accord-
ingly it will be found, that there are many errors
which are adopted, many truths which are over-
looked, or not clearly understood, and many difficulties
which stagger and perplex them, for want, properly
speaking, of the exercise of their common sense; i. e.
in cases precisely analogous to such as daily occur in
the ordinary affairs of life, in which those very same
persons would form a correct, clear, prompt, and de-
cisive judgment. It is well worthy of consideration,
how far the tendency to this habit might be diminished
by the use of a diction conformable to the suggestions
which have been here thrown out.
With respect to the Number of words employed,
“ it is certain,” as Dr. Campbell observes, “ that of
whatever kind the sentiment be, witty, humorous,
grave, animated, or sublime, the more briefly it is
expressed, the Energy is the greater.”—“As when
the rays of the sun are collected into the focus of a
burning-glass, the smaller the spot is which receives
them, compared with the surface of the glass, the
greater is the splendour, so, in exhibiting our senti-
ments by speech, the narrower the compass of words
is, wherein the thought is coinprised, the more
energetic is the expression. Accordingly, we find
that the very same sentiment expressed diffusely, will
be admitted barely to be just ;-expressed concisely,
will be admired as spirited.” He afterwards remarks,
that though a languid redundancy of words is in all
cases to be avoided, the energetic brevity which is
the most contrary to it, is not adapted alike to every
subject and occasion. “ The kinds of writing which
are less susceptible of this ornament, are, the De-
scriptive, the Pathetic, the Declamatory,t especially
* “It must indeed be acknowledged, that in many cases
innovations have been introduced, partly by the ceasing to em-
ploy the words designating those doctrines which were designed
to be set aside : but it is probable they may have been still more
frequently and successfully introduced under the advantage of
retaining the terms, while the principles were gradually sub-
verted. And therefore, since the peculiar words can be kept to one
invariable signification only by keeping that signification clearly
in sight, by means of something separate from these words them-
selves, it might be wise in Christian authors and speakers some-
times to express the ideas in common words, either in connexion
with the peculiar terms, or, occasionally, instead of them.
Common words might less frequently be applied, as affected de-
nominations of things, which have their own direct and common
denominations, and be less frequently combined into uncouth
phrases. Many peculiar and antique words might be exchanged
for other single words of equivalent signification, and in com-
mon use. And the small number of peculiar terms acknowledged
and established, as of permanent use and necessity, night, even
separately from the consideration of modifying the diction, be,
occasionally, with advantage to the explicit declaration and clear
comprehension of Christian truth, made to give place to a fuller
expression, in a number of common words, of those ideas of
which they are the single signs.” Foster, Essay iv. p. 304.
t This remark is made, and the principle of it (which Dr.
Campbell has omitted) subjoined, in chap. ii. Sec. 2, of this
Article, p. 262. -
WOLs, I.
the last. It is, besides, much more suitable in writing Chap. III.
than in speaking. A reader has the command of his -º/-
time ; he may read fast or slow, as he finds conve-
nient ; he can peruse a sentence a second time when
necessary, or lay down the book and think. But if,
in haranguing the people, you comprise a great deal in
few words, the hearer must have uncommon quick-
ness of apprehension to catch the meaning, before
you have put it out of his power, by engaging his
attention to something else.” The mode in which
this inconvenience should be obviated, and in which
the requisite expansion may be given to any thing
which the persons addressed cannot comprehend in a
very small compass, is, as we have already remarked,
not so much by increasing the number of words in
which the sentiment is conveyed in each sentence,
(though in this some variation must of course be ad-
mitted,) as by repeating it in various forms. The
uncultivated and the dull will, require greater expan-
Sion, and more copious illustration of the same
thought, than the educated and the acute ; but they
are even still more liable to be wearied or bewildered
by prolixity. If the material is too stubborn to be
speedily cleft, we must patiently continue our efforts
for a longer time, in order to accomplish it : but
this is to be done, not by making each blow fall
more slowly, which would only enfeeble them, but by
often-repeated blows.
It is needful to insist the more on the energetic
effect of Conciseness, because so many, especially
young writers and speakers, are apt to fall into a
style of pompous verbosity, not from negligence, but
from an idea that they are adding both Perspicuity
and Force to what is said, when they are only incum-
bering the sense with a needless load of words. And
they are the more likely to commit this mistake, be-
cause such a style will often appear not only to the
author, but to the vulgar (i. e. the vulgar in intellect,)
among his hearers, to be very majestic and impres-
sive. It is not uncommon to hear a speaker or writer
of this class, mentioned as having a “very fine com-
mand of language," when, perhaps, it might be said
with more correctness, that “ his language has a
command of him ;" i. e. that he follows a train of
words rather than of thought, and strings together
all the striking expressions that occur to him on the
subject, instead of first forming a clear notion of the
sense he wishes to convey, and then seeking for the
most appropriate vehicle in which to convey it. If,
indeed, any class of men are found to be the most
effectually convinced, persuaded, or instructed, by a
turgid anyplification, it is the Orator's business, true
to his object, not to criticise or seek to improve their
taste, but to accommodate himself to it. But it will
be found that this is not near so often the case as
many suppose. The Orator may often by this kind
of style gain great admiration, without being the
nearer to his proper end, which is to carry his point.
It will frequently happen that not only the ap-
probation, but the whole attention of the hearers
will have been confined to the Style, which will have
drawn their minds, not to the subject, but from it.
In those spurious kinds of Oratory, indeed, which have
been above mentioned, (p. 272,273,) in which the in-
culcation of the Subject-matter is not the principal
object proposed, a redundancy of words may often
be very suitable; but in all that comes within the
2 P
282
R H E TO R. I. C.
ference is, that in a proper Pleonasm, a complete cor- Chap. III.
rection is always made by razing. This will not S-N-y
Rhetoric. legitimate province of Rhetoric, there is no fault to
be more carefully avoided.*
It will therefore be advisable for a tiro in composi-
tion to look over what he has written, and to strike
out every word and clause which he finds will leave the
passage neither less perspicuous nor less forcible than
it was before ; “ quamvis invita recedant;" remem-
bering that, as has been aptly observed, “ nobody
knows what good things you keave out :" if the
general effect is improved, that advantage is enjoyed
by the reader unalloyed by the regret which the
author may feel at the omission of any thing which
he may think in, itself excellent. But this is not
enough ; he must study contraction, as well as omis
sion. There are many sentences which would not
bear the omission of a single word consistently with
perspicuity, which yet may be much more concisely
expressed, with equal clearness, by the employment of
different words, and by recasting a great part of the
expression. Take for example such a sentence as the
following : “A severe and tyrannical exercise of
power must become a matter of necessary policy with
Kings, when their subjects are imbued with such
principles as justify and authorize rebellion;” this
sentence could not be advantageously, nor to any
considerable degree, abridged, by the mere omission of
any of the words ; but it may be expressed in a much
shorter compass, with equal clearness and far greater
energy; thus, “ Kings will be tyrants from policy,
when subjects are rebels from principle.”f The
hints we have thrown out on this point coincide pretty
nearly with Dr. Campbell's remark on “ Perbosity,” as
contra-distinguished from “ Tautology,"f and from
** Pleonasm.” “ The third and last fault I shall
mention against vivid Conciseness is Verbosity. This
it may be thought coincides with the Pleonasm already
discussed. One difference however is this ; in the
Pleonasm there are words which add nothing to the
sense; in the Verbose manner, not only single words,
but whole clauses, may have a meaning, and yet it
were better to omit them, because what they mean is
uniuyportant. Instead, therefore, of enlivening the
expression, they make it languish. Another dif-
* “By a multiplicity of words, the sentiment is not set off
and accommodated, but like David, in Saul's armour, it is in-
cumbered and oppressed.
“Yet this is not the only, or perhaps the worst, consequence re-
sulting from this manner of treating Sacred writ,” [paraphrasing]
“we are told of the torpedo, that it has the wonderful quality
of numbing every thing it touches; a paraphrase is a torpedo.
By its influence the most vivid sentiments become lifeless,
the most sublime are flattened, the most fervid chilled, the
most vigorous enervated. In the very best compositions of this
kind that can be expected, the Gospel may be compared to a
rich wine of a high flavour, diluted in such a quantity of water
as renders it extremely vapid.” Campbell, Rhetoric, book iii.
ch. ii. Sec. 2.
+ Burke.
† Tautology, which he describes as “either a repetition of the
same sense in different words, or a representation of any thing
as the cause, condition, or consequence, of itself,” is, in most
instances, (of the latter kind at least,) accounted an offence
rather against correctness than brevity; the example he gives
from Bolingbroke, “how many are there by whom these tidings
of good news were never heard,” would usually be reckoned a
blunder rather than an instance of prolivity; like the expression
of “Sinecure places which have no duty annexed to them.” “The
Pleonasm,” he observes, “implies merely superfluity. Though
the words do not, as in the Tautology, repeat the sense, they
add nothing to it; e. g. They returned [back again] to the [same]
city [from] whence they came [forth.]” Book iii. ch. ii. sec, 2.
always answer in the Verbose style; it is often neces-
sary to alter as well as blot.”*
It is of course impossible to lay down precise rules
as to the degree of Conciseness which is, on each oc-
casion that may arise, allowable and desirable ; but
to an author who is, in his expression of any senti-
ment, wavering between the demands of Perspicuity
and of Energy, (of which the former of course re-
quires the first care, lest he should fail of both,) and
doubting whether the phrase which has the most
forcible brevity will be readily taken in, it may be re-
commended to use both expressions;–first to expand
the sense, sufficiently to be clearly understood, and
then to contract it into the most compendious and
striking form. This expedient might seem at first
sight the most decidedly adverse to the brevity re-
commended; but it will be found in practice that the
addition of a compressed and pithy expression of the
sentiment, which has been already stated at greater
length, will produce the effect of brevity. For it is
to be remembered that it is not on account of the
actual Number of words that diffuseness is to be con-
demned, (unless one were limited to a certain space,
or time,) but to avoid the flatness and tediousness
resulting from it; so that if this appearance can be
obviated by the insertion of such an abridged repe-
tition as is here recommended, which adds poignancy
and spirit to the whole, Conciseness will be, practically,
promoted by the addition. The hearers will be struck
by the forcibleness of the sentence which they will
have been prepared to comprehend ; they will under-
stand the longer expression, and remember the shorter.
But the force will, in general, be totally destroyed, or
much enfeebled, if the order be reversed ; –if the
brief expression be put first, and afterwards expanded
and explained ; for it loses much of its force if it be
not clearly understood the moment it is uttered ; and
if it be, there is no need of the subsequent expansion.
The sentence recently quoted from Burke, as an
instance of Energetic brevity, is in this manner
brought in at the close of a more expanded exhibition
of the sentiment, as a condensed conclusion of the
whole. ‘‘ Power, of some kind or other, will survive
the shock in which manners and opinions perish ;
and it will find other and worse means for its support.
The usurpation which, in order to subvert ancient
institutions, has destroyed ancient principles,\ will
hold power by arts similar to those by which it has
acquired it. When the old feudal and chivalrous
spirit of fealty, which, by freeing kings from fear,
freed both kings and subjects from the precaution
of tyranny, shall be extinct in the minds of men,
plots and assassinations will be anticipated by pre-
ventive murder and preventive confiscation, and that
long roll of grim and bloody maxims, which form
the political code of all Power, not standing on its
own honour, and the honour of those who are to
obey it. Kings will be tyrants from policy when
subjects are rebels from principle.” Burke, Reflections
on the Revolution in France, Works, vol. v. p. 153.
The same writer, in another passage of the same
work, has a paragraph in like manner closed and sum-
med up by a striking metaphor, (which will often
-y
* Campbell, Rhetoric, book iii. ch. ii. Sec. 2. part iii.
R H E To R I c. 283
ourselves; and the act of composition fills and de- Chap. III.
Rhetoric, prove the most concise, as well as in other respects,
- lights the mind with change of language and succes- >TVT
\-V-2 striking, form of expression,) such as would not have
been so readily taken in if placed at the beginning.
“To avoid therefore the evils of inconstancy and
versatility, ten thousand times worse than those of
obstinacy and the blindest prejudice, we have conse-
crated the State, that no man should approach to look
into its defects or corruptions but with due caution;
that he should never dream of beginning its reforma-
tion by its subversion ; that he should approach to
the faults of the State as to the wounds of a fathér,
with pious awe and trembling solicitude. By this
wise prejudice we are taught to look with horror on
those children of their country who are prompt rashly
to hack that aged parent in pieces, and put him into
the kettle of magicians, in hopes that by their poison-
ous weeds, and wild incantations, they may regenerate
the paternal constitution, and renovate their father's
life.”* Burke, Reflections on the Revolution in France,
Works, vol. v. p. 183. -
So great, indeed, is the effect of a skilful inter-
spersion of short, pointed, forcible sentences, that even
a considerable violation of some of the foregoing rules
may be by this means, in a great degree, concealed ;
and vigour may thus be communicated (if vigour of
thought be not wanting) to a Style chargeable even
with Tautology. This is the case with much of the
language of Dr. Johnson, who is certainly, on the
whole, an Energetic writer, though he would have
been much more so, had not an over attention to the
roundness and majestic sound of his sentences, and
a delight in balancing one clause against another, led
him so frequently into a faulty redundancy. Take,
as an instance, a passage in his life of Prior, which
may be considered as a favourable specimen of
his style : “Solomon is the work to which he in-
trusted the protection of his name, and which he
expected succeeding ages to regard with veneration.
His affection was natural ; it had undoubtedly been
written with great labour; and who is willing to
think that he has been labouring in vain He had
infused into it much knowledge, and much thought ;
had often polished it to elegance, often dignified it with
splendour, and sometimes heightened it to sublimity ;
he perceived in it many excellences, and did not dis-
cover that it wanted that without which all others are
of small avail, the power of engaging attention and
alluring curiosity. Tediousness is the most fatal of all
faults; negligences or errors are single and local ;
but tediousness pervades the whole; other faults are
censured and forgotten, but the power of tediousness
propagates itself. He that is weary the first hour, is
more weary the second ; as bodies forced into motion
contrary to their tendency, pass more and more slowly
through every successive interval of space. Unhap-
pily this pernicious failure is that which an author is
least able to discover. We are seldom tiresome to
* This, however, being an instance of what may be called the
classical Metaphor, no preparation or explanation, even though
sufficient to make it intelligib, e, could render it very striking to
those not thoroughly and early familiar with the ancient fables
of Medea.
The Preacher has a considerable resource, of an analogous
kind, in similar allusions to the history, description, parables, &c.
of Scripture, which will often furnish useful illustrations and
forcible metaphors, in an address to those well acquainted with the
Bible; though these would be frequently unintelligible, and al-
ways comparatively feeble, to persons not familiar with Scripture.
f
sion of images; every couplet when produced is new,
and novelty is the great source of pleasure. Perhaps
no man ever thought a line superfluous when he first
wrote it, or contracted his work till his ebullitions of
invention had subsided.” It would not have been
just to the author, nor even so suitable to the present
purpose, to cite less than the whole of this passage,
which exhibits the characteristic merits, even more
strikingly than the defects, of the writer. Few could
be found in the works of Johnson, and still fewer in
those of any other writer, more happily and forcibly
expressed ; yet it can hardly be denied that the parts
here distinguished by italics are chargeable, more or
less, with Tautology. -
It happens, unfortunately, that Johnson's Style is
particularly easy of imitation, even by writers utterly
destitute of his vigour of thought ; and such imitators
are intolerable. They bear the same resemblance to
their model, that the armour of the Chinese, as de-
scribed by travellers, consisting of thick quilted cotton
covered with stiff glazed paper, does to that of the
ancient knights; equally glittering, bulky, and cumber-
some, but destitute of the temper and firmness which
was its sole advantage. At first sight, indeed, this
kind of Style appears far from easy of attainment ;
on account of its being remote from the colloquial,
and having an elaborately artificial appearance ; but
in reality, there is none less difficult to acquire. To
string together substantives, connected by conjunctions,
which is the characteristic of Johnson's Style, is, in
fact, the rudest and clumsiest mode of expressing our
thoughts : we have only to find names for our ideas,
and then put them together by connectives, instead
of interweaving, or rather felting them together, by a
due admixture of verbs, participles, prepositions, &c.
So that this way of writing, as contrasted with the
other, may be likened to the primitive rude carpentry,
in which the materials were united by coarse external
implements, pins, nails, and cramps, when compared
with that art in its most improved state, after the
invention of dovetail joints, grooves, and mortices,
when the junctions are effected by forming properly
the extremities of the pieces to be joined, so as at
once to consolidate and conceal the juncture.
If any one will be at the pains to compare a few
pages, taken from almost any part of Johnson's works,
with the same quantity from any other of our admired
writers, noting down the number of substantives in
each, he will be struck with the disproportion. This
would be still greater, if he were to examine with the
same view an equal portion of Cicero ; but it must
be acknowledged that the genius of the Latin lan-
guage allows and requires a much smaller proportion
of substantives than are necessary in our own.
In aiming at a Concise Style, however, care must
of course be taken that it be not crowded; the frequent
recurrence of considerable ellipses, even when ob-
scurity does not result from them, will produce an
appearance of affected and laborious compression,
which is offensive. The author who is studious of
Energetic brevity, should aim at what may be called
a Suggestive Style; such, that is, as, without making a
distinct, though brief, mention of a multitude of par-
ticulars, shall put the hearer's mind into the same
train of thought as the speaker's, and suggest to him
* Aº
284 R H E T O R. I. C.
be the better furnished for being stored with ten times Chap. III.
Rhetoric. more than is actually expressed.* Aristotle's Style,
as many of some kinds of articles as were needed, S-N-
\-y-Z which is frequently so elliptical as to be dry and ob-
scure, is yet often, at the very same time, unneces-
sarily diffuse, from his enumerating much that the
reader would easily have supplied, if the rest had been
fully and forcibly stated. He seems to have regarded
his readers as capable of going along with him readily,
in the deepest discussions, but not, of going beyond
him, in the most simple; i. e. of filling up his mean-
ing, and inferring what he does not actually express;
so that in many passages a free translator might con-
vey his sense in a shorter compass, and yet in a less
cramped and elliptical diction. A particular statement,
of which the general application is obvious, will often
save a long abstract rule, which needs much expla-
nation and limitation ; and will thus suggest much
that is not actually said ; thus answering the purpose
of a mathematical diagram, which though itself an
individual, serves as a representative of a class.
Slight hints also respecting the subordinate branches
of any subject, and notices of the principles that will
apply to them, &c. may often be substituted for
digressive discussions, which, though laboriously
compressed, would yet occupy a much greater space.
Judicious divisions likewise and classifications, save
much tedious enumeration ; and, as has been formerly
remarked, a well-chosen epithet may often suggest,
and therefore supply the place of, an entire argument.
It would not be possible, within a moderate compass,
to lay down precise rules for the Suggestive kind of
writing we are speaking of ; but if the slight hints
here given are sufficient to convey an idea of the
object to be aimed at, practice will enable a writer
gradually to form the habit recommended. It may
be worth while, however, to add, that those accus-
tomed to rational conversation, will find in that a very
useful exercise, with a view to this point, (as well as
to almost every other connected with Rhetoric ;) since,
in conversation, a man naturally tries first one and then
another mode of expressing his thoughts, and stops as
soon as he perceives that his companion fully compre-
hends his sentiments, and is sufficiently impressed with
them.
We have dwelt the more earnestly on the head of
Conciseness, because it is a quality in which young
writers (who are the most likely to seek for practical
benefit in a Treatise of this kind,) are usually most
deficient; and because it is commonly said that, in
them, exuberance is a promising sign ; without suf-
ficient care being taken to qualify this remark, by
adding, that this over-luxuriance must be checked by
judicious pruning. If an early proneness to redun-
dancy be an indication of natural genius, those who
possess this genius should be the more sedulously on
their guard against it; and those who do not, should
be admonished that the want of a natural gift cannot
be supplied by copying its attendant defects. The
praises which have been bestowed on Copiousness of
diction, have probably tended to mislead authors into
a cumbrous verbosity. It should be remembered,
that there is no real Copiousness in a multitude of
synonymes and circumlocutions. A house would not
* Such a Style may be compared to a good map, which marks
distinctly the great outlines, setting down the principal rivers,
towns, mountains, &c. and leaving the imagination to supply the
villages, hillocks, and streamlets; which if they were all in-
serted in their due proportions would crowd the map, though,
after all, they could not be discerned without a microscope,
while it was perhaps destitute of those required for
other purposes; nor was Lucullus's wardrobe which,
according to Horace, boasted five thousand mantles,
necessarily well stocked, if other articles of dress were
wanting. The completeness of a library does not
consist in the number of volumes, especially if many
of them are duplicates; but in its containing copies of
all the most valuable works. And in like manner,
true Copiousness of language consists in having at
command, as far as possible, a suitable expression for
each different modification of thought. This, conse-
quently, will often save much circumlocution ; so that
the greater our command of language, the more
concisely we shall be enabled to write. In an author
who is attentive to these principles, diffuseness may
be accounted no dangerous fault of Style, because
practice will gradually correct it : but it is otherwise
with one who pleases himself in stringing together
well-sounding words into an easy, flowing, and
(falsely-called) Copious Style, destitute of nerve ;
and who is satisfied with a small portion of matter ;
seeking to increase, as it were, the appearance of his
wealth by hammering out his metal thin. This is far
from a curable fault. When the Style is fully formed
in other respects, pregnant fulness of meaning is
seldom superadded ; but when there is a basis of
Energetic condensation of thought, the faults of harsh-
ness, baldness, or even obscurity, are much more
likely to be remedied. Solid gold may be new-
moulded and polished ; but what can give solidity to
gilding 2 -
Lastly, the Arrangement of words may be made
highly conducive to Energy. The importance of an
attention to this point, with a view to Perspicuity,
has been already noticed : but of two sentences equally
perspicuous, and consisting of the very same words,
the one may be a feeble and languid, the other a
striking and Energetic expression, merely from the
difference of Arrangement.
Some, among the moderns, are accustomed to speak
of the Natural order of the words in a sentence, and to
consider, each, the established Arrangement of his own
language as the nearest to such a natural order; re-
garding that which prevails in Latin and in Greek as
a sort of deranged and irregular structure. We are
apt to consider that as most natural and intrinsically
proper, which is the most familiar to ourselves ; but
there seems no good ground for asserting, that the
customary structure of sentences in the ancient lan-
guages is less natural, or less suitable for the purposes
for which language is employed, than in the modern.
Supposing the established order in English or in
French, for instance, to be more closely conformed to
the grammatical or logical analysis of a sentence,
than that of Latin or Greek, because we place the
Subject first, the Copula next, and the Predicate last,
&c. it does not follow that such an Arrangement is
necessarily the best fitted in every case to excite the at-
tention,--to direct it to the most essential points,
to gratify the imagination,--or to affect the feelings :
it is, Surely, the natural object of language to express
as strongly as possible the speaker's sentiments, and to
convey the same to the hearers ; and that Arrangement
of words may fairly be accounted the most natural by
R H E T O R. I. C. 285
Rhetoric, which all men are naturally led, as far as the rules of
S-V-' their respective languages allow them, to accomplish
are really needed ; which is no less absurd than to Chap. III.
attempt remedying the intricacies of a road by re- -2-
this object. The rules of many of the modern lan-
guages do indeed frequently confine an author to an
order which he would otherwise never have chosen ;
but what translator of any taste would ever volun-
tarily alter the Arrangement of the words in such a
sentence, as MeyāAn "Apreputs ‘Eq}eadwu, which our
language allows us to render exactly, “Great is Diana
of the Ephesians !” How feeble in comparison is
the translation of Le Clerc, “ La Diane des Ephesiens
est une grande Déesse !” How imperfect that of Beau-
sobre, “La grande Diane des Ephesiens !” How un-
dignified that of Saci, “ Vive la grande Diane des
Ephesiens !”
Our language indeed is, though to a less degree,
very much hampered by the same restrictions; it be-
ing in general necessary, for the expression of the sense,
to adhere to an order which may not be in other re-
spects the most eligible : “Cicero praised Caesar,”
and “Caesar praised Cicero,” would be two very dif-
ferent propositions ; the situation of the words being
all that indicates, (from our want of Cases,) which is
to be taken as the nominative, and which as the accu-
sative ; but such a restriction is far from being an
advantage. The transposition of words which the
ancient languages admit of, conduces, not merely to
variety, but to Energy, and even to Precision. If, for
instance, a Roman had been directing the attention of
his hearers to the circumstance that Caesar had been
the object of Cicero's praise, he would, most likely,
have put “Caesarem " first ; but he would have put
“ Cicero " first, if he had been remarking that not
only others, but even he, had praised Caesar.
It is for want of this liberty of Arrangement that
we are often compelled to mark the emphatic words of
our sentences by the voice, in speaking, and by italies,
in writing ; which would, in Greek or in Latin, be
plainly indicated, in most instances, by the collocation
alone. The sentence which has been often brought
forward as an example of the varieties of expression
which may be given to the same words, “Will you
ride to London to-morrow 7" and which may be pro-
nounced and understood in, at least, five different
ways, according as the first, second, &c. of the words
is printed in italics, would be, by a Latin or Greek
writer, arranged in as many different orders, to answer
these several intentions. The advantage thus gained
must be evident to any one who considers how im-
portant the object is which is thus accomplished, and
for the sake of which we are often compelled to resort
to such clumsy expedients ; it is like the proper dis-
tribution of the lights in a picture; which is hardly of
less consequence than the correct and lively represen-
tation of the objects. -
It must be the aim then of an author, who would
write with Energy, to avail himself of all the liberty
which our language does allow, so to arrange his
words that there shall be the least possible occasion
for under-scoring and italics; and this, of course, must
be more carefully attended to by the writer than by
the speaker, who may, by his mode of utterance, con-
ceal, in great measure, a defect in this point. It
may be worth observing, however, that some writers,
having been taught that it is a fault of Style to require
many of the words to be in italics, fancy they avoid
the fault, by olnitting those indications where they
moving the direction-posts.” The proper remedy is,
to endeavour so to construct the Style, that the col-
location of the words may, as far as is possible, direct
the attention to those which are emphatic. And the
general maxim that should chiefly guide us, is, as
Dr. Campbell observes, the homely saying, “Nearest
the heart, nearest the mouth ;” the idea, which is the
most forcibly impressed on the author's mind, will
naturally claim the first utterance, as nearly as the
rules of the language will permit. And it will be
found that, in a majority of instances, the most Em-
phatic word will be the Predicate ; contrary to the
rule which the nature of our language compels us, in
most instances, to observe. It will often happen,
however, that we do place the Predicate first, and
obtain a great increase of -Energy by this Arrange-
ment. Of this licence our translators of the Bible
have, in many instances, very happily availed them-
selves ; as, e. g. in the sentence lately cited, “Great
is Diana of the Ephesians;” so also, “Blessed is
he that cometh in the name of the Lord:” it is
evident how much this would be enfeebled by alter-
ing the Arrangement into “He that cometh in the
name of the Lord is blessed.” And, again, “To Him
give all the prophets witness :” here, indeed, it may
be said that that is properly the Subject which comes
first ; since that of which we are speaking is He, of
whom we assert, that all the prophets bear Him wit-
ness; but still, the placing of the oblique case first,
is a departure from the most common, and, what
many call, the Grammatical order of our language.
And, again, “Silver and Gold have I none; but what
I have, that give I unto thee."f Another passage, in
which they might advantageously have adhered to
the order of the original, is, “"Ezregev, & Teae Babu)\tºv,
# penſixm,” + which would certainly have been ren-
dered as correctly, and more forcibly, as well as
more closely, “Fallen, fallen is Babylon, that great
city,” than, “Babylon is fallen, is fallen.”
The word “IT" is frequently very serviceable in
enabling us to alter the Arrangement : thus, the sen-
tence, “ Cicero praised Caesar,” which admits of at
least two modifications of sense, may be altered so as
to express either of them, by thus varying the order:
“ It was Cicero that praised Caesar,” or, “It was:
Caesar that Cicero praised.” “ IT '' is, in this mode
of using it, the representative of the Subject, § which
it thus enables us to place, if we will, after the Pre-
dicate.
With respect to Periods, it would be neither prac-
* The censure of frequent and long Parentheses also leads
some writers into the like preposterous expedient of leaving out
the marks ( ) by which they are indicated, and substituting
commas; instead of so framing each sentence that tiley shall not
be needed. . It is no cure to a lame man, to take away his
crutches.
f 24cts, ch. v. ver. 6.
i Rev. ch. xviii. ver. 2.
§ Of whatever gender or number the subject referred to may
be, “IT’’ may, with equal propriety, be employed to represent
it. Our translators of the Bible have not scrupled to make
“IT’ refer to a masculine noun : “It is I, be not afraid;” but
they secm to have thought it not allowable, as perhaps it was
not, at the time when they wrote, to make such a reference to a
plural noun. “Search the Scriptures they are they which
testify of Me :”, we should now say, without any impropriety.
“IZ is they, &c.”
286
R H E T O R. I. C.
Rhetoric tically useful, nor even suitable to the present object,
S-N-2 to enter into an examination of the different senses in
which various authors have employed, the word. A
technical term may allowably be employed, in a
scientific work, in any sense not very remote from
common usage (especially when common usage is
not uniform, and invariable, in the meaning affixed to
it,) provided it be clearly defined, and the definition
strictly adhered to. By a Period, then, is to be un-
derstood in this place, any sentence, whether simple
or complex, which is so framed that the Grammatical
construction will not admit of a close, before the end
of it ; in which, in short, the meaning remains sus-
pended, as it were, till the whole is finished. A loose
sentence, on the contrary, is, any that is not a Period;—
any, whose construction will allow of a stop, so as to
form a perfect sentence, at one or more places, before
we arrive at the end. E. g. “We came to our jour-
ney's end—at last—with no small difficulty—after
much fatigue—through deep roads—and bad weather.”
This is an instance of a very loose sentence ; (for it is
evident that this kind of structure admits of degrees,)
there being no less than five places, marked by
dashes, at any one of which the sentence might have
terminated, so as to be grammatically perfect. The
same words may be formed into a Period, thus :
“ At last, after much fatigue, through deep roads,
and bad weather, we came, with no small difficulty,
to our journey's end.” Here, no stop can be made at
any part, so that the preceding words shall form a
sentence before the final close. These are both of
them simple sentences ; i. e. not consisting of several
clauses, but having only a single verb ; so that it is
plain we ought not, according to this view, to confine
the name of Period to complea, sentences ; as Dr.
Campbell has done, notwithstanding his having adopted
the same definition as has been here laid down.
Periods, or sentences nearly approaching to Periods,
have certainly, when other things are equal, the ad-
vantage in point of Energy. An unexpected conti-
nuation of a sentence which the reader had supposed
to be concluded, especially if in reading aloud, he
had, under that supposition, dropped his voice, is apt
to produce a sensation in the mind of being disagree-
ably balked ; analogous to the unpleasant jar which
is felt, when in ascending or descending stairs, we
meet with a step more than we expected : and if this
be often repeated, as in a very loose sentence, a kind
of weary impatience results from the uncertainty
when the sentence' is to close. This, however, must
have been much more the case in the ancient lan-
guages, than in the modern ; because the variety of
Arrangement which they permitted, and, in particular,
the liberty of reserving the verb, on which the whole
sense depends, to the end, made that structure natural
and easy, in many instances in which, in our language,
it would appear forced, unnatural, and affected. But
the agreeableness of a certain degree, at least, of
Periodic structure, in all languages, is apparent from
this ; that they all contain words which may be said
to have no other use or signification but to suspend
the sense, and lead the hearer of the first part of the
sentence to expect the remainder. He who says,
** the world is not eternal, nor the work of chance,”
expresses the same sense as if he said, “The world is
neither eternal, nor the work of chance;" yet the
latter would be generally preferred. So also, “ The
vines afforded both a refreshing shade, and a delicious Chap. III.
fruit;” the word “ both,” would be missed, though S-N-"
it adds nothing to the sense. Again, “While all the
Pagan nations consider Religion as one part of Virtue,
the Jews, on the contrary, regard Virtue as a part of
Religion;”* the omission of the first word would
not alter the sense, but would destroy the Period; to
produce which is its only use. The MEN, AE, and TE
of the Greek are, in many places, subservient to this
use alone. -
The modern languages do not indeed admit, as was
observed above, of so Periodic a Style as the ancient
do : but an author, who does but clearly understand
what a Period is, and who applies the test we have laid
down, will find it very easy, after a little practice, to
compose in Periods, even to a greater degree than, in
an English writer, good taste will warrant. His
skill and care will be chiefly called for in avoiding
all appearance of stiffness and affectation in the con-
struction of them,-in not departing, for the sake of
a l’eriod, too far from colloquial usage, and in ob-
serving such moderation in the employment of this
Style, as shall prevent any betrayal of artifice,—any
thing savouring of elaborate stateliness, which is
always to be regarded as a worse fault than the
Slovenliness and languor which accompany a very
loose Style.
It should be observed, however, that, as a sentence
which is not strictly a Period, according to the fore-
going definition, may yet approach indefinitely near
to it, so as to produce nearly the same effect, so on
the other hand, Periods may be so constructed as to
produce much of the same feeling of weariness and
impatience which results from an excess of loose
sentences. If the clauses be very long, and contain
an enumeration of many circumstances, though the
sentence be so framed, that we are still kept in ex-
pectation of the conclusion, yet it will be an impatient
expectation ; and the reader will feel the same kind of
uneasy uncertainty when the clause is to be finished,
as would be felt respecting the sentence, if it were
loose. And this will especially be the case, if the rule
formerly given with a view to Perspicuity be not
observed,f of taking care that each part of the sen-
tence be understood, as it proceeds. Each clause, if
it consist of several parts, should be continued with
the same attention to their mutual connection, so as
to suspend the sense, as is employed in the whole
sentence ; that it may be, as it were a Periodic clause;
and if one clause be long and another short, the shorter
should, if possible, be put last. Universally indeed a
sentence will often be, practically, too long, i.e. will have
a tedious, dragging effect, merely from its concluding
with a much longer clause than it began with ; so
that a composition which most would censure as
abounding too much in long sentences, may often
have its defect, in great measure, remedied without
shortening any of them ; merely by reversing the
order of each. This of course holds good with respect
to all complex sentences of any considerable length,
whether Periods or not. An instance of the difference
of effect produced by this means, may be seen in such
a sentence as the following : “The State was made,
under the pretence of serving it, in reality, the prize
of their contention, to each of those opposite parties,
* Josephus. + P. 272.
R. H. E. T. O. R. I. C.
287
ducive to the improvement of Style, to practise casting Chap. III.
a sentence into a variety of different forms. ~~~
Rhetoric, who professed in specious terms, the one, a prefer-
S-' ence for moderate Aristocracy, the other, a desire of
admitting the people at large, to an equality of
civil privileges.” This may be regarded as a complete
Period ; and yet, for the reason just mentioned, has
a tedious and cumbrous effect. Many critics might
recommend, and perhaps with reason, to break it into
two or three ; but it is to our present purpose to
remark that it might be, in some degree at least,
decidedly improved, by merely reversing the clauses ;
as thus : “The two opposite parties, who professed
in specious terms, the one, a preference for moderate
Aristocracy, the other, a desire of admitting the people
at large to an equality of civil privileges, made the
State, which they pretended to serve, in reality the
prize of their contention.”* Another instance may
be cited from a work, in which any occasional awk-
wardness of expression is the more conspicuous, on
account of its general excellence, the Church Liturgy;
the style of which is so justly admired for its remark-
able union of energy with simplicity, smoothness, and
elegance : the following passage from the Exhortation
is one of the very few, which, from the fault just
noticed, it is difficult for a good reader to deliver with
spirit : “And although we ought at all times humbly
to acknowledge our sins before God, yet ought we
most chiefly so to do, when we assemble and meet
together—to render thanks for the great benefits that
we have received at his hands, – to set forth his most
worthy praise, to hear his most holy word, and to ask
those things which are requisite and necessary,+as
well for the body as the soul.” This is evidently a very
loose sentence, as it might be supposed to conclude
at any one of the three places which are marked by
dashes (—); this disadvantage, however, may easily be
obviated by the suspension of voice, by which a good
reader, acquainted with the passage, would indicate that
the sentence was not concluded ; but the great fault
is the length of the last of the three principal clauses,
in comparison of the former two ; (the conclusions
of which we have marked ||) by which a dragging
and heavy effect is produced, and the sentence is made
to appear longer than it really is. This would be more
manifest to any one not familiar, as most are, with
the passage ; but a good reader of the Liturgy will
find hardly any sentence in it so difficult to deliver to
his own satisfaction. It is perhaps the more profitable
to notice a blemish occurring in a composition so
well known, and so deservedly valued for the excel-
lence, not only of its sentiments, but of its language.
It is a useful admonition to, young writers, with a
view to what has lately been said, that they should
always attempt to recast a sentence which does not
please; altering the Arrangement and entire construc-
tion of it, instead of merely seeking to change one
word for another. This will give a great advantage
in point of Copiousness also : for there may be, suppose,
a substantive, which, either because it does not fully
express our meaning, or for some other reason, we
wish to remove, but can find no other to supply its
place ; but the object Inay perhaps be easily accom-
plished by means of a verb, adverb, or some other
part of speech, the substitution of which implies an
alteration of the construction. It is an exercise ac-
cordingly which may be recommended as highly con-
* Thucydides, on the Corcyrean sedition.
It is evident, from what has been said, that in
compositions intended to be delivered, the Periodic
Style is much less necessary, and therefore much
less suitable, than in those designed for the closet.
The speaker may, in most instances, by the skilful
suspension of his voice, give to a loose sentence the
effect of a Period : and though, in both species of
composition the display of art is to be guarded against,
a more unstudied air is looked for in such as are
spoken. º
The study of the best Greek and Latin writers may
be of great advantage towards the improvement of the
Style in the point concerning which we have now
been treating, (for the reason lately mentioned,) as
well as in most others : and there is this additional
advantage, (which, at first sight, might appear a dis-
advantage,) that the Style of a foreign writer cannot
be so closely imitated as that of one in our own lan-
guage : for this reason there will be the less danger of
falling into an obvious and servile imitation. Boling-
broke may be noted as one of the most Periodic of
English writers; Swift and Addison, (though in other.
respects very different,) are among the most loose.
Antithesis has been sometimes reckoned as one
form of the Period ; but it is evident that, accord-
ing to the view here taken, it has no necessary con-
nection with it. One clause may be opposed to another,
by means of some contrast between corresponding
words in each, whether or not the clauses be so con-
nected that the former could not, by itself, be a corn-
plete sentence. Tacitus, who is one of the most
Antithetical, is at the same time one of the least
Periodic, of all the Latin writers.
There can be no doubt that this figure is calculated
to add greatly to Energy. Every thing is rendered
more striking by contrast; and almost every kind of
subject-matter affords materials for contrasted expres-
sions. Truth is opposed to error ; wise conduct to
foolish; different causes often produce opposite effects;
different circumstances dictate to prudence opposite
conduct; opposite impressions may be made by the
same object, on different minds ; and every extreme
is opposed both to the Mean, and to the other ex-
treme. If, therefore, the language be so constructed
as to contrast together these opposites, they throw
light on each other by a kind of mutual reflexion,
and the view thus presented will be the more striking.
By this means also we may obtain, consistently with
Perspicuity, a much greater degree of Conciseness;
which in itself is so conducive to Energy; e. g.
“When Reason is against a man, he will be against
Reason ;”* it would be hardly possible to express
this sentiment, rot Antithetically, so as to be clearly
intelligible, except in a much longer sentence. Again,
“Words are the Counters of wise men, and the
Money of fools;” here we have an instance of the
combined effect of Antithesis and Metaphor in pro-
ducing increased Energy, both directly, and at the
same time, (by the Conciseness resulting from them,)
indirectly; and accordingly, in such pointed and pithy
expressions, we obtain the gratification which, as
Aristotle remarks, results from “the act of learning
quickly and easily.” It is a remark of the same au-
* Hobbes,
288
R H E T O R. I. C.
Rhetoric, thor, that, in Antithesis, either “ contraries are joined
S-N-" to contraries,” in the two clauses respectively, or
been expressed simply, are expanded into complex Chap. III.
ones, by the addition of clauses, which add little or S-N-
“ the same thing is joined to contraries;” of this last,
the former of the two examples, just cited, is an
insiance ;-the condemnation pronounced on any
one's principles or conduct, by Reason, being con-
trasted with his dislike and defiance of it : the other
example is an instance of the former kind ; “ Coun-
ters ” being opposed to “ Money,” and “wise men"
to “fools.” Of the same nature is the Antithetical
expression, “ Party is the madness of many, for the
gain of a few ;” which affords, likewise, an instance
of this construction in a sentence which does not
contain two distinct clauses. Frequently the same
words, placed in different relations with each other,
will stand in contrast to themselves; as in the expres-
sion, “A fool with judges; among fools, a judge;”*
and in that given by Quinctilian, “ non utedam vivo,
sed ut vivam edo ;” “ I do not live to eat, but eat to
live;” both of these are instances also of perfect
Antithesis, without Period ; for each of these sentences
might, grammatically, be concluded in the middle.
Of the same kind is an expression in a Speech of
Mr. Wyndham's, “Some contend that I disapprove
of this plan, because it is not my own ; it would be
more correct to say, that it is not my own, because
I disapprove it.”
The use of Antithesis has been censured by some,
as if it were a paltry and affected decoration, unsuit-
able to a chaste, natural, and masculine Style. Pope,
accordingly, himself one of the most Antithetical of
our writers, speaks of it in the Dunciad with con-
tempt :
“ I see a Chief who leads my chosen sons,
All arm'd with Points, Antitheses, and Puns.”
The excess, indeed, of this Style, by betraying arti-
fice, effectually destroys Energy ; and draws off the
attention, even of those who are pleased with effemi-
nate glitter, from the matter to the Style. But, as
Dr. Campbell observes, “ the excess itself into which
some writers have fallen, is an evidence of its value—
of the lustre and emphasis which Antithesis is calcu-
lated to give to the expression. There is no risk of
intemperance in using a liquor which has neither spirit
nor flavour.”
It is, of course, impossible to lay down precise rules
for determining, what will amount to excess, in the use
of this, or of any other figure : the great safeguard
will be the formation of a pure taste, by the study of
the most chaste writers, and unsparing self-correction.
But one rule always to be observed in respect to the
antithetical construction, is to remember that in a
true Antithesis the opposition is always in the ideas
expressed. Some writers abound with a kind of mock-
antithesis, in which the same, , or nearly the same
sentiment which is expressed by the first clause, is
repeated in a second ; or at least, in which there is
but little of real contrast between the clauses which
are expressed in a contrasted form. This kind of
style not only produces disgust instead of pleasure,
when once the artifice is detected, which it soon must
be, but also, instead of the brevity and vigour result-
ing from true Antithesis, laiours under the fault of
prolixity and heaviness. Sentences which might have
* Cowper.
nothing to the sense ; and which have been compared
to the false handles and keyholes with which furniture
is decorated, that serve no other purpose than to
correspond to the real ones. Much of Dr. Johnson's
writing is chargeable with this fault.
Bacon, in his Rhetoric, furnishes, in his common-
places, (i.e. heads of Arguments, pro and contra, on a
variety of subjects,) some admirable specimens of
compressed and striking Antitheses; many of which
are worthy of being enrolled among the most ap-
proved proverbs : e. g. “He who dreads new reme-
dies, must abide old evils.” “ Since things alter for
the worse spontaneously, if they be not altered for the
better designedly, what end will there be of the evil?”
“The humblest of the virtues the vulgar praise, the
middle ones they admire, of the highest, they have
no perception.” &c.
It will not unfrequently happen that an Antithesis
may be even more happily expressed by the sacrifice
of the Period, if the clauses are by this means made
of a more convenient length, and a resting-place
provided at the most suitable point : e. g. “The per-
secutions undergone by the Apostles, furnished both
a trial to their faith, and a confirmation to our's :-
a trial to them, because if human honours and rewards
had attended them, they could not, even themselves,
have been certain that these were not their object;
and a confirmation to us, because they would not
have encountered such sufferings in the cause of im-
posture.” If this sentence were not broken as it is,
but colmpacted into a Period, it would have more
heaviness of effect, though it would be rather shorter:
e.g. “ The persecutions undergone by the Apostles,
furnished both a trial of their faith, since if human
honours, &c. &c. and also a confirmation of ours, be-
cause,” &c. Universally, indeed, a complex sentence,
whether Antithetical or not, will often have a degree
of spirit and liveliness from the latter clause heing
made to turn back, as it were, upon the former, by
containing, or referring to, some word that had there
been mentioned : e. g. “ The introducers of the now-
established principles of political economy may fairly
be considered to have made a great discovery ; a dis-
covery the more creditable, from the circumstance
that the facts on which it was founded had long been
well known to all.” This kind of Style also may, as
well as the Antithetical, prove offensive if carried to
such an excess as to produce an appearance of af-
fectation or mannerism.
Lastly, to the Speaker especially, the occasional
employment of the Interrogative form, will often prove
serviceable with a view to Energy. It calls the hearer's
attention more forcibly to some important point, by
a personal appeal to each, either to assent to what is
urged, or to frame a reasonable objection ; and it
often carries with it an air of triumphant defiance of
an opponent to refute the argument if he, can.
Either the Premiss or the Conclusion, or both, of any
argument, may be stated in this form ; but it is evi-
dent that if it be introduced too frequently, it will
necessarily fail of the object of directing a particular
attention to the most important points. To attempt
to make every thing emphatic, is to make nothing
emphatic. The utility, however, of this figure, to the
Orator at least, is sufficiently established by the single
R H E TO R. I. C. 289
tainly nothing is more adverse to this appearance than Chap. III.
Rhetoric, consideration, that it abounds in the Speeches of
over-refinement. Any expression indeed that is vul- \-N-
S-V-7 Demosthenes. -
§ 3. On the last quality of Style to be noticed,
Elegance or Beauty, it is the less necessary to en-
large, both because the most appropriate and cha-
racteristic excellence of the class of compositions
here treated of, is, that Energy of which we have
been speaking, and also because many of the rules
laid down under that head, are equally applicable
with a view to Elegance ; the same Choice, Number,
and Arrangement of words, will, for the most part,
conduce both to Energy and to Beauty. The , two
qualities however are by no means undistinguishable :
a Metaphor, for instance, may be apt, and striking,
and consequently conducive to Energy of expression,
even though the new image, introduced by it, have
no intrinsic beauty, or be even unpleasant ; in which
case it would be at variance with Elegance, or at least
would not conduce to it. Elegance requires that all
homely and coarse words and phrases should be
avoided, even at the expense of circumlocution; though
they may be the most apt and forcible that language
can supply. ' And Elegance implies a smooth and
easy flow of words in respect of the sound of the
sentences; though a more harsh and abrupt mode of
expression may often be, at least equally, energetic.
Accordingly, many are generally acknowledged to
be forcible writers, to whom no one would give the
credit of Elegance; and many others, who are allowed
to be elegant, are yet by no means vigorous and
energetic.
When the two excellencies of Style are at vari-
ance, the general rule to be observed by the Orator,
is to prefer the energetic to the elegant. Sometimes,
indeed, a plain, or even a somewhat homely expres-
sion, may have even a more energetie effect, from that
very circumstance, than one of more studied refine-
ment, since it may convey the idea of the Speaker's
being thoroughly in earnest, and anxious to convey
His sentiments, where he uses an expression that can
have no other recommendation; whereas a strikingly
elegant expression may sometimes convey a suspicion
that it was introduced for the sake of its Elegance;
which will greatly diminish the force of what is said.
Universally, a writer or speaker should endeavour to
maintain the appearance of expressing himself, not,
as if he wanted to say something, but as if he had
something to say: i. e. not as if he had a subject set
him, and was anxious to compose the best essay or
declamation on it that he could ; but as if he had some
ideas to which he was anxious to give utterance;—
not as if he wanted to compose (for instance) a sermon,
and was desirous of performing that task satisfactorily,
but as if there was something in his mind which he was
desirous of communicating to his hearers. This is
probably what Dr. Butler means when he speaks of a
man's writing “with simplicity and in earnest.” His
manner has this advantage, though it is not only in-
elegant, but often obscure : Dr. Paley's is equally
earnest, and very perspicuous; and though often
homely, is more impressive than that of many of our
most polished writers. It is easy to discern the preva-
lence of these two different manners in different
authors, respectively, and to perceive the very different
effects produced by them ; it is not so easy for one
who is not really writing “ with simplicity and in
earnest,” to assume the appearance of it. But cer-
WOL. I.
gar, in bad taste, and unsuitable to the dignity of
the subject, or of the occasion, is to be avoided ;
since, though it might have, with some hearers, an
energetic effect, this would be more than counter-
balanced by the disgust produced in others; and where
a small accession of Energy is to be gained at the
expense of a great sacrifice of Elegance, the latter
will demand a preference. But still, the general rule
is not to be lost sight of by him who is in earnest
aiming at the true ultimate end of the Orator, to
which all others are to be made subservient ; viz. not
the amusement of his hearers, nor their admiration of
himself, but their Conviction or Persuasion. It is
from this view of the subject that we have dwelt most
on that quality of Style which seems most especially
adapted to that object. Perspicuity is required in all
compositions; and may even be considered as the
ultimate end of a Scientific writer, considered as such ;
he may indeed practically increase his utility by writ-
ing so as to excite curiosity, and recommend his
subject to general attention ; but in doing so, he is,
in some degree, superadding the office of the Orator
to his own ; as a Philosopher, he may assume the
existence in his reader of a desire for knowledge, and
has only to convey that knowledge in language that
may be clearly understood. Of the Style of the Orator,
(in the wide sense in which we have been using this
appellation, as including all who are aiming at Con-
viction,) the appropriate object is to impress the mean-
ing strongly upon men's minds. Of the Poet, as such,
the ultimate end is to give pleasure; and accordingly
Elegance or Beauty (in the most extensive sense of
those terms,) will be the appropriate qualities of his
language.
Some indeed have contended, that to give pleasure
is not the ultimate end of Poetry;* not distinguish-
ing between the object which the Poet may have in
view, as a man, and that which is the object of Poetry, as
Poetry. Many, no doubt, may have proposed to them-
selves the far more important object of producing
moral improvement in their hearers through the
medium of Poetry; and so have others, the inculca-
tion of their own political or philosophical tenets, or,
(as is supposed in the case of the Georgics,) the en-
couragement of Agriculture: but if the views of the
individual are to be taken into account, it should be
considered that the personal fame or emolument of
the author is very frequently his ultimate object. The
true test is easily applied : that which to competent
judges affords the appropriate pleasure of Poetry, is
good poetry, whether it answer any other purpose or
not ; that which does not afford this pleasure, however
instructive it may be, is not good Poetry, though it
may be a valuable work.
It may be doubted, however, how far these remarks
apply to the question respecting Beauty of Style; since
the chief gratification afforded by Poetry, arises, it
may be said, from the Beauty of the thoughts; and
undoubtedly if these be mean and common-place, the
Poetry will be worth little ; but still, it is not any
quality of the thoughts that constitutes Poetry. Not-
withstanding all that has been advanced by some
* Supported in some degree by the authority of Horace :
Aut prodesse volunt, aut delectare Poeta.
2 Q
290 R H E. T. O. R. I. C.
Rhetoric. French critics,” to prove that a work, not in metre, good Prose composition, * such and such thoughts ex- Chap. III.
v=-V-V may be a Poem, (which doctrine was partly derived pressed in good language;” that which is primary in S-N-
from a misinterpretation of a passage in Aristotle's
Poetics,t) universal opinion has always given a con-
trary decision. Any composition in verse, (and none
that is not,) is always called, whether good or bad, a
Poem, by all who have no favourite hypothesis to
maintain. It is indeed a common figure of speech to
say, in speaking of any work that is deficient in
the qualities which Poetry ought to exhibit, that it is
not a Poem ; just as we say of one who wants the
characteristic excellences of the species, or the sex,
that he is not a man : ; and thus some have been led
to confound together the appropriate excellence of the
thing in question, with its essence : but the use of
such an expression as, an “indifferent,” or “ a dull
Poem,” shows plainly that the title of Poetry does not
necessarily imply the requisite Beauties of Poetry.
Poetry is not distinguished from Prose by superior
Beauty of thought or of expression, but is a distinct
kind of composition ; $ and they produce, when each
is excellent in its kind, distinct kinds of pleasure.
Try the experiment, of merely breaking up the me-
trical structure of a fine Poem, and you will find it
inflated and bombastic Prose: remove this defect by
altering the words and the Arrangement, and it will
be better Prose than before ; then arrange this again
into metre, without any other change, and it will be
tame and dull Poetry; but still it will be Poetry, as is
indicated by the very censure it will incur ; for if it
were not, there would be no fault to be found with it ;
since, while it remained Prose, it was (as we have
supposed,) unexceptionable. The circumstance that
the same Style which was even required in one kind
of composition, proved offensive in the other, shows
that a different kind of language is suitable for a com-
position in metre.
Another indication of the essential difference be-
tween the two kinds of composition, and of the
superior importance of the expression in Poetry, is,
that a good translation of a Poem, (though, perhaps,
strictly speaking, what is so called is rather an imita-
tion,) is read with equal, or even superior pleasure
by one well acquainted with the original; whereas
the best translation of a Prose work, (at least of one
not principally valued for beauty of Style,) will sel-
dom be read by one familiar with the original. And
for the same reason, a-fine passage of Poetry will be
reperused, with unabated pleasure, for the twentieth
time, even by one who knows it by heart -
According to the views here taken, good Poetry
might be defined, “ Elegant and decorated language
in metre, expressing such and such thoughts;” and
* See'Preface to Telemaque.
+ Wixot Aéryot has been erroneously interpreted language with-
out metre, in a passage where it certainly means Metre without
music; or, as he calls it in another passage of the same work,
thixopergia. .
f “I dare bo all that may become a man
Who dares do more is none.”—Macbeth.
§ It is hardly necessary to remark, that we are not defending
or seeking to introduce any unusual or new sense of the word
Poetry; but, on the contrary, explaining and vindicating that
which is the most customary among all men who have no parti-
cular theory to support. The mass of mankind often need, in-
deed, to have the meaning of a word (i. e. their own meaning,3
eaplaimed and developed.; but not, to have it determined what it
shall mean, since that is determined by their use ; the true sense
of each word being, that which is understood by it.
each being subordinate in the other. - -
What has been said may be illustrated as fully, not
as it might be, but as is suitable to the present occa-
sion, by the following passages from Dr. A. Smith's
admirable fragment of an Essay on the Imitative Arts:
“Were I to attempt to discriminate between Dancing
and any other kind of movement, I should observe,
that though in performing any ordinary action,-in
walking, for example, across the room, a person may
manifest both grace and agility, yet if he betrays the
least intention of showing either, he is sure of offend-
ing more or less, and we never fail to accuse him of some
degree of vanity and affectation. In the performance of
any such ordinary action, every one wishes to appear
to be solely occupied about the proper purpose of
the action; if he means to show either grace or
agility, he is careful to conceal that meaning ; and
in proportion as he betrays it, which he almost always
does, he offends. In Dancing, on the contrary, every
one professes and avows, as it were, the intention
of displaying some degree either of grace or of .
agility, or of both. The display of one or other, or
both of these qualities, is, in reality, the proper pur-
pose of the action ; and there can never be any dis-
agreeable vanity or affectation in following out the
proper purpose of any action. When we say of any par-
ticular person, that he gives himself many affected airs
and graces in Dancing, we mean either that he exhibits
airs and graces unsuitable to the nature of the Dance,
or that he exaggerates those which are suitable. Every
Dance is, in reality, a succession of airs and graces of
some kind or other, which, if I may say so, profess
themselves to be such. The steps, gestures, and
motions which, as it were, avow the intention of ex-
hibiting a succession of such airs and graces, are the
steps, gestures, and motions which are peculiar to
Dancing. * * .# * The distinction
between the sounds or tones of Singing, and those of
Speaking, seems to be of the same kind with that
between the step, &c. of Dancing, and those of any
other ordinary action. Though in Speaking a person
may show a very agreeable tone of voice, yet if he
seems to intend to show it, if he appears to listen
to the sound of his own voice, and as it were to tune
it into a pleasing modulation, he never fails to offend,
as guilty of a most disagreeable affectation. In Speak-
ing, as in every other ordinary action, we expect and
require that the speaker should attend only to the
proper purpose of the action,-the clear and distinct
expression of what he haste say. In Singing, on the
contrary, every one professes the intention to please
by the tone and cadence of his voice; and he not
only appears to be guilty of no disagreeable affectation
in doing so, but we expect and require that he should
do so. To please by the Choice and Arrangement of
agreeable sounds, is the proper purpose of all music,
vocal, as well as instrumental; and we always expect
that every one should attend to the proper purpose of
whatever action heis performing. A person may appear
to sing, as well as to dance, affectedly; he may endea-
vour to please by sounds and tones which are unsuit-
able to the nature of the song ; or he may dwell too
much on those which are suitable to it. The dis-
agreeable affectation appears to consist always, not
in attempting to please by a proper, but by some
," R. H. E. T. O. R. I. C.
29:
Rhetorie. improper modulation of the voice.” It is only necessary
S-N-7 to add, (what seems evidently to have been in the
author's mind, though the Dissertation is left un-
finished,) that Poetry has the same relation to Prose,
as Dancing to Walking, and Singing to Speaking ;
and that what has been said of them, will apply exactly,
Tmutatis mutandis, to the other. It is needless to state
this at length, as any one, by going over the pas-
sages just cited, merely substituting for “Singing,”
“Poetry,” for “Speaking,” “Prose,” for “Voice,”
“Language,” &c. will at once perceive the coln-
cidence.
What has been said will not be thought an unneces-
sary digression, by any one who considers, (not to
mention the direct application of Dr. Smith's remarks,
to Elocution) the important principle thus established
in respect of the decorations of Style: viz. that though
it is possible for a poetical Style to be affectedly and
offensively ornamented, yet the same degree and kind
of decoration which is not only allowed, but required,
in Verse, would in Prose be disgusting ; and that the
appearance of attention to the Beauty of the expression,
and to the Arrangement of the words, which in Verse
is essential, is to be carefully avoided in Prose.
And since, as Dr. Smith observes, “ such a design,
when it exists, is almost always betrayed; the safest
rule is, never, during the act of composition, to study
Ebegance, or think about it at all. Let an author
study the best models—mark their beauties of Style,
and dwell upon them, that he may insensibly catch
the habit of expressing himself with Elegance ; and
when he has completed any composition, he may re-
vise it, and cautiously alter any passage that is awk-
ward and harsh, as well as those that are feeble and
obscure : but let him never, while writing, think of
any beauties of Style; but content himself with such
as may occur spontaneously. He should carefully
study Perspicuity as he goes along ; he may also,
though more cautiously, aim, in like manner, at
Energy; but if he is endeavouring after Elegance,
he will hardly fail to betray that endeavour; and in
proportion as he does this, he will be so far from
giving pleasure, to good judges, that he will offend
more than by the rudest simplicity.
CHAPTER IV.
OF ELO CUTION.
ON the importance of this branch, it is hardly neces-
sary to offer any remark. Few need to be told that
the effect of the most perfect composition may be
entirely destroyed, even by a Delivery which does not
render it unintelligible ;-that one, which is inferior
both in matter and style, may produce, if better
spoken, a more powerful effect than another which
surpasses it in both those points; and that even such
an Elocution as does not spoil the effect of what is
said, may yet fall far short of doing full justice to it.
* What would you have said,” observed AEschines,
when his recital of his great rival's celebrated Speech
on the Crown was received with a burst of admira-
tion,--" what would you have said had you heard
Him speak it 7”
The subject is far from having failed to engage
attention : of the prevailing deficiency of this, more
than of any other qualification of a perfect Orator,
many have complained; and several have laboured to
remove it : but it may safely be asserted, that their
endeavours have been, at the very best, entirely un-
successful. Probably not a single instance could be
found of any one who has attained by the study of any
system of instruction that has appeared, a really good
Delivery; but there are many, probably nearly as many
as have fully tried the experiment, who have by this
means been totally spoiled ;-who have fallen irre-
coverably into an affected style of spouting, worse, in
all respects, than their original mode of Delivery.
Many accordingly have, not unreasonably, conceived
a disgust for the subject altogether; considering
it hopeless that Elocution should be taught by any
rules; and acquiescing in the conclusion that it is
to be regarded as entirely a gift of nature, or an acci-
dental acquirement of practice. It is to counteract
the prejudice which may result from these feelings,
that we profess in the outset a dissent from the
principles generally adopted, and lay claim to some
degree of originality in our own. Novelty affords at
least an opening for hope, and the only opening, when
former attempts have met with total failure.
Chap, ſiſ.
chap, iv.
•º’
The requisites of Elocution correspond in great Requisites
measure with those of Style: Correct Enumciation, in
opposition both to indistinct utterance, and to vulgar
and dialectic pronunciation, may be considered as
answering to Purity, and Grammatical Propriety. These
qualities of Style and of Elocution, being equally re-
quired in common conversation, do not properly fall
within the province of Rhetoric. The three qualities,
again, which have been treated of, under the head
of Style, Perspicuity, Energy, and Elegance, may be
regarded as equally requisites of Elocution ; which, in
order to be perfect, must convey the meaning clearly,
forcibly, and agreeably. -
Before however we enter upon any separate ex-
amination of these requisites, it will be necessary to
premise a few remarks on the distinction between
of Elocu-
tion.
the two branches of Delivery, viz. Reading aloud, and Reading
Speaking.
The object of correct Reading is, to convey and Speak-
to the hearers, through the medium of the ear, what *š'
is conveyed to the reader by the eye;—to put them
in the same situation with him who has the book be-
fore him ;--to exhibit to them, in short, by the voice,
not only each word, but also all the stops, para-
graphs, italic characters, notes of interrogation, &c.”
* It may be said, indeed, that even tolerable Reading aloud
supplies more than is exhibited by a book to the eye; since
though italics, e. g. indicate which word is to receive the em-
phasis, they do not point out the tone in which it is to be pro-
nounced; which may be essential to the right understanding of
the sentence; e. g. in such a sentence as in Genesis, ch. i. “God
said, let there be light, and there was light:” here we can indi-
cate indeed that the stress is to be upon “was;” but it may be
pronounced in different tones; one of which would alter the
sense, by implying that there was light already. This is true
indeed; and it is also true, that the very words themselves are
2 Q 2
292
R H E T O R. I. C.
of whom we are weary, and to occupy the mind with Chap. Iv.
reflections of its own. *~~
Rhetoric, which his sight presents to him, His voice seems to
-y- indicate to them, “thus and thus it is written in the
book or manuscript before me.”. Impressive reading
superadds to this, some degree of adaptation of the
tones of voice to the character of the subject, and of
the Style. What is usually termed fine Reading seems
to convey, in addition to this, a kind of admonition to
the hearers respecting the feelings which the com-
position ought to excite in them : it appears to say,
“ this deserves your admiration;–this is sublime ;-
this is pathetic, &c.” But Speaking, i. e. natural
speaking, when the Speaker is uttering his own sen-
timents, and is thinking exclusively of them, has some-
thing in it distinct from all this: it conveys, by the
sounds which reach the ear, the idea, that what is
said is the effusion of the Speaker's own mind, which
he is desirous of imparting to others. A decisive
proof of which is, that if any one overhears the voice
of another, to whom he is an utter stranger—suppose
in the next room—without being able to catch the
sense of what is said, he will hardly ever be for a
moment at a loss to decide whether he is Reading or
Speaking; and this, though the hearer may not be one
who has ever paid any critical attention to the various
modulations of the human voice. So wide is the dif-
ference of the tones employed on these two occasions,
be the subject what it may.” *
The difference of effect produced is proportionably
great : the personal sympathy felt towards one who
appears to be delivering his own sentiments is such,
that it usually rivets the attention, even involuntarily,
though to a discourse which appears hardly worthy of
it. It is not easy for an auditor to fall asleep while
he is hearing even perhaps feeble reasoning clothed
in indifferent language, delivered extemporaneously,
and in an unaffected style; whereas it is common for
men to find a difficulty in keeping themselves awake,
while listening even to a good dissertation, of the
same length, or even shorter, on a subject, not unin-
teresting to them, when read, though with propriety,
'and not in a languid manner. And the thoughts, even
of those not disposed to be drowsy, are apt to wander,
unless they use an effort from time to time to prevent
it; while, on the other hand, it is notoriously difficult
to withdraw our attention, even from a trifling talker,
not always presented to the eye with the same distinctions as are
to be conveyed to the ear; as e.g. “abuse,” “refuse,” “pro-
ject,” and many others are pronounced differently, as nouns and
as verbs. This ambiguity however in our written signs, as well
as the other, relative to the emphatic words, are imperfections
which will not mislead a moderately practised reader. Our
meaning in saying that such Reading as we are speaking of, puts
the hearers in the same situation as if the book were before them,
is to be understood on the supposition of their being able not only
to read, but to read so as to take in the full sense of what is
written.
* “ At every sentence let them ask themselves this question,
How should I utter this, were I Speaking it as my own immediate
sentiments 2 I have often tried an experiment to show the
great difference, between these two modes of utterance, the
natural and the artificial; which was, that when I found a person
of vivacity delivering his sentiments with energy, and of course
with all that variety of tones which nature furnishes, I have
taken occasion to put something into his hand to read, as rela-
tive to the topic of conversation; and it was surprising to see
what an immediate change there was in his Delivery, from the
moment he began to read. A different pitch of voice took place
of his natural one, and a tedious uniformity of cadence succeeded
to a spirited variety; insomuch that a blind man could hardly
conceive the person who Read to be the same who had just been
Speaking.” Sheridan, Art of Reading,
Of the two branches of Elocution which have been
just mentioned, it might at first sight appear as if one
only, that of the Speaker, came under the province of
Rhetoric. But it will be evident, on consideration,
that both must be, to a certain extent, regarded as
connected with our present subject ; not merely be-
cause many of the same principles are applicable to
both, but because any one who delivers (as is so com-
monly the case) a written composition of his own,
may be reckoned as belonging to either class; as a
Reader who is the author of what he reads, or as a
Speaker who supplies the deficiency of his memory by
writing. And again, in the (less common) case where
a Speaker is delivering without book, and from
memory alone, a written composition, either his own
or another's, though this cannot in strictness be called
Reading, yet the tone of it will be very likely to re-
semble that of Reading. In the other case, that where
the author is actually reading his own composition,
he will be still more likely, notwithstanding its being
his own, to approach, in the Delivery of it, to the
Elocution of a Reader; and, on the other hand, it is
possible for him, even without actually deceiving the
hearers into the belief that he is speaking extempore,
to approach indefinitely near to that Style.
The difficulty however of doing this to one who has
the writing actually before him, is considerable; and
it is of course far greater when the composition is not
his own. And as it is evident from what has been
said, that this, as it may be called, Extemporaneous
style of Elocution, is much the more impressive, it
becomes an interesting inquiry, how the difficulty in
question may best be surmounted.
Little, if any, attention has been bestowed on this Artificial
point by the writers on Elocution; the distinction style of
above pointed out between Reading and Speaking,
having seldom or never been precisely stated and
dwelt on. Several however have written elaborately
on “good Reading,” or on Elocution, generally; and
it is not to be denied, that some ingenious and (in
themselves) valuable remarks have been thrown out
relative to such qualities in Elocution as might be
classed under the three heads we have laid down, of
Perspicuity, Energy, and Elegance: but there is one
principle running through all their precepts, which
being, according to our views, radically erroneous,
must (if those views be correct) vitiate every system,
founded on it. The principle we mean is, that in
order to acquire the best style of Delivery, it is requi-
site to study analytically the emphases, tones, pauses,
degrees of loudness, &c. which give the proper effect
to each passage that is well delivered—to frame
rules founded on the observation of these—and then,
in practice, deliberately and carefully to conform the
utterance to these rules, so as to form a complete arti-
ficial system of Elocution. r
That such a plan not only directs us into a circuitous
and difficult path, towards an object which may be
reached by a shorter and straighter, but also, in most
instances, completely fails of that very object, and
even produces, oftener than not, effects the very re-
verse of what is designed, is a doctrine for which it
will be necessary to offer some reasons; especially as
it is undeniable that the system here reprobated, as
employed in the case of Elocution, is precisely that
Elocution,
R H E TO R. I. C. 293
When however we protest against all artificial sys- Chap. IV.
tems of Elocution, and all direct attention to Delivery, at S-V-
the time, it must not be supposed that a general inat- Natural
tention to that point is recommended; or that the most style of
Rhetoric. recommended and taught in this very article, in re-
\-N-' spect of the conduct of Arguments. By analyzing the
best compositions, and observing what kinds of argu-
ments, and what modes of arranging them, in each * >
Elocution.
case, prove most successful, general rules have been
formed, which an author is recommended studiously
to observe in Composition: and this is precisely the
procedure which, in Elocution, we deprecate. The
reason for making such a difference in these two cases
is this ; whoever (as Dr. A. Smith remarks in the
passage lately cited”) appears to be attending to his
own utterance, which will almost inevitably be the
case with every one who is doing so, is sure to give
offence, and to be censured for an affected delivery;
because every one is expected to attend exclusively to
the proper object of the action he is engaged in; which,
in this case, is the expression of the thoughts—not the
sound of the expressions. Whoever therefore learns
and endeavours to apply in practice, any artificial
rules of Elocution, so as deliberately to modulate his
voice conformably to the principles he has adopted,
(however sound they may be in themselves) will
hardly ever fail to betray his intention; which always
gives offence when perceived. Arguments, on the
contrary, must be deliberately framed : whether any
one's course of reasoning be sound and judicious, or
not, it is necessary, and it is expected, that it should
be the result of thought. No one, as Dr. Smith ob-
serves, is charged with affectation for giving his atten-
tion to the proper object of the action he is engaged
in. As therefore the proper object of the Orator is to
adduce convincing arguments, and topics of Persua-
sion, there is nothing offensive in his appearing deli-
berately to aim at this object. He may indeed weaken
the force of what is urged, by too great an appearance
of elaborate composition, or by exciting suspicion of
Rhetorical trick ; but he is so far from being ex-
pected to pay no attention to the sense of what he
says, that the most powerful argument would lose
much of its force, if it were supposed to have been
thrown out casually, and at random. Here therefore
the employment of a regular system (if founded on
just principles) ean produce no such ill effect as in the
case of Elocution ; since the habitual attention which
that implies to the choice and arrangement of argu-
ments, is such as must take place, at any rate; whether
it be conducted on any settled principles or not. The
only difference is, that he who proceeds on a correct
system, will think and deliberate concerning the
course of his Reasoning to better purpose than he who
does not : he will do well and easily, what the other
does ill, and with more labour. Both alike must be-
stow their attention on the Matter of what they say,
if they would produce any effect; both are not only
allowed, but expected to do so.
The two opposite modes of procedure therefore
which are recommended in respect of these two
points, (the Argument and the Delivery,) are, in fact,
both the result of the same circumstance; viz. that
the Speaker is expected to bestow his attention on the
proper ultimate object of his Speech, which is, not the
Elocution, but the Matter.t
* See ch. iii. sec. 3. p. 290.
+ Style occupies in some respects an intermediate place be-
tween these two ; in what degree each quality of it should or
should not be made an object of attention at the time of composing,
raised, lowered, &c.
perfect Elocution is to be attained by never thinking
at all on the subject; though it may safely be affirmed
that even this negative plan would succeed far better
than a studied modulation. But it is evident that if
any one wishes to assume the Speaker as far as pos-
sible ; i. e. to deliver a written composition with
some degree of the manner and effect of one that is
extemporaneous, he will have a considerable difficulty
to surmount : since though this may be called, in a
certain sense, the NATURAL MANNER, it is far from
being what he will naturally, i. e. spontaneously,
fall into. It is by no means natural for any one to
read as if he were not reading, but speaking. And
again, even when any one is reading what he does not
wish to deliver as his own composition, as, for instance,
a portion of the Scriptures, or the Liturgy, it is evident
that this may be done better or worse, in infinite de-
grees; and that though (according to the views here
taken) a studied attention to the sounds uttered, at
the time of uttering them, leads to an affected and
offensive delivery, yet, on the other hand, an utterly
careless reader cannot be a good one.
With a view to Perspicuity then, the first requisite Reading.
in all Delivery, viz. that quality which makes the
meaning fully understood by the hearers, the great
point is that the Reader (to confine our attention for
the present to that branch) should appear to under-
stand what he reads. If the composition be, in itself,
intelligible to the persons addressed, he will make
them fully understand it, by so delivering it. But to
this end, it is not enough that he should himself
actually understand it; it is possible, notwithstanding,
to read it as if he did not. And in like manner with
a view to the quality, which has been here called
Energy, it is not sufficient that he should himself feel,
and be impressed with the force of what he utters;
he may, notwithstanding, deliver it as if he were
unimpressed.
The remedy that has been commonly proposed for
these defects, is to point out in such a work, for instance,
as the Liturgy, which words ought to be marked as em-
phatic,-in what places the voice is to be suspended,
One of the best writers on the
subject, Sheridan, in his Lectures on the Art of Read-
ing,” (whose remarks on many points coincide with
the principles here laid down, though he differs from
us on the main question—as to the System to be prac-
tically followed with a view to the proposed object,)
adopts a peculiar set of marks for denoting the differ-
ent pauses, emphases, &c. and applies these, with
accompanying explanatory observations, to the greater
part of the Liturgy, and to an Essay subjoined;t
and how far the appearance of such attention is tolerated, has
been already treated of in the preceding chapter.
* See note *, p. 292.
+ “For the benefit of those who are desirous of getting over
their bad habits, and discharging that important part of the
Sacred office, the Reading the Liturgy with due decorum, I shall
first enter into a minute examination of some parts of the Ser-
vice, and afterwards deliver the rest, accompanied by such marks
as will enable the Reader, in a short time, and with moderate
pains, to make himself master of the whole.
“But first it will be necessary to explain the marks which you
294
JR H E TO R H C.
Rhetoric. recommending that the habit should be formed of
*~~" regulating the voice by his marks; and that afterwards
readers should “write out such parts as they want to
deliver properly, without any of the usual stops; and,
after having considered them well, mark the pauses
and emphases by the new signs which have been
annexed to them, according to the best of their judg-
ment,” &c.
To the adoption of any such artificial scheme there
are three weighty objections; Ist, that the proposed
system must necessarily be imperfect; 2dly, that if it
will hereafter see throughout the rest of this course. They are
of two kinds; one, to point out the emphatic words, for which I
shall use the Grave accent of the Greek, [*]. -
“The other, to point out the different pauses or stops, for
which I shall use the following marks :
“ For the shortest pause, marking an incomplete line, thus'.
“For the second, double the time of the former, two ".
“And for the third or full stop, three ".
“When I would mark a pause longer than any belonging to
the usual stops, it shall be by two horizontal lines, as thus =.
“When I would point out a Syllable that is to be dwelt on
some time, I shall use this —, or a short horizontal over the Syl-
Habie.
“When a Syllable should be rapidly uttered, thus *, or a
curve turned upwards; the usual marks of long and short in
Prosody.
“The Exhortation I have often heard delivered in the follow-
ing manner : s
“‘Dearly beloved brethren, the Scripture moveth us in sundry
places to acknowledge and confess our manifold sins and wick-
edness. And that we should not dissemble nor cloke them be-
fore the face of Almighty God our Heavenly Father, but confess
-them with an humble Fowly penitent and obedient heart, to the
end that we may obtain, forgiveness of the same, by his infinite
goodness and mercy. And although we ought at all times
humbly to acknöwledge our sins before God, yet ought we most
chiefly so to do, when we assemble and meet together. To
render thanks for the great benefits we have received at his
hands, to set forth his most worthy praise, to hear his most holy
word, and to ask those things that are requisite and necessary,
as well for the body as the soul. Wherefore I pray and beseech
you, as many as are here present, to accompany me with a pure
heart and humble voice to the throne of the heavenly grace, say-
ing after me.”
“In the latter part of the first period, “but confess them with
an humble lowly penitent and obedient heart, to the end that we
may obtain, forgiveness of the same, by his infinite goodness and
mercy, there are several faults committed. In the first place,
the four epithets preceding the word “heart,’ are huddled together,
and pronounced in a monotone, disagreeable to the ear, and
enervating to the sense; whereas each word rising in force above
the other, ought to be marked by a proportional rising of the
notes in the voice; and, in the last, there should be such a note
used as would declare it at the same time to be the last—‘with
an humbleſ lowly' penitent and obedient heart,’ &c. At first view
it may appear, that the words ‘humble’ and “lowly,’ are synony-
mous ; but the word “lowly,” certainly implies a greater degree of
humiliation than the word “humble.” The word ‘penitent” that
follows, is of stronger import than either; and the word “obe-
dient, signifying a perfect resignation to the will of God, in con-
sequence of our humiliation and repentance, furnishes the climax.
But if the climax in the words be not accompanied by a suitable
chimax in the notes of the voice, it cannot be made manifest.
In the following part of the sentence, * to the end that we may
obtain' forgiveness of the same’’ there are usually three emphases
laid on the words, end, obtain, same, where there should not be
‘any, and the only emphatic word, forgiveness, is slightly passed
over; whereas it should be read—" to the end that we may ob-
tain forgiveness of the same,' keeping the words, obtain, and
forgiveness, closely together, and not disuniting them, both to
the prejudice of the Sense and Cadence, &c. &c.
“I shall now read the whole, in the manner I have recom-
mended; and if you will give attention to the marks, you will
be reminded of the manner, when you come to practise in your
private reading. ‘ Dearly beloved brethren :=The Scripture
moveth us' in sundry places' to acknowledge and confess our
manifold sins and wickedness'ſ and that we should not dissèmble
were perfect, it would be a circuitous path to the
object in view ; and 3dly, that even if both those
objections were removed, the object would not be
effectually obtained.
Chap. IV.
1st, Such a system must necessarily be imperfect, Imperfec-
because though the emphatic word in each sentence tion of the
f artificial
may easily be pointed out in writing, no variety o
marks that could be invented,—not even musical nota-
tion,--would suffice to indicate the different tones *
in which the different emphatic words should be pro-
nounced ; though on this depends frequently the
whole force, and even sense of the expression. Take
the word “covet,” in the tenth Commandment.
nor clöke them’ before the face of Almighty Godſ our Heavenly
Father'ſ but confess them' with an humbleſ lowly' penitent' and
obedient heartſ to the end that we may obtain forgiveness of the
same by his infinite goodness and mercy” And although we
ought at all times' humbly to acknowledge our sins before Göd”
yet ought we most chiefly so to do' when we assemble and meet
together' to render thanks' for the great benefits we have received
at his hands” to set forth' his most worthy präise’’ to hearſ his
most holy word and to ask those things' which are requisite and
necessary’ as well for the body' as the soul” Wherefore I pray
and beseech yöu' as many as are here presentſ to accompany mê'
with a pure heart' and humble voiceſ to the throne of the heavenly
grace’ saying,’ &c.”
The generality of the remarks respecting the way in which
each passage of the Liturgy should be read, are correct ;
though the mode recommended for attaining the proposed end,
is totally different from what is suggested in the present chapter.
In some points, however, the author is mistaken as to the em-
phatic words: e. g. in the Lord’s Prayer, he directs the following
passage to be read thus; “thy will be doneſ on carth' as it is'
in Heaven,” with the emphasis on the words “be " and “ is;”
these, however, are not the emphatic words, and do not even
exist, in the original Greek, but are supplied by the translator;
the latter of them might, indeed, be omitted altogether without
any detriment to the sense; “thy will be done, as in Heaven,
so also on earth,” which is a more literal translation, is perfectly
intelligible. A passage in the second Commandment again, he
directs to be read, according indeed to the usual mode, both of
reading and pointing it, “ visit the sins of the fathers' upon
the children' unto the third and fourth generation of them that
hate me ;” which mode of reading destroys the sense, by making
a pause at “children,” and none at “ generation;” for this im-
plies that the third and fourth generations, who suffer these
judgments, are themselves such as hate the Lord, instead of being
merely, as is meant to be expressed, the children of such ; “ of
them that hate me,” is a genitive not governed by “generation,”
but by “children :” it should be read (according to Sheridan's
marks) “visit the sins of the fathers' upon the children unto the
third and fourth generation' of them that hate me :” i.e. “visit
the sins of the fathers who hate me, upon the third and fourth
generations of their descendants.” The same sanction is given
to an equally common fault in reading the fifth Commandment;
“ that thy days' may be long in the land' which the Lord thy
God giveth thee:” the pause should evidently be at “-long" not
at “ land.” No one would say in ordinary conversation, “I
hope you will find enjoyment in the garden' which you
have planted.” He has also strangely omitted an emphasis on
He has, how-
ever, in the negative or prohibitory commands avoided the
common fault of accenting the word “ not.” And here it may
be worth while to remark, that in some cases the Copula ought
to be made the emphatic word ; (i. e. the “ is ” if the proposition
be affirmative, the “ not,” if negative) viz. where the proposition
may be considered as in opposition to its contradictory. If, e.g.
it had been a question, whether we ought to steal or not, the
commandment, in answer to that, wonld have been rightly pro-
nounced, “ thou shalt not steal :” but the question being, what
things we are forbidden to do, the answer is, that “to steal” is
one of them, “ thou shalt not steal.” In such a case as this, the
proposition is considered as opposed, not to its contradictory, but
to one with a different Predicate. The question being not,
which Copula (negative or affirmative) shall be employed, but
what shall be affirmed or denied of the subject : e.g. “it is law-
ful to beg, but not to steal :” in such a case, the Predicate will
usually be the emphatic word, not the Copula.
* See note *, p. 291.
system.
R H E TO RIC.
295
Rhetoric... as an instance the words of Macbeth in the witches
\-y-' cave, when he is addressed by one of the Spirits which
Circuitous-
ness of the
artificial
system.
not leave nature to do her own work
they raise, “Macbeth ! Macbeth Macbeth !” on which
he exclaims, “Had I three ears I'd hear thee :” no
one would dispute that the stress is to be laid on the
word “three ;” and thus much might be indicated
to the reader's eye; but if he had nothing else to
trust to, he might chance to deliver the passage in
such a manner as to be utterly absurd ; for it is pos-
sible to pronounce the emphatic word “ three,” in
such a tone as to indicate that “ since he has but
two ears, he cannot hear.” It would be nearly as
hopeless a task to attempt adequately to convey, by
any written marks, precise directions as to the rate,
the degree of rapidity or slowness, with which each
sentence and clause should be delivered. Longer
and shorter pauses may indeed be easily denoted ; and
marks may be used, similar to those in music, to in-
dicate, generally, quick, slow, or moderate time ; but
it is evident that the variations which actually take
place are infinite;—far beyond what any marks could
suggest; and that much of the force of what is said
depends on the degree of rapidity with which it is
uttered; chiefly on the relative rapidity of one part in
comparison of another : for instance in such a sentence,
as the following in one of the Psalms, which one may
usually hear read at one uniform rate ; ‘‘ all men
that see it shall say, this hath God done ; for they
shall perceive that it is his work;” the four words,
“ this hath God done,” though monosyllables, ought
to occupy very little less time in utterance than all
the rest of the verse together.
2dly, But were it even possible to bring to the highest
perfection the proposed system of marks, it would
still be a circuitous road to the desired end. Suppose
it could be completely indicated to the eye, in what
tone each word and sentence should be pronounced
according to the several occasions, the learner might
ask, “but why should this tone suit the awful,
this, the pathetic, this, the narrative style 3 why is
this mode of delivery adopted for a command,-
this for an exhortation,-this, for a supplication?” &c.
The only answer that could be given, is, that these
tones, emphases, &c. are a part of the language;—
that nature, or custom, which is a second nature,
suggests, spontaneously, these different modes of
giving expression to the different thoughts, feelings,
and designs, which are present to the mind of any
one who, without study, is speaking in earnest
his own sentiments. Then, if this be the case, why
Impress but
the mind fully with the sentiments, &c. to be uttered;
withdraw the attention from the sound, and fix it on
the sense; and nature, or habit, will spontaneously
suggest the proper Delivery. That this will be the
case, is not only true, but is the very supposition on
which the artificial system proceeds; for it professes
to teach the mode of delivery maturally adapted to
each toocasion. It is surely, therefore, a circuitous
path that is proposed, when the learner is directed,
first to consider how each passage ought to be read ;
i.e. what mode of delivering each part of it would spon-
#aneously occur to him, if he were attending exclusively
to the matter of it ; then to observe all the modula-
tions, &c. of voice, which take place in such a Deli-
very; then, to note these down by established marks,
in writing ; and, lastly, to pronounce according to
these marks. This seems like recommending, for
the purpose of raising the hand to the mouth, that he
should first observe, when performing that action,
without thought of any thing else, what muscles are
contracted,—in what degrees, and in what order;
then, that he should note down these observations ;
and, lastly, that he should, in conformity with these
notes, contract each muscle in due degree, and in pro-
per order; to the end that he may be enabled, after
all, to—lift his hand to his mouth ; which, by suppo-
sition, he had already done. Such instruction is like
that bestowed by Moliere's pedantic tutor upon his
Bourgeois Gentilhomme, who was taught, to his in-
finite surprise and delight, what configurations of the
mouth he employed in pronouncing the several letters
of the alphabet, which he had been accustomed to
utter all his life, without knowing how.”
3dly, Lastly, waving both the above objections, if a
person could learn thus to read and speak, as it were,
by note, with the same fluency and accuracy as are
attainable in the case of singing, still the desired ob-
ject of a perfectly natural as well as correct Elocu-
tion, would never be in this way attained. The
reader's attention being fixed on his own voice, the
inevitable consequence would be that he would betray
more or less, his studied and artificial Delivery ; and
would, in the same degree, manifest an offensive
affectation. T
The practical rule then to be adopted, in conformity
with the principles here maintained, is, not only to
pay no studied attention to the voice, but studiously
to withdraw the thoughts from it, and to dwell as in-
tently as possible on the Sense; trusting to nature to
suggest spontaneously the proper emphases and tones.:
He who not only understands fully what he is read-
ing, but is earnestly occupying his mind with the
matter of it, will be likely to read as if he under-
stood it, and thus, to make others understand it ; Ś.
and in like manner, he who not only feels it, but is
exclusively absorbed with that feeling, will be likely
to read as if he felt it, and to communicate the im-
* “Qu'est ce que vous faites quand vous promonce2 O & Mais,
je dis, O !”
An answer which, if not savouring of Philosophical analysis,
gave at least a good practical solution of the problem.
+ It should be observed, however, that, in the reading of the
Liturgy especially, so many gross faults are become quite fami-
liar to many, from what they are accustomed to hear, if not
from their own practice, as to render it peculiarly difficult to
unlearn, or even detect them ; and as an aid towards the exposure
of such faults, there may be great advantage in studying Sheri.
dan's observations and directions respecting the delivery of it;
provided care be taken, in practice, to keep clear of his faulty
principle, by withdrawing the attention from the sound of the
voice, as carefully as he recommends it to be directed to that
point. -
† Many persons are so far impressed with the truth of the
doctrine here inculcated, as to acknowledge that “it is a great
fault for a reader to be too much occupied with thoughts respect-
ing his own voice;” and thus they think to steer a middle course
between opposite extremes : but it should be remembered that
this middle course entirely nullifies the whole advantage pro-
posed by the plan recommended. A reader is sure to pay too
much attention to his voice, not only if he pays any at all, but if
he does not strenuously labour to withdraw his attention from it
altogether.
$ Who, for instance, that was really thinking of a resurrection
from the dead, would ever tell any one that our Lord “rose
again from the dead,” (which is so common a mode of reading
the Creed,) as if He had done so more than once 2
Chap. IV.
\– V-'
Appearance
of affecta-
tion result-
ing from
the artifi-
cial system.
Natural
Iſla Il Ile I’.
296
R H E T O R. I C.
indeed, so far imitate him with advantage, as to adopt Chap. IV.
his plan, of fixing his attention on the matter, and not ‘-N-
Rhetoric. pression to his hearers. But this cannot be the case
\-y-e’
tion he hopes for from the hearers, &c.
Advantages
if he is occupied with the thought of what their opi-
nion will be of his reading, and, how - his voice
ought to be regulated;—if, in short, he is thinking of
himself, and, of course, in the same degree, abstract-
ing his attention from that which ought to occupy it
exclusively.
It is not, indeed, desirable, that in reading the Bible,
for example, or any thing which is not intended to appear
as his own composition, he should deliver what are,
avowedly, another's sentiments, in the same style, as
if they were such as arose in his own mind; but it is
desirable that he should deliver them as if he were
reporting another's sentiments, which were both fully
understood and felt in all their force by the reporter;
and the only way to do this effectually,–with such
modulations of voice, &c. as are suitable to each word
and passage, is to fix his mind earnestly on the
meaning, and leave nature and habit to suggest the
utterance.
Some may, perhaps, suppose that this amounts to
the same thing as taking no pains at all ; and if, with
this impression, they attempt to try the experiment
of a natural Delivery, their ill-success will probably
lead them to censure the proposed method, for the
failure resulting from their own mistake. In truth,
it is by no means a very easy task, to fix the attention
on the meaning, in the manner, and to the degree, now
proposed. The thoughts of one who is reading any
thing very familiar to him, are apt to wander to other
subjects, though perhaps such as are connected with
that which is before him ; if, again, it be something
new to him, he is apt (not indeed to wander to another
subject,) but to get the start, as it were, of his readers,
and to be thinking, while uttering each sentence, not
of that, but of the sentence which comes next. And
in both cases, if he is careful to , avoid those faults,
and is desirous of reading well, it is a matter of no
small difficulty, and calls for a constant effort, to pre-
vent the mind from wandering in another direction;
viz. into thoughts respecting his own voice,—respect-
ing the effect produced by each sound,—the approba-
And this is
the prevailing fault of those who are commonly said
to take great pains in their reading ; pains which will
always be taken in vain, with a view to the true ob-
ject to be aimed at, as long as the effort is thus
applied in a wrong direction. With a view, indeed,
to a very different object, the approbation bestowed
on the reading, this artificial delivery will often be
more successful than the natural. Pompous spout-
ing, and many other descriptions of unnatural tone
and measured cadence, are frequently admired by
many as excellent reading ; which admiration is
itself a proof that it is not deserved ; for when the
Delivery is really good, the hearers (except any one
who may deliberately set himself to observe and
criticise,) never think about it, but are exclusively
occupied with the sense it conveys, and the feelings
it excites. -
Still more to increase the difficulty of the method
of imitation here recommended, (for it is no less wise than honest
thinking about his voice; but this very plan, evidently
by its nature, precludes any further imitation; for if,
while reading, he is thinking of copying the manner
of his model, he will, for that very reason, be unlike
that model; the main principle of the proposed me-
thod being, carefully to exclude every such thought.
Whereas, any artificial system may as easily be
learned by imitation as the notes of a song. Practice
also, (i. e. private practice for the sake of learning,)
is much more difficult in the proposed method; be-
cause the rule being to use such a Delivery as is
suited, not only to the matter of what is said, but also,
of course, to the place, and occasion, and this, not by
any studied modulations, but according to the spon-
taneous suggestions of the matter, place, and occasion,
to one whose mind is fully and exclusively occupied
with these, it follows, that he who would practise
this method in private, must, by a strong effort of a
vivid imagination, figure to himself a place and an
occasion which are not present; otherwise, he will
either be thinking of his Delivery, (which is fatal to his
proposed object,) or else will use a Delivery suited to
the situation in which he actually is, and not, to that
for which he would prepare himself. Any system, on
the contrary, of studied emphasis and regulation of
the voice, may be learned in private practice, as easily
as singing. • .
Some additional objections to the method recom-
mended, and some further remarks on the counter-
balancing advantages of it, will be introduced presently,
when we shall have first offered some observations on
Speaking, and on that branch of Reading which the
most nearly approaches to it.
When any one delivers a written composition, of
which he is, or is supposed to profess himself the au-
thor, he has peculiar difficulties to encounter,” if his
object be to approach as nearly as possible to the ex-
* It must be admitted, however, that the difficulty of reading
the Liturgy with spirit, and even with propriety, is something
peculiar, on account of (what has been already remarked) the
inveterate and long-established faults to which almost every one's
ears are become familiar ; so that such a delivery as would
shock any one of even moderate taste, in any other composition,
he will, in this, be likely to tolerate, and to practise. Some, e. g.
in the Liturgy, read, “ have mercy upon us, miserable sinners;”
and others, “ have mercy upon us, miserable sinners;” both,
laying the stress on a wrong word, and making the pause in the
wrong place, so as to disconnect “us” and “miserable sinners,”
which the context requires us to combine. Every one, in expres-
sing his own natural sentiments, would say “ have mercy, upon
us-miserable-sinners.”
Many are apt even to commit so gross an error, as to lay the
chief stress on the words which denote the most important things;
without any consideration of the emphatic word of each sentence:
e.g. in the Absolution many read, “let us beseech Him to grant us
true repentance;” because “forsooth true repentance” is an im-
portant thing; not considering that, as it has been just mentioned,
it is not the new idea, to which the attention should be directed by
the emphasis; the sense being, that since God pardoneth all that
have true repentance, therefore, we should “beseech Him to
grant it to us.” -
In addition to the other difficulties of reading the Liturgy
well, it should be mentioned, that prayer, thanksgiving, and the
#. to take a fair view of difficulties) this circumstance is like, even when avowedly not of our own composition, should
Hººd to be noticed, that he who is endeavouring to bring be delivered as (what in truth they ought to be) the genuine
p it into practice, is in a great d e precluded from **** of our own minds at the moment of utterance ;
by the I p 5 8 egree prec which is not the case with the Scriptures, or with any thing
adoption of the advantages of imitation.
A person who hears and
*Natural approves a good reader in the Natural manner, may,
Ill:11] Il CT,
else that is read, not professing to be the speaker's own corºs
position,
R H E T O R. I. C.
297
Rhetoric, temporaneous style. It is indeed impossible to produce for the reader to draw off his mind as much as pos- Chap. IV.
\-N- the full effect of that style, while the audience are
aware that the words he utters are before him : but
he may approach indefinitely near to such an effect;
and in proportion as he succeeds in this object, the
impression produced will be the greater. It has been
already remarked, how easy it is for the hearers to
keep up their attention,-indeed, how difficult for
them to withdraw it, when they are addressed by one
who is really speaking to them in a natural and earnest
manner; though perhaps the discourse may be in-
cumbered with a good deal of the repetition, awk-
wardness of expression, and other faults incident
to extemporaneous language; and though it be pro-
longed for an hour or two, and yet contain no more
matter than a good writer could have clearly expressed
in a discourse of half an hour ; which last, if read to
them, would not, without some effort on their part,
have so fully detained their attention. The advantage
in point of Style, Arrangement, &c. of written, over
extemporaneous, discourses, (such at least as any but
the most accomplished orators can produce,) is suffi-
ciently evident : * and it is evident also that other
advantages, such as have been just alluded to, belong
to the latter. Which is to be preferred on each occa-
sion, and by each orator, it does not belong to the
present discussion to inquire: but it is evidently of the
highest importance to combine, as far as possible, in
each case, the advantages of both. A perfect fami-
liarity with the rules laid down in the first Chapter of
this Essay, would be likely, it is hoped, to give the
extemporaneous orator that habit of quickly me-
thodizing his thoughts on a given subject, which is
essential (at least where no very long premeditation
is allowed,) to give to a speech something of the
weight of argument and clearness of arrangement
which characterise good Writing. In order to
attain the corresponding advantage,_to impart to
the delivery of a written discourse something of the
vivacity and interesting effect of real, earnest,
speaking, the plan to be pursued, conformably
with the principles we have been maintaining, is,
* Practice in public speaking, generally,–practice in speaking
on the particular subject in hand,-and (on each occasion) pre-
meditation of the matter and arrangement, are, all, circumstances
of great consequence to a speaker. Nothing but a miraculous
gift can supersede these advantages. The Apostles accordingly
were forbidden to use any premeditation, being assured that “it
should be given them, in that same hour, what they should
say ” and when they found, in effect, this promise fulfilled to
them, they had experience, within themselves, of a sensible
miracle. This circumstance may furnish a person of sincerity
with a useful test for distinguishing (in his own case) the emo-
tions of a fervid imagination, from actual inspiration. It is
evident that an inspired preacher can have nothing to gain from
practice, or study of any kind ; he therefore who finds himself
inprove by practice, either in Argument, Style, or Delivery,
or who observes that he speaks more fluently and better on
subjects on which he has been accustomed to speak, or better,
with premeditation, than on a sudden, may indeed deceive his
hearers by a pretence to inspiration, but can hardly deceive
kinself.
t. Accordingly, it may be remarked, that, (contrary to what
might at first sight be supposed,) though the preceding Chapters,
as well as the present, are intended for general application, yet
it is to the eartemporary speaker that the rules laid down in the
former part (supposing them correet,) will be the most pecu-
liarly useful; while the suggestions offered in this last, respect-
ing Elocution, are more especially designed for the use of the
reader.
*WOL, I. - f
sible from the thought that he is reading, as well
as from thought respecting his own utterance;—
to fix his mind as earnestly as possible on the matter,
and to strive to adopt as his own, and as his own at
the moment of utterance, every sentiment he delivers;
—and to say it to the audience, in the manner which
the occasion and subject spontaneously suggest to
him who has abstracted his mind both from all con-
sideration of himself, and from the consideration that
he is reading. *
The advantage of this NATURAL MANNER, (i. e. the
manner which one naturally falls into who is really
speaking, in earnest, and with a mind exclusively in-
tent on what he has to say,) may be estimated from
this consideration; that there are few who do not speak
so as to give effect to what they are saying: some,
indeed, do this much better than others :-some have,
in ordinary conversation, an indistinct or incorrect
pronunciation,--an embarrassed and hesitating utter-
ance, or a bad choice of words : but hardly any one
fails to deliver,” (when speaking earnestly) what he
does say, so as to convey the sense and the force of it,
much more completely than even a good reader would,
if those same words were written down and read.
The latter might, indeed, be more approved ; but that
is not the present question ; which is, concerning the
impression made on the hearers' minds. It is not the
polish of the blade, that is to be considered, nor the
grace with which it is brandished, but the keenness
of the edge, and the weight of the stroke.
On the contrary, it can hardly be denied that the
elocution of most readers, when delivering their own
compositions, is such as to convey the notion, at the
very best, not that the preacher is expressing his own
sentiments, but that he is making known to his au-
dience what is written in the book before him : and,
whether the composition is professedly the reader's
own, or not, the usual mode of delivery, though,
grave and decent, is so remote from the energetic
style of real Natural Speech, as to furnish, if one
may so speak, a kind of running comment on all that
is uttered, which says, “I do not mean, think, or
feel, all this; I only mean to recite it with propriety
and decorum :” and what is usually called fine Read-
ing, only superadds to this, (as has been above re-
marked,) a kind of admonition to the hearers, that
they ought to believe, to feel, and to admire, what is
read.
It is easy to anticipate an objection which many
will urge against, what they will call, a colloquial
style of delivery ; viz. that it is indecorous, and un-
suitable to the solemnity of a serious, and especially,
of a Religious discourse. The objection is founded
on a mistake. Those who urge it, derive all their
notions of a Natural Delivery from two, irrelevant,
instances ; that of ordinary conversation, the usual
subjects of which, and consequently its usual tone,
* There is, indeed, a wide difference between different men,
in respect of the degrees of impressiveness with which, in earnest
conversation, they deliver their sentiments; but it may safely
be laid down, that he who delivers a written composition with
the same degree of spirit and energy, with which he would na-.
turally speak on the same subject, has attained, hot indeed,
necessarily, absolute perfection, but the utmost excellence at-
tainable by him. Any attempt to out-do his own Natural man-
ner, will inevitably lead to something worse than failure,
298 y R. H. E. T. O. R. I. C.
Rhetoric, are comparatively light ;—and, that of the coarse and in short, all thoughts of self, which, in proportion as
Chap. IV.
extravagant rant of vulgar fanatical preachers. But , they intrude, will not fail to diminish the effect. º "
\-N-
to conclude that the objections against either of these
styles, would apply to the Natural Delivery of a man
of sense and taste, speaking earnestly, on a serious
subject, and on a solemn occasion, or that he would
naturally adopt, and is advised to adopt, such a style
as those objected to, is no less absurd than if any one,
being recommended to walk in a natural and un-
studied manner, rather than in a dancing step, (to
employ Dr. A. Smith's illustration,) or a formal march,
should infer that the natural gait of a clown follow-
ing the plough, or of a child in its gambols, were
proposed as models to be imitated in walking across
a room. It is evident, that what is natural in one
case, or for one person, may be, in a different one,
very unnatural. It would not be by any means natu-
ral, to an educated and sober-minded man, to speak
like an illiterate enthusiast; nor to discourse on the
most important matters in the tone of familiar con-
versation respecting the triſling occurrences of the
day. Any one who does but notice the style in which
a man of ability, and of good choice of words, and
utterance, delivers his sentiments in private, when he
is, for instance, earnestly and seriously admonishing a
friend,-defending the doctrines of Religion,-- or
speaking on any other grave subject on which he is
intent, may easily observe how different his tone is
from that of light and familiar conversation,-how
far from deficient in the decent seriousness which
befits the case : even a stranger to the language might
guess that he was not engaged in any frivolous topic :
and when an opportunity occurs of observing how he
delivers a written discourse, of his own composition,
on perhaps the very same, or a similar subject, one
may generally perceive how comparatively stiff, lan-
guid, and unimpressive is the effect. It may be said,
indeed, that a sermon should not be preached before
a congregation assembled in a place of worship, in
the same style as one would employ in conversing
across a table, with equal seriousness, on the same
subject : this is undoubtedly true : and it is evident
that it has been implied in what has here been said; the
Natural manner having been described as accommo-
dated, not only to the subject but to the place, occasion,
and all other circumstances : so that he who should
preach exactly as if he were speaking 'in private,
though with the utmost earnestness, on the same
subject, would so far be departing from the genuine
Natural manner; but it may be safely asserted, that
even this would be by far the less fault of the two. He
who appears unmindful, indeed, of the place and
occasion, but deeply impressed with the subject, and
utterly forgetful of himself, would produce a much
stronger effect than one, who, going into the opposite
extreme, is, indeed, mindful of the place and the
occasion, but not fully occupied with the subject,
(though he may strive to appear so ;) being partly
engaged in thoughts respecting his own voice. The
latter would, indeed, be less likely to incur censure ;
but the other would produce the deeper impression.
The object, however, to be aimed at, (and it is not
unattainable,) is to avoid both faults ;-to keep the
mind impressed both with the matter spoken, and
with all the circumstances also of each case, so that
the voice may spontaneously accommodate itself to
all ; carefully avoiding all studied modulations, and,
It must be admitted, indeed, that the different
kinds of Natural Delivery of any one, on different
subjects and occasions, various as they are, do yet
bear a much greater resemblance to each other, than
any of them does to the Artificial style usually em-
ployed in reading: a proof of which is, that a person
familiarly acquainted with the Speaker, will seldom
fail to recognise his voice, amidst all the variations
of it, when he is speaking naturally and earnestly;
though it will often happen that, if he have never
before heard him read, he will be at a loss, when
he happens accidentally to hear without seeing him,
to know who it is that is reading ; so widely does
the artificial cadence and intonation differ in many
instances from the natural. And a consequence of
this is, that the Natural manner, however perfect, -
however exactly accommodated to the subject, place,
and occasion, will, even when these are the most
solemn, in some degree remind the hearers of the tone
of conversation : amidst all the differences that will
exist, this one point of resemblance, that of the
delivery being unforced and unstudied, will be likely,
in some degree, to strike them. Those who are good
judges will perceive at once, and the rest, after being
a little accustomed to the Natural manner, that there
is not necessarily any thing irreverent or indecorous
in it ; but that, on the contrary, it conveys the idea
of the speaker's being deeply impressed with that
which is his proper business. But, for a time, many
will be disposed to find fault with such a kind of
elocution. But even while this disadvantage con-
tinues, a preacher of this kind may be assured that
the doctrine he delivers is much more forcibly im-
pressed, even on those who censure his style of
delivering it, than it could be in the other way. A
discourse delivered in this style has been known to
elicit the remark, from one of the lower orders, who
had never been accustomed to any thing of the kind,
that “ it was an excellent sermon, and it was great
pity it had not been preached :” a censure which ought
to have been very satisfactory to the preacher : had
he employed a pompous spout, or modulated whine,
it is probable such an auditor would have admired his
preaching, but would have known and thought little
or nothing about the matter of what was taught.
Which of the two objects ought to be preferred by
a Christian minister, on Christian principles, is a
question not hard to decide, but foreign to the pre-
sent discussion : it is important, however, to remark,
that an orator is bound, as such, not merely on
moral, but, if such an expression may be used, on
rhetorical principles, to be mainly, and indeed exclu-
sively, intent on carrying his point ; not, on gaining
approbation, or even avoiding censure, except, with
a view to that point. He should, as it were,
adopt as a motto, the reply of Themistocles to the
Spartan commander, Eury biades, who lifted his staff
to chastise the earnestness with which his own
opinion was controverted : “Strike, but hear me.”
Besides the inconvenience just mentioned,—the
censure to which the proposed style of elocution will
be liable from perhaps the majority of hearers, till
they shall have become somewhat accustomed to it,-
this circumstance also ought to be mentioned, among
what many, perhaps, would reckon, (or at least feel.)
H. H. E. T. O. R. I. C. 299
Rhetoric, as the disadvantages of it; that, after all, even when
\-- no disapprobation is incurred, no praise will be be-
into the causes of that remarkable phenomenon, as Chºp. V.
it may justly be accounted, that a person who is able S-V-'
stowed, (except by observant critics,) on a truly
Natural Delivery : on the contrary, the more perfect
it is, the more will it withdraw, from itself, to the
arguments and sentiments delivered, the attention of
all but those who are studiously directing their view
to the mode of utterance, with a design to criticize
or to learn. The credit, on the contrary, of having a
very fine elocution, is to be obtained at the expense
of a very moderate share of pains ; though at the
expense also, inevitably, of much of the force of what
is said.
One inconvenience, which will at first be expe-
rienced by a person who, after having been long
accustomed to the Artificial Delivery, begins to adopt
the Natural, is, that he will be likely suddenly to feel
an embarrassed, bashful, and, as it is frequently called,
nervous sensation, to which he had before been com-
paratively a stranger. He will find himself in a new
situation,-standing before his audience in a different
character—stripped, as it were, of the sheltering veil
of a conventional and Artificial Delivery 5–in short,
delivering to them his thoughts, as one man speaking
to other men ; not, as before, merely reading in
public. And he will feel that he attracts a much
greater share of their attention, not only by the
novelty of a manner to which most congregations are
little accustomed, but also, (even supposing them to
have been accustomed to extemporary discourses,)
from their perceiving themselves to be personally
addressed, and feeling that he is not merely reciting
Something before them, but saying it to them. The
speaker and the hearers will thus be brought into a
new, and closer relation to each other ; and the
increased interest thus excited in the audience, will
cause the Speaker to feel himself in a different situ-
ation,--in one which is a greater trial of his confidence,
and which renders it more difficult than before to
withdraw his attention from himself. It is hardly
necessary to observe that this very change of feelings
experienced by the speaker, ought to convince him
the more, if the causes of it (to which we have just
alluded,) be attentively considered, how much greater
impression this manner is likely to produce. As he
will be likely to feel much of the bashfulness which
a really extemporary speaker has to struggle against,
so, he may produce much of a similar effect,
After all, however, the effect will never be com-
pletely the same. A composition delivered from
writing, and one actually extemporaneous, will always
produce feelings, both in the hearer and the speaker,
considerably different; even on the supposition of
their being word for word the same, and delivered so
exactly in the same tone, that by the ear alone no
difference could be detected : still the audience will
be differently affected, according to their knowledge
that the words uttered, are, or are not, written down
and before the speaker's eyes : and the consciousness
of this, will produce a corresponding effect on the
mind of the speaker. For were this not so, any one
who, on any subject, can speak (as many can,)
fluently and correctly in private conversation, would
find no greater difficulty in saying the same things
before a large congregation, than in reading to them
a written discourse.
And here it may be worth while briefly to inquire
with facility to express his sentiments in private to a
friend, in such language, and in such a manner, as
would be perfectly suitable to a certain andience, yet
finds it extremely difficult to address to that audience
the very same words, in the same manner; and is, in
many instances, either completely struck dumb, or
greatly embarrassed, when he attempts it.* It can-
not be from any superior deference which he thinks
it right to feel for their judgment; for it will often
happen that the single friend, to whom he is able to.
speak fluently, shall be one whose good opinion he
more values, and to whose wisdom he is more dis-
posed to look up, than that of all the others together.
The speaker may even feel that he himself has a
decided and acknowledged superiority over every one
of the audience ; and that he should not be the least
abashed in addressing any two or three of them,
separately ; yet still all of them, collectively, will often
inspire him with a kind of dread.
Closely allied in its causes with the phenomenon we
are considering, is, that other curious fact, that the
very same sentiments expressed in the same manner,
will often have a far more powerful effect on a large
audience than they would have, on any one or two
of these very persons, separately. That is in a great
degree true of all men, which was said of the Athe-
nians, that they were like sheep, of which a flock is
more easily driven than a single one.
Another remarkable circumstance, connected with
the foregoing, is the difference in respect of the style
which is suitable, respectively, in addressing a mul-
titude, and two or three even of the same persons.
A much bolder, as well as less accurate, kind of
language is both allowable and advisable, in speaking
to a considerable number; as Aristotle has remarked,t
in speaking of the Graphic and Agonistic styles, the
former suited to the closet, the latter to public speak-
ing before a large assembly. And he ingeniously
compares them to the different styles of painting ;
the greater the crowd, he says, the more distant is the
view; so that in scene-painting, for instance, coarser
and bolder touches are required, and the nice finish,
which would delight a close spectator, would be lost.
He does not, however, account for the phenomena in
question.
The solution of them will be found by attention
to a very curious and complex play of sympathies
which takes place in a large assembly; and, (within
certain limits,) the more, in proportion to its num-
bers. First, it is to be observed that we are disposed
to sympathize with any emotion which we believe to
exist in the mind of any one present ; and hence, if
we are at the same time otherwise disposed to feel
that emotion, such disposition is in consequence
heightened. In the next place, we not only ourselves
feel this tendency, but we are sensible that others do
the same ; and thus, we sympathize not only with the
other emotions of the rest, but also, with their sym-
pathy towards us. Any emotion accordingly which we
* Most persons are so familiar with the fact, as hardly to
have ever considered that it requires explanation : but attentive
consideration shows it to be a very curious, as well as important
OJ162.
+ Rhetoric, book iii.
2 R 2
300
R H E T O R. I. C.
passage which, in the closet, might just at the first Chap. Iv.
glance tend to excite awe, compassion, indignation, \-N-"
Rhetoric feel, is still further heightened by the knowledge that
STN-" there are others present who not only feel the same,
but feel it the more strongly in consequence of their
sympathy with ourselves. Lastly, we are sensible that
those around us sympathize not only with ourselves,
but with each other also ; and as we enter into this
heightened feeling of theirs likewise, the stimulus to
our own minds is thereby still further increased.
The case of the Ludicrous affords the most obvious
illustration of these principles, from the circumstance
that the effects produced are so open and palpable.
If any thing of this nature occurs, a man is disposed,
by the character of the thing itself, to laugh: but much
more, if any one else is known to be present whom
he thinks likely to be diverted with it; even though
that other should not know of the presence of the
first ; but much more still, if he does know it; be-
cause his companion is then aware that sympathy
with his own emotion heightens that of the other:
and most of all will the disposition to laugh be in-
creased, if many are present, because each is then
aware that they all sympathize with each other, as
well as with himself. It is hardly necessary to men-
tion the exact correspondence of the fact with the
above explanation. So important, in this case, is the
operation of the causes here noticed, that hardly and
one ever laughs when he is quite alone : or if he does,
he will find on consideration, that it is from a conception
of the presence of some companion whom he thinks
likely to have been amused, had he been present, and
to whom he thinks of describing, or repeating, what
had diverted himself. Indeed, in other cases, as well
as the one just instanced, almost every one is aware
of the infectious nature of any emotion excited in a
large assembly. It may be compared to the increase
of sound by a number of echoes, or of light, by a
number of mirrors; or to the blaze of a heap of fire-
brands, each of which would have speedily gone out, if
kindled separately, but which, when thrown together,
help to kindle each other.
The application of what has been said to the case
before us, is sufficiently obvious. The speaker who
is addressing a large assembly, knows that each of
them sympathizes both with his own anxiety to acquit
himself well, and also with the same feeling in the minds
of the rest. He knows also, that every slip he may
be guilty of, that may tend to excite ridicule, pity,
disgust, &c. makes the stronger impression on each
of the hearers, from their mutual sympathy, and their
consciousness of it. This augments his anxiety.
Next, he knows that each hearer, putting himself,
mentally, in the speaker's place,” sympathizes with
this augmented anxiety; which is by this thought
increased still further. And if he becomes at all
embarrassed, the knowledge that there are so many to
sympathize, not only with that embarrassment, but
also with each other's feelings, on the perception of
it, heightens the speaker's confusion to the utmost.
The same causes will account for a skilful orator's
being able to rouse so much more easily, and more
powerfully, the passions of a multitude : they in-
flame each other by mutual sympathy, and mutual
consciousness of it. And hence it is that a bolder
kind of language is suitable to such an audience: a
* Hence it is that shy persons are, as is matter of common
remark, the more distressed by this infirmity when in company
with those who are subject to the same,
or any other such emotion, but which would, on a
moment's cool reflection, appear extravagant, may be
very suitable for the Agonistic Style; because before that
moment's reflection could take place in each hearer's
mind, he would be aware that every one around him
sympathized in that first emotion; which would thus
become so much heightened as to preclude, in a great
degree,' the ingress of any counteracting sentiment.
If one could suppose such a case as that of a
speaker, (himself aware of the circumstances,) ad-
dressing a multitude, each of whom believed himself
to be the sole hearer, it is probable that little or no
embarrassment would be felt, and a much more sober,
calm, and finished style of language would be adopted.
The impossibility of bringing the delivery of a
written composition completely to a level with real
extemporary speaking, (though, as has been said, it
may approach indefinitely near to such an effect,)
is explained on the same principle. Besides that the
audience are more sure that the thoughts they hear
expressed, are the genuine emanation of the speaker's
mind at the moment, their attention and interest are
the more excited by their sympathy with one whom
they perceive to be carried forward solely by his own
unaided and unremitted efforts, without having any book
to refer to : they view him as a swimmer supported
by his own constant exertions; and in every such
case, if the feat be well accomplished, the surmounting
of the difficulty affords great gratification ; especially
to those who are conscious that they could not do the
Same. And one proof, that part of the pleasure con-
veyed does arise from this source, is, that as the spec-
tators of an exhibition of supposed unusual skill in
Swimming, would instantly withdraw most of their
interest and admiration, if they perceived that the
performer was supported by corks, or the like; so
would the feelings alter of the hearers of a supposed
extemporaneous discourse, as soon as they should
perceive, or even suspect, that the orator had it writ-
ten down before him. -
The way in which the respective inconveniences of
both kinds of discourses may best be avoided, is
evident from what has been already said. Let both
the extemporary Speaker, and the Reader of his own
composition, study to avoid, as far as possible, all
thoughts of self, earnestly fixing the mind on the mat-
ter of what is delivered ; and the one will feel the less
of that embarrassment which arises from the thought
of what opinion the hearers will form of him ; while
the other will appear to be speaking, because he ac-
tually will be speaking, the sentiments, not indeed
which at that time first arise in his own mind, but,
which are then really present to, and occupy his
mind. :
One of the consequences of the adoption of the
mode of elocution here recommended, is that he who
endeavours to employ it will find a growing reluct-
ance to the delivery, as his own, of any but his own
compositions. Doctrines, indeed, and arguments he
will freely borrow; but he will be led to compose
his own discourses, from finding that he cannot
deliver those of another to his own satisfaction, with-
out laboriously studying them, as an actor does his
part, so as to make them, in some measure, his own.
And with this view, he will generally find it advisable
R H E T O R. I. C. 301
One important practical maxim resulting from the Chap. IV.
#hetoric, to introduce many alterations in the expression, not
views here taken, is the decided condemnation of all S-V-'
S-N-2 with any thought of improving the style, absolutely,
but only with a view to his own delivery. And in-
deed, even his own former compositions, he will be
led to alter, almost as much, in point of expression, in
order to accommodate them to the Natural manner
of delivery.” Much that would please in the closet,-
much of the Graphic style described by Aristotle, will
be laid aside for the Agonistic ;—for a style somewhat
more blunt and homely,–more simple and, ap-
parently, unstudied in its structure, and, at the same
time, more daringly energetic. And if again he is
desirous of fitting his discourses for the press, he
will find it expedient to reverse this process, and
alter the style afresh. A mere sermon-reader, on the
contrary, will avoid this inconvenience, and this
labour ; he will be able to preach another's dis-
courses nearly as well as his own ; and may send his
own to the press, without the necessity of any great
preparation : but to these advantages he will sacri-
fice more than half the force which might have
been given to the sentiments uttered. And he will
have no right to complain that his discourses,
though replete perhaps with good sense, learning,
and eloquence, are received with languid apathy,
or that many are seduced from their attendance on
his teaching, by the vapid rant of an illiterate fana-
tic. Much of these evils must, indeed, be expected,
after all, to remain : but he does not give himself
a fair chance for diminishing them, unless he does
justice to his own arguments, instructions, and ex-
hortations, by speaking them, in the only effectual
way, to the hearts of his hearers, that is, as uttered
naturally from his own.t
* In many instances accordingly, the perusal of a manuscript
sermon, would afford, from the observation of its style, a toler-
ably good ground of conjecture as to the author's customary
elocution.
† The principles here laid down may help to explain a re-
markable fact, which is usually attributed to other than the true
causes. The powerful effects often produced by some fanatical
preachers, not superior in pious and sincere zeal, and inferior in
learning, in good sense, and in taste, to men who are listened
to with comparative apathy, are frequently considered as proofs
of superior eloquence; though an eloquence tarnished by bar-
barism, and extravagant mannerism. But may not such effects
result, not from any superior powers in the preacher, but merely
from the intrinsic beauty and sublimity, and the measureless
importance of the subject 2 Why then, it may be replied, does
Ilot the other preacher, whose subject is the same, produce the
same effect 2 The answer is, because he is but half-attended to.
The ordinary measured cadence of reading, is not only in itself
dull, but is what men are familiarly accustomed to : Religion
itself also, is a subject so familiar, in a certain sense, (fami-
liar, that is, to the ear,) as to be trite, even to those who know
and think little about it. Let but the attention be thoroughly
roused, and intently fixed on such a stupendous subject, and
that subject itself will produce the most overpowering emotion.
And not only unaffected earnestness of manner, but, perhaps,
even still more, any uncouth oddity, and even ridiculous extra-
vagance, will, by the stimulus of novelty, have the effect of thus
rousing the hearers from their ordinary lethargy. So that a
preacher of little or no real eloquence, will sometimes, on such
a subject, produce the effects of the greatest eloquence, by
merely forcing the hearers (often, even by the excessively glar-
ing faults of his style and delivery,) to attend, to a subject
which no one can really attend to unmoved. *
It will not of course be supposed that our intention is to
recommend the adoption of extravagant rant. The good effects
which it undoubtedly does sometimes produce, incidentally, in
some, is more than counterbalanced by the mischievous conse
quences to others. •.
recitation of speeches by school-boys ; a practice so
much approved and recommended by many, with a
view to preparing youths for public Speaking in after-
life. It is to be condemned, however, (supposing the
foregoing principle correct,) not as useless merely,
but absolutely pernicious, with a view to that object.
The justness, indeed, of this opinion will, doubtless, be
disputed ; but its consistency with the plan we have
been recommending, is almost too obvious to be in-
sisted on. In any one who should think a Natural
Delivery desirable, it would be an obvious absurdity
to think of attaining it by practising that which is the
most completely artificial. If there is, as is evident,
much difficulty to be surmounted, even by one who
is delivering, on a serious occasion, bis own compo-
sition, before he can completely succeed in abstracting
his mind from all thoughts of his own voice,—of the
judgment of the audience on his performance, &c.
and in fixing it on the Matter, Occasion, and Place, —
on every circumstance which ought to give the cha-
racter to his elocution,--how much must this difficulty
be enhanced, when neither the sentiments he is to
utter, nor the character he is to assume, are his own, or
even supposed to be so, or in anywise connected with
him :—when neither the place, the occasion, nor the
audience, which are actually present, have any thing to
do with the substance of what is said. It is therefore
almost inevitable, that he will studiously form to him-
self an Artificial manner; * which, especially if he
succeeds in it, will probably cling to him through
life, even when he is delivering his own compositions
on real occasions. The very best that can be ex-
pected, is, that he should become an accomplished
actor, possessing the plastic power of putting him-
self, in imagination, so completely into the situation.
of him whom he personates, and of adopting, for the
moment, so perfectly, all the sentiments and views of
that character, as to express himself exactly as such
a person would have done, in the supposed situation.
Few are likely to attain such perfection; but he who
shall have succeeded in accomplishing this, will have
taken a most circuitous rout to his proposed object,
if that object be, not to qualify himself for the stage,
but to deliver in public, on real and important occa-
sions, his own sentiments. He will have been care-
fully learning to assume, what, when the real occasion
occurs, need not be assumed, but only expressed.
Nothing surely can be more preposterous than la-
bouring to acquire the art of pretending to be, what
he is not, and, to feel, what he does not feel, in order
that he may be enabled, on a real emergency, to
pretend to be and to feel just what the occasion itself
requires and suggests.f
* Some have used the expression of “a conscious manner,” to
denote that which results, either in conversation,-in the ordi-
mary actions of life, or in public Speaking, from the anxious
attention which some persons feel to the opinion which the com-
pany may form of them ;—a consciousness of being watched and
scrutinized in every word and gesture, together with an extreme
anxiety for approbation, and dread of censure. -
ºf The Barmecide, in the Arabian Nights, who amused himself
by setting down his guest to an imaginary feast, and trying his
skill in imitating, at an empty table, the actions of eating and
drinking, did not propose this as an advisable mode of instruct-
ing him how to perform those actions in reality.
302
R H E T O R. I. C.
fulness; but it will be more speedily and more eom- Chap. IV.
pletely subdued : the very system pursued, since it S-N-2
Rhetoric,
\-y-'
Let all studied recitation therefore, every kind of
speaking which, from its nature, must necessarily be
artificial,—be carefully avoided, by one whose object
is to attain the only truly impressive, the Natural
Delivery.* l
The last circumstances to be noticed among the
results of the mode of delivery recommended,
is, that the speaker will find it much easier, in
this Natural manner, to make himself heard; he
will be heard, that is, much more distinctly,–at a
greater distance,—and with far less exertion and
fatigue to himself. This is the more necessary to be
mentioned, because it is a common, if not a prevail-
ing opinion, that the reverse of this is the fact. There
are not a few who assign as a reason for their adop-
tion of a certain unnatural tone and measured cadence,
that it is necessary, in order to be heard by a large
congregation. But though such an artificial voice
and utterance will often appear to produce a louder
sound, (which is the circumstance that probably de-
ceives such persons,) yet a natural voice and delivery,
provided it be clear, though it be less laboured, and
may even seem low to those who are near at hand,
will be distinctly heard at a much greater distance.
The only decisive proof of this must be sought in
experience; which will not fail to convince of the
truth of the assertion, any one who will fairly make
the trial.
The requisite degree of loudness will be best ob-
tained, conformably with the principles here incul-
cated, not by thinking about the voice, but by looking
at the most distant of the hearers, and addressing one's
self especially to him. The voice rises spontaneously,
when we are speaking to a person who is not very
Ilear
And that the organs of voice are much less strained
and fatigued by the Natural action which takes place
in real speaking, than by any other, (besides that it is,
what might be expected, a priori,) is evident from daily
experience. An extemporary Speaker will usually be
much less exhausted in two hours, than an elaborate
reciter, (though less distinctly heard,) will be, in one.
Even the ordinary tone of reading aloud is so much
more fatiguing than that of conversation, that feeble
patients are frequently unable to continue it for a
quarter of an hour without great exhaustion ; even
though they may feel no inconvenience from talking,
with few or no pauses, and in no lower voice, for more
than double that time. -
He then who shall determine to aim at the Natural
manner, though he will have to contend with con-
siderable difficulties and discouragements, will not be
without corresponding advantages in the course he is
pursuing. He will be at first, indeed, repressed to
a greater degree than another, by emotions of bash-
* It should be observed, that, the censure here pronounced on
school-recitations, and all exercises of the like nature, re-
lates, exclusively, to the effect produced on the style of Elocution.
With any other objects that may be proposed, the present argu-
ment has, obviously, no concern. Nor can it be doubted that a
familiarity with the purest forms of the Latin and Greek lan-
guages, may be greatly promoted by committing to memory,
and studying, not only to understand, but to recite with pro-
priety, the best orations and plays in those languages. But
let no one seek to attain a natural, simple, and forcible Elocu-
tion, by a practice which, the more he applies to it, will carry
him still the farther from the object he aims at.
forbids all thoughts of self, striking at the root of
the evil. He will, indeed, on the outset, incur censure,
not only critical but moral ;-he will be blamed for
using a colloquial delivery; and the censure will very
likely be, as far as relates to his earliest efforts, not
wholly undeserved ; his manner will probably at first
too much resemble that of conversation, though of
serious and earnest conversation : but by perseverance
he may be sure of avoiding deserved, and of mitigat-
ing, and ultimately overcoming, undeserved, censure.
He will, indeed, never be praised for a very fine de-
livery ; but his matter will not lose the approbation
it may deserve; as he will be the more sure of being
heard and attended to. He will not, indeed, meet
with many who can be regarded as models of the
Natural manner; and those he does meet with, he
will be precluded, by the nature of the system, from
‘minutely imitating ; but he will have the advantage
of carrying within him an INFALLIBLE GUIDE, as long
as he is careful to follow the suggestions of nature,
abstaining from all thoughts respecting his own
utterance, and fixing his mind intensely on the bu-
siness he is engaged in. And though he must not
expect to attain perfection at once, he may be assured
that, while he steadily adheres to this plan, he is in
the right road to it; instead of becoming, as on the
other plan, more and more artificial, the longer he
studies: and every advance he makes will produce
a proportional effect: it will give him more and more
of that hold on the attention, the understanding, and
the feelings, of the audience, which no studied mo-
dulation can ever attain. And though others may be
more successful in escaping censure, and insuring
admiration, he will far more surpass them, in respect
of the proper object of the Orator, which is, to carry
his point.
Much need not be said on the subject of Action,
which is at present so little approved, or, designedly,
employed, in this country, that it is hardly to be
reckoned as any part of the Orator's art.
Action, however, seems to be natural to man, when
speaking earnestly : but the state of the case at pre-
sent seems to be, that the disgust excited, on the one
hand, by awkward and ungraceful motions, and, on
the other, by studied gesticulations, has led to the
general disuse of Action altogether; and has induced
men to form the habit (for it certainly is a formed
habit,) of keeping themselves quite still, or nearly so,
when speaking. This is supposed to be, and perhaps
is, the more rational and dignified way of speaking :
but so strong is the tendency to indicate strong inter-
nal emotion by some kind of outward gesture, that
those who do not encourage or allow themselves in
any, frequently fall unconsciously into some awkward
trick of swinging the body,” folding a paper, twist-
ing a string, or the like. But when any one is reading,
or even speaking, in the Artificial manner, there is
* Of one of the ancient Roman Orators it was satirically
remarked (on account of his having this habit,) that he must
have learned to speak in a boat. Of some other Orators, whose
favourite action is rising on tiptoe, it would perhaps have
been said, that they had been accustomed to address their audi-
ence over a high wall. -
R H E T O R. I. C. 303
is no reason that he should study to repress this Chap. IV.
tendency. \ =\/~
Rhetoric, little or nothing of this tendency; precisely, because
~~' the mind is not occupied by that strong internal emo-
tion which occasions it. And the prevalence of this
manner may reasonably be conjectured to have led to
the disuse of all gesticulation, even in extemporary
speakers; because if any one, whose delivery is arti-
ficial, does use action, it will of course be, like his
voice, studied and artificial; and savouring still more
of disgusting affectation, from the circumstance that
it evidently might be entirely omitted.” And hence,
the practice came to be generally disapproved, and
exploded. -
It need only be observed, that in conformity with
the principles maintained throughout this Chapter, no
care should, in any case, be taken to use graceful or
appropriate action ; which, if not perfectly unstudied,
will always be, (as has been just remarked,) intoler-
able. But if any one spontaneously falls into any
gestures that are unbecoming, care should then be
taken to break the habit ; and that, not only in pub-
lic speaking, but on all occasions. The case, indeed,
is the same with utterance: if any one has, in com-
mon discourse, an indistinct, hesitating, dialectic, or
otherwise faulty, delivery, his Natural manner cer-
tainly is not what he should adopt in public speaking;
but he should endeavour, by care, to remedy the
defect, not in public speaking only, but in ordinary
conversation also.
titudes and gestures. It is in these points, principally,
if not exclusively, that the remarks of an intelligent
friend will be beneficial. -
If, again, any one finds himself naturally and spon-
taneously led to use, in speaking, a moderate degree
of action, which he finds from the observation of
others, not to be ungraceful or inappropriate, there
*—
4;
Gratas inter mensas symphonia discors,
Et crassum unguentum, et Sardo cum melle papaver
Offendunt ; poterat duci quia coena sine istis.
Horace, Ars Poet.
And so also, with respect to at-
It would be inconsistent with the principle just
laid down, to deliver any precepts for gesture; be-
cause the observance of even the best conceivable
precepts, would, by destroying the natural appearance,
be fatal to their object: but there is a remark, which
is worthy of attention, from the illustration it affords
of the erroneousness, in detail, as well as in principle,
of the ordinary systems of instruction in this point.
Boys are generally taught to employ the prescribed
action either after, or during the utterance of the
words it is to enforce. The best and most appro-
priate action, must, from this circumstance alone,
necessarily appear a feeble affectation. It suggests
the idea of a person speaking to those who do not
fully understand the language, and striving by signs
to explain the meaning of what he has been saying.
The very same gesture, had it come at the proper,
that is, the natural, point of time, might perhaps have
added greatly to the effect; viz. had it preceded some-
what the utterance of the words. That is always the
natural order of action. An emotion,” struggling
for utterance, produces a tendency to a bodily ges-
ture, to express that emotion more quickly than words
can be framed ; the words follow, as soon as they can
be spoken. And this being always the case with a
real, earnest, unstudied speaker, this mode of placing
the action foremost, gives (if it be otherwise appro-
priate,) the appearance of earnest emotion actually
present in the mind. And the reverse of this natural
order would alone be sufficient to convert the action
of Demosthenes himself into unsuccessful and ridi-
culous mimicry.
* Format emin Natura priils nos intus ad omnem
Fortunarum habitum ; juvat, aut impellit ad iram :
Auf ad humwin maerore gravi deducit, et angit:
Post effert animi motus interprete linguá.
- Horace, Ars Poet.
G E O M E T R Y.
mitted to the son ; the son again with new acquisi- History.
Geometry. History of the Science. +
\- tº º tº tions, passed them down to his children ; each suc-S-N-'
poºl. The origin of Geometry, like that of the other an: ceeding generation added improvements to the obser-
origin of cient sciences, is involved in obscurity. Herodotus and vations and experience of that which preceded it; till
Geometry. Strabo inform us, that we owe the invention of it to the at length arose some superior genius, who collecting
annual overflowings of the Nile; which, inundating the
lands of Lower Egypt, and frequently carrying away
the marks and boundaries by which every man's par-
ticular property was assigned, rendered it necessary to
have some means of ascertaining the respective por-
tions of land belonging to each individual, after the
subsiding of the waters. In many cases also, the land
was swallowed up in the Nile itself, which by in-
creasing its boundaries in certain places, abstracted
every year some portion of land from cultivation ;
and, according to the former historian, Sesostris,
into one mass all the traditionary knowledge of his
predecessors, formed them from the efforts of his
own mind into a rude system ; this was afterwards
remodelled and improved by others; and thus by
degrees, Geometry, which had, originally, nothing
further in view than the mere division of property,
became an independent and highly important science.
And we think it very probable, that it had already
assumed this first form of a system when it was
employed by the Egyptians for the purposes that have
been stated in the leading part of this article.
who had divided the country amongst his people, at a
certain annual rent, in such cases sent proper persons
to measure and value the property thus lost, that a
corresponding reduction might be made in the yearly
At all events, it is in Egypt the first traces of the First traces
science are found ; and whence it was transplanted of Geome-
into Greece by the celebrated philosopher, Thales. º
tribute. It has been however very properly observed,
that supposing this to be the true state of the case,
yet it by no méans points out the origin of Geometry, it
rather shows that this science had already attained to
a certain state of maturity, and that it was merely
This distinguished sage was born about 640 years
before the Christian era, and being unable to gratify
his ardent desire for knowledge in his native country,
he travelled into Egypt at an advanced period of life,
where he conversed with the priests, who, in them-
selves, embodied all the learning of that country.
employed then, as it would be now, in similar cases.
At the same time it must be admitted, that the deriv-
ation of the word Geometry, which is from Yī, earth,
and uetpéu, measure, shows clearly that its principal
Diogenes Laertius relates, that Thales measured Thales, the
the height of the pyramids, or probably of the obelisks, first Gre-
by means of their shadows; and Plutarch says, that º:
the king Amasis was astonished at this instance of A. C. 640.
application in the early ages of the world, was the
measurement and the division of lands ; and there is
no doubt, whether Geometry had its origin in the cir-
cumstances alluded to or not, that they furnished a
motive for its cultivation, and gave rise to various
useful and important propositions. But with respect
to its first origin, we can scarcely conceive a state of
society, however rude, in which something like the
first principles of Geometry did not exist. As soon
as man began to relinquish his wandering and savage
life, and taste the pleasures of social intercourse ; as
soon as laws were framed to secure to each individual
the reward of his own industry and labour, the lands,
which had before yielded spontaneously all that he
required in his barbarous state, stood now in need of
cultivation, in order to render their productions sub-
servient to his more refined appetites, and to the neces-
sity of his family, or the little society over which
he presided ; this refinement necessarily gave rise
to the division of lands, and the partition of flocks
and herds, and this again, to comparison of quantity
and magnitude; which comparison on the one hand,
laid the foundation of Arithmetic, and on the other,
that of Geometry, and formed the first links in the
chain of propositions which now constitute these two
abstract sciences.
In the first instance, there can be little doubt that
the attempts were rude and frequently inaccurate, but
the science, even in this state, must be said to have
commenced; the observations of the father were trans-
sagacity in the Grecian philosopher. It would seem
therefore, by this account, that if Thales actually went
to Egypt as a student, he very soon surpassed his
masters, whose knowledge of the science of geometry
could be but little advanced, if this statement be cor-
rect. But whether this philosopher taught the Egyp-
tians, or the latter taught him the method of measuring
the heights of objects by their shadows, we see, at all
events, that he returned to his own country, furnished
at least with some elementary knowledge of geometry;
and that it was he who laid the foundation of that
science in Greece, and inspired his countrymen with
a taste for its study. Various discoveries are attributed
to Thales concerning the circle and the comparison
of triangles, and in particular he is mentioned as the
first who found that all angles in a semicircle are
right angles: this discovery is said to have excited in
his mind the most lively emotions, and foreseeing,
probably, the many important consequences to which
it might lead, he is said to have expressed his
gratitude to the muses by a sacrifice. He is also
stated to have first employed the circumference of the
circle for the measure of angles ; but this, from what
we have stated relative to Archimedes in our History
of ASTRoNoMY, seems to be incorrect.
The next Grecian geometer of importance was Pythagoras.
Pythagoras, who ilourished about 550 years before A. c. 550,
Christ, and who had been a pupil of Thales. Like
his master he travelled into Egypt, and afterwards
into India, and acquired from the priests of the former
G E O M E. T. R. Y.
305
Geometry country, and from the Brahmins in the latter, a great
\-v- stock of learning, both in geometry and in astronomy;
Hippo-
crates,
CEnopides,
&c.
he did not however immediately transplant this acqui-
sition of learned lore into his native country, but
opened his first school in Italy, which was afterwards
the most celebrated in antiquity. To this philosopher
we are indebted for the discovery of that remarkable
property in right angled triangles, which constitutes
the forty-seventh proposition in the first book of
Euclid's Elements of Geometry; namely, that the square
described upon the hypothenuse is equal to the sum
of the squares described upon the other two sides; a
proposition equally curious from the peculiarity of the
result, and important for the numerous applications it
finds in every branch of mathematical science. This
property of the sides of a right angled triangle gave
rise to investigations relative to the incommensura-
bility of certain lines, as for example the side of a
square and its diagonal ; and other properties, again
laid the foundation of that part of solid geometry
which relates to the five regular bodies. Pythagoras
is also said to have first demonstrated that of all plane
bodies, the circle is that which has the greatest area
under a given circumference.
From this time, at least, therefore geometry had
assumed the character of a regular science, and it was
cultivated with more or less success, from this date
to the destruction of the Alexandrian school, by all
the most learned of the Grecian philosophers; we
have indeed evident proof of the progress made in the
science by the Elements of Geometry of Euclid; a work
which has stood the test of so many ages without a
rival, or at least without an equal for the closeness of
its logical reasoning, and the accuracy of its demon-
strations.
Before this time, however, some geometers of note
had cultivated the science in Greece, of whom OEno-
pides, of Chios, Zenodorus, and Hippocrates, are the
most distinguished : to the two former we are said to
be indebted for some practical geometrical problems,
and to the latter, for the celebrated quadrature of the
lunes which still bear his name. Having described
on the three sides of an isoceles right angled triangle
as diameters, three semicircles, placed all in the same
direction, he observed, that the sum of the two
equal lunes comprised between the two quadrants
of the circumference on the hypothen use, and the
circumferences on the two equal sides, was equal in
area to the triangle, and therefore each equal to half
the triangle ; and this was the first instance in which
a curvilineal space had been shown to be equal to a
rectilineal area. Hippocrates also attempted the
quadrature of the circle, and seems to have deceived
himself, with the belief that he had effected it ; he
was more successful, however, in some other points,
and was the first to show that the duplication of the
cube required the finding of two mean proportionals
between two given lines. He wrote also Elements of
Geometry, much esteemed at that time, but they are
lost; and the only regret that can be entertained for
the circumstance is, that they would enable us to
understand what the state of that science then was.
The date of Hippocrates is generally stated at about
450 years before Christ. Aristotle also mentions two
other distinguished geometers of this period, viz.
Brison and Antiphon, but we have no records of their
particular discoveries.
WOL. I.
We come next to the school of Plato, founded about History.
390, A. c. This philosopher, as Thales and Pythagoras
had done before, travelled into Egypt, and having
acquired a great store of knowledge on various sub-
jects, and particularly on geometry, he returned to
Greece, and there established his school, over which
was placed the celebrated inscription, “Let no one
enter here who is ignorant of Geometry;” he, in
fact, considered this as the first of all human sciences,
and although we have no express work of his on the
subject, there is every reason to believe that he was
very profound in his geometrical knowledge. We
have already mentioned the problem of the duplica-
tion of the cube, which about this time engaged so
much attention, and which Hippocrates had, as we
have seen, reduced to the finding of two geometrical
means between the side of the given cube, and another
line double of the same. Plato took up the problem
at this point, and having in vain attempted to solve it
geometrically, (viz. by the help of the ruler and com-
passes only,) he invented a method of solution by
two rulers ; but being a mechanical construction it
could not be admitted as a geometrical solution, which
indeed we now know to be impossible. The most
important discovery, however, attributed to Plato was
that of the geometrical analysis, to which we may
also add, as very little inferior, the invention of what
is now termed geometrical loci ; but there is perhaps
some doubt to what extent Plato himself advanced
these doctrines, they, doubtless, both had their origin
in his school, as had also the conic sections, but
whether any of these were originally due to this phi-
losopher is uncertain, although it is very usual to
attribute the merit of the discoveries to him, particu-
larly of the first.
Geometry had now made so great a progress that
a new course of its elements became necessary, a task
which was undertaken by Leon, a scholar of Neoclis
or Neoclide, a philosopher, whº had studied under
Plato. To this author has been ascribed the inven-
tion of that part of the solution of a problem called
its determination; that is to say, the part which
points out the limits of possibility, or impossibility.
Eudoxus, who was also one of the most celebrated
friends of Plato, generalized many theorems, and thereby
contributed greatly to the advancement of the science.
To him has indeed been attributed the invention of
\-N-7
Plato.
A. C. 390
Leon,
Neoclis,
and Eudo-
A. c. 368.
the conic sections, which, at all events, he cultivated
with great success ; he has been also stated as the
author of the doctrine of proportions, given in the
fifth book of Euclid's Elements ; and it seems unques-
tionable that he was the first who discovered that a
cone or pyramid is equal to one-third of the prism of
equal base and altitude. Some other important geo-
metrical inventions and discoveries are attributed to
Eudoxus, amongst which is that of the theory of curved
lines generally. This distinguished geometer died in
the year 368, A. c.
The school of Plato was now divided into two,
which upon some points maintained different opinions,
but they both agreed in regarding the knowledge of
mathematics, as absolutely necessary to every one
who was desirous of studying philosophy. Thus the
geometrical thecries which had been so much cul-
tivated during the life-time of the celebrated founder
of this school, still continued to make great progress.
Amongst those who most contributed to the advance-
- 3 s
Division of
the Pla-
tonic
306 G E O M
E T R Y.
furnished with a brief sketch of this ingenious process. History.
Having seen, that if he inscribed in and circumscribed S-2-2
Geometry. ment of the science at this periou was Aristaeus, who
\-y—’ composed five books on the conic sections, and of
Euclid.
A. c. 280, about 280 years before Christ, and soon after the consequently, by computing the perimeter of the two
founding of the Alexandrian school. The place of his polygons, whatever may be the number of their sides,
birth is not certainly known, but it appears that he we shall be certain that the circumference of the
had studied at Athens previously to his settling at circle is comprised between these two limits. Archi-
Alexandria. Pappus, in the introduction to the medes first employed polygons of six sides ; then by
seventh book of his Collections, gives him an excel- bisecting each, he obtained two others of twelve, then
Hent moral character, gentle and modest towards all, of twenty-four, forty-eight, and lastly of ninety-six,
and particularly to those who cultivated the mathe- where he stopped ; the exterior and interior polygons
matical sciences. He composed treatises on various already approaching towards each, very nearly ; and
subjects, but he is best known by his Elements, a work here, by taking the mean of the two, he found that
on geometry and arithmetic, in thirteen books, which the diameter was to the circumference as, seven to
still exist; but of these, the first six, and the eleventh some number between twenty-one and twenty-two,
and twelfth, are those only which are now consulted, but much nearer to the latter; and in short, the
the other books on numbers being of no value in the approximation of seven to twenty-two, is near enough
present state of arithmetic ; but of the other eight, even in the present day, for most practical cases. The
it may be said, that notwithstanding the various most interesting part of this process, however, was
attempts that have been made, either to improve or that by which he made every successive approxima-
to surplant them, they have stood the test of more tion a step towards the next, and which considering
than 2000 years, and still maintain their preeminence the very defective state of the Greek numeral notation
in the schools and universities, not only in this coun- at this time, displays an effort of genius which has
try, but in every part of the world where the science certainly never been surpassed. The fluxional analysis
of geometry is cultivated, which is such an instance has enabled us now to approach towards the actual
of excellence and unvaried approbation as cannot be ratio much more nearly, but the results are more
paralleled in any other scientific treatise whatever. curious than useful : such is the present approxima-
Comment- The Elements of Euclid have had a great number tion, that we might with the necessary data state cor-
ariº.ºn of commentators, from the time of Theon, who was rectly to the nearest unit, the number of grains of
Euclid, the first, to the present day; after Theon, who flou- sand that would compose a sphere equal in diameter
rished about the middle of the fourth century, the to the orbit of Saturn; a refinement which no prac-
Elements of Euclid, as well as most of the other sci- tice can ever require.
entific works of the Greeks, passed first under the This, however, is only one of the numerous dis-
persecution, and afterwards under the patronage of coveries with which Archimedes enriched the Grecian
the Arabs, to whom we are mostly indebted for those geometry; he wrote also treatises On the Sphere and
that have been preserved. To an Arabic version of Cylinder, that is to say, on the ratio between these two .
this work, we owe our first Latin editions by Athelard, solids, when their diameters and altitudes were equal,
..in England, and by Campanus, in Italy, about the and on the relation of their surfaces. He was the
same time; that is, during the twelfth or thirteenth first to discover the elegant deduction, that the
century. The former remains only in manuscript in solidity of the sphere is to that of the cylinder as
some libraries, but the latter was made the foundation 2 to 3 ; and that their curvilinear surfaces are equal,
of some other Latin translations about the beginning or, which is the same thing, that the surface of the
of the sixteenth century, or rather at the latter end of sphere is equal to four of its great circles.
the fifteenth. The Greek text appeared for the first His treatise On Conoids and Spheroids relates to the
time at Basle, in 1533, edited by Simon Grynaeus; solids generated by the conic sections revolving about
and this has been made the foundation of various other their axes; those produced by the rotation of the
editions that have since appeared, particularly of the parabola and hyperbola, he called conoids ; and such
celebrated one of Commandine, in 1572, and again in as are generated by the revolution of the ellipse about
1619. It was this also that Gregory used in preparing either axis, are his spheroids. Here he compares the
the Oxford edition; and lastly, Simson's translation area of an ellipse with that of a circle; he also proves
in 1756, is also drawn principally from the same that the sections of conoids and spheroids are conic
- Source. . sections, and he treats of their tangent planes. He
Archi- We have now arrived at the period of our history proves, for the first time, that a parabolic conoid is
medes,..., which introduces us to the prince of Grecian mathe- equal to three times the half of a cone of the same
A. c. 359 maticians, Archimedes, who lived about 250 years base and altitude; and he also investigates the
which the ancients have spoken in the highest terms
of approbation, but which are unfortunately lost.
He composed likewise five books on solid loci, which
shared the fate of his conic sections; this philosopher
is said to have been the friend and preceptor of
Euclid. - - -
Euclid flourished under the first of the Ptolemies,
before Christ. He was the first who discovered an
approximate ratio between the diameter and the cir-
cumference of a circle, and which has been made the
foundation of the numerous modern approximations
which are not dependent on the doctrine of fluxions. It
may therefore be interesting to many of our readers to be
about a circle two regular polygons of the same num-
ber of sides, the circumference of the circle, which
will fall between their perimeters, will be greater than
the one, and less than the other; and by continually
augmenting the number of sides, the circle will at
length differ less from the actual perimeter of either,
by a quantity less than any that can be assigned;
ratio of any segment of a hyperbolic conoid, or of a
spheroid to a cone of the same base and altitude.
His reasoning is a model of accuracy; and it exhibits
the true spirit of the ancient synthetic method; it is,
however, exceedingly prolix and difficult, so much so,
indeed, that few will have patience to follow the steps
…” G E O M ET R Y.
307
three following have been only handed down to our History.
time through the medium of an Arabic version, made S-V-
Geometry, of the venerable mathematician, more especially as
S-N-' the same conclusion may be found with equal cer-
tainty by the modern analysis, at an infinitely less
expense of thought and labour. His work On Spirals
treats of a curve, which was the invention of his
friend Conon, who, it seems, had found its properties,
but he died before he had time to complete their de-
monstrations; these Archimedes has supplied; the
whole subject is, however, so much his own, that
what is properly the spiral of Conon, is usually
called the spiral of Arghimedes. He has also treated
Of the Equilibrium ºf Planes, or of their Centres of
Gravity, in two books; and next Of the Quadrature of
the Parabola. This is the first complete quadrature
of a curve that was ever found. He here shews that
the area of any segment of a parabola cut off by a
chord, is two-thirds of the circumscribing parallelo-
gram ; and this he proves by two different methods.
His Arenarius was written to evince the possibility
of expressing, by numbers, the grains of sand that
might fill the whole space of the universe. Here he
introduces a property of a geometrical progression,
that has since been made the foundation of the theory
of logarithms; but it would be going too far to sup-
pose that Archimedes had made any approach to that
noble invention. This tract is valuable, not on ac-
count of the subject on which he treats, but because
of the information it contains respecting the ancient
astronomy, and the application which it gives of the
Greek arithmetic. In addition to the works we have
enumerated, there is a treatise On Bodies which are
carried on a Fluid, in two books, and a book of
Lemmas, which is a collection of theorems and
problems, curious in themselves, and useful in the
geometrical analysis. These are all the writings of
Archimedes now extant, but many have been lost.
The works of Archimedes are the most precious
relict of ancient geometry; they shew to what an
extent such a genius as his could carry its method of
demonstration ; but they likewise prove, that there
were certain limits beyond which it became inappli-
cable, on account of the unwieldiness of the machinery.
In general, the progress of discovery is slow ; but
Archimedes took up the subject where men of ordi-
mary capacities were at a stand, and by the vigour of
his mind, anticipated the labour of ages: he was,
about the year 1250, A. D. and which was rendered
into Latin about the middle of the seventeenth cen-
tury. The eighth book is entirely lost, but attempts
have been made to supply it, by following out the
plans of the author as far as they could be ascer-
tained from the first seven. This task was first
undertaken by the celebrated Dr. Halley, who also
revised and corrected the translation that had been
before made of the leading part ; and in 1710 pub-
lished the splendid Oxford edition of this noble
monument of Grecian geometry. The first four
books of Apollonius treat of the generation of the
conic sections, and of their principal properties,
with reference to their axes, foci, and diameters The
greater part of these properties were, indeed, known
before the time of this author, and are merely given
as preliminaries to his general and extended view of
the subject. Before this time the right cone only had
been considered ; but Apollonius treats generally of
every cone having a circular base, and presented many
new theorems, or rendered those already known more
general. The following books contain a great number
of elegant and interesting propositions entirely new,
but which it would be inconsistent with our plan to
describe in detail. The most important of his other
works were : 1. On the Section of a Ratio; 2. On the
Sections of a Space; 3. On Determinate Sections ; 4. On
Tangencies ; 5. On Inclinations; and, 6. On Plane
Loci.
We must here pass over, with very brief notices,
the names of several other distinguished geometers
who lived about this time. We have already men-
tioned Eratosthenes and Nicomedes; the former was
most distinguished as a geometer for his construction
of the duplication of the cube, and for two books,
entitled, De Locis ad Medietates, and the latter for
the invention of the conchoid, a curve which still
carries his name; and for the application that he made
of it to the finding two mean proportionals between
two given lines or numbers.
Eratos-
thenes and
Nico–
medes,
Conon, Trasideus, Nicoteles, and Dositheus, were
also distinguished geometers about this period ; but
their labours have not been handed down to our
time.
We have now, unquestionably, passed the zenith of Decline of
Grecian science, we find, indeed, many authors, but Grecian
they added little, perhaps nothing, whatever to the *
undoubtedly, the Newton of antiquity.
Apollonius. This was the most brilliant epoch in the history of
A. c. 240. Grecian science ; such a philosopher as Archimedes
would alone have given a character and eclat to the
period when he flourished; but nearly at the same
time we meet with Eratosthenes, Apollonius, Nico-
medes, and some others, who are still admired for
the elegance, depth, and ingenuity of their geometrical
compositions; of these, however, Apollonius, un-
doubtedly stands next in fame to Archimedes.
This Great Geometer, as he was deservedly surnamed
by his contemporaries, flourished about 240 years before
the commencement of the Christian era. He composed
a great number of works upon the higher branches
of the science, most of which are unfortunately
lost, or only small fragments of them remain ;
but we have, at least, nearly entire, his treatise On
the Conte Sections, which is alone sufficient to justify
the high reputation that he has acquired. This treatise
was divided into eight books, of which the first four
have reached us in their original language ; but the
discoveries of Archimedes and Apollonius. We must,
however, except Theodosius, the author of an excel-
lent treatise On Spherics, in three books, which have
been preserved and justly admired ; and Menelaus of
Alexandria, who lived in the second century of the
Christian aera ; he was the author of a treatise On
Trigonometry, in six books; and another On Spherics,
in three books, which are still extant. He appears,
also, to have treated of the geometry of curved lines.
Ptolemy, also, the author of the Almagest, born in 70
A. D. must be considered, if not as an original genius, at
least as a valuable promoter of geometrical science;
his treatise On Optics, which is lost, is supposed to
have contained some beautiful specimens and appli-
cations of geometry.
The next two or three centuries are entirely barren
of any names, which in this brief sketch of the
History of Geometry require to be particularized.
9 S 2
308
G E O M E. T. R. Y.
Geometry. Science in general was, indeed, now fast declining,
S-N-' and the only names of distinction between this time
Pappus.
A, D, 380.
oblivion many analytical works
and the fall of Alexandria, which totally extinguished .
the faint light that still remained of Grecian learning,
are very few. Pappus, Theon, and his accomplished
daughter Hypatia, Diocles, and Procłus, are, perhaps,
the only names to which it will, in our case, be re-
quisite to call the attention of the reader.
Pappus flourished about the year 380, A. D., and was
the author of a work which, although it does not
possess so much originality as some we have referred
to, is still extremely curious and interesting. We
allude to his Mathematical Collections, in eight books,
of which, however, the first and half of the second
are lost. He seems to have intended to collect, into
one body, several scattered discoveries, and to illus-
trate and complete, in many places, the writings of the
most celebrated mathematicians, in particular, those of
Apollonius, Archimedes, Euclid, and Theodosius; for
this purpose he has given a multitude of lemmas,
and curious theorems, which they had supposed
known ; and he has also described the different at-
tempts which had been made to resolve the most
difficult problems, as the duplication of the cube, and
the trisection of an angle. The preface to his seventh
book is highly valuable ; having preserved from
on geometry, of
which we should otherwise have been entirely igno-
rant. The abridgement which he has given of these
is all that remains of the greater number ; yet it has
served to give a continuity to the History of Geome-
try, and to inspire modern mathematicians with a
high opinion of the theories of the ancients. In fact,
such of their geometrical writings as have descended
to our times, are merely elementary; their more
recondite works have either been entirely lost, or are
only known by the account which Pappus has given
of them. The books that remain of this author, have
suffered much from the injuries of time ; there are
many inaccuracies, and some passages so mutilated
as to be hardly intelligible. The original Greek,
except some extracts, has never been published. The
only translation that has been given, which is by
Commandine, was published at Pesara in 1558, and
again, with little variation, in 1660, at Bologna.
Commandine appears to have had access to only one
manuscript, which wanted the first two books, and
which was, throughout, very faulty. There are, how-
ever, several manuscripts of Pappus in the libraries
of some public institutions. The University of Ox-
ford possesses two, one of which has half the second
book : this part, which, treats of arithmetic, was
published by Dr. Wallis in 1688; it is, therefore,
probable, that both these books treated on this sub-
ject. Amongst many other curious problems con-
tained in this work, Pappus has some perfectly original,
such as that of finding quadrable spaces on the sur-
by analysis, is far from elementary, and shews that History.
the author, with the means of investigation which he \--
possessed, must have been a very profound geome-
trician. This problem has since been generalised, it
having been shown that, if instead of the quadrant
making a complete revolution, it makes only a given
part of a revolution, while the moveable point descends
through it. The spherical space described between
the quadrant, the corresponding arc of the base, and
the spiral, is to the square of the radius, as the arc
of the base to a quarter of the circumference. We
shall have again to refer to this species of problems
in speaking of the geometry of the moderns.
We shall only further add respecting this work of
Pappus, that in the preface to the seventh book is
given a sufficiently distinct idea of that beautiful
theorem, commonly ascribed to the Pere Guldin, and
which English mathematicians commonly call the
centrobaryc problem ; viz, the solidity of any solid,
or the area of any surface described by the motion of
an area or line, is equal to the product of the area, or
length of the generatrix into the path of its centre
of gravity.
Theon is principally distinguished for his Com- Theon,
mentaries, or Scholia on Euclid, although, according Hypatia.
to the statements and corrections of Dr. Simson in
his translation, he rather darkened and bewildered
the subject, than elucidated it. Theon was the father
of the accomplished and unfortunate Hypatia, who
had so much distinguished herself by her cultiva-
tion of the mathematical sciences generally, that she
was deemed worthy to succeed her father in the
Alexandrian school, where she shone a distinguished
ornament to her sex and her country, till she fell a
sacrifice to the blind fury of a bigoted and fanatical
mob, about the beginning of the fifth century.
After Theon and his daughter we meet with only Proclus,
Proclus, who was Sporus, &c.
two or three names of any note.
the chief of the Platonists at Athens, signalized himself
by his Commentaries on Euclid ; and Diocles has been
principally remembered as the author of the cissoid,
a curve still named after him. Eutocius also attri-
butes to him the solution of a problem concerning
the division of the sphere ; Sporus and Philo also
lived about this period; the former gave a solution to
the problem of finding two mean proportionals, and
the latter extended the approximation of the ratio
between the diameter and the circumference of the
circle to the ten thousandths part, or to four places
of decimals, the diameter being unity. Some other
names might also be mentioned, but they possess
little interest, and we must now consider the light of
Grecian science as about to be extinguished. What
little remained up to this period, the commencement
of the seventh century, had long taken refuge in the
museum of Alexandria, where, destitute of support
and encouragement, they could not fail to degenerate.
Still, however, they preserved, at least by tradition or Destruc-
imitation, that strict and correct character bestowed tion of the
upon them by the early Greeks; but before the date Alexandri-
above mentioned, a tremendous political and religious * ".
storm arose which threatened their total destruction. * * *
Filled with all the enthusiasm a militant religion is cal-
culated to inspire, the successors of Mohammed ra-
vaged that vast extent of country which stretches
from the east to the southern confines of Europe.
All the cultivators of the arts and sciences, who
faces of a sphere. He demonstrates, by means of the
theorems of Archimedes, that if a moveable point,
proceeding from the vertex of a hemisphere, passes
over a quarter of the circumference, while this quadrant
makes an entire revolution about the vertical axis of
the hemisphere, the space included between the cir-
cumference of the base and the spiral of double cur-
wature, described on the hemisphere by the moving
point, is equal to the square of the diameter. Such
a proposition as this, even with all the aid afforded
G. E. O. M. E. T. R. Y.
309
Geometry, from every part had taken refuge in Alexandria, were earlier date than the Greeks, and whether the first History.
\—V-2 driven away with ignominy, or fell by the swords of knowledge which the latter nation obtained was not
their conquerors: the former fled into remote coun- of Hindoo or Chinese origin. Opinions on this sub-
tries, to drag out the remainder of their lives in ject are much divided. The researches of the learned
poverty and distress. The places, and the instruments have brought to light tables in India which must have
which had been so useful in making observations on been constructed by geometry; but the period at
astronomy, which was then scarcely distinguishable which they were formed, although unquestional, a
from geometry, were involved with the records in one very early one, has not been completely ascertained.
common ruin. The whole of the valuable library, The Hindoos have a treatise called the Suryd Sid- Geometry
which contained the works of so many eminent phi- h'anta, which they profess to be a revelation from ºf the ºn-
losophers and geometers, and which was the common heaven to Maya, a man of great sanctity, about four ..."
depository of every species of learning which does million years ago; but notwithstanding the extrava-
honour to the human mind, was devoted to the flames gance of this fable, there seems no question that it is
by the Arabs; the Caliph Omar observing, “ that if of a very remote date ; and although interwoven with
they agreed with the Koran, they were useless, and if many absurdities, it contains a rational system of
they did not they ought to be destroyed,” a sentiment trigonometry, which differs entirely from that first
worthy of such a leader and of the cause in which he known in Greece and Arabia. It is, in fact, founded
was engaged. This event happened in the year 640 on theorems not known in Europe before the time of
of the Christian era. Victa, not more than two centuries back ; and it em-
The Arabs It has been said that a few were enabled to escape ploys the sines of arcs, and not the chords of the
promote the blind fury of Omar and his followers by flight, double arcs, which was the practice of the Greeks.
the science, and of course these carried with them some remnant It is, therefore, questionable, whether the introduction
of that general learning for which this school had
been so celebrated ; but still, destitute of books and
instruments, and probably of the means of subsistence
without manual labour, very little of that great mass
of learning could have been preserved, and still less
accumulated, had not the Arabians themselves, within
less than two centuries of this fatal conflagration,
become the admirers and supporters of those very
sciences they had before, in their bigoted fury, so
nearly annihilated. Fortunately for geometry and for
the sciences in general, these men now studied the
works of the Greeks with the greatest assiduity, and
if they added little to the general stock of knowledge
which they found contained in the few manuscripts
which escaped from the general wreck, they became
at least sufficiently masters of many of the subjects
to comment upon them, and to set a due estimation
lupon these valuable relics of ancient science. It is
by this means so many of them have been preserved,
and that we are enabled to bestow our admiration on
the transcendant talents and genius of Archimedes,
Apollonius, and the other distinguished Greeks, whose
names we have recorded. It is, however, principally
for the preservation of the Greek authors that we
are indebted to the Arabs, and not for any important
improvements or discovery in geometry; for if we
except the simplification they gave to trigonometry,
we owe to them very little, and even this is by some
of the sines into trigonometry, which is generally
considered as an Arabic invention, may not have been,
as well as their numerals, of Indian origin. The
Chinese also, according to their romantic historians,
were very early promoters of geometry and astronomy;
but whatever may be the antiquity of these sciences
amongst them, their extent has been very limited,
and they have been long perfectly sterile in their
hands. -
Before we enter upon the geometry of modern Geometry
Europe, it may be proper to allude slightly to the
state of geometry amongst the Romans.
like people were at no time distinguished by their
knowledge in what have been termed the exact
sciences ; they studied astronomy, but not so much
for the love of the science itself, as for its supposed
relation with astrology, and their desire to pry into
the secrets of futurity. With such ideas geometry
was not likely to be much extended in their hands,
and, in fact, the only authors of any note amongst
them, were Boetius the senator and consul, and Wi-
truvius; which latter has displayed considerable know-
ledge of geometry, particularly in the ninth book of
his architecture; and he seems to have had some ge-
neral knowledge of most other mathematical sub-
jects. A few other names might be mentioned, but
they would answer no purpose but needlessly to
lengthen this historical sketch.
* I s Ill&
This war-
We are arrived now at what have been properly State of
termed the dark ages ; for from the fatal catastrophe geometry
which extinguished the last faint glimmerings of during the
supposed to have been derived by them from India
with the numeral figures which we now employ in
arithmetic; one perhaps of the most useful discoveries
dark ages.
that was ever made, and that to which the mathema-
tical sciences are more indebted than to any other
whatever. It would be useless to quote here the
names of the several Arabs who have translated, or
ordered the translation, of the different Greek authors
to whom we have referred, and still less so, those of
the Persians and Turks ; because in these two coun-
tries nothing appears to have been attended to but
the most elementary parts; we shall therefore pass to
a slight mention of the geometry of the Hindoos and
Chinese, not that they, any more than the Persians
and Turks, have pursued this science to any great
length, but because it is a question whether they did
not possess their knowledge on the subject at an
Grecian science in the middle of the seventh century, we
pass over a space of nearly six hundred years without
meeting with any discovery to arrest our attention
for a moment, except those we have already spoken
of as due to the Arabs ; we might, indeed, mention
the venerable Beda, 700 A.D. and Roger Bacon, 1240
A. D. as individuals who, during this long period, dis-
played some knowledge of the sciences; but we owe
to them no discoveries. During the thirteenth century,
indeed, we meet with several names of some note; in
fact, the sun of science, which had been so long set,
was now gradually advancing towards the horizon of
Europe, and the twilight had already commenced of
that brilliant day which now illuminates so great a
310
G E O M ET R Y.
Geometry, portion of the globe. Amongst the mathematicians
\-N-2 of this time, may be mentioned John de Sacro-Bosco,
Geometers or John of Halifax, who wrote a treatise On the Sphere,
tºº º: and Campanus of Navarre, who translated Euclid,
and composed a treatise On the quadrature of the
tury.
y Circle ; Albertus Magnus wrote also on geometry
during this century.
Of the four- The fourteenth century is still further distinguished
teenth cen- by its geometers, and particularly in England; amongst
tury. whom we may mention Wallingfort and the poet
Chaucer; but it is only in the fifteenth century that
geometry shone forth with that splendour which was
indicative of the sublime discoveries that were to fol-
Of the fif- low. The principal promoters during this century were
teenth cen-Purbach and Muller, or Regiomontanus, Lucus de
tury. Burgo; and the celebrated Copernicus, although he
never wrote on this subject, was a learned geometri-
cian. Purbach's first essay was to amend the Latin
translation of Ptolemy's Almagest : he wrote a tract
which he entitled, An Introduction to Arithmetic ; a
treatise On Gnomonics and Dialing ; he corrected by the
Greek text the ancient version of Archimedes made
by Gerrard of Cremona ; he translated the Comics of
Apollonius; the Cylinders of Serenus; and gave a Latin
version of the Spherics of Theodosius and Menelaus.
He commented on certain books of Archimedes, which
Eutocius had passed over; refuted a pretended quadra-
ture of the circle by Cardinal Cusa ; besides various
important labours connected with astronomy, which
was, indeed, his favourite science ; one of the most
useful of which was his rejection of the ancient
sexagesimal division of the radius, instead of which he
divided it, or supposed it divided, into 600,000 parts.
Regionontanus, who out-lived his friend and preceptor
Purbach, made a still further improvement in this
case, by carrying the division to 100,000, and calcu-
lating new tables for every degree and minute of the
quadrant.
Lucus de Burgo revived Campanus's translation of
Euclid, which, however, was only published in 1509.
His work, Summa de Arithmetica, Geometria, &c. 1494,
contains a treatise On Geometry. The progress which
had now been made in the Greek tongue, and the
invention of printing, contributed greatly to the dis-
semination of geometrical knowledge. The Greek
mathematicians began to be known in Europe, and
Euclid was printed for the first time at Venice in
1482, in a folio volume, by Erhard Ratdolt, one of
the first printers of that age.
Of the six- About the beginning the sixteenth century several
teenth cen- of the Greek authors were translated and published,
tury. as the Spherics of . Theodosius, and such books of
Apollonius as were then known ; but the translators,
although good Greek scholars, had but little know-
ledge of geometry, so that these translations were
in many respects defective ; at length Commandine,
about the middle of the century, who possessed both
the requisite qualifications, undertook a similar task.
He translated into Latin, and published in 1558, a part
of the works of Archimedes, with a commentary. He
published, also, a translation of the first four books
of Apollonius's Conics, with the Commentary of Eu-
tocius, and the Lemmas of Pappus. His Latin
translation of Euclid appeared in 1572. We owe to
him also a treatise On Geodisia, or the division of
figures, the work of an Arabian geometer. But his
last and most important labour was his translation of
the Mathematical Collections of Pappus, the only one . History.
that has yet appeared, and it is probable that but for
the mathematical zeal of the author, this interesting
work, so highly curious and valuable, might still have
been nearly unknown to modern geometers.
John Dee, a singular and eccentric English writer,
wrote some mathematical works about this time,
many of them connected with astrology and alchemy,
and some on geometry. In 1570 he published a
Preface Mathematical to the English Euclid by Henry
Billingsley, “ which,” says Dr. Hutton, “ is certainly
a very curious and elaborate composition ;” and the
same year Divers and many Annotations and Inventions
dispersed and added after the tenth Book of the English
Euclid. During this century, Maurolycus published
some works which were much esteemed at that time ;
and it was also in the same century that Tartaglia,
who had translated Euclid into Italian, discovered
the method of solving cubic equations, which were
clandestinely published by Cardan, and still bear his
name. He also translated a part of Archimedes, and
demonstrated the rule for finding the area of a tri-
angle when the three sides are given ; but the rule
itself was discovered by Hero the younger, some
centuries before. We might, if our limits admitted
of it, particularize the works of a number of other
ingenious mathematicians of this period, but we
can only name a few of the most distinguished; as
. Clavius, whose translation and commentary on Euclid
is still esteemed ; Metius, a mathematician of the
Low Countries, the author of a very convenient ap-
proximation to the ratio between the diameter and
circumference of a circle, viz. 113 to 355. This was
soon after extended by Romanus to seventeen places
of decimals. Nonius distinguished himself by the inven-
tion of a method of reading angles to a great degree of
accuracy, something resembling what we still, some-
times, improperly attribute to him, but which is more
properly called a vernier, or vernier scale. Wright,
an English mathematician, was the author of the
chart which we always improperly attribute to
Mercator. But, perhaps, the man of most original
genius, who wrote on mathematical subjects during
this age, was Vieta, who flourished in France just Vieta, born
before the commencement of the seventeenth century; 1540.
his writings abound with marks of great originality’
and the finest genius ; and his inventions and im-
provements in all parts of mathematics, were very
considerable. He was, to a certain degree, the in-
ventor and introducer of literal algebra ; that is, in
which letters are used instead of numbers, as well
as of many beautiful theorems in that science. He
made also very considerable improvements in geo-
metry and trigonometry; his Angular Sections is a
very ingenious and masterly performance; by these
he was enabled to resolve the problem of Adrianus
Romanus, proposed to all mathematicians, amounting
to an equation of the 45th degree. His Apollonius
Galus, being a restoration of Apollonius's tract On
Tangencies; and many other geometrical pieces to be
found in his works, show the truest and finest taste
for geometrical investigations. He gave some mas-
terly tracts on trigonometry, both plane and spherical,
which may be found in the collection of his works
published at Leyden in 1646, by Schooten; besides
another larger and separate volume in folio, published
in the author's life-time at Paris in 1579; containing
G E O M E T R Y. 311
Geometry, extensive trigonometrical tables, with the construction conic sections about an axis, and his investigations. History.
*—V- and use of the same ; these are particularly described were limited to such bodies; but Kepler treated of S-v-'
in the introduction to Dr. Hutton's Logarithms. To solids generated by the rotation of these curves about
this complete treatise on trigonometry, plane and
spherical, are subjoined several miscellaneous problems
and observations; such as on the quadrature of the
circle, the duplication of the cube, &c. Computa-
tions are here given of the ratio of the diameter of
the circle to its circumference, and of the length of
the sine of one minute, both to a great many places
of figures.
Geometry The seventeenth century gave birth to many
of the illustrious geometers; but it was now found that
seventeenth analysis was a much more powerful and expeditious
*y instrument, and many who commenced their mathe-
matical career as geometers, were turned from their
pursuit to follow the new analysis, which had its
origin about this period; our business is, however,
only with the geometrical writings of these authors.
One of the earliest geometers of this century was
Lucas Valerius, an Italian; he distinguished himself
by his determination of the situation of the centre
of gravity in conoids, spheroids, and their segments.
Marinus Ghetaldus was well acquainted with the
ancient geometry, and, guided by the indications of
Pappus, attempted a restoration of the lost book
of Apollonius On Inclinations ; he also wrote a sup-
plement to the Apollonius Galus of Vieta. Lodolph
Van Ceulen distinguished himself by his laborious
approximation to the circumference of a circle,
when the diameter is unity, stating it to be
3.14159,2,6535,897.93,23846,26433,83279,50238, or
rather that this number is in defect ; but that with
the last number increased by unity, it is in excess,
the true ratio lying between these two numbers.
Willebrod Snellius was another Dutch mathema-
tician of this period; at an early age he undertook
to restore the work of Apollonius on determinate
sections, which was published under the title of Apal-
lonius Batavus. He published also a work, Cyclometria,
where he treated of the approximation between the
diameter and circumference, and displayed in it some
ingenuity and dexterity in his numerical operations.
Albert Girard, also a Fleming, possessed great
originality and genius. He first gave a rule for find-
ing the area of a spherical triangle, or of a polygon
bounded by great circles on a sphere; he also offered
some general theorems for measuring and comparing
solid angles, and endeavoured to restore the porisms
of Euclid.
Kepler, Hitherto no new principle had been introduced into
born 1571. geometrical investigations ; the models laid down by
the Greek mathematicians were considered as stand-
ards of perfection, and no one had yet been bold
enough to break the charm, till the celebrated Kepler,
in his Nova Stereometria, ventured on this dangerous
ground, and first introduced considerations of infinity
into geometry: according to these new views a circle
was conceived to be composed of an infinite number
of indefinitely small triangles, having their vertex at
the centre, and their bases at the circumference;
cones, in like manner, were supposed to consist of an
infinite number of small pyramids, &c. By this inge-
nious way of treating his subject, Kepler was enabled
to go far beyond Archimedes with infinitely less
labour. The latter conceived all the bodies that he
had treated of, as formed by the rotation of different
any line whatever in their planes, and thus gave, as it
were, to the problems of Archimedes, an almost inde-
finite extent; and what is of more importance, he thus
laid the foundation of the modern doctrine of infini-
tesimals.
The next important innovation in the method of Cavallerius,
handling geometrical subjects, was made by Cavalle- born 1998.
rius in his work, Geometria Indivisibilibus, published
in 1635. Here a line is conceived to be made up
of an infinite number of points; a surface of an infi-
nite number of lines ; and a solid, as composed
of an infinite number of surfaces, which elements of
magnitude he called indivisibles. So bold an innova-
tion was not likely to be received with universal
approbation by men who had devotcºl themselves to
the study of the ancients, and who knew no other
standard for forming their taste and judgment; in fact,
this work met with great opposition, and led to vari-
ous controversies. In answer to some of the objec-
tions that had been urged, Cavallerius maintained that
the hypothesis he had advanced, was by no means an
essential part of his theory, which, in fact, was the
same as the ancient method of exhaustions, but free
from its tedious and indirect mode of reasoning. To
effect their purposes, the ancients were under the
necessity of inscribing and circumscribing polygons
about circles, and polyhedra in the same way about
spheres; and although with great ingenuity, it was also
with great labour that they arrived at their conclusion.
Cavallerius advanced more directly to his object.
He considered, as we have stated, surfaces as com-
posed of an infinite number of lines, and solids as
made up of an infinite number of planes; and the prin-
ciple he assumed was, that the ratio of these infinite
sums of lines or planes, as compared with the unit of
numeration, in each case, was the same as that of the
surfaces or solids of which they were the measure.
This work of Cavallerius is divided into seven books;
in the first six the author applies his new theory to
the quadrature of the conic sections, and the solidity
of their solids of revolutions, and to other questions
of a similar nature relative to spirals ; the seventh is
employed in demonstrating the same things by prin-
ciples independent of indivisibles, and establishing by
the agreement of the results, the exactitude of the
new method.
The French geometers, during this time, were no Fermat,
less intent upon improving and extending geometry, born 1665.
The dates of the letters of Fermat, published in the
Commerce Epistolaire, of this author, shew that his
investigations preceded the year 1636, and therefore
that his discoveries were independent of those of the
Italian geometer. Archimedes had measured the area
of the common parabola, and found the solidity of the
conoid produced by the rotation of the plane about its
axis. Fermat, by a new method, solved both these
problems with great facility, and determined moreover
the situation of the centre of gravity of the paraboloid
as well as that of the solid generated by the parabola
revolving about its base; and what was still more
difficult, he found the quadrature of parabolas of all
orders, and the value of their solids of revolution,
made about either an absciss or an ordinate ; he ascer-
tained likewise the centres of gravity of these solids,
312
G E O M E. T. R. Y.
Geometry. and solved a number of other problems which marked
S-N-Z him as a most profound geometer
Roberval,
born 1602.
Descartes,
:born 1596.
Roberval was also a geometer of high reputation,
although inferior to Fermat ; and he solved as soon as
the problems were proposed to him by the latter, all
the cases of the parabolas above mentioned. He em-
ployed considerations similar to those of Cavallerius,
but under more guarded language; that is, he assumed
surfaces to be made up with other surfaces of little
breadth, and solids as composed of a number of inde-
finitely thin prisms, instead of calling them lines and
sections, as Cavallerius had done. On these principles
he solved a number of very difficult and curious pro-
blems in a work, entitled Traité des Indivisibles, which
was not printed till after his death in 1693. Geom,etry
is also indebted to Roberval for several curious inves-
tigations relative to the cycloid, and particularly for his
method of tangents, which was an exceedingly near
approach to the principles of fluxions, and will be
more particularly noticed in our History of ANALYSIs.
The celebrated Descartes was a contemporary with
Fermat and Roberval, and much rivalry and, unfor-
tunately, much of envy and petty jealousies subsisted
between these great masters. They proposed to each
other difficult problems, and both the question and
answer were frequently couched in, or accompanied
with language, which we are sorry to see employed
between men whose talents it is impossible not to
respect ; this spirit however was very common at this
period, and was cherished till the time of the Bernoullis,
between whom even the fraternal relation, in which
these great geometricians stood towards each other,
was forgotten in their eharacters of scientific rivals.
Descartes was unquestionably a man of distinguished
talents; and as the author of a system of philosophy,
which found able defenders for many years, he will
always stand conspicuous in the annals of mathe-
matical science ; but in geometry, more is certainly
attributed to him than is justly his due. He is, for
example, always cited as the first who invented the
application of algebra to geometry, which is not
strictly the case. He certainly considerably extended
the nature of this application, but the foundation had
been already laid by Vieta, and practised to a certain
extent by others. It was Descartes, however, who
first solved, in general terms, the problem that had
been proposed by the ancient geometers; tıamely,
having any number of right lines given in position on
a plane, to find a point, from which we may viraw as
many other right lines, one to each of the given lines,
making with them given angles, and under the fol-
lowing conditions, viz. that the product of the two
lines thus drawn shall have a given ratio, with the
square of the third, if there be only three, or with the
product of the two others, if there be four; or if there
be five, that the product of the three shall have the
given ratio with the product of the two lines remain-
ing, and a third given line, &c. &c. Descartes was
the author also of several other highly interesting
geometrical problems which led the way to the esta-
blishment of the new analysis, and will therefore be
more appropriately treated of in the history of that
science. His work, containing the investigations
alluded to above, was published in 1637.
Our limits will only admit of noticing in very
concise terms the distinguished geometers who suc-
ceeded those last mentioned, in fact we are now
.*
nearly arrived at that period when the entire current History.
of mathematical science took a new direction ; every -v-
discovery in geometry is now leading us nearer to the
invention of the new analysis, and they are so blended
with it, that it is almost impossible to notice the one
without referring also to the other ; we shall there-
fore, in this place, confine our observations within a
very limited space, referring the reader who is desir-
ous of examining the progress of geometry at this
time, to the History of Analysis to which we have
already referred in the preceding page. The names
which intervene between this time and the full deve-
lopement of the new analysis, by Newton and Leib-
nitz, were Gregory St. Vincent, a Flemish mathemati-
cian, whose object was the quadrature of the circle,
in which he thought he had succeeded; but, although
mistaken in this, he arrived at such a multitude of
curious and interesting properties and theorems, as
fully to recompense him for his laborious research.
Another name which will ever be highly esteemed guygº,
by every admirer of the exact sciences, occurs at this born 1629
period. Huygens was one of the brightest ornaments
of the seventeenth century; at a very early age he
published his Theoremata de Circuli et hyp. quad., and
he afterwards found the surfaces of conoids and sphe-
roids, a problem which had not been attempted before
his time. He determined the measure of the cissoid,
and showed how the problem of the rectification of
curves might be reduced to that of their quadratures.
It is also to him that we are indebted for the theory
of evolutes and involutes. His treatise De Horologia
Oscillatorio is a work of the highest merit, and con-
tains some of the most beautiful applications of geo-
metry to mechanics that had ever been made before
his time.
Dr. Barrow, an English mathematician, and the Dr.Barrow,
tutor of the illustrious Newton, was highly distin- born 1630.
guished at this period by his geometrical writings :
his Geometrical Lectures are composed partly in the
style of the ancient, and partly in that of the modern
geometry. To him we are indebted for another step
towards the new analysis.
For the rest it will be sufficient to state the names
of Tacquet, James Gregory, Borelli, Viviani, Simson,
Stewart, and Horsley, each of whom has distinguished
himself by his taste for geometrical pursuits, has
added some perfections, and rendered some service to
the science, but not such as to claim from us any par-
ticular notice in this brief sketch.
It only now remains for us to add a few remarks Descriptive
relative to a new species of geometry introduced into geometry
notice in France, by Monge, during the period of the
revolution, under the designation of descriptive geo-
metry. When any surface whatever penetrates ano-
ther, there most frequently results from their inter-
sections, curves of double curvature, the determina-
tion of which is necessary in many arts, as in groined
vault work, cutting arch-stones, wood-cutting, for
ornamental work, &c., the form of which is frequently
very singular and complicated : it is in the solution
of problems appertaining to these subjects that de-
scriptive geometry is especially useful.
Some architects, more versed in geometry than
persons of that profession commonly are, have long
ago thrown some light on the first principles of this
kind of geometry. There is, for example, a work by
a jesuit, named Courcier, who examined and showed.
G E O M E T R Y.
3.13
Geometry, how to describe the curves resulting from the mu-
-v2–’tual penetration of cylindrical, spherical, and coni-
Fig. l y
cal surfaces : this work was published at Paris in
1663. P. Deraud, Matheurin, Frezier, &c. had like-
wise contributed a little towards the promotion of
this branch of geometry. But Monge has given it
very great extension, not only by proposing and re-
solving various problems both curious and difficult,
but by the invention of several new and interesting
theorems. We can only mention in this place one or
two of the problems and theorems. Among the pro-
blems are the following ; first, Two right lines being
given in space, and which are neither parallel nor in
the same plane, to find in both of them the points of
their least distance, and the position of the line joining
these points; second, Three spheres being given in
space, to determine the position of the plane which
touches them. There are also some curious problems
relative to lines of double curvature, and to surfaces,
resulting from the application of a right line that leans
continually upon two or three lines given in posi-
tion in space. Among the theorems the following
may be mentioned ; if a plane surface given in space
be projected upon three planes, the one horizontal,
and the two others vertical, and perpendicular to each
other, the square of that surface will be equal to the
sum of the squares of the three surfaces of projection.
A few other works possessing some novelty in their
manner of treating the subject, may be also here
enumerated, as Développement de Géométrie, and Appli-
cations de Géométrie, by Baron Dupin, the celebrated
author of Travels in England ; the Polygonometrie, of
L'Huillier; the Géométrie du Compas, by Mascheroni, in
which no instrument but the compasses is employed ;
the Géométrie de Position, by Carnot; and Cresswell on
Geometrical Maarima et Minima.
As to the elementary works of the present day they
are very numerous ; those most approved of, however,
are the translation of Euclid's Elements, by Simson ;
and the Geometries of Ingram, Playfair, Bonnycastle,
and Leslie ; and amongst the French writers we may
mention the Treatises of Geometry, by La Croix, and
Le Gendre ; to which latter work we have been much
indebted, in compiling the following treatise, although
we have, in some instances, deviated widely from it.
BOOK I.
Properties of lines, angles, and triangles.
JDEFINITIONs.
1. GEOMETRY is that science which is applied to
the measure of extension. Extension is comprised
under three dimensions; namely, length, breadth,
and depth or thickness.
2. A line is length, without breadth or thickness.
The extremities of a line are called points. So that a
point has no dimensions, but position only.
3. A right or straight line is the nearest distance
between two points. In the following treatise when
the word line is used, a right line is to be understood.
4. Every line which is not a right line, or com-
posed of right lines, is a curve. Thus A B is a right
line, A D a compound or crooked line, and A E a curve,
or curved line, fig. 1.
5. A surface is that which has length and breadth,
without thickness.
WOL. I.
6. A plane is a surface in which any two points Book I.
being taken within it, the right line which joins those S-N-2
points will be every where in the surface.
7. Every surface, which is neither a plane nor com-
posed of planes, is a curved surface.
8. A solid is a body comprised under three dimen-
sions ; length, breadth, and thickness.
9. When two right lines, not in the same right line,
meet each other, they form an angle, which is greater
or less as the lines are more or less inclined or
opened. The point of their meeting is called the
summit, or angular point, and the two lines are its
sides, fig. 2. -
An angle may be designated by the single letter at
its summit, or by three letters; in which latter case
that letter which is at the summit or angular point,
is to be read in the middle. Thus the above angle
may be called the angle A, or B A C, or C A B-
Angles, like other quantities, may be added, sub .
tracted, multiplied, and divided.
10. When one line, as C D, meets another, as A B,
So that the angles on each side are equal to one another,
each of them is called a right angle; and the line C D is
said to be perpendicular to A B, fig. 3; and CD is said Fig. 3.
to be the perpendicular distance of the point C from the
line A. B.
11. An acute angle is less than a right angle, as
A B C ; and an obtuse angle is greater than a right
angle, as CB D, fig. 4. •
12. Parallel lines are those in which any point being
taken in the one, and any point being taken in the
other, the perpendicular distance of these points from
the other line shall be equal to each other, fig. 5.
The usual definition of parallel lines: that they
are those, “ which produced to any distance what-
ever will never meet,” is not sufficiently specific.
For in order to demonstrate the properties of those
lines, as given in the 29th Proposition of the Elements
of Euclid, or our 19th proposition, it is not sufficient
to know that parallel lines will never meet, but also
that they will never approach ; and it cannot be de-
monstrated in this part of Geometry, that two right
lines may not approach, although they never meet;
a condition which Euclid takes for granted in his
twelfth axiom. It is essential to the demonstration
of the above proposition, that it be first shown that
parallel lines do not approach towards each other,
and it is therefore necessary to demonstrate the
twelfth axiom by means of the previous proposition;
or to give a definition of parallel lines which will
comprehend their essential property of never approach-
ing towards each other, or of being every where at
the same perpendicular distance.
Simson, in his translation, has endeavoured, by
means of two other definitions, five propositions, and
corollaries, to demonstrate the twelfth axiom of
Euclid; and after all he has failed, because he has not
shown that two lines cannot approach without ulti-
mately intersecting. He has shown that they cannot
approach, and then recede again ; but he has taken
for granted, as Euclid himself has done, that if they
do approach they will meet if produced, which is the
very point in question. -
We have, therefore, preferred the definition above
given, and have made the property of paralled lines
never meeting, a proposition instead of a definition.
13. A plane figure is a plane terminated on all sides
- 2 T
314
G E O M E T R Y.
Definitions of other terms employed in Geometry.
Book I.
\-N-
Geometry, by lines. If the sides are right lines, it is called a
\–V-2 rectilineal figure ; it receives also particular denomina-
tion according to the number of its sides.
14. A rectilineal figure of three sides is a triangle,
fig. 6; of four sides a quadrilateral ; of five sides a
An ariom is a self evident truth, and which there-
fore requires no demonstration.
A proposition is any thing proposed to be done or
Fig. 6. d
- trated.
pentagon; but generally a figure of more than four º ". is a proposition proposed to be demon-
sides is called a polygon. d propos prop
15. A triangle, whose sides are all equal to each * oblem sition in which hing i
Fig. 7 other, is called an equilateral triangle, fig. 7 ; when prod º d proposition in which something is
only two of its sides are equal, it is an isosceles tri- Prº º a. ºrimina: roposition intended to
& , ſº IIIll
Fig. 8. i. fig. º and yº º: y are all unequal, it is render what folio: II) Ol'e ...
Fig. 9. called a scalene triangle, fig. 9. •2. & g g
Triangles also receive specific denominations from . A *******Tººnt truth drawn immediately
the nature of their angles. from a preceding proposition. tº
16. When a triangle A B C, has one of its angles, A º is a º: º: "...º.º.
as A, a right angle, it is called a right angled triangle, . 1OnS, sº i. i. P. º i. º ative COn-
and the side B C opposite the right angle is called , or gener y and application. . . tº
Fig. 10 the hypothen use, fig. 10. An hypothesis is a supposition advanced either in
* ii. When one of the angles, as B, fig. II, is obtuse, . *:::::: a proposition, or in the course of
* “ it is an obtuse angled triangle; and when all the e demonStration.
angles are acute, it is an acute angled triangle, g
Fig. 12. fig. 12. Illustration of the symbols to be employed.
Quadrilateral figures receive also particular deno- In order to render the demonstration as concise as
minations, as follow . . . o g = º possible, mathematicians have agreed in the adoption
17. A square is a quadrilateral, having all its sides of certain symbols, to signify particular terms of
Fig. 13. equal, and all its angles right angles, fig. 13. ... frequent recurrence, which, without in any degree
g 18. A rectangle has its opposite sides parallel, and its weakenin g the force of the argument, bring the
Fig. 14, angles right angles, fig. 14. . . © e whole subject more immediately under the eye of the
19. Every quadrilateral having its opposite sides reader: thus,
Fig. 15, parallel, is a parallelogram, fig. 15. º g The sign + signifies addition, and is read plus,
* 20. A parallelogram which has all its sides equal, so that A H B is read A plus B, and signifies that the
but its angles not right angles, is called a rhombus, quantity B is to be added to the quantity A.
Fig. 16. fig, 16. When only the opposite sides are equal, it The sign — signifies subtraction, and is read minus:
Fig. 17. is a rhomboid, fig. 17. * ſº . . thus A – B is read A minus B, and implies that the
21. A trapezoid is a quadrilateral, in which two quantity B is to be taken from the quantity A.
Fig. 18. only of the opposite sides are parallel, fig. 18. The sign × signifies multiplication, and is read
22. The diagonal of any rectilineal figure, is a right
line joining any two of its angles which are not ad-
jacent. In fig. 18, A C is the diagonal.
23. An equilateral polygon is one in which the sides
are all equal ; and an equiangular polygon is one
which has all its angles equal. -
24. Two polygons are said to be equilateral to each
other, when the sides of the one are equal to those of
the other, each to each, and are placed in the same
order; that is, so that in following the perimeters in
the same direction, the first side of the one is equal
to the first side of the other, the second side of the
one to the second side of the other, and so on ; and
in like manner polygons are said to be equiangular
when their angles are equal, each to each, taken also
in the same order.
In both the above cases the equal sides and angles
which are alike situated, are called homologous.
Regular polygons, whose number of sides do not
exceed twelve, receive specific denominations, as
follow : -
A polygon of three sides is called a triangle.
nultiplied by ; thus A × B, is read A multiplied by B,
and implies that the quantity A is to be multiplied by
the quantity B.
The parenthesis or vinculum, is used to reduce a
quantity compounded of several others into one only:
thus A + B – C is sometimes included in a paren-
thesis thus, (A + B – C); and in this form it may be
considered as a single quantity, and then (A + B – C)
× D, and (A + B) × (C + D), signify that the quan-
tity expressed by (A + B – C), is to be multiplied
by D ; and that the quantity (A + B) is to be multi-
plied by (C + D).
A number placed before any quantity as 3 B, or
5 (A–B), signifies that the quantity is to be multi-
plied by that number, or that it is such a multiple
of the quantity as is expressed by the number : thus,
the above signify three times B, and five times (A–B),
although in this case the sign of multiplication does
not appear. In the same way we express any part of
a quantity by prefixing to the quantity the fraction
expressing the part. As # A, 4 (A + B), &c. which
signify half A, one-third of (A + B), &c.
º º: e is e "2-4 E & e .. pºon. The square of any line A B, is denoted by AB = ,
six sides ..........a hexagon. the cube of a line by A B*, and so on.
seven sides . . . . . . . . a heptagon. The sign v signifies the square root of a quantity :
eight sides . . . . . . . . an octagon. thus w/2, v.A X B, &c. denote the square root of the
nine sides . . . . . . . . . . a nonagon. number 2, or of the product A x B, or which is the
ten sides . . . . . . . . . . a decagon. same, the mean proportional between A and B.
twelve sides. . . . . . . . a dodecagon. The sign = placed between any two quantities,
denotes that these quantities are equal to each other :
G E O M E T R Y. 315
right angles, then will CB, BD be in one and the Book I.
thus A = B and (A – B) = B x C, signify that A is -
same right line. ~~~
Geometry.
*— y equal to B, and that the difference A — B, is equal to
Fig. 19. - & A E D = two right angles, and C E B + D E B =
A B C together, are equal to two right angles. two right º nº the four angles C E A +
Let E B be perpendicular to CD, then the two AI, pºſſ (; ; ; + D E B = four right angles.
angles E BC and E B D are both right angles, (def. 10:) Cor. 2. Hence, also, the sum of all the angles that
and if A B coincide with BE, the two angles A B C, ca. . . about any given point, is equal to four
A B D, will also be both right angles; but if not, and right l
A B falls otherwise, as in the figure, then, since A BD gnt angles.
tº 2 Cor. 3. When one of the four angles formed by the
is equal to the sum of E B A and E B D, the two intersection of two right lines is a right angle, the
angles E B C and E B D, are equal to the three ABC, other three angles are also right angles
E B A and E B D ; but C B E is equal to the two gnL angles.
A B C and E BA, therefore the two angles C B A and l mammºgº
A B D are equal to the two E B C and E B D ; but PROPos ITION IV. Theorem.
these are both right angles, therefore C B A and ABD If two triangles have two sides of the one equal to two
are, together, equal to two right angles. sides of the other, each to each, and have the angles
Otherwise, by employing the conventional symbols. included by these sides also equal ; the iriangles will be
Let E B be perpendicular to CD, then E B C and equal, and have all the corresponding sides and angles
E B D are each right angles; consequently E B C + equal, fig. 22.
EBD = two right angles; and if AB coincide with E B, Let the two triangles A B C, D E F have the side Fig. 22.
then p A B C + A B D = two right angles. A C = D F, CB = EF, and the angle C = the
But if not, because angle F.; then will the side A B = D E, &c. as stated
E B C = E B A + A B C in the proposition.
we shall have For the triangle A B C may be conceived to be
E B D + E B A + A B C = E B C + E B D, applied to DEF, so that the point C falls upon F,
but E B D +- E B A = A B D ; and the side A C upon F D to which it is equal ; con-
therefore A BC + AB D = E B C + E B D, sequently the point A will coincide with the point D.
but E B C + E B D = two right angles; And, because the angle C = F, the side C B will fall
therefore A B C + A B D = two right angles. upon FE, and being equal to it, the point B will
Corollary. Hence, also, the sum of all the angles coincide with E, and therefore the side A B with
made by any number of lines meeting CD in B on D E (ax. 5;) thus the two triangles coinciding, will
the same side, is equal to two right angles. be equal to each other (ax. 6,) and have A B = D E,
PRoPosition II.—Theorem. A = D and B = E.
If two right lines meet the extremity of another right line, PROPoSITION V.—Theorem.
so as to make the adjacent angles equal to two right angles, g e
these two lines are . one and the * right ine, fig. 20. If two triangles have two angles of the one equal to
ſº - G two angles of the other, each to each, and the side adjacent
Fig. 20. Let the lines C B, BD meet the line A B at the
the product B × C.
The sign Z placed between two quantities, denotes
that the first of those quantities is less than the second:
thus A Z B, is read A is less than B; but when the
sign is inverted, as A A B, it signifies and is to be read
A is greater than B.
The above are all the conventional signs employed
in the following book; what further symbols of this
kind may be required as we proceed, will be explained
in their proper places.
Avioms.
1. Things which are equal to the same thing, are
equal to one another.
2. If equals be added to equals, the wholes are equal.
3. The whole is greater than its part.
4. The whole is equal to the sum of all its parts.
5. A right line may be drawn from any point to
another point, and there can be but one such right line
joining those two points.
6. Magnitudes, whether lines, surfaces, or solids,
which coincide or fill the same space, are equal.
7. All right angles are equal to each other.
PRoPosition I.-Theorem.
If one right line meet another right line, it makes the
two adjacent angles taken together equal to two right
angles, fig. 19.
Let the line A B meet C D, the two angles A B D,
point B, so as to make A B C + AB D equal to two
For if B D be not in the same right line with CB,
let B E* be in a right line with it; then by prop. 1, the
two angles A B C + AB E = two right angles ; but
by hypothesis A B C + A B D = two right angles ;
therefore A B C + AB E = A B C + AB D ; taking
away the common angle ABC we shall have A B E =
A BID; a part equal to the whole, which is impossible;
therefore B E is not in the same right line with CB;
and the same may be demonstrated of every line but
BD. Therefore BD is in the same right line with
C B.
PROPosLTIon III.—Theorem.
If two right lines cut each other, the vertical or opposite
angles are equal, fig. 21.
Let A B and C D cut each other in E, then will Fig. 21.
A E C = B E D and C E B = A E D. Because the
right line C E meets the right line A B, the two
angles,
AEC -- CEB = two right angles, (prop. 1,)
so also C E B + BED = two right angles;
therefore A E C + C E B = C E B + B E D ;
taking away the common angle C E B, there remains
the angle A E C equal to the angle B E D ; and in
the same way it may be shown that C E B is equal
to A. E. D.
Cor. I. The sum of the four angles formed about
the point E is equal to four right angles; for CEA +
* The line B E is omitted in the plate by the engraver.
2 T 2
316 G E O M E T R Y.
Geometry, to these angles also equal, the triangles will be equal, Produce A O to E.; then by the above proposition Book I.
S-N- and have the other corresponding sides and angles equal,
Fig. 23. That is, if A C = C B, then will A = B. Conceive the triangle A B C, or in the side B C, or within the
the angle C to be bisected by the line C D ; then in triangle A B C.
the two triangles A C D and C B D, there are two First let it fall without the triangle A B C, as in
sides A C, CD, equal to the two CB, CD, each to fig. 26, then (A I + B I) 7 A B \,,. S :
each, and the included angles equal ; consequently and (C I + I G) 7 G #} prop. y
the two triangles A CD and B C D are also equal, and therefore (A I + B I + C I + IG), or (A G + BC) 7
the angle A = the angle B, (prop. 4.) (A B + G C), but A G = AIB ; therefore B C 7 G C,
Cor. 1. The triangles A CD, D C B are equal, and or B C 7 E F.
the side A D = D B, and the angle CD A = CD B, If the point G fall in B C, as in fig. 27, it is obvi-
(prop. 4 :) hence the line which bisects the vertical ous that B C 7 G C, or greater than its equal E F.
angle of an isosceles triangle also bisects the base, and Lastly, if G fall within the triangle A B C, as in
is perpendicular to it, (def. 10.) fig. 28, (B A + B C) 7 (A G + G C) (prop. 9;)
Cor. 2. If the three sides of a triangle are equal to but B A = A G, therefore B C 7 G C, or greater than
each other, the three angles will also be equal to each its equal E. F.
other.
PR oposition VII.-Theorem. PRoPosition XI.-Theorem.
If a triangle have two of its angles equal, the sides The greater side of every triangle is opposite the greater
opposite to those angles will also be equal, fig. 45. angle; and the greater angle is opposite the greater side,
Fig. 45. That is, if the angle A = B, then will A C = BC. **
First let it be granted that a point may be found in Let the angle C B A of the triangle ABC be 7 A, Fig. 29.
BC, or B C produced, fig. 45, such that a line drawn then will AC 7 BC. Make the angle A B D = B.A. D.
from it to A shall be equal to its distance from B, and then will A D = BD, (prop. 7.) Now (B D + D C)
if C be not that point, let it be some other point as D; 7 BC, (prop. 8,) but (B D + DC) = (A D + D C) =
join D A, then because A D = D B; the angle D A B= A C ; therefore A C 7 BC.
D B A ; but D B A or C B A = C A B; therefore Next, let C A be greater than B C, then A B C 7
D A B = C A B, al, part to the whole, which is impos- B A C. For if it be not greater, it must be either
sible; and the same may be shown of every point in equal to it or less ; but it is not less, because then
E C except C : therefore C A = B C. B C 7 A C, (by the above,) which it is not, neither
Cor. Hence if the three angles of a triangle be equal can it be equal; because then A C = B C, (prop. 7,)
to each other, the three sides will also be equal. which it is not; being therefore neither equal nor less
it must be greater.
PRoposition VIII.—Theorem.
Any two sides of a triangle are greater than the third PRoPosLTION XII.-Theorem.
side, fig. 24. If the three sides of one triangle are equal to the throe
Fig. 24. Let A B C be a triangle, any two of its sides (A C+ sides of another triangle, each to each, the triangles will
CB) 7 A B. For A B being a right line, it is the be equal, fig. 30. -
shortest distance between the two points A and B, Let A B = DE, A C = D F, and B C = E F ; then
(def. 3;) therefore (AC -- CB) z AB; and the will A - D; for if A 7 D, then BC 7 E F, (prop.
same may be demonstrated of any other two sides. 10,) but it is not ; and if A Z D, then B C Z FE,
PROPos ITI on IX. — Theorem. but it is not ; therefore A being neither greater nor
If from a point within a triangle, there be drawn two less than D, it must be equal to it; and since A B,
igit lin pou * l tº º, & g’sº gº id - } A C are equal to DE, D F, each to each, and the
; ſº to the extremities of %. º *.*.*.* included angles being also equal, the triangles are
º º tºgether will be less than the sum of the other equal, and have all their corresponding angles also
wo sides of the triangle, fig. 25. equal, (prop. 4.)
Fig. 25 Let A B C be a triangle, and O a point taken within
fig. 22.
Let the angle A = D, B = E, and the side A B =
DE, then will the triangle A B C = DE F. For the
side A B may be applied to the side D E, so that A
falls on D, and B on E ; and since the angle A = D,
the side A C will fall upon D F and B C on E F, and
consequently the point C will fall upon F, and the
two triangles will coincide or fill the same space,
and will therefore be equal to each other, (ax. 6 ;)
that is, the side A C = D F, B C = EF, and the angle
C = F.
Proposition VI.—Theorem.
If two of the sides of a triangle are equal to each other,
the angles opposite those sides will also be equal to each
other, fig. 23.
it; join A O, OB, then will (AO + O B) Z (A C + .
CB).
(O E + E B) 7 OB; add to each A O, then (A E + S-N--
E B) 7 (AO + O.B), but (A C + C E) 7 A E. Much
more therefore is (AC + C E + E B) or (AC + C B)
7 (AO + O.B). -
PR oposition X.-Theorem.
If there be two triangles which have two sides of the
one equal to two sides of the other, each to each, but the
angle contained by the two sides of the one, greater than
the angle contained by the two sides of the other, that
which has the greater angle will have the greater base,
fig. 26, 27, 28. -
Let AB = DE, and A C = D F, but BAC 7 & 26.27,
EDF; then will BC 7 E. F. Make the angle C A G 28.
= F D E and A G = DE ; then will G C = E F,
(prop. 4.) Now the point G will either fall without
- PROPosition XIII.-Theorem.
If one side of a triangle be produced, the exterior angle
G E O M E. T. R. Y. ,
317
Geometry, will be greater than either of the interior and opposite
\-a-N- angles, fig. 31.
Fig. 31.
Fig. 32.
Let the side A B be produced to D,” the angle CBD
is greater than either of the angles A C B or C A B.
Conceive C B to be bisected in E ; join A E, and
produce it to F, making E F = E A, and join FB;
then the two sides C E, E A are equal to the two B E,
E F, each to each, and the angle A E C = F E B,
(prop. 3;) therefore the angle E B F = E CA, (prop.
4;) but C B D 7 E B F, therefore it is also greater
than E C A or B C A ; and in the same way if C B be
produced, and A B bisected, it may be shown that
AB G, or its equal CBD, is greater than C A B.
PR opos ITION XIV.--Theorem.
Any two angles of a triangle are together less than
two right angles, fig. 32.
That is, A + C, or A + B, or B + C, are together,
less than two right angles. Let A C be produced to
D, then by the last proposition, the angle C B A 4
IB C D ; to each add BC A, then (C B A + B C A) z
(B C D + B C A); but B C D + B C A = two right
angles, (prop. l ;) therefore C B A + B C A Z two
right angles; and the same may be shown of any
other two angles of the triangle A B C.
PRoposition XV.-Theorem.
Of all lines that can be drawn from a point to a line,
the perpendicular is the shortest, and of the others, that
which is nearer to the perpendicular is less than the one
more remote, and from the same point to the same line
there can be drawn but two lines equal to each other, one
on each side of the perpendicular, fig. 33.
Let IB be any right line, and C a point beyond it;
and let C D be perpendicular to IB ; let also CE,
C G be any other lines, then will C D be the shortest,
and EC less than C. G. Produce CD to H, making
D H = CD, and join E H, G H. Because C D E,
is a right angle, E D H is a right angle, (prop. 3,
cor. 3 ;) and in the triangles C E D, H E D, the two
sides E. D, CD are equal to the two E D, DH, each
to each, and the angle CD E = H DE ; therefore
E H = EC, (prop. 4,) and in the same manner it may
be shown that G H = G C. Now (EC + E H) 7
C H., (prop. 6;) and (C G + G H) 7 (E C + E H)
(prop. 9;) or since C D = D H, E C = E H, and C G
= G H ; 2 E C 7 2 CD, and 2 C G 7 2 E C ; con-
sequently E C 7 CD, and C G 7 E C ; but E C is
any line except the perpendicular, therefore the per-
pendicular is shorter than any other line drawn from
C to the line I B ; and of the rest, E C is less than
C G : C G than C I, and so on. Take FD = D E,
and join CF, then C F = CE, (prop. 4,) and it is
the only line that can be drawn from C to IB, that is
equal to CE. For any line falling between D and F,
will be less than C F or C E, (by the foregoing,) and
any line falling beyond F, will be greater than C F or
C E ; therefore C F is the only line that can be drawn
from C to the line I B, that is equal to C E ; that is,
there can be but two equal lines, one on each side of
the perpendicular.
* The line should have been produced on the side towards B,
instead of A as in the figure.
PRoposition XVI.-Theorem.
If two right angled triangles have their hypothenuses,
and one of their other sides equal, each to each, the tri-
angles will be equal, or have their other sides and angles
equal, fig. 34.
Book I.
\-N/-
Let the triangles A B C, D, E F be right angled at B Fig. 34.
and E, and have A C = D F, and C B = EF, then will
the triangles be equal. For apply ABC to DEF, so that
A B may fall on DE, and the point B upon E.; because
the angle B = E, the line B C will fall upon E F, and
because C B = EF, the point C will fall on F, and
the line C A upon F D. For if A C do not fall on DF,
let it fall in some other direction, and meet the base
DE, then will this line be equal to D F : therefore
we shall have two lines drawn from a point above
a line, to that line, on the same side of the per-
pendicular, equal to each other, which is impossi-
ble by the last proposition ; A C therefore cannot
but coincide with FD, and consequently the two
triangles are equal to each other, or they have all
their corresponding sides and angles equal.
PRoPosition XVII.—Theorem.
Parallel lines will not meet when produced to any dis-
tance whatever, fig. 35. -
Let A B and CD be two parallel lines; they will
not meet when produced. In one of them as A B,
take any two points E, F, and let fall the perpendicu-
lars EG, FH: then E G will be equal to FH, (def. 12;)
in the same way it may be shown, that any point
whatever being taken in A B, its perpendicular dis-
tance from C B will be equal to F H ; consequently
no point in A B can fall in C D ; that is these lines
can never meet, however far they may be produced.
PROPosſTroN XVIII.--Theorem.
A line which is perpendicular to one of two parallel
lines is also perpendicular to the other, fig. 36.
Let E F be perpendicular to A B, one of two paral-
lel lines A B and C D ; it will also be perpendicular
to the other : for if E F be not perpendicular to C D,
let F G* be perpendicular to it; then EF and FG are
equal to each other, (def. 12.) Hence a line drawn from
a point to a line, and perpendicular to it as E F, is
equal to F G more remote, which is impossible,
(prop. 16.) F. G therefore is not perpendicular to
C D, and the same may be shown of every line
except FE ; therefore FE, which is perpendicular to
A B, is also perpendicular to CD.
PRoposition XIX-Theorem.
If two parallel lines be cut by a third line, the two
alternate angles will be equal to each other, and the out-
ward angle will be equal to the inward angle on the same
side, and the two interior angles on the same side will be
together equal to two right angles, fig. 37.
Fig. 35.
Fig. 36.
Let the parallel lines A B, C D be cut by the line Fig. 37.
G H, then will E FC = B E F, and G E B = E F D,
also B E F + E F D = two right angles. First if G H
be perpendicular to A B, then the truth of the propo-
sition is manifest from the last ; and if it be not, draw
E I perpendicular to CD, and FK perpendicular to
* The letter G is omitted by the engraver,
3.18 G E O M E T R Y.
Geometry. A B : then will E IF and E KF be two right angled
S-N-' triangles, in which the hypothenuse E F is common,
and, because I H and CD are parallel, the angle Book I.
H F K = F K C ; but HFK = EF I, (prop. 3;) \-y-
Fig. 38. mºº .. †e * º º º º less than two right angles, these lines produced will
D b C . : for if A B be en j l cº meet and form a triangle, of which the third angle A,
C e parallel ; for it. e not para e to ° shall be equal to the difference between two right
let some other line P G be parallel to CP.3, then, anºles, and the sum of the two interior an lesſ B
because E G is parallel to CD, the angle G E F = j C. 8
E FC, (prop. 19;) but BEF = E FC; therefore G E F s
= BEF, a part equal to a whole, which is impossible; PRopos ITION XXV.-Theorem.
therefore E G is not parallel to CD, and the same may In every polygon the sum of all the interior angles is
be shown of every line passing through E, except AB; equal to twice as many right angles as the figure has
consequently AB is parallel to C. D. sides, wanting four right angles, fig. 41.
PRoPosition XXI.—Theorem. Let A B C D E be any polygon ; from a point O Fig. 41.
If a line falling upon two other lines make the out- wº º draw the lines O A, O B, O C, &c., to every
* * = & º angle of the figure which will divide the polygon into
ward angle equal to the interior angle on the same side, anv triangles as the figure has sides : now the sum
these two lines are parallel, fig. 38. . many ºriangle as heigure has sººn
of the three angles of every triangle being equal to
Let H I fall upon A B, CD, and make the angle two right angles, (prop. 24;) the sum of all the angles
HEB = EFD, then AB is parallel to CD. For if of all the triangles is equal to twice as many right
not, let some line, as E G be parallel to CD, then angles as the figure has sides; but of these angles
H E G = E F D, (prop. 19 ;) but H E B = E F D ; those about the point O, which are equal to four right
therefore H E B = H E G, a part to the whole, which angles, (prop. 3, cor. 1,) are angles of the triangles,
is impossible ; consequently E G is not parallel to but are not angles of the polygon ; therefore the
C D, and the same may be shown of every other line angles of the polygon alone are equal to twice as
passing through E, except A B ; therefore A B is many right angles as the figure has sides, wanting
parallel to CD. four right angles.
PROPosition XXII.--Theorem. PRoposition XXVI.--Theorem.
If one line falling upon two others make the sum of the If each of the sides of any polygon be produced, the
two interior angles upon the same side equal to two right sun. of all the outward angles is equal to four right angles,
angles, these lines are parallel, fig. 38. fig. 42. o ©
Since, by hypothesis B.E.F.-- EFD = two right Let the sides A B, BC, CD, &c. of the polygon Fig. 42.
angles, and since B E F + A E F = two right angles: A B C D, &c. be produced, then the sum of each in-
(prop. l ;) it follows that EFD = AEF, which are ward and outward angle is equal to two right angles,
alternate angles; therefore A B is parallel to CD, (prop. 1;) therefore the sum of all the outward and
(prop. 20.) inward angles, is equal to twice as many right angles
PR oposition XXIII.-Theorem. as the figure has sides. But the sum of all the inward
e te º - angles and four right angles, is equal to twice as
Lines which are parallel to the same line are parallel many right angles as the figure has sides, (by the
to each other, fig. 39. last proposition ;) therefore the sum of all the inward
Fig. 39. Let A B and CD be both parallel to I H ; they will and outward angles is equal to all the inward angles,
and the side E I of the one equal to K F of the other,
(def. 12 :) therefore these triangles are equal, and the
angle IFE = K E F, (prop. 16;) but these are alter-
nate angles: also I E F = EF K, to each of these add
IEA and KFD, which are equal, being both right
angles, and we shall have A E I + IE F = E FK +
KFD, or A E F = E FD, which are the two other
alternate angles.
Again A E F = G E B, (prop. 3;) therefore
G E B = E FD, that is, the outward angle is equal
to the inward angle on the same side : to each of
these add B E F, then will the two G E B + B E F =
B E F + E FD; but G E B + BE F = two right
angles, (prop. 1;) therefore BE F + EF ID = two
right angles; that is, the interior angles on the same
side are together equal to two right angles.
PRoPosition XX. —Theorem.
If a line falling upon two other lines make the alter-
nate angles equal to each other, those lines are parallel,
fig. 38.
be parallel to each other : draw any line cutting each
of the three lines as E. F. K; because A B is parallel
to I H, the angle B E F = E FI, (prop. 19 ;)
therefore F KC and B E F are both equal to E FI:
they are therefore equal to each other, and they are
alternate angles, therefore A B and C D are parallel,
(prop. 20.)
PROPosLTION XXIV.-Theorem.
The three angles of every triangle taken together are
equal to two right angles, fig. 40.
Let A B C be a triangle, the three angles A + Fig. 40.
B + C = two right angles. Produce AC to D, and
draw C E parallel to A B : then since A B, C E are
parallel and B C meets them, the alternate angles
A B C and B C E are equal, (prop. 19 ;) and because
these parallels are also cut by A C, the angle B A C =
E CD, (prop. 19 ;) consequently. A B C + B A C =
B C E + E CID = B C D ; to each of these equals,
add B C A ; then A B C + B A C + BC A = B C D +
BC A ; but B C D +B C A = two right angles; there-
fore ABC + B A C + B C A = two right angles.
Cor. It follows from this, that if two lines A B, AC
are cut by a third line B C, so as to make the two
interior angles on the same side as A B C + B C A
and four right angles ; taking away all the inward
angles from each sum, there remains the sum of all
the outward angles equal to four right angles.
G. E. O. M. E. T. R. Y.
3.19
'Geometry;
--~~
Fig. 43.
Fig. 44.
PRoPosition XXVII.-Theorem.
The opposite sides and angles of any parallelogram are
Tespectively equal to each other: that is, the angles to the
angles, and the sides to the sides; and the diagonal divides
the parallelogram into two equal triangles, fig. 43.
Let A B C D be a parallelogram, then will A D =
BC, AB = D C, the angle A = C, the angle A B C =
A D C ; and the triangle B A D = B C D. Draw the
diagonal D B : because A D is parallel to B C, the
angle A D B = C B D : and for the same reason the
angle C D B = A B D, (prop. 19 ;) therefore A DB
+ B D C = AB D + D B C, or the angle A B C = AD C
which are two of the opposite angles. Again, because
the two triangles A B D, and C D B, have two angles
equal, each to each, and the side adjacent to them
common ; they will be equal in all respects, (prop.
5,) and will have the angle A = C : which are the
other two opposite angles, also the side A D = BC,
A B = D C ; and the triangle A D B = the triangle
D C B.
Cor. Hence two parallels comprised between two
other parallels are equal to each other.
PR oposition XXVIII.—Theorem.
Lines which join the extremities of two equal and
parallel lines towards the same parts, are themselves equal
and parallel, fig. 43.
Let A B, D C join the extremities of the equal and
parallel lines A D, B C, then will D C be equal and
parallel to A B. Draw the diagonal D B, then the
angle A D B = D B C, (prop. 19,) and A D = BC, and
B D is common ; therefore A B = DC, (prop. 4,) and
the angle A B D = CBD, (prop. 4. ;) but these last
are alternate angles, therefore A B is parallel to DC,
(prop. 20.)
PRoposition XXIX. —Theorem.
A quadrilateral whose opposite sides are equal, is a
parallelogram, that is, if A D = B C and A B = D C ;
the figure A B C D is a parallelogram, fig, 43.
Draw the diagonal B D : then in the two triangles
A B D and D B C, the three sides of the one are equal
to the three sides of the other, each to each ; there-
fore the corresponding angles are equal, (prop. 12;)
that is, A D B = D B C, and C D B = A B D ; therefore
A D is parallel to B C, and A B to DC, (prop. 20.)
PRoposition XXX.-Theorem.
The two diagonals of any parallelogram bisect each
other, fig. 44.
For the diagonals being drawn, the angle D A O =
BC O, and A D O = C B O, (prop. 19 ;) also A D =
R C : therefore AO = O C, and DO = O B, (prop. 5,)
the diagonals are therefore bisected in O.*
BOOK II.
On Ratios and Proportions.
DEFINITIons.
1. RATIo is the relation of two magnitudes of the
same kind to each other, with respect to quantity.
---sº
* The letter O is omitted in the figure.
quantities are said to be commensurable.
The relations of magnitudes, with respect to quantity, Book II. $
may be expressed by numbers, either exactly or ap-
proximatively; and in the latter case, the approxima-
tion may be brought within less than any assignable
difference.
Thus, of two magnitudes, one of them may be
conceived to be divided into some number of equal
parts, each of the same kind as the whole; and one of
those parts being considered as an unit, of measure,
the magnitude may be expressed by the number of
units it contains. If then the other magnitude con-
tain a certain number of those units, this also may be
expressed by the number of its units, and the two
But if, what-
ever unit be assumed for the measure of the first mag-
nitude, the second magnitude do not contain an exact
number of such units, then the two magnitudes are
said to be incommensurable, and their relation, with
respect to quantity, cannot be correctly expressed in
numbers; but the relation between the first magni-
tude and a third, may be expressed in numbers, and
the third magnitude be such as to differ from the
second, by a quantity less than any that can be
assigned ; for it is obvious, that a third magnitude
may be found commensurable with the first, which
shall differ from the second, by less than the measur-
ing unit; and as the measuring unit may be less than
any assignable quantity, the difference between the
Second magnitude, (which is incommensurable with
the first,) and the third, (which is commensurable with
it,) may be so taken as to differ from each other, by
less than any assignable quantity.* f
Hence, it appears, that when magnitudes are com-
mensurable, we may always express their relation, or
ratio, numerically; and that when they are incom-
mensurable, we may still approximate so nearly to
their correct ratio, by means of numbers, that the
ratio assumed shall differ from the actual ratio of the
incommensurable magnitudes, by a quantity less than
any that can be assigned. Therefore, of two magni-
tudes, A and B, we shall conceive A to be divided
into some number, M units, each equal to A', or A =
MAſ; and B as equal to N such units, or B = N A ,
M and N being integral numbers; and, consequently,
the ratio of A to B will be expressed by the ratio of
M Aſ to NA’. In the same manner the ratio of any
other two magnitudes C and D, may be expressed by
PC/ to Q C', P and Q being also integral numbers;
* In order to connect the doctrine of commensurable quan-
tities with incommensurables, or magnitudes generally with
numbers, it must be assumed that whatever relations subsists
between A, B, C, D (in which A and B, C and D are com-
mensurables;) subsists also between A, M, C, N, (in which
A and M, and C and N are incommensurables) provided B and
D be such as to differ from M and N respectively, by quanti-
tics which are less than any quantity that can be assigned.
Authors have invented a variety of ingenious devices to hide
this transition; but, however the defect may be concealed on a
superficial view of the subject, it will always be found, upon a
closer investigation, to be admitted or taken for granted, and
we have preferred stating the full amount of the defect to hiding
it under, a specious disguise. Euclid's doctrine of ratios and
propositions is perhaps unobjectionable, and applies equally to
commensurables and incommensurables; but as soon as we have
occasion to apply it to numbers, the difficulty again appears. It
cannot, for example, be shown that the proper numerical measure
of a rectangle is the product of its two sides, without admitting
the principle advanced above, or one tantamount to it, and
equally objectionable.
320 G E O M E T R Y.
Geometry, and since A' and C/ are each units of their respective
\–v. - kinds, these ratios are simply those of M to N, and of
P to Q.
2. Ratios are said to be equal to each other, when
the number expressing the second term divided by
the first, is equal to the number expressing the fourth
tº º g ... N Q
term divided by the third ; thus, if M = F
ratio of M to N is said to be equal to the ratio of P
to Q ; and these four quantities are then said to be
proportional.
3. When magnitudes or quantities are in proportion,
they are expressed thus, M . N. . . P : Q, and they
are read, “ M is to N as P is to Q.”
4. Of four proportional quantities, the first and third
are called antecedents, and the second and fourth con-
sequents.
5. Three magnitudes are proportionals, when the
first has the same ratio to the second, that the second
has to the third, and then the middle term is said to
be a mean proportional between the other two.
6. Of four proportional quantities, the last is said
to be a fourth proportional to the other three taken in
their order. … -
7. Magnitudes are said to be in proportion, by
inversion or inversely, when the consequents are
taken as antecedents, and the antecedents as conse-
quents. -
S. Magnitudes are in proportion, by alternation or
alternately, when the antecedent is compared with
the antecedent, and the consequent with the conse-
quent.
5 then the
PRoPosition I.-Theorem.
When four quantities are in proportion, the product
of the two extremes is equal to the product of the two
7)?6 Ol)?S.
Let A, B, C, D be four quantities in proportion,
and M . N. . . P : Q be their numerical representa-
tives; then will M × Q = N × P ; for since they are
T
in proportion ‘. F ; therefore Q = ***, and
M × Q = N × P.
Cor. Hence if there be three proportional quan-
tities, the product of the extremes is equal to the
square of the mean, (def. 5.)
PRoPoSITIon II.-Theorem.
If the product of two quantities be equal to the product
of two other quantities, two of theºn will be the eatremes,
and the other two the means of a proportion.
Let M × Q = N × P ; then will M : N :: P. Q.
For if P have not to Q the ratio which M has to N,
let P have to Q', (a number less than Q,) the same
ratio that M has to N ; that is, let M . N. : : P : Q';
then M × Q = N × P, or Q = ***. but Q =
N × P
M
same way it may be shown, that it is not greater;
consequently Q' = Q, and the four quantities are pro-
portional ; that is, M . N. . . P : Q.
, therefore Q' is not less than Q ; and in the
PROPosition III.-Theorem. Book II.
If four quantities be in proportion; they will be in pro- TYT
portion when taken alternately. -
Let M, N, P, Q be the numerical representatives
of the four quantities in proportion ; so that
M : N . . P : Q, then will also
M : P : . N : Q. -
Because M. : N : : P : Q, M × Q = N × P, or
M × Q = P × N ; but M × Q, and P × N., are the
products of the extremes and means of the terms M,
P, N, Q ; and they are equal to each other ; therefore
- M : P : : N : Q.
PRoposition IV.—Theorem.
If four quantities be in proportion, they will be in pro-
portion when taken inversely.
Let M : N . . P : Q, then will also
N : M : : Q P;
for the first four terms being in proportion,
M × Q = N × P, or N × P = M × Q.
But N × P, and M × Q, are the products of the ex-
tremes and means of the four quantities N, M, Q, P.;
and these products being equal,
N. : M . . Q . P.
PR oposition V.—Theorem.
If four quantities be in proportion, they will be in pro-
portion by composition or division.
Let, as before, M, N, P, Q be the numerical repre-
sentatives of the four quantities, so that
M : N . . P : Q, then will
MIN : M :: P + Q : P;
for by the first M × Q = N × P, or N × P = M × Q,
to each add M × P ; then
Mix F + N × P = M × P + M. G.
or MIN × P = P + Q x M.
But MI IN and P, are the extremes, and P + Q and
M, the means, of the four quantities in the second line,
and the product of these being equal, the quantities
are in proportion ; that is,
MIN : M :: PIE G : P.
PRoposition VI.-Theorem.
Equimultiples of any two quantities, have the same
ratio as the quantities.
Let M and N be any two quantities, and m any
integral number ; then will
m M. : m N. : M : N :
for m M × N = m N × M = m, M. N.
PROPos ITION VII.-Theorem.
Of four proportional quantities, if there be taken any
equimultiples of the two antecedents, and any equimulti-
ples of the two consequents, the four resulting quantities
will be proportionals.
Let M, N, P, Q be the numerical representatives
of four quantities in proportion ; and let m and n be
any numbers whatever, then will
m M. : n N . . m P : n Q.
Because M . N. : P : Q, M × Q = N × P
therefore m M x n Q = n N × m P;
G E O M E T R Y
321
Geometry. and these being the product of the extremes and
S-N-" means, of m M, n N., m P., n Q, they are proportionals,
Or m M . 7, N . . m. P. m. Q.
PR oposition VIII.-Theorem.
Of four proportional quantities, if the two consequents
be either augmented or diminished by quantities that have
the same ratio as the antecedents, the resulting quantities
and the antecedents will be proportionals.
Let M : N : : P : Q be the four quantities; and
let M ; P : . m . m., then will
M : N + m :: P : Q -- n.
Pecause M ; N . . P : Q, M × Q = N × P :
and because M . P . . m . n, M × n = m × P ;
therefore M × Q -- M × n = N × P + m × P;
M x Q -i- m = P × N + m ;
M : N + m :: P : Q -Em.
Or
hence
PROPOSITIon IX.-Theorem.
If any number of quantities be proportionals, any one
antecedent will be to its consequent as the sum of all the
antecedents is to all the consequents.
Let M . N. . . P : Q . : R . S., &c. be quantities in
proportion, then will
M : N : : M + P -- R - N + Q -- S.
Because M. : N : : P : Q, M × Q = N × P ;
and because M : N : : R . S, M × S = N × R. ;
therefore M × Q -- M × S = N × P + N × R,
to each add M × N., or N × M,
then M × N + M × Q + M × S = N × M +
- N × P + N × R,
or M × N + Q-FS = N × M-F PER;
therefore M . N. : M + P + R : N + Q + S.
PR oposition X.-Theorem.
If two magnitudes be each increased or diminished by
like parts of each, the resulting quantities will have the
same ratios as the first two.
& M N
Let Mand N be any magnitudes, and 7m and ºn be
like parts of each, then will
M
M+* : N + š, ; M ; N.
772. - m
For it is obvious that
(M + ...) × N = (N +}) × M,
each being equal to M × N + Mºs Consequently
the four quantities are proportional.
PRoposition XI.-Theorem.
If four quantities be proportionals, their squares or
cubes will also be proportionals.
then will , Mº ; Nº :: Pº : Q3,
and * Ms : N3 : P3 : Q3.
For since M × Q = N × P
M? X Q & —- N2 X P2
M3 × Q3 = N 3 × P9, &c.
YOL. I.
and, therefore, - - - Book II,
M? : N2 : ; P2 : Q2 tº-
M3 : N 3 :: P3 : Q3, &c. Book III,
Cor. In the same way it may be shown, that any \-N-
power or roots of proportional quantities are pro-
portionals.
PRoPosition XII.-Theorem.
If there be four proportional quantities, and four other
proportional quantities, the product of the corresponding
terms will be proportionals. .
Let M : N : : P : Q
and R : S : : T : V,
then will VIXTR N × S : : P ×T: Q XV;
for since M × Q = N × P, and R × V = S X T
M × Q x R × V = N × P × S x T.
Or M. XTR × Qx W = N × S X Pºx T.
therefore M × R N × S . . P × T : Q x V,
BOOK III.
Of the circle, and the measure of angles.
DEFINITIONs.
1. THE circumference of a circle is a curved line
A B D, every where equally distant from a point
within C, called the centre, fig. 46.
2. The circle is the superficial space, included within
the circumference. These terms are frequently con-
founded ; the circumference being sometimes called
the circle. Thus, we say, describe a circle from a
given point, &c., and not describe the circumference
of a circle ; but the distinction is easily made, the one
being a line, and the other the space included
within it.
3. The radius of a circle is any right line drawn
from the centre and terminated in the circumference,
as CA, C B, C D ; consequently all the radii of the
same circle are equal to each other : and
The diameter of a circle is any right line passing
through the centre and terminating at each extremity
in the circumference, as A. D. Hence, a diameter
is equal to double the radius ; and hence the radius
is sometimes called the semi-diameter. -
4. An arc of a circle is any portion of the circum-
ference, as A B or B D. º,
5. The chord or subtence of an arc is any right line,
as A B, joining the extremities of the arc ; and the
space included within the chord and the arc is called
a segment. The same chord is common to two arcs
and two segments; but unless the contrary be stated,
it is always to be understood that the less arc, or less
segment, is spoken of in these cases.
6. A sector of a circle is the space included be-
tween any two radii and the arc comprised between
them, as A C B or B C D.
7. A line is said to be inscribed in a circle when its .
two extremities are in the circumference, as A B.
8. An angle is inscribed in a circle, or contained in it,
when it is comprised between two chords meeting at
a point in the circumference, as B.A. D.
9. A triangle, or any right lined figure, is said to be
inscribed in a circle, when all the angular points of the
2 U.
Fig. 46.
322 G E O M E T R Y.
Geometry, former are in the circumference of the latter, as the line A B with DF, and arc A. M. B with DNF. Book III.
For if these latter do not coincide let them be \-y-
A B C, A B C D, fig. 48.
Fig. 48. 10. A secant is any line which cuts the circumfe- situated in some other way, as in the figure, and join
Fig. 49. rence of the circle in two points, as A B, fig. 49. EG, cutting the arc DNF in H. Then DE = E G,
11. A tangent is any right line which touches the being radii of the same circle; and for the same reason
circumference in one point only, as CD, fig. 49; and DE = E H ; therefore E G=EH a part to the whole,
the touching point M is called the point of contact. which is absurd : and the same may be shown of any
12. A rectilineal figure is said to be circumscribed point that is not in the arc DNF : that is, no point in
about a circle, or the circle inscribed in it, when all the arc AMB, falls out of the arc DNF; consequently
the sides of the former are tangents to the circle, these arcs coincide and are equal to each other. Next
Fig. 50. fig. 50. let # arc AMB = D NF; then will the chord A B
PROPosition I.—Theorem. For if A B be not equal to D F, let A I be equal to
A diameter divides the circle and its circumference into D F : then, because DF and AI are equal chords in
two equal parts, fig. 51. equal circles, the arcs Mºdel,'. *: º a Fe
Fig. 51. Let AB be a diameter, it divides the circle into two ºr ***, *.*.*.* - º but A M B =
§ equal parts. For conceive the semicircle A B D to be D i. . º: A}} -: à *: the ºi. greater,
applied upon A B C, so that the diameter A B may be "...”.” ‘’”." erefore the arc A. is not un-
common ; then will the circumferences also coincide. equal to DNF ; that is, it is equal to it.
Por if they do not, from the centre O draw the line
O EF; then OF 7 OE ; but OF and OE, being both Proposition v.–Theorem.
radii, are equal, (def. 3, book iii. :) they are therefore In equal circles equal angles at the centre are subtended
both equal and unequal at the same time. Which is by equal arcs, and equal arºs subtend equal angles; and
impossible; that is, OP is not unequal to OF3, and when the arcs are unequal the angles will have the same
the º . º of . º º à'. iºn natio to each other which the arcs have, fig. 54.
cumferences therefore colncide, and are equal to eac . .
other, as are also the two segments; and each of them ºf AMººn. PNF be equal arºs of equal cirºle;
is called a semicircle. and let C and E be the centres; then if the angle C
= E, the arc A. M. B = D N F. Because the circles
PRoposition II.-Theorem. are equal, A C = DE, and C B = EF, and the angle
g & º A C B is equal to D E F, therefore the base or chord
Any chord 2% (1, circle which does not pass through the A B = DF, (prop. 4, book i.;) and therefore also the
centre is less than the diameter, fig, 52. arc AB = DF, (prop. 4, book iii.;) that is equal angles
Fig. 52 Let A B be a diameter, and DE a chord not passing at the centre, in equal circles, are subtended by equal
through the centre, DE Z A B. Let C be the centre arcs.
of the circle; and join CD, C E : then D E Z (DC Again, if the arc AB = DF, then will the angle
+ CE) (prop. 8, book i.) but D E + C E = A B; C = E.
therefore D E Z A B. Because the arc AB = D F, the chord A B = DF,
- (prop. 4, book iii.;) and the three sides of the triangle
PRoPositroN III.-Theorem. ACB are equal to three sides of the triangle DEF, each
A right line cannot cut a circle in more than two tº º º º * * º
oints. circles equal arcs Subtend equal angles. Next, let the
p If it were possible for a right line to cut a circle arcs M N and PQ of the equal circles M ON, PQ R.
in more than two points, lines drawn from the centre be unequal, then will the arc M N be to PQ, as the
g 3. angle M O N to P. R. Q. Conceive the arc M N to be
to each of these points would be equal to each other, di ; d into an ber of 1 parts M b, b. c. c N
(def. 3, book iii.) which is impossible; because from .. . º numbero º º th Cl, ** N. C d
a point there cannot be drawn to a line more than two ma º ô", 3. * VIIll º, € "I ai .
lines which are equal to one another, (prop. 15, book i.) JOIn * th º . º: * a TCS º Cl d i.
Therefore a right line cannot cut a circle in more than are equal, the angles M. o.º. 400, s.c. are all equal to
two points each other, and any one of them may be taken as the
P ſº measuring unit of the angle MON. From P towards Q,
PRoposition IV.-Theorem. on the arc PQ, apply the measuring unit Pa = M a
In the same, or in equal circles, equal arcs are subtended º}. º º . º lº.
by equal chords, and equal chords by equal arcs, fig. 53. Ra, Rb, R c, &c., dividing PR finto the equal angles
Fig. 53. Let A M B, D NF, be equal circles, and A B, D F PR a, a R b, &c. each equal to the angle MO a.
equal chords; then will the arc AMB = DNF. Let
C and E be the centres of the two circles, and join
AC, CB, D E and E F ; then in the two triangles
A CB, D E F, the three sides of the one are equal to
the three sides of the other, each to each, and conse-
quently the triangles also are equal, (prop. 12, booki.;)
and if the circle A M B be applied to the circle
DNF, so that the point or centre C falls upon E,
and the line or radius A C upon E D, the radius
C B will fall upon E F, (because the angle C = E,)
and the points A and B will coincide with E and F;
Thus the angles MON will be the same multiple of
M Oa, as the arc M N is of M a ; and in the same
manner the angle PR f is the same multiple of MO a
as PF is of M a ; these quantities will therefore be to
each other as the number of units in each; that is,
MN : P f :: M ON : P R f.
But the arc Plf may be made to approach nearer to
PQ, and the angle PR f nearer to P R Q than any
assignable difference, by reducing the magnitude of
the measuring unit; and hence it follows, that what-
ever ratio subsists between M N and Pf, and MO N.
G E O M E. T. R. Y. 323
equal to the two DB, BE, each to each, and E D is Book III.
Geometry, and PR f, subsists also between M N and PQ, and
common; therefore these two triangles are equal, and -N-"
S–SV-7 M O N and PR Q ; *
Fig. 55. Let A B be any chord in a circle, and CD a line point, as G.; join FG, and produce it to meet A B C
- drawn from the centre C, bisecting A B in D, then in B, and join also A. G. Then G being the centre of
will C D be perpendicular to A B. the circle A E D, A G = G D ; but A G + F G 7 AF,
Draw the two radii A. C., C B : in the two triangles (prop. 8, book i.;) therefore G D + F G, or F D 7
A CD, B C D, the two sides A C, AD, are equal to AF; but A F = F B; hence also F D 7 F B, a part
the two, B C, B D, and C D is common ; hence the greater than the whole, which is absurd ; therefore G
triangles are equal, and have their corresponding is not the centre of the circle A E D, and the same may
angles equal, (prop. 12, book i. ;) therefore each at be shown of every point that is not in A.C. The
D is a right angle, and CD is perpendicular to A B, centre of the circle AED is therefore in A C; that is,
(def. 10, book i.) the centres of the circles and the point of contact are
Again, let C D be perpendicular to A B, then will in the same right line.
A B be bisected in D. For in the two right angled
triangles A CD, B C D, the hypothenuses are equal, PRoPosition IX.—Theorem.
and the side CD is common; therefore the third sides e
A D, D B are also equal, (prop. 16, book i. :) that is If two circles touch each other externally, the centres
the chord A B is bisected in D. - of the circles and the points of contact are in the same
Cor. 1. Hence a line bisecting any chord in a circle "ght ºne, fig. 58.
at right angles passes through the centre. Let A E D and A C B touch each other externally Fig. 58.
Cor. 2. It follows also from the above, that the line in A ; then will the centres of the circles, and the
which bisects and is perpendicular to a chord, bisects point A be in the same right line.
also the arc of that chord ; for the angles at C being Let F be the centre of A B C, join AF and produce
equal, the arcs which subtend them, A E, E B, are it to E; the centre of the circle AED is in this line.
also equal, (prop. 4, book iii.) or the arc AB is bisected For if it be not, let it be in some other point as G,
in E. and join A G, F G : then A F + A G 7 GF, (prop. 8,
g w book i.;) but A G = G D, and A F = F B; therefore
PR oposition VII.-Theorem. G D + F B 7 G E ; a part greater than the whole,
If more than two equal lines can be drawn from any which is impossible ; and the same may be shown of
point within a circle to the circumference, that point will any, Point not in FE : therefore the centre of the
be the centre, fig. 56. | circle EA D is not out of the line FE ; that is, it is
Fig. 55, Let ABC be a circle, and D a point within it; then "".
that is M N : P Q : : M O N : P R Q.
Scholium. Since the arcs have always to each other
the ratio which the angles at the centres have, it
follows that the arcs may be assumed as the mea-
sure of the angles at the centre ; and as all the an-
gles that can be formed about the centre of a circle,
or any other point, are together equal to four right
angles, (prop.3, cor. 1, booki.) the whole circumference
will be the measure of four right angles ; the semi-
circle the measure of two right angles, and a quadrant
or quarter of the circumference the measure of one
right angle.
PRoposition VI.-Theorem.
If a right line drawn through or from the centre of a
circle bisect a chord, it will be perpendicular to it, or if it
be perpendicular to the chord it will bisect it, fig. 55.
if any three lines DA, D B, D C, drawn from the
point D to the circumference, be equal to each other,
that point will be the centre. Join A B, BC, bisect
A B in E, and B C in F, and join E D, D F. In the
triangles A E D, BE D, the two sides AD, A E, are
* It is here taken for granted, that if four quantities, A B C D,
be proportionals, and that N and M be two other quantities in-
commensurable with B and D, but which latter are still such
that they may be made to approach nearer to N and M than
any assignable quantities, that then also A : N : : B : M. It
must be acknowledged, that this conclusion is not so strictly
geometrical as could be wished, but it is a defect which necessa-
rily attends the transition from magnitude to number; and which,
however it may be disguised, is still to be found upon a minute
and strict inquiry. In the first six books of Euclid, magnitudes
only are considered, and the difficulty does not appear; but it
presents itself the moment we attempt to apply his propositions to
the purposes of mensuration. See note to Definitions, Book II.
the angles at D are equal, (prop. 12, book i.;) conse-
quently each of them is a right angle, (def. 10, book i.;)
ED therefore bisects the chord E D at right angles,
and therefore passes through the centre, (prop.6, cor. 1,
book iii.) In the same way D F passes through the
centre, consequently the point D is the centre.
PROPOSITION VIII.-Theorem.
If two circles touch each other internally, the centres of
the circles and the point of contact are in the same right
line, fig. 57.
Let the two circles A C B, E A D, touch each other Fig. 57
internally in the point A ; then will the point A and
the centres of the circles be in the same right line.
Let F be the centre of the circle A B C, and draw the
diameter A F C ; the centre of the circle A D E will
be also in this line. For if not, let it be in some other
PRoPosition X.-Theorem.
Chords in a circle which are equally distant from the
centre are equal to each other; and if they are equal to
each other they are equally distant from the centre,
fig. 59.
Let the chord A B = CD; they are equally distant Fig. 59.
from the centre. Let G be the centre of the circle,
and G F, G E two perpendiculars from the centre upon
the chords A B, CD; then E G = G F : join A G, C G.
Now E G, being perpendicular to A B, it bisects it in
E, (prop. 6, book iii.;) and for the same reason G F
bisects CD in F : therefore A E = CF; also A G =
C G : hence the two right angled triangles A E G,
G F C are equal to each other, (prop. 16, book i.;) and
consequently E G = F G : that is the equal chords
A B, CD are equally distant from the centre. -
2 U 2
324
G. E. O. M. E. T. R. Y.
been proved equal, there remains the angle C E G Book III.
equal to the angle A C H. But the former of these, S-V-
Geometry. Next let them be equally distant from the centre;
\-V-' that is, let E G = F G; then will also AB = C D : for
Fig. 60. Let the line A B be perpendicular to the extremity g
of the radius CD ; then will A B be a tangent to the For let the tangent DE pass through the point of
circle, or touch it in the point D only. contact A ; then the angle D A C, being measured by
For take any other point E in A B, and join CE, C half the arc A B C, and the angle D A B by half the arc
being the centre : then will CE (prop. 15, book i.) 7 A B, (Prop. 13, book iii.;) it follows by equal subtrac-
DC, or than CF; therefore the point E is beyond the tion, that the difference or angle BAC must be mea-
circumference, and the same may be shown of every sured by half the arc BC which it stands upon. .
point in the line AB, except the point D; consequently
A B touches the circle in no one point except at D ; PRoposition XV.--Theorem.
and is therefore a tangent (def. 11, book iii.) to it at All angles in the same segment of a circle, or standing
that point. wpon the same arc, are equal to each other, fig. 64.
PROPosLTION XII.—Theorem. Let A CB, A D B be two angles in the same seg- Fig. 64.
- e ment AC, DB, or which is the same, standing upon the
If a right line be a tangent to a circle, a radius drawn same arc A. E. B; then will the angle ACB be equal to
to the point of contact will be perpendicular to the the angle ADB.
tangent, fig. 61. For each of these angles is measured by half the
Fig. 61. Take any point E, as before : then it is obvious, arc AEB, (prop. 14, book iii. :) and thus having equal
since the line is wholly without the circle, that CE 7 measures, they are equal to each other.
C F, or than CD ; consequently CD is the shortest º
line from the centre C to A B ; therefore C D is per- PROPOSITIon XVI.—Theorem.
pendicular to A B, (prop. 15, book i.) An angle at the centre of a circle is double the angle at
the circumference, when both of them stand upon the same
\ IPR oposition XIII.-Theorem. arc, fig, 65.
The angle formed by a tangent and chord is measured Let A C B be an angle at the centre C, and A D B an Fig. 65.
by half the arc of that chord, fig. 62. angle at the circumference, both standing upon the
ig. 62 Let A B be a tangent to a circle, and CD a chord ***.*.* * chord A B, then will the angle C be
Fig. 62. e e a tange Ircle, double of the angle D, or the angle D equal to half the
drawing the lines as above; in the two right angled
triangles A E G, C F G : the hypothenuses are equal,
and the side E G = F G : therefore also E A = CF,
(prop. 16, book i.;) but A B is double of Ale, and
CD is double of CF; consequently AB = CD.
PROPosition XI.-Theorem.
A right line perpendicular to the extremity of a radius
ãs a tangent to the circle, fig. 60,
drawn from the point of contact C ; then is the angle
B C D measured by half the arc C F D, and the angle
A CD by half the arc A G D.
For draw the radius EC to the point of contact, and
the radius E F perpendicular to the chord at H. Then
the radius E F, being perpendicular to the chord CD,
bisects the arc C F D, (prop. 6, cor, book iii.;) there-
fore C F is half the arc C F D.
In the triangle C E H, the angle H being a right
angle, the sum of the two remaining angles E and ECH
is equal to a right angle, (prop. 24, book i.) which is
equal to the angle B CE, because the radius C E is
perpendicular to the tangent, (prop. 12, book iii.)
From each of the equals take away the common part
or angle E CH, and there remains the angle C E F
equal to the angle B C D. But the angle E is measured
by the arc CF, (prop. 5, book iii.) which is half CFD;
therefore the equal angle B C D must also have the
same measure, half the arc C F D of the chord C.D.
Again the line G E F, being perpendicular to the
chord CD, bisects the arc C G D. Therefore C G is
half the are C G D. Now since the line C E meeting
FG makes the sum of the two angles at E equal to
two right angles, and the line C D makes with A B
the sum of the two angles at C equal to two right
angles ; if from these two equal sums there be taken
away the parts or angles E CH and BCH, which have
C E G, being an angle at the centre, is measured by
the arc CG, (see prop. 5, book iii.;) consequently
the equal angle A CD must also have the same mea-
sure C G, which is half the arc C G D.
PRoposition XIV.—Theorem.
An angle at the circumference of a circle is measured by
half the arc that subtends it, fig. 63.
Let B A C be an angle, at the circumference it has Fig. 63.
for its measure half the arc BC which subtends it.
angle C.
For the angle at the centre C is measured by the
whole arc A. E. B., (prop. 5, book iii.;) and the angle
at the circumference D is measured by half the same
arc A E B, (prop. 14;) the angle D is only half the
angle C, or the angle C double the angle D.
PROPOSITION XVII.-Theorem.
An angle in a semicircle is a right angle, fig. 66.
Let A B C or A D C be a semicircle, then any angle Fig. 66.
A B C in that segment is a right angle.
For the angle B at the circumference is measured
by half the arc A DC, (prop. 14, book iii.;) that is by
a quadrant of the circumference. But a quadrant is
the measure of a right angle ; therefore the angle B
is a right angle.
Cor. It follows from this, that an angle in an arc
that is greater than a semicircle, is less than a right
angle; and an angle in an arc less than a semicircle is
greater than a right angle.
PROPosition XVIII.-Theorem.
The angle formed by a tangent to a circle and a chord
drawn from the point of contact, is equal to the angle in
the alternate segment, fig. 67.
. G E O M E T R Y.
325
Geometry.
~~~
Fig. 67.
Fig. 68.
If A B be a tangent, A C a chord, and D any angle
in the alternate segment A DC; then will the angle D
be equal to the angle B A C made by the tangent and
the chord of the arc A. E. C.
For the angle D at the circumference is measured by
half the arc A E C, (prop. 13 and 14, book iii.;) and the
angle B A C, made by the tangent and chord, is also
measured by the same half arc A. E. C.: therefore these
two angles are equal. -
PRoposition XIX.-Theorem.
The sum of any two opposite angles of a quadran-
gle inscribed un a circle is equal to two right angles,
fig. 68.
Let A B C D be a quadrangle inscribed in a circle;
then shall the sum of the two opposite angles, A and C,
or B and D, be equal to two right angles.
For the angle A is measured by half the arc D C B,
which it stands upon, and the angle C by half the arc
Fig. 69.
D A B, (prop. 14, book iii.;) therefore the sum of the
two angles, A and C, is measured by half the sum of
these two arcs, that is by half the circumference. But
half the circumference is the measure of the two right
angles, (prop. 5, schol, book iii.;) therefore the sum of
the two opposite angles, A and C, is equal to two
right angles. And in like manner it is shown the sum
of the other two opposite angles, B and D, is equal to
two right angles.
PRoposition XX.-Theorem.
If any side of a quadrangle inscribed in a circle be pro-
duced out, the outward angle will be equal to the inward
opposite angle, fig. 69.
If the side A B of the quadrangle ABCD, inscribed
in a circle, be produced to E, the outward angle DAE
will be equal to the inward opposite angle C.
IFor the sum of the two adjacent angles D A E, DAB
is equal to two right angles, (prop. 1, book i.;) and
the sum of the two opposite angles, C and D A B, is
Fig. 70.
Fig. 71,
equal to two right angles, (prop. 19, book iii.;) there-
fore the sum of the two right angles, DAE and D A B,
is equal to the sum of the two, C and D A B ; from
each of these equals, taking away the common angle
DAB,there remains the angle D A E equal the angle C.
PRoPosition XXI.—Theorem.
Two parallel chords intercept equal arcs, fig. 70.
Let the chords A B, C D be parallel, then will the
arcs A B, C D be equal, or AB = CD.
For draw the line B C ; then because the lines A B
CD are parallel, the alternate angles B and C are
equal, (prop. 20, book i.) But the angle at the circum-
ference B is measured by half the arc AC, (prop. 14,
book ii.;) and the other angle at the circumference C
is measured by the arc B D ; hence the halves of the
arcs A C, B D, and consequently the arcs themselves
are equal.
PRoposition XXII.-Theorem.
If a tangent and chord be parallel to each other, they
intercept equal arcs, fig. 71.
Let the tangent A B C be parallel to the chord
DE; then are the arcs BD, BE equal; that is,
B D = B.E.
For draw the chord B D ; then because the lines Book III.”
A B, DE are parallel, the alternate angles D and B \-N-
are equal ; but the angle B, formed by a tangent and
a chord, is measured by half the arc BD, (prop. 18,
book iii.;) and the angle at the circumference D, is
measured by half the arc B E ; the arcs B E, BD are
therefore equal.
PROPosition XXIII.—Theorem.
The angle formed within a circle by the intersection
of two chords, is measured by half the sum of the two arcs
intercepted by those chords, fig. 72.
Let the two chords A B, CD intersect at the point Fig. 72.
E.; the angle A E C, or D E B, is measured by half the
sum of the two arcs A C, D B.
Eor draw the chord AF parallel to C D ; then be-
cause the linés A F, CD are parallel, and A B cuts
them, the angles on the same side, A and DEB, are
equal; but the angle at the circumference A is mea-
sured by half the arc BF, (prop. 14, book iii.) or of
the sum of F D and D B ; therefore the angle F is
also measured by half the sum of F D and D.B. Again,
because the chords AF, CD are parallel, the arcs
A C, F D are equal, (prop. 21 ;) therefore the sum
of the two arcs A C, D B is equal to the sum of the two
FD, D B ; and consequently the angle E, which is
measured by half the latter Sum, is also measured
by half the former.
PROPosLTION XXIV.-Theorem.
The angle formed without a circle by two secants, is
measured by half the difference of the intercepted arcs,
fig. 73.
Let the angle E be formed by two secants, A B and
CD. This angle is measured by half the difference
of the two arcs, A C, D B, intercepted by the two
Secants. - -
Draw the chord AF parallel to CD ; then because
the lines AF, CD are parallel, and A B cuts them, the
angles on the same side, A and D E B, are equal,
(prop. 21, book i.) But the angle A, at the circum-
ference, is measured by half the arc B F, or of the
difference of D F and D B ; therefore the equal angle
E is also measured by half the difference of D.F, D B.
Again because the chords A F, CD are parallel,
the arcs A C, F D are equal, (prop. 21, book iii.;)
therefore the difference of the two arcs, A C, D B, is
equal to the difference of the two D.F, D B ; con-
sequently the angle E, which is measured by half
the latter difference, is also measured by half the
former.
PROPOSITION XXV.-Theorem.
The angle formed by two tangents, is measured by half
the difference of the two intercepted arcs, fig. 74.
Fig. 73.
Let EB, ED be two tangents to a circle at the points Fig. 74.
A, C : then the angle E is measured by half the dif-
ference of the two arcs C F A, C G A.
For draw the chord AF parallel to E D ; then be-
cause the lines A F, E D are parallel, and E B meets
them, the angles on the same side, A and E, are equal,
(prop. 21, book iii.;) but the angle A, formed by the
chord A F and tangent A B, is measured by half the
arc AF: therefore the equal angle E is also mea-
326,
G E O M E T R Y.
Geometry, sured by half the same arc A F, or half the difference
~~" of the arcs C F A and C.F.
Fig. 75.
Fig. 76.
Fig. 77.
Again, because the tangent E D and chord A F are
parallel, the intercepted arcs (prop. 21, book iii.)
C G. E., C F are equal; the arc AF therefore is equal to
the difference of C F A and C G A ; consequently the
angle E, which is measured by half the former, is also
measured by half the latter.
Cor. In like manner it is proved that the angle E
(fig. 74) formed by a tangent E CD, and a secant
E AB, is measured by half the difference of the two
intercepted arcs, CA and CF B.
Problems relative to Books II. and III.
PROBLEM I.
To divide a given right line A B into two equal parts,
fig. 75.
From the two extremities, A and B, and with any
equal radii greater than half A B, describe arcs of
circles intersecting each other in C and D, and draw
the line CD, which will bisect the given line A B in
the point E.
Join A C, CB, A D, DB, which are all equal to each
other; consequently the triangles D A C, D B C, which
have the three sides of the one equal to the three
sides of the other, each to each, will have their corres-
ponding angles also equal ; therefore the angle ACE
= BCE. And because A C = C B, the angle C A E
= CBE; hence the angles A CE, CAE, being equal
to the two EC B, C B E, each to each, and the side
A C = BC, the two triangles A C E, BCE, are equal;
and will have the base AE = EB, that is the right line
A B has been bisected in E, as was required to be
done.
PROBLEM II.
To bisect a given angle, BAC, fig. 76.
From the summit A, with any radius, describe an
arc cutting off the equal parts A D, A E ; and from D
and E, with any radius greater than half DE, describe
the two arcs intersecting in F; and join A F, which
will bisect the angle A, as required. Join D F, EF,
then the two triangles ADF, AEF, will have the sides
A D, D F equal to A E, E F, each to each, and the base
A F common; therefore the triangles will be equal,
and the angle DAF = E A F; that is, the angle A has
been bisected by the line A F.
PROBLEM III.
At a given point C in a line A B to raise a perpendi-
cular, fig. 77.
From the given point C, set off the equal distances
CD, CE, on the line A B, and from D and E as centres,
with any radius greater than D C or EC, describe arcs
intersecting each other in F; join CF, which will be
the perpendicular required.
Join D F, FE, then in the two triangles D F C,
E FC, the sides D F, D C are equal to E F, E C, each
to each, and the base FC is common ; therefore the
triangles are equal, and the angle D C F = E C F :
they are therefore right angles, and FC is perpendi-
cular to A B. g -
Scholium. As it is assumed that a given line may be
perpendicular to it.
Problems
produced, if the point C were at the extremity of the Prº
relative to
line AB, the line might be produced and the construc-‘i.
Q gº tº a tº tº ooks II,
tion remain as above; but it is sometimes a conve- ... iii.
nience in practice to erect a perpendicular without S-N-
producing the line beyond the point at which it is
to be erected. In such cases we may proceed as
follows :
Take any point D (fig. 78) out of the line A B, and Fig. 78.
from D as a centre, and with the radius DC, describe
a circle, E CF, cutting A B in E ; join ED, and pro-
duce it, to cut the circumference in F, draw F C ; it
will be the perpendicular required. For E C F being
a semicircle, the angle C in it is a right angle, and
consequently C F is perpendicular to A B.
PROBLEM IV.
From a given point A, to let fall a perpendicular upon
a given line B C, fig. 79.
From the point A, with any radius greater than the Fig. 79
perpendicular distance, describe an arc cutting B C in
two points, D and E ; from D and E as centres, with
any radius, describe arcs intersécting in F; join A F,
cutting B C in G, then will C G be the perpendicular
sought.
For join D A, D F, A E, E F : the triangles ADF,
A G F, having the three sides equal, each to each ; the
angle D A F = E A F : and the triangle D A E, being
isosceles, the angle A D E = A E D : hence in two
triangles D A G, EA G, the two angles A D G, D A G
are equal to the two AEG, E A G, each to each, and
the side A D = AE; therefore the triangles are equal,
and the angles at G are equal ; they are therefore
right angles, and A G is perpendicular to A B.
Scholium. As in the last problem this construction
supposes the line A B (fig. 80) of unlimited length. Fig. 80.
If the point be nearly opposite the end of the line the
following construction may be employed: From any
point D in A B, and with the radius D C, describe an
arc C A F ; and from A, with the radius AC, describe
an arc cutting the former in C and F ; join C F, and it
will be the perpendicular sought.
Join A C, AF, which being equal chords, the arcs
A C, AF will be also equal: hence D A bisects the are
A F, and consequently also the chord of the arc : but
the line drawn from the centre to bisect a chord is
Hence C G is perpendicular to
AG, and consequently A G is perpendicular to D G or
to AB. &
PROBLEM V.
At a point A, in a given line A B, to make an angle
equal to a given rectilineal angle C, fig. 81.
From the centres A and C, with any radius, describe Fig. 81.
the arcs DE and FG ; join E D, and from F, with
the distance DE, describe an arc cutting FG in G ;
draw AG, so will the angle A = C.
For the chords DE, FG, being equal, the arcs DE
and FG are also equal; and consequently the angles
C and A.
PROBLEM VI.
Through a given point A, to draw a line parallel to a
given line, B C, fig. 82.
From the given point A, draw any line A D to the Fig. 82.
line A B ; and at the point A make the angle DAF =
A DC, produce AF, and it will be parallel to BC.
G E O M E T R Y 327
beometry.
For the alternate angles ADC, and FAD being
\-y-' equal, the lines E.F and B C are parallel.
Fig. 83.
Fig. 85.
Fig. 86.
ProBLEM VII.
To describe a triangle when there are given the two
sides and the included angle, fig. 83.
Draw the indefinite line A D, and at the point A
make the angle B A C equal to the given angle; take
also A B, and A C equal to the given sides, and join
C B, and A B C will be the triangle required, as is
obvious. -
PROBLEM VIII.
Given two angles, and any side of a triangle to con-
struct the triangle, fig. 83.
There are two cases to this problem, accordingly as
the given side is adjacent to one only, or to both, the
given angles.
1. When the given side is adjacent to both the
given angles. |
Let A B be the given side, and A and B the given
angles. At A and B, make angles equal to the given
angles, and produce the lines till they intersect in C,
A BC will be the triangle required. -
2. Let A B be the given side, and A and C the given
angles. Produce A B to D, and at B make the angle
C B D equal to the sum of the two angles A and C;
and at A make the angle A equal to one of the given
angles, meeting B C in C, then will A B C be the
triangle required.
The first case requires no demonstration ; and in
the second, since CBD is equal to the sum of the given
angles, and since the three angles are equal to two
right angles, A B C must be equal to the third angle;
which reduces the problem to the former case.
tº PROBLEM IX.
Given two sides of a triangle, and an angle opposite to
one of them to construct the triangle, fig. 85.
Let A B be one of the given sides, and C A the
other, and B the given angle. At the point B make
the angle A B C equal to the given angle; and from A,
with A C as a radius, describe an arc cutting BD in C
and C/; join A C, AC", and A B C or A B Cº will be the
triangle required, as is obvious.
Scholium. It appears from the above, that when A C
is greater than the perpendicular A E, let fall from A to
BD, there are two triangles answering the required
conditions. If A C be equal to that perpendicular
distance, there is but one, and in that case the triangle
will be right angled; and if A C is less than the per-
pendicular distance A E, the construction is im-
possible.
PROBLEM X.
To describe a triangle that shall have its three sides
equal to three given lines, A, B, C, fig. 86.
Traw DE equal to C, and from D and E as centres,
and with radii equal to A and C, describe arcs inter-
secting in F; join D F, ET, and DE F will have its
three sides equal to the three given lines A, B, and C,
as is obvious.
It is necessary in this case that any two of the sides
be greater than the third.
Problems
PRoBLEM XI. relative to
Books II.
Given the two adjacent sides, A and B, of a paral- and iii.
lelogram, and the angle they include, to describe the paral- S-N-
lelogram, fig. 87.
Draw DE equal to B, one of the given sides, and at Fig. 87.
D make the angle FD E equal to the given angle;
take D F = A, and through F draw FG parallel to
D F, and through E, E G parallel to DF; so shall
E G F D be the parallelogram required. For DE = B,
and DF = A; by the construction and the sides being
parallel the opposite sides are equal, and the figure is
a parallelogram.
Cor. This construction comprehends the construc-
tion of the square and rectangle. It is only necessary
in these cases, that the angle D be made equal to a
rectangle.
PROBLEM XII.
To make a square equal to the sum of two given squares,
fig. 88.
Let A B, C B be the sides of the given squares : on Fig. 88.
A B, at the point B, erect the perpendicular B C, equal
to the other given line, and join A C, so will A C be
the side of the square required. For
A C2 = A B 2 + B C2.
Cor. Hence also we may make a square equal to
three or more squares : for produce B A and B C
towards D and E, (fig. 89,) and let G H be the side
of a third square; take B E = G H, and B D = A C,
and join DE; so shall DE* = A B* + B C* + G Hº ;
for D E * = D B 3 + BE2; and DB2 = A C2 = AB 2
+ B C2 and B E 2 = G H 2 : therefore D E 2 = A B*
+ B C * + G Hº ; and we may proceed in like man-
ner with any number of squares.
PROBLEMI XIII.
To make a square equal to the difference of two given
squares, fig. 90.
Let AB, BC be the sides of the given squares : on Fig. 90.
A B, the greater, describe the semicircle A B C ; and
from B, with the radius C B, describe the arc m n,
cutting the semicircle in C ; join C B, C A ; and CA
will be the side of the square required. For by the
construction C B is equal to the lesser given side B C,
and A B to A B ; and the angle C, being in a semi-
circle, is a right angle : therefore
A C 2 = A B2 – B C 2.
PROBLEM XIV.
To describe a circle through any three given points,
A, B, C, not in a right line, fig. 91.
From the middle point B draw the lines B.A, B C Fig. 91.
to the other two given points; and bisect these by
the perpendiculars DO, EO, which will intersect in
some point O; then from the centre O, and with the
distance OB, describe a circle which will pass through
the other two points A and C. For the two right
angled triangles O A.D, O B D, having the side A D,
D B equal, and OD common ; also the angles at D
right angles, will have their third sides likewise equal,
that is O A = O B ; and in the same way it may be
shown, that O C = O B; hence the three lines OA,
OB, O C, being all equal, are radii of the same circle.
328
G E O M E T R Y.
Geometry.
Fig. 92.
Fig. 93.
Fig. 94.
Fig. 95
Fig. 96. ,
ProBLEM xv.
To find the centre of any given circle, or of any arc of a
given circle, fig. 92. -
Take any three points in the given arc or circle, and
find the centre of the circle passing through them by
the last problem, and it will be the centre sought, as
is obvious.
PROBLEM XVI.
To draw a tangent to a given circle, through a given
point A, either in or beyond the circumference, fig.93.
Find the centre of the circle, and then first, if the
given point is in the circumference, join A and the
centre O, and at A draw B C perpendicular to AO, and
it will be the tangent required.
But if A be beyond the circumference, then also join
A and the centre O, and upon AO describe the semi-
circle A DO ; then from A, through D, draw the line
BC, and it will be the tangent sought. For AD O,
being an angle in a semicircle, is a right angle ; con-
sequently B C is perpendicular to DO, and is therefore
a tangent to the circle.
PROBLEM XVII.
Upon a given line A B, to describe a segment that may
contain a given angle C, fig. 94. -
At the ends of the given line make the angles DAB,
D BA, each equal to the given angle C.; and draw
A E, BE, perpendicular to A D, BD, and with the
centre E and radius E A, or B.E., describe a circle, so
shall AFB be the segment required; that is, any
angle F in it will be equal to the given angle C.
• For the two lines A D, BD, being perpendicular to
the radii EA, E B, are tangents to the circle ; and the
angle A or B, which is made equal to the given
angle C, is equal to the angle in the alternate segment
A F B. -
PROBLEM XVIII.
To cut off a segment from a given circle that shall
contain an angle equal to a given angle C, fig. 95.
Draw any tangent A B, to the given circle; and a
chord A D, making the angle D A B = C ; so shall
D E A be the segment required. -
For the angle A, made by the tangent and chord,
being equal to the angle C ; the angle E in the alter-
nate segment is also equal to the angle C.
PROBLEM XIX.
To inscribe a circle in a given triangle ABC, fig. 96.
Bisect the angles A and B with the two lines AD,
D; from the intersection D, draw the perpendicu-
lars DE, D F, D G, and they will be radii of the circle
required. For in the two triangles A D G, AED, the
angle DAG = E A G, and the angle D G A = DEA ;
therefore also G D A = A DE, because the sum of
the three angles of every triangle is equal to two
right angles. Hence the side A D, being common,
and the angles adjacent to it equal, the triangles are
equal, and the side D G = DE; in the same manner
it may be shown, that D F = DE ; consequently a
circle described from D, with the radius DE, will pass
through G and F ; and the sides AB, BC, CA, being
perpendiculars to these radii, will be tangents to the
circle; which is therefore inscribed in the triangle.
PRoBIEM XX.
To circumscribe a circle about a given triangle A B C,
fig. 97.
Bisect any two sides with two perpendiculars, as Fig. 97.
DF, DE, and D will be the centre : from D, with
the radius D A, describe a circle, which will pass
through A B C. The demonstration is the same as in
the last problem. * tº
BOOK IV.
of the proportions of figures, and the measure of areas.
1. SIMILAR FIGUREs, are those which have the angles
of the one equal to the angles of the other, each to
each, and the sides about the equal angles in each
proportional.
2. Homologous sides and angles, are those sides and
angles which have the same situation in any two
similar figures. -
3. In different circles similar arcs, similar segments,
and similar sectors, are those which correspond to
equal angles at the centre.
4. The base of any rectilineal figure is any side on
which the figure is supposed to stand.
5. The altitude of a parallelogram, or trapezoid, is
‘the perpendicular distance between the side taken for
a base and the side opposite.
6. The altitude of a triangle is the perpendicular
distance of its vertex from the base.
7. The area, or surface of a figure, is its superficial
content : and it is estimated numerically by the num-
ber of times it contains some other area which is
assumed for its measuring unit. -
8. Figures having equal areas, that is figures which
contain the same measuring unit the same number of
times, are said to be equal.
Hence figures may be equal to each other, although
they are not similar.
Some authors distinguish between figures, which are
both equal and similar, and those which are only equal
according to the above definition. In this case the
former are called identical, and the latter equal; or the
former equal, and the latter equivalent.
PRoPosition I.-Theorem.
The complements about the diagonal of any parallelo-
gram are equal to each other, fig. 98.
Let A C be a parallelogram and B D its diagonal : Fig. 98.
and let E F be parallel to D C, and G H to AD,
both passing through any common point I in the
diagonal ; then the figures A I, IC are called the
complements of the parallelograms E G,”H F, and it
is to be demonstrated that they are equal to each other.
Because the diagonals of parallelograms bisect them,
(prop. 27, book i.;) the triangles D G I, and D E I are
equal; for the same reason I H B and IFB are
equal ; as are likewise D A B and D C B : if therefore
from these last equal triangles there be taken on one
side the two triangles D GI and I FB, and on the
other the two triangles DEI and IH B, there will
remain the complement IC equal to the complement
A I. * *. •
... Book IV
G E O M E. T. R. Y. 329 -
base is equal to the sum of the two parallel sides, and its Book IV.
Geometry. - &
altitude the perpendicular distance between them, fig. 100, \-N->
Proposition II.-Theorem.
Parallelograms on the same base, and between the same
Fig. 99.
parallels are equal to each other, fig. 99. -
Let A B C D, A BE F be two parallelograms on
the common base A B, and between the same parallels
A B, D E, then will the parallelogram A B C D =
A BE F; because A B = D C, and A B = FE, (prop.
27, book i.) to each add CF, then will D F = C E ;
also D A = C B, and A F = BE, (prop. 27, book i.;)
therefore, in the two triangles DAF and CBE, the
three sides D F, DA, and A F are equal to the three
CE, CB, and BE, each to each ; therefore the
triangles themselves are also equal, (prop. 12, book i.;)
and if each of them be taken from the whole figure
A BDE, there will remain the parallelogram AlB CD
in the one case, equal to the parallelogram A B E F in
the other. -
Cor. 1. Parallelograms on equal bases, and between
the same parallels are equal; because the bases being
equal, the one figure may be applied to the other, so
that their bases shall coincide; and they may then be
considered as standing on the same base; which thus
reduces itself to the case above demonstrated.
Cor. 2. Because parallel lines have every where the
same perpendicular distance, which in this case is the
altitude of the parallelograms; it follows then that
parallelograms of equal bases and altitudes are equal
to each other.
Cor. 3. Every parallelogram is equal to a rectangle.
of the same base and altitude.
PRoPosition III.-Theorem.
Triangles on the same base and between the same paral-
lels are equal to each other, fig. 99.
Let A B C, ABF, be triangles upon the same base
and between the same parallels; the triangle ABC
= A B F ; produce CF, and draw A D parallel to BC
and B E to A F ; then will A B C D and A B E F, be
parallelograms upon the same base and between the
same parallels; therefore,by the last prop. A B C D =
A BEF; but the triangle A B C is half the paral-
lelogram A B C D, and the triangle A B F is half
the parallelogram A BEF, (prop. 27, book i.;) there-
fore the triangle A B C = A B F.
Cor. 1. Hence also triangles on equal bases and
between the same parallels are equal, for the equal
bases may be made to coincide, and the case thus
reduced to the above.
Cor. 2. Because the perpendicular distance from C
and F to the base A B, or A B produced, are equal,
(def. 12, book i.) which are the altitudes of the
triangles: it follows that triangles of equal bases
and altitudes are equal to each other.
Cor. 3. Since the triangle A B C is half the paral-
lelogram AlB C D ; or A B F half the parallelogram
A BE F; and that these are parallelograms of equal
bases and altitudes with the triangle ; it follows
that every triangle is equal to half a parallelogram of
the same base and altitude. .
Cor. 4. Hence a triangle is equal to half the rect-
angle of equal base and altitude.
PRoPosition IV.-Theorem.
A trapezoid is equal to half a parallelogram, whose
WOL. I. -
Let A B C D be a trapezoid whose two parallel sides Fig. 100.
are A B, D C ; produce A B to E, till B E = DC, and
D C to F, till CF = AB, and join FE, so shall ABCD
be equal to half the parallelogram AEFD, which
has for its base the sum of the two parallel sides, and
for its altitude the perpendicular distance between
them. Draw C G, B H parallel to AD or F. E.;
then because B E = DC, or AG, the two parallelo-
grams A C, B F, are upon equal bases, and between
the same parallels, therefore they are equal, (by cor. I,
last prop.) and because C B is the diagonal of the
parallelogram G. H.; the triangle C G B = C H B,
(prop. 27, book i.) consequently A C + C G B = BF4-
C HB, or A B C D = BE FC, or A B C D = half
the parallelogram. A E FD.
PRoposition V.—Theorem.
Triangles having the same altitude, are to each other in
the same ratio as their bases, fig. 101.
Let the two triangles ADC, DEF have the same Fig. 101.
altitude, they will have to each other the ratio of their
bases; that is, A D C : D E F : : A D : D E. -
Conceive the base AD of the triangle ADC divided
into any number of equal parts, or units of measure,
as A B, BD ; and let the same unit be repeated on the
base DE, till it either coincides with E, or fall beyond
it by a quantity less than the measuring unit, as at M,
and join CB, FG, FH, FM; thus dividing the trian-
gles A CD and D FM into a number of triangles, ACB,
B CD, D F G, G F H, &c. which are equal to each
other, having equal bases and altitudes, (prop. 3,
cor. 2, book iv.;) therefore the same number of
units which there are in the base A D, the same
number of equal triangles, or units, are there in the
triangle A C D ; and the same number of units there
are in the base DM, equal to those in A D, the
same number of triangles are there in D FM, each
also equal to those in A DC, therefore as
A D : D M :: A C D : D F M.
But DM may be made to differ from DE, by a quan-
tity less than the measuring unit; and the unit itself
may be taken less than any assignable quantity; there-
fore D M may be made to differ from DE, by a quan-
tity less than any that can be assigned, and at the same
time the triangle D F M will differ from DFE by less
than any quantity that can be assigned; consequently,
(see note to def. 1, book ii.)
- A D : D E :: A D C : D FE
Or ADC : D FE . . A D : D E.
Proposition VI.-Theorem.
Parallelograms of equal allitude, are to each other as
their bases, fig. 102.
Let A D KI, DE FK be parallelograms of equal Fig. 102,
altitude, they are to each other as their bases: for join
A K, D F ; then by the last proposition,
AKD : DE F : : A D : DE;
5ut the parallelogram A K is double of the triangle
A KD, and the parallelogram D F is double of the
triangle DEF, (prop. 27, book i.;) and equimultiples
of quantities have the same ratio as the quantities;
therefere as -
- AD KI; D E FK :: AD : D E.
2 x
330
G E O M ET. R. Y.
Geometry.
v
Fig. 103.
CG, in which let there be taken BL
Paoposition VII-Theorem.
Triangles and parallelograms having
to each other as their altitudes, fig, 103.
Let A B C, BEF be two triangles, having the bases
AB, BE equal, and whose altitudes are the perpendi-
culars CG, F H ; then will the triangle A B C : the
triangle BEF ::: C G : FH.
For, let B K be perpendicular to A B, and equal to
= F H ; and
equal bases, are
draw A K and A. L.
Then triangles of equal bases and altitudes being
equal, the triangle A B K = A B C, and A B L = BEF.
But considering now A B K, A BL as two triangles
on the bases BK, BL, and having the same altitude
A B, these will be as their bases, namely, the triangle
A B K . A B L : : B K : B L. But A B K = A B C,
and A B L =lb E F, also B K = C G, and B L = F H.
Therefore A B C : BEF : : C G : F H.
And since parallelograms are the doubles of triangles,
having the same bases and altitudes, these when their
bases are equal, will likewise have to each other the
same ratios as their altitudes.
Fig. 104.
PRoPosition VIII.-Theorem.
Triangles and parallelograms are to each other in the
ratio of the products of their bases and altitudes, fig. 104
Let ABC, EFG be any two triangles whose alti-
tudes are CD, G H, and bases A B, E F, then will
trian. AIBC : trian. EFG : ; AB x CD : EF × G.H.
Let KLM be another triangle whose base KL = AB,
and altitude M N = G. H. -
Because A BC and KLM have equal bases, they are
to each other as their altitudes ; and because E FG
and KLM have equal altitudes, they are to each other
as their bases: that is, in the former, AIBC : K.L.M
2 : DC : MN (prop. 5, book iv.;) in the latter, KL M
: EIF G : : KL : E F (prop. 7, book iv.) Hence by
(prop. 12, book ii.) -
ABC x KLM : EFG x KLM::DCXKL : MN × E.F.
Or since quantities have the same ratio, their equi-
multiples have ..º
A B C : E FG : : D C x K.L. : M N × E. F.
But K L = A B, and M N = G H ; therefore
ABC : E FG : : A B X D C : E F x G H.
And since every parallelogram is double of a triangle
of equal base and altitude, and that equimultiples of
quantities have the same ratio as the quantities,
(prop. 6, book ii.) it follows that parallelograms are
also to each other as the product of their bases and
altitudes.
Scholium. Since the area of parallelograms, and con-
sequently of rectangles, are to each other as the
product of their bases and altitudes, this product may
be assumed as the proper measure of such areas; by
which is to be understood, that as many units as there
are in the product of the base and altitude of any rect-
angle, the same number of units are there in the area
of the rectangle; the latter unit being the square
described upon the linear unit, by which the sides of
the figure are measured.
In the same way the area of a triangle is measured
by half the product of its base and altitude; and the
area of a trapezoid by the product of its altitude, by
half the sum of its two parallel sides.
. ſh -
Paoposition IX—Theorem.
The sum of all the rectangles contained under one whole
line, and the several parts of another line is equal to the
rectangle contained under the two whole lines, fig. 105.
Bobk IV,
... Let A D be one line, and AB another, divided into Fig. 105.
the parts A E, E F, FB; the rectanglés contained
under D A and A E, DA and EF, DA and FB are
together equal to the rectangle DA, A B. f
Let D A be perpendicular to A B, and A C, the
rectangle contained under D’A, A B ; conceive also
E G, F H, to be perpendicular to A B ; then because
D C is parallel to AIB, (def. 18, booki.) AD, EG, F H
and C B are all equal to each other, and the whole
figure or the rectangle of A B, and A D is divided into
the three rectangles A G, E. H., F C ; of which AG is
equal to the rectangle of A D and A E ; E H = the
rectangle of E F and E G, or EF and AD, because
E G = AD ; and F C = the rectangle of F B and
F H, or F B and A D, because H F = AD ; therefore
the rectangle AB x AD = A E x AD + E F x AD
+FB X AD. (schol. to last prop.)
PRoPosition X. —Theorem.
The square of the sum of two lines is greater than the
sum of their squares, by twice the rectangle of those lines,
fig. 106.
Let A B be the sum of any two lines A C and B C,
or AB = A C + B C ; then will A B* = AC 2 + BC*
+ 2 AC × B C. Let A B D E be the square on the
line A B, and A C F G the square on the line A C.
Produce C F and G F to the other sides at H and I.
From C H and G. I which are equal, being each equal
to the side of the square A B, or BD, (prop. 27,
cor. 1, book i.) take the parts CF, G F which are
also equal, being sides of the square on A C, and
there remains F H = FI, which are equal to DI,
H D, being opposite sides of a parallelogram,
(prop. 27, book i. :) the figure F I D H has there-
fore all its sides equal, and its angles are right angles;
it is therefore a square on the line FI, or on its equal
CB, (def. 17.) Again I C is a rectangle contained by
A C and C B, for C F = AC; and G H is a rectangle
contained by A C and B C ; for G F = AC, and F H
= FI = BC; therefore the whole square ABDE,
which is made up of the four figures, that is of the
two squares A F, FD, and the two rectangles FB,
Fig. 106.
and GH, is equal to the squares on AC and B C
and twice the rectangle A C × B C. *
Cor. Hence if a line be divided into two equal parts,
the square of the whole line is equal to four times
the square of half the line.
Proposition XI.-Theorem.
The square of the difference of two lines is less than the
sum of their squares, by twice the rectangle of the said
lines, fig. 107. . - -
Let A C, B C be any two lines, and A B their
difference, then will AB4 = A C* + BC” – 2 AC x CB.
For let ABDE be the square on the difference AB,
and ACFG the square on the line A. C. Produce
ED to H, also produce D B and H C, and draw KI,
making B I the square of the other line B.C. . .
Fig. 107.
, G E o M ET. R. Y. 331
= CAD half the rectangle A K, and as the doubles of Book IV.
Geometry. Now the square A D is less than the two squares
equal things are equal, the square A G is equal to the S-->
Sº-Yº - A F, BI, by the two rectangles EF, DI ; but GF =
Fig.
197. A C, and G E or FH=BC; consequently the rectangle rectangle A K ; and in like manner it may be shown,
E F contained under E G and G F is equal to the that the square CI is equal: to the rectangle BK;
rectangle of BC and A C. Again, FH being equal consequently the two squares AG, CI, are together
to CI or B C or DH, by adding the common part equal to the whole square on A B ; that is, A B*=
H C, the whole H I will be equal to the whole FC, A C* + B Cº. - - - - -
or equal to AC, and consequently the figure DI is Cor. 1. Hence the square on either side of a right
equal to the rectangle contained by A C and B C. angled triangle, is equal to the difference of the
Hence the two figures E F, DI are two rectangles squares on the hypothenuse and other side ; that is,
on the lines A C, B C, and consequently the square of A C* = A B* – B Cº, or B C* = A B*— A C*.
A B is less than the square of A C, B C by twice the Cor. 2. Because the rectangle under the sum and
rectangle A C x B.C. - difference of any two unequal lines, is equal to the
difference of their squares ; therefore the square on
PROPosition XII.-Theorem. either side of a right angled triangle is equal to the
The difference of the squares of any two unequal lines rectangle under the sum and difference of the hypo-
is equal to the rectangle under # . and #: of thenuse and the other side.
the same lines, fig. 108. - - 4.
Fig. 108, Let A B, A C be any two unequal lines, then will - PRoposition XIV.-Theorem.
A C* = AB+ BC. × AB-AC, In any triangle the difference of the squares of the
For let A BDE be the square of AB, and AC FG two sides is equal to the difference of the square of the
the square of AC, produce D B till Błł is equal to two lines or distances, included between the extremes of
Ac, and let Hibe parallel to AB or ED, and pro- the base and perpendicular, fig, 110.
duce FC both ways to I and K. Let A B C be any triangle, having CD perpendi- Fig. 110:
Then the difference of the two squares AD, AF, cular to AB, then will the difference of A Cº, B Cº, be
is evidently the two rectangles E F, KB ; but the equal to the difference of A D*, B D → ; that is,
rectangles E.F, BI are equal, being contained under A C*- B C* = A D* – B D*. -
equal lines; for E K and BH are each equal to AC, For since A C* = AD 3 + § (prop. 13, book iv.)
and G E is equal to C B, being each equal to the and . B C* = B D* + C D* prop. 13, bo .
difference between AB and AC, or their equals A E the difference between AD* + C D* and BD” + C D*
and AG; therefore the two EF, KB are equal to is equal to the difference between A D* and B D*
the two K B and B I, or to the whole KH; and by taking away the common square C D*. That is,
consequently KH is equal to the difference of the A C*- B C* = A D* – B D*. -
squares AD, AF; but K H is a rectangle contained Cor. Since A C* – B C2 = A C + B C x
under D H (or the sum of AB and A C,) and KD XCTEC - 2, book i -
(or the difference of A B and A. C.) Therefore the — (prop. 12, book iv.) — —
difference of the squares A B and AC, is equal to and, A Pº- B D* = AD + BD x AD – BP
the rectangle contained under the sum and difference of ºf follows that the rectangle under the sum and dif-
those lines. ference of the sides of a triangle, is equal to the
rectangle under the sum and difference of the seg-
PROPosition XIII.-Theorem. ments of the base, or the whole base and the sum
e e - and difference of the segments, according as the per-
In, every ſight ºngºd ºriangle; the square, ºf the pendicular faii without ºr within the bas.
hypothenuse is equal to the sum of the square of the other
two sides, fig. 109. . |P W XV.-Th
Fig. 109. Let A B C be a right angled triangle, having the ROPOSITION © €O7°67??,
right angle C ; then will the square on the hypothenuse
A B, be equal to the two squares on AC and B C, or
A B2 = A C 2 + B C2.
Let A E be the square on AB, A G the square on
A C, and C I the square on CB, and let C K be paral-
lel to A D or BE; join AI, BF, CD, and CE. Then
because the lines CG, CB meet the line AC, so as
to make two right angles, these two C G, C B are in
the same right line (prop. 2, book i.) and for the same
reason A C, C H are in the same right line. Because
the angle FA C = DAB, being each a right angle ;
to ... add the angle BAC, then will the angle
FA B be equal to the angle CAD ; and the two
triangles F A B, C A D will have the sides FA, AB,
equal to the two CA, AD, each to each, and the angles
included by these sides equal, therefore the triangles
are equal (prop. 4, book i.;) that is, the triangle
FAB = CAD ; but FAB = half the square AG,
being on the same base and between the same parallels
(prop, 3, cor. 3, book iv.;) and, for the same reason
In an obtuse angled triangle, the square of the side
subtending the obtuse angle, is greater than the sum of
the square of the other two sides, by twice the rectangle
of the base and distance of the perpendicular from the
obtuse angle, fig. 111.
Let A B C be a triangle obtuse angled at B, and Fig. Irri
CD perpendicular to A B; then will A C* = A B* +
B C 3 -H 2 A.B. × B D. * * -
For AD 9–AB* + 2 AB x B D (prop. 10, book. iv.)
and if we add C D 2 to each, their results s
AD 2 + C D* = A B* + BID3 + C D*-ī-2 A B x B.D.
But AD 3 + C D* = A C* and BD” + C D* = BC2;
therefore A C* = A B* + B Cº -H 2 A B x B. D.
Psorosition XVI—Theorem. - t
In any triangle, the square of the side subtending art
acute angle is less than the squares of the other two sides
by twice the rectangle of the base, and the distance of the
perpendicular from the acute angle, fig. 112.
2 x 2
332
G E O M E T R Y,
Geometry.
Let ABC be a triangle, having the angle at A
S-N-' acute, and C D perpendicular to A B, then will
Fig. 112.
Fig. 113.
Fig. 114.
Fig, 115,
B C2 = A B* + A C* – 2 A. B. × A D.
For in fig. 1, A C* = B C* + A B* + 2 AB x BD
by the last proposition, to each of these equals add the
square of AIB, " . .
then A B* + A C* = B C* + 2 A B* + 2 A. B. × B D
or = BC*-H 2 AB X AB -- B D = BC” + 2A B x AD
that is B Cº = A B2 + A C2 – 2 A B x AD. -
Again, in fig. 2, A C* = A D* + DC” (prop. 13,
book iv.) and A B* = A D* + DB2 + 2 AD x BD
(prop. 10, book iv.) therefore A B* + A C2 = BD”
+ D C* + 2 A D 2 + 2 A D x D B ;
Thut BD2 + D C2 = B C*;
therefore AB* + AC* = BC* + 2 AD24-2 AD x DB,
Or = BC*-ī- 2 A B x AID;
that is, B C* = A B* + A C* – 2 AB x AD,
Proposition XVII.-Theorem.
In any triangle, the double of the square of a line
drawn from the verter to the middle of the base, together
with double the square of the half base, is equal to the
sum of the squares of the other two sides, fig. 113.
Let A B C be a triangle, and C D the line drawn
from the vetex to the middle of the base, dividing it
into two equal parts A D, D, B, then will A C* +
C B2 – 2 C D 2 + 2 D B 2.
For AC* = DC” + AD” + 2 AD x DE, (prop. 15,
book iv.) = D C* + BID* + 2.BD × DE
and B Cº = DC* + BD” – 2 DB x DE, (prop. 16,
book iv.) therefore, by equal additions,
A C* + B C2 = 2 D C* + 2 DB2.
Paoposition XVIII.-Theorem."
In an isosceles triangle, the square of the line drawn
from the vertex to any point in the base, together with the
rectangle of the segments of the base, is equal to the square
of one of the equal sides of the triangle, fig. 114.
Let A B C be an isosceles triangle, and CD a line
drawn from, the vertex to any point in the base ;
then will the square on A C be equal to the square on
CD, together with the rectangle of AD and D B;
that is, A C* = CDs + A D x D B.
Let C E bisect the vertical angle, and it also bisect
the base perpendicularly (prop. 6, cor. I, book i.)
making AE = B E.
Now in the triangle A CD obtuse angled, as D, we
have A C* = C D* + A D* + 2 A D x DE (prop. 15)
= C D* + A D x AD + 2 DE
= C D* + A D x A E + DE
= C D2 + A D x B E + D E
= CD 2 + A D x D B.
PRoposition XIX.-Theorem:
In any parallelogram the sum of the squares of the two
diagonals is equal to the sum of the squares of the four
sides, fig. 115. - -
Let A B C D be a parallelogram, and A C, D B its
diagonals, then will -
A C2 + D B2 = AD 2 + D C2 + C B2 + AB 2.
For since the diagonals of parallelograms bisect each
other (prop. 30, book i.) DE = E B, and A E = EC;
therefore 2 EC2 + 2 E B2 = D C2 + C B3
and 2 A E2 + 2 E B2 = D A2 + A B*.
But A C2 = E C3, and 2E C2 = 2 AE°, therefore .
4 AE2 + 4 BE2 = A D2 + D C2 + B C* + A B*.
But 4 AE2 = A C2 and 4 BE2 = D Bº ; therefore
A C2 + DB2 = AD 3 + DC2 + B Cº + A B*.
PRoPosLTION XX.-Theorem.
A line drawn parallel to the base of a triangle, di-
vides the other two sides proportionally, fig, 116.
Let D E be drawn parallel to the side B C of the Fig. 116.
triangle A B C, then * *
9 A D : D B : . A E : E C.
Join B E and D C. . The two triangles BDE,
DEC having the same base D E, and the same alti-
tude, since both their vertices lie in a plain parallel
to the base, are equal, (prop. 3, book iv.)
The triangles AD E, BDE, whose common vertex
is E, have the same altitude, and are to each other as
their bases, (prop. 5, book iv.;) hence we have
r AD E : B D E . . A D : D B.
The triangles A DE, DE C, whose common vertex
is D, have also the same altitude, and are to each other
as their bases; hence -
A DE : D E C : : A E : E C.
But the triangles B DE, DEC are equal; and
therefore, since those proportions have a common
ratio, we obtain
A D : D B . . A E : E C.
Cor. 1. Hence, (prop. 5, book ii.) we have A D +
D B : AD :: AE + E C : AE, or A B : A D : : A C
: AE ; and also A B : B D : : A C : C E.
Cor. 2. If between two straight lines A B, CD,
(fig. 117,) any number of parallels A C, EF, G H,
BD, &c. be drawn, those straight lines will be cut
proportionally, and we shall have AF : C F : : E G
; F H :: G B : HD.
For, let O be the point where AB and CD meet. In
the triangle O EF, the line AC being drawn parallel
to the base E F, we shall have O E . A E . . O F :
CF, or OE : O F :: AE : C F. In the triangle
O G. H., we shall likewise have O E : E G : ... O F :
EH, or O E : O F : : E G : FH. And by reason of
the common ratio O E : O F, those two proportions
give A E : C F : : E G : FH. It may be proved in
the same manner, that EG : F PH . . GIB : H D, and
so on; hence the lines A B, CD are cut proportionally
by the parallels A C, EF, G H, &c.
Proposition XXI-Theorem.
If the sides of a triangle are cut proportionally by any
line DE, so that we have A D : D B : . A E : E C, the
line D E will be parallel to the base B C, fig, 118.
For if D E is not parallel to B C, suppose that DO Fig. 118.
is parallel to it. Then, by the preceding theorem, we
shall have A D : B D : : A O : O C. But, by hypo-
thesis, we have AD : D B : : A E : E C ; hence we
must have A O ... O C : . A E : E C, or A O : A E
:: O C : E C ; an impossible result, since AO, the
one antecedent, is less than its consequent. A E, and
O C, the other antecedent, is greater than its 'conse-
quent E C. Hence the parallel to B C, drawn from
the point D, cannot differ from DE ; hence D E is
that parallel.
Scholium. The same conclusion would be true, if
Book IV.
S-N-
G E O M E T R Y.
333
Geometry, the proportion : AB AD :: A C : AE were the pro-
>-N- posed one.
Fig. 119.
Fig. 120.
For this proportion, would give us A B —
AD : A D :: A C–A E : A E, or B D : A D . . C. E.
: A E, (prop. 5, book ii.)
PRoPosition XXII.—Theorem.
The line which bisects any angle of a triangle, divides
the base into two segments, which are proportional to the
adjacent sides, fig. 119. -
Let A D bisect the angle B A C of the triangle
A B C, then A D divide C B in the proportion of CA
to B.A., or - - -
C A : B A : : C D : D B.
Through the point C, draw C E parallel to A D till
it meet B A produced.
In the triangle BC E, the line A D is parallel to
the base CE; hence (prop. 20, book iv.) we have the
proportion B D : D C : . A B : A E.
But the triangle ACE is isosceles: for since AD, CE
are parallel, we have the angle A C E = D A C, and
the angle A E C = B.A.D., (prop. 19, book i.;) and,
by hypothesis, D A C = B.A. D; hence the angle ACE
= AEC, and consequently A E = AC. In place of
A E in the above proportion, substitute A C, and we
shall have B D : D C . . A B : A C,
PRoPosſTIon XXIII.—Theorem.
Two equiangular triangles have their homologous sides
proportional, and are similar, fig. 120.
Let A B C, CD E be two triangles which have
their angles equal, each to each, namely, B A C =
CD E, A B C = D CE, and A C B = DE C; then the
homologous sides, or the sides adjacent to the equal
angles, will be proportional, so that we shall have
B C : C E :: A B : C D : : A C : D E.
Place the homologous sides BC, C E in the same
straight line; and produce the sides B.A., ED till they
meet in F.
Since B C E is a straight line, and the angle BCA is
equal to CED, it follows (prop. 19, book i.) that AC
is parallel to D.E. In like manner, since the angle
A B C is equal to D CE, the line A B is parallel to
D.C. Hence the figure A CD F is a parallelogram.
. In the triangle BFE, the line A C is parallel to the
base FE ; hence (prop. 20, book iv.) we have B C :
CE : : B A : A F; or, putting CD in the place of its
equal A F,
- B C : C E : : B A : CD.
In the same triangle B E F, if B F be considered as
the base, CD is parallel to it; and we have the pro-
portion B C : C E : ; FD : D E ; or putting A C in
the place of its equal F D, *
B C : C E :: A C : D E.
And finally, since both those proportions contain the
same ratio B C : C E, we have
A C : DE 4: B.A. : CD.
Thus the equiangular triangles B A C, CDE have
their homologous sides proportional. But two figures
are similar when they have their angles respectively
equal, and their homologous sides proportional; con-
sequently the equiangular triangles BAC, CDE, are
two similar figures.
Cor. For the similarity of two triangles it is enough
that they have two angles equal, each to each ; since
the third will also be equal in both, and the two
triangles will be equiangular -
PROPosition XXIV.-Theorem.
Two triangles which have their homologous sides pro-
portional, are equiangular and similar, fig. 121.
Let B C : E F :: A B : DE : A C : D F : then Fig. 121.
will the triangles ABC, DEF have their angles equal,
namely, A = D, B = E, C = F.
At the point E, make the angle FE G = B, and at
F, the angle EFG = C; the third G will be equal
to the third A, and the two triangles ABC, EFG will
be equiangular. Therefore, by the last theorem, we
shall have B C : E F : : A B : E G ; but, by hypo-
thesis, B C : E F : : A B : D E ; hence E G = D E.
By the same theorem, we shall also have B C : E F
:: A C : FG ; and, by hypothesis, B C : E F : A C
: D F; hence FG = DF. Hence (prop. 12, book i.)
the triangles E G F, DE F, having their three sides
respectively equal, are themselves equal. But, by
construction, the triangles E G F and A B C are equi-
angular ; hence DEF and ABC are also equiangular
and similar. & -
Scholium. By the last two propositions, it appears
that in triangles, equality among the angles is a con-
sequence of proportionality among the sides, and
conversely; so that one of those conditions suffi-
ciently determines the similarity of two triangles,
The case is different with regard to figures of more
than three sides . even in quadrilaterals, the propor-
tion between the sides may be altered without alter-
ing the angles, or the angles be altered without alter-
ing the proportion between the sides ; and thus pro-
portionality among the sides cannot be a consequence
of equality among the angles of two quadrilaterals,
or vice versa. It is evident, for example, that by
drawing E F (fig. 122) parallel to B C, the angles of Fig. 122.
the quadrilateral A E FD, are made equal to those of
A B C D, though the proportion between the sides is
different ; and, in like manner, without changing the
four sides A B, BC, CD, AID, we can make the point
B approach D or recede from it, which will change
the angles. - .
Proposition XXV-Theorem.
Two triangles which have an equal angle included be-
tween proportional sides, are similar, fig. 123.
, Let the angles A and D be equal ; if A B : D E . . Fig. 123.
A C : D F, the triangle ABC is similar to DE F.
Take A G = DE, and draw GH parallel to B C.
The angle A G H (prop. 19, book i.) will be equal to
the angle A B C ; and the triangles A G H, A B C will
be equiangular: hence AB : A G :: AC : AH. But,
by hypothesis, A B : D E : : A C : D F ; and, by
construction, A G = DE : hence A H = DiF. The
two triangles A G H, DEF have an equal angle in-
cluded between equal sides; therefore they are equal;
but the triangle A G H is similar to A B C ; therefore
DE F is also similar to ABC.
PRoposition XXVI.—Theorem.
Two triangles which have their homologous sides parallel,
or perpendicular to each other, are similar, fig. 124 and
125. - Y
First. If the side A B is parallel to DE, and B C to Fig. 124
EF, the angle ABC will be equal to DEF; for ABC and 125,
= A H C = DE C, (prop. 19, book i.;) and if A C is
Book IV.
S-N-
334
G E O M ET R Y.
The triangles B.A.D and BAC have the common Book IV.
angle B, the right angle BDA = BAC, and therefore S-Sºy
Geometry, parallel to DF, the angle ACB will be equal to D FE,
S-N-' and also B A C to EDF ; hence the triangles A B C,
Fig. 136.
DEF are equiangular; hence they are similar. ,
Secondly. If the side DE is perpendicular to AB,
and the side D F to AC, the two angles I and H of the
quadrilateral AID H will be right; and since all the
four angles are together equal to four right angles,
the remaining two IAH, IDH will be together equal
to two right angles. But the two angles EDF,
ID H are also equal to two right angles: hence the
angle EDF is equal to IAH or B.A. C. In like man-
ner, if the third side EF is perpendicular to the third
BC, it may be shown that the angle DFE is equal to
C, and DE F to B; hence the triangles A B C, D E F,
which have the sides of the one perpendicular to the
corresponding sides of the other, are equiangular and
similar.
Scholium. In the case of the sides being parallel, the
homologous sides are the parallel ones : in the case
of their being perpendicular, the homologous sides
are the perpendicular ones. Thus in the latter case
DE is homologous with AB, D F with A C, and E F
with B C. -
The case of the perpendicular sides might present
a relative position of the two triangles different from
that exhibited in the diagram ; but the equality of
the respective angles might still be demonstrated,
either by means of quadrilaterals like AID H having
two right angles, or by the comparison of two trian-
gles having two of their angles vertical, and a right
angle in each. Besides, we may always conceive a
triangle DE F to be constructed within the triangle
A BC, and such that its sides shall be parallel to those
of the triangle compared with A B C ; and then the
demonstration given in the text will apply. -
PaopositroN XXVII.-Theorem.
Any lines drawn through the verter of a triangle, will
divide the base, and a line parallel to the base, in the same
proportion, fig. 126.
Let AF, A G, A H be drawn from the vertex A to the
base B C of the triangle A B C, and let D E be parallel
to B C ; then will DI : D F ::: I K : FG ::: KL :
G H, &c. -
For since DI is parallel to BF, the triangles A D F :
and A B F are equiangular; and D I : B F :: A I :
AF; also, since IK is parallel to FG, we have in like
manner AI : A F : ; IK : FG ; hence, the ratio AI:
AF being common, DI : B F : ; IK : FG. In the
same manner I K . F.G. . . K L : G H ; and so with
the other segments : hence the line DE is divided at
the points I, K, L, as the base B C at the points
F, G, H. -
Cor. Therefore if B C were divided into equal parts
at the points F, G, H, the parallel D E would also be
divided into equal parts at the points I, K, L,
PRoPosition XXVIII–Theorem.
If from the right angle of a right angled triangle, a
perpendicular be let fall on the hypothenuse; the two trian-
gles thereby made, will be similar to the whole triangle,
and to one another. Each side of the triangle will be a
mean proportional between the whole base and the adjacent
segment, and the perpendiculars will be a mean propor-
tional between the two segments, fig. 127. . -
the third angle BAD of the one equal to the third c Fig. 27.
of the other; hence those two triangles are equian-
gular and similar. In the same manner it may be
shown, that the triangles D A C and BAC are similar;
hence all the three triangles are similar and equian-
ular. - 4.
§ Again, the triangles BAD, BAC being similar, their
homologous sides are proportional. But BID in the
triangle A BD, and BA in the triangle A B C are ho-
mologous, because they lie opposite the equal angles
B AD, B C A ; the hypothen use B.A. of the former is
homologous with the hypothenuse B C of the latter :
hence the proportion B D : B A : B A : B C. By
the same reasoning, we should find D C : A C : : A C
: BC; hence each of the sides A B, A C is a mean
proportional between the hypothenuse and the seg-
ment adjacent to that side. -
Further, since the triangles ABI), ADC are similar,
by comparing their homologous sides, we have B D
: A D : : A D : D C ; hence, the perpendicular A D
is a mean proportional between the segments D B, DC
of the hypothenuse.
Scholium. Since B D : A B : : A B : B C, the pro-
duct of the extremes will be equal to that of the
means, or A B* = B D . B C. For the same reason
we have A C* = D.C.B C ; therefore A B2 + A C2 =
B D. BC + DC. BC = (BD+D C.) B C=B C. BC
= B C*; or the square described on the hypothenuse
B C is equal to the squares described on the two sides
AB, A.C. Thus we again arrive at the property of the
square of the hypothenuse, by a path very different from
that which formerly conducted us to it; and thus it
appears, that the property of the square of the hypo-
thenuse is a consequence of the more general property,
that the sides of equiangular triangles are propor-
tional. Thus the fundamental propositions of geo-
metry are reduced, as it were, to this single one, that
equiangular triangles have their homologous sides
proportional. -
It happens frequently, as in this instance, that by
deducing consequences from one or more propositions,
we are led back to some proposition already proved.
In fact, the chief characteristic of geometrical theo-
rems, and one indubitable proof of their certainty is,
that, however we combine them together, provided
only our reasoning be correct, the results we obtain
are always perfectly accurate. The case would be
different, if any proposition were false or only approx-
imately true ; it would frequently happen that on
combining the propositions together, the error would
increase and become perceptible. Examples of this
are to be seen in all the demonstrations, in which the
'reductio ad absurdum method is employed. In such
demonstrations, where the object is to show that two
quantities are equal, we proceed by showing that if
there existed the smallest inequality between the
quantities, a train of accurate reasoning would lead
us to a manifest and palpable absurdity; from which,
we are forced to conclude that the two quantities are
equal.
Cor. If from a point A, (fig. 128,) in the circum- Fig. 128.
ference of a circle, two chords A B, A C be drawn to
the extremities of a diameter B C, the triangle BAC
(prop, 17, book iii.) will be right angled at A ; hence,
first, the perpendicular A D is a mean proportional be-
G. E. O. M. E. T. R. Y.
335
Geometry, tween the two segments B D, DC, of the diameter, or
\-y-' what amounts to the same, A D* = B. D. D. C.
Fig. 129.
Fig. 130,
Fig. 131,
Hence also, in the second place, the chord A B is a
mean proportional between the diameter B C and the
adjacent segment B D, or what amounts to the same,
A B* = B.D .B.C. In like manner, we have A C * =
CD. BC; hence A B* : A C* : : B D : D C ; and
comparing A B* and A C2 to B C*, we have A B* :
B Cº.: : B D : B C, and A C* : B C2 : : D C : B C.
Proposition XXIX.—Theorem.
Two triangles having an equal angle, are to each other
as the rectangles of the sides which contain that angle,
fig. 129. - -
That is, the triangle ABC is to the triangle ADE,
as the rectangle A B. A C is to the rectangle A D.
A E.
Draw B.E. The triangles A BE, ADE, having
, the common vertex E, have the same altitude, and
consequently (prop. 5, book iv.) are to each other as
their bases : that is,
A BE : AD E : : A B : A D.
In like manner,
A B C : A BE :: A C : A E.
Multiply together the corresponding terms of those
proportions, omitting the common term A B E ; we
have A B C : A D E . . A B . A C : A D. A. E.
PRoPosition XXX.-Theorem.
Two similar triangles are to each other as the squares of
their homologous sides, fig. 130.
Let the angle A be equal to D, and the angle B = E.
Then, first, by reason of the equal angles A and D,
according to the last proposition, we shall have
ABC : D EF :: A B. A C : D E. D F.
Also, because the triangles are similar,
AB : DE :: A C : D F.
And multiplying the terms of this proportion by the
corresponding terms of the identical proportion,
A C : D F : A C : D F,
there will result
A B. A C : D E. D.F. : : A C2 : D F2.
Consequently,
- A B C : DEF : : A C2 : D F 9.
Therefore two similar triangles A B C, DEF are
to each other as the squares of the homologous sides
A C, D F, or as the squares of any other two homo-
logous sides.
PROPosition XXXI.-Theorem.
Two similar polygons are composed of the same number
of triangles similar, each to each, and similarly situated,
fig. 131. *
From any angle A, in the polygon A B C DE, draw
diagonals A C, A D to the other angles. From the
corresponding angle F, in the other polygon FG HIK,
draw diagonals FH, FI to the other angles.
These polygons being similar, the angles A B C,
FG H, which are homologous, will be equal, (def. 1
and 2,) and the sides A B, BC will also be propor-
tional to FG, GH; that is, A B : FG : : B C : G. H.
Wherefore the triangles A B C, F G H have each an
equal angle, contained between proportional sides;
they are therefore similar ; hence the angle BCA is Book Iv.
equal to GHF. Take away these equal angles from S-N-'
the equal angles B C D, G HI; there remains ACD
= F HI. But since the triangles A B C, F G H are
similar, A C : FH : : B C : G. H.; and (def. 1) since
the polygons are similar, B C : G H . : C D : HI;
hence A C : FH : CD : H L. But the angle A CD
is equal to FHI; hence the triangles A CD, F HI
have an equal angle in each, included between pro-
portional sides, and are consequently similar. In the
same manner all the remaining triangles may be
shown to be similar ; therefore two similar polygons
are composed of the same number of triangles similar
and similarly situated.
Scholium. The converse of the proposition is equally
true : If two polygons are composed of the same number
of triangles similar and similarly situated, those two
polygons will be similar.
For the similarity of the respective triangles will
give the angles A B C = FG H, BC A=G HF, A CD
= FH I; hence B C D = G H I, likewise CD E =
HIK, &c. Moreover we shall have A B : FG : : BC
: G H ! : A C : FH ! : C D : H I, &c. hence the
two polygons have their angies equal and their sides
proportional; hence they are similar.
PRoPosition XXXII.-Theorem.
The contours or perimeters of similar polygons are to
each other as the homologous sides ; and the surfaces are
to each other as the squares of those sides, fig. 131.
By the nature of similar figures, we have A B : Fig. 131.
F G. : : B C : G.H. : : C D : H I, &c.; therefore
(prop. 9, book ii.) the sum of the antecedents A B +
B C + C D, &c. (the perimeter of the first polygon)
is to the sum of the consequents F G + G H + HI,
&c. (the perimeter of the second polygon) as any
one antecedent is to its consequent, that is as the side
A B is to its corresponding side F G.
Again, since the triangles A B C, F G H are similar,
we have (prop. 30, book iv.) the triangle A B C :
R G H ! : A C 2 : F Hº ; and in like manner, from the
similar triangles A CD, F HI, we shall have A CD :
FH I : : A C* : F PI*; therefore, by reason of the
common ratio, A C* : F PH%, it follows that
A B C : FG H . . A C D : F HI.
By the same mode of reasoning,
A CD : F H I : : A DE : F I K;
and so on, if there were more triangles. Consequently
(prop. 9, book ii.) the sum of the antecedents A B C
+ A CD + A DE, or the polygon A B C D E, is to
the sum of the consequents FG H + F HI + F I K,
or to the polygon FG HIK, as one antecedent A B C
is to its consequent FG H, or as A B* is to FG% ;
hence the surfaces of similar polygons are to each
other as the squares of the homologous sides.
Cor. If three similar figures were constructed, on
the three sides of a right angled triangle, the figure
on the hypothenuse would be equal to the sum of the
other two : for the three figures are proportional to
the squares of their homologous sides; but the square
of the hypothenuse is equal to the sum of the squares
of the two other sides; hence, &c. * - a
Proposition XXXIII.-Theorem. ,
The segments of two chords which intersect each other
in a circle, are reciprocally proportional, fig. 139.
336
G E O M ET R Y.
Fig. 132.
Fig. 133,
Pig. 134.
Fig. 135.
AO : D O :: CO ... O B.
Join A C and BD. In the triangles A CO, BOD
the angles at O are equal, being vertical ; the angle
A is equal to the angle D, because both are inscribed
in the same segment, (prop. 15, book iii.;) for the
same reason the angle C=B; the triangles are there-
fore similar, and the homologous sides give the pro-
portion, AO : D O :: CO : O B. .
Cor. Therefore A O. O B = D O. CO; hence the
rectangle under the two segments of the one chord is
equal to the rectangle under the two segments of the
other
PROPositroN XXXIV.-Theorem.
If from the same point without a circle secants be drawn
terminating in the concave arc, the whole secants will be
Teciprocally proportional to their external segments,
fig. 133. -
That is, O B : O C : : OID : O A.
IFor, join A C, B D, then the triangles OAC, O BID
have the angle O common; likewise the angle B = C
(prop, 15, book iii.;) these triangles are therefore
similar; and their homologous sides give the propor-
tion, O B : O C : : O D : O A.
Cor. The rectangle O A. OB is hence equal to the
rectangle O C. O D.
Scholium. This proposition bears a great analogy to
the preceding, and differs from it only as the two
chords A B, C D, instead of intersecting each other
within the circle, cut each other externally. The fol-
lowing proposition may also be regarded as a particu-
lar case of the proposition just demonstrated.
PRoPosition XXXV.-Theorem.
If from a point without a circle, a tangent and a secant
be drawn, the tangent will be a mean proportional between
the secant and its external segment, fig. 134.
That is, O C : O A : : O A : O D,
Or O A2 = O C . O D.
For, joining A D and A C, the triangles O A D,
O A C have the angle O common ; also the angle
O A.D, formed by a tangent and a chord, has for its
measure (prop. 18, book iii.) half of the arc A. D.; and
the angle C has the same measure : hence the angle
O A D = C ; and the two triangles are similar, and
we have the proportion,
O C : O A : : O A : O D,
which gives O A2 = O C. O D.
PRoPosition XXXVI.-Theorem.
If any angle of a triangle be bisected by a line which
cuts the base ; the rectangle of the segments of the base,
together with the square of the bisecting line, is equal to
the rectangle of the sides, including the bisected angle,
fig. 135.
Let A D bisect the angle BAC of the triangle ABC;
then B A . A C = B D . D C + D A2. -
Describe a circle through the three points A, B, C ;
produce AID till it meets the circumference, and join
C E. -
The triangle BAD is similar to the triangle E A C;
for, by hypothesis, the angle B A D = E A C; also the
angle B = E, since they both have for measure half
of the arc A. C.; hence these triangles are similar, and
the homologous sides give the proportion, BA : AE Book. IV.
; : A D : A C ; hence B.A. A C = D E. A D ; but A E \-N--
= A D + D E, and multiplying each of these equals
by AD, we have A E. A D = A D* + AID. DE; now
AD. DE = B. D. DC, (prop. 33, book iv.;) hence
finally, B.A. A C = A D* + B D. D. C.
Paorosition XXXVII.-Theorem.
In any triangle, the rectangle of any two of its sides is
equal to the rectangle of the perpendicular let fall on its
third side, and the diameter of its circumscribing circle,
fig. 136,
Let AD be the perpendicular upon B C, and EC Fig. 136.
the diameter of the circumscribing circle; then
: - B A . A C = A D . E. C.
For, joining A E, the triangles A BD, A E C are
right angled, the one at D, the other at A ; also the
angle B-E; these triangles are therefore similar, and
they give the proportion, A B : C E :: A D : A C ; and
Hence A B . A C = C E . A D.
Cor. If these equal quantities be multiplied by the
same quantity BC there will result A B. A. C. B C =
C E. A. D. BC; now A D. B C is double of the sur-
face of the triangle, (prop. 8, book iv.;) therefore
the product of the three sides of a triangle is equal to its
surface multiplied by twice the diameter of the circum-
scribed circle.
Scholium. It may also be demonstrated, that the
surface of a triangle is equal to its perimeter multi-
plied by half the radius of the inscribed circle.
For the triangles A. OB, B O C, A O C, (fig. 137) Fig. 137.
which have a common vertex at O, have for their
common altitude the radius of the inscribed circle ;
hence the sum of these triangles will be equal to the
sum of the bases A B, BC, A C, multiplied by half the
radius O D; hence the surface of the triangle ABC
is equal to the perimeter multiplied by half the radius
of the inscribed circle. .
PRoPosition XXXVIII.-Theorem.
In every quadrilateral inscribed in a circle, the rectangle
of the two diagonals is equal to the sum of the rectangles
of the opposite sides, fig. 138. -
That is, A. C. B D = A B. C D + A D. B. C.
Take the arc CO = A D, and draw B O meeting
the diagonal A C in I.
The angle A B D = C B I, since the one has for its
measure half of the arc A D, and the other half of CO
equal to A D ; the angle A D B = B C I, because they
are both inscribed in the same segment AO B; hence
the triangle A B D is similar to the triangle I B C, and
we have the proportion AD : C I : : B D : BC; hence
A D. B C = C I. BD. Again, the triangle A B I is
similar to the triangle B D C ; for the arc A D being
equal to C O, if O D be added to each of them, we
shall have the arc A.O = D C ; hence the angle A BI
is equal to DB C ; also the angle BA I to B D C,
because they are inscribed in the same segment; hence
the triangles A B I, D B C are similar, and the homo-
logous sides give the proportion, A B : B D : : A I :
C D ; hence A B. C D = A I. B. D. -
Adding the two results obtained, and observing that
A I. BD+ C I. B D = (A I + C I.) B D = A C. B.D,
we shall have A D. B C + A B. C D = A C : B D.
Scholium. Another theorem concerning the in-
G E O M E T R Y. 337
Problems
Again, let it be proposed to divide the line AB relative to
(fig. 141) into parts proportional to the given lines P, *.*.*
Q, R. Through A, draw the indefinite line A G ;
make A C = P, CD = Q, DE = R ; join the extre- Fig. 141.
Geometry, Scribed quadrilateral may be demonstrated in the same
S-N-" manner : - - -
The similarity of the triangles A B D and B IC gives
the proportion B D : B C : : A B : B I; hence B I.
I} D = BC. A. B. If C O be joined, the triangle
ICO, similar to A B I, will be similar to B D C, and
will give the proportion B D : C O : : D C : O I;
hence O I . B D = CO. DC, or, because C O =
A D; O I. BD = AD . D. C. Adding the two re-
sults, and observing that B I. B D + O.I. B D is the
same as B O - B D, we shall have B O . B D = AIB .
B C + A D. D. C.
If B P. had been taken equal to AD, and C K P
been drawn, a similar train of reasoning would have
given us -
C P. C A = A B. A D + B C . C. D.
But the arc B P being equal to CO, if B C be added
to each of them, it will follow that C BP = BC O ;
the chord C P is therefore equal to the chord BO, and
consequently B O . BD and C.P. CA are to each other
as BD is to CA; hence, e t -
BD: CA: ; AB . BC + AID. DC : AID. AB+ BC.C.D.
Therefore the two diagonals of an inscribed quadrila-
teral are to each other, as the sums of the rectangles under
the sides which meet at their extremities.
These two theorems may serve to find the diagonals
when the sides are given.
Proposition XXXIX-Theorem.
Let P be a given point within a circle upon the radius
A C, and let a point Q be taken externally upon the same
radius produced, so that C P : C A : : C A : C Q ; if
from any point M of the circumference straight lines
MP, M Q be drawn to the two points P and Q, these
straight lines will every where have the same ratio, or
MP : M Q : A P : A Q, fig. 139.
mities E and B; and through the points C, D draw
CI, D F parallel to E B ; the line A B will be divided
into parts AI, IF, FB proportional to the given lines
P, Q, R. º
For, by reason of the parallels C I, DF, E B, the
parts A I, IF, F B are proportional to the parts AC,
CD, DE; and, by construction, these are equal to the
given lines P, Q, R.
ProBLEM II.
To find a fourth proportional to three given lines
A, B, G, fig. 142.
Draw the two indefinite lines DE, DF, forming any Fig. 142.
angle with each other. Upon DE take DA = A, and
DB = B; upon D F take D C = G ; join A C ; and
through the point B, draw B X parallel to A C ; D X
will be the fourth proportional required : for, since
B X is parallel to A C, we have the proportion D A :
D B : : D C : D X ; now the three first terms of that
proportion are equal to the three given lines; conse-
quently DX is the fourth proportional required.
Cor. A third proportional to two given lines A, B,
may be found in the same manner, for it will be the
same as a fourth proportional to the three lines,
A, B, B.
PROBLEM III.
To find a mean proportional between two given lines
A and B, fig. 143.
Upon the indefinite line DF, take DE = A, and Fig. 143.
E F = B; upon the whole line DF, as a diameter,
Fig. 139. For by hypothesis, C P : C A : : C A : CO; or sub- describe the semicircle D G F : at the point E, erect
stituting CM for CA, C P : CM : : CM: CG); hence the upon the diameter the perpendicular E G meeting the
triangles C PM, C Q M, have each an equal angle O circumference in G.; EG will be the mean propor-
contained by proportional sides ; hence they are tional required. º Fºl frº tº a g
similar ; and hence the third side MP is to the third For the perpendicular E G, let fall from a point in
side M Q, as CP is to CM or CA. But (prop.5, book ii.) the circumference upon the diameter, is a mean pro-
the proportion C P : C A : : C A : C Q gives C P : CA portional between DE, DF, the two segments of the
; : C A – C P : C Q – CA, or CP : C & : ; AP: A Q; diameter, (prop. 28, book iv. 3) and these segments
therefore MP : M Q - : A P : A Q. are equal to the given lines A and B. -
PROBLEM IV. -
Problems relating to book IV. To divide a line in extreme and mean ratio, that is into
- g two parts, such that the greater part shall be a mean
- PRoBLEM I. w proportional between the whole line and the other part,
To divide a given straight line into any number of equal fig. 144.
parts, or into parts proportional to given lines, fig. 140. At the extremity B of the line A B, erect the per- Fig. 144,
Fig. 140. Let it, for example, be proposed to divide the line pendicular B C equal to the half of A B; from the
A B into five equal parts. Through the extremity A,
draw the indefinite straight line A G ; and taking A C
of any magnitude, apply it five times upon A G ; join
the last point of division G, and the extremity B, by
the straight line G B ; then draw C I parallel to G R :
AI will be the fifth part of the line AIB ; and thus, by
applying AI five times upon AB, the line A B will be
divided into five equal parts. -
For, since C I is parallel to GB, the sides AG, A B
(prop. 20, book iv.) are cut proportionally in C and I.
But AC is the fifth part of A G, hence AI is the fifth
part of A B.
WOL. I.
point C as a centre, with the radius CB describe a
semicircle; draw AC cutting the circumference in D;
and take A F = A D : the lime A B will be divided at
the point F in the manner required ; that is, we shall
have A B : A F : : A F : F B. •
For A B, being perpendicular to the radius at its
extremity, is a tangent ; and if A C be produced till
it again meets the circumference in E, we shall have
(prop., 35, book iv.) AE : A B : : A B : AD; hence,
(prop. 5, book ii.) A E – A B : A B ; : A B — A D :
A D. But since the radius is the half of A B, the
diameter D E is equal 'to A B, and consequently AE
2 Y
338 G E G M E. T. E. Y. *
Geometry. — AB = AD = AF; also, because AF = AD, we proportional X to the lines A and C so that A =C= - Problems
~~ have A B — A D = FB; hence AF : A B : : FB : C : X; then A* : C*; : A : X. º
AD or AF; whence (prop. 4, book ii.) A B : A F T w - , OOK I W.
Fig. 145, Through the point A draw AE parallel to C D, the three given lines, P, Q, R. The two lines X, Y
make B E = CE, and through the points B and A will be to each other as the products A. B. C.,
draw BAID; this will be the line required. P. Q . R.
For, A E being parallel to CD, we have BE : EC For, since P : A : : B : X, it follows that A : B =
: : B A : A D ; but BE = EC; therefore B A = AD. P.X; and multiplying each of these equals by C, we
- - have A. B. C = C. P. X. In like manner, since C :
PROBLEM VI. Q : : R : Y, it follows that Q. R = C. Y.; and multi-
- sy ºr T - - Jºe lying each of these equals by P, we have P. Q. R.
º. 6. º that #. * equal to f given Pºc . Y.: hence i. product A. B. C is to the
Parallelºgram, or to a given triangle, fig. 146 and 147 product P. Q.R as C. F. x is to P. C. y, or as x
Fig. 146, Let AB C D be the given parallelogram, A B its is to Y. - . - . .
base, DE its altitude ; between A B and D E find a * ,
mean proportional XY ; then will the square con- - |PROBLEM X. - -
º, upon X, Y be equal to the parallelogram To find a triangle that shall be equal to a given poly-
For, by construction, A B : X Y : : X Y : D E ; gon, fig. 151. - s - - . .
therefore x Y2 = A B.D E ; but AB. DE is thé . Let A B C DE be the given polygon. Draw first Fig. 151.
measure of the parallelogram, and XY8 that of the the diagonal CF, cutting off the triangle C. DE ;
square; consequently they are equal. through the point D, draw D F parallel to CE, and
Fig. 147: Again, let A B C (fig. 147) be the given triangle, meeting AE produced ; join CF; the polygon A
B C its base, AD its altitude: find a mean propor- B C D E will be equal to the polygon A B C F, which
tional between BC and the half of AD, and let XY has one side less than the original polygon. *
be that mean ; the square constructed upon XY will For the triangles CDE, CFE have the base CE
be equal to the triangle ABC, - - - common ; they have also the same altitude, since
For, since B C : XY : : XY : 4 AD, it follows that their vertices D, F, are situated in a line DF paraller
XY 2 = BC. # A D ; hence the square constructed to the base; these triangles are therefore equal.
upon XY is equal to the triangle A B C. Add to each of them the figure A B CE, and there
* - - will result the polygon A B C D E equal to the poly-
* { } gon A B C F. -
PROBLEM VII - The angle B may in like manner be cut off, by sub-
Upon a given line to describe a rectangle that shall be stituting for the triangle ABC the equal triangle
equal to a given rectangle, fig. 148. A G C, and thus the pentagon ABDE will be changed
Fig. 148. Let AD be the given line, and ABFC the given intºn equal triangle G C F. . . . .
rectangle. -4 , - - The same process may be applied to every other
Find a fourth proportional to the three lines AID, figure; for, by successively diminishing the number
A B, A C, and let A X be that fourth proportional; a of its sides, one being retrenched at each step of the
rectangle constructed with the lines AD and AX will prºcess, the equal triangle will at last be found.
be equal to the rectangle ABFC. - ... -- Scholium. We have already seen that every triangle
For, since A D : A B : : A C : AX, it follows that may be changed into an equal square; and thus
AD. AX = AB.A.C.; hence the rectangle ADEX a square, may always be found equal to a given
is equal to the rectangle A BE C. - - rectilineal figure, which operation is called squaring
- - the rectilineal figure, or finding the quadrature of it.
PR. VIII. The problem of the quadrature of the circle consists
- raoniº, viii. - in finding a square equal to a circle whose diameter
To find two lines which shall have the same ratio to is given. -
each other, as the rectangle of the two given lines A and p -
#.". : the rectangle of the two given lines C and D, - - Problew XI.
Fig. 149. Let x be a. fourth proportional to the three 1ines B, To find the side of a square which shall be equal to the
; : AF : F.B. p
- PROBLEM V.
Through a given point A, in the given angle B.CD, to
draw the line B D, so that the segments AB, A D, com-
prehended between the point A and the two sides of the
angle, shall be equal, fig. 145. .
C, D ; then will the two lines A and X have the same
ratio to each other as the rectangles A B and CD.
For, since B : C : : D : X, it follows that C ; D =
B. X ; hence A ...B : C - D : : A - B : B. X : :
A : X: " . - .
Cor. Hence to obtain the ratio of the squares con-
structed upon the given lines A and C, find a third
- ProBLEM IX.
Tofind two lines that shall have the sameratio to each
other as the product of the three given lines A, B, C,
has to the product of the three given lines, P, Q, R,
fig. 150. . . g
Find a fourth proportional X to the three given Fig. 150,
lines A, B, C : find also a fourth proportional Y to
sum or the difference of two given squares, fig. 152.
Jet A and B be the sides of the given squares.
First. If it is required to find a square equal to
the sum of these squares, draw the two indefinite
lines E. D, EF at right angles to each other; take
ED = A, and E C = B; join D G ; this will be the
side of the square required. * .
G. E. O. M. E. T. H. Y.
339
Geometry. For the triangle DEG being right angled, the square
S-N-' constructed upon DG is equal to the sum
Fig, 153.
Fig. 154,
of the
squares upon E D and E G. -
* Secondly. If it is required to find a square equal to
the difference of the given squares, form in the same
manner the right angle FE H ; take GE equal to the
shorter of the sides A and B ; from the point G as a
centre, with a radius GH, equal to the other side,
describe an arc cutting E.H. in H: the square described
upon E H will be equal to the difference of the squares
described upon the lines A and B. -
For the triangle GE H is right angled, the hypo-
thenuse: G H = A, and the side GE = B; hence the
square constructed upon EH, &c.
Scholium. A square may thus be found equal to the
sum of any number of squares; for the construction
which reduces two of them to one, will reduce three
of them to two, and these two to one, and so of others.
It would be the same, if any of the squares were to be
subtracted from the sum of the others. r
Phoblem XII.
To construct a square which shall be to a given square
ABCD as the line M is to the line N, fig. 153.
Upon the indefinite line E G, take E F = M, and
F G = N ; upon E G as a diameter describe a semi-
circle, and at the point F erect the perpendicular F.H.
From the point H, draw the chords H. G., HE, which
produce indefinitely: upon the first take. H K equal to
the side A B of the given square, and through the
point K draw KI parallel to E G ; H I will be the
side of the square required. ,
For, by reason of the parallels KI, GE, we have
H.I : H K : : H E : H G ; hence HI2 : H K* : : HE”
: H. G*; but in the right angled triangle E H G
(prop. 28, book iv.) the square of H E is to the
square of H G as the segment E F is to the segment
EG, or as M is to N ; hence HI* : H K*:: M : N.
But H K = A B; therefore the square described upon
HI is to the square described upon AB as M is to N.
Problem XIII.
Upon the side FG, homologous to A B, to describe a
polygon similar to the given polygon A B C D E,
fig. 154. - -
In the given polygon, draw the diagonals AC, AD;
at the point F make the angle G F H = BAC, and at
the point G the angle FG H = A BC; the lines FG,
G H will cut each other in H, and FG H will be a
triangle similar to A B C. In the same manner upon
FH, homologous to A C, construct the triangle FIH
similar to A D.C.; and upon F I, homologous to AD,
construct the triangle FIK similar to A D E. The
polygon FG HIK will be similar to A B C D E, as
required.
For, these two polygons are composed of the same
number of triangles, which are similar and similarly
situated, (prop. 3, book iv.)
PaobleM XIV. - .
Two similar figures, being given, to construct a figure
which shall be similar to one of them, and equal to their
sum or their différence.
P : Y :: M* :
Let A and B be homologous sides of the two given
figures. Find a square equal to the sum or to the dif-
ference of the squares, described upon A and B; let
X be the side of that square ; then will X in the
figure required, be the side which is homologous te
the sides A and B in the given figures. The figure
itself may then be constructed on X, by the last
problem. ~ *
For, the similar figures are as the squares of their
homologous sides ; now the square of the side X is
equal to the sum, or to the difference, of the squares
described upon the homologous sides A and B; there-
fore the figure described upon the side X is equal to
the sum, or to the difference; of the similar figures
described upon the sides A and B.
PROBLEM XV.
To construct a figure similar to a given one, and
bearing to it any given ratio of M to N.
Let A be a side of the given figure, X the homolo-
gous side of the figure required. The square of X
must be to the square of A as M is to N ; hence X
will be found by problem 12; and knowing X, the
rest will be accomplished by problem 13.
Problem XVI.
To construct a figure similar to one given figure, and
equal to another, fig. 156. f -
Find M the side of a square equal to the figure P,
and N the side of a square equal to the figure Q. Let
X be a fourth proportional to the three given lines M,
N, A B ; upon the side X, homologous to A B,
Froblems
relative to.
Book IV.
W , - -
Fig. 156.
describe a figure similar to the figure P; it will also :
be equal to the figure Q. -
For, calling Y the figure described upon the side X,
we have P : Y : : A B* : X*; but, by construction,
AB : X : : M : N, or A B* : X2 : : M* : N*; hence
N*. But by construction also, M* =
P and N* = Q ; therefore. P : Y : : P : Q ; conse-
quently Y = Q ; hence the figure Y is similar to the
figure P, and equal to the figure Q.
Paoniº XVII.
To construct a rectangle equal to a given square C,
and having its adjacent sides together equal to a given
line A B, fig. 157.
Upon A B as a diameter, describe a semicircle; draw
the line DE parallel to the diameter, at a distance A D
equal to the side of the given square C ; from the
point E, where the parallel cuts the circumference,
draw EF perpendicular to the diameter; A.F and FB
will be the sides of the rectangle required. . . . . .
For their sum is equal to AB, and their rectangle
A F. F.B. is equal to the square of EF, or to the
square of A D ; hence that rectangle is equal to the
given square C. t -
Scholium. To render the problem possible, the
distance A D must not exceed the radius; that is, the
side of the square C must not exceed the half of the
line A. B. 1 ' . . . . . . . . "
Paoninx xviii.
To construct a rectangle that shall be equal to a given
Fig. 157.
2 Y 2
340
G. E. O. M. E. T. R. Y.
angles is one-third of two right angles, (prop. 24, Booky.
book i.) or one-sixth of four right angles; conse-º-º-'
Geometry, square C, and the difference of whose adjacent sides shall
~~' be equal ta a given line A B, fig. 158. d
quently the arc A B is one-sixth of the whole circum-
Fig. 158. Upon the given line A B as a diameter, describe a Q 6 tº
se: ; . the extremity of the diameter draw the ference, * it is º º : i. º A º B,
tangent AID, equal to the side of the square C; through (pr op. 14, boo iii.) f ere . t "A.E. fº .#ied
the point D and the centre o draw the secant Dí; six times in the circumference from A to B, from B to
then-will DE and D F be the adjacent sides of the C, from C to D, will be the regular hexagon required.
rectangle required. - - Join now A C, C E, E A, and AIE c will be the equi-
For, first, the difference of their sides is equal to lateral triangle, as is obvious. &
the diameter EF or AB; secondly, the rectangle DE, , §cholium. The figure ABCO is a parallelogº.
DF is equal to A. D*, (prop. 35, book iv.;) hence that º: i..', *...*.*, ...'. º .*.
rectangle is equal to the given square C. diagonals A C* + B O 2 is equal to the sum of the
squares of the sides; that is, to 4 A B*, or 4 B O’;
* Yy . and taking away B O from both, there will remain
BOOK W. A C 2 = 3 B O 2 ; hence A C* : B O2 : : 3 : 1, or AC
Or *An imaºrwººmsºn us - : B O : : A/3 : 1; hence the side of the inscribed equila-
Of regular polygons, and the measure of the circle. teral triangle is to the radius, as º: root ji.
DEFINITIoN. is to unity.
A REGULAR polygon is one having all its angles and Proposition IV-Problem.
sides equal. W - - -
In a given circle, to inscribe a regular decagon ; then a
PRoposition I.-Theorem. pentagon, and a pentedecagon, fig. 162. 1 *
All regular polygons of the same number of sides are Divide the radius A O in extreme and mean ratio Fig. 162.
similar, fig. 159. - º: tº. the point M; ** º:
ig. 159. Let ABCDEF, a b c def, be two regular polygons, equal to 0. e greater segment; wi €
Fig. 159 (in this case hexagons.) #: sum of . j. is ... of the º . will require to be
the same in both figures, being each equal to eight º led ten :*::::: e ºn. ion. A O
right angles, (prop. 25, book i.) and the number of ow. lºsA M : we nave, §,”;º & Å;
angles in each are also equal, and equal to each other; . AB AM, hen: º º gi AB 6, AMB hav
that is, each is equal to one-sixth of eight right angles. a common an º jº. pro sº
Again, since the polygons are regular, by hypothesis, sides : hence ; roº 25, book iv.) the º
the sides A B, BC, CD, &c. are all equal, as are also N s; , nence (prop,49, pook iv. y gº
. . . . . . .”W. A.I. E. C. . . . . . Now the triangle OA B, being isosceles, AM B must
5 * ~ 5 tº Lv3 º . be isosceles also. and A B = B M ; besides A B =
CD : c d, &c. That is, the two figures have their O M ; h lso M'B = ovſ, hé the triangl
angles equal, and the sides about those angles propor- BMo j Wºms ; nence riangle
tional; they are therefore similar. ºf Again, the angle A M B being exterior to the isos-
- - c. - - t celes triangle BMO, is double of the interior angle O,
PRoposition II.-Theorem. ~\ (prop. 24, book i.;) but the angle A M B = M. A B ;
To inscribe a square in a given circle, fig. 160. hence the triangle OAB is such, that each of the
tº r º angles at its base, O A B or OBA, is double of O the
Fig. 160. Draw two diameters AC, BP, cutting, each other angle at its vertex; hence the three angles of the
at right angles; join their extremities, A, B, C, P; triangles are together equal to five times the angle O,
the figure A B C D will be the square required. For which consequently is the fifth part of the two right
the angle A OB, B O C, &c. being equal, the chords angles, or the tenth part of four; hence the arc A B
A B, BC, &c. are also equal ; and the angles A B C, is the tenth part of the circumference, and the chord
B C D, &c. being in semicircles, are right angles. The AB is the side of the regular decagon.
figure is therefore equilateral, and its angles right Cor. 1. By joining the alternate angles of the re-
angles; it is therefore a square. gular decagon, the regular pentagon A C E GI will
Scholium. Since the triangle is right angles, B D*= also be formed.
B Cº - D.C." or 2D gº = BD" or D C / 2 = BD Cor. 2. A B being still the side of the decagon, let
or D C : B D : : 1 : A/2 ; and in the same Way, since AL be the side of the hexagon; the arc B L will
B C2 = 2 B O 2 ; B C : B O : : y 2 : 1; that is, the then, with reference to the whole circumference, be
side of the inscribed square is to radius; as the dia- † — ºr, or pr; hence the chord B L will be the side
meter is to the side of the inscribed Square, the ratio of the pentedecagon Or regular polygon of fifteen sides.
in both cases being as the square root of 2 to unity. It is evident, also, that the arc C L is the third of CB.
- e - = * * . . s' Scholium. Any regular polygon being inscribed, if
PROPosition III.-Theorem. the arcs subtended by its sides be severally bisected,
. . . . . . . the chords of those semi-arcs will form a new regular
_To inscribe an equilateral triangle and a regular her polygon of double the number of sides: thus, it is plain,
agon in a given circle, fig. 161. - - the square may enable us successively to inscribe re-
Fig. 161. First. To inscribe the regular hexagon in a circle. gular polygons of 8, 16, 32, &c. sides. And in like
From any point A in the circle apply the line A B equal
to the radius, and join BO, O being the centre; then
because ABO is an equilateral triangle, each of its
. . . . . . . . . . .” --- rº- - 2 . .” * e . . . . . . . . . . .
manner, by means of the hexagon, regular polygons
of 12, 24, 48, &c. sides may be inscribed; by means
of the decagon, polygons of 20, 40, 80, &c. sides; by
G E O M E. T. R. Y. 341
Geometry. means of the pentedecagon, polygons of 30, 60, 120, tion for the other triangles, it will appear that the sum Book W.
\–V-’ &c. sides.* -
Fig. 163. At T, the middle point of the arc AB, apply the apothem of the polygon.
tangent GH, which (prop. 22, book iii.) will be - -
parallel to AB; do the i. at the middle point of RRoposition VII.-Theorem.
each of the arcs B C, CD, &c.; those tangents, by © & •
their intersections, will form the regular jº. The p erimeters. of two regular p olygons, having the
tº g e q " i " . A same number of sides, are to each other as the radii of the
polygon G H IR, &c., similar to the inscribed one. . . ibed circl nd also as the radii of the inscribed
It is evident, in the first place, that the three points circumscribed circles, and also as the radii of the inscribe
O, B, H, lie in the same straight line : for the right circles ; and their areas are to each other as the squares
• *, *- : § - > É' of those radii, fig. 163. - w
angled triangles OTH, OH N, having the common g g
hypothenuse O H, and the side OT = ON, must be Let A B be a side of the one polygon, O the centre, Fig. 163.
equal; and consequently the angle TO H = H ON, and consequently OA the radius of the circumscribed
wherefore the line OH passes through the middle circle, and QD, perpendicular to AB, the radius of
point B of the arc TN. For a like reason, the point the inscribed circle; let a b, in like manner, be a side
i is in the prolongation of O C ; and so with the rest. of the other polygon, o its centre, o a and od the radii
But since B H is parallel to AB, and H I to B C, the of the circumscribed and the inscribed circle. The
angle G. H. I = A B C; in like manner, H.I K = perimeters of the two polygons are to each other as
B C D ; and so with all the rest : hence the angles of the sides A B and a b : but the angles A and a are
the circumscribed polygon are equal to those of the equal, being each half of the angle of the polygon;
inscribed one. And farther, by reason of the same so also are the angles, B and b; hence the triangles
parallels, we have G H : A B : : O H. : O B, and H I : A BO, a b o are similar, as are likewise the right
#3 C : : Ó H : O B; therefore G H : A B : : H I : B C. angled triangles A DO, a do also A B : a b : : A O :
But AB = BC, therefore G H = HI. For the same a 0 : : DO : do; therefore the perimeters of the poly-
reason, HI = IK, &c.; hence the sides of the cir- gons are to each other as the radii A. O, a o of the cir-
cumscribed polygon are all equal ; and this polygon cumscribed circles, and also as the radii D O, do of
is regular, and similar to the inscribed one. the inscribed circles. -
Cor. 1. Reciprocally, if the circumscribed polygon Again the areas of those polygons are to each other
G H I K, &c. were given, and the inscribed one A B C, as the squares of the homologous sides A B, a b : they
&c. were required to be deduced from it, it would only are therefore likewise to each other as the squares of
be necessary to draw from the angles G, H, I, &c. of A. O., a 0 the radii of the circumscribed circles, or
the given polygon, straight lines 5 G, OH, &c. meet- as the squares of OD, 0 d the radii of the inscribed
ing the circumference in the points A, B, C, &c.; then circles. r - -
to join those points by the chords AB, BC, &c.; which PRoposition VIII.-Lemma,
would form the inscribed polygon. An easier solution - -
of this problem would be simply to join the points of Any curve, or any polygonal line, which envelopes the
contact T, N, P, &c. by the chords TN, N P, &c. convex line A M B from one end to the other, is longer
which likewise would form an inscribed polygon than A M B the enveloped line, fig. 164.
similar to the circumscribed one. , By the term convex line is to be understood a line, Fig. 164.
Cor. 2., Hence we may circumscribe abºut a circle polygonal or curve, or partly curve and partly polygo.
any regular polygon, which can be inscribed within hal, such that a straight line cannot cut it in more
it ; and conversely. than two points. If in the line AMB there were any
- - sinuosities or re-entring portions, it would cease to be
PRoposition VI.-Theorem. convex, because a straight line might evidently cut it
The area of a regular polygon is equal to its perimeter º.º. than two points. The arcs of a circle are
multiplied by half the radius of the inscribed circle, essentially convex; but the present proposition. ex-
fig, 163. - tends to any line which fulfils the required condition.
g - . e g This being premised, if the line A MB be not
Fig. 163. , ſº the egular Polygon be G.H. Tºše the triangle shorter than any of those which envelope it, there will
PRoPosition V.-Problem.
A regular inscribed polygon A B C D, &c. being given,
to circumscribe a similar polygon about the same circle,
fig. 163.
G O H will be measured by G H x + OT; the triangle
O H I by HI 2 + O N : but ON=OT; hence the two
triangles taken together will be measured by (G H +
HI) x + O.T. And, by continuing the same opera-
• It was long supposed, that, besides the polygons here men-
tioned, no other could be inscribed by the operations of elemen-
tary geometry, or what amounts to the same, by the resolution of
equations of the first and second degree. But M. Gauss, of
Goettingen, at length proved, in a work entitled Disquisitiones
Arithmeticae, Lipsiae, 1801, that by the method in question, a re-
gular polygon of 17 sides might be inscribed, and generally a
regular polygon of 2n + 1 sides, provided 2n + 1 be a prime
number. See also Barlow's Essay on the Theory of Numbers,
of them all, or the whole polygon, is measured by the S-N-'
sum of the bases G. H., HI, I K, &c. or the perimeter
of the polygon, multiplied into 4 OT, or half the
radius of the inscribed circle. - -
Scholium. The radius O T of the inscribed circle is
obviously the perpendicular let fall from the centre
to one of the sides ; and is sometimes named the
be found among the latter a line shorter than all the
rest, which is shorter than A MB, or, at most, equal
to it. Let A CD E B be this enveloping line; any
where between those two lines draw the straight line
PQ, not meeting, or at least only touching, the line
A M B. The straight line P. Q is shorter than
PCDEQ; hence if, instead of the part PCDEQ, we
substitute the straight line PQ, the enveloping line
A PQ B will be shorter than APD Q B. But, by
hypothesis, this latter was shorter than any other;
hence that hypothesis was false; consequently all of
the enveloping lines are longer than A MB,
342
G E O M E T R Y.
Geometry:
Fig. 165.
Paoposition IX. —Lemma.
Two concentric circles being given, a regular polygon
may always be inscribed within the greater, the sides of
which shall not meet the circumference of the less ; and
likewise, a regular polygon may always be described about
the less, the sides of which shall not meet the circumference
of the greater, fig. 165.
Let C A, C B be radii of the given circles." At the
point A, apply the tangent DE, terminating in the
greater circumference at D and E ; inscribe within
this greater circumference any regular polygons, by
the methods already explained; next bisect the arcs
subtended by its sides, and draw the chords of those
half arcs; a polygon will thus be found, having twice
as many sides. Continue the bisection, till an arc is
obtained less than D B E. Let MBN be that arc, the
middle point of it being supposed to lie at B: it is
plain that the chord M N will be farther from the
centre than DE ; and that consequently the regular
polygon, of which M N is a side, cannot meet the
circumference, of which CA is the radius.
Now, the same construction remaining, join CM
and CN, meeting the tangent D E in P and Q ; PQ
will be the side of a polygon described about the less
circumference, similar to that polygon inscribed
within the greater, of which the side is M. N. And it
is evident, that this circumscribed polygon having PQ
for its side, can never meet the greater circumference,
CP being less than C M.
Hence, by the same operation, a regular polygon
may be inscribed within the greater circumference,
and a similar one described about the lèss, both of
which shall have their sides included between the
two circumferences.
Scholium. If two concentric sectors FC G, I C H
be given, a portion of a regular polygon may, in like
manner, be inscribed in the greater, or circumscribed
about the less, so that the perimeters of the two
polygons shall be included between the two circum-
ferences. For this purpose, it will be sufficient to
divide the arc FB G successively into 2, 4, 8, 16, &c.
equal parts, till a part smaller than D B E is obtained.
By the expression, portion of a regular polygon, is
here meant the figure terminated by a series of equal
chords inscribed in the arc FG, from one of its ex-
tremities to the other. . This portion has all the prin-
cipal properties of regular polygons; it has its angles
equal, and its sides equal, it can be inscribed in a
circle, or circumscribed about one : yet, properly
speaking, it forms part of a regular polygon only in
Fig. 166.
those cases where the arc subtended by one of its
sides is an aliquot part of the circumference.
Proposition X. —Theorem.
The circumferences of circles are to each other as their
radii, and the surfaces as the squares of those radii,
fig. 166. -
For the sake of brevity, let us designate the cir-
cumference whose radius is C A by circ. CA; we are
to; show that circ. C. A : circ. O.B. : : C A : Olb. .
If this proposition is not true, C A must be to O B
as circ. CA is to a fourth term less or greater than .
circ O B : suppose it less ; and that, if possible, CA :
O B : ; circ. C. A. : circ. O.D. -
In the circle of which O B is the radius inscribe a Book V.
regular polygon EFG KLE, such that the sides of ~~
it shall not meet the circumference of which O D is
the radius by the last proposition; inscribe a similar
polygon, MN PFM, in the circle of which A C is
the radius. - -
Then, since those polygons are similar, their peri-
meters MN P'S M, EFG K E will be to each other
(prop. 7, book v.) as CA, O B, the radii of the cir-
cumscribed circles, that is M N P S M : E FG KE
; : C A : OB. But, by hypothesis, C A : O B : : circ.
CA : circ. O D; therefore M N PSM : E FG K E : :
circ, CA: circ. O D; which proportion is false, be-
cause (prop. 8, book v.) the perimeter M N S P M is
less than circ. CA, while on the contrary EFG K E is
greater than circ. OD; therefore it is impossible that
C A can be to O B as circ, CA is to a circumference
less than circ. O B : or, in more general terms, it is
impossible that one radius can be to another, as the
circumference described with the former radius is to
a circumference less than the one described with the
latter radius. - -
Hence, too, we conclude it to be equally impossible
that C A can be to O B as circ. C A is to a circumfe-
rence greater than circ. O B ; for if this were the case,
by reversing the ratios, we should have O B to CA as
a circumference greater than circ. O B is to circ. CA;
or, what amounts to the same thing, as circ. O B is to
a circumference less than circ. C. A ; and therefore one
radius would be to another as the circumference
described with the former radius is to a circumference
less than the one described with the latter radius; a
conclusion shown above to be erroneous.
And since the fourth term of this proportion C A :
O B : : circ. C A : a can neither be greater nor less
than circ. O B, it must be equal to circ. O B : conse-
quently the circumference of circles are to each other
as their radii.
By the same construction, a similar train of reason-
ing would show, that the surfaces of circles are to
each other as the squares of their radii. We need not
enter upon any farther details respecting this propo-
sition, particularly as it forms a corollary of the fol-
lowing theorem :
Cor. The similar arcs A B, DE (fig. 167) are to Fig. 167.
each other as their radii A C, D O ; and the similar
sectors A C B, D O E are to each other as the squares
of those radii.
For, since the arcs are similar, the angle C (def. I,
book iv.) is equal to the angle O ; but C is to four
right angles (prop. 5, book. iii.) as the arc AB is to the
whole circumference described with the radius A C ;
and O is to four right angles, as the arc D F is to the
circumference described with the radius O D ; hence
the arcs AB, D E are to each other as the circumfe-
rences of which they form part ; but these circum-
ferences are to each other as their radii A.C, DO ;
therefore arc A B : arc D. E. : : A C : D O.
For a like reason, the sectors A C B, D O E are to
each other as the whole circles; which again are as
the squares of their radii; therefore sect. A C B :
sect. DOE : : A C2 : D O2
PROPosſTION XI.-Theorem.
The area of a circle is equal to the product of its cir-
cumference by half the radius, fig, 168.
G E O M E T R Y.
343
equal to the product of the square of its radius by the Book V
Geometry. Let us designate the surface of the circle whose
V-V-' radius is C A by surf. CA; we shall have surf. C.A =
Fig. 168.
Fig. 169.
+ C A x circ. C.A.
For if + C A x circ. C A be not the area of the circle
whose radius is CA, it must be the area of a circle
either greater or less. Let us first suppose it to be
the area of a greater circle; and, if possible, that
+ C A x circ. C A = surf. C B.
About the circle whose radius is C A describe a
regular polygon DEFG, &c. such (prop. 9, book v.)
that its sides shall not meet the circumference whose
radius is C.B.: 'The surface of this polygon will be
equal (prop.6, book v.) to its perimeter D E + EF
+ F G + &c. multiplied by 4. A C ; but the perimeter
of the polygon is greater than the inscribed circum-
ference enveloped by it on all sides; hence the sur-
face of the polygon DE FG, &c. is greater than
# A C x circ. A C, which by the supposition is the
measure of the circle whose radius is C B ; thus the
polygon must be greater than that circle. But in
reality it is less, being contained wholly within the
circumference ; hence it is impossible that + C A ×
circ. A C can be greater than surf. CA; in other words,
it is impossible that the circumference of a circle mul-
tiplied by half its radius can be the measure of a
greater circle. - -
In the second place, we assert it to be equally im-
possible that this product can be the measure of a
smaller circle. To avoid the trouble of changing our
figure, let us suppose that the circle in question is the
one whose radius is C B ; we are to show that + C B
× circ, C B cannot be the measure of a smaller circle,
of the circle, for instance, whose radius is C. A. Grant
it to be so; and that, if possible, 4 C B x circ. C B =
surf. C.A.
Having made the same construction as before, the
surface of the polygon DE FG, &c. will be measured
by (DE + E F + F G + &c.) x + C A ; but the peri-
meter D E + E F + F G + &c. is less than circ. C B,
being enveloped by it on all sides ; hence the
area of the polygon is less than + C A x circ. C B,
and still more than + C B x circ. C B. Now, by
the supposition, this last quantity is the measure of
the circle whose radius is C A ; hence the polygon
must be less than the inscribed circle, which is absurd;
it is therefore impossible that the circumference of a
circle multiplied by half its radius, ean be the mea-
sure of a smaller circle.
Hence, finally, the circumference of a circle multi-
plied by half its radius is the measure of that circle
itself.
Cor. 1. The surface of a sector is equal to the arc
of that sector multiplied by half its radius.
For (fig. 169) the sector A C B is to the whole
circle as the arc A. MB is to the whole circumference
A B D, or as A M B x + A C is to A B D x + A C. But
the whole circle is equal to ABD x + A C ; hence the
sector A C B is measured by A MB x + A C.
Cor. 2. Let the circumference of the circle whose
diameter is unity be denoted by ºr ; then, because cir-
cumferences are to each other as their radii or dia-
meters, we shall have the diameter I, to its circum-
ference T as the diameter 2 CA is to the circumference
whose radius is C A, that is, 1 : 7 : ; 2 C A : circ. CA,
therefore circ. C A = 2 + x C A. Multiply both terms
by + CA; we have 3 C A × circ. C A = 7 x C A*, or
surf, C A = 7 x C A2; hence the surface of a circle is
constant number 7, which represents the circum- \-y-
ference whose diameter is 1, or the ratio of the cir-
cumference to the diameter. -
In like manner, the surface of the circle, whose
radius is O B, will be equal to 7 x OB%; but ºr x
C A* : T × OB? : : C A*: O 3° ; hence the surfaces
of cireles are to each other as the squares of their radii,
which agrees with the preceding theorem. .
Scholium. It is of course understood, that the
problem of the quadrature of the circle consists in
finding a square equal in surface to a circle, the radius
of which is known. Now it has just been proved;
that a circle is equal to the rectangle contained by its
circumference and half its radius; and this rectangle
may be changed into a square, by finding (prop. 3,
book v.) a mean proportional between its length and
its breadth. To square the circle, therefore, is to find
the circumference when the radius is given; and for
effecting this, it is enough to know the ratio of the
circumference to its radius or its diameter. R
Hitherto the ratio in question has never been de-
termined except approximately; but the approxima-
tion has been carried so far, that a knowledge of the
exact ratio would afford no real advantage whatever
beyond that of the approximate ratio. Accordingly,
this problem, which engaged geometers so deeply,
when their methods of approximation were less
perfect, is now sunk to the rank of those useless ques-
tions, with which no one possessing the slightest
tincture of geometrical science will occupy any por-
tion of his time.
Archimedes showed that the ratio of the circumfe-
rence to the diameter is included between 3+3 and
33-? ; hence 3+ or 3, affords at once a pretty accurate
approximation to the number above designated by ºr ;
and the simplicity of this first approximation has
brought it into very general use. Metius, for the same
number, found the much more accurate value ###.
At last the value of Tr, developed to a certain order
of decimals, was found by other calculators to be
3.1415926535897932, &c.; and some have had pa-
tience enough to continue these decimals to the hun-
dred and twenty-seventh, or even to the hundred and
fortieth place. Such an approximation is evidently
equivalent to perfect correctness : the root of an im-
perfect power is in no case more accurately known.
The following problems will exhibit two of the
simplest elementary methods of obtaining those
approximations.
PRoposition XII.—Problem,
The surface of a regular inscribed polygon, and that of
a similar polygon circumscribed, being given ; to find the
surfaces of the regular inscribed and circumscribed poly-
gons having double the number of sides, fig. 170.
Let A B be a side of the given inscribed polygon; Fig. 170.
E F, parallel to AB, a side of the circumscribed
polygon ; C the centre of the circle. If the chord
A M and the tangents A P, B Q be drawn, A M will
be a side of the inscribed polygon, having twice the
number of sides ; and (prop. 5, book v.) PQ, double
of PM, will be a side of the similar circumscribed
polygon. Now, as the same construction will take
place at each of the angles equal to A CM, it will be
sufficient to consider A C M by itself, the triangles
344
G E O M E T R Y.
between the inscribed and the circumscribed polygon, Book v.
and since those polygons agree as far as a certain S-N-2
Geometry, connected with it being evidently to each other as the
\-e-N-' whole polygons of which they form part. Let A,
then, be the surface of the inscribed polygon whose
side is AB, B that of the similar circumscribed poly-
gon; Aſ the surface of the polygon whose side is
A M, B' that of the similar circumscribed polygon;
A and B are given; we have to find A' and B'. . .
. First. The triangles A CD, A CM, having the
common vertex A, are to each other as their bases
CD, CM; they are likewise to each other as the
polygons A and A', of which they form part ; hence
A : A' : : C D : C M. Again, the triangles CAM,
CME, having the common vertex M, are to each other
as their bases CA, C E ; they are likewise to each
other as the polygons Aſ and B of which they form
part; hence Aſ : B : : C A : C E. But since AID and
ME are parallel, we have C D : C M : : C A : C E ;
hence A : A' : : A' : B; hence the polygon A/, one of
those required, is a mean proportional between the
two given polygons A and B, and consequently
Aſ = y A x B. -
Secondly. The altitude C M being common, the
triangle CPM is to the triangle C P E as PM is to
PE; but (prop. 21, book iv.) since C P bisects the angle
M CE, we have PM : PE : : C M : C E : : CD : C A
: : A : A'; hence C PM : C PE : : A : A'; and conse-
quently C P M : C P M + C P E or CME : : A : A
-H A/. But C M P A or 2 C M P and C M E are to
each other as the polygons B and B, of which they
form part ; hence B^: B : : 2 A : A + A/. Now Aſ
has already been determined; this new proportion will
- 2 A • B .
- - . A + A* *
and thus by means of the polygons A and B, it is
easy to find the polygons Aſ and B', which have
double the number of sides.
serve for determining B", and give us B" =
PRoPosition XIII.-Problem.
To find the approximate ratio of the circumference to
the diameter. - .
Let the radius of the circle be 1; the side of the
inscribed square will be v 2, (prop. 2, book v.) that
of the circumscribed square will be equal to the
diameter 2; hence the surface of the inscribed square
is 2, and that of the circumscribed square is 4. Let
us therefore put A = 2, and B = 4; by the last pro-
position, we shall find the inscribed octagon A' =
A/ 8 = 2.8284271, and the circumscribed octagon
I6 gº g
T 2 + y 8T 3.3137085. The inscribed and the
circumscribed octagon being thus determined, we shall
easily, by means of them, determine the polygons
having twice the number of sides. We have only in
this case to put A = 2.8284271, B = 3.3137085 ;
we shall find Aſ = .A/ A . B = 3.0614674, and
B=; *: # = 3.1825979. These polygons of 16 sides
will in their turn enable us to find the polygons of 32;
and the process may be continued, till their remains
no longer any difference between the inscribed and the
circumscribed polygon, at least so far as that place of
decimals where the computation stops, so far as the
seventh place, in this example. Being arrived at this
point, we shall infer that the last result expresses the
surface of the circle, which, since it must always lie
place of decimals, must also agree with both as far as
the same place. t -
We have subjoined the computation of those
polygons, carried on till they agree as far as the
seventh place of decimals.
Circumscribed polygon.
Number of sides. Inscribed polygon.
4 . . . . . . . . 2.0000000 . . . . . . . . 4.0000000
8 . . . . . . . . 2.8284271 . . . . . . . . 3.3137085
16 . . . . . . . . 3.0614674 . . . . . . . . 3.1825979
32 . . . . . . . . 3.1214451 . . . . . . . . 3.1517249
64 . . . . . . . . 3.1365485 . . . . . . . . 3.1441184
138 . . . . . . . . 3.1403311 . . . . . . . . 3.1422236
256 . . . . . . . . 3.1412772 . . . . . . . . 3.14.17504
512 . . . . . . . . 3.1415 138 . . . . . . . . 3.1416391
1024 . . . . . . . . 3.1415729 . . . . . . . . 3.1416025
2048 . . . . . . . . 3.14.15877 . . . . . . . . 3.1415951
4096 . . . . . . . . 3.1415914 . . . . . . . . 3.1415933
8192 . . . . . . . . 3.1415923 . . . . . . . . 3.1415928
16384 . . . . . . . . 3.1415925 . . . . . . . . 3.1415927
32768 . . . . . . . . 3.1415926 . . . . . . . . 3.1415925
The area of the circle, therefore, is equal to
3.1415926. Some doubt may exist perhaps about the
last decimal figure, owing to errors proceeding from
the parts omitted ; but the calculation was carried
on with an additional figure, that the final result
here given might be absolutely correct even to the
last decimal place. i
Since the surface of the circle is equal to half the
circumference multiplied by the radius, the half cir-
cumference must be 3.1415926, when the radius is l;
or the whole circumference must be 3.1415926, when
the diameter is l; hence the ratio of the circumference
to the diameter, formerly expressed by T, is equal to
3.1415926.
PRoposition XIV.-Lemma.
The triangle C A B is equal to the isosceles triangle
D CE, which has the same angle C, and one of its equal
sides C E or CD a mean proportional between C A and
CB. And if the angle C A B is right, the perpendicular
CF, drawn to the base of the isosceles triangle will be a
mean proportional between the side CA and half the sum
of the sides CA, C B, fig. 171.
First. Because of the common angle C, the triangle Fig. 171.
A B C is to the isosceles triangle D C E as A C x C B
is to D C x C E or DC” (prop. 29, book iv.;) hence
those triangles will be equal, if D C* = A C x C B,
or if D C is a mean proportional between A C and
C B. - - -
Secondly. Because the perpendicular C G F bisects
the angle A C B, we shall have A G : G B : : A C : C B
(prop. 21, book iv.;) and therefore (prop. 5, book ii.)
A G : A G + G B or A B : : A C : A C + C B ; but
A G is to A B as the triangle A C G is to the triangle
A C B, or 2 CD F : besides if the angle A is right,
the right angled triangles AC G, C D F must be
similar, and give AC G : CD F : : A C* : C F*; hence,
. A C* : 2 CF2 : : A C : A C + C B. •
Multiply the second pair by A C ; the antecedents
will be equal, and consequently we shall have 2 C F*
= AC. (AC + CB) or cF. - A c. (*****)
2
gººmºsº
G E O M E T R Y.
345
Hence, putting a = 1, b = 1.4142136, we shall find Book V.
Geometry, hence if the angle A is right, the perpendicular CF
*—y- will be a mean proportional between the side A C and
Fig. 172.
and
the half sum of the sides A C, CB.
Proposition xv.–Problem.
To find a circle differing as little as we please from a
given regular polygon, fig, 172. - -
Let the square B M N P be the proposed polygon.
From the centre C, draw C A perpendicular to MB,
and join C B.
The circle described with the radius CA is inscribed
in the square, and the circle described with the radius
CB circumscribes this same square; the first will in
consequence be less than it, the second greater : it is
now required to compress those limits.
Take C A and CE, each equal to the mean propor-
tional between C A and C B, and join E D ; the
1sosceles triangle CD E will, by the last proposition,
be equal to the triangle C A B. Perform the same
operation on each of the eight triangles which com-
pose the square; you will thus form a regular octagon
equal to the square B MNP. The circle described
with the radius CF, a mean proportional between CA
C A + C B
2
the circle whose radius is CD will circumscribe it.
The first of them will therefore be less than the given
square, the second greater.
If the right angled triangle C D F be, in like
manner, changed into an equal isosceles triangle, we
shall by this means form a regular polygon of sixteen
sides, equal to the proposed square.
, will be inscribed in this octagon, and
the circumscribed circle will be greater.
The same process may be continued, till the ratio
between the radius of the inscribed and that of the
circumscribed circle, approach as near to equality as
we please. In that case, both circles may be regarded
as equal to the square.
Scholium. The investigation of the successive radii
is reduced to this. Let a be the radius of the circle
inscribed in one of the polygons, b the radius of the
circle circumscribing the same polygon ; let a' and bº
be the corresponding radii for the next polygon, which
is to have twice the number of sides. From what has
been demonstrated, b' is a mean proportional between
a and b, and a' is a mean proportional between a and
0. ; !, so that b'= w/ a.b, and a' = va.º b
a and b the radii of one polygon being known, we may
easily discover the radii aſ and bº of the next polygon;
and the process may be continued till the difference
between the two radii become insensible ; then either
of those radii will be the radius of the circle equal to
the proposed square or polygon.
This method is easily practised with regard to lines;
for it implies nothing but the finding of successive
: hence
mean proportionals between lines which are given : it
is still more easily practised with regard to numbers,
and forms one of the most commodious plans which
elementary geometry can furnish, for discovering
speedily the approximate ratio of the circumference
to the diameter.
first inscribed radius C A will be one, and the first
circumscribed radius CB will be y 2 or 1.4142136,
WOL. Is
- The circle in-
scribed in this polygon will be less than the square ;
Let the side of the square be 2; the
b’ = 1.1892071, and a' = 1.0986841. These numbers .
will serve for computing the rest, the law of their
combination being known. -
Radii of the circumscribed circles. Radii of the inscribed circles.
1.4142136 . . . . . . . . . . . . . . 1.0000000
l. 1892O7I. . . . . . . . . . . . . . . I.O98684 L
1.1430500 . . . . . . . . . . . . . . I. 1210863
1.1320,149 . . . . . . . . . . . . . . 1.1265639.
1.1292862 . . . . . . . . . . . . . . 1.1279257
1.1986063 . . . . . . . . . . . . . . 1.1282657
Since the first half of these ciphers is now become
the same on both sides, it will occasion little error to
assume the arithmetical means instead of the mean
proportionals or geometrical means, which differ from
the former only in their last figures. By this method,
the operation is greatly abridged ; the results are :
1.1984360 . . . . . . . . . . . . . . 1.128.3508
1.1283934 . . . . . . . . . . . . . . 1.1283721
1.1283827 . . . . . . . . . . . . . . 1.1283774
1.1283801 . . . . . . . . . . . . . . 1.1283787
1.1283794 . . . . . . . . . . . . . . 1.12837.91
I. 12837.92 . . . . . . . . . . . . . . 1.12837.9%
Thus 1.1283792 is very nearly the radius of a circle
equal in surface to the square whose side is 2. From
this, it is easy to find the ratio of the circumference to
the diameter: for it has already been shown that the
surface of the circle is equal to the square of its radius
multiplied by the number r; hence if the surface 4
be divided by the square of 1.1283792 the radius, we
shall get the value of T, which by this computation is
found to be 3.1415926, &c. as was formerly determined
by another method.
POOK WI.
Of planes and solid angles.
DEFINITIons.
I. THE common section of two planes is the line in
which they meet to cut each other.
2. A line is perpendicular to a plane, when it is per-
pendicular to any two lines in that plane which
meet it.
3. One plane is perpendicular to another, when every
line in the one which is perpendicular to their common
section is perpendicular to the other plane.
4. The inclination of two planes to each other, or the
angle they form between them, is the angle contained
by two lines drawn from any point in the common.
section, and at right angles to the same, one of these
lines in each plane. *..
5. A line is parallel to a plane, when, if both are
produced to any distance, they do not meet; and con-
versely, the plane is then also parallel to the line.
6. Two planes are parallel to each other, when both
being produced to any distance they do not meet.
7. A solid angle is the angular space included be-
tween three or more planes which meet at the same
point.
2 z
346. G E O M E T R Y.
Geometry.
S-N-' PRoPosition I.-Theorem.
4 straight line cannot be partly in a plane and partly
out of it, fig. 173.
Fig. 173. . . For the part of the line which is in the plane may
be produced in the plane, as for example to D ; and if
a part of the line were also out of the plane, then two
straight lines might have a common segment A B,
which is impossible. - -
PROPosition II.-Theorem.
Two straight lines which intersect each other lie in the
same plane, and determine its position, fig. 174.
Fig. 174, Let AB, AC be two straight lines which intersect
each other in A ; and conceive some plane passing
through one of the lines as AB, and if also A C be in
this plane, then it is clear that the two lines, accord-
ing to the terms of the proposition, are in the same
plane ; but if not, let the plane passing through A B
be supposed to be turned round A B till it passes
through the point C, then the line AC, which has two
of its points A and C in this plane, lies wholly in it;
and hence the position of the plane is determined by
the single condition of containing the two straight
lines A. B. A. C.
. Cor. 1. A triangle A B C, or any three points not
in a straight line, detérmines the position of a plane.
Cor. 2. Hence, also, two parallels AB, CD (fig. 2)
determines the position of a plane. For drawing the
secant EF, the plane of the two straight lines A E.
E F is that of the parallels AB. C.D.
we shall have - - -
A P* + A P2 = 2 A. Q2 – 2 P Q?.
Therefore, by taking the halves of both, we have
A P2 = A Q2 — PQ2, or A Q* = A P* + PQ?; hence
the triangle A PQ is right angled at P; and there-
fore AP is perpendicular to PQ.
Scholium. Thus it is evident, not only that a straight
line may be perpendicular to all the straight lines
which pass through its foot in a plane, but that it
always must be so, whenever it is perpendicular to
two straight lines drawn in the plane. -
Cor. 1. The perpendicular A P is shorter than any
oblique line A Q ; therefore it measures the true
distance from the point A to the plane PQ. -
Cor. 2. At a given point P on a plane, it is impos-
sible to erect more than one perpendicular to that
plane ; for if there could be two perpendiculars at the
same point P, draw along these two perpendiculars a
plane, whose intersection with the plane M N is PQ;
then those two perpendiculars would be perpendicu-
lar to the line PQ, at the same point, and in the same
plane, which is impossible.
It is also impossible to let fall from a given point
out of a plane two perpendiculars to that plane ; for
let A P, A Q be these two perpendiculars ; then the
triangle A PQ would have two right angles A PQ,
A QP, which is impossible.
PRoposition V.-Theorem.
Oblique lines equally distant from the perpendicular to:
a plane are equal ; and, of two oblique lines unequally
distant from the perpendicular, that which is nearer is less
than that more remote, fig. 177.
PRoPosition III.-Theorem.
The common section of two planes is a right line,
fig. 175. - -
Fig. 175. Let A CBDA, and A E B F A be two planes cutting
each other, and A, B two points in which the planes
1meet. Draw the line A B, this line is the common
intersection of the two planes. *
For, because the right line touches the two planes
in the points A and B, it lies wholly in both these
planes, or is common to both of them. That is, the
common intersection of the two planes is in a right
line. -
PRoposition IV.-Theorem.
If a straight line AP be perpendicular to two other
straight lines PB, PC, which cross each other at its foot
in the plain MN, it will be perpendicular to any straight
line PQ drawn through its foot in the same plane, and
thus it will be perpendicular to the plane MN, fig. 176.
Fig. 176, Through any point Q in PQ, draw (prop. 5, book iv.)
the straight line BC in the angle B PC, so that B Q=
Q C ; join A B, A Q, A.C.
The base B C being divided into two equal parts at
the point L, the triangle B P C (prop. 17, book iv.)
will give -
- PC2 + PE2 = 2 PQ2 + 2 QC2.
The triangle BAC will, in like manner, give
A C* + A B2 = 2 A Q* + 2 Q Cº.
Taking the first equation from the second, and ob-
serving that the triangles A PC, APB, which are both
right angled at P, give -
A C* – P C* = A Pº, and A B* –PB% = AP2;
For the angles APB, APC, APD being right, if Fig.177
we suppose the distances P B, PC, PD to be equal
to each other, the triangles A PB, A PC, APD will
have each an equal angle contained by equal sides;
therefore they will be equal ; therefore the hypothe-
nuses, or the oblique lines A B, A C, A D will be
equal to each other. In like manner, if the distance
FE be greater than PD or its equal PB, the oblique
line A E will evidently be greater than AIB, or its
equal AD ; that is A B will be less than AE.
Cor. All the equal oblique lines AIB, AC, Al), &c.
terminate in the circumference of a circle B C D,
described from P the foot of the perpendicular as a
centre ; therefore a point A being given out of a
plane, the point P at which the perpendicular let fall
from A would meet that plane, may be found by
marking upon that plane three points B, C, D, equally
distant from the point A, and then finding the centre
of the circle which passes through these points; this
centre will be P, the point sought.
Scholium. The angle A B P is called the inclination
of the oblique line A B to the plane MN; which in-
clination is evidently equal with respect to all such
lines A B, AC, A D, as are equally distant from the
perpendicular ; for all the triangles ABP, ACP,
AIDP, &c. are equal to each other.
PROPosition VI.—Theorem.
Let AP be a perpendicular to the plane M. N, and
B C a line situated in that plane; if from...P, the foot of
the perpendicular, PD be drawn at right angles to B C,
and A D joined, A D will be perpendicular to B C,
fig. 178.
G E O M ET R Y.
347
Geometry. Take DB = DC, and join PB, PC, AB, AC; since
Fig. 178,
Fig. 179.
D B = D C, the oblique line P B = P C ; and with
regard to the perpendicular AP, since PB = PC, the
oblique line AB = AC by the last proposition; there-
fore the line AID has two of its points A and D equally
distant from the extremities B and C ; therefore A D
is a perpendicular at the middle of BC. .
Cor. It is evident likewise, that BC is perpendicular
to the plane APD, since BC is at once perpendicular
to the two straight lines AD, PD.
Scholium. The two straight lines A E, B C afford an
instance of two lines which do not meet, because they
are not situated in the same plane. The shortest
distance between these lines is the straight line PD,
which is perpendicular both to the line AIP and to the
line B. C. The distance PD is the shortest between
these two lines; for if we join any other two points,
such as A and B, we shall have AB 7 AD, AD 7 PD;
therefore A B 7 P.D. *.
The two lines AE, CB, though not situated in the
same plane, are conceived as forming a right angle
with each other, because A D and the line drawn
through one of its points parallel to B C would make
with each other a right angle. In the same manner,
the line AB and the line PD, which represent any two
straight lines not situated in the same plane, are sup-
posed to form with each other the same angle, which
would be formed by A B and a straight line parallel to
PD drawn through one of the points of A B.
PRoposition VII.-Theorem.
If the line A P be perpendicular to the plane MN,
any line DE parallel to AP will be perpendicular to the
same plane, fig. 179.
Along the parallels A P, D E, extend a plane; its
intersection with the plane M N will be PD; in the
plane M N draw B C perpendicular to PD, and join
A D.
By the corollary of the preceding theorem, B C is
perpendicular to the plane A PDE ; therefore the
angle B D E is right; but the angle E D P is right
also, since A P is perpendicular to PD, and D E
parallel to AP; therefore the line DE is perpendicular
to the two straight lines D P, D B ; therefore it is
perpendicular to their plane M.N.
Cor. 1. Conversely, if the straight lines A P, DE
are perpendicular to the same plane MN, they will be
parallel; for if they be not so, draw through the point
D a line parallel to AP, this parallel will be perpen-
dicular to the plane MN; therefore through the same
point D more than one perpendicular might be erected
in the same plane, which (prop. 4, book vi.) is im-
possible.
Cor. 2. Two lines A and B, parallel to a third C,
are parallel to each other; for, conceive a plane per-
pendicular to the line C, the lines A and B, being
parallel to C, will be perpendicular to the same plane;
therefore, by the preceding corollary, they will be
parallel to each other.
When the three lines are in the same plane the case
falls under prop. 23, book i. -
PRoPosition VIII.—Theorem. .
If the line AB be parallel to a straight line C D
drawn in the plane MN, it will be parallel to that plane,
fig. 180, * *
For if the line AB, which lies in the plane A B C D, Book VI.
could meet the plane MN, this could only be in some
point of the line C B, the common intersection of the Fig. 180.
two planes; but A B cannot meet CD, since they are
parallel; hence it will not meet the plane MN; hence
(def. 5) it is parallel to that plane. - -
PRoposition IX-Theorem.
Two planes M. N., P Q perpendicular to the same
straight line A B, are parallel to each other, fig. 18l.
For, if they can meet anywhere, let O be one of Fig. 181
their common points, and join O A, O B; the line
AB, which is perpendicular to the plane MN, is per-
pendicular to the straight line O A drawn through its
foot in that plane; for the same reason AB is perpen-
dicular to BO; therefore OA and OB are two per-
pendiculars let fall, from the same point O, upon the
same straight line; which is impossible : therefore
the planes MN, PQ, cannot meet each other; there-
fore they are parallel. -
PRoPosition X.-Theorem.
The intersections E F, G H of two parallel planes
MN, PQ, with a third plane FG, are parallel,
fig. 182.
For, if the lines E.F, GH, lying in the same plane, Fig. 1g2.
were not parallel, they would meet each other when
produced ; therefore the planes MN, PQ, in which
those lines lie, would also meet ; therefore the planes
would not be parallel.
Proposition XI.-Theorem.
The line AB, which is perpendicular to the plane MN,
is also perpendicular to the plane PQ, parallel to MN,
fig. 181.
Having drawn any line BC in the plane PQ, by the
last proposition, along the lines A B and B C, extend
a plane A B C, intersecting the plane MN in AD; the
intersection AD will be parallel to B C ; but the line
A B, being perpendicular to the plane MN, is perpen-
dicular to the straight line A D ; therefore also to its
parallel B C : hence the line A B being perpendicular
to any line BC drawn through its foot in the plane
PQ, is consequently perpendicular to that plane.
PRoPosition XII.-Theorem.
The parallels E G, FH, comprehended between two
parallel planes MN, PQ, are equal, fig. 182.
Through the parallels E G, F H, draw the plane
E G H F to meet the parallel planès in E F and G H.
The intersections E F, G H (prop. 10, book vi.) are
parallel to each other ; so likewise are E G, F H ;
therefore the figure E G H F is a parallelogram ; and
E G = F H. - . .
Cor. Hence it follows that two parallel planes are
every where equidistant; for if E G and F H are per-
pendicular to the two planes MN, PQ, they will be
parallel to each other, (prop. 7, cor. 1, book vi. 5) and
therefore equal. .
2 z º.
gº
348
Y.
G E O M E T T.
Geometry. '
Fig. 183.
Fig. 184.
PRoposition XIII.-Theorem.
If two angles C A E, D B F, not situated in the same
plane, have their sides parallel and lying in the same
direction, those angles will be equal, and their planes will
be parallel, fig. 183. -
Make A C = BD, AE = BF; and join CE, DF,
AB, CD, E F. Since AC is equal and parallel to BD,
the figure A B D C is a parallelogram, (prop. 28,
book i. ;) therefore C D is equal and parallel to A B.
Eor a similar reason, E F is equal and parallel to A B ;
hence also CD is equal and parallel to EF; the figure
C E F D is therefore a parallelogram, and the side
C E is equal and parallel to D F ; therefore the trian-
gles C A E, D B F have their corresponding sides
equal; consequently the angle C A E = D B F.
Again, the plane ACE is parallel to the plane BDF.
For suppose the plane parallel to BDF, drawn through
the point A, were to meet the lines CD, EF, in points
different from C and E, for instance in G and H ; then,
(prop. 12, book vi.) the three lines A B, G. D, F H
would be equal: but the lines A B, C D, E F are
already known to be equal; hence C D = G D, and
F H = EF, which is absurd ; hence the plane A C E
is parallel to B D F.
Cor. If two parallel planes MN, PQ are met by two
other planes CADB, EABF, the angles CAE, DEF,
formed by the intersections of the parallel planes will
be equal; for (prop. 10, book vi.) the intersection
A C is parallel to BD, and A E to BF, therefore the
angle C A E = D B F. -
PRoposition XIV.-Theorem.
If three straight lines A B, CD, E F, not situated in
the same plane, are equal and parallel, the triangles
A C E, B D F formed by joining the extremities of these
straight lines will be equal, and their planes will be
parallel, fig. 183.
For since A B is equal and parallel to CD, the
figure AB C D is a parallelogram; hence the side A C
is equal and parallel to B. D. For a like reason the
sides A E, B F are equal and parallel, as also CE,
DF; therefore the two triangles A C E, BDF, are
*qual; and, consequently, as in the last proposition,
their planes are parallel.
PRoPosLTION XV.—Theorem.
Two straight lines, included between three parallel
planes, are cut proportionally, fig. 184.
Suppose the line A B to meet the parallel planes
MN, PQ, R.S, at the points A, E, B; and the line
CD to meet the same planes at the points C, F, D;
then A. E. : E B : : C F : F D. -
Draw A D meeting the plane PQ in G, and join
A C, E G, G F, B.D ; the intersections E G, B D, of
the parallel planes PQ, R. S., in the plane A B D, are
parallel, (prop. 10, book vi.;) therefore A E : E B : :
A G : G D ; in like manner, the intersections A C,
G F being parallel, A.G : G D : : C F : FD ; the ratio
A G : G D is the same in both ; hence -
AE : E B ; ; C F : FD.
PRoposition XVI.-Theorem.
If the line AP be perpendicular to the plane MN, any
plane APB drawn along A P will be perpendicular to the
plane MN, fig. 185. i
Let the two planes A B, M N intersect each other
in the line B.C. In the plane M N draw DE perpen-
dicular to BP; then the line AP, being perpendicular
to the plane MN, will be perpendicular to each of
the two straight lines BC, DE; but the angle APD,
formed by the two perpendiculars PA, PD at their
common intersection B P, is the measure of the angle
of the two planes, (def. 4;) and since in the present
case the angle is a right angle, the two planes are
perpendicular to each other.
Scholium. When the three lines such as AP, BP,
DP are perpendicular to each other, each of these
lines is perpendicular to the plane of the other two ;
and the planes themselves are perpendicular to each
other.
PROPOSITION XVII.-Theorem.
If the plane A B be perpendicular to the plane MN,
and if in the plane A B the line PA be perpendicular to
the common intersection BP, then will A P be perpendi-
cular to the plane MN, fig. 185.
For in the plane M N draw PD perpendicular to
PB; then because the planes are perpendicular, the
angle A P D is a right angle ; therefore the line A P
is perpendicular to the two straight lines PB, PD ;
and is therefore perpendicular to their plane M. N. .
Cor. If the plane A B be perpendicular to the plane
MN, and if at a point P of the common intersection
a perpendicular be erected to the plane MN, that
perpendicular will be in the plane AB; for if not, then
in the plane A B we might draw AP perpendicular to
B, their common intersection, and this A P at the
same time would be perpendicular to the plane M N ;
therefore at the same point P there would be two
perpendiculars to the plane MN, which is impossible.
PRoposition XVIII.-Theorem.
If two planes be perpendicular to a third plane, their
common intersection will be also perpendicular to the third
plane, fig. 185.
Let AB, A D be perpendicular to MN, then will
their common intersection AP be perpendicular to the
same plane M. N. -
For at the point P erect the perpendicular to the
plane MN; then that perpendicular must be in the
plane AID, and also in A B, (by the last prop. ;) there-
fore it is their common intersection A P.
PROPosition XIX. — Theorem.
If a solid angle is formed by three plane angles, the
sum of any two of these angles will be greater than the
third, fig. 186.
The proposition requires demonstration only when
the plane angle, which is compared to the sum of the
other two, is greater than either of them. Therefore
suppose the solid angle S to be formed by three plane
angles A SB, ASC, BSC, whereof the angle ASB is
; gates: ; we are to show that ASB 4 A SC +
Book W.K.
~~
Fig. 185.
Fig. 186
G E O M E T R Y. 349
D TE, we have SB A = TE D. Likewise SB = Book VI.
Geometry. In the plane ASB make the angle BSD = B S C,
TE; therefore the triangle SA B is equal to the S-V-
*-v-7 draw the straight line A D B at pleasure; and having
taken S C = SD, join A C, B C. -
The two sides B.S, SD are equal to the two BS,
S C ; the angle BSD = B SC; therefore the trian-
gles BSD, BSC are equal; therefore B D = BC.
But AB Z AC + BC; taking BD from the one side,'
and from the other its equal BC, there remains AD Z
A C. The two sides A S, SD are equal to the two
A S, S C ; the third side A D is less than the third
side A C ; therefore (prop. 8, book i.) the angle
A SD Z A S C. Adding BSD = BSC, we shall have
A S D + B.S D or A S B Z A S C + B S C.
PROPosſTIon XX.-Theorem.
The sum of the plane angles which form a solid angle,
is always less than four right angles, fig. 187.
triangle TD E.; therefore SA = TID, and AB = DE.
In like manner it may be shown, that SC = TF, and
B C = E F. That granted, the quadrilateral SAO C
is equal to the quadrilateral T DPF; for, place the
angle A SC upon its equal D TF ; because SA =
TD, and S C = TF, the point A will fall on W, and
the point C on F; and at the same time, A O, which
is perpendicular to SA, will fall on PD which is per-
pendicular to TD, and in like manner O C on PF;
wherefore the point O will fall on the point P, and
A O will be equal to D P. But the triangles A. O.B.
D PE, are right angled at O and P; the hypothe-
nuse A B = DE, and the side AO = DP; hence those
triangles are equal; therefore the angle O Alb = PDE.
The angle O A B is the inclination of the two planes
ASB, ASC; the angle PDE is that of the two planes
Fig. 187, Conceive the solid angle S to be cut by any plane DTE, D TF; hence those two inclinations are equal
A B C DE ; from O, a point in that plane, draw to the to each other. r T -
several-angles straight lines AO, OB, OC, OD, OE. It must, however, be observed, that the angle A of
The sum of the angles of the triangles A SB, BSC, the right angled triangle O A B is properly the incli-
&c. formed about the vertex S, is equivalent to the nation of the two planes A SB, A SC, only when the
sum of the angles of an equal number of triangles perpendicular B O falls on the same side of SA as SC
A OB, B O C, &c. formed about the point O. But at falls; for if it fell on the other side, the angle of the
the point B the angles A BO, OBC, taken together two planes would be obtuse, and joined to the angle
make the angle A B C (prop. 19, book vi.) less than A of the triangle O A B it would make two right
the sum of the angles A B S, S B C ; in the same angles. But in the same case, the angle of the two
manner, at the point C we have BC O + O CD Z planes T DE, TDF would also be obtuse, and joined
B C S + S CD ; and so with all the angles of the to the angle D of the triangle D PE, it would make
polygon A B C D E : whence it follows, that the sum two right angles; and the angle A being thus always
of all the angles at the bases of the triangles whose equal to the angle at D, it would follow in the same
vertex is in O, is less than the sum of the angles at manner that the inclination of the two planes A SB,
the bases of the triangles whose vertex is in S; hence A S C, must be equal to that of the two planes TDE,
to make up the deficiency, the sum of the angles TD F.
formed about the point O, is greater than the sum of g --- t * ~ *
the angles about the point S. But the sum of the Scholium 2, relative to the measure of solid angles.
angles about the point O is equal to four right angles, A more general definition of solid angles than that
(prop. 3, book i.;) therefore the sum of the plane given at the commencement of this book is, that a
angles, which form the solid angle S, is less than four solid angle is the angular space included between
right angles. • , - several plane surfaces, or one or more curved surface
Scholium l. This demonstration is founded on the meeting in the point which forms the summit of the
supposition that the solid angle is convex, or that the angle.
plane of no one surface produced can ever meet the According to this definition, solid angles bear just
solid angle; if it were otherwise, the sum of the plane the same relation to the surfaces which comprise
angles would no longer be limited, and might be of them, as plane angles do to the lines by which they
any magnitude. are included ; so that, as in the latter, it is not the
magnitude of the lines, but their mutual inclination
PRoPosition XXI.-Theorem. which determines the angles; so, in the former, it is
º not the magnitude of the planes, but their mutual in-
If ºolid ºgles are cºmposed of three ſºle angles clination which determine the solid angles. Accord-
respectively equal to each other, the planes which contain ing to this view of the subject, the spherical surface
the equal angles will be equally inclined to each other, described about the summit of any solid angle as a
fig. 188. centre, will become a measure of that angle; as the
Fig. 188. Let the angle A S C = D TF, the angle A SB = circular arc is employed to measure and to compare
DTE, and the angle BS C = ETF ; then will the
inclination of the planes A SC, AS B, be equal to
that of the planes D TF, DT E.
Having taken S B at pleasure, draw B O perpendi-
cular to the plane ASC; from the point O, at which
that perpendicular meets the plane, draw OA, O C
perpendicular to SA, SC; join A B, BC; next take
TE=SB; draw EP perpendicular to the plane DTF;
from the point P draw PD, PF, perpendicular to TD,
T F; lastly, join D E, E F.
The triangle S A B is right angled at A, and the
triangle TDE at D; and since the angle ASB =
rectilinear angles. Let us imagine, in the first place,
such a sphere to be described about any given solid
angle comprised under three plane angles, and that
those planes are produced till they cut the surface of
the sphere ; then will the surface of the spherical
triangle included between those planes be the measure,
or may be assumed as the measure, of the solid angle,
made by the planes at the common point of meeting;
for no change can be conceived in the relative position
of the bounding planes, that is, in the magnitude of
the solid angle, without a corresponding and pro-
portional mutation in the surface of the spherical
350
G E O M E T R Y.
A VB, BV C, &c. taken together, form the conver or Book VII.
lateral surface of the pyramid. \-N-"
Geometry, triangle; and if, in like manner, the three or more
\-N-" plane surfaces comprising another solid angle be pro-
grams. To construct this solid, let A B C D E,
Fig. 189, (fig. 189.) be any rectilineal figure. In a plane parallel tuated ; therefore the solid angles B and C are equal,
to ABC draw the lines FG, GH, HI, &c. parallel to the and therefore B G will fall on its equal bg; and it is
sides AB, BC, CD, &c.; thus there will be formed a likewise evident, because the parallelograms A B G F
figure FGHI K, similar to A B C D E. Now let the and a b g f are equal, that the side G F will fall on its
vertices of the corresponding angles be joined by the equal g f. and in the same manner G H on g h :
lines AF, BG, CH, &c. the faces AB G F, B C H G, therefore the upper base FG H IR will coincide with
&c. will evidently be parallelograms, and the solid its equal fgh ik, and the two solids will be identical,
thus formed will be a prism. . . - since their vertices are the same. -
3. The equal and parallel plane figures A B C D E, Cor. Two right prisms which have equal bases and
FGHIK are called the bases of the prism. The other equal altitudes, are equal. For, since the side A B is
planes or parallelograms taken together constitute the equal to a b, and the altitude B G to bg, the rectangle
lateral or conver surface of the prism. A B G F will be equal to a bgf; and in the same way
4. The altitude of a prism is the perpendicular the rectangle B G H C will be equal to b g h c, and
distance between its bases; and its length is a line thus the three planes, which form the Solid angle B,
equal to any one of its lateral edges, as A F, or will be equal to the three which form the solid angle b.
B G, &c. - Hence the two prisms are equal.
5. A right prism is one in which the lateral edges - .
A F, B.G., &c. are perpendicular to the planes of its PRoposition II.-Theorem.
bases; then each of them is equal to the altitude of - g
the prism ; in every other case the prism is oblique. In every parallelopipedon the opposite planes are equal
6. A prism is triangular, quadrangular, pentagonal, and parallel, fig. 193.
&c. according as the base is a triangle, a quadrilateral, By the definition of this solid, the bases A B CD, Fig. 193.
a pentagon, &c. E F G H are equal parallelograms, and their sides are
7. A prism which has a parallelogram for its base parallel: it remains only to show, that the same is
has all its faces parallelograms, and is called a parallel- true of any two opposite lateral faces, such as AE HD,
Fig. 190, opiped, or parallelopipedon, (fig. 190.) A parallelopiped BFG C. Now AD is equal and parallel to B C, be-
is rectangular, when all its faces are rectangles. cause the figure ABCD is a parallelogram ; for a like
8. When the faces of a rectangular parallelopiped reason, A E is parallel to BF; hence the angle D A E
are squares, it is called a cube. - is equal to the angle CBF, and the planes D A E, CBF
9. A pyramid is a solid formed by several triangu- are parallel ; hence also the parallelogram D A E H is
Fig. 191, lar planes which meet in a point, as V, (fig. 191,) equal to the parallelogram CBF G. In the same way
duced till they cut the surface of the same, or if an
equal sphere, whose centre coincides with the summit
of the angle, the surface of the spherical triangle or
polygon included between the planes which determine
the angle, will in like manner be a correct measure
of that angle ; and the ratio which subsists between
the areas of these triangles and polygons, or other
surfaces thus formed, will be accurately the ratio
which subsists between the solid angles, constituted
by the meeting of the several planes or surfaces at the
centre of the sphere.
It will, of course, be understood, that this measure-
ment has only a relation to the magnitude of the
angles. It has no reference to their geometrical
properties, which may be very different, although their
magnitudes, as above estimated, may be the same.
BOOK VII.
Of solids bounded by planes.
DEFINITIons.
1. A solid is that which has length, breadth, and
thickness. -
2. A prism is a solid 'contained by plane figures, of
which two that are opposite are equal, similar, and
parallel to one another; and the others are parallelo-
and terminate in the same plane rectilineal figures
A B C D E. -
The plane figure ABC DE is called the base of the
pyramid; the point V is its vertea : and the triangles
10. The altitude of a pyramid is the perpendicular
drawn from the vertex to the plane of its base, pro
duced if necessary. - . . .
11. A pyramid is triangular, quadrangular, &c, ac-
cording as its base is a triangle, a quadrangle, &c.
12. A pyramid is regular, when its base is a regular
figure, and the perpendicular from its vertex passes
through the centre of its base; that is, through the
centre of a circle which may be conceived to circum-
scribe its base. * ,
13. Two solids are similar, when they are contained
by the same number of similar planes, similarly
situated, and having like inclinations to one another.
PRoposition I.-Theorem.
. Two prisms are equal when a solid angle in each is
contained by three planes, which are equal in both and
similarly situated, fig. 192. -
Let the base ABCDE he equal to the base a b c de; Fig. 192.
the parallelogram AlBGF equal to the parallelogram
a bgf; and the parallelogram B C H G equal to the
parallelogram b c h g ; then will the prism A B C I be
equal to the prism a b c i.
For apply the base ABC DE upon its equal ab cae,
so that the bases (being equal) may coincide. But
the three plane angles which form the solid angle B,
are respectively equal to the three plane angles which
form the solid angle b, that is, A B C = a b c, A BG =
ab g, and G B C = g b c, and they are also similarly si-
it might be shown that the opposite parallelograms
A B F E, D C G H are equal and parallel. -
Cor. Since the parallelopipedon is a solid bounded by
six planes, whereof those lying opposite to each othek
G E O M ET R Y
.*
Geometry, are equal and parallel, it follows that any face and the
—v-, one opposite to it may be assumed as the bases of the
Fig. 194,
Fig. 195.
parallelopipedon. -
Scholium. If three straight lines AB, A E, A D,
passing through the same point A, and making given
angles with each other, are known, a parallelopipedon
may be formed on those lines. For this purpose, a
plane must be extended through the extremity of each
line, and parallel to the plane of the other two ; that
is, through the point B a plane parallel to D A E,
through D a plane parallel to BAE, and through E a
plane parallel to B A D. The mutual intersections of
those planes will form the parallelopipedon required.
PROPosition III.-Lemma.
In every prism A B C I, the sections N OPQR,
STVXY, formed by parallel planes, are equal polygons,
fig. 194.
For the sides ST, NO are parallel, being the inter-
sections of two parallel planes with a third plane
A B G F ; moreover the sides ST, N O, are in-
cluded between the parallels NS, OT, which are sides
of the prism ; hence N O is equal to ST. For like
reasons, the sides O P, P Q, Q R, &c. of the section
N OPQR, are respectively equal to the sides TV,
V X, X Y, &c. of the section S T V X Y. And since
the equal sides are at the same time parallel, it follows
that the angles N OP, OPQ, &c. of the first section
are respectively equal to the angles S TV, TV X of
the second. Hence the two sections N OPQR,
STV XY are equal polygons.
Cor. Every section in a prism, if drawn parallel to
the base, is also equal to that base.
PRoPosition IV.-Theorem.
The two symmetrical triangular prisms A B D H E F,
B C D F GH, into which the parallelopipedon A G may be
decomposed, are equal to each other, fig. 195.
Through the vertices B and F, draw the planes
B a dc, Fe h g at right angles to the side B F, and
meeting A B, D H, C G, the three other sides of the
parallelopipedon, in the points a, d, c towards one
direction, and in e, h, g towards the other ; then the
sections B a d c, Fe h g will be equal parallelograms;
being equal because they are formed by planes per-
pendicular to the same straight line, and consequently
parallel; and being parallelograms, because a B, d c,
two opposite sides of the same section, are formed
by the meeting of one plane with two parallel planes
A B FE, D C G H. -
For a like reason, the figure Bae F is a parallel-
ogram; so also are B F g c, c d h g, and a d he, the
other lateral faces of the solid B a dc Fe h g ; hence
that solid is a prism, (def. 5;) and that prism is a
right one, because the side B F is perpendicular to its
base.
This being proved, if the right prism Bh be divided
by the plane BFHD into two right triangular prisms
a B de Fh, B d clºh g; it will remain to be shown that
the oblique triangular prism ABDEFH will be equal
to the right triangular prism a B de F h. And since
those two prisms have a part A B D he F in common,
it will only be requisite to prove that the remaining
. namely, the solids B a AD d, Fele H h are
equal.
Now, by reason of the parallelograms A B F E,
a BFe, the sides AE, a e, being equal to their Book VII.
parallel BF, are equal to each other; and taking's
away the common part A. e., there remains A a = E e.
In the same manner we could prove D d = H h.
Let us now place the base Fe h on its equal B a d ;
the point e falling on u, and the point h on d, the sides
e E, h H will fall on their equals a A, d D, because
they are perpendicular to the same plane B a d. Hence
the two solids in question will coincide exactly with
each other, and the oblique prism B A D FE H is
therefore equal to the right one B a d Feh.
In the same manner might the oblique prism
B D CIF H G be proved equal to the right prism
B d c Fh g. But (prop. 1, book vii.) the two right
prisms B a d Feh, B d clºhg are equal, since they have
the same altitude B F, and since their bases Bad, B de
are halves of the same parallelogram. Hence the two
triangular prisms BA D FE H, B D C F H G, being
equal to the equal oblique prisms, are equal to each
other. -
Cor. Every triangular prism "A BD HEF is half of
the parallelopipedon A G described on the same solid
angle A, with the same edges A B, A D, AE.
PRoposition V-Theorem.
If two parallelopipedons AG, A L have a common base
A B CD, and if their upper bases E FG H, I KL M lie
in the same plane and between the same parallels E K,
H L, those two parallelopipedons will be equal to each
other, fig. 196.
There may be three cases to this proposition, ac- Fig. 196,
cording as EI is greater, less than, or equal to EF;
but the demonstration is the same for all. In the first
place, then, we shall show that the triangular prism
A E I D H M is equal to the triangular prism
B FK C G L. - -
Since A E is parallel to BF, and HE to G F, the
angle AEI = B F K, H E I = G F K, and H E A =
G F B. Of these six angles the first three form the
solid angle E, the last three the solid angle F; there-
fore, the plane angles being respectively equal, and
similarly arranged, the solid angles F and E must be
equal. Now, if the prism A E M be laid on the prism
BFL, the base A E I being placed on the base B F K
will coincide with it because they are equal; and since
the solid angle E is equal to the solid angle F, the side
E H will fall on its equal FG ; and nothing more is
required to prove the coincidence of the two prisms
throughout their whole extent, for (prop. 1, book vii.)
the base A E I and the edge E H determine the prism
A EM, as the base B F K and the edge FG determine
the prism B F L ; hence these prisms are equal.
But if the prism A E M is taken away from the solid
A L, there will remain the parallelopipedon A IL; and
if the prism BFL is taken away from the same solid,
there will remain the parallelopipedon A E G ; hence
those two parallelopipedons A IL, AEG are equal.
PRoposition VI.-Theorem.
Two parallelopipedons having the same base and the
same altitude are equal to each other, fig. 197.
Let A B C D be the common base of the two paral- Fig. 197,
Ielopipedons AG, A L; since they have the same alti-
tude, their upper bases EFG H, I KLM will be in
the same plane. Also the sides E F and A B will be
352
G E O M E T R Y.
But if the two altitudes are incommensurable with Book VII.
each other, divide one of them into any number of S-N-'
Geometry, equal and parallel, as well as IK and AB; hence EF
~~' is equal and parallel to IK ; for a like reason G F is
Fig. 197.
the points A, B, C, D, draw A I, B K, C L., D M, may be compared with each of the parallelopipedons
perpendicular to the plane of the base ; and we shall A G, A K.. The two solids AG, A Q, having the same
thus form the parallelopipedon AL equal to AG, base A E H D, are to each other as their altitudes A B,
and having its lateral faces A K, BL, &c. rectangular. A O ; in like manner the two solids A Q, A K, having
Hence if the base A B C D be a rectangle, AL will be the same base AOLE, are to each other as their alti-
the rectangular parallelopipedon equal to A G, the tudes AD, A M. Hence we have the two proportions,
Fig. 198. parallelopipedon proposed. But if A B C D (fig. 198) sol. A G : sol. A Q ; : A B : A O,
is not a rectangle, draw AO and B N perpendicular to sol. A Q : sol. A K : : A D : A M.
QP, and 99.3%. Nº Pºndicular to the base; then Multiply together the corresponding terms of those
the solid ABNOIPQ will be a rectangular parallelo- jºi. in the . fie common mul-
pipedon : for, by construction, the base ABN O and #: l. A Q : hall h
e º plier sol. ; we snail nave
its opposite I K P Q are rectangles; so also are the
lateral faces, the edges A I, O Q, &c. being perpendi- sol. A G : Sol. A K. : : A B x AD : A O X AM.
cular to the plane of the base; hence the solid AP is But AB X A D represents the base A B CD; and A O.
a rectangular parallelopipedon. But the two parallelo- X AM represents the base A M N O ; hence two rec-
pipedons AP, AL may be conceived as having the tangular parallelopipedons of the same altitude are to
same base AB K I and the same altitude A O ; hence each other as their bases.
the parallelopipedon A G, which was at first changed - :
into an equal parallelopipedon A L, is again changed Proposition X-Theorem.
into an equal rectangular parallelopipedon A P, having 4mu two rectangular parallelopipedons are to each
i."; º A I, and a base A BNO equal to the other ". the products of their bases by their altitudes,
g that is to say, as the products of their three dimensions,
fig. 201. -
PRoposition VIII.-Theorem. For, having placed the two solids AG, A Z, so that Fig.201.
Two rectangular parallelopipedons A G, A L, which their surfaces have the common angle BAE, produce
have the same base A B C D, are to each other as their the interior planes necessary for completing the third
altitudes A E, AI, fig. 199. parallelopipedon A K, having the same altitude with
fig, 199 First, suppose the altitudes A E, AI, to be to each ºption A G. By the last proposition, we
equal and parallel to L. K. : Let the sides E. F., H G be
produced, and likewise L K, IM, till by their inter-
sections they form the parallelogram NOP Q; this
parallelogram will evidently be equal to either of the
bases EFG H, IKL M. Now if a third parallelopi-
pedon be conceived, having A B CD for its lower base,
and N OPQ for its upper, this third parallelopipedon
will (prop. 5, book vii.) be equal to the parallelopipe-
don A G, since with the same lower base, their upper
bases lie in the same plane and between the same
parallels G Q, FN. For the same reason this third
parallelopipedon will also be equal to the parallelopi-
pedon A L; hence the two parallelopipedons AG,
A L, which have the same base and the same altitude,
are equal.
PR opositron VII.—Theorem.
Any parallelopipedon may be changed into an equal
rectangular parallelopipedon having the same altitude and
an equal base, fig. 197 and 198.
Let A G be the parallelopipedon proposed. From
other as two whole numbers, for example as 15 is
to 8. Divide A E into 15 equal parts; whereof AI
will contain 8; and through it, y, z, &c. the points of
division, draw planes parallel to the base. These
planes will cut the solid A G into 15 partial parallelo-
pipedons, all equal to each other, having equal bases
and equal altitudes,—equal bases, because every
section MIKL, made parallel to the base A B C D of
a prism, is equal to that base, equal altitudes because
these altitudes are the same divisions A ar, a y, yz, &c.
But of those 15 equal parallelopipedons, 8 are con-
tained in A L; hence the solid A G is to the solid
A L as 15 is to 8, or generally, as the altitude A E
is to the altitude AI, -
equal parts or units, and the other into parts equal to
the former ; then, as is shown in our second book,
the remainder (if the second altitude be not exactly
commensurable with the first) will be less than the
measuring unit; and this unit may be taken less than
any assignable quantity. Whatever ratio therefore
obtains between the commensurable parts, differing by
less than any assignable quantity from the incommen-
Surable, obtains also between the incommensurable ;
but when the altitudes are commensurable, the prisms
are as the altitudes ; they are therefore so also when
the altitudes are incommensurable.
PRoPostTIon DX-Theorem.
Two rectangular parallelopipedons A G, A K, having
the same altitude A E, are to each other as their bases
A B C D, A M N O, fig. 200.
Having placed the two solids by the side of each Fig. 200,
other, as the figure represents, produce the plane
O N KL till it meets the plane D C G H in PQ ; we
shall thus have a third parallelopipedon A Q, which
sol. A G : sol. A K : : A B C D : AMN O.
But the two parallelopipedons A K, A Z having the
same base AMNO, are to each other as their alti-
tudes A E, A X; hence we have g
k sol, A K : Sol. A Z : : A E : A X.
Multiply together the corresponding terms of those
proportions, omitting in the result the common mul-
tiplier sol. A K ; we shall have
sol. A G : sol. AZ : : A B C D X A E : A M N Ox A X.
Instead of the bases A B C D and A MNO, put A Bx.
AD and A Ox AM; it will give -
sol. A G : sol, AZ; ; ABxA Dx AE : A Ox AMxAX.
G E O M ET R Y. 353
its height; hence the solidity of the former is, in Book VII.
Geometry. Hence any two rectangular parallelopipedons are to -
- like manner, equal to the product of its base by its \-N-
*-y-' each other, &c. -
Scholium. We are consequently authorized to assume,
as the measure of a rectangular parallelopipedon, the
product of its base by its altitude, in other words, the
product of its three dimensions.
In order to comprehend the nature of this measure-
ment, it is necessary to reflect, that by the product of
two or more lines is always meant the product of the
numbers which represent them, those numbers them-
selves being determined by their linear unit, which
may be assumed at pleasure. Upon this principle, the
product of the three dimensions of a parallelopipedon
is a number, which signifies nothing of itself, and
would be different if a different linear unit had been
assumed. But if the three dimensions of another
parallelopipedon are valued according to the same
linear unit, and multiplied together in the same man-
ner, the two products will be to each other as the
solids, and will serve to express their relative mag-
nitude. • .
The magnitude, of a solid, its volume or extent,
form what is called its solidity; and this word is ex-
clusively employed to designate the measure of a
solid : thus we say the solidity of a rectangular paral-
lelopipedon is equal to the product of its base by its
altitude, or to the product of its three dimensions.
As the cube has all its three dimensions equal, if
the side is 1, the solidity will be 1 × 1 × 1 = 1; if the
side is 2, the solidity will be 2 × 2 × 2–8; if the side
is 3, the solidity will be 3 × 3 × 3 = 27 ; and so on :
hence, if the sides of a series of cubes are to each
other as the numbers 1, 2, 3, &c. the cubes them-
selves or their solidities will be as the numbers 1, 8,
27, &c. Hence it is, that in arithmetic, the cube of a
number is the name given to the product which results
from three factors each equal to this number.
If it were proposed to find a cube double of a given
cube, the side of the required cube would have to be to
that of the given one, as the cube root of 2 is to unity.
Now, by a geometrical construction, it is easy to find
the square root of 2; but the cube root of it cannot
be so found, at least not by the simple operations of
elementary geometry, which consist in employing
nothing but straight lines, two points of which are
known, and circles whose centres and radii are de-
termined. -
Owing to this difficulty the problem of the duplica-
tion of the cube became celebrated among the ancient
geometers, as well as that of the trisection of an angle,
which is nearly of the same species. The solutions
of which such problems are susceptible have, how-
ever, long since been discovered; and though less
simple than the constructions of elementary geometry,
they are not, on that account, less rigorous or less
satisfactory.
PRoposition XI.-Theorem.
The solidity of a parallelopipedon, and generally of
any prism, is equal to the product of its base by its
altitude.
For, in the first place, any parallelopipedon (prop. 7,
book vii.) is equal to a rectangular parallelopipedon,
having the same altitude and an equal base. Now the
solidity of the latter is equal to its base multiplied by
WOL., I,
altitude. . - . -
In the second *::: and for a like reason, any rectan-
gular prism is half of the parallelopipedon so constructed
as to have the same altitude and a double base. But
the solidity of the latter is equal to its base multiplied.
by its altitude ; hence that of a triangular prism is
also equal to the product of its base (half that of the
parallelopipedon) multiplied into its altitude.
In the third place, any prism may be divided into
as many triangular prisms of the same altitude, as
there are triangles capable of being formed in the
polygon which constitutes its base. But the solidity
of each triangular prism is equal to its base multi-
plied by its altitude; and since the altitude is the same
for all, it follows that the sum of all the partial prisms
must be equal to the sum of all the partial triangles,
which constitute their bases, multiplied by the
common altitude. -
Hence the solidity of any polygonal prism is equal
to the product of its base by its altitude.
Cor. Comparing two prisms, which have the same
altitude, the products of their bases by their altitudes
will be as the bases simply ; hence two prisms of the
same altitude are to each other as their bases. For a like
reason, two prisms of the same base are to each other as
their altitudes. \ .
PRoposition XII.-Theorem.
Similar prisms are to one another as the cube of their
homologous sides, fig. 203. -
Let P and p be two prisms of which B C, b e are Fig. 203,
homologous sides; the prism P is to the prism p as
the cube of B C to the cube of b c. From A and a,
homologous angles of the two prisms, draw. A H, a h
perpendicular to the bases B.C.D, b c d. Join B H,
take Ba = b a, and in the plane B H A draw a h per-
pendicular to B H; then a h shall be perpendicular to
the plane C B D, (prop. 16, book vi.) and equal to a h,
the altitude of the other prism; for if the solid angles
B and b were applied the one to the other, the planes
which contain them, and consequently the perpendi-
culars a h, a h would coincide. .
Now because of the similar triangles A.B H, a b h,
and the similar figures A C, a c we have
- A H : a h : : A B : a b : : B C : b c ;
and because of these similar bases, the -
base B C D : base b c d : : B C* : b cº, (prop. 32, book iv.)
From these two proportions, by considering all the
quantities as represented by numbers, we get
(prop. 12, book ii.)
A H x base B C D : a h x base BCD :: BC 3: b c xB C*;
a h x base B C D : a h x base b c d : : b c x BC? : b cº;
therefore (prop. 12, book ii.) and cancelling the like
terms. -
A H x base B C D : a h x base b c d : : BC * : b c 3.
but A H x base B C D expresses the solidity of the
prism P; and a h x base B C D expresses the solidity
of the other prism p'; therefore -
prism P: prism p': : B C* : b c 3.
Cor. Similar prisms are to one another in the tri-
plicate ratio of their homologous sides. For let Y and
Z be two lines, such that B C : b c : : b c : Y, and b c :
Z : : Y : Z; then the ratio of B C to Z is triplicate of
the ratio of BC to b c,
3 A
354
G. E. O. M. E. T. R. Y.
Geometry.
But since B C : be : ; b c : Y,
\-v- therefore B C* : b cº : : b cº : Y*, (prop. 11, book ii.;)
Fig. 204,
Fig. 205.
Fig. 206.
and, multiplying the antecedents by BC, and the con-
sequents by be, B C*: b cº : : B C × b c 3: b c x Y”
: : B C x b c : Y”; but Y? = b c x Z ; therefore
B C S : b c s : : B C X b c : b c x Z : : B C : Z. But
B C* : b cº :: prism P : prism p ; therefore the prisms
have to each other the ratio of B C to Z, that is the
triplicate ratio of B C to b c. - -
PRoPosition XIII.-Theorem.
If a triangular pyramid A – B C D be cut by a plane
parallel to its base, the section FG H is similar to the
base, fig. 204.
For because the parallel planes BCD, F G H are
cut by a third plane A B C, the sections FG, B C are
parallel, (prop. 10, book vi.) In like manner it ap-
pears, that FH is parallel to BD ; therefore the angle
HFG is equal to the angle D B C, (prop. 13, book. vi.;)
and because the triangle ABC is similar to the triangle
AFG, and the triangle ABD is similar to the triangle
A FH, we have
B C : B A : : FG : FA,
and B A : B D : : E A : F H.
Therefore B C : B D : : FG : F PH ; now the angle
DBC has been shown to be equal to the angle HFG ;
therefore the triangle D B C, H. F G are equiangular,
(prop. 25, book iv.) -
PRoposition XIV.-Theorem.
If two triangular pyramids A–BCD, a-bcd, which
have equal bases, and equal altitudes, be cut by planes
that are parallel to the bases, and at equal distances from
them; the sections FG H, fgh will be equal, fig. 205.
Draw AKE, ake perpendicular to the bases B C D,
b c d, meeting the cutting planes in K and k ; then, be-
cause of the parallel planes, we have (prop. 15, bookvi.)
AE : A K : : A B : A F, and a e : a k : : a b : a f.
but, by hypothesis, A E = a e, and A K = a k ;
therefore A B : A F : : a b : a f; again, because of
similar triangles, A B : A F : : B C : FG, and ab : af
: ; b c : fg; therefore, B C : FG : : b c : fg; and
hence, BC * : ; FG% :: b c 2 : fg”, (prop. 11, book ii.;)
but because of the similar triangles B D C, F G H,
B C2 : FG% : : trian. B D C : trian. FH G, and in like
manner b cº: fg”: : trian. b c d : trian. fgh, therefore
trian. B C D : trian. FG H : : trian. b c d : trian. fgh.
Now trian. B C D = trian. b c d, (by hypothesis,) there-
fore the triangle FH G is equal to the triangle f h g.
Scholium. It is easy to see, that what is proved in
this and the preceding proposition, is also true of poly-
gonal pyramids.
PRoPosition XV.-Theorem.
A series of prisms of the same altitude may be inscribed
in a pyramid, and another series may be circumscribed
about it, which shall exceed the other by less than any
given solid, fig. 206. r -
Let A B C D be a pyramid, and let A C, one of its
lateral edges, be divided into some number of equal
parts, at the points F, G, H ; through these let planes
pass parallel to the base B C D, making with the sides
of the pyramids the sections Q PF, SR G, UTH;
which will be similar to one another and to the base,
(prop. 13, book vii.) From B. in the plane of the
triangle A B C, draw B K parallel to CF, meeting Book VII.
F P produced in K; in like manner from D draw Di. S.-->
parallel to CF meeting F Q produced in L; join KL,
and the solid C B D — FK L will evidently be a prisma.
By the same construction let the prisms PM, R.O,
TV be described : also let the straight line IP,
which is in the plane of the triangle A B C, be pro-
duced till it meet B C in h, and let M Q be produced
till it meet D C in g; join hg, then C h g, F PQ will
be a prism, and be equal to the prism PM. In the
same manner is described the prism m S equal to the
prism R.O, and the prism q U equal to the prism.
TV. Therefore the sum of all the inscribed prisms
h Q, m S, and q U is equal to the sum of the prisms
PM, R.O, and TV ; that is, to the sum of all the
circumscribed prisms, except the prism B L ; where-
fore B L is the excess of the prisms circumscribed
about the pyramid above the prisms inscribed with-
in it. t
Let us now suppose that Z denotes some given solid
equal to a prism, which has the same base C B D as
the pyramid, and its altitude equal to a perpendicular
from E (a point in A C) upon the base. Then, how-
ever near E may be to C, it will evidently be possible
to divide A C into such a number of equal parts, that
one of them, CF, shall be less than C E ; and this
being the case, the prism B L will evidently be less
than the prism whose base is the triangle C B D and
altitude, a perpendicular from E on the base B C D ;
that is less than the given solid Z ; therefore the
excess of the circumscribed above the inscribed prisms
may be less than the solid Z. *
Cor. Since the difference between the circumscribed
and inscribed prisms may be less than any given mag-
nitude, and the pyramid is greater than the latter, and
less than the former, it follows that a series of prisms
may be circumscribed about the pyramid, and also a
series of prisms may be inscribed in it, which shall
differ from the pyramid itself by less than any given
solid. --
Proposition XVI—Theorem. .
Pyramids that have equal bases and altitudes are equal
to one another, fig. 207.
Let A–B C D, a - bc d be two pyramids that have Fig. 207.
equal bases B C D, b c d, and equal altitudes ; viz. the
perpendiculars drawn from the vertices A and a upon
the planes BCD, b c d, the pyramid A B C D is equal
to the pyramid a bed.
For, if they are not equal, let Z represent the solid
which is equal to the excess of one of them, a - bc d
above the other A–B CD ; and let a series of prisms
C E, FG, H K, L M, of the same altitude, be circum-
scribed about the pyramid A B C D, so as to exceed
it by a solid less than Z, which is always possible ;
(prop. 15, book vii.) also let a series of prisms c e fgh
k l m, equal in number to the other and of the same al-
titude, be circumscribed about the pyramid a-b c d.
And because the pyramids have equal altitudes, and the
number of prisms described about each is the same,
the altitudes of the prisms will be all equal, and the
bases of the corresponding prisms in the two pyra-
mids, in EF, ef, will be sections of the pyramids at
equal distances from their bases; therefore they are
equal (prop. 14, book vii.) and the prisms themselves
are equal, (prop. 1, book vii.) and the sum of all
the prisms described about the one pyramid is equal
G E G M ET. R. Y. 355
- Geometry. to the sum of all the prisms described about the other
*~~' pyramid. For the sake of abridging, let P and p
Fig. 208,
Fig. 209.
denote the pyramids A B CD, and a b c d, respectively,
and Q and q express the sums of the prism described
about them. Then, because by the hypothesis Z = p
-P, and by construction Z 7 Q—P, therefore (p-P) 7
(Q-P); hence p must be greater than Q; but Q is
equal to q; therefore p must be greater than q; that is
the pyramid p is greater than q, the sum of the prisms
described about it, which is impossible; therefore the
pyramids P, p are not unequal, that is they are equal
to each other, - -
* .
PRoposition XVII.-Theorem.
Every triangular pyramid is the third of the triangular
Prism having the same base and altitude, fig. 208.
Let F A B C be a triangular pyramid, A B C D EF
a triangular prism of the same base and altitude : the
pyramid will be equal to one-third of the prism.
Conceive the pyramid FA B C to be cut off from the
prism by a section made along the plane FA C, and
there will remain the solid FA CD E, which may be
considered as a quadrangular pyramid whose vertex
is F, and whose base is the parallelogram AC D E.
Draw the diagonal CE, and extend the plane F C E,
which will cut the quadrangular pyramid into two
triangular ones FA C E, F C D E. These two trian-
gular pyramids have for their common altitude the
perpendicular let fall from F on the plane A CD E ;
they have equal bases, the triangles A C E, CDE
being halves of the same parallelogram; hence the
two pyramids FA C E, F C D E are equal.
the pyramid F C D E and the pyramid-FA B C, have
Eut
equal bases A B C, D EF; they have also the same
altitude, namely, the distance of the parallel planes
A B C, D EF; hence the two pyramids are equal.
Now the pyramid FC DE has already been proved
equal to FACE; hence the three pyramids FABC,
FC DE, FA CE, which compose the prism A BID are
all equal. Hence the pyramid F A B C is the third
part of the prism A B D, which has the same base
and the same altitude.
Cor. The solidity of a triangular pyramid is equal
to a third part of the product of its base by its
altitude.
PRoposition XVIII.-Theorem:
Any pyramid SA B C D E is measured by the third part
of the product of its base by its altitude, fig. 209.
For, extending the planes SEB, S E C through the
diagonals E B, E C, the polygonal pyramid SABCDE
will be divided into several triangular pyramids all
having the same altitude S.O. But (prop. 17, book vii.)
each of these pyramids is measured by multiplying its
base A B.E., B C E, or CD E by the third part of its
altitude S O ; hence the sum of these triangular
pyramids, or the polygonal pyramid S A B C D E will
be measured by the sum of the triangles A.B.E., BCE,
C DE, or the polygon A B C D E, multiplied by SO ;
hence every pyramid is measured by a third part of
the product of its base by its altitude.
Cor. 1. Every pyramid is the third part of the prism
which has the same base and the same altitude.
Cor. 2. Two pyramids having the same altitude are
to each other as their bases,
Scholium. The solidity of any polyedral body may Book VII.
be computed, by dividing the body into pyramids; and Book viii
this division may be accomplished in various ways. .
One of the simplest is to make all the planes of
division pass through the vertex of one solid angle;
in that case, there will be formed as many partial
pyramids as the polyedron has faces, minus those faces
which form the solid angle whence the planes of
division proceed.
Paoposition XIX-Theorem.
Two similar pyramids are to each other as the cubes of
their homologous sides, fig. 210.
For two pyramids being similar, the smaller may Fig. 210.
be placed within the greater, so that the solid angle S
shall be common to both. In that position the bases
A B C D E, a b c de will be parallel ; because, since
the homologous faces are similar, the angle S a b is .
equal to S A B, and S b c to S B C ; hence the plane
A B C is parallel to the plane a b c. This granted,
let S.O be the perpendicular drawn from the vertex
S to the plane A B C, and o the point where this
perpendicular meets the plane a b c ; from what has
already been shown we shall have S O : So :: S.A.:
S a : A B : a b ; and consequently,
+ SO : 3. So : : A B : a b. -
Let H represent the altitude of the frustum of a
pyramid, having parallel bases A and B; v A B will
be the mean proportion. But the bases AB C D E,
a b c de being similar figures, we have
A B C D E : a b c de : : A B* : a b%.
Multiply the corresponding terms of these two propo-
sitions; there results the proportion, -
A B C D E x + SO : : a b c de x 4. So : : A B* : a bº.
Now ABCDE x + SO is the solidity of the pyramid
S A B C DE, and a b c de x + So is that of the
pyramid Sabcde, (prop. 17 and 18, book vii.;) hence
two similar pyramids are to each other as the cubes
of their homologous sides.
JBOOK VIII.
The three round bodies.
DEFINITIONs.
1. A cylinDER is a solid produced by the revolution
of a rectangle A B C D, conceived to turn about the
immovable side A B, fig. 211.
In this rotation, the sides AD, B C, continuing
always perpendicular to A B, describe equal circular
planes D H P, C G Q, which are called the bases of the
cylinder, the side CD at the same time describing the
convex surface.
The immovable line A B is called the axis of the
cylinder. .
Every section KLM, made in the cylinder, at right
angles to the axis, is a circle equal to either of the
bases; for, whilst the rectangle A B C D revolves
about A B, the line K.I, perpendicular to AB, describes
a circular plane, equal to the base, which is a section
made perpendicular to the axis at the point I.
Every section PQ G H, passing through the axiss
3 A 2
356
G E O M E. T. R. Y.
Geometry. is a rectangle, and is double of the generating rect- About the circle whose radius is CD, circumscribe Book VIII.
\-y– angle A B C D. . a regular polygon G H IP, (prop. 9, book v.) the S-N-
Fig. 212, the immovable side S.A, fig. 212. the regular polygon G HIP for its base, and H for its
In this rotation, the side A B describes a circular altitude; this prism will be circumscribed about the
plane B D C E, named the base of the cone; and the cylinder, whose base has CD for its radius. Now,
hypothenuse SB its convex surface. - (prop. 11, book vii.) the solidity of the prism is equal
The point S is named the verter of the cone, SA its to its base G H IP, multiplied by the altitude H ; the
aris or altitude. base G H II* is less than the circle, whose radius is
Every section HKFI, formed at right angles to the CA; hence the solidity of the prism is less than surf.
axis, is a circle; every section SD E passing through CA X H. But, by hypothesis, surf, C A x H is the
the axis is an isosceles triangle double of the gene- solidity of the cylinder inscribed in the prism; hence
rating triangle SAB. the prism must be less than the cylinder ; whereas in
3. If from the cone S CD B, the cone S FK H be reality it is greater, because it contains the cylinder;
cut off by a section parallel to the base, the remaining hence it is impossible that surf, C A x H can be the
solid C B H F is called a truncated cone, or the frustum measure of the cylinder whose base has CD for its
of a cone. . . radius, H being the altitude; or, in more general
We may conceive it to be described by the revolu- terms, the product of the base, by the altitude of a cylin-
tion of a trapezium A B HG, whose angles A and C der, cannot measure a less cylinder. -
are right, about the side A G. The immovable line We must now prove that the same product cannot
A G is called the axis or altitude of the frustum, the measure a greater cylinder. To avoid the necessity
circles B D C, H F K are its bases, and B H is its of changing our figure, let C D be a radius of the
side. - .. given cylinder's base ; and, if possible, let surf CD
-4. Two cylinders, or two cones, are similar, when x H, be the measure of a greater cylinder, for ex-
their axes are to each other as the diameters of their ample, of the cylinder whose base has C A for its
bases. - : radius, H being the altitude. :
Fig. 213. , 5. If in the circle ACD, (fig.213,) which forms the The same construction being performed as in the
base of a cylinder, a polygon ABCDE is inscribed, a first case, the prism, circumscribed about the given
right prism, constructed on this base A B C D E, and cylinder, will have G H IP × H for its measure; the
equal in altitude to the cylinder, is said to be inscribed area G H IP is greater than surf CD; hence the
in the cylinder, or the cylinder to be circumscribed about solidity of this prism is greater than surf CD x H:
the prism. - hence the prism must be greater than the cylinder,
The edges AF, BG, CH, &c. of the prism, being having the same altitude, and surf. C. A for its base.
perpendicular to the plane of the base, are evidently But on the contrary the prism is less than the cylin-
included in the convex surface of the cylinder; hence der, being contained in it; therefore the base of G
the prism and the cylinder touch one another along cylinder, multiplied by its altitude, cannot be the measure
these edges. . of a greater cylinder.
Fig. 214. 6. In like manner, if ABCD (fig.214) is a polygon, Hence, finally, the solidity of a cylinder is equal to
Fig. 215.
2. A come is a solid produced by the revolution of a
right angled triangle S A B, conceived to turn about
circumscribed about the base of a cylinder, a right
prism, constructed on this base ABCD, and equal in
altitude to the cylinder, is said to be circumscribed
about the cylinder, or the cylinder to be inscribed in the
prism. *
Let M, N, &c. be the points of contact in the sides
A B, BC, &c.; and through the points M, N, &c. let
MX, NY, &c. be drawn perpendicular to the plane
of the base : those perpendiculars will evidently lie
both in the surface of the cylinder, and in that of the
circumscribed prism; hence they will be their lines
of contact.
Note. The cylinder, the cone, and the sphere, are
the three round bodies treated of in the elements of
geometry.
PROPosition I.-Theorem.
The solidity of a cylinder is equal to the product of its
base by its altitude, fig. 215.
Let CA be a radius of the given cylinder's base;
H the altitude; let surf. CA, represent the area of the
circle whose radius is CA; we are to show that the
solidity of the cylinder is surf. CA x H. For, if surf.
C A x H is not the measure of the given cylinder, it
must be the measure of a greater cylinder, or of a
smaller one. Suppose it first to be the measure of a
smaller one ; of a cylinder, for example, which has
CD for the radius of its base, H being the altitude.
sides of which shall not meet the circumference
whose radius is CA. Imagine a right prism, having
the product of its base by its altitude.
Cor. 1. Cylinders of the same altitude are to each
other as their bases; and cylinders of the same base
are to each other as their altitudes.
... Cor. 2. Similar cylinders are to each other as the
cubes of their altitudes, or as the cubes of the dia-
meters of their bases. For the bases are as the
squares of their diameters; and the cylinders being
similar, the diameters of their bases (def. 4) are to
each other as the altitudes: hence the bases are as the
squares of the altitudes ; hence the bases, multiplied
by the altitudes, or the cylinders themselves, are as
the cubes of the altitudes. -
Scholium. Let R be the radius of a cylinder's base ;
H the altitude : the surface of the base (prop. 11,
book v.) will be TR2 ; and the solidity of the cylin-
der will be ºr R2 x H, or T R2 H. r |
PRóPosition II.—Lemma.
The convex surface of a right prism is equal to the
perimeter of its base multiplied by its altitude, fig. 213.
For this surface is equal to the sum of the rect-
angles A FG B, B G H C, CH ID, &c. (fig. 213)
which compose it. Now the altitudes A F, , B.G.,
C H., &c. of those rectangles, are equal to the alti-
tude of the prism ; their bases A B, B C, CD, &c.
taken together, make up the perimeter of the prism's
G E O M ET. R. Y.
357
Geometry, base. Hence the sum of these rectangles, or the
S-N-2 convex surface of the prism, is equal to the perimeter
Fig. 214.
Fig. 216.
of its base, multiplied by its altitude. -
Cor. If two right prisms have the same altitude,
their convex surfaces will be to each other as the peri-
meters of their bases.
Proposition III.-Lemma.
The conver surface of a cylinder is greater than the
conver surface of any inscribed prism, and less than the
convex surface of any circumscribed prism, fig. 213.
For (fig. 213) the convex surface of the cylinder and
that of the prism may be considered as having the
same length, since every section made in either
parallel to AF is equal to AF; and if these surfaces
be cut, in order to obtain the breadths of them, by
planes parallel to the base, or perpendicular to the
edge AF, the one section will be equal to the circum-
ference of the base, the other to the contour of the
polygon A B C DE, which is less than that circumfe-
rence : hence, with an equal length, the cylindrical
surface is broader than the prismatic surface; hence
the former is greater than the latter. - -
By a similar demonstration, the convex surface of
the cylinder might be shown to be less than that
of any circumscribed prism B C D KL KH, fig. 214.
PRoPosition IV.-Theorem.
The conver surface of a cylinder is equal to the circum-
ference of its base multiplied by its altitude, fig. 216.
Det C A be the radius of the given cylinder's base,
Hits altitude; the circumference whose radius is CA,
being represented by circ, CA, we are to show that
circ, C A x H will be the convex surface of the cylin-
der. For, if this proposition be not true, then circ.
CA x H must be the surface of a greater cylinder, or
of a less one. Suppose it first to be the surface of a
less cylinder; of the cylinder, for example, the radius
of whose base is CD, and whose altitude is H. .
About the circle whose radius is CD, circumscribe
a regular polygon G H IP, the sides of which shall not
meet the circle whose radius is CA; conceive a right
prism having H for its altitude, and the polygon
G HIP for its base. The convex surface of this prism
will be equal (prop. 2, book viii.) to the contour of
the polygon G H IP multiplied by the altitude H :
this contour is less than the circumference whose ra-
dius is C A ; hence the convex surface of the prism is
less than circ. C A x H. But, by hypothesis, circ. CA
× H is the convex surface of the cylinder whose base
has CD for its radius; which cylinder is inscribed in
the prism: hence the convex surface of the prism
must be less than that of the inscribed cylinder; but,
by hypothesis (prop. 3, book viii.) it is greater: hence,
in the first place, the circumference of a cylinder's base
multiplied by its altitude cannot be the measure of a smaller
cylinder. |
Neither can this product be the measure of a
greater cylinder. For, retaining the present figure,
let C D be the radius of the given cylinder's base;
and; if possible, let circ. CD x H be the convex
surface of a cylinder, which with the same altitude
has for its base a greater circle, the circle, for in-
stance, whose radius is C.A. The same construction
being performed as above, the convex surface of the Book VIII:
prism will again be equal to the contour of the poly-
gon G H IP multiplied by the altitude H. But this
contour is greater than circ. CD; therefore the surface
of the prism must be greater than circ. CD x H,
which, by hypothesis, is the surface of the cylinder
having the same altitude, and C A for the radius of
its base. Hence the surface of the prism must be
greater than that of the prism. Now if this prism
were inscribed in the cylinder, its surface (prop. 3,
book viii.) would be less than the cylinder's; much
more then is it less when the prism does not reach so
far as to touch the cylinder. Consequently also, in
the second place, the circumference of a cylinder's base
multiplied by the altitude cannot measure the surface of a
greater cylinder.
The product in question being, therefore, neither the
measure of the convex surface of a less nor greater
cylinder, must be the measure of the cylinder itself.
IPRoposition V.—Theorem.
The solidity of a cone is equal to the product of its
base by the third of its altitude, fig. 217. . r
Let S O be the altitude of the given cone, AO the Fig. 217.
radius of its base ; the surface of the base being de-
signated by surf. A O, it is to be demonstrated that
surf. A Ox 3 SO is equal to the solidity of the cone.
Suppose, first, that surf. A O x 3 SO, is the solidity
of a greater cone; for example, of the cone whose
altitude is also S O, but whose base has O B, greater
than AO, for its radius. - -
About the circle whose radius is AO, circumscribe
a regular polygon MNPT (prop. 9, book v.) so as
not to meet the circumference whose radius is O B ;
imagine a pyramid having this polygon for its base,
and the point S for its vertex. The solidity of this
pyramid (prop. 18, book vii.) is equal to the area of
the polygon MNPT multiplied by a third of the alti-
tude S O. But the polygon is greater than the in-
scribed circle represented by surf. A O; hence the
pyramid is greater than surf. A O x 3 S O, which, by
hypothesis, measures the cone having S for its vertex
and O B for the radius of its base : whereas, in reality,
the pyramid is less than the cone, being contained in
it ; hence, first, the base of a cone multiplied by a
third of its altitude cannot be the measure of a
greater cone. s’
Neither can this same product be the measure of a
smaller cone. For now let O B be the radius of the
given cone's base; and, if possible, let surf. OB x
+ SO be the solidity of the cone having SO for its
altitude, and for its base the circle whose radius is
AO. The same construction being made, the pyramid
S M N P T will have for its measure the area. Mſ NPT
multiplied by 4 S O. But the area M N P T is less
than surf. O B ; hence the measure of the pyramid
must be less than surf. O B x 3 SO, and consequently
it must be less than the cone whose altitude is SO
and whose base has AO for its radius. But, on the
contrary, the pyramid is greater than the cone, be-
cause the cone is contained in it ; hence, in the
second place, the base of a cone multiplied by a
third of its altitude cannot be the measure of a
smaller one. . . . . . -
Consequently the solidity of a cone is equal to the
product of its base by the third of its altitude.
358
G E O M ET R Y
and two such cones, base to base, the surface of the Book VIII.
double pyramid will envelope that of the double cone, S-N-
Geometry. Cor. A cone is the third of a cylinder having the
*Y* same base and the same altitude; whence it follows,
Fig. 218,
1. That cones of equal altitudes are to each other
-as their bases; - - -
2. That cones of equal bases are to each other as
their altitudes ; • . -
3, That similar cones are as the cubes of the dia-
meters of their bases, or as the cubes of their
altitudes. - -
Scholium. Let R be the radius of a cone's base,
H its altitude ; the solidity of the cone will be ºr R.”
x +H, or + T R*H.
Proposition VI.-Theorem.
The convey surface of a come is equal to the circumfe-
fence of its base multiplied by half its side, fig. 218.
Let A O be a radius of the base of the given cone,
S its vertex, and S A its side ; the surface will be
circ. A O x + S.A. For, if possible, let circ. A Ox SO
be the surface of a cone having S for its vertex, and
for its base a circle whose radius O B is greater
than A. O.
About the smaller circle describe a regular polygon
MNPT, the sides of which shall not meet the circle
whose radius is OB; and let SMNPT be the regular
pyramid, having this polygon for its base, and the
point S for its vertex. The triangle SMN, one of
those which compose the convex surface of the
pyramid, has for measure its base M N multiplied by
half its altitude SA, or half the side of the given
cone; and since this altitude is the same in all the
other triangles S N P, SPQ, &c. the convex surface
of the pyramid must be equal to the perimeter
MN PT M multiplied by 4 S A. But the contour
MNPTM is greater than circ. A O; hence the convex
surface of the pyramid is greater than circ. A O x +
SA, and consequently greater than the convex surface
of the cone having the same vertex S, and the circle
whose radius is O B for its base. But the surface of
this cone is greater than that of the pyramid ; be-
cause, if two such pyramids are adjusted to each other
base to base, and two such cones base to base, the
surface of the double cone will envelope on all sides
that of the double pyramid, and therefore be greater
than it, as is evident; hence the surface of the cone
is greater than that of the pyramid, whereas by the
hypothesis it is less : hence, in the first place, the
circumference of the cone's base multiplied by half
the side cannot measure the surface of a greater cone.
Neither can it measure the surface of a smaller
cone ; for let B O be the radius of the base of the
given cone; and, if possible, let circ. B O x 3 SE be
the surface of a cone having S for its vertex, and AO
less than O B for the radius of its base.
The same construction being made as above, the
surface of the pyramid SMNPT will still be equal to
the perimeter MNPT x + S.A. Now this perimeter
MNPT is less than circ. O B ; likewise SA is less
than SB; consequently, for a double reason, the
convex surface of the pyramid is less than circ. O B x
+ SB, which, by hypothesis, is the surface of the cone
having SA for the radius of its base; henee the
surface of the pyramid must be less than that of the
inscribed cone. But it is obviously greater; for, ad-
justing two such pyramids to each other, base to base,
and will be greater than it. Hence, in the second
place, the circumference of the base of the given cone
multiplied by half the side cannot be the measure of
the surface of a smaller cone. . - - - -
Therefore, finally, the convex surface of a cone is
equal to the circumference of its base multiplied by
half its side. -
Scholium. Let L be the side of a cone, R. the radius
of its base ; the circumference of this base will be
27 R, and the surface of the cone will be 2 ºr R x +L,
or "r R. L. - g
PRoposition VII.—Theorem.
The conver surface of a truncated cone ADE B is
equal to its side A D multiplied by half the sum of
A B, D E, the circumferences of its two bases, fig. 219
In the plane S A B which passes through the axis Fig. 219.
SO, draw the line AF perpendicular to SA, and
equal to the circumference having A O for its radius;
join SF ; and draw D H parallel to A.F.
From the similar triangles SAO, SD C we have
A O : D C : : S A : SD ; and by the similar triangles
SA F, SD H, A F : D H :: SA : SD ; hence A F :
D H : : A O : D C, or (prop. 10, book v.) as circ. A O
is to circ. D. C. But, by construction, A F = circ, AO ;
hence D H = circ, DC. Hence the triangle S A F,
measured by A F x + SA, is equal to the surface of
the cone SA B which is measured by circ. AO x + S.A.
For a like reason, the triangle S D H is equal to the
surface of the cone S D E. Therefore the surface of
the truncated cone A DE B is equal to that of the
trapezium AID H F. But the latter (prop. 4, book iv.)
is measured by AD x (ºr PH ) ; hence the sur-
face of the truncated cone ADEB, is equal to its side
AD multiplied by half the sum of the circumferences
of its two bases.
Cor. Through I, the middle point of A D, draw
IKL parallel to AB, and IM parallel to AF; it may
be shown as above that IM = circ. I K. But the tra-
pezium ADHF = AD x IM = AD x circ. IK. Hence
it may also be asserted, that the surface of a truncated
cone is equal to its side multiplied by the circumference of
a section at equal distances from the two bases.
Scholium. If a line AID, lying wholly on one side of
the line O C, and in the same plane, make a revolu-
tion around O C, the surface described by AID will
irc. irc. D. C.
have for its measure AD x (ºr: A O ; circ. D ), Orº
AD x circ. I K ; the lines A O, DC, I K being per-
pendiculars, let fall from the extremities and from the
middle of the axis O C.
For, if A D and O C are produced till they meet in
S, the surface described by A D is evidently that of a
truncated cone having A O and D C for the radii of
its bases, the vertex of the whole cone being S.
Hence this surface will be measured as we have said.
This measure will always hold good, even when the
point D falls on S, and thus forms a whole cone; and
also when the line AID is parallel to the axis, and thus
forms a cylinder. In the first case D C would be
nothing ; in the second, D C would be equal to AO
and to J. K. - -
G E O M ET. R. Y.
359
Geometry.
\ *** 30
Fig. 220.
|
Fig. 221,
less than circ. CD; hence, for these two reasons, the Book VHT,
surface of the solid described by the polygon must be S-ºr-
PRoposition VIII.-Lemma.
Let AB, BC, CD be several successive sides of a re-
gular polygon, O its centre, and O I the radius of the
inscribed circle; if that portion of the polygon A B C D,
which lies wholly on one side of the diameter FG, be sup-
posed to make a revolution about this diameter, the surface
described by A B C D will have for its measure M Q x
circ. O I, M Q being the altitude of that surface, or the
aris included between AM and DQ the extreme perpen-
diculars, fig. 220. g - -
The point I being the middle of A B, and IK a
perpendicular let fall from the point I upon the axis,
the surface described by A B by the last proposition
will have for its measure AIB x circ. I K. Draw A X
parallel to the axis; the triangles A B X, O I K will
have their sides perpendicular, each to each, namely,
OI to AB, IK to AX, and O K to B X; hence these
triangles are similar, and give the proportion A B :
AX, or MN :: O I : IK, or as circ. O I to circ. IK ;
Hence A B X circ. I K = M. N. × circ. OI. Whence it is
plain that the surface described by the partial polygon
A B C D is measured by (MN + N P + PQ) × circ.
O I, or by M Q x circ. O I; hence it is equal to the
altitude multiplied by the circumference of the in-
scribed circle.
Cor. If the whole polygon has an even number of
sides, and if the axis FG passes through two oppo-
site vertices F and G, the whole surface described by
the revolution of the half polygon FA C G will be
equal to its axis FG multiplied by the circumference
of the inscribed circle. This axis FG will at the same
time be the diameter of the circumscribed circle.
PRopositroN IX,-Theorem.
The surface of a sphere is equal to its diameter multi-
plied by the circumference of a great circle, fig. 221.
It is first to be shown, that the diameter of a
sphere multiplied by the circumference of its great
circle cannot measure the surface of a larger sphere.
If possible, let A B x circ. A C be the surface of the
sphere whose radius is CD.
About the circle whose radius is C A, circumscribe
a regular polygon having an even number of sides, so
as not to meet the circumference whose radius is CD;
let M and S be the two opposite vertices of this poly-
gon; and about the diameter M S let the half polygon
MPS be made to revolve. The surface described by
this polygon will be measured (prop. 7, book viii.)
by M S x circ. A C ; but M S is greater than A B ;
hence the surface described by this polygon is greater
than A.B. × circ. AC, and consequently greater than the
surface of the sphere whose radius is C D ; but the
surface of the sphere is greater than the surface de-
scribed by the polygon, since the former envelopes
the latter on all sides. Hence, in the first place, the
diameter of a sphere multiplied by the circumference
of its great circle cannot measure the surface of a
larger sphere.
Neither can this same product measure the surface
of a smaller sphere. For, if possible, let D E x circ.
C D be the surface of that sphere whose radius is C.A.
The same construction being made as in the former
case, the surface of the solid generated by the revolu-
tion of the half polygon will still be equal to M S X
zirc. A C. But MS is less than DE, and circ. A C is
less than DE x circ. CD, and therefore less than the
surface of the sphere whose radius is A. C. But the
surface described by the polygon is greater than the
surface of the sphere whose radius is AC, because the
former envelopes the latter; hence, in the second
place, the diameter of a sphere multiplied by the cir-
cumference of its great circle, cannot measure the
surface of a smaller sphere.
Therefore the surface of a sphere is equal to its
diameter multiplied by the circumference of its great
circle. - -
Cor. The surface of the great circle is measured by
multiplying its circumference by half the radius, or by
a fourth of the diameter; hence the surface of a sphere
is four times that of its great circle.
Proposition X.—Theorem.
The surface of any spherical zone is equal to its alti-
tude multiplied by the circumference of a great circle,
fig. 222 and 223. -- -
Let E F be any arc less or greater than a quadrant;
and let FG be drawn perpendicular to the radius EC;
the zone with one base, described by the revolution
of the arc E F about E C, will be measured by EG x
circ. E. C. -
For, suppose, first, that this zone is measured by
something less; if possible, by E G x circ. CA. In
the arc E F, inseribe a portion of a regular polygon
E M N OPF, whose sides shall not reach the circum-
ference described with the radius C A ; and draw C I
perpendicular to EM. By proposition 8, book viii.
the surface described by the polygon E MF turning
about EC will be measured by E G x circ. CI. This
quantity is greater than E G x circ. A C, which by hy-
pothesis is the measure of the zone described by the
arc E F. Hence the surface described by the polygon
EMNOPF must be greater than the surface described
by E F the circumscribed are ; whereas this latter
surface is greater than the former, which it envelopes
on all sides ; hence, in the first place, the measure of
any spherical zone with one base cannot be less than
the altitude multiplied by the circumference of a great
circle. . a
Secondly, the measure of this zone cannot be greater
than its altitude multiplied by the circumference of a
great circle. For suppose the zone described by the
revolution of the arc AB about AC to be the proposed
one ; and, if possible, let zone A B 7 AD x circ. A C.
The whole surface of the sphere composed of the two
zones A B, BH, is measured by A H x circ. A C,
(prop. 9, book viii.) or by A D x circ. A C + D H x
circ. A C ; hence, if we have zone AIB 7 D H x circ. A C,
we must also have zone B H 7 D H x circ. A C ;
which cannot be the case, as is shown above. There-
fore, in the second place, the measure of a spherical
zone with one base, cannot be greater than the alti-
tude of this zone multiplied by the circumference of
a great circle. .
Hence, finally, every spherical zone with one base
is measured by its altitude multiplied by the circum-
ference of a great circle. t
Let us now examine any zone with two bases, de-
seribed by the revolution of the arc FH (fig. 223) Fig. 223,
about the diameter D E. Draw FO, H Q perpendi-
360
G E O M E T R Y.
Geometry.cular to this diameter. The zone described by the arc
S-V-' FH is the difference of the two zones described by
Fig. 224.
Fig. 225.
Fig. 226,
the arcs D H and D F ; the latter are respectively
measured by DQ x circ. CD and DO x circ. CD;
hence the zone described by FH has for its measure
(DQ – DO) × circ. CD, or O Q x circ. CD.
That is, any spherical zone, with one or two bases,
is measured by its altitude multiplied by the circum-
ference of a great circle. * * :
Cor. Two zones, taken in the same sphere or in
equal spheres, are to each other as their altitude ; and
any zone is to the surface of the sphere as the alti-
tude of that zone is to the diameter.
PRoPosition XI.—Theorem:
If the triangle BAC and the rectangle B CEF, having
the same base and the same altitude, turn simultaneously
about the common base B C, the solid described by the
revolution of the triangle will be a third of the cylinder
described by the revolution of the rectangle, fig. 224
and 225. .
On the axis, let fall the perpendicular AD; the
cone described by the triangle A B D is the third part
of the cylinder described by the rectangle A F B D
(prop. 5, book viii.;) also the cone described by the
triangle A D C is the third part of the cylinder de-
scribed by the rectangle A D C E ; hence the sum of
the two cones, or the solid described by A B C, is the
third part of the two cylinders taken together, or of
the cylinder described by the rectangle B C E F.
If the perpendicular A D (fig. 225) falls without the
triangle; the solid described by A B C will, in that
case, be the difference of the two cones described by
A B D and A CB; but, at the same time, the cylinder
described by BCEF will be the difference of the two
cylinders described by AFB D and A E C D. Hence
the solid, described by the revolution of the triangle,
will still be a third part of the cylinder described by
the revolution of the rectangle having the same base
and the same altitude.
Scholium. The circle of which A D is radius has for
its measure ºr × A D*; hence T × A D* x B C mea-
sures the cylinder described by B C E F, and + ºr x
A D* x B C measures the solid described by the
triangle A B C. -
PRoposition XII-Problem.
The triangle C A B being supposed to perform a revo-
lution about the line C D, drawn at will without the
triangle through its vertea C, to find the measure of the
solid so produced, fig. 226.
Produce the side A B till it meets the axis C D in
D; from the points A and B, draw A M, B.N perpen-
dicular to the axis. -
The solid described by the triangle C A D is mea-
sured (prop. 11, book viii.) by 3 ºr x A M” x C D ;
the solid described by the triangle CBD is measured
by ; ºr × B N* × CD; hence the difference of those
solids, or the solid described by A B C, will have for
its measure 4 ºr (AM? — B N*) × CD.
To this expression another form may be given.
From I the middle point of A B, draw IK perpendi-
cular to CD ; and through B, draw B O parallel to
CD; we shall have A M + B N = 2 I K, (prop. 4,
book iv.) and A M – BN = AO ; hence (AM+ BN)
x (AM – NB), or AM? – B N* = 2 I K x AO, Book VIII.
(prop. 12, book iv.) Hence the measure of the solid S-N-7
in question is expressed by 4 ºr x IK x AO x C D.
But if C P is drawn perpendicular to AB, the triangles
A B O, D C P will be similar, and give the proportion
A O : C P : : A B : CD ; hence A O x C D = C P ×
A B ; which C P × AB is double the area of the
triangle A B C ; hence we have A O x C D = 2 ABC;
hence the solid described by the triangle A B C is also
measured by 4 ºr x A B C x IK, or which is the same
thing, by A B C x 3 circ. I K, circ. I K being equal to
2 T × I K. Hence the solid described by the revolution
of the triangle A B C, has for its measure the area of this
triangle multiplied by two-thirds of the circumference
traced by I, the middle point of the base.
Cor. If the side A C = CB, (fig. 227,) the line C I Fig. 227.
will be perpendicular to A B, the area A B C will
be equal to A B X + C I, and the solidity & T + ABC
+ I K will become 3 T × A B x IK x C I. But the
triangles ABO, CIK are similar, and give the propor-
tion A B : B O or M N : : C I : I K; hence A B x
IK = M N × C.I; hence the solid described by the
isosceles triangle A BC will have for its measure 3 T ×
M N × CI2. . - -
Scholium. The general solution appears to include
the supposition that AB produced will meet the axis;
but the results would be equally true, though A B
were parallel to the axis. - -
Thus, the cylinder described by AMNB (fig. 228) Fig. 228,
is equal to T. A M*. M N ; the cone described by
A C M is equal to 4 ºr . A M 9. CM, and the cone
described by B C N to 4 ºr . A M*. CN. Add the first
two solids and take away the third; we shall have the
solid described by A B C equal to T. A M*. (M N +
1 C M – 3 C N) : and since C N – C M = MN, this
expression is reducible to T. A M*. # MN, or 4 CP’.
M. N. ; which agrees with the conclusion drawn
above. - -
PRoPosition XIII.-Theorem.
Let A B, BC, CD be several successive sides of a re-
gular polygon, O its centre, and O I the radius of the
wnscribed circle; if the polygonal sector A O D, lying all
on one side of the diameter FG be supposed to perform a
nevolution about this diameter, the solid so described will
have for its measure & T. O I*. M. Q, M Q being that
portion of the axis which is included by the extreme perpen-
diculars A M, D Q, fig. 229,
For, since the polygon is regular, all the triangles Fig. 229,
A O B, B O C, &c. are equal and isosceles. Now, by
the last corollary, the solid produced by the isosceles
triangle AOB has for its measure 3. T.O I*. MN; the
solid described by the triangle B O C has for its
measure 3 ºr . O I*. NP; and the solid described by
the triangle COD has for its measure 4. T. O I*.
PQ ; hence the sum of those solids, or the whole
solid described by the polygonal sector A O D, will
have for its measure 3. T. O I*. (M N + N P + I Q)
or 3. T O I*. M Q.
PRoPosition XIV.-Theorem.
Every spherical sector is measured by the zone which
forms its base, multiplied by a third of the radius ; and
the whole sphere has for its measure a third of the radius,’
multiplied by its surface fig. 230. .
Let ABC be the circular sector, which, by its re- Fig. 230,
G E O M ET R Y. 361
surface will be 4 ºr R2 ; its solidity 4 m Rº x 3. R, or Book VIII,
4. tr. R*. If the diameter is named D, we shall have
Book IX.
Geometry, volution about AC, describes the spherical sector; the
S—y-' zone described by A B being A D x circ. A C, or 2 ºr .
**
A C. AID, and it is to be shown that this zone multi-
plied by 3 of AC, or that 3 r . A C°. A D, will measure
the sector. - - -
First, suppose, if possible, that 3 T. A C*. A D is
the measure of a greater spherical sector, say of the
spherical sector described by the circular sector
similar to A C B. - .
In the arc E F, inscribe ECF, a portion of a regular
polygon, such that its sides shall not meet the arc AB;
then imagine the polygonal sector E N F C to turn
about E C, at the same time with the circular sector
E C F. Let CI be a radius of the circle inscribed in
the polygon ; and let F C be drawn perpendicular to
EC. The solid described by the polygonal sector will,
by the last proposition, have for its measure 3 C I*.
E G ; but C I is greater than A C by construction ;
and E G is greater than AD ; for joining A D, E F,
the similar triangles E FG, A B D give the proportion
E G : A D :: FG : B D : : C F : C B ; hence E G 7.
A D. . . . . - - -
For this double reason, 4. T C I*. E G is greater than
3. Tr. CA”. A D. The first is the measure of the solid
described by the polygonal sector; the second, by
hypothesis, is that of the spherical sector described by
the circular sector E C F ; hence the solid described
by the polygonal sector must be greater than the sphe-
rical sector ; whereas, in reality, it is less, being con-
tained in the latter: hence our hypothesis was false;
therefore, in the first place, the zone or base of a sphe-
rical sector multiplied by a third of the radius, cannot
measure a greater spherical sector. . . . .
Secondly, it is to be shown, that it cannot measure
a less spherical sector. Let C E F be the circular
sector, which, by its revolution, generates the given
spherical sector; and suppose, if possible, that + T .
C E°. E G is the measure of some smaller sphe-
rical sector, say of that produced by the circular
sector A C B. -
The construction remaining as above, the solid
described by the polygonal sector will still have for
its measure 4 tr. C I*. E. G. But C I is less than C E ;
hence the solid is less than 3 ºr . C. E°. EG, which,
according to the supposition, is the measure of the
spherical sector described by the circular sector ACB.
Hence the solid described by the polygonal sector
must be less than the spherical sector described by
A C B ; whereas, in reality, it is greater, the latter
being contained in the former ; therefore, in the
second place, it is impossible that the zone of a sphe-
rical sector, multiplied by a third of the radius, can be
the measure of a smaller spherical sector.
Hence every spherical sector is measured by the
zone which forms its base, multiplied by a third of
the radius. -
A circular sector A C B may increase till it becomes
equal to a semicircle: in which case, the spherical
sector described by its revolution is the whole sphere.
Hence the solidity of a sphere is equal to its surface mul-
tiplied by a third of the radius
Cor. The surfaces of spheres being as the squares of
their radii, these surfaces multiplied by the squares of
the radii must be as the cubes of the latter. Hence
the solidity of two spheres are as the cubes of their radii,
or as the cubes of their diameters. . . -
Scholium. Let : R be the radius of a sphere, its
WOL. I. -
E C F :
circumscribed cylinder (including its bases) as 2 is to 3.;
R = +D, and R*= + D*; hence the solidity may like-
wise be expressed by + T x + D*, or + T D9.
PRóposition XV-Theorem.
The surface of a sphere is to the whole surface of the
and the solidities of these two bodies are to each other in
, the same ratio, fig. 231. *
Let M N P Q be a great circle of the sphere ; Fig. 231.
A B C D the circumscribed square ; if the semicircle
PM Q and the half square P A D Q are at the same
time made to revolve about the diameter PQ, the
semicircle will generate the sphere, while the half-
square will generate the cylinder circumscribed about
that sphere.
The altitude A D of that cylinder is equal to the
diaméter PQ ; the base of the cylinder is equal to the
great circle, its diameter A B being equal to M. N. ;
hence (prop. 4, book viii.) the convex surface of the
cylinder is equal to the circumference of the great
circle multiplied by its diameter. This measure
(prop. 9, book viii.) is the same as that of the surface
of the sphere ; hence the surface of the sphere is equal
to the conver surface of the circumscribed cylinder.
But the surface of the sphere is equal to four great
circles; hence the convex surface of the cylinder is
also equal to four great circles; and adding the two
bases, each equal to a great circle, the total surface of
the circumscribed cylinder will be equal to six great
circles; hence the surface of the sphere is to the total
surface of the circumscribed cylinder as 4 is to 6,
or as 2 is to 3; which is the first branch of the pro-
position. -
In the next place, since the base of the circumscribed
cylinder is equal to a great circle, and its altitude
to the diameter, the solidity of the cylinder (prop. 1,
book viii.) will be equal to a great circle multiplied by
its diameter. But (prop. 14, book viii.) the solidity of
the sphere is equal to four great circles multiplied by
a third of the radius; in other terms, to one great circle
multiplied by 4 of the radius, or by 3 of the diameter;
hence the sphere is to the circumscribed cylinder as 2
to 3, and consequently the solidities of these two
bodies are as their surfaces.
BOOK IX.
Of the sphere, and spherical triangles.
DEFINITIONs. -
1. THE sphere is a solid terminated by a curve surface,
all the points of which are equally distant from a point
..within, called the centre.
The sphere may be conceived to be generated by the
revolution of a semicircle D A E (fig. 223) about its Fig. 223.
diameter DE; for the surface described in this move-
ment, by the curve D A E, will have all its points
equally distant from the centre C.
2. The radius of a sphere is a straight line drawn
from the centre to any point in the surface ; the
: diameter or aris is a line passing through this centre,
and terminated on both sides by the surface.
- 3 B
362 G E O M E T R Y.
Geometry. All the radii of a sphere are equal; all the diameters
v-y— are equal, and double of the radius.
Cor. 2. Two great circles always bisect each other; Book IX.
for their common intersection, passing through the S-N-
Fig. 234.
bases of the zone or segment.
3. A great circle of the sphere is a section which
passes through the centre; a small circle, one which
does not pass through it.
4. A plane is a tangent to a sphere, when their
surfaces have but one point in common. -
5. The pole of a circle of a sphere is a point in the
'surface equally distant from all the points in the cir-
cumference of this circle.
6. A spherical triangle is a portion of the surface of a
sphere, bounded by three arcs of great circles.
Those arcs, named the sides of the triangle, are
always supposed to be each less than a semicircumfe-
rence. The angles, which their planes form with each
other, are the angles of the triangle.
7. A spherical triangle takes the name of right-
angled, isosceles, equilateral, in the same cases as a rec-
tilineal triangle.
8. A spherical polygon is a portion of the surface of
a sphere terminated by several arcs of great circles.
9. A lune is that portion of the surface of a sphere,
which is included between two great semicircles meet-
ing in a common diameter.
10. A spherical wedge or ungula is that portion of
the solid sphere, which is included between the same
great semicircles, and has the June for its base. . .
11. A spherical pyramid is a portion of the solid
sphere, included between the planes of a solid angle
whose vertex is the centre. The base of the pyramid is
the spherical polygon intercepted by the same planes.
12. A zone is the portion of the surface of the
sphere, included between two parallel planes, which
form its bases. One of those planes may be a tangent
to the sphere; in which case, the zone has only a
single base.
13. A spherical segment is the portion of the solid
sphere, included between two parallel planes which
form its bases. -
One of those planes may be a tangent to the
sphere ; in which case, the segment has only a single
base. " : -
14. The altitude of a zone or of a segment is the
distance of the two parallel planes, which form the
15. Whilst the semicircle D A E (see def. 1) re-
volving round its diameter DE, describes the sphere;
any circular sector, as DCF or FC H, describes a solid,
which is named a spherical sector.
PRoposition I.-Theorem.
Every section of a sphere made by a plane is a circle,
fig. 234.
Let AM B be the section, made by a plane in the
sphere whose centre is C. From the point C, draw
CO perpendicularly to the plane AMB; and drew
lines CM, C M to different points of the curve AMB,
which terminates the section.
The oblique lines CM, CM, CB being equal, being
radii of the sphere, they are equally distant from the
perpendicular C O, (prop. 5, book vi. ;) hence all the
lines OM, MO, O B are equal ; hence the section
A MB is a circle, whose centre is O.
Cor. 1. If the section passes through the centre of
the sphere, its radius will be the radius of the sphere;
hence all great circles are equal.
centre, is a diameter.
Cor. 3. Every great circle divides the sphere and its
surface into two equal parts; for, if the two hemis-
pheres were separated, and afterwards placed on the
common base, with their convexities turned the same
way, the two surfaces would exactly coincide, no
point of the one being nearer the centre than any point
of the other.
Cor. 4. The centre of a small circle, and that of the
sphere, are in the same straight line perpendicular to
the plane of the little circle.
Cor. 5. Small circles are the less the farther they
lie from the centre of the sphere; for the greater C O
is, the less is the chord A B, the diameter of the small
circle A M B. -
Cor. 6. An arc of a great circle may always be made
to pass through any two given points in the surface
of the sphere ; for the two given points and the
centre of the sphere make three points, which deter-
mine the position of a plane. But if the two given
points were at the extremities of a diameter, these two
points and the centre would then lie in one straight
line, and an infinite number of great circles might be
made to pass through the two given points.
Proposition II.-Theorem.
In every spherical triangle A B C, any side is less than
the sum of the other two, fig. 235.
Let O be the centre of the sphere; and draw the Fig. 235.
radii O A, O B, O C.
Imagine the planes A O B,
A O C, CO B ; those planes will form a solid angle
at the point O; and the angles A O B, AOC, CO B
will be measured by AIB, A C, B C, the sides of the
spherical triangle. But (prop. 19, book vi.) each of
the three plane angles composing a solid angle is less
than the sum of the other two ; hence any side of the
triangle A B C is less than the sum of the other two.
PRoposition III.-Theorem.
The shortest distance between one point to another, on
the surface of a sphere, is the arc of the great circle
which joins the two given points, fig. 236.
Let A NB be the arc of the great circle which joins Fig. 236.
the points A and B; and without this line, if possible, .
let M be a point in the line of the shortest distance
between A and B. Through the point M, draw M.A.,
MB, arcs of great circles; and take B N = M.B.
Py the last theorem, the arc A.N B is shorter than
A M + M.B.; take BN = BM respectively from both ;
there will remain A N Z A M. Now, the distance of
B from M, whether it be the same with the arc BM
or with any other line, is equal to the distance of B
from N ; for by making the plane of the great circle
B M to revolve about the diameter which passes
through B, the point M may be brought into the posi-
tion of the point N ; and the shortest line between
M and B, whatever it may be, will then be identical
with that between N and B ; hence the two lines from
A to B, one passing through M, the other through N,
have an equal part in each, the part from M to B equal
to the part from N to B. The first line is the shorter,
by hypothesis; hence the distance from A to M must
G. E. O. M. E. T. R. Y. 363
same time makes a right angle with the arc A.M. For Book Dº.
Geometry, be shorter than the distance from A to N ; which is
(prop. 16, book vi.) the line D C being perpendicular S-N-"
~~~' absurd, the arc A M being proved greater than AN:
Fig. 237, Let A B C be any spherical triangle ; produce the to A. M. ſº g int ion D will be the pol
sides AB, AC till they meet again in 'D. The arcs ºã. their point of intersection D will be the pole
A BD, A CD will be semicircumferences, since Cor. 3. Conversely, if the distance of the point D
(prop. 1, book ix.), two great circles always bisect from each of the points A and M be equal to a qua-
each other. But in the triangle BCD, We *Y* drant, the point D will be the pole of the arc AM,
(prop. 2, book ix.) the side B C Z B D + CD; add and also the angles D A M, A M D will be right
A B+ AC to both ; we shall have AB + A C + BC angles.
4 AB D + A CD, that is to say, less than a circum- For, let C be the centre of the sphere; and draw
ference. - - the fadii CA, CD, CM. Since the angles A CD,
MCD are right, the line C D is perpendicular to the
PRoposition V.-Theorem. two straight lines CA, CM; it is therefore perpendi-
The sum of all the sides of any spherical polygon is less ºlar to the Planº ; hence the Point P is the Pole of
than the circumference of a great circle, fig. 238. Yº... * consequently the angles DAM,
Fig. * , Pºt us take for example, the Pentagon. A HCPP. scholium. The properties of these poles enable us to
Produce the sides A B, PC, till they meet in F; then describe arcs of a circle on the surface of a sphere,
since B C is less than B F + CF, the perimeter of the with the same facility as on a plane surface. It is
pentagon ABCDE will be less than that of the qual evident, for instance, that by turning the arc D.F, or
drilateral A EDF. Again, produce the sides A E, any other line extending to the same distance, round
FD, till they meet in G; we shall have E D4E G + th: point D, the extremity F will describe the small
DG ; hence the perimeter of the quadrilateral A EDF circle in N G, and by turning the quadrant D FA
is less than that of the triangle AFG ; which last is round the point D, its extremity A will describe the
itself less than the circumference of a great circle ; are of the great circle A M. -
hence a fortiori the perimeter of the polygon ABCDE If the arc AM were required to be produced, and
is less than this same circumference. nothing were given but the points A and M through
which it was to pass, we should first have to deter-
PROPosition VI.—Theorem. mine the pole D, by the intersection of two arcs des-
The diameter D E being drawn perpendicular to the .....". º * º: %. i.
plane of the great circle A MB, the extremities D and E . might describe the arc AM and its prolongation
of this diameter will be the poles of the circle AM B, and from D as a centre, and with the same distance as i
of all the little circles, as FN G, which are parallel to it, before 2 -
fig. 223. © - Lastlyx if it be required from a given point P to let
Fig, 223. For, D C being perpendicular to the plane A MB, fall a perpendicular on the given arc A.M.; produce
hence no point of the shortest line from A to B can
lie out of the arc A NB; consequently this arc is
itself the shortest distance between its two extre-
mities. -
Proposition IV.-Theorem.
The sum of all the three sides of a spherical triangle
is less than the circumference of a great circle, fig. 237.
is perpendicular to all the straight lines C A,
CM, CB, &c. drawn through its foot in this plane;
hence all the arcs DA, D M, DB, &c. are quarters of
the circumference. So likewise are all the arcs E A,
EM, E B, &c.; hence the points D and E are each
equally distant from all the points of the circumference
A M B ; therefore (def. 5) they are the poles of that
circumference.
Again, the radius DC, perpendicular to the plane
A M B, is perpendicular to its parallel FN G ; hence
(prop. 1, book ix.) it passes through O the centre of
the circle FN G ; therefore, if the oblique lines D F,
DN, D G be drawn, these oblique lines will diverge
equally from the perpendicular DO, and will them-
selves be equal. But, the chords being equal, the
arcs are equal; hence the point D is the pole of the
small circle FN G; and for like reasons the point E
is the other pole. >7 -
Cor. 1. Every arc DM, drawn from a point in the
arc of a great circle A MB to its pole, is a quarter of
the circumference, which, for the sake of brevity, is
usually named a quadrant; and this quadrant at the
to the plane AMC, every plane DMC passing through
the line D C is perpendicular to the plane A M C ;
hence the angle of these planes, or the angle AMD,
is a right angle. -
Cor. 2. To find the pole of a given arc A M, draw
the indefinite arc MD perpendicular to A M ; take
MD equal to a quadrant; the point D will be one of
the poles of the arc A M D ; or thus, at the two points
A and M, draw the arcs A D and M D perpendicular
this arc to S, till the distance P S be equal to a qua-
drant ; then from the pole S, and with the same
distance, describe the arc PM, which will be the per-
pendicular required.
PROPosition VII.-Theorem.
Every plane perpendicular to a radius at its extremity
is a tangent to the sphere, fig. 240.
Let F A G be a plane perpendicular to the radius Fig. 240.
O A. Any point M in this plane being assumed, and
OM, AM being joined, the angle O AM will be right,
and hence the distance O M will be greater than O A.
Hence the point M lies without the sphere; and as
the case is similar with every other point in the plane
F AG, this plane can have no point but A common to
it with the surface of the sphere ; it is therefore a
tangent, def. 4. -
Scholium. In the same way it may be shown, that
two spheres have but one point in common, and there-
fore touch each other, when the distance between
their centres is equal to the sum or the difference of
- 3 B 2
364
G E O M E T R Y.
Geometry, their radii; in which case, the centres and the point of E H + GF is equal to a semicircumference. Now, Book IX.
\—V-' contact lie in the same straight line.
Fig. 241.
Fig. 242
Proposition VIII.-Theorem.
The angle BAC, formed by AB, A C two arcs of
great circles, is equal to the angle F A G formed by the
tangents of these arcs at the point A ; and is therefore
measured by the arc DE described from the point A as a
pole between the sides AB, A C, produced if necessary,
fig. 240 and 241. -
For the tangent A F, drawn in the plane of the arc
A B, is perpendicular to the radius A O; and the
tangent AG, drawn in the plane of the arc AC, is perpen-
dicular to the same radius AO. Hence (book vi. def.4)
the angle FA G is equal to the angle contained by the
planes O A B, O A C ; which is that of the arcs A B,
A C, and is named B A C. t
In like manner, if the arcs A D and A E are both
quadrants, the lines OD, OE will be perpendicular to
A O, and the angle DOE will still be equal to the
-angle of the planes AO D, A OE ; hence the arc DE
is the measure of the angle contained by these planes,
or of the angle C A B. *
. . Cor. The angles of spherical triangles may be com-
pared together, by means of the arcs of great circles
described from their vertices as poles and included
between their sides; hence it is easy to make an angle
of this kind equal to a given angle. - * *
Scholium. Vertical angles, such as A CO and B CN
(fig. 241) are equal; for either of them is still the
angle formed by the two planes A CB, O C N.
It is farther evident, that, in the intersection of two
arcs ACB, O CN, the two adjacent angles A CO,
O C B taken together are equal to two right angles.
PROPosition IX.-Theorem.
The triangle A B C being given, if from the points A,
B, C as poles, the arcs E F, FD, DE be described to
form the triangle D E F ; then, conversely, the three
points D, E, F will be the poles of the sides B C, A C,
A B, fig. 242. - -
For, the point A being the pole of the arc E F, the
distance A E is a quadrant; the point C being the
pole of the arc D E, the distance C E is likewise a
quadrant : hence the point E is removed the length
of a quadrant from each of the points A and C ; hence
(prop. 6, cor. 3, book ix.) it is the pole of the arc A. C.
It might be shown, by the same method, that D
is the pole of the arc B C, and F that of the arc
A B.
Cor. Hence the triangle A B C may be described by
means of DE F, as DE F may by means of A B C.
PRoPosition X-Theorem.
The same supposition being made as in the last theorem,
each angle in the one of the triangles, A B C, D E F will
be measured by the semicircumference minus the side lying
opposite to it in the other triangle, fig. 242 and 243.
Produce the sides AB, A C, if necessary, till they
meet E F in G and H. The point A being the pole
of the arc G H, the angle A will be measured by that
arc. But the arc E H is a quadrant, and likewise
G F, E being the pole of A H, and F of A G ; hence
E H + G F is the same as EF + G H ; hence the S-V-
arc G H, which measures the angle A, is equal to a
semicircumference minus the side EF. In like manner,
the angle B will be measured by + circ. — D F; the
angle C by + circ. — D.E. -
And this property must be reciprocal in the two
triangles, since each of them is described in a similar
manner by means of the other. Thus we shall find
the angles D, E, F of the triangle DEF to be mea-
sured respectively by + circ.—BC, ºr circ. — A C, 4 circ.
– A B. Accordingly the angle D, for example, is
measured by the arc MI; but MI + B C = M C +
B I = + circ.; hence the arc M I, the measure of D, is
equal to + circ.—B C ; and so of all the rest.
Scholium. It must farther be observed, that besides
the triangle DEF (fig. 243) three others might be Fig. 243.
formed by the intersection of the three arcs DE, E F,
D F. But the proposition immediately before us is
applicable only to the central triangle, which is dis-
tinguished from the other three by the circumstance
(see fig. 242) that the two angles A and D lie on the
same side of BC, the two B and E on the same side of
AC, and the two C and F on the same side of A B.
... Various names have been given to the triangles
A B C, DCF; but they are now more generally deno-
minated polar triangles.
PRoPosLTION XI.-Lemma.
The triangle A B C being given, if from the pole. A,
with a distance AC, the arc D E C of a small circle be
described ; if from the pole B, with a distance B C, the
arc D F C be described in like manner; and if from the
point D, where the arcs DEC, D FC intersect each other,
AD, D B two arcs of great circles be drawn; then will
A D B, the triangle thus formed, have all its parts equal
to those of the triangle ACB, fig. 244. .
For, by construction, the side A D = A C, D B =
B C, and A B is common ; hence those two triangles
have their sides equal, each to each, and it is to be
shown that the angles opposite these equal sides are
also equal.
If the centre of the sphere is supposed to be at O,
a solid angle may be conceived as formed at O by the
three plane angles A O B, A O C, B O C ; likewise
another solid angle may be conceived as formed by
the three plane angles A O B, A O D, BOI). And
because the sides of the triangle A B C are equal to
those of the triangle A D B, the plane angles forming
the one of these solid angles must be equal to the
plane angles forming the other, each to each. But in
this case the planes, in which the equal angles lie, are
equally inclined to each other ; hence all the angles
of the spherical triangle D A B are respectively equal
to those of the triangle C A B, namely, DAB = BAC,
D B A = A B C, and A D B = A CB; therefore the
sides and the angles of the triangle ADB are equal to
the sides and the angles of the triangle A C B.
PRoposition XII.-Theorem.
Two triangles on the same sphere, or on equal spheres,
are equal in all their parts, when they have each an equal
angle included between equal sides, fig. 245.
Fig. 244.
Suppose the side A B = EF, the side A C = E G, Fig. 245.
and the angle BAC = FEG; the triangle EFG may
G E O M E T R Y. 365
Geometry, be placed on the triangle A B C, or on A B D symme- gle to the middle of the base, is at right angles to that Book IX)
\—- trical with ABC, just as two rectilineal triangles are \-N-
placed upon each other, when they have an equal
angle included between equal sides. Hence all the
parts of the triangle EFG will be equal to all the
parts of the triangle A B C ; that is, besides the three
parts equal by hypothesis, we shall have the side BC
= FG, the angle A B C = EFG, and the angle A CB
= EFG.
PropositroN XIII.-Theorem.
Two triangles on the same sphere, or on equal spheres,
are equal in all their parts, when two angles and the in-
cluded side of the one are equal to two angles and the
included side of the other.
For one of those triangles, or the triangle symme-
trical with it, may be placed on the other, and be made
to coincide with it, as is obvious.
PRoposition XIV.—Theorem,
If two triangles on the same sphere, or on equal spheres,
have all their sides respectively equal, their angles will
likewise be all respectively equal, the equal angles lying
opposite the equal sides, fig. 246. -
The truth is evident by prop. 11, book ix., where it
base, and bisects the opposite angle.
PRoposition XVI—Theorem.
In a spherical triangle ABC, if the angle A is greater
than the angle B, the side B C opposite to A will be
greater than the side A C opposite to B; and conversely,
$f the side B C is greater than A C, the angle A will be
greater than the angle B, fig. 248.
First. Suppose the angle A 7 B ; make the angle Fig. 248.
BA D = B; then (prop. 15, book ix.) we shall have
A D = D B ; but AD + D C is greater than A C ;
hence, putting DB. in place of A D, we shall have
D B + D C or B C 7 AC.
Secondly. If we suppose B C 7 AC, the angle B A C
will be greater than A B C. For, if B A C were equal
to A B C, we should have B C = A C ; if B A C were
less than A B C, we should then, as has just been
shown, find B C Z A C. Both these conclusions are
false; hence the angle BAC is greater than ABC.
Proposition XVII— Theorem.
If the two sides AB, A C of the spherical triangle
A BC, are equal to the two sides DE, D F of the triangle
DEF, drawn upon an equal sphere; and if at the same
Fig. 246,
was shown that, with three given sides A B, A C, B C, time the angle A is greater than the angle D, then will
there can only be two triangles ACB, ABD, different the third side B C of the first triangle be greater than the
as to the position of their parts, and equal as to the third side E F of the second, fig. 249.
magnitude of those parts. Hence those two trian- The demonstration is every way similar to that of Fig. 249.
gles, having all their sides respectively equal in both, prop. 10, book i.
must either be absolutely equal, or at least symmetri- * 2
cally so ; in both of which cases, their correspond- &
ing angles must be equal, and lie opposite to equal PRoposition XVIII.—Theorem: .
sides. - - If two triangles on the same sphere, or on equal spheres,
PRoposition XV-Theorem. are mutually equiangular, they will also be mutually equi-
lateral, fig. 250. . -
In every isosceles spherical triangle, the angles opposite Let A and B be the two given triangles; P and Q Fig. 250.
the equal sides are equal; and conversely, if two angles of their polar triangles. Since the angles are equal in
a spherical triangle are equal, the triangle will be isosceles, the triangles A and B, the sides will be equal in the
fig. 247. polar triangles P and Q, (prop. 10, book ix.;) but
Fig. 247. First. Suppose the side A B = A C ; we shall have since the triangles P and Q are mutually equilateral,
the angle C = B. For, if the arc A D be drawn from
the vertex A to the middle point D of the base, the
two triangles A BD, A CD will have all the sides of
the one respectively equal to the corresponding sides
of the other, namely, A D common, B D = DC, and
A B = A C ; hence, by the last proposition, their
angles will be equal; therefore B = C.
Secondly. Suppose the angle B = C ; we shall have
the side A C = A B. For, if not, let A B be the
greater of the two ; take B O = AC, and join O C.
The two sides BO, B C are equal to the two AC, BC;
the angle O B C, contained by the first two, is equal to
ABC contained by the second two.
book ix.) the two triangles B O C, A C B have all their
other parts equal ; hence the angle O C B = A B C ;
but, by hypothesis, the angle A B C = A C B ; hence ...
we have O C B = A C B, which is absurd ; therefore
A B is not different from A C ; that is, the sides A B,
A C, opposite to the equal angles B and C, are equal.
Scholium. The same demonstration proves that the
angle. B A D = D.A C, and the angle B D A = A D C.
Hence the two last are right angles; consequently the
are drawn from the verter of an isosceles spherical trian-
Hence (prop. 12, ..
they must also (prop. 14, book ix.) be mutually equi-
angular ; and, lastly, the angles being equal in the
triangles P and Q, it follows (prop. 10, book ix.) that
the sides are equal in their polar triangles A and B.
Hence the mutually equiangular triangles A and B are
at the same time mutually equilateral.
This proposition may also be demonstrated, without
the aid of polar triangles, as follows:
Let A B C, D E F be two triangles mutually
equiangular, having A = D, B = E, C = F; we are
to show that AB = DE, A C = D F, A C = DF,
B C = E F. - ,
On the prolongations of the sides A B, A C, take
AG = DE, and A H = DF; join G. H.; and produce
the arcs B C, GH, till they meet in I and K. §
... The two sides A G, A H are equal, by construction,
to the two D.F, DE; the included angle G A H =
P A C = E D F ; hence (prop. 12, book ix.) the trian-
gles AG H, DEF, are equal in all their parts; hence
the angle A G H = DE F = A B C, and the angle
A G H = DFE = A CJB.
in the triangles fºg, KB G, the side B G is
common; the angle IBG = G B K ; and, since IGB
366
G E O M E T R Y.
Cor. 2. A spherical triangle may have two or even Book IX.
Geometry. -- B G K is equal to two right angles, and likewise
S-N- G B K + IBG, it follows that B G K = IB.G. Hence
Fig. 215,
(prop. 13, book ix.) the triangles IB G, G B K are
equal; hence I G = BK, and IB = G. K.
In like manner, the angle A H G being equal to
A C B, we can show that the triangles I C H, HC K
have two angles and the interjacent side in each equal;
they are therefore themselves equal ; and IH = C K,
and H K = I C. .
Now if the equals CK, I H be taken away from the
equals BK, I G, the remainders B C, G H will be
equal. Besides, the angle B C A = A H G, and the
angle A B C = A G H. Hence the triangles ABC,
A H G have two angles and the interjacent side in
each equal ; and are therefore themselves equal. But
the triangle DE F is equal in all its parts to A H G ;
hence it is also equal to the triangle A B C, and we
have AB = DE, AC = D F, B C = EF; therefore if
two spherical triangles are mutually equiangular, the
sides opposite their equal angles will also be equal.
Scholium. This proposition is not applicable to rec-
tilineal triangles; in which, equality among the angles
indicates only proportionality among the sides. Nor
is it difficult to account for the difference observable,
in this respect, between spherical and rectilineal
triangles. In the proposition now before us, as well
as in the four last, which treat of the comparison of
triangles, it is expressly required that the arcs be
traced on the same sphere, or on equal spheres. Now
similar arcs are to each other as their radii; hence, on
equal spheres, two triangles cannot be similar without
being equal. Therefore it is not strange that equal-
ity among the angles should produce equality among
the sides.
The case would be different, if the triangles were
drawn upon unequal spheres; there, the angles being
equal, the triangles would be similar, and the homo-
logous sides would be to each other as the radii of
their spheres.
PRoposition XIX-Theorem.
The sum of all the angles in any spherical triangle is
less than six right angles, and greater than two, fig. 251.
For, in the first place, every angle of a spherical
triangle is less than two right angles, (see the follow-
ing scholium;) hence the sum of all the three is less
than six right angles.
Secondly, the measure of each angle in a spherical
triangle (prop. 10, book ix.) is equal to the semicir-
cumference minus the corresponding side of the polar
triangle ; hence the sum of all the three is measured
by three semicircumferences minus the sum of all the
sides of the polar triangle. Now (prop. 4, book ix.)
this latter sum is less than a circumference; therefore,
taking it away from three semicircumferences, the re-
mainder will be greater than one semicircumference,
which is the measure of two right angles ; hence, in
the second place, the sum of all the angles in a
spherical triangle is greater than two right angles.
Cor. 1. The sum of all the angles in a spherical
triangle is not constant, like that of all the angles in
a rectilineal triangle; it varies between two right
angles and six, without , ever arriving at either of
these limits. Two given angles therefore do not serve
to determine the third.
definition.
three angles right, two or three obtuse. -
If the triangle ABC have two right angles B and C,
the vertex A will (prop. 6, book ix.) be the pole
of the base B C ; and the sides A B, A C will be
quadrants. - .
If the angle A is also right, the triangle A B C will
have all its angles right, and its sides quadrants. The
tri-rectangular triangle is contained eight times in the
surface of the sphere; as is evident by fig. 252, Sup-
posing the arc M N to be a quadrant.
Scholium. In all the preceding observations, we
have supposed, in conformity with definition 6, that
our spherical triangles have always each of their sides
less than a semicircumference; from which it follows
that any one of their angles is always less than two
right angles. For (see fig. 237) if the side A B is less
than a semicircumference, and AC is so likewise, both
those arcs will require to be produced before they can
meet in D. Now the two angles A B C, CBD, taken
together, are equal to two right angles; hence the
angle ABC itself is less than two right angles.
We may observe, however, that some spherical
triangles do exist, in which certain of the sides are
greater than a semicircumference, and certain of the
angles greater than two right angles. Thus, if the
side A C is produced, so as to form a whole circumfe-
rence ACE, the part which remains, after subtracting
the triangle A B C from the hemisphere, is a new
triangle also designated by A B C, and having A B,
BC, A E D C for its sides. Here, it is plain, the side
A E D C is greater than the semicircumference AED;
and, at the same time, the angle B opposite to it ex-
ceeds two right angles, by the quantity CB D.
The triangles whose sides and angles are so large
have been excluded from our definition; but the only
reason was, that the solution of them, or the determi-
nation of their parts, is always reducible to the solu-
tion of such triangles as are comprehended by the
Indeed, it is evident enough, that if the
sides and angles of the triangle A B C are known, it
will be easy to discover the angles and sides of the
triangle which bears the same name, and is the dif-
ference between a hemisphere and the former triangle.
PRoposition XX.-Theorem.
The lune AM BNA is to the surface of the sphere, as
M A N, the angle of this lune, is to four right angles, or
as the arc MN, which measures that angle, is to the
circumference, fig. 252.
Suppose, in the first place, the arc M N to be to
the circumference M N P Q as some one rational
number is to another, as 5 to 48, for example. The
circumference M N P Q being divided into 48 equal
parts, M N will contain 5 of them ; and if the pole
A were joined with the several points of division, by
as many quadrants, we should in the hemisphere
AMNPQ have 48 triangles, all equal, because having
all their parts equal. Hence the whole sphere must
contain 96 of those partial triangles, the lune A M B
N A will contain 13 of them ; hence the lune is to
the sphere as 10 is to 96, or as 5 to 48, in other
words, as the arc M N is to the circumference.
If the arc M N is not commensurable with the cir-
cumference, we may still show, by the mode of
S-N-2
Fig. 252
G E O M ET. R. Y. 367
FQE = CPB, and the surface DQE = APB ; Book IX.
Geometry, reasoning exemplified in book ii., that in this case
hence we have D Q F + F Q E – D QE = APC + S-a-’
\–V-’also, the lune is to the sphere as MN is to the cir-
Fig. 253.
cumference. - -
Cor. 1. Two lunes are to each other as their res-
pective angles. g • -
Cor. 2. It was shown (prop. 19, book ix.) that the
whole surface of the sphere is equal to eight tri-rect-
angular triangles; hence, if the area of one such
triangle is taken for unity, the surface of the sphere
will be represented by 8. This granted, the surface
of the lune, whose angle is A, will be expressed by
2 A (the angle A being always estimated from the
right angle assumed as unity) since 2 A: 8 : : A : 4.
Thus we have here two different unities ; one for
angles, being the right angle, the other for surfaces,
being the tri-rectangular spherical triangle, or the
triangle whose angles are all right, and whose sides
are quadrants.
Scholium. The spherical ungula, bounded by the
planes A MB, A NB, is to the whole solid sphere as
the angle A is to four right angles. For, the lunes
being equal, the spherical ungulas will also be
equal; hence two spherical ungulas are to each
other, as the angles formed by the planes which bound
them.
I’Roposition XXI.-Theorem.
Two symmetrical spherical triangles are equal in surface,
fig. 253.
Let A B C, D E F be two symmetrical triangles,
that is to say, two triangles having their sides AIB =
DE, A C = DF, CB = EF, and yet incapable of
coinciding with each other; we are to show that the
surface A B C is equal to the surface D E F.
Let P be the pole of the little circle passing
through the three points A, B, C; * from this point
draw (prop. 6, book ix.) the equal arcs PA, PB,
PC ; at the point F, make the angle D F Q=A CP,
the arc FQ = CP; and join D Q, EQ.
The sides D F, FQ are equal to the sides AC, CP;
the angle D F Q = A C P ; hence (prop. 12, book ix.)
the two triangles D F Q, A C P are equal in all their
parts; hence the side D Q = AP, and the angle
D QF = AP C.
In the proposed triangles D FE, A B C, the angles
D FE, A CB, opposite to the equal sides DE, A B,
being equal, (prop. 11, book ix.) if the angles D F Q,
A CP, which are equal by construction, be taken
away from them, there will remain the angle QFE,
equal to PC B. Also the sides Q.F, FE are equal to
the sides PC, CB; hence the two triangles FQ E,
C PB are equal in all their parts; hence the side
Q E = PB, and the angle FQ E = CPB.
Now, observing that the triangles D F Q, A C P,
which have their sides respectively equal, are at the
same time isosceles, we shall see them to be capable
of mutual adaptation, when applied to each other;
for, having placed PA on its equal Q F, the side PC
will fall on its equal Q D, and thus the two triangles
will exactly coincide: hence they are equal, and the
surface D Q F = A PC. For a like reason, the surface
* The circle which passes through the three points A, B, C,
or which circumscribes the triangle A B C, can only be a little
circle of the sphere; for if it were a great circle, the three sides
AB, BC, A C would lie in one plane, and the triangle A B C
would be reduced to one of its sides. * -
C P B – A PB, or DFE = ABC; therefore the two
symmetrical triangles A B C, DE, F are equal in
surface.
Scholium. The poles P and Q might lie within the
triangles AL C, DEF; in which case it would be
requisite to add the three triangles D Q F, FQ E,
D Q E together, in order to make up the triangle
DEF; and in like manner, to add the three triangles
APC, CPB, APB together, in order to make up the
triangle A B C ; in all other respects, the demonstra-
tion and the result would still be the same.
PRoPosition XXII.-Theorem.
If two great circles A O B, C OD intersect each other
anyhow in the hemisphere AO CBD, the sum of the oppo-
site triangles A O C, B O D will be equal to the lune
whose angle is B O D, fig. 241. **
For, producing the arcs O B, O D in the other hemis- Fig. 241.
phere, till they meet in N, the arc O B N will be a
semicircumference, and A O B one also ; and taking
O B from both, we shall have B N = AO. For a like
reason, we have D N = CO, and BD = A C. Hence
the two triangles AOC, B D N have their three sides
respectively equal; besides, they are so placed as to
be symmetrical ; hence (prop. 21, book ix.) they are
equal in surface, and the sum of the triangles AOC,
BOD is equal to the lune O B N DO whose angle
is B O D.
Scholium. It is likewise evident that the two sphe-
rical pyramids, which have the triangles AOC,
B O D for bases, are together equal to the spherical
ungula whose angle is B O D.
PROPOSITION XXIII.--Theorem.
The surface of any spherical triangle is measured by
the excess of the sum of its three angles above two right
angles, fig. 254.
Let A B C be the proposed triangle : produce its Fig.254.
sides till they meet the great circle DEFG drawn
anywhere without the triangle. By the last theorem,
the two triangles A DE, A G H are together equal to
the lune whose angle is A, and which is measured
(prop. 20, book ix.) by 2 A. Hence we have A DE
+ A G H = 2 A ; and for a like reason, B G F + BID
= 2 B, and C IH + C FE = 2 C. But the sum of
those six triangles exceeds the hemisphere by twice
the triangle A B C, and the hemisphere is represented
by 4 ; therefore twice the triangle A B C is equal to
2 A + 2 B + 2 C–4; and consequently once ABC =
A + B + C – 2 ; hence every spherical triangle is
measured by the sum of all its angles minus two right
angles.
Cor. 1. However many right angles there be con-
tained in this measure, just so many tri-rectangular
triangles, or eighths of the sphere, which (prop. 20,
book ix.) are the unit of surface, will the proposed
triangle contain. If the angles, for example, are each
equal to 4 of a right angle, the three angles will amount
to 4 right angles, and the proposed triangle will be
represented by 4 – 2 or 2; therefore it will be equal
to two tri-rectangular triangles, or to the fourth part
of the whole surface of the sphere.
368
G E O M E T R Y.
Geometry. Cor. 2. The spherical triangles A B C is equal to
\-y-/ A + B + C
The vertical angle of the tri-rectangular pyramid is Book IX.
formed by three planes at right angles to each other; \-y-'
the lune whose angle is —l; likewise the
spherical pyramid, which has ABC for its base, is equal
. . A + B + C
to the spherical ungula whose angle is −– 1.
Scholium. While the spherical triangle A B C is
compared with the tri-rectangular triangle, the sphe-
rical pyramid, which has A B C for its base, is com-
pared with the tri-rectangular pyramid, and a similar
proportion is found to subsist between them. The
solid angle at the vertex of the pyramid is, in like
manner, compared with the solid angle at the vertex
of the tri-rectangular pyramid. These comparisons
are founded on the coincidence of the corresponding
parts. If the bases of the pyramids coincide, the
pyramids themselves will evidently coincide, and like-
wise the solid angles at their vertices. From this, the
following consequences are deduced.
First. Two triangular spherical pyramids are to each
other as their bases ; and since a polygonal pyramid
may always be divided into a certain number of trian-
gular ones, it follows that any two spherical pyramids
are to each other, as the polygons which form their
bases.
Second. The solid angles at the vertices of those
pyramids are also as their bases; hence, for com-
paring any two solid angles, we have merely to place
their vertices at the centres of two equal spheres, and
the solid angles will be to each other as the spherical
polygons intercepted between their planes or faces;
see Scholium 2, prop. 21, book vi.
this angle, which may be called a right solid angle, will
serve as a very natural unit of measure for all other
Solid angles. And if so, the same number that ex-
hibits, the area of a spherical polygon, will exhibit
the measure of the corresponding solid angle. If the
area of the polygon is 3, for example, in other words,
if the polygon is 3 of the tri-rectangular polygon,
then the corresponding solid angle will also be # of
the right solid angle.
PROPOSITIon XXIV.—Theorem.
The surface of a spherical polygon is measured by the
sum of all its angles, minus the product of two right
angles by the number of sides in the polygon minus two,
fig. 255. w.
From one of the vertices A, let diagonals AC, AD
be drawn to all the other vertices; the polygon
A B CD E will be divided into as many triangles minus
two as it has sides. But the surface of each triangle
is measured by the sum of all its angles minus two
right angles; and the sum of the angles in all the
triangles is evidently the same as that of all the angles
in the polygon ; hence the surface of the polygon is
equal to the sum of all its angles diminished by twice
as many right angles as it has sides minus two.
Scholium. Let s be the sum of all the angles in a
spherical polygon, n the number of its sides; the
right angle being taken for unity, the surface of the
polygon will be measured by s – 2 (n − 2,) or
s — 2 m + 4,
Fig. 255.
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• → ∞, ∞， ∞）.
± • • • •
Arithmetic,
S-N-'
Definition.
Idea of
number,
how ac-
quired 2
rations of the mind.
A R H T H M E T I C.
*
History of the Science.
(4.) ARITHMETIc may be defined to be the science of
numbers and their notation, and of the different ope
rations to which they are subject. -
With the exception of the theory of arithmetical
notation, we shall not include under the head of Arith-
metic any portion of what is commonly called the
Theory of Numbers, the complete discussion of which
would require a very extensive knowledge of Algebra,
and which will be afterwards considered in a separate
treatise. We shall confine ourselves in the following
treatise to the consideration of the common operations
of Arithmetic, and to those common rules for the
solution of numerical questions, which are of such
frequent occurrence in the ordinary business of life,
and whose principles may be established and under-
stood without the aid of algebraical investigations.
(2.) The idea of number is one of those which are
first presented to the mind, and which indeed may be
considered as nearly coexistent with the exercise of
our natural faculties; and the mode in which it is ac-
quired, considered as a metaphysical question, forms a
natural introduction to an historical notice of the
different methods of numeration, which have been
adopted by different nations at different periods of the
world.
If objects of various kinds be placed before a child,
he will be struck with the more marked peculiarities
by which they are severally distinguished; but the
idea of their multitude will probably escape his obser-
vation, or, if in any way excited, will leave no distinct
impression on the mind : the case would be somewhat
different, if different objects of the same kind were
placed before him, as under such circumstances the
very first idea which would succeed to his perception
of their resemblance, would be that of their multi-
tude. But the passage from the vague idea of multi-
tude to the more definite one of number, is one of
great difficulty in the infant state of the reflex ope-
It requires an analysis of the
individual units of which a number is composed, which
can only be effected by the comparison of different
numbers with each other; and the process of the mind
by which such comparisons are made is slow and diffi-
cult, unless the numbers are small, and our attention
powerfully directed to them by the excitement of our
appetites, or other circumstances: thus place before a
child different sets of toys, or fruits, or other objects
naturally desirable, in the selection of which a choice
is left to him, and he will rapidly acquire the habit of
comparing them with each other; and, as the result
of such a comparison, and of the examination of the
individuals of which each set is composed, he will
gradually acquire the idea of number.
Abstraction is the creature of language, and without
the aid of language he will never separate the idea of
any number from the qualities of the objects with which
it is associated. He will have a distinct idea of four
WOL. I.
cows, as distinguished from five cows; but it by no History.
means follows, that the idea of the number four, as S-V-2
connected with four cows, will be perfectly identical
in his mind with the idea of the number four as con-
nected with four horses; as they would in both cases
be blended with his ideas of the individual qualities
of the objects themselves: but if his idea of the
number four be registered in the memory by a specific
word, independent of the qualities of the objects with
which it was in the first instance associated, he will
become accustomed, after a more enlarged experience,
to pronounce the word without reference to such
associations, though they must necessarily spring up
in one form or other in the mind, upon a farther
analysis of the idea, of which the word is the general
symbol.”
* We are thus lead to the distinction of numbers into abstract
and concrete, though the abstraction exist merely in the word
by which any number is designated, or in the equivalent symbol
by which it is represented in different arithmetical systems. In
Arithmetic we consider both kinds of numbers, though the ope-
rations are in all cases the same as if the numbers were perfectly
abstract; the association of qualities being merely of use in
directing us to the particular operations or reductions to be per-
formed, and in assisting us in the proper interpretation of the
result: thus in the statement of the Rule of Three question, “If
1 lb. 6 oz. of tea cost 8s. 4d. what is the price of 3 lb. 8 oz.”
We say, lb. oz. lb. oz. s, d.
1 6 : 3 8 : : , 8 4 : ºr,
And after reducing the two first terms to units of one denomina-
tion, and the third term to units of another denomination, we have
0Z. OZ, g
20 50 100 : ar.
The last term is 250, and the same number results, whether we
suppose the terms of the proportion to be abstract or concrete;
but there is an obvious advantage in considering them as concrete,
as we are thus guided not merely to the previous reduction of the
terms of the ratios, but likewise to the interpretation and reduc-
tion of the result. -
* Most writers on Arithmetic would state this question in the
following manner :
• OZ. s. d. Ib. oz.
1 6 : 8 4 - : 3 8 : «.
In this statement, however, there is a manifest violation of pro-
priety, as the terms of the ratios are not homogeneous; and the
practice is not justified by any corresponding advantage. It is
obvious, however, from the preceding observations, as well as
from other considerations, that the result will be the same as is
obtained from the correct proportion.
Some authors have defined abstract or discrete numbers to be
those which have no denomination annexed to them; considering
all others as contract or concrete. Upon this definition, a diffi-
culty arose about the class to which fractional numbers were to
be referred; the units of the numerator being limited in value
by the denominator, and consequently being in this respect diffe-
rent from abstract whole numbers. The solution of the difficulty is
to be found in the meaning of the word denomination, which in our
definition would be confined to designate a quality of the subject
which the unit is supposed to denote. This question is very well
discussed in the Whetstone of Witte, the first work on Algebra
published in England, by Robert Recorde, in 1557.
This latter distinction of abstract and concrete would nearly
answer to their meaning in ordinary language, when applied to
any general term and its corresponding adjective : thus, in the
well known epigram,
Mentitur qui te vitiosum, Zoile, dicit,
Non vitiosus homo es, Zoile, sed vitium.
3 C 369
370 A R IT H M ET I C.
Arithmetic. (3.) We might suppose this process for the forma-
S-V-'tion of abstract numbers, to be completely effected by
*...* attaching names to the series of natural numbers, be-
numbers tº e tº : g
... ginning from unity; but if such names were perfectly
fectly arbi- arbitrary and independent of each other, our progress
trary. in numeration would be extremely limited, as the
memory would be overwhelmed with a multitude of
disconnected words; and the performance of the most
simple operations of addition, subtraction, multiplica-
tion, or division, would require an insight into the
constitution of numbers, to which the mind, particu-
larly in the infancy of society, would be altogether
unequal: under such circumstances, we might readily
credit the narrations of travellers who have limited
the powers of numeration of some savage tribes to
five, or to ten ; but it will be found, upon an ex-
amination of the numerical words of different lan-
guages, that they have been formed upon regular
principles, subordinate to those methods of numeration
which have been suggested by nature herself, and
which we may suppose to have been more or less
practised amongst all primitive people; for in what
other manner can we account for the very general
adoption of the decimal system of notation, and what
other origin can we assign to it than the very natural
practice of numbering by the fingers on the two
hands.” Q -
Theoretical (4.) Assuming such an hypothesis as true, it would
origin of not be difficult to give a probable theory of the for-
the decimal mation of the decimal scale of numeration, and of
... the adaptation of language to it; for suppose a num-
Inotation, º
ber of counters, or pebbles, or objects of any other
kind were placed before a person accustomed to count
upon his fingers; in making his tale, he would first
place his fingers in succession upon ten counters; and
let us suppose him to reject nine of them, and to put
the tenth apart as a register of the completion of one
operation. Again, let him repeat the same operation,
rejecting nine counters each time, and preserving the
tenth, until the number of counters remaining is less
than ten : let them be preserved by themselves in a
place, which, for greater distinctness, we will call A,
whilst the place for the counters which were separated
from the original heap, to mark the completion of each
operation, is called B. We may now suppose the same
process to be repeated upon the counters in the place
B, rejecting nine and preserving the tenth in a place C,
until the number of counters remaining in B is less
than ten : if the number of counters in C exceed ten,
the same process may be repeated upon them, and
every tenth counter may be placed in D; and so on,
until the number of counters remaining in the last
place is less than ten. We shall thus get a series of
sets of counters A, B, C, D, &c. where each counter in
—ºw-
Witium is a general abstract term for every species of vice ; but
vitiosus, though equally general in its application, is concrete, as
designating a quality of a subject.
* There is a curious passage in Ovid on the origin of the
decimal scale of numeration. Speaking of the ancient Roman
year, he says,
Annus erat decimum cum Luna replenerat orbem,
Hic numerus magno tunc in honore fuit ;
ASeu quia tot digiti, per quos numerare solemus,
Sew quia bis quino femina mense parit;
Seu quod ab usque decem numero crescente venitur
Principium spatiis sumitur inde novis.
Fasti, lib. iii. 124.
B corresponds to ten counters in A; every counter
superior place corresponding to ten in a place next
inferior : in this arrangement the counters acquire a
representative value dependent upon their position,
and the number itself may be considered as expressed
by a comparatively small number of counters, particu-
larly when the number is large. -
By a little variation of the process we should be
enabled, by means of nine counters only correspond-
ing to each place, to effect a similar resolution of any
number whatever of objects, and consequently to ex-
press it: it would be merely necessary, whenever ten
counters were required for any one place, to remove
the nine which were previously there, and to place one
counter in the next superior place ; we shall thus
possess a natural abacus, representing very distinctly
the principles and formation of the decimal scale of
numeration.
(5.) The discovery of this mode of breaking up Nomencla.
numbers into classes, the units in each class increasing
in a decuple proportion, would lead, very naturally, to
the invention of a nomenclature for numbers thus
resolved, which is more simple and equally compre-
hensive. By giving names to the first nine natural
numbers, or digits,” and also to the units of each class
in the ascending series by ten, we shall be enabled,
by combining the names of the digits with those of
the units possessing local or representative value, to
express in words any number whatsoever : thus the
number resolved by means of counters in the follow-
ing manner,
D C B
A.
would be expressed, (supposing seven, six, five, and four,
denote the numbers of the counters, in A, B, C, D, and
ten, hundred, and thousand, the value of each unit in
D, C, and D,) by seven, six tens, five hundreds, four
thousands; or, inverting the order, and making the
slight changes required by the existing form of the
language, by four thousand, five hundred, and sixty-
SęVen.
It is quite unnecessary for us to exhibit this transition
from the expression of a number by artificial methods,
to its expression in words either for other numbers or
for other languages than our own ; the one just given
being abundantly sufficient for the illustration of our
hypothesis. *
The advantages of this resolution of numbers are
not confined to the expression of large numbers by
* The earlier writers on Arithmetic distinguished numbers into
digital, articulate, and compound; the first denoting the first
nine natural numbers, which were counted upon the digiti, or
fingers; the second multiples of ten, of a hundred, &c. which
might be counted upon the articuli, or joints of the fingers; the
third, all numbers which arise from adding digital and articulate
numbers together. The Arabs denoted the second class of num-
bers by a word, which means knots, . -
History.
in C to ten in B, and so on; every counter in a \-N-
ture of
numbers in
the decimal
scale.
A R I, T H M ET I C.
37I
Arithmetic. few words which are easily remembered ; for we thus
~~' become familiar with the Superior units, such as ten,
a hundred, a thousand, as well from frequent repetition
as from our knowledge of their relation to each other
and to unity; and are thus enabled to form clear and
distinct conceptions of large numbers, whose compo-
sition we discover, in the words by which they are
expressed, or in the symbols by which they are re-
presented. .
(7.) But the decimal scale of numeration is not the
only one which may be properly characterised as a
matural scale. In numbering with the fingers we might
very naturally pause at the completion of the fingers
on one hand; and registering this result by a counter,
Other natu-
ral scales
of nume-
ration.
or by any other means, we might proceed over the
fingers of the same hand again, or with the fingers of
the second hand, and register the result by another
counter, or replace the former by a new counter, which
should become the representative of ten. If the first
process were adopted, we should be led to the formation
of a scale of numeration which is strictly quinary : by
pursuing the second process, we should end in the
formation of the denary scale, with the quinary scale
subordinate to it; and in adopting language to such a
practical mode of numeration, we should give inde-
pendent names to the first five digits, and subsequently
express the digits between five and ten by combining
the name of five, considered as a superior unit with
the names of the first four digits: in the first system
the name for ten would be expressed by a word equi-
valent to twice five ; in the second, it would be ex-
pressed by a simple and independent word.
Again, the scale of numeration by twenties has its foun-
dation in nature, equally with the quinary and demary
scales. In a rude state of society, before the discovery of
other methods of numeration, men might avail them-
selves for this purpose, not merely of the fingers on the
hands but likewise of the toes of the naked feet; such
a practice would naturally lead to the formation of a vi-
cenary scale of numeration, to which the denary, or the
denary with the quinary, or the quinary alone, might
be subordinate : in the first case we must have single
and independent names for the first nine digits, for ten,
and for twenty; in the seeond for the first four digits,
for five, for ten, and for twenty; in the last, for the
first four digits, for five, and for twenty. Such are the
principles of a philosophical nomenclature adapted to
suit these different scales of numeration, subject of
course to such variations as may be required by the
genius of the language to which they are applied.
Of other systems of numeration the binary might
be considered as natural, from the use of the two
hands in Separating objects into pairs, and from the
prevalence of binary combinations in the members of
the human body; * but the scale of its superior units
increases too slowly, to embrace within moderate
limits the numbers which are required for the ordinary
wants of life, even in the infancy of society. It re-
quires twelve orders of superior units in the binary
scale for numbers, which are expressible in the qui-
nary scale by five orders, in the denary by three, and
in the vicenary by two; the adoption of this scale there-
fore would require a more complete knowledge of the
classification of numbers, upon which numerical
* Monboddo, on the Origin and Progress of Language, p. 55i.
systems depend, than we could expect to find at the . History.
period of society when such systems are formed.
There are no members of the human body, and no
use of those members, which could naturally lead to
the adoption of any other scale of numeration than
those above mentioned; the Senary scale possesses
some advantages over the quinary, and the duodenary
over the denary, but the perception of those advan-
tages belongs to an advanced state of arithmetical
knowledge, and they form, therefore, no argument for
the adoption of such scales at the period of Society to
which our argument refers.
(8.) As the necessity of numeration is one of the Methods
earliest and most urgent of those wants, which are not of numera-
essential to the support and protection of life, we might tº
w . a preceded
naturally expect that the discovery of expedients for .
that purpose should precede the epoch of civilisation, tion of nu-
and the full developement and fixing of language. merical
That such has been the case, we shall find very fully language-
and clearly established, by an examination of the
numerical words of different languages; for without
any exception, which can be well authenticated, they
have been formed upon regular principles, having re-
ference to some one of those three systems of nume-
ration, which we have characterised as natural; the
quinary scale, whenever any traces of it appear, being
generally subordinate to the denary, and in some
cases both the quinary and denary scales being sub-
ordinate to the vicenary. In some cases also we
shall find from an examination of primitive numeri-
cal words, conveying traces of obsolete methods of
numeration, that the quinary, and even the vicenary
scales have been superseded altogether by the denary,
either from a sense of its superior advantages in the
progress of society and civilisation, or introduced
from other nations through commercial intercourse,
colonization, or conquest.
Besides the general proposition contained in the
preceding statement, that the natural scales of numera-
tion alone have ever met with general adoption ; there is
another proposition which is in some degree a conse-
quence of the former, but which an examination of
the structure of numerical language will in many
cases more completely establish ; which is, that amongst
all nations practical methods of numeration have preceded
the formation of numerical language. • *
(9.) It is in the language of people far removed from Numerical
civilized life, that the connection existing between words are
practical methods of numeration and numerical words º' i. ays
will generally be most clearly exhibited; for in such “
languages words are more immediately the transcripts
of things, and are less diverted from their primitive
meaning and application, than in those which have
been expanded by the culture necessary to fit them
for the multiplied wants of civilized life, and to
enable them to express the infinite variety of ideas
introduced by an enlarged exercise of the reflex ope-
rations of the mind; it might be contended, indeed,
that numerical words, being of early use, and there-
fore primitive, are likely to remain unaltered amidst
the fluctuations to which all languages are subject,
before they become fixed and permanent by literature;
but the changes of language depend less upon in-
ternal than upon external causes, being less affected.
by the progress of the arts of life, than by the intro-
duction of new words, or even of portions of new
languages, through intercourse with other nations,
3 C 2
372 A R I, T H M E T I C.
Arithmetic, colonization, or conquest: different languages become Italy, of the former existence of many of which we History.
S-N- in this manner incorporated with each other, and pri- have evidence, have been reduced to mere dialects of \-N-
mitive languages either altogether disappear, or lose two ; the only trace of any of the languages which
much of their original character. In this union of we know from the authority of Strabo existed for-.
the languages of different people with each other, merly in Spain, is to be found in the mountains of
possessing different numerical systems, as well as Biscay : it is only at the base of the Pyrenees, and in
different numerical words, it is natural to suppose the remote parts of Brittany, where the influence and .
that the most perfect system of numeration, or the authority of the Romans were little felt or known,
best constructed numerical language, should be adop- that we can discover any remains of the languages of
ted in whole or in part; and in those cases where a the numerous tribes of ancient Gaul : the mountains
change of grammatical structure is a consequence of of Wales and of Scotland have alone prevented the
this union, the numerals, particularly such as are exclusive use of a common language in Great Britain:
compound, may be different from those of either of the Arabic and its derivatives have nearly superseded,
the component languages, and may become more or or greatly affected all other languages, where the au-
less expressive of practical methods of numeration, thority of the Koran has been long acknowledged : the
whether primitive or not ; it is the combination of all commercial activity and enterprising character of the
these circumstances, that renders it extremely difficult Malays, have propagated their language, in whole or
in such languages to trace the existence of primitive in part, throughout the islands of the Indian Archipe-
methods of numeration in numerical words, and to lago and the South Sea: in North America the nume-
show the connection which subsists between them. rous tribes who were driven from their settlements
Use ºf "" (10) Extensive collections have been made of the nu- by European colonists, have disappeared with their
merals for als of different le. for th f ascertain- 1 : and the same effect, in perhaps a still
tracing the merals of different people, for the purpose of ascertain- languages; an e S , in permap
affinities of ing the affinities of languages, and perhaps few classes greater degree, has attended the progress of the
languages,
of words could be selected which are better calculated
to answer this object; but the preceding, as well as
other considerations, show that their authority is not in
all cases to be depended upon. The more philosophical
of modern Philologists, indeed, have ceased to regard
affinity of roots as a decisive proof of the affinity of
languages; it may arise from the mere mixture of lan-
guages, or from the intercourse of the people by whom
they are spoken, but it by no means demonstrates them
to be of common origin, unless accompanied also by
a corresponding affinity of grammatical structure.
Thus the numerals of nearly all the languages of
Europe, and of many of those of Asia, are nearly the
same, or very slightly different from each other; and
some authors have attempted from this circumstance,
supported by the analogy of other roots, to refer all
those languages to a common origin ; * the essential
diversity, however, of their grammatical structure,
would show such a classification to be much too
comprehensive; and even after referring them to three
great classes, the Indo Pelasgic,f including the San-
Scrit, Greek, and Latin, the Persian and German, with
their immediate derivatives ; the Slavonic, including
the Armenian, Russian, Polish, and Bohemian ; and
the Celtic, including the Welsh, the Erse, the Gaelic,
the Armorican, and the Basque of Biscay; we shall
still find some reasons for thinking that we have
associated together, and particularly in the last of
these classes, some languages which are essentially
distinguished from each other.
It has long been a favourite theory of Philologists
to trace up all existing languages to a small number
of others, which they consider as primitive; but the
reasonings by which such theories have commonly
been supported, are founded upon an assumption of
an order in the occurrence of facts which is directly
contrary to experience; it being the constant tendency
of civilisation, and the certain influence of extensive
empires to diminish, and not to increase the number
of languages; the numerous languages of Greece and
* Parsons, Remains of Japhet; Wallancey, Collectanea de Rebus.
Hibernicis, vol. iii. No. 11.
f Frederick Schlegel, Ueber die Sprache and Weisheit der
Indier ; Vater, Mithridates.
Spanish dominion in the South.
The immense number of languages, radically dif-
ferent from each other, which are spoken by the
tribes of barbarous countries which have never been
Subject to a common empire, establishes the same pro-
position in a still more striking manner. The Jesuit
Missionary Dobrizhoffer” says, that there are upwards
of thirty known languages spoken in Paraguay alone.
Father Lasuent observed no fewer than seventeen lan-
guages in an extent of only 500 miles on the coast of
California. More than 150 other languages have been
observed in other parts of that vast continent, and
farther researches would probably greatly increase
that number. Mr. Bowdicht has given the numerals
of thirty-one languages, most of which are spoken
within a district of small extent upon the western
coast of Africa; and Mr. SaltS has given those of fifteen
others on the eastern coast, between Mozambique
and Abyssinia. That continent, indeed, may be
almost said to Swarm with languages, so numerous
do they appear in almost every part of the small
portion of it, which has hitherto been subject to ex-
amination.
In judging of the proper uses of numerals for ascer-
taining the affinity of languages, it is particularly neces-
sary to consider whether they exist under their original
and unaltered form, or have been mixed up with others
without a more intimate union, or have become mere
dialects of a predominant language. In the languages
of barbarous and primitive people, possessing a general
affinity of grammatical structure, as in those of the
tribes of South America,[] they will generally form
a just measure of the affinities of the languages
themselves ; in the absence of such a common struc-
ture, and in cases where languages from different
causes are greatly altered from their primitive form,
the affinity of numerals may serve as a monument of
the communication of the people by whom they are
used, and even of the present intermixture of their
*. History of the Ahipones.
+ Humboldt, Essai politique sur le royaume de Nouvelle Espagne.
: Mission to Ashantee, Appendix.
§ 7'ravels in Abyssinia, Appendix.
|| Humboldt, Personal Narrative, vol. iii. p. 244. English Trans.
A R I T H M E T I C.
373
Arithmetic, languages, but furnishes no proof of their primitive
affinity with each other.
There are some circumstances, particularly in the
numerals of African languages, which are extremely
difficult to explain.
Cashna,” two neighbouring African tribes, okoo is the
name for five in the first, and for three in the second,
all the other numerals being different from each other;
and Mr. Bowdicht has remarked other instances of a
similar interchange of the names for four and five in
the numerals of tribes, geographically remote from
each other, in which all the rest are different; again,
the name for four in the Inta language is the same as
that for the same number in the language of Empoon-
ga, at the distance of 1000 miles; and the name for
five in the first of these languages, is the same as that
for five in the language of Kamsallahoo. Barton f has
given from the records of the first settlers in North
America, the numerals of the Nanticocks, an extinct
tribe, who inhabited the south bank of the Chesapeak,
which are nearly identical with those of the Mandin-
goes of Africa, as will be immediately seen upon
examination of them.
Nanticocks. Mandingoes. S
1. Killi. 1. Killim.
2. Filli. 2. Foola.
3. Sabo. 3. Sabba.
4. Nano. 4. Nani.
5. Turo. 5. Loolo:
6. Woro. 6. Woro.
7. Wollango. 7. Oronglo.
8. Secki. 8. Sec.
9. Collango. 9. Conanto.
1O. Ta. 10. Tang.
The resemblance of these numerals is apparently
too remarkable to be accidental, yet the people by
whom they were used, belonged to races essentially
different, and between whom it is difficult to imagine
that any intercourse could have taken place; the ex-
amination of their languages, if it were now possible,
might perhaps throw some light upon this very
curious and very embarrassing fact.
There are perhaps some cases, where an affinity
exists between languages, which is in no respect
borne out by the affinity of their numerals. Voyagers
and others have remarked the resemblance between
the languages of Nootka Sound and its neighbourhood
on the north-west coast of America and the Aztec of
Ancient Mexico; and Humboldt, though he sup-
=w
* Horneman, Proceedings of the African Society, p. 148–156.
‘h Mission to Ashantee.
: On the Origin of the American Tribes and Indians.
§ Park's First Travels in Africa, p. 61.
| There is considerable difficulty in collecting materials for
an inquiry of this kind, as travellers have usually contented
themselves with giving the simple numerals as far as ten, with-
out noticing the formation of the expressions for higher numbers;
or where such are given, they have seldom added an explanation
either of the meaning or grammatical connection of the terms
in compound expressions, which is highly necessary, in order to
deduce from them an idea of their arithmetical systems. Amongst
other exceptions to this remark, (which is only generally true,)
we ought in justice to mention Mr. Crawford, who has given an
excellent account of the numeral systems of the Islanders in the
Indian-Archipelago, in drawing up which he professes to have
been guided by the excellent observations of Professor Leslie in
his Philosophy of Arithmetic.
There is a work of the Abbé Hervas, expressly on this subject,
In the languages of Bornou and
poses that the resemblance is more apparent than real,
the same very peculiar combination of consonants, yet
admits the existence of some affinity between them ;
it will be found, however, that they have not one
numeral in common, or between which the most dis-
tant resemblance can be traced.
In the classification of the languages of Europe, the
Lapponian, Finnish, Esthonian, and Hungarian, have
been usually associated together, as belonging to the
same family.* The following are the numerals in the
first and last of these languages:
Lapponian.i. Hungarian.
1. Auft. 1. Egi.
2. Gouft. 2. Ketto.
3. Golm. 3. Harum.
4. Nielja. 4. Negy.
5. Wit. 5. Et.
6. Gut. 6. Hai.
7. Zhicczhia. 7. Hét.
8. Kantze. 8. Nyoltz.
9. Antze. 9. Kilentz.
10. Laage. 10, Tiz.
If the affinity of these languages, which so many
authors have attempted to prove, really exist, it is
quite clear that little or no trace of it is discoverable
in a comparison of their numerals.
The extraordinary coincidences as well as diversities
of numerals, which are given above, show how dan-
gerous it is to form any general conclusions respecting
the relations of languages from the comparison of a
small number of their roots, however apparently well
chosen for the purpose. §
(11.) We shall now quit the philological discussion of
numerals, and proceed to the consideration of them as
records of systems of numeration ; in this inquiry we
shall not pretend to embrace those systems in all
known languages, which would lead into very exten-
sive details, but shall confine ourselves to such as may
be requisite to establish our two general propositions,
(Art. 8 ;) noticing occasionally remarkable examples
of the adaptation of language to systems of numera-
tion, and other facts which may illustrate the process
History.
arising in a great measure from the frequent use of \-y-
Scales and
methods of
numeration.
followed by the human mind in the formation of such
systems; imperfect as this notice must necessarily be,
it will enable us to give some degree of arrangement.
to a great multitude of very interesting facts, and will
show in a very remarkable manner, how near an ap-
proach is made, in a great many instances, to the sim-
plicity of the most philosophical language. -
Of all the systems of numerical words with which
we are acquainted, that of Thibet possesses the most
simple structure, and makes the nearest approach to
arithmetical notation by local value; the first twenty-
nine numerals are as follows : -
W
frequently referred to by Humboldt and Vater, entitled Idea del'
Arithmetica di tufte le Wazioni conosciºte, a copy of which we
have not been able to procure. The materials of this work
must be of great interest and value, as the author was in possession
of a large collection of American vocabularies, which only exist
in manuscript. .
* Schubert's Reise durch Schweden, Norwegen, Lappland,
Finnland, &c. vol. iii. p. 453. -
+ Knud Lecms, De Lapponibus Finmarchiae.
f Kalmar, Prodromos Idiomatis Scythico-Mogorico, Chuno-
Avarici, sive apparatus Criticus in Linguam Hungaricam, p. 79.
§. Klaproth, 4sia Polyglotta, p. 40.
Numerals
of Thibet.
z_-
374
A R I T H M E T I C.
Arithmetic. 1. Cheic. 16. Chutru.
\-y- 2. Gnea. 17. Chutoon.
3. Soom. 18. Chughe.
4. Zea. 19. Chugoo.
5. Gna. 20. Gnea chutumbha.
6. Tru. 21. Gnea cheic.
7. Toon. 22. Gnea gnea.
8. Ghe. 23. Gnea soom.
9. Goo. 24. Gnea zea.
10. Chutumbha. 25. Gnea gna.
11. Chucheic. 26. Gnea tru.
12. Chugnea. 27. Gnea toon.
$ 13. Chusum. 28. Gnea ghe.
14. Chuzea. 29. Gnea goo.
15. Chugna.
In this system, the numerals from ten to nineteen, are
formed by the combination of the first syllable of the
word for ten, with the names of the first nine numbers,
in the same manner as in most of the languages adapted
to the decimal scale; but from twenty-one to twenty-
nine, the name for two acquires a value from position,
in a manner which bears the closest analogy to our ordi-
nary arithmetical notation. Turner,” who has only
given these numerals incidentally in his observations
on the Thibetan month and calendar, has added no ex-
planation whatever of their mode of expressing higher
numbers.f
Our arith- If the same simplicity of structure prevails through-
metical no- out the numerical language of Thibet, (and it is diffi-
fation pro- cult to imagine that this happy idea when once started
bably due
... should not have been pursued to a much greater ex-
tent,) it would give great weight to the opinion that
we are indebted to this country for our system of
arithmetical notation; as of the two great difficulties
attending its invention, namely, local value and the
zero, one at least was overcome, at the period when
their numerical language was fixed.:
Considered (12.) The Hindoos consider this method of numera-
by the Hin-tion as of livine origin,” the invention of nine figures with
doos of Di- device of place being ascribed to the beneficent Creator of
Vine orišin the universe.”$ Of its great antiquity amongst them
there can be no doubt, having been used at a period
certainly anterior to all existing records. Most other
memorable inventions they have attributed to human
authors; but this, in common with the invention of
letters, they have ascribed to the Divinity, agreeably
to the practice of the Egyptians, Greeks, and most
other nations, with respect to the more important in-
ventions in the arts of life, whose origin is lost in the
remoteness of antiquity.
The intimate analogy in the grammatical structure,
and in many of the roots of the classical languages of
Europe with the Sanskrit, combined with the evidence
furnished by historical and other monuments, point
out the East as the origin of those tribes, whose pro-
gress to the west was attended by civilisation and
empire, and amongst whom the powers of the human
* Embassy to Thibet, p. 321.
+ See also Klaproth, Asia Polyglotta, p. 353, where the nu-
merals are given under a somewhat different form ; and Remusat,
Récherches sur les Langues Tartares, p. 364.
† The numerals in a kindred language, and where the com-
pound expressions for numbers are nearly similar to those of
Thibet, may be seen in Kirkpatrick’s Vocabulary of the Newar
Dialect of Nepaul, p. 243. :
§ Bhascara; Vasara, and Crishna's Commentary on the Vija.
Ganita, quoted by Mr. Colebrooke in his Hindoo Algebra, p. 4.
mind have received their highest degree of develope- History.
ment ; and it may, perhaps, be not altogether unfair \-N-2
to form some inference respecting the extent of the
arithmetical system of those tribes at the period of
their separation, by the numeral words which those
languages possess in common. The Sanskrit names of Sanskrit
the ten numerals, which are numerals.
1. Eca, 6. Shata,
2 Dwau, 7. Sapta,
3. Traya, 8. Ashta,
4. Chatur, 9. Nova,
5. Ponga, 10. T)asa,
have been adopted with slight variations, as we have
before remarked, not merely in all languages of the
same class and origin, but likewise in many others
which are radically different from them. If we pro-
ceed to the expressions for higher numbers, we find
the same general law of their formation, by the com-
bination of the names of the articulate numbers with
those of the nine digits. In the Sanskrit also, as well
as in its immediate descendant the Hindostanee, it is
more elegant to make use of a word which is equiva-
lent to less twenty, rather than of the one which would
naturally express nineteen, and similarly for other num-
bers in the next series below the articulate numbers:*
precisely as in the Latin, we say unus de viginti for
movemdecim, unus de triginta for viginti movem; and the
same form of expression is observable in the Greek : t
these are points of resemblance in the construction of
their numerical terms which deserve to be remarked,
though not without example in other languages. If
we pursue our comparison of the other and higher
numerical terms of those languages, we shall find few
other points of resemblanee; the names for twenty, a
hundred, a thousand, are completely different: making
it probable at least, that at the epoch of which we are
speaking, their Arithmetic was confined within very
narrow limits.
The Sanskrit numeral' language assigns names to Great ex
seventeen orders of superior units in the decimal scale, tent of
as will be immediately seen from the following list : i.
1. Eca. 109. Abja or padma. language.
10. Dasā. IO10. C'harva. -
10°. Sáta. 1011. Nic'harva.
10°. Sahasra. 101*. Mahadpadma.
10*. Ayuta. 1019. Sáncu.
10°. Lacsha. 101*. Jaludhi or samudra.
10°. Prayuta. 101*. Antya.
107. Coti. 10°. Madhya,
10°. Arbuda. 107. Parard'ha.f
This luxury of names for numbers, much greater
than what are required for the ordinary uses of life,
or even for the most extended astronomical calcula-
tions, is entirely without example in any other lan-
guage, whether ancient or modern ; and implies a
familiarity with the classification of numbers accord-
ing to the decimal scale and the power of indefinite
extension which it possesses, which could only arise
from some very perfect system of numeration, such
as that “ with device of place.” Indeed there is no
circumstance which so strongly characterises Hindoo
science, as this very extraordinary facility of dealing
with high numbers: witness their enormous astrono-
* Haihed's Grammar of the Bengal Language, p. 160. *
† Matthiae, Greek Grammar, vol. i. p. 174.
# Colebrooke, Hindoo Algebra, p. 4.
A R H T H M E T I C. '375
simplicity of form : they are likewise characters to History.
Arithmetic, mical periods, and the extravagant dates of their
- . . . . which other meanings are attached, and which are only ~~
S-N-' chronology; and this at a period when the most
Chinese (13.) There is another eastern people, remarkable at number seventy, the symbol for seven is placed over
numerals, once for the great antiquity and unchangeable character that for ten, and similarly in other cases. There is
of their existing institutions, who possess a numeral lan- also another character denominated ling, which means
guage of great extent, connected with a very perfect residue or remainder, which in some respects may be
system of numeration. The following is the list of considered as filling the place of a zero in notation by
Chinese numerals :* w local value : thus, in expressing 1001, the symbols
AJ denominated yih, ts' hyen, ling, ling, yih, are written
: . 1. i. ‘successively underneath each other. It is clear that if
3. San looo. Tsº. €11 the symbol called tº hyen for 1000 were omitted, this
4. ss." loooo. wº. notation would strictly coincide with our ordinary
5. Ngoo. io. Es." arithmetical notation; the use of this character,
6. Lyell. io, Chao however, is certainly superfluous, though it affords a
7. Ts’hih. ioſ. King very remarkable approximation to a more perfect
8. Páh. ios. Kwai. system of numeration. In the use of the symbols of
9. Kyed rºyal. the third series, obviously founded upon the principle
º o of approximating the system of Chinese Arithmetic
The very peculiar character of the Chinese language, to that of the Hindoos, the symbols are written from
al language, in short, of symbols and their combina- right to left, and the character ling is replaced by the
tions, which is addressed to the eye and not to the ear, European zero. . . . . e ſº d
connects these numeral terms inseparably with the The essential distinction of Chinese arithmetical Their great
seventeen figures, or eharacters, which are made use of notation and our own, clearly consists in the use of antiquity.
in Chinese Arithmetic. In alphabetical languages, symbols for the superior units in one case, which are
there is no connection between numerical words and expressed by position alone in the other. In the last
numerical symbols, the latter being, in almost all cases of their three methods of notation, those systems
- where they exist, of subsequent invention to the would become identical by the entire omission of the
former; but the Chinese numeral symbols, being second of the two lines of symbols.
either simple elements, or keys, or composed of them The Chinese consider the symbols of the first class
like other characters, are transferred to the oral lan- for numbers which are below 10,000, as coeval with
guage upon those arbitrary yet regular principles by the invention of their other characters, and conse-
which monosyllabic sounds are attached to all their quently as possessing an antiquity of at least 3000
characters, however complicated they may be. years. The symbols for higher numbers are of later
Threekinds In Plate I., we have given three series of Chinese date, having been introduced at different times to meet
scientific people of the western world were incapable
by any refinement of arithmetical notation, of express-
ing numbers beyond one hundred millions.
There is an epoch in the languages of all civilized
people, at which they acquire a fixed and permanent
character, and after which the admission of new
terms, not arising from those natural combinations
which the genius of the language sanctions, is effected
with great difficulty: this takes place whenever a
national literature, whether oral or written, is so gene-
rally diffused, as to form a standard of reference or a
test of purity, which, whilst it enforces a legitimate
character upon all existing terms, watches over the
introduction of all others with extreme jealousy; from
this consideration alone, independently of other evi-
dence, we should be inclined to assign to the Sanskrit
terms for high numbers, and consequently to the
system of numeration, upon which they are founded,
an antiquity at least as great as their most ancient
literary monuments; as the arbitrary imposition of so
Imany new names, for the most part independent of
each other, and in number also so much greater than
could possibly be required for any ordinary application
of them, would be a circumstance entirely without
example in any language which had already acquired
a settled and generally recognised character.
of Chinese numeral charaeters, the first being those which com-
figures.
monly occur in historical and scientific works; the
second are the characters made use of in bonds and
formal instruments, in order to avoid frauds, to which
the first series of numerals are very liable, from their
* Marshman, Clavis Sinica, p. 299.
conventionally used for the purpose of numerals. Thus
the character used in such documents for one, means
perfection ; that for two, is a verb meaning to assist,
to separate; for three, an accusation ; for four, to earpose
publicly; for five, to associate; for six, a mound of earth;
for seven, a certain tree; for eight, to divide ; for nine,
a peculiar stone; for ten, to collect ; and similarly for
the characters of the other superior units. The use
of such characters for numbers corresponds to our use
of numeral words at full length instead of figures, for
such purposes; but the analogy exists in the applica-
tion only, the Chinese expressions for numbers being
in all cases symbolical. The third set of figures are
used for mercantile purposes, and are said to have been
introduced by the Catholic Missionaries; they have
been adopted in consequence of their greater simplicity
of form, and from their admitting of being rapidly and
easily written. -
In the same Plate, (I.) the reader will find ex-
amples of the actual mode of expressing particular
numbers ; such as 22, 100, 1100, 1010, 1001, and
1923000, according to these three methods of nota-
tion : * in the two first, the numbers are written in
vertical columns, the value decreasing downwards,
the digital symbol being placed immediately over the
symbol of the Superior unit : thus, to express the
the increasing wants of their. Arithmetic : it would
follow, therefore, if full credit can be attached to their
annals, that the claims of the Chinese to the first in-
vention of arithmetical figures, are equal, if not supe-
rior, to those of any other people. Independently,
*—a k
* Morrison's Chinese Grammar, p. 84.
376
A R IT H M E T I C.
Arithmetic, indeed, of direct historical evidence, we might venture
S-N-" to infer, from the universal prevalence of the decimal
Not the in-
ventors of
notation by
local value.
scale throughout the empire, not merely in the classi-
fication of numbers, but also in the divisions of their
coins, their weights, and their measures; from the
great number of superior units, expressed by their
symbols; and from the great perfection of their
practical Arithmetic, for which they have long been
celebrated throughout the neighbouring countries, and
the Indian Archipelago, that they have been in posses-
sion of a very perfect system of numeration during
many ages: an opinion which derives additional Sup-
port from observing, that amongst them literature,
science and the arts of life have long reached a sta-
tionary point; and that, from the very nature of their
government and institutions, a limit is put to the
progress of improvement, and, apparently, even to the
powers and speculations of the human mind.
As the Chinese are not in possession of the
method of arithmetical notation by nine figures and
zero, they clearly can have no proper claim to its in-
vention, however nearly in some respects they may
have approximated to it; for it is next to impossible
that a system of numeration, so much more perfect and
commodious than their own, if once generally known
or practised, could ever have been lost or abandoned.
In considering the claims of other nations to this
great invention, which is, unquestionably, of eastern
origin, if our decision is to be determined by the known
antiquity of possession, we must certainly refer it to
Hindostan ; though some circumstances in the con-
struction of the numerical language of Thibet have
induced us to express a suspicion, that it may have
originated in that country; an opinion which derives
some support from the frequent and intimate commu-
nication between these countries from very early
periods: and whilst from Hindostan they derived the
doctrines of Bouddha, the Sanskrit alphabet, under the
form in which it is seen in the most ancient inscrip-
tions, and the polysyllabic portion of their language,
which is otherwise intimately allied with the mono-
syllabic colloquial medium of China, it is not impro-
bable that they may have communicated in return the
elements of the system of arithmetical notation by
local value.
Economy
of numeri-
cal words.
(14.) The economy of numerical words, which is ob-
servable in most languages, affords a very strong confir-
mation of the truth of our proposition, that they have
been in all cases adapted to systems of numeration
previously in use; thus, it is a very rare case in any
language to find two different words to express the
same number ; and when such do occur, they are
usually the vestiges of primitive methods of nume-
ration which have been superseded by others adapted
to the denary scale, where the new terms which have
accompanied its introduction are either of foreign
origin, or formed from the natural combinations of the
language in such a manner as to be more expressive of
the process of numeration itself.
Malay and We shall find many examples of this circumstance
Javanese
numerals.
in the languages of the islanders of the Indian
Archipelago; their primitive systems of numeration,
which were in ancient times for the most part quinary,
subordinate to the vicenary, have been superseded by
the more perfect arithmetical system of Hindostan,
transmitted to them, either immediately or indirectly,
through the Malays of Malacca and Sumatra. The
following list of Malay numerals, with those corres- History,
ponding to them of the ordinary language of Java, will
assist us in generalizing some remarks, not merely on
this subject,” but likewise others which arise imme-
diately from an examination of them :
Malay numerals. Javanese numerals.
1. Sa, Satu, süatu. 1. Sa, siji.
2. Düa. 2. Loro.
3. Tiga. 3. Tàlu.
4. Ampat. . 4. Papat.
5. Lima. 5. Limo.
6. Anām. 6. Nänäm.
7. Tūjah. 7. Pitu.
8. Düläpan, Salāpan. 8. Wolu.
9. Sambilan. 9. Songo.
10. Püluh, Sa-puluh. 10. Púluh.
ll. Sa blas. ll. Sawālas.
12. Düa blas. 12. Rolas.
13. Tiga blas. 13. Tălulas.
20. Düa púluh. 20. Rong påluh, or likur.
21. Düa puluh Satu. 21. Rong puluh siji, or
- sa-likur.
25. Dúa púluh lima, or 25. Rong púluh limo, or
taugah tiga puluh. limo likur, or lawe.
30. Tiga púluh. 30. Tălung puluh.
35. Tiga púluh lima, or 35. Tălung puluh limo.
- taugah ampat påluh.
50. Lima púluh. 50. Limo puluh, sekāt.
60. Anám puluh. 60. Nänäm puluh, Swidak
65. Anám puluh lima, or 65. Pitusasor.
taugah anám puluh.
Rātus, Sa-rātus.
Düa rātus.
100. Hatus.
200. Rongātus.
400. Papat-ratus, Samas.
100.
200.
400. Ampat rātus.
800, Dülāpan rātus. 800. Wolung-atus,domas.
1000. Ribu. 1000. Hewu.
10*. Laksa. 10*. Lākso.
10°. Sa-puluh laksa, 10°. Káti.
10°. Sa-yüta. 100. Yūto.
107. — 107. Wändro.
IO8. 108. Boro.
109. —. 109. Pärti.
1049. — 1019. Partomo.
IO!!. — 1011. Gulmo.
101*. — 10°. Kerno.
IO13. 1013. Wurdo.
In most of the islands of the Indian Archipelago,
there is a ceremonial dialect, as well as the one in
ordinary use, and as might be expected, the numerals
are not always the same in both : thus in the ceremo-
nial dialect of Java, the term for one is satungil, com-
pounded of sa, one, and tungil, alone by itself; for two,
the word kaleh is used, which is the preposition with ;
for three, tigo ; for four, kawan, a fock or herd of
animals ; for five, gangsal, a term of unknown deri-
vation; and for ten, the Sanskrit term doso; the other
terms, excepting where those above mentioned are
used in expressions for compound and articulate num-
bers, are the same as in the ordinary dialects. .
The influence of the Malays in the Indian Archipe-
lago is, comparatively, of modern date; and we conse-
quently find, every where, remains of ancient dialects,
very different from those at present in use. That of
Java is an immediate derivative of the Sanskrit, pos-
-*
—s.
* Crawford's Indian Archipelago, vol. i. p. 264 ; Marsden's
Malay Grammar and Dictionary, p. 37
A R IT H M ET I C.
377
languages generally, which neither in the perfection . History.
of their grammar, nor even in their copiousness, appear S-v-
Arithmetic. Sessing likewise the Sanskrit numerals, with slight
Q-N-'variations, and those chiefly in the names of the
superior units, which have been transmitted un-
changed from the ancient to the modern dialects.
The great number of the names of those units, un-
exampled in the languages of any other of those
islands, which possess no native term for a num-
ber beyond onethousand, and no borrowed term for a
number beyond one million, is a circumstance strongly
confirmatory of our argument respecting the great
antiquity of those names and of the arithmetical
system connected with them, amongst the people from
whence they were derived.
Confusion (15.) Throughout theislands of the Indian Archipelago,
in borrow- with the exception of the Lampungs, an inland people
ed numeri- of Sumatra, the Sanskrit term laksha for 100,000 has
cal terms.
been borrowed to express, not the same number, but
10,000; a circumstance which frequently causes mis-
takes in their commercial transactions with the people
of Hindostan. In a similar manner the Javanese use
the term kāti for 10°, which is the same as the Sanskrit
term koti for 107, and the term yūto for 10° the same as
the Sanskrit ayuta for 10*: this confusion of the terms
for high numbers, which are evidently borrowed from
each other, is a very remarkable circumstance, and
can only be accounted for by supposing that amongst
a rude people, little accustomed to the use and con-
templation of such numbers, the terms by which they
were expressed would convey no distinct impression
to the mind, and consequently in making use of them
more reliance would be placed upon the uncertain
testimony of the memory, than the surer guidance of
the understanding.
Other examples may easily be produced of a similar
change in the value of borrowed numerical terms.
In the Newar dialect of Nepaul, we find lak-sehee bor-
rowed to express a million ;” in the language of the
Mantischeou Tatars, immediately bordering on the
north of China, in which the numerals are taken
generally from the Chinese, though they have lost
their monosyllabic form, we find the term iwuan for
1000, obviously derived from the Chinese term wan,
which expresses 10,000.f Again, alp, the term for
1000 in most of the languages which modern philo-
logists have agreed to call Semitic,f and which pre-
vails in those of Upper Egypt, Abyssinia, and Darfur,
signifies 10,000 in the Amharic, a language intimately
to bear any certain relation to the state of civilisation
of the people by whom they are spoken.
Some authors have asserted, that many nations pos-
sess a numerical language more extensive than their
powers of numeration, and have referred, in proof of
their assertion, to the numeral words of many South
American tribes, which are sufficiently comprehensive,
though the people by whom they are used cannot
without great difficulty count beyond twenty. When
people are descended from a people more civilized than
themselves, from whose monuments of whatever
nature such terms are collected, such an opinion may
be entitled to credit; but in all other cases it seems
to involve its own refutation, as the very existence and
interpretation of the word implies that its meaning is
understood by some one at least, if not generally.
The statements of travellers respecting the lan-
guages and customs of people with which they have
not become familiar from long intercourse, must
always be received with extreme caution ; and there
are few subjects upon which greater mistakes have
been made, than on those which respect the extent
and methods of numeration of barbarous nations. In
most instances such errors have consisted in greatly
understating the extent to which such people are able
to count; but in other cases, they have been of a com-
pletely opposite character, as the following example
will show : we had long been embarrassed with the
account given by Labillardière,” of the enormous ex-
tent of the numeral language of the natives of Ton-
gataboo, one of the Friendly Islands, proceeding as far
as 10*, a fact in apparent contradiction to our theory,
and not to be explained by their intercourse with the
Malays, from whom much of their numeral language
is derived, but who possessed no terms for numbers
equally great; and it was only by referring to the
account given of these islands by Mariner,t that we
found that their highest numerical term was mano for
100,000, and that the other terms which he has put
down for higher numbers, have significations of a very
different nature, imposed upon the poor Naturalist
from a species of revenge, more remarkable for its
humour than decency, for the persevering and annoy-
ing efforts which he made to extract from them the
names of numbers of which they had no knowledge.
If we examine the limits of the numeral terms of Names for
different languages, we shall find few which possess superior
terms for numbers beyond a thousand; and the cases *
allied with them ; the term she for 1000 having been
interpolated between it and the term meto for 100,
derived immediately from the common Semitic term
Small num-
for that number. §
(16.) The poverty of languages innative terms for high
ber of na numbers, arises either from the limited extent of their
!...” Arithmetic, or from the difficulty in all established
for high
numbers,
languages of inventing new words, even when the
want of them is felt : it is from this latter reason
chiefly, that the extent of numerical language is no
just measure of advancement in the arts of life, or
even in the art of numeration itself; and we shall find
many examples of barbarous people who possess
terms for higher numbers than the Greeks or Romans,
or of other nations incomparably more civilized than
themselves. The same remark may be extended to
are extremely rare in which they reach a million. The
instances are still rarer where such terms are native,
having been introduced, as in some cases we have seen
already, by intercourse with other nations; and we
frequently find the same terms for such numbers where
the lower numerals, as well as the languages to which
they belong, are essentially different from each other :
such examples are not without a considerable histori-
cal interest, as monuments of the communications of
nations with each other, and as indicating the channels
through which improvements not merely in arithme-
tic, but likewise in the other arts of life, have been
conveyed. We have already given examples of facts
* Kirkpatrick's Nepaul, p. 243.
f Klaproth, Asia Polyglotta, p. 300.
§ Salt's Travels in Abyssinia, App.
vol. I.
: Ibid. p. 107.
* Voyage in Search of La Perouse, vol. ii. p. 408. English
edition. -
f Mariner's Account of the Tonga Islands.
3 D
378
A R I T H M ET I C.
Arithmetic. of this kind among Eastern languages, and it would be
S-V-7 very easy to multiply their number : a few more in-
Greek.
Latin.
Italian.
German.
Spanish.
stances will establish the truth of our assertions, res-
pecting the invention and transmission of numeral
terms in a still more striking manner.
The Greeks possessed a term, uvpia, for 10,000;
and, notwithstanding the increasing wants of their
Arithmetic, they never attempted to proceed beyond
it: it appears originally to have signified an indefinite
number, and in this sense it is always used in Homer;
but in later times they gave it a new and restricted
meaning without abandoning the old, and distinguished
between its definite and indefinite signification by a
difference of accent or tone. This term in its later
sense, at least, was unknown to the AEolic tribes at
the time of the colonization of Latium, as no traces
of it appear in the Latin language, though the terms
for 100 and 1000 were transmitted through them with
very slight alterations. The characteristic contempt
of the Romans for whatever was connected with
science or the arts, may sufficiently account for their
not attempting to extend their numeral language as
far as the Greek, by borrowing or inventing an addi-
tional word ; and the improvement which was not
effected during the zenith of their empire, could not
be looked for during its decline, and that long period
of darkness and barbarism, which ended in the ex-
tinction of the Latin as a living language. At the
beginning of the fourteenth century, when the modern
Italian, its legitimate successor, was beginning from
the revival of learning and the writings of native
authors, to assume a settled character, and when the
introduction of the Hindoo arithmetical notation,
through the Arabians, was bringing into familiar use
numbers much greater than were expressible by the
Roman numerical symbols, we find a great addition
to their former numerical language, by the use of the
word millione, which properly signifies great thousand,
to denote the square of one thousand, and which was
followed by the words billione, trillione, deduced imme-
diately from the former by pursuing the natural analo-
gies of the language: a series of numeral terms were
thus formed, proceeding not by tens, but by millions,
like the monads of Archimedes, which proceed by
myriads of myriads. In a numeral language thus con-
stituted there is clearly no limit to the expression of
numbers, the composition of the names for the monads
or superior units being once understood.
These terms were at different periods adopted in
almost every language of Europe : the Germans, who
adhere as much as possible to the combinations of their
own language in the formation of new words, resisted
the introduction of the term million, forming no na-
tural succession to their native words hundert and
tausend, until the commencement of the sixteenth
century. The Poles,” the most cultivated of Sclavonic
nations, admitted it at a still later period; and it was
introduced into Russia, along with the Hindoo nota-
tion, by Peter the Great, at the commencement of the
last century.
The Spanish term for a million is cuento, which in
ordinary language means a tale or fable for children ;
it most probably originated from cubo ciento, the cube
* The Polish word for 100 is sto, and for 1000 tisiacz; the
Russian word for 100 is also sto, but there is no native word for
1000, which is expressed by disset sto, or ten hundred.
of a hundred. Though without any certain means of
History.
judging of its antiquity, we have probable reasons for \-º-º:
thinking it nearly, if not quite, as old as the corres-
ponding Italian word; for the Arabian notation was
known at least as early in Spain as in Italy; and as
the consequent necessity for an extended numeral lan-
guage, must have been equally felt in both countries,
it is not likely that one word could have long existed
in one language without being communicated to the
other, particularly when the intimate alliance of the
languages with each other, and the frequent inter-
course at that period of the people by whom they are
spoken, is considered.
There are two different series of names for superior Welsh:
units in the Welsh language; one ancient, and the other
used in its more modern and latinized form : * in the
last of these, we have cant, 100 ; mil, 1000 ; myrz,
10,000; can inil, 100,000; myrzcan, or milvil, 1,000,000;
milcan mil, 10,000,000; and similarly for higher num-
bers. The selection of the word myrz for 10,000, which
is clearly the Greek uvptès, and the deriving of the rest
from the Latin, would appear to show that they had been
introduced at a late date by some monk or other person
who was familiar with the classical languages. The
ancient and more native superior numerals are chiefly
remarkable for their redundancy, and an extent greater
than amongst any other European people : thus we
have three names, mwnt, catyrva, rhiolla, for 100,000 ;
and other three, mynta, bewa, catyrva vawr, for
1,000,000. The appearance of the Latin word catyrva
as a numeral is a very extraordinary circumstance, and
we are not aware of any hypothesis by which it may
be explained.
Of other Celic languages the Erse,t and its descend-
ant the Gaelic, have no native term beyond ciad, or
100; the expression for 1000 being deichciad, ten hun-
dred, or more commonly the Latin term mile. We
believe the same remark applies to the Armoric lan-
guage, and the Basque of Biscay.
In Rabbinical Hebrew only, we find the term ribbo
for 10,000, which is never found in any of the kindred
dialects. The term eleph, or alph, for 1000, as we
have before remarked, prevails very extensively;
being found, with slight variations, in Arabic, Persian,
Abyssinian, the ancient Punico-Maltese, and in many
of the languages in the north of Africa; and we very
frequently find it, as well as the term meah, for 100,
terminating the numeral systems of languages which
have no other terms in common.
In the Amharic, and some neighbouring dialects,
where the term alph has been misapplied to denote
10,000, we find likewise the term ilef denoting
1,000,000, a solitary example amongst Semitic lan-
guages of a term for so great a number.
Neither the Arabians nor Persians, though the nota-
tion by numeral figures possessing local value was
known amongst the former at least as early as the ninth
or tenth centuries, have attempted to make any addi-
tion to their numeral language; a circumstance which
may be accounted for, partly by the advanced state of
their literature at the period when it was first known,
and partly by the genius of those languages not admit-
ting the formation of terms like the millione and cuento
of the Italian and the Spanish : thus to express a
* Owen's Welsh Dictionary.
ºf Wallancey's Irish Grammar.
Erse.
Hebrew.
Amharic
Arabian
and Persian
A R. I T H M E T 1 C 379
Arithmetic. million, they are obliged to repeat the term for a thou-
^-N-' sand twice; a thousand millions, to repeat it three times,
Gothic,
North
American
numeral
systems.
and similarly for other numbers in the same series.*
We recollect in an old German author on Arithmetic to
have seen a similar expedient adopted to express the
number 10°, made use of by Archimedes in his
Arenarius, which is given as follows :
Ein tausend.
tau tau tau tau tau tau tau tau tau tau tau tau
tau tau tau tau tau tau tausend mahl tausend.t
There are many other examples of the formation of
expressions for superior units by the repetition of the
names for its factors, as often as they are contained
in it. In the Codeſt Argenteus, preserved at Upsal, and
which is a translation of the four Gospels made by
Bishop Ulphilas in the fourth century, into Moeso-
Gothic, we find taihun taihund, † or ten ten for 100.
In the language of the Knisteneaux, one of the principal
hunting tribes of North America, § who inhabit the
northern shores of Lake Superior, we find 100 ex-
pressed by mitana mitenah, or ten ten ; and 1000 by
mitana mitena mitanah, or ten ten ten. The Sapibo-
cones, a South American tribe, express 10, 100, 1000,
by tunca, tunca tunca, tunca tunca tunca, respectively:“I
such a mode of expression, indeed, is one of the most
simple and obvious expedients for denoting numbers,
which are not immediately within the compass of any
numerical language.
(17.) We shall find in general, that the numeral lan-
guages of the tribes in the central parts of North America
are more complete, both in structure and extent, than
could be expected from their low state of civilisation :
they are almost universally adapted to the decimal
scale, and in most instances extend as far as 1000.
The Algonquins a kindred tribe of the Knisteneaux,
speaking a dialect of the same language, and possess-
ing many numerals in common, have simple terms
ning outwack and kitchiwack, both for 100 and 1000.
The Hurons, once a numerous and powerful tribe,
living in Upper Canada, around the lake of that name,
who speak a language” singularly rude and inartificial,
without adjectives, abstract nouns, or verbs of action,
and incapable of expressing a negation, without an
absolute change of the word, possess a numeral
language sufficiently regular; the name for 10 being
* Chardin, Voyages en Perse, par Langlés, tom. iv. p 293.
+ Rechenbuch auf den Linien und mit zifern durch Simon
Jacob von Coburgk, Rechenmeister zu Frankfurt am. Mayn, 1559.
† Hickes, in his Thesaurus Linguarum Veterum Septemiriona-
Jium, considers our term hundred to have originated in the custom
df writing the last syllable hund of this expression only for
greater brevity, particularly when combined with other numbers.
From the same principle of abbreviation, we have got the term
thousand, contracted from taihun hund, or tigos hund, ten hundred.
The reader may see other etymologies of these words, many of
;them extremely absurd, in the Etymologicum Anglicanum of
Junius.
§ Dr. Richardson, in Franklin's Journey.
| Mackenzie's Journey to the North Sea, Introduction.
- Humboldt, Vues des Cordillères et des Monumens de l'Amerique,
p. 251.
** Monboddo, Origin and Progress of Language, p. 543. The
numerals are given in a very curious and rare work by a Francis-
can monk, G. Sagard, published in 1632, entitled Le Grand
Voyage des Hurons, situé en Amerique vers la mer douce es dernières
confins de la nouvelle France; with a dedication, “Au roy des
roys et taut puissant monarque du ciel et de la terre Jesus Christ,
Sauveur du monule,” written in a very quaint style, but describing
with considerable force and eloquence the efforts of the mission-
aries to bring these rude people under the dominion of Christ.
assou, for 100 egyo-tiuoissan, and for 1000 assou attevoig- History.
navoy. We shall find numeral systems equally com-
plete among the Iroquois, and the rest of the tribes of
Upper Canada; amongst the Indians on the Delaware,
and those who formerly occupied the neighbourhood
of New York ; amongst the ancient inhabitants of
Virginia; * and most of the tribes of Central North
America of whose languages we possess any records.
The decimal scale is much less generally prevalent The deci-
among the numerous tribes of South America than mal scale
among those of North, and their numeral systems lº in.
much less perfect, rarely proceeding beyond a hundred, South
and frequently limited to much smaller numbers: there America,
are not wanting, however, numeral systems adapted to
the decimal scale, which are sufficiently complete and
comprehensive ; but in most cases the names for
numbers, particularly for those which are compound,
are of such extraordinary length and complexity, as
to appear to exceed the powers of human utterance.
In one language, however, namely the Qquichua, or
ancient Peruvian, we find a numeral system equally
simple and more extensive than that of the Greeks or
Romans, as the following names or expressions for the
series of superior units will show:
10. Chunca,
100. Pachac.
1,000. Huaranca.
l O,OOO. Chunca huaranca.
100,000. Pachac huaranca.
1,000,000. Hunu.f
The New World is not without its examples also of
names for superior units borrowed by one people from
another more civilized than themselves : thus the
Molluches, a tribe who inhabit a district to the South
of Chili, have adopted the Peruvian term pataca for
100, and huaranca for 1000 ; though the languages
of these people, as well as their other numerals, have
nothing further in common. †
It is quite unnecessary to pursue this inquiry into
the extent and developement of numeral systems
farther, as the examples which we have adduced will
sufficiently demonstrate the truth of the assertions
which we made at its commencement. We shall now
proceed to the consideration of some peculiarities in
the expression of numbers, which illustrate in a very
striking manner the very regular and artificial manner
in which numeral language has in most cases been
constructed.
(18.) We have before noticed the method of expressing Numbers
some numbers, such as nineteen, twenty-nine, &c. by *.*
gº º ... • expressed
their defect from the next superior articulate numbers, ...-
which is usual in the Sanskrit, Greek, and Latin ; and ence to the
we shall find the same peculiarity in the Malay and next supe-
other languages. Thus, instead of saying sambilan ºr *-
püluh sambilan, or ninety-nine, $ they more frequently i. Ilkli Di-
use the expression körang asa sa-rátus, or wanting one "
of a hundred. The word sambilan, || or nine itself,
means one taken, that is, taken from the heap or
* Account of Virginia, by Captain Smith, 1624. Their names
for 100 and 1000 are of very formidable length ; for the first
being mecattaughysinough, and for the second, mecattweun-
guavagh. .
+ Humboldt, Wues des Cordillères, &c. p. 252.
f There is a grammar of the language of this tribe published
by Robert Falkner, an English Jesuit, who resided as a mis-
sionary in Patagonia for upwards of forty years.
§ Marsden's Malay Grammar, p. 39.
| Crawfurd's Indian Archipelago, vol. i. p. 256.
3 D 2
380 A R I T H
M E T I C.
Arithmetic, whole ; and sakörang, which in Malay means one
wanting, is the term for nine in the Achinese dialects.
Numerals The numeral language of the Oedh-Ostiaks, or
* Sable Fur Ostiaks, a Siberian tribe living on the banks
* of the Jenesei, exhibits this peculiarity of construction
in a very remarkable manner, and we shall therefore
give it, with more than ordinary detail, as follows:*
1. Chusem,
2. Ynem.
3. Dógom.
4. Syjem.
5. Chöjem.
6. Ahjem, or Chöjem-chusem, 5 and 1.
7. Ohnem, or Chôjem-ynem, 5 and 2. t
8. Chöjem-dògom, 5 and 3; or ynem botsch
chojum, 2 from 10.
9. Chöjem-syjem, 5 and 4; or Chusem botsche
chojum, 1 from 10.
10. Chojum.
11. Chusem chojum.
18. Ynem botsche agem, 2 from 20.
20. Agem.
50. Cholepky-scha.
70. Ohna-chojum.
80. Ynem botsche chojum chojum, 2 from IO
times 10. - :
90. Chusem botsche chojum chojum, 1 from 10
times 10. -
100. Kyschash, or ky.
1000. Chojum-kyschash, 10 times 100.
Such is the numeral language which we might ex-
pect to be formed by a people labouring under extreme
poverty of numeral words, and who endeavoured to
adapt them to a system of numeration previously
known.
The other tribes who inhabit the banks of the
Jenesei and its tributary streams, whose languages
constitute a distinct class, being intimately allied with
each other, but different from those of other Siberian
people, whether of the Samoeid, Tatar, or Mongol
race, possess numeral systems which are generally
formed in the same manner. T -
Kamts- The same construction is observable in the lan-
*..., guages of the Kamtschatkans, and the inhabitants of
i. jºie the Kurile Islands, which are opposite to the mouth of
Sla Il C. S. & g º ſº
the Amur, as will be readily seen from an examination
of their expressions for one, two, eight, nine, and
ten. :
Kamtschatka. Kurile Islands.
1. Syhnāp. vs. 1. Sinezb.
2. Düpk. 2. Zuzb.
8. Dühpyhs, 2 from 10. 8. Zujemambe, 2 from 10,
9. Syhnāppyhs, 1 from 10. 9. Sinesambe, 1 from 10.
10. Upyhs. 10. Fambe.
Other pe- (19.)There is another peculiarity in the construction
culiarities.
of numeral language, of very general prevalence both
in Asiatic and European languages, which we shall
now proceed to notice. Every student in Greek lite-
rature is acquainted with the phrase, apparently so
remarkable, of égéopov ºutta Mavrov, which, literally
translated, means the seventh half talent, but which in
* Klaproth, Asia Polyglotta, p. 171.
+ Ibid.
f Ibid. p. 315. La Perouse's Voyage, vol. ii. p. 85. English
edition.
all cases denotes sir talents and a half.” Vestiges . History.
of the same construction are observable in the Latin S-N-'
word sestertius, which is the contracted form of semis
tertius, and signifies two whole asses and a half, a meaning
distinctly expressed in its original symbol L.L.S. which
in later times became H.S. Of a similar description
is the Anglo-Saxon phrase, threo healf, or thridde
healfe, two and a half;t and the German, anderthalb,
for one and a half; viertehalb, for three and a half;
elfiehalb, for ten and a half; and similarly in other
cases. : 's
So prevalent was this mode of expressing numbers
amongst the ancient Cimbri and their Danish descen-
dants, that we find it combined with the vicenary
scale, for the expression of the alternate articulate
numbers between forty and a hundred, as will be im-
mediately seen from what follows:
10. Tie. -
20. Tyve.
30. Tredeve, 3 times I.O.
40. Fyrteve, 4 times 10. -
50. Halv tredie sinds tyve, half the third time 20.
60. Tre sinds tyve, 3 times 20.
70. Halv fierte sinds tyve, half the fourth time 20.
80. Fire sinds tyve, 4 times 20. -
90. Halv femte sinds tyve, half the fifth time 20.
10O. Hundrede. § .
We find examples of expressions precisely similar
to those for 50, 70, and 90, in the Icelandic language.
Thus halft fiorda hundrada means three hundred and
fifty ; and in expressing the age of a person, half way
between two articulate numbers, instead of saying
thirty-five, fifty-five, &c. they use the phrase halft
fertugr, which means half the fourth ten ; halft sextogr,
or half the sixth ten ; and similarly in other cases.| -
Exact parallels to such expressions are to be found
in the Malay, Javanese, and other Eastern languages.
Thus in the first of these languages, instead of dila
pilluh lima, or twenty-five, it is more usual to say tan-
gah tiga pilluh, or, literally, half of thirty; and similarly
for thirty-five, forty-five, fifty-five, and so on. Again,
for one hundred and fifty, they use the expression
tangah dila rātus, which is half of two hundred ; that is,
of the second hundred." In the same manner in Java-
nese, ewidak sasor, or half sixty, means fifty-five ; pitu-
sasor, or half seventy, means sixty-five ; and similarly in
other cases.** --
It is needless to add instances from other languages
of a mode of expression which is so common that it
hardly can be considered as peculiar, but which ex-
hibits evidence, in the latter cases at least, of that
constant reference to the articulate numbers, which is
so generally characteristic of numeral language.
(20.) The mode of expressing numbers intermediate to
articulate numbers, in the language of Lapland, is very
peculiar and very significant ; “the first ten numerals
In Danish
numerals.
Icelandic.
In Javanese
and Malay.
Expres-
sions for
compound
numbers.
* Matthiae's Grcek Grammar, p. 176.
† Hickesii Thesaurus Linguarum Septentriomaliun, Gramma-
tica Maeso-Gothica, p. 33. -
† Noehden's German Grammar, p. 198.
§ Parson’s Remains of Japhet, c. x. p. 317. .
|| Hickesii Thesaurus : Grammatica Islandica, p. 42. Hickes
says that the Scotch, when asked the hour of the day, instead of
saying half past nine, half past eleven, prefer answering it is half
ten, it is half tunelve ; and he considers this mode of expression
as a vestige of the Danish dominion in that country.
* Marsden's Malay Grammar, p. 40.
** Crawfurd’s Indian Archipelago, vol. i. p. 268.
© A R. I. T H M E T I C.
381.
Arithmetic, are given above, (Art. 10:) to express 11, they say
auft nubbe lokkai, which is one to the second ten ; for 12,
In the * gouft nubbe lokkai, two to the second ten ; for 23, golm
#. goaalmad lokkai, three to the third ten. They proceed in
this manner, combining the cardinal with the ordinal
numbers, as far as zhioette, or 100, which is the limit
of their numeral system.* z -
In the numeral language of the Knisteneaux, the
numbers from 10 to 20 are expressed by the first nine
numerals with the preposition osap, or with, the term for
ten being omitted.t
Of the Knis-
teneaux.
1. Peyac. 11. Peyac osap.
2. Nishew. 12. Nishew osap.
3. Nishtou. 13. Nishtou osap.
4. Neway. 14. Neway osap.
5. Ni-annan. 15. Niannan osap.
6. Negoutawoisic. 16. Negoutawoisic osap.
7. Nishwoisic. 17. Nishwoisic osap.
8. Jannawew. 18. Jannawewosap.
9. Shack. 19. Shack osap.
10. Mitatat. 20. Nishew mitenah.
The expression for 21 is nishew mitenah peyac osap,
the omission of the preceding articulate number being
no longer allowable, on account of the ambiguity
which it would occasion.
In Malay In the Malay and Javanese languages the expres-
and Java- sions for numbers between 10 and 20, as may be seen
InêSC,
from our list of their numerals, are formed by adding
to the digit the particle blas in one case, and wiilas in
the other ; probably identical with the Javanese term
tälas, which means done or finished, that is with re-
ference to the end of the scale. For numbers beyond
20, the expressions are formed in the regular way, ex-
cepting those cases which are included in some of the
peculiarities above mentioned, or in which absolute
terms, the remains of former methods of numeration,
are used : thus lawe is used to denote 25, and denotes
also a thread or string ; and sekút or ekút, which means
a skein of thread, also denotes 50 ; swidak, a term of
unknown derivation, is used for 60 ; and samas and
domas, denoting respectively ‘‘ one bit of gold and two
bits of gold,” are used, the first for 400 and the last
for 800.
Our term eleven, and the Anglo-Saxon endlufon
means leave one, that is above ten, the point from
which the numeration commences again as it were
anew ; and in the same manner twelve means leave
two, with reference to the same number; beyond this
numbert the terms are formed in the way which is
usual in most languages, by the combination of
the nine digits with the preceding articulate number;
and this departure from a very general rule in the
expression of these two numbers, which is observable
in all languages of Gothic origin, is, as far as we
know, peculiar to them. It must be considered,
however, as a variation and not as a violation of a
general principle; the point of departure from which
the numeration recommences being equally kept in
view in both cases.
It might be imagined that this distinction in the
formation of the expression for eleven and twelve, had
its origin in the frequent use of the latter number
amongst Scandinavian nations.' Thus amongst the
In English.
* Knud (Canutus) Leems, De Lapponibus Finmarchiæ.
+ Mackenzie's Travels, Introduction. :
† Junii Etymologicum Anglicanum, on word eleven.
§ Hickesii Thesaurus : Grammatica Islandica, p. 43.
inhabitants of Iceland and Norway, the addition of History.
the word tolfraid (that is duodena ratio) to the symbols S-N-"
or expressions for ten and a hundred, made the one .
signify twelve units, and the other twelve decads, and . .
similarly for higher numbers: thus, CC vatra tolfraid, amongst
or ducenti anni tolfraid, means 240 years ; CCC daga Scandina-
tolfraid ac fim dagar, or three hundred days tolfraid and Yian na-
five days, means 365 days, and similarly in other cases. *
Traces of this preference of the number 12 amongst
ourselves, as well as amongst other Gothic nations,
are to be found not merely in the very frequent use
of the term dozen in the classification and parcelling
out of many objects of barter and trade, but likewise
in our primary divisions of money, weights, and mea-
sures. In some cases, even the technical meaning
attached by merchants to the word hundred, associated
with certain objects, is six score; a usage which is
commemorated, though perhaps in too sweeping and
general a form, in the popular distich,
Five score of men, money, and pins,
Six score of all other things.
Though the influence of this division by twelve upon
the customs and languages of northern nations is very
remarkable, yet it hardly can be considered as indi-
cating the existence of a duodenary scale of notation,
properly so called ; for, in the first place, the name
for twelve is dependent upon the radix of the decimal
scale; and in the second place, though there is a simple
name gross for 12° or 144, yet in no case is that number,
or even the former considered as an articulate number,
or as a point of departure for a new numeration. The
partition indeed of numbers and concrete units by 12,
probably suggested in the first instance by the natural
divisions of the year, is of very general use ; but has
no natural connection, in its origin at least, with the
methods of classifying whole numbers : it being a
refinement long posterior to the formation of numeral
systems, to consider an abstract unit as capable of
division at all, and still less that the results of such
successive divisions should constitute a series of
inferior units, admitting of classification in the same
manner as abstract whole numbers themselves.
There are few other circumstances in the formation
of expressions for compound numbers as distinguished
from those which are articulate, which deserve to be
remarked. In no one respect is the general economy
of numeral language more strikingly exemplified, than
in the terms for such numbers; for we not only hardly
ever find two names for 11, 12, 13, and so on, but in
no one instance do we find them expressed by an
arbitrary and independent name, that is by a name which
has no reference to the radix of the scale of numera-
tion; a proof amounting nearly to demonstration,
that words have been expressly adapted to such scales
and are consequently subsequent to them.
(21.) The names of the articulate numbers are usually Formation
formed by the incorporation of the term, for the radix of expres-
of the scale, with the names of the nine digits; and . º
in almost all cases the etymology of such names is .
sufficiently obvious. We frequently, however, find
two names for 20, one of them arbitrary and inde-
pendent, and the other adapted in the usual manner
to the decimal scale; the former are very generally
vestiges of the vicenary scale, which has been super-
seded by the denary, and will be noticed hereafter ;
the latter commonly admit of a resolution into their
382
A R I T H M E T I C.
Arithmetic.
\---"
In Greek.
In Latin,
Turkish
and Mand
scheu
numerals
component parts, as readily as the terms for the other
articulate numbers. Thus, in our own language, score
is a term of the former kind, reminding us of an
ancient and extinct method of numeration ; whilst
twenty is immediately derived from the Gothic twentig,
compounded of twa and tig, the latter signifying ten
equally with taihun,” and generally used in preference
to the latter in all compound words, n
The Greek word eicoatt seems to defy all probable
etymology, and may therefore most properly be re-
ferred to the class of arbitrary terms; whilst the terms
Tpad coura, reaſoapákovta, &c. are regular and simple in
their formation, though the origin of the term kovita
for ten, corresponding to the Latin ginta, in triginta,
quadraginta, &c. is extremely difficult to explain. The
Latin word viginti is equivalent to biginti, or twice ten,
and is not derived, as some authors have imagined,
from the Celtic term ugent, or ugain, for the same
number. :
An accurate etymological examination of the expres-
sions for articulate numbers in different languages, would
frequently lead to results of great interest, not merely
as exhibiting traces of ancient methods of numeration,
but likewise as showing the limits to which they have
proceeded, or the changes which they have undergone.
In many cases, however, the etymologies of such
words are extremely difficult, exhibiting very obscure
traces of the digital numbers merely, with no discover-
able reference to the radix of the scale; and in others,
they may be considered as arbitrary and independent
terms, which it is impossible in any way to connect
with any system of numeration.
In the Oigour, of the elevated plain of Turfan,
the most pure of the numerous class of Turkish
dialects, and in the Mandscheu, one of the principal
of the languages denominated Tungusic, we shall find
examples which illustrate these observations, as will
be seen from the following list of their numerals :
Oigour, or Eastern Turkish.Ş Mandscheu.}
1. Bir. 1. Emu.
2. Iki. 2. Dschus.
3. Utsch. 3. Ilan.
4. Töst. 4. Duin.
5. Bisch. 5. Sundscha.
6. Alty. 6. Ningan.
7. Yidi. 7. Nadan.
8. Sekis. 8. Dschakūn.
9. Toehus. 9. Ujun.
IO. On. 10. Dschuan.
20. Igirmi. 2O. Orin.
30. Otus. 30, Gutschin.
40. Chirch. 40. Dechi.
50. Ellik. 50. Sousai.
* Our word ten is derived from the word taihun or tehan, or
perhaps from the old German word (Franco-Theotiscan,) zehen, to
draw, i.e. one from the heap or number ; and the participle tig
or tigos in one case, and zogh or zug in the other, are used in
compound terms, as in those for 20, 30, &c. which signify drawn
twice, drawn thrice, and so on ; thus sibontig and sibonzug are
used indifferently in ancient German for 70, in which language
we also find zehenzogh, or zehenzug for 100, equivalent to taihun
taihund, noticed above.
f Jamieson's Hermes Scythicus, p. 199.
: Parson's Remains of Japhet.
$ Klaproth, Sprache und Schrift der Uiguren, Paris, 1820;
Asia Polyglotta, p. 214. Remusat, Récherches sur les Langues
Tatares, p. 267. /
| Klaproth, Sprachatlas; Asia Polyglotta.
Nindscheu.
Nadandscheu.
60. Altmisch. 60.
7 O. Yitmisch. 7 O.
SO. Sekis on. 80. Dschakſindscheu.
90. Tochus on. 90. Ujundscheu.
100. Yus. 100. Tanu.
1000. Ming. 1000. Mingan. Asº
10*. Tâmen.
10°. Kuldy.
IO6. Niut. d
In the first of these systems, the names for 20, 30,
40, and 50, have no common principle of formation,
and with the exception of the first may be considered
as perfectly arbitrary : those for 60 and 70, involve
the names of the digital numbers 6 and 7, without any
apparent reference to the radix of the scale; whilst
those for 80 and 90 are formed in the ordinary manner.
The name for 100 prevails not merely amongst all
Turkish tribes, but has likewise been borrowed by
some Siberian people, * who speak languages belong-
ing to an actually different class; whilst the term ming
for 1000 has been communicated not merely to the
Mandscheu, but to the Mongol and all Tungusic lan-
guages, from one extremity of the continent of Asia to
the other, though their numeral systems have nothing
more in common. The other terms, as far as a million,
are apparently arbitrary, and certainly native ; there
being no terms for such high numbers amongst any
neighbouring or kindred nations.
In the second system of numerals which we have
given above, the names for 20, 30, 40, and 50 are
arbitrary, whilst those for the subsequent articulate
numbers are formed in the ordinary manner. In the
other Tungusic dialects, we generally find the greatest
regularity in the formation of their numeral systems;
the names for the articulate numbers being formed by
the combination of the name for ten with that of the
digital number, excepting in the one which follows
from the dialect of Nertschinsk, where we find the
Mandscheu names for 20 and 30, and all the others,
expressed by a modified form of the names of the
corresponding digits.t
. Omin.
. Dschur.
. Ilán.
. Dygin.
. Dschön.
. Orin.
. Gotin.
. Dyginni.
. Tonna. 50. Tonnanni.
Njanun. 60. Njannunni.
. Nôddan. 70. Nodanni.
. Dschöpkun. 80. Dschöpkunni.
. Jagyn. 90. Jaginni.
The same principle of formation of the expressions
for articulate numbers is observable in the Semitic and
many Asiatic languages. Where examples are so
numerous, we shall content ourselves with the follow-
ing list of Mongol numerals, which are found with
slight variations amongst all Tatar nations, from the
Wolga to the Wall of China.f
. Nige.
Gojer.
. Churban,
. Dürbän.
. Taban.
. Djirohn.
7. Dolohn.
i
8. Naiman.
9. Jisun.
10. Arban.
20. Chorin.
30. Chutschin.
* Klaproth, Asia Polyglotta, p. 159.
+ Ibid. Sprachattas.
: Ibid. Asia Polyglotta, p. 284.
History.
"-N-
Nerts-
chink nu-
merals.
Mongol
numerals.
A R I T H M ET I C.
383
sº
Arithmetic, 40, Dütschin 80. Najan.
\-y-Z 50. Tabin. - 90. Jaran.
60. Djiran. 100. Djan.
7 O. Dalan.
Systems (22) We could very easily extend to a much greater
adapted to length our observations upon numeral systems adapted
the quinary to the decimal scale, as there are few cases in which
*.*.* they may not be made the foundation of some remarks
ry scales. • ' gº * & *
• 2 ºn of interest and importance, illustrative of general
principles concerned in their formation; but the limits
to which we are confined by the very nature of this
work, compel us to bring them to a conclusion. We
shall now proceed, therefore, to the consideration of
the other natural scales of notation, the quinary and the
vicenary, which, though incomparably less generally
prevalent than the denary, yet are very frequently met
with amongst savage and rude people, and sometimes
also amongst people considerably advanced in the arts of
life; and even amongst civilized people we find traces
of their former existence, though subsequently they
have been partly or wholly superseded by systems
adapted to the decimal scale.
“Aristotle,” says Sir Thomas Herbert, “not with-
out good reason admired, that both Greeks and bar-
barians used a like numeration unto ten; which, seeing
it was so universal, could not rationally be concluded
accidental, but rather a number that had its foundation
in nature.” The passage of the Greek Philosopher, to
which this admirable old traveller refers, is found in
his Problems ; and is in every respect so curious, and
contains so correct a description of what constitutes a
! scale of numeration, that we shall give it entire :
Reasons ac- Ataré Távtes dutpwrot kai Bápéapot kai éNNºves, eis Tà
cording to Śēka katapuduo?gu, kai oilk els áNNov ćpioudu, otov B, Y,
sº º f º * fy p g y
Aristotle of 8, e. e?ra TráNtv ćrava&m Nodaw, Čv Trévre, 8vo Trévre,
the univer- .
ey tº V * S > * º f f *
tº ūatrep &včeka, buièeka oijä’ av čğwtépw Travo duevot Tüºv
sality of the . º y r. e ex * y • */ an
decimal 66ka, eita éke?0ev étrava&TNoogºv" &ott Yap Éraotos Tüv
asº ry rs
apudućv č, ćutpoooev, kal &v # 8vo, kat eit’ &\\os tés :
&pténoga, 6’ &ntos épio'avtes àxpt Túv 6éka oi Yāp 8)
dTo Tüxms ye āvto ‘paivovtat, kai
*. R → * * * * y s y \ f 5 V - w
Tö 86 dei kai éiri Tävtwy, oùk &To Twyms, d\\d. Øvoukov.
seale.
Toto ovtes del :
IIórepov 8t, Tă 6éka Tévetos épt0aos, éxwv Yàp Távta rā.
too diplôa.00 eið), āption, Tepitrov, tetpdºwvov, küşov,
Aïcos, Čzárečov, Tptºrov giveerov; ) 6tt apx" | 8ekās ;
év Yāp ka? 8vo kai tpia ca: Têttapa, Yêvetat &eicās' # 6tt
tà 'pepdueva gºuata évvea; ) 6tt év čáka čva)\otićats
7érrapes kigucot àptówo: āroreWoovrat &# ſov (pāot
àpièuſºv ći IIvøayópelot to rav avveotávat ; # 8tt Tävres
in fibéav &v6pwrot &xovres 6éka ŚaktüNovs; otov ovu
Tràºovs éxovtes to 0 oikstov čplēuo?, rövtw tige TM#6et
kai TâNNa āpubuogat.t
The universality of the decimal scale proves, ac-
cording to Aristotle, that its adoption was not acci-
dental, but had its foundation in some general law of
nature: to 6é ač, Kai émi Távtwv, oùk atro Túxms, a\\2.
ºpworticov. This is a most philosophical principle of
reasoning, which leads in the present instance to the
correct conclusion, notwithstanding the Pythagorea:
and Platonic dreams about the perfection and proper-
ties of the number ten, which are thrown out as con-
jectures to account otherwise for its general adoption.
J3ut were there any traces in the time of Aristotle, in
the Greek language itself, (we speak not of others,)
* Some Years Travels into Africa and Asia the Great, &c. 1677.
* Apia rot exovs IIgoğAmplatov Tampa Togº te,
of the quinary scale, a case to which he alludes 2 We History.
shall state some reasons for answering this question in S——
the affirmative. -
In the Odyssey of Homer we find the word repara-Traces of
Çeoréat, to count by fives (quasi per quinos digitos,) used the quina-
as equivalent to apúnetº... Calypso, speaking of *
Proteus, making the tale of his phocae, says, the Greeks.
qubicas wév to Tpútov ćpt0affoet ka? &retaw
Aöråp &mijv rāgas repairdogetat, #6é 67tat
Aéºétat €v uéogotas, vogels ēs ºrtheat uſixtev.
* Oèvoro. 6'. 4II.
The familiar use of this word, whose derivation is
so very obvious, would seem to indicate that the
method of counting by fives was common, at least in
the time of Homer ; and the introduction of the same
word by Apollonius in his Argonautics,” would prove
that the use of it had continued in the poetical, if not
in the ordinary, language of Greece to a much later
period. -
But we have other evidence besides the existence of
a word, to show their tendency at least to follow this
quinary classification of numbers. In ancient Greek
inscriptions, (and some authors assign an antiquity to
this practice as remote as the laws of Solon,)f we have
5 and IO expressed by II and A, the initials of the
words IIevte and Aeka ; 50 was denoted by inscribing
the A within the II; and 500, by inscribing within it
H, the initial of Hekarov. In other respects this sym-
bolical notation corresponded entirely with the Latin,
and in common with it constituted a system for the
representation of numbers, which might be considered
as quinary subordinate to the denary. With the
Greeks this rude method of notation was superseded,
except for inscriptions, at a very early period by the
more perfect system derived from the Hebrews; but
with the latter it remained unchanged to the end of
their empire.
We can discover no other trace of the existence of
the vicenary scale amongst the Greeks and Romans, and Palmy-
either in their numeral language or symbols. If, rene nume-
however, we refer to the East, from whence their ral symbo”
alphabets originated, we shall find amongst the Phoeni-
cians a system of numerals, first ascertained by Dr.
Swinton; from coins found at Sidon, which possess
simple symbols for ten and twenty ; by the latter of
which they proceed as for 100. An examination of
Palmyrene inscriptions § furnishes likewise a system
of numerals of great extent, with simple symbols for
five, ten, and twenty ; but in other respects intimately
allied with the former, and proceeding like it according
to the vicemary Scale, within the same limits. The
reader will find both these systems in Plate I. Nos. 2
and 3, which are of great interest, not merely from
their analogy to the Roman numeral symbols, but
likewise as furnishing the key to the numeral systems
of the Celtic nations.
The intercourse of the Phoenicians with Spain,
Cornwall, and Wales, and more particularly with
Romans,
Phoenician
* Speaking of the streams which flow from the Thermodon,
he says, -
Terpäicis eis Eicarov 8sbotto kev, eit is karra
IIeutra; ot. Ap'yovaurikov, 8.976.
+ Gatterer, Artis Diplomatica, Elementa, p. 64. Beveridge,
Arithmetices Chronologicae, lib. i. 1705; Rose, Inscriptiones Graeca.”
vetustissimae. -
# Philosophical Transactions, 1758, p. 791.
§ Ibid. 1754, p. 690.
384
A R I T H M E T I C.
over twice twenty; and similarly in other cases. The History.
origin of this solitary vestige of the quinary scale in 's-N-
Arithmetic. Ireland, is an historical fact attested by innumerable
>~~' monuments; and the general affinity of structure be-
. * tween the Celtic and Semitic languages, however this class of languages is extremely difficult to explain,
i."... altered by subsequent intercourse with other people, is unless we suppose that their primitive methods of nu-
nary. of all monuments of their ancient communication with meration were quinary, subordinate to the vicenary,
each other, the most permanent and unquestionable.
Amongst all the nations of the Celtic race, the numeral
language is constructed in conformity with the Phoe-
nician numerals, proceeding by twenties as far as 100,
and no farther. The following is a list of Welsh, Erse,
and Gaelic numerals: •
and that this was a monument of the resistance made
by popular habits or prejudices to the partial introduc-
tion of the denary scale, from a people more civilized
than themselves.
The numeral systems in the Armorican and Basque
languages possess a general conformity with those
above given, as a small number of their numerals will
Welsh. Erse. Gaelic. º
1. Un. 1. Aon. 1. Aon. readily show :
2. Dau. 2. Do. 2. Da. Armorican. Basque.
3. Tri. 3. Tri. 3. Tri. 1. Unan. 1. Bat.
4. Pedwar. 4. Ceatair, or 4. Ceithar. 2. Daou. 2. Bi.
ceitre. 3. Tri. 3. Iru.
5. Pump. 5. Cúig. 5. Coig. 20. Hugent. 20. Oguei.
6. Cwec. 6. Sè. 6. Sia. 40. Daou hugent. 40. Berroguei.
7. Saith. 7. Scact. 7. Seachd. 60. Tri hugent. 60. Iruroguei.
; Nº. ; º 8. gºl The first of these systems resembles the Welsh, a
... IN &lll. 9. Naol. 9. Nai. language with which the Armorican is closely allied :
10. Deg. 10. Deic. IO. Deich. the second, though differing considerably from the
11. Unarzeg. !. Aon deag. J.Agn leag. former, yet possesses a greater analogy to it than
15. Pymtheg. 15. Quig deag. 15. Coig deag. could be expected from the peculiar and insulated
16. Unarpyintheg. 16. Seact deag. 16. Sia dºg nature of this language, so difficult to associate even
20. Ugain or 2O, Fitce. 2O. Fichid. with the Celtic languages, and still less with those of
ugaint. . G any other class. -
21. Un arugain. 21. Aon is fitce. 21. º thar The vicenary scale appears to have prevailed very Amongst
e !”. extensively amongst Scandinavian nations, if we may Scandi-
30. Deg ar u- 30. Deic ar fi- 30. Deich thar judge º the º vestiges of it, not º; navian
gain. cead. - fichid. amongst them, but likewise amongst those people *
36. Unarpym. 36. Sºº, deag 36. Sia deg whose languages are partly derived from them. We
theg ar u- is fitce. * * have before noticed the curious construction of the
gain. . chid. ſº Danish numerals between 40 and 100, adapted to this
40. Deugain. 40. Da fitcead. 40. Da fichid. system ; and also the preference given to the numbers
50. Deg ar deu- 50. Deic is da 50. Deich thar tºols, and twenty by the inhabitants of Iceland. In
gain. fitcead. da fighid, our own language also, the word score, which origi-
60. Trigain. . . 60. Tri fitcead. 60. Tri fichid. nally meant a notch or incision, has become equivalent
70. Deg ar tri- 70. Deic is tri 70. Deich that to twenty, a long mark being made on a tally to sig-
gain. fiteead. *...* nify the successive completion of such a number;
80. Pedwar u- 80. Ceitre fit- 80. Ceithar fi- a plain indication that such a mode of scoring* or
gain. cead. . chid. counting, was of all others the most familiar to the
90. Deg ar Ped- 90. Deie is cei- 90. Peigh thºr habits of our ancestors. In expressing numbers be-
war ugain. tre fitcead. º * yond 40, though we do not copy the banish form of
inia. . . expression for 50, 70, 90, yet in popular language we
100. Cant. 100. Cead. locº º, . readily say three º i.". 9 i. and
g ſº ; :---- ten than seventy, four score than eighty, and so on, par-
IOOO. Mil. * 1000. Mile.t. 1000. Pºiº ticularly º # numbers are ſº in j,
* +
All these systems possess much of a common cha-
racter, and the two last are nearly identical; a circum-
stance which might be expected, as the Gaelic is a
mere dialect of the Erse and an immediate descendant
of it. Amongst the Welsh numerals we find a pecu-
liarity, without any corresponding example in any
other Celtic dialect; which consists in making pymtheg
(15) an articulate number, and a point of departure
for a new numeration : thus 16 is un ar pymtheg, one
over fifteen ; 17 is dau ar pymtheg, two over fifteen ; 38
is tri ar pymtheg ar ugain, three over fifteen over twenty ;
59 is pedwar ar pymtheg ar deugain, four over fifteen
* Owen's Welsh Grammar and Dictionary. -
t Vallancey’s Irish Grammar. Neilson's Irish Grammar.
* Shaw's Analysis of the Gaelic Language. -
a manner, as to be frequently and familiarly used by
the humbler and less latinized classes of society. The
French have given a still more striking proof of
the influence of national habits of thinking and
acting upon language; they have made soirante a
point of departure for a new system of numeration by
twenties, expressing 70 by soirante div, 80 by quatre
vingt, and 90 by quatre vingt dia, instead of septante,
octante, nonante, the terms which sometimes have been,
and which in conformity with the general analogies of
the language should be used to express those numbers.
* Amongst other reproaches to Lord Say, which Shakspeare
has put into the mouth of Jack Cade, it is said, “and whereas,
before, our forefathers had no other books but the score and the
tally, thou bast caused printing to be used : and contrary to the
king, his crown and dignity, thou hast built a paper mill.”
Henry VI. Second Part. s
A R I T H M ET I C.
385
Arithmetic. The examples which we have given, are not the
S-,- only ones in which the decimal scale has not entirely
Other in- succeeded in obliterating all traces of the primitive
stances, existence of quinary and vicenary systems of nume-
ration, which are so extensively used amongst people
in a rude state of civilisation. The Persian term
pendje signifies five, and pentcha, the eananded hand;
and the corresponding terms in the Sanskrit are said
to have a similar meaning. The term lima, which
with very slight modifications is used for five through-
out the Indian Archipelago and the Islands of the South
Sea, means hand in the language of the Celebes, For-
mosa, Otaheite, and many other Islands. Among the
ancient Javanese numerals, we find very distinct traces
of both these scales; for besides the Sanskrit term poncho
for five, we find also a simple term lawe for twenty-
five, the only instance with which we are acquainted
of a secondary articulate number in the quinary scale,
it being usually superseded before it reaches that
point by one or other of the other natural scales,
again, in the same ancient dialect, we find likor, an
arbitrary term for twenty, which is frequently used in
expressions for compound numbers; and also terms for
two secondary articulate numbers in the vicenary scale;
namely, sa-mas, one four hundred, do-mas, two four
hundred, a circumstance of rather unusual occurrence :
the only instance of a ternary articulate number in this
scale, is to be found in the Azteck, or ancient language
of Mexico.
The following numerals in the Ende language, a
dialect of the Flores in the same group of Islands,
shows the operation of the same principle in their
formation, though partly derived from the ordinary
Polynesian numerals.”
Sa 7. Limazua.
. Zua. 8. Ruabútu.
Télu. 9. Trāsa.
. Watu. . Sabiálu.
Lima. . Buluzua.
6. Limasa. . Sang'asu.
Ende
numerals.
:
The terms for six and seven, are equivalent to five one,
five two, in strict conformity with the quinary scale;
the term for eight is two four, a remarkable circum-
stance, which ought rather to be attributed to the
poverty of the language of a rude people, who felt
great difficulties in the numeration and expression of
very small numbers, than to any natural tendency to
proceed by the quatenary scale.f
(23.) In examining the numerals of the islanders of
the South Seas, we shall find that they very generally
exhibit traces of a Malay origin, and that in some cases
the denary scale has completely prevailed, and super-
seded the other natural scales ; of this kind are the
numerals of the Friendly or Tonga Islands, which are
otherwise remarkable for their great extent ; in
general, however, we shall find that their systems of
numeration are denary, subordinate to the vicenary,
as may be seen from the following numerals of Ota-
heite : {
1, Tahai.
2. Rua.
Numerals
of the
Islands of
the South
Seas.
3. Torou.
4. Ita.
In Otaheite,
* Raffles, History of Java, vol. ii. App. F.
+ Crawfurd's Indian Archipelago, vol. i. p. 256.
# Monboddo, Origin and Progress of Language, vol. i. p. 544;
Cook's Voyages.
WOL. I.
5. Rima. 30. Tahai-taou-mara-hourou. History.
6. Wheneu. 32. Tahai-taou-ma-rua. \-N--
7. Hetu. 40. Rua-taou.
S. Warou. 50. Rua-taou-mara-hourou.
9. Iva. 60. Torou-taou.
10. Hourou. 80. Ita-taou.
ll. Ma-tahai. 100. Rima-taou.
12. Ma-rua. 200. Aou-manna.
20. Tahai-taou. 2,000. Manna-time.
21. Tahai-taou- 20,000. Torou-time.
mara-tahai.
The expression for eleven means one more, for twelve
two more, and so on as far as twenty, which is the
true basis of their numeral system. The names for
200, 2000, 20,000, were given by Sir Joseph Banks to
Lord Monboddo, and would indicate the resumption
of the denary scale beyond 200. But Forster,” the
most judicious and philosophical of the observers of
the South Sea Islanders, declares that the teachers alone
can count as far as 200, and that few others can pro-
ceed beyond 10; we shall hereafter notice many
examples of powers of numeration which are equally
confined.
The inhabitants of Otaheite and the Society Islands,
the Sandwich Islands, the Friendly Islands, the Mar-
quesas, the Easter Islands and New Zealand, New Gui-
nea, and other Islands in the neighbourhood, belong to
the same race, and possess nearly the same numerals,
at least for low numbers, differing chiefly in the extent
to which the decimal scale has superseded the vice-
nary. If we turn our attention to the different and
less favoured race who inhabit New Caledonia, Tanna, In New Ca-
Mallicollo, and the other Islands of the New Hebrides,t ledonia, &c.
we find a difference in their languages and numerical
systems, which are chiefly quinary, as will be seen
from the following examples :
New Caledonia. Tannã. Mallicollo.
l. Pārai. 1. Rettec. 1. Thkai.
2. På-röo. 9. Carroo. 2. Ery.
3. Par-ghen. 3. Kähär. 3. Erey.
4. Par-bai. 4. Kafā. 4. Ebats.
5. Pā-nim. 5. Karirrom. 5. Erihm.
6. Pânim-gha. 6. Ma-riddee. 6. Tsukai.
7. Pânim-roo. 7. Ma-carroo. 7. Goory.
8. Pánim-ghen. 8. Ma-kāhār. 8. Goorey.
9. Pânim-bai. 9. Ma-kafā. 9. Goodbats.
10. Pärooneek. 10. Karirrom- 10. Seneatn.
karirrom.
In the first of these systems, six, seven, eight, and
nine, are expressed by five one, five two, five three, and
five four ; in the second, by more one, more two, more
three, more four ; in the third, by the combination of
the word goo, of which we do not know the meaning,
with one, two, three, four ; in the second, ten is ex-
pressed by the repetition of the term for five, an
example of which we recollect to have seen some-
where in the numerals of a tribe in Africa. In every
respect, indeed, the formation of these quinary
systems, as far as they proceed, is as regular and sys-
tematic, as any of the denary systems which we have
examined; and are equally, if not more completely
derived from practical methods of numeration.
Labillardière f has given the numerals of New
* Observations made during a Voyage Round the World, by
John Reinhold Forster, p. 528. •
† Ibid. p. 284.
3. Voyages d'Entrecasteau.r, vol. ii. App.
3 E
386
A R I T H M E T I C.
Arithmetic. Caledonia as far as forty, though it is evident from an
Numerals
in North-
eaStern
Asia,
In Kamts-
chatka.
examination of them, that they are little more than a
repetition of the first ten numerals ; the form also
under which they appear in his work, is so very dif-
ferent from that given above, that it is extremely dif-
ficult to recognise in them a common character,
farther than that of being adapted to the same scale;
an instance, amongst a thousand others which might
be produced, of the impossibility of forming correct
vocabularies of languages, by persons who have not
been habituated, from long intercourse, with the
native sounds.
(24.) We shall find many examples of numerals adapted
to this scale amongst the miserable tribes who inhabit
the north-eastern parts of Asia. Of the following ex-
amples, the first are the numerals of the continental
Roriaks to the north of Kamtschatka ; the second, of
the Koriaks of the Island of Karaga ; the third, of
the Tschutki, on the Anadyr, who inhabit the western
part of the north-eastern angle of the continent of
Asia.”
1. Onnen. 1. Ingsing. 1. Innen.
2. Hyttaka. 2. Gnitag. 2. Nirach.
3. Ngroka. 3. Gnasog. 3. N'roch.
4. Ngraka. 4. Gnasag. 4. N'rach.
5. Myllanga. 5. Monlon. 5. Myllygen.
6. Onnan-myl- 6. Ingsinagasit. 6. Innan-mylly-
langa. gen.
7. Njettan-myl- 7. Gnitagasit. 7. Nirach-myl-
langa. lygen.
8. Ngrok-myl- 8. Gnasogasit. 8. Anwrotkin.
langa. -
9. Ngrak-myl- 9. Gnasagasit. 9. Chonatschin-
langa. - ki.
10. Myngytkan. 10. Damalagnos. 10. Myngyten.
Of these numeral systems, which possess much of
a common character, the first is formed in the most
regular manner ; in the second the name monlon for
five is replaced by gasit in the compound words ; in
the last, the expressions for numbers according to the
quinary scale, is interrupted after 7, and 8 and 9 are
expressed by words which have no connection with
those which precede them ; in all these cases the
name for ten is an independent word; in these in-
stances, as well as in many which will follow, we are
deprived of much interesting information respecting
the methods of numeration of these primitive people,
by our entire ignorance of the etymology and gram-
matical construction of their languages.
The following numerals of the inhabitants of the
north and south of the peninsula of Kamtschatka are
remarkable, as the names for 8 and 9 alone are
adapted to the quinary scale, whilst those for other
numbers, with the exception perhaps of that for 7, in
the first decad, are apparently independent.
I. Konni. I. Dischak.
2. Kascha. 2. Kascha.
3. Tschok. 3. Tschook.
4. Tschak. 4. Tschaaka.
5. Koshleh. 5. Kumnaka.
6. Kylkoch. 6. Ky’lkoka.
7. Ngtonok. 7. Itätyk.
8. Tschook-tonok. 8. Tschookotuk.
* Klaproth, Sprachatlas, '56,
9. Tschaak-tonok. 9, Tschaktuk.
10. Tuta. 10. Kumechtuk.”
If the following account of the method of counting
of these people be correct, it would appear that they
adopt the method which would naturally lead to the
vicenary scale, and which in every instance may be
considered as its foundation. “It is very amusing to
see them attempt to reckon above ten : for having
reckoned the fingers of both hands, they clasp them
together, which signifies ten : they then begin at their
toes and count to twenty ; after which they are quite
confounded, and cry matcha, that is, where shall I take
more.”f
(25.) The Greenlanders, the Esquimaux, the inha-
bitants of Norton Sound, of the Aleutian Islands, of
Kadjak and the other Fox Islands, and of the sea coast
of the north-east angle of Asia, bordering on the
Tschutki of the Anadyr, constitute a distinct and com-
mon race, who may be properly termed Polar
Americans, equally remarkable for their very limited
powers of numeration, and for the extreme poverty of
their numeral language. The Greenlanders, according
to the relation of the Moravian Missionary Crantz, ;
who resided for many years amongst them, in count-
ing commence with the fingers on the ſeft hand, and
thence proceed to those of the right, naming the first
ten numerals as follows:
1. Attausek.
2. Arlaek.
3. Pingajuah.
4. Sissamat. 9. Sissamat.
5. Tellimat. 10. Tellimat, or Kollit.
They afterwards proceed to the toes of the feet,
and the second series as far as 19 are expressed as
follows :
6. Arbennek.
7. Arlak.
8. Pingajuah.
11. Arkanget. 16. Arbasanget.
12. Arlack. 17. Arlack.
13. Pingajuah. 18. Pingajuah.
14. Sissamat. 19. Sissamat.
15. Tellimat. .
These names are mere repetitions of the names of the
first five digits, with a slight variation in those of six,
eleven, sixteen, to distinguish the series of which
they form successively the commencement: the term
for 20, the completion of those members of the human
body which are employed in this natural process of
numeration, is innuk or man ; for 40, they use the ex-
pression innuk arlaak, two men ; for 100, innuk tellimat,
five men; but beyond 20 they proceed with great
difficulty and reluctance, and generally apply to such
numbers a term which signifies innumerable.
There are other examples of the identity of the
terms for man and for twenty amongst the tribes of
South America, originating in the same method of
numeration. Thus, in the numerals of the Jaruroes
canipume, man, is the term for 20, and noenipume
(noeni 2) two men is the term for 40.8
The Esquimaux, according to the relation of Captain
Parry, are still more limited in their power of nume-
* Klaproth, Sprachatlas, p. 16. -
+ Account of Russian Discoveries in Annual Register for 1764,
App. 4.
† Account of Greenland, vol. i. p. 208.
§ Humboldt, Wues des Cordillères, p. 253.
| Second Voyage, p. 556.
#.
History.
\-vº-'
Numerals
of the
Polar Ame
rican
tribes.
In Green-
land.
A R. I T H M E T I C.
387
Arithmetic, ration than the inhabitants of Greenland ; the first
Y-N-Z five mumerals are,
à. 1. Attöwsak.
* 2. Mädleroke, or Ardlek.
3. Pingabake.
4. Sittamat.
5. Těd-lee-mâ.
They usually express the remaining numerals of the
decad by the repetition of the first five ; in some
cases they use the term Argwénrāk for 6, and Argwén-
rāk towa for 7; and when reference is made to the
fingers on the right hand, they express 8, 9, and 10,
by
Kittuklee-moot,
Mikkeelukka-moot,
Eërkit-koke,
which are derived from the names for 2d, 3d, and 4th
fingers, which are,
2d, Keituk-lie-rak.
3d, Mikkēe-lie-rak.
4th, Irkit-köh.
In counting as far as three, they make use of their
fingers, and generally make some mistake before they
reach 7; beyond 9, they hold up both hands; and if
15 or 20 are required, they make another person do
the same, but never resort to the toes of the feet;
they feel greatly distressed to go beyond 10, and
generally cry out 00nooktoot, which may mean any
number between 10 and 10,000.
The numerals of the Eastern Tschutki, of the inha-
bitants of Kadjak, the principal of the Fox Islands, and
of Norton Sound, sufficiently resemble the preceding
to prove them to be the same people.
Kadjak.f. Norton Sound.
. Atauldsen. 1. Adowjak.
As loka. 2. Arba.
. Pingaswak. 3. Pingashook.
4
5
Eastern Tschutki.”
1. Atashek.
. Malgok.
. Pigajut.
. Ishtamat.
. Tatlinnat.
. Itamik. . Sissamat.
. Talimik. . Dallamik.
find many examples of quinary numeral systems ter- History.
minating, as they always do, in the denary or vicenary S-N-
scales ; of the first kind are the numerals of the
Jaloffs, one of the nations visited by Park in his first
journey. :
Fook agh juorom.
1. Ben, or Benna. 15. Jaloffs.
2. Niar. 16. Fook agh juorom ben.
3. Nyet. 2O. Nitt, or Niar fook.
4. Nianet. 30. Fanever, or Nyet fook.
5. Juorom. 40. Nianet fook.
6. Juorom ben. 50. Juorom fook.
7. Juorom niar. 100. Temier.
8. Juorom nyet. 200. Niar temier.
9. Juorom nianet. 1000. Djoone.
10. Fook. 1100. Djoone agh temier.”
l
I
. Fook agh ben.
The word for 5, juorom, likewise signifies hand, and
the system is in every respect a perfect example of the
union of the quinary and denary scales, the first being
subordinate to the other.
The numerals of the Foulahs, a neighbouring tribe,
though essentially different from the preceding, are
of the same character.
Foulahs.
1. Go. 6. Jego.
2. Deeddee. 7. Jedeeddee.
3. Tettee. S. Je tettee.
4. Nee. 9. Je nee.
5. Jouee. 10. Sappo.
In ordinary cases, says Winterbottom, f they reckon
by the fingers of the hands, first on the right hand,
and secondly on the left; but in trading and in other
occasions, where accurate numeration is important,
they use small pebbles, gun flints, or the kernels of
the palm nut, which they dispose in heaps of 5 and 10 ;
thus showing that their practical methods of counting
accurately coincide with their numeral language.
Of the same kind are the numerals of the Jallonkas
and Fellups, two tribes visited by Park, and of the in-
habitants of the coast of Lagoa Bay.
. Malgok.
. Pigajuk.
. Kulla.
I
Numerals
Sewinlak.
. Aglinlikm.
. Aghoiljujun.
. Mall'chonghin.
. Pengtjujun.
. Kulm'ghaen.
Kulen.
l
(26.) If we advance southwards from the Pole, from
of the
central
tribes of
North
America.
the fishing to the hunting tribes of North America, we
shall find, as we have before remarked, the decimal
scale generally prevalent, and in most cases their
numeral systems perfectly regular, and comprehending
large numbers; in some instances, however, we may
discover traces of the quinary scale in the formation
of the numerals between five and ten ; thus, amongst
the following numerals of the Delaware Indians, those
for 6, 7, 8, are modified forms of those for 1, 2, 3.
6. Ciuttas.
7. Nissas.
8. Naas.
1. Ciutta.
2. Nissa.
3. Naha.
4. Nuee-oo. 9. Paes-chun.
5. Pa-reen-ach. 10. Thae-raen. . .
(27.) Amongst the innumerable languages of Africa, we
Numerals
of African
Tribes.
* Klaproth, Sprachatlas, p. 56.
t Ibid. Asia Polyglotta, p. 325.
Jalionka. Fellups. Lagoa Bay.
1. Kidding. 1. Enory. 1. Chingea.
2. Fidding. 2. Cookaba. 2. Seberey.
3. Sarra. 3. Sisajee. 3. Triarou.
4. Nani. 4. Sibakeer. 4. Moonau.
5. Soolo. 5. Footuck. 5. Thanou.
6. Seni. 6. Footuck enory. 6. Thanou-na-
* chingea.
7. Soolo ma fid- 7. Footuck 7. Thanou-na-
ding. cookaba. seberey.
8. Soolo ma Sarra. 8. Footuck si- 8. Thanou-na-
Sajee. triarou.
9. Soolo ma nani. 9. Footuck si- 9. Thanou-na-
bakeer. F]] OOH] all,
10. Foo. 10. Sibankonyen, 10. Koomoo.
It is very seldom that their numerals are given to a
sufficient extent to enable us to judge whether they
proceed by the denary or vicenary scale. We know
but of one case of the latter kind, in the numerals of
the Mandingoes, the first ten of which we have given
before. (Art. 10.)
* Classical Journal, vol. v.
t Account of Sierra Leone, vol. i. p. 174.
3 E 2
388
A R I T H M E. T.I. C.
Arithmetic.
>~~
Mandin-
goes.
Azteck
numerals.
Their
numerical
lierogly-
phics.
ll.
2O.
30.
40.
5O.
6O.
7O.
Tang killm.
Mulu.
Mulu nintang.
Mulu foola.
Mulu foola nintang.
Mulu Sabba.
Mulu Sabba nintang.
80. Mulu nani.
90. Mulu nani nintang.
100. Kemi.
IOOO. Ali. *
(28.) Of all numeral systems adapted to the vicenary
scale, the most perfectly developed is the Azteck, or
ancient Mexican, proceeding as far as an articulate
number of the third order ; the numerals are as
follow :
1. Ce. 15. Matlactli oz chicuace.
2. Ome. 16. Matlactli oz chicome.
3. Jei. 20. Pohualli, or cem-po-
4. Nahui. hualli.
5. Macuilli. 30. Cem-pohualli oz mat-
6. Chicuace. lactli.
7. Chicome. 40. Om-pohualli.
8. Chicuei. 50. Om-pohualli oz mat-
9. Chicuhnahui. lactli.
10. Matlactli. 60. Jei-pohualli.
11. Matlactli oz ce. 80. Nahui-pohualli.
12. Matlactli omome. 100. Macuilli-pohualli.
. Matlactli ozjei. 400.
. Matlactli oz nahui. 800. Xiquipilli.f
We are obliged to omit the name for four hundred, as
it is not mentioned by Humboldt, from whose splendid
works these numerals are taken ; and we have in vain
searched for a Mexican grammar, or vocabulary,
in many of the principal libraries of this country.
In the same author we find an account of the
symbols employed for numbers in their hierogly-
phical writing, which exactly corresponded with their
numeral language. A small standard, or flag, denoted
20 ; if divided by two cross lines, and half coloured,
it represented half twenty, or 10; and if three
quarters coloured, it denoted 15. The square of
twenty, or 400, was denoted by a feather, because
grains of gold enclosed in a quill, were used in some
places as money, or a sign for the purposes of ex-
change. The figure of a sack indicated the cube of
twenty, or 8000, and bore the name of Xiquipilli, given
also to a kind of purse that contained 8000 grains of
cacao. These symbols were repeated twice, thrice,
four times, &c. to denote multiples of them by 2, 3, 4,
&c.; and grouped together, like the common symbols,
to denote any compound number.
(29.) The Chibcha or Muysca language, of the Indians
of Bogota, in New Grenada, exhibits a numeral system
adapted to the same scale, to which the denary alone
is subordinate, and which merits consideration
on more accounts than one. The following are the
numerals:
Muysca
numerals.
1. Ata 4. Muyhica.
2. Bosa. 5. Hisca.
3. Mica. 6. Ta.
* Jackson's Account of Marocco, p. 226.
f Humboldt, Wues des Cordillères, p. 141 and 25f.
7. Cahupgua. 2}. Guetas asaqui ata. History.
8. Suhuzu. 22. Guetas asaqui bosa. S-N-"
9. Aca. 30. Guetas asaqui ub- - , º,
10. Ubchica. chica.
11. Quicha ata. 40. Gue-bosa,
12. Quicha bosa. 60. Gue-mica.
13. Quicha mica. 80. Gue-muyhica.
15. Quicha hişca. 100. Gue-hisca.
20. Quicha ubchica, or -
guetá.
The term ubchica, after the first decad of numerals,
is replaced by quicha in the second decad, which
means foot ; thus the expressions for 1 I, 19, &c. mean
foot one, foot two, &c. being accurately significant of
their primitive methods of numeration. Twenty is
expressed either by quicha ubchica, foot ten, or by gueta,
which signifies' house ; forty, by two houses ; sixty, by
three houses; and similarly for higher articulate num-
bers in the same series.
Humboldt has given from the researches of Du- Their
quesne, a Canon of the Metropolitan Church of Santa meaning.
Fè de Bogota, the etymological significations of most
of these numerals. Thus ata signifies water; bosa, an
enclosure ; mica, changeable; muyhica, a cloud threaten-
ing d tempest ; hisca, repose; ta, harvest ; cahupguſt,
deaf; suhuzu, a tail; and ubchica, resplendent moon. No
meaning has been discovered of aca, the numeral for 9.
It is impossible amidst meanings so various, to recog-
nise any principle which may seem to have pointed
out the use of these terms as numerals ; and it is
making little advance towards an explanation of the
difficulty to say, with Duquesne, that the words relate
either to the phases of the moon in its increase or
wane, or to objects of agriculture or worship ; as far
as their signification as numerals are concerned, they
may be considered as perfectly arbitrary; and it is in
vain to attempt any probable theory for the explana-
tion of a fact, where there is no analogy to guide us,
except perhaps the very imperfect one which is fur-
nished by the ordinary meanings of the second series
of Chinese numeral symbols.
The same people possessed hieroglyphical symbols Muysca
for the first ten numbers, and for twenty, which are numerical
given in Plate I. fig. 5. In the Mexican numeral sym-
bols, there is an intelligible connection between the
sign and the thing signified ; but if the following
explanations given to Duquesne, by some Indians
who were instructed in the calendar of their ancestors,
be correct, it is impossible to conceive any associa-
tion which is more perfectly arbitrary. Thus the
hieroglyphic for one, is a frog ; for two, a nose with
extended nostrils, part of the lunar disk, figured as a
face ; for three, two eyes open, another part of the
lunar disk; for four, two eyes closed ; for five, two
figures united, the nuptials of the Sun and moon,
conjunction; for sir, a stake with a cord, alluding to the
sacrifice of Guesa tied to a pillar; for seven, two ears ;
for eight, no meaning assigned ; for nine, two frogs
coupled ; for ten, an ear; for twenty, a frog extended.
‘It would be difficult, for a common observer, to dis-
cover in these symbols the objects mentioned in the
preceding explanations of them ; but, in answer, it
may be said, that their forms have degenerated from
long use, and consequently furnish no decisive argu-
ment against the correctness of their traditional inter-
pretation; and that the same objections would apply
hierogly -
phics:
A R I T H M E T I C.
389
Arithmetic. to the present explanation of the keys of the Chinese
~~' symbols, however certainly derived, in many instances
chandra at least, from rude imitations of natural objects.
Sangkala It might be imagined that there existed some
of Java. analogy between this use of words as numerals, which
have other significations, and the custom which has
prevailed among the Javanese from very remote
antiquity, denominated chdindra sangkala, “ reflections
of royal times,” or the light of royal dates.” It consists
in attaching the names of various objects, or things,
or their representations, to the nine digits and zero,
twenty or more being assigned to each of them ; and
in expressing a date, to select such of them as may form
a sentence, significant of the event which it comme-
morates. Thus the date (1400) of one of the most
calamitous events of their history, is expressed thus:
Sirna ilang Kertaning Bámi.
Lost and gone is the pride of the land.
O O 4 l
Thus Būmi is one of the words significant of
unity ; kertaming, of four ; ilang and sirna, of zero.
Again, the date (1313) on the tomb of the Princess
Chermai is thus stated :
Ráya wulan putri ſku.
Like unto the moon was that princess.
3 l 3 I
Where iku and wulan are significant of unity, and
putri and kaya of three. This practice constitutes a
technical memory of a very elegant and amusing
nature, and reminds us rather of the literary luxury
of a refined people, than of the efforts of a primitive
nation, to pass from practical methods of numeration
to numerical language.
The Mexicans, Muyscas, and Peruvians, constituted
the only three nations of ancient America, who
possessed governments regularly organized, and who
had made considerable progress in many of the arts
of civilized life, in architecture, sculpture, and paint-
ing. They were the only people, in short, in that vast
continent, who could be considered as possessing
literary or historical monuments. On this account
alone their numeral systems would merit very par-
ticular attention ; but still more so from their perfect
developement. The first presents the most complete
example that we possess of the vicenary scale, with
the quinary and denary subordinate to it. The
second, of the same scale, with the denary alone sub-
ordinate to it ; whilst the third, or Peruvian, is
strictly denary, and is equally remarkable for its great
extent and regularity of construction.
(30.) It is the latter scale which is of rare occurrence
amongst American tribes, the vicenary being much
more generally prevalent in their numeral systems; so
much so indeed as to be almost characteristic of them.
In proceeding to a farther consideration of them, we
must again lament our inability to procure access to
vocabularies, or grammars, of these languages, in con-
sequence of which we are compelled to pass over a sub-
ject of very great interest in a very cursory and imperfect
manner, having been only able to collect a very small
number of disconnected facts which have reference
to it.
Dobrizhoffert has given an account of the numeral
systems of the Abipones and Guaranies, amongst whom
Of othcr
South
American
tribes.
* Raſhes, Java, vol. i. p. 372, and vol. ii. App. G.
† History ºf the Ahipones.
he resided for many years, and with whose habits and History,
language he was intimately acquainted : the first are S-y-
an equestrian people of Paraguay, whose predatory
habits long made them formidable to the Spaniards
and neighbouring tribes. The first five numerals are
expressed by 1. Imitara. Abipones.
2. Iſioaka.
3. Inoaka yekaim.
4. Geyenk fiate.
5. Neenhalek.
The names for 1, 2, 3, have no reference to natural
objects; the expression for 4, means the fingers of
the emu, a bird extremely common in ‘Paraguay, pos-
sessing four claws on each foot, three before and one
turned back ; whilst that for five is the name of a
beautiful skin with five different colours. The same
number, however, is more commonly expressed by
hanam begem, the fingers of one hand; to express num-
bers between five and ten, they combine the name for
five with the inferior units; ten is expressed by lanam
rihegem, the fingers of both hands; and for twenty, they
say hanam rihegem cat grachahaka anomicheri hegem, the
fingers of both hands and feet. *
The Guaranies are another tribe of Paraguay, who Guaranies.
speaka language which is the mother of many other dia-
lects, yet they possess only four independent numerals.
1. Petey.
2. Mokoy.
3. Inbohossi.
4. Irundy.
If we pass further north to the Tupi, a very numerous Tupi.
tribe in Brazil, speaking a kindred language to the
former, we find only five independent numerals.”
1. Auge-pe
2. Mocouein.
3. Mossaput.
4. Oioicoudie.
5. Ecoinbo.
Humboldt interrogated a native of the Maco Macoes, Maco
a tribe on the Orinoco, who knew no names for num- Macoes,
bers beyond four.
1. Niante.
2. Tojus.
3. Percotahuja.
4. Inantegroa.t
The Caribbees who constituted the native population Caribbees
of Barbadoes, St. Christopher's, Antigua, and the other and Gali'
Islands of the Caribbean Sea, and who, under the
name of Galibi, are dispersed extensively over the
adjoining continent, and form one of the finest of the
American tribes, are equally limited in their names for
numbers.; -
1. Aban.
2. Bean.
3. Eleona.
4. Beambouri.
In all these cases, the numeration beyond five is car-
ried on by means of the fingers and toes, and their
numeral language becomes generally, as in the case of
=rrºr-
* Southey's History of Brazil, vol. i. p. 226.
+ Humboldt's Personal Narrative, vol. v. p. 125. English
edition.
# Raymond, Histoire des Caraibes, 1665.
390
A R H T H M E T I C.
of numeration equally perfect with those of the History.
Greeks and Romans, and incomparably superior to S-N-2
those of any other American nation : the Quipus were Peruvian
knots, nine in number, movable upon a string like the Quipus,
Arithmetic. the Abipones, descriptive of their practical methods of
\-v-' counting ; thus amongst the last mentioned people,
to express five, they show the fingers of one hand, and
Achaguas.
Zamucoes.
for ten, the fingers of both hands; “for twenty, their
expression is pleasant,” says Davies,” “being obliged
to show all the fingers of their hands and the toes of
their feet.” -
In the languages of these rude tribes, abstract
terms are almost entirely unknown, and their expres-
sions from mere poverty, in many cases assume a
highly figurative form, being obliged to refer to natu-
ral objects and the most common relations of life, to
xpress ideas which do not otherwise come within
the compass of their languages; thus in the Caribbean
language, the fingers are termed the children of the
hand, and the toes the children of the feet; and the
phrase for ten, chon oucabo raim, all the children of the
hands.
There is no difficulty in producing other examples
of numeral language constructed in this manner, and
equally descriptive of practical methods of numeration.
The Achaguas, a tribe on the Orinoco, express five by
abacaje, or the fingers of one hand; ten, by tucha macaje,
all the fingers ; twenty, by abacaytacay, or all the fingers
and loes; forty, by incha matacacay, or the fingers and
toes of two men ; and so on for very large numbers;f
among the Zamucoes, as well as the Muyscas, five,
is the hand finished ; siv, one of the other hand; ten, the
two hands finished; eleven, foot one ; twelve, foot two ;
twenty, the feet finished. # It is evident that this absence
beads of a rosary, which was attached by one end
to a rod; of these strings there was one for units, and
one for each of the successive orders of superior units
as far as one hundred millions. The use of the quipus
was nearly the same as that of the Roman abacus ;
and it not only enabled them to express any number,
but likewise to perform the ordinary arithmetical
operations of addition, subtraction, multiplication, and
division. Knots of peculiar and different colours ap-
pear to have been used in the numeration of different
objects, whether of gold, silver, &c. and to have been
appropriated to them.*
The whole business of calculation
been confided to the Quinpucamoya,
the quipus ; and the reports of the early historians
of this empire bear testimony to the rapidity and ac-
curacy of their operations. We are not aware of the
existence of any similar practice among other Ame-
rican nations. Marsden, in his account of Sumatra,
has noticed a practice which bears some analogy to it,
where it is usual to denote the completion of a tale of
one hundred, by making a knot in a string, which is
repeated as often as necessary; such knots, or quipus,
are made use of not merely as an assistance to the
memory in the process of numeration, but likewise as
records or accounts of numbers.f
appears to have
or guardians of
(32.) It was an opinion maintained by that singularly The arith-
paradoxical writer De Pauw, that no indigenous metic of
nation of America could reckon in their own idiom $9üth.
beyond three ;f the facts, however, given above, are sººn
more than sufficient to refute such an assertion; extremely
though it must be allowed, that the numeral systems limited.
of abstract and independent terms for numbers, and
the tedious circumlocutions which it occasions, must
form an insuperable obstacle to the expression of large
numbers in such languages.
In the collection of Theodore de Bry, there is an
account of the inhabitants in the neighbourhood of
Brazilians
of Pernam-
buco. Pernambuco in Brazil, by a German Jesuit of the of the South American tribes are remarkably limited
name of Stadius, containing the following statement of in absolute extent, and still more so in arbitrary and
their methods of numeration, which is applicable to independent words: it is to the latter chiefly that De
many other American tribes: numeros mon ultra quina- Pauw refers, and there are some examples which
rium notant : si res numerandae quinarium excedant, in- might appear to bear out his assertion : of this kind
dicant eos digitis pedum et manuum pro numeris demon- are the numerals of the Abipones mentioned above,
stratis : quod si numeros et horum multitudinem excedat, and the celebrated example of the Yancos on the
conjungunt aliquot personas et pro multitudine digitorum in Amazon, whose name for three is
ſº illis res notant et numerant. § - Poettarrarorincoaroac,
º: (31.) The practical methods of counting of American of a length sufficiently formidable to justify the re-
... tribes, however, are not in all cases restricted to the mark of La Condamine : Heureusement pour ceuw qui
ing among fingers and toes, and their numeration is not neces- ont a faire avec euz, leur Arithmetique me va pas plus
the Guara- sarily confined to twenty, the radix of their scale, loin. §
IſilëS, when destitute of the aid of names, whether arbitrary All travellers have borne testimony to the extreme
or not, for higher numbers, or when they cannot call
in the assistance of other persons. The Guaranies
make heaps of maize, each consisting of twenty
grains, two, three, four, &c. of which are used to
denote 40, 60, SO, &c. the excess above any one of this
series of articulate numbers being reckoned in the
ordinary way: the same custom prevails in other parts
of that continent, and we are reminded of it in the
Mexican hieroglyphical symbols.
The ancient Peruvians possessed practical methods
* History of Barbadoes, St. Christopher's, Antego, Martinico,
Montserrat, and the rest of the Caribby Islands: Englished by
John Davies, of Kedwilly, 1666.
+ Southey's History of Brazil, note, p. 638.
# Humboldt, Vues (les Cordillères, &c. p. 253.
§ America. Descriptio, vol. i. part iii. p. 128.
difficulty which these South American tribes usually
experience in attempting to count even small num-
bers; they are indolent from constitution and habit,
and are reluctant to enter upon any exercise of
the mind which requires the least effort of abstrac-
tion. Dobrizhoffer relates of the Abipones, that they
could rarely count as far as ten. When attempting,
upon their return from their expeditions, to give an
idea of the number of their enemies, or of the horses
they had captured, they would mark out a space, and
say that they were as many as could stand within it.
* Histoire des Yucays Roys de Peru, p. 680, 1633.
+ Marsden's Sumatra, p. 192. In counting money, each tenth
and sometimes also each liundredth piece is put aside.
: Récherches Philosophiques sur les Americains, vol. ii. p. 162.
§ La Condamine, Voyage de la Riviere des Amazons, p. 64.
A R. I T H M E T I C. 391
after long and diligent teaching that he counted as far , History,
as five ; and Aristotle, at the conclusion of the passage
which we have quoted above, on the universality of ...".”
aft tribe men-
Arithmetic. On one occasion, when he accompanied a party of ten
S-N-2 upon a defensive expedition, he mentions the following
dialogue as having taken place between them: “Are
we many ?” “Yes, you are many.” “Are we innume-
rable 2" “ Yes, you are innumerable.”
indeed, were the Missionaries throughout Paraguay
and Brazil, of this deficiency of the natives, that it
is a general practice in the churches of the several
Reductions, to teach, or attempt to teach them to
count as far as two hundred in the Spanish or Portu-
guese language.
In the account of the Caribbees which we have
referred to above, it is said, that in counting numbers
beyond ten, they generally get confused, and exclaim,
“ in their gibberish,” as Davies expresses it, tamigati
cati mitibouri bali, they are as many as the hairs of my
So sensible,
the decimal scale, says that a certain tribe of Thrace ...d by
formed the only exception, whose numeration was Aristotle,
limited to four : Mövot be äpt016volt Tiêv epakāv Yévos
Ti eis Tétrapa, 6td Tö, Öa Tep Tū Tatēēa, ſwij čvvågøat
plumuovečew étaroxy, unèë xpija tv pºmbevös cºval ToxNoë
ãvtóws. This passage is curious, as showing that even
amongst the Greeks some attention was paid to the
methods of numeration of barbarous nations; and
though we might admit the fact, however contrary to
modern observation, yet we certainly must dispute the
correctness of the conclusion, that their powers of
numeration were limited to four, because they never
felt either the want or the use of higher numbers.
(33.) Themention of this passageof Aristotle naturally The natural
leads us to the consideration of the question, whether scales alone
in any modern instance, any other than the natural *
head, or the sand on the sea shore.
The general testimony of Humboldt is decisive
of the same fact; he declares that he never met with
&-
a native Indian who, if asked his age, would not an-
swer indifferently 16 or 60 :* he at the same time ob-
serves, that this is the case even amongst tribes who
possess a numeral language which embraces very high
numbers; may we not, however, reasonably suspect,
that the existence of such terms rests in general upon
very insufficient authority or that the individuals
whom he interrogated were less skilled than others of
their countrymen in the practice and language of
numeration ? For it is absurd to suppose, that terms
exist among such rude people to which they can
attach no meaning.
We have given examples of people whose powers of
scales of notation have ever prevailed in any nation “
whatsoever ? whether, in short, there is any limitation
to the first of the general propositions which are
stated in Art. 8: The examples which we have
|hitherto produced, are strongly confirmatory of its
being universally true ; and show, that though in
Some cases numerical language may fail in reaching
even the radix of the lowest of these scales, yet that
there is no exception to the existence of practical
methods by which the numeration is extended, at least
as far as ten, if not much farther ; and that these
methods are essentially adapted to the natural scales,
and furnish indeed the foundation of them.
In parcelling out certain objects, it very commonly Alleged
happens, that a particular number of them are united instances
numeration are equally confined with those who are
the subject of our present discussion, particularly f oth
Of Other
Numerals
amongst the Polar Americans; and it would not be
difficult to produce other instances which are equally
remarkable. The natives of New South Wales possess
no numerals beyond those which follow :
or associated together, and the lot designated by a
peculiar name: thus, pair, couple, brace are synony-
mous terms : but the associations which our habits
have long connected with them, would not allow of
of natives
of New 1. Wagul. their being interchanged with propriety in the expres–
Holland. 2. Boola. sions, a pair of horses, a couple of dogs, and a brace of
3. Brewy.f partridges. The term leash is of still more restricted
When a number exceeds three, they use the phrase
murray-loolo, which signifies an indefinite number.
We know, however, from the authority of a gentleman
who has long filled an official situation in that colony,
that they count to higher numbers by means of the
fingers. For five, they hold up the expanded hand ;
for ten, both the hands; for greater numbers, they
avail themselves of the hands of another person, in the
same manner as the Esquimaux, and in this manner
they are enabled to proceed as far as twenty or thirty.
application; whilst warf, (from the German wurfen,
to cast,) or cast, is appropriated to the four herrings
which the fisherman throws at a time, two in each
hand, in making his tale.* Terms of this kind, which
are not perfectly abstract, afford no proper evidence
of the existence of the binary, ternary, or quaternary
scales of notation, as the process of classification is
generally terminated at the very first step, and does
not proceed to articulate numbers of the second or
higher orders. We may sometimes hear such an ex-
Koussa . The Koussa Caffres, as well as the Hottentots, accord- pression as pair of pair, couple of couple, but never
$. ing to the authority of Lichtenstein,t have no numeral brace of brace, leash of leash, a warf of warf, as the last
Hottentots. + 3.
beyond ten, though some authors have extended it to
100 ; whenever they express a number, they raise up
the like number of fingers; so indistinct and im-
perfect is the impression conveyed to the minds of
these rude people by an abstract term, unaided by an
appeal to the senses.
It is mentioned by Suidas, Š that the ancient comic
poets, amongst other marks of stupidity which they
attributed to one Melitides, asserted that it was only
* Personal Narrative, vol. v. p. 125. English edition,
+ Collins's New South Wales, App.
† Travels in Southern Africa, vol. i. App.
5 In voce yéâotos.
set of expressions would indicate a degree of abstrac-
tion in the terms which they never possess. If men
were all sportsmen or fishermen, and the only objects
which required numeration were birds or fish, one
might possibly conceive that the accidental circum-
stances which lead to this primary classification of such
objects, might have been followed to a sufficient extent
to form a ternary or quaternary scale; but in no other
manner could we conceive such scales to be generally
adopted, which have no foundation in those practical
methods of numeration which are pointed out by
nature herself.
* Leslie's Philosophy of Arithmetic, p. 3.
392 - A. R. I T H M E T I C.
Arithmetic. It is mentioned by Crawfurd,” that the woolly
S-S.--' haired races who inhabit the mountains of the penin-
sula of Malacca, have no native terms for numbers
beyond two ; that for one, being nai, and for two, bu,
which likewise signifies second born ; for higher num-
bers they use the common Polynesian numerals ; such
an example furnishes no proof of the existence of the
binary scale amongst these people; and even granting
that native terms for higher numbers never existed,
and were not superseded by those of a predominant
language, the case is merely analogous to many others
which we have mentioned, where numeral language
had not kept pace with practical methods of nume-
ration.
Binary (34.) Though it is in vain to look for the binary Arith-
Arithmetic metic amongst the primitive institutions of nations,
** yet its adoption has been recommended in later times
by the celebrated Leibnitz, as presenting many advan-
tages, from its enabling us to perform all the opera-
tions in Symbolical Arithmetic, by mere addition and
subtraction : it requires the use but of two symbols
for zero and unity, which are adequate to the expres-
sion of all numbers.
As unity was considered the symbol of the Deity,
this formation of all numbers from zero and unity was
considered in that age of metaphysical dreaming, as
an apt image of the creation of the world by God from
chaos. It was with reference to this view of the
binary Arithmetic, that a medal was struck bearing
on its obverse, as an inscription, the Pythagorean
distich,
Numero Deus (1) impari guadet;
and on its reverse, the appropriate verse descriptive of
the system which it celebrated,
Omnibus ea nihilo ducendis sufficit Unum.t
This invention was studiously circulated by its
author by means of the scientific journals, and his ex-
tensive correspondence; ; it was communicated by him
to Bouvet, a Jesuit Missionary at Pekin, at that time
engaged in the study of Chinese antiquities, and who
imagined that he had discovered in it a key to the ex-
planation of the Cova, or lineations of Fohi, the founder
of the Empire. They consist of eight sets of three
lines, either entire or broken, arranged in the follow-
ing manner, or in a circle.
(1) (2.) (3.) (4.) (5.) (6.) (7.) (8.)
The cova or — — — — — — — — — — — —
suspended — — — — — * *-* = ºm-
** — — — — — — — — -wº
Fohi.
If we suppose the broken lines to represent zero, and
the entire line unity, and that it possesses value from
its position, increasing as it descends, these lineaţions,
would severally become in the binary arithmetical
notation, O, 1, 10, 11, 100, 101, 110, 111, or 0, 1, 2, 3,
4, 5, 6, 7, respectively. The explanation of this system
is certainly thus far consistent; and if the assertion
made by Leibnitz be true, that it applies likewise to
the great Cova of Fohi, consisting of 64 characters,
and 384 lines, embracing six places of figures in this
system, and representing therefore all the natural
numbers in order between 0 and 63, it would afford a
strong presumption that this theory was correct, and
* Indian Archipelago, vol. i. p. 255.
+ Leibnitzii Opera, tom. iii. p. 346.
f Ibid. tom. ii. p. 349, 391, tom. iv. p. 152, 207.
would thus furnish an example of a species of Arith- History.
metic with device of place, possessing an antiquity of ~~~~
more than three thousand years.
These figures of eight cova are held in great vene-
ration, being suspended in all their temples, and though
not understood, are supposed to conceal great mysteries,
and the true principles of all philosophy both human
and divine. The good Jesuit who seems to have
caught the very spirit of Chinese belief, is trium-
phant at his discovery, and seems to consider these
symbols of the binary Arithmetic of Fohi, as a most
mysterious testimony to the unity of the Deity, and as
containing within it the germ of all the sciences. Cette
gure, says he, est une des figures de Fohi, qui par l'art
admirable d'une science consommée, avoit scu renfermer,
comme sous deux symboles, généraux et magiques, les prin-
cipes de toutes les sciences de la vraie sagesse ; et ce grand
Philosophe, dont laphysiognomie n'a rien de Chinois, quoigue
cette nation le réconnoisse powr l'auteur des sciences et
pour le fondateur de la monarchie, avoit bàti ce systeme
de sa figure circulaire, ce semble, pour calculer et recon-
noitre eractement toutes les periodes et les mouvemens des
corps célestes et dommer les connoissances claires de tous
les changemens, qui par leur moyen arrivent continuelle-
ment et successivement dams la mature.*
We have been induced to make this digression on
the subject of the binary Arithmetic, chiefly for the
purpose of noticing this very curious and very ancient
monument of its existence; if, however, we make
every concession in favour of the explanation above
given, and many serious doubts might easily be
started, we can at most consider it but as a solitary
instance of its adoption not by a nation, but by an
individual who surpassed his contemporaries in know-
ledge, and who left this, amongst other memorable in-
ventions, to his successors, who begun by venerating
it as a relic of the founder of their science and their
monarchy, and concluded by regarding it as a mysti-
cal symbol, which contained the hidden principles of
the most sublime and important truths.
(35.) Of scales, different from those which are properly Duodenary
called natural, the existence of the binary and duode- scale.
nary alone have been supported by probable argu-
ments ; the first, under any circumstances, could
claim a philosophical existence only, and could hardly
therefore be considered as militating against the uni-
versality of our proposition ; the second we have
noticed before, and have stated our reasons for thinking
that the preference shown amongst Scandinavian
nations for the number twelve, and its very general
use in the division of concrete numbers, furnish no
sufficient ground for considering it as having been used
as the radix of a scale of notation, however nearly in
some respects it may have approximated to it.
(36.) We shall now conclude this examination of nu- Conclusion.
meral systems, which has perhaps proceeded to a greater.
length than is consistent with the design of a work of
this nature. We think we have fully established the
propositions which we proposed as the objects of our
investigation ; and have shown that the principles
which are concerned both in the origin and formation
of numeral systems and numeral languages, are not
only remarkably consistent with the most philosophical
theory, but possess an universality of application,
which is seldom to be met with, except in the physical
* Leibnitzii Opera, tom. iv. p. 153.
A R IT H M ET I c.
393
Arithmetic. Sciences.
It is quite necessary to refer to this method of nu- History,
meration, in order to explain many passages in classical \-y-
We shall add one more instance of this
S-N-2 extraordinary accordance between theory and obser-
Methods
of indigi-
tation.
vation.
In Art. 4, we have given what we considered a pro-
bable theory of the origin of the classification of
numbers by successive decimation, and we have since
discovered the following passage in a history of the
Island of Madagascar, by which it is illustrated in a
very remarkable manner. After noticing their numeral
language, which coincides with that of the Indian
Archipelago, and refuting the assertions of some
authors who have limited their powers of numeration
to ten, he adds the following account of their mode of
counting. “ Lorsqu'ils veulent compter les hommes d'une
armée, ils obligent les hommes de passer un a un par un pas-
sage etroit en presence des principauw chefs et de poser une
pierre chacun en une place; et quand ils ont tous passés, ils
comptent toutes les pierres de dia, en dir, qu'ils adjoutent
ensemble : puis les divaines de dia: en dia, et les centaines
jusques à ce qu'ils soient à la fin de leur nombre.”
(37.) Before we proceed to give an account of symbo-
lical Arithmetic, as it exists, or has existed amongst dif-
ferent nations, we shall notice a species of digital Arith-
metic very generally practised amongst the ancients, and
to which frequent allusion is made in classical authors.
It consisted in denoting the nine digits and the articu-
late numbers as far as 100, by inflections of the fin-
gers of the left hand, whilst the hundreds were marked
on the right hand, by the same inflections which were
used to denote the articulate numbers on the left, and
the thousands were a repetition on the right hand of
the inflections used for the digits; they were thus
enabled to denote all numbers which were less than
ten thousand. This is the extent to which this system
of digital Arithmetic appears to have been carried in
ancient times, at least if we may judge from the work
of Nicholas, a Monk of Smyrna,t the earliest of all
those with which we are acquainted, in which it is
distinctly described. But the venerable Bede, in a
short Tract, de Computo vel de Loquela per Gestum
Digitorum, has extended this method of numeration as
far as a million, by placing the left hand for lower
numbers and the right hand for higher, either expanded
or closed, with the fingers upwards or downwards,
upon the breast, thighs, and other parts of the body;
ten variations only being required to answer this
purpose. The same illustrious author has proposed
another application of this system, for the purpose of
holding conversations by means of the fingers of one
hand, and which may be done by making the natural
numbers in their order the representatives of the
successive letters of the alphabet, when the indication
of the number would likewise be made the indication
of the letter; thus, to convey the caution “caute age” to
a friend amongst thieves or sharpers, it would be
merely requisite to make the signs of the numbers 3,
1, 20, 19, 5, 1, 7, 5. -
* Histoire de la grande Isle de Madagascar, par de Flacourt,
ch. xxviii. 1661.
+ Nikoxdou Xuvpudwov repl SakrvXſkov uérpov. It is published
in the Spicilegium Evangelicum of Possinus; an Appendix to, or
rather a Commentary on, the Catena Graecorum Patrum, Rome,
1683, where representations are given of hands with the fingers
in the several positions which are required : the same may be
seen also in Hemischius, de Numeratione Multiplici, and with the
additional positions of Bede, in the Theatrum Arithmeticum of
Leopold, 1727.
WOL, I.
authors. Juvenal states it as a peculiar felicity of
Nestor, that he counted the years of his age on the
right hand : - * .
Felix nimirum, qui tot per saecula mortem
Distulit, atque suos jam deatra computat annos.
Sat. x. 248.
The image of Janus was represented, according to
Pliny, with his fingers so placed as to represent 365,
the number of days in the year:
Janus geminus a Wumá rege dicatus, qui pacis bellique argumento
colitur, digitis itº figuratis, ut trecentorum sea aginta quinque
dierum nota per significationem anni temporis et avi se Deum
indicaret. Hist. Wat, lib. xxxiv. 7.*
The same custom must be kept in view in order to
comprehend the sarcastic exaggeration in the Greek
epigram of Nicarchus, in vetulam annosam : -
Hºdos 30pjaaga *Adºpov TAéov, #xepi Aaff
Tºpas aptóweto 6at 3e étepov apčapevil.t
The following passages are a few out of a great
number which contain similar allusions : -
Alii igitur digitis complicatis numerum, alii constrictis significa-
bantur. r Quinctilian, lib. ii. ch. iii.
Componit vultum, intendit oculos, movet labra, agitat digitos,
computat nihil. Caii Plinii Epist. 20. lib. ii.
Numerum docet me arithmetica, avaritiae accomodare digitos. ,
p - Seneca, Epist. 88. Iib. i.
Ecce autem avortit niacus laeva, in femore habet, manum
Deatera digitis rationem computat, ferients femur. -
Plauti Miles Gloriosus, act ii. sc. 2.;
From the first and last of these passages, we
should be inclined to suspect, that, however general
this practice may have been among the ancients, it
varied both at different times and with different
persons, in the particular mode in which the numbers
were denoted.
Henischius' and other authors have discovered some
reference to this practice, in the description of Wisdom
in the Proverbs of Solomon :
Length of days is in her right hand, and in her left hand riches
and honour. Prov. iii. 16.
However fanciful such an explanation may appear
to be, it is both simple and natural, compared with
that which has been given of the following verse in
the Parable of the Seed, and which Bede has quoted
with approbation :
But others fell into good ground and brought forth fruit, some
an hundred fold, some sixty fold, some thirty fold.
Matt. xiii. 8.
“ Centesimus,” says St. Jerome, “ et sewagesimus' et
trigesimus fructus, quanqam de una terra et de una semente
nascatur, tamen multum differt in numero. Triginta re-
feruntur ad nuptias, nam et insa digitorum conjunctio,
quasi molli se complexans osculo et faderans, maritum.
pingit et conjugem. Sevaginta vero ad viduas, eo quod in
angustia et tribulatione sunt positae, unde et superiori digito
deprimuntur quantoque major est difficultas experta,
quondam voluptatis illecebris abstinere, tanto majus est
* Henischius, de Wumeratione Multiplici, 1605; Leslie's
Philosophy of Arithmetic, p 223. - .
‘F Host, de Numeratione emendatá veteribus Latini's et Graecis
wsitatá, Antwerp, 1582. ; :
f Wallancey, Collectanea de rebus Hibernicis, vol. iii. p. 567.
§ De Numeratione Multiplici.
3 F
394
A. R. I T H M E T I C.
Arithmetic, praemium. Porro centesimus numerus (diligenter quaeso,
S-V-' Lector, attende) de sinistra transfertur ad deateram ; et
iisdem quidem digitis, quibus in lavá nupta significantur
et vidua, circulum faciens exprimet Virginitatis coronam.”
It is necessary to refer to the configurations of the
fingers themselves, in order to understand the allusions
to the numbers in this very singular commentary,
which, at all events, shows how very familiar and
common this practice must have been at the time it
was written.
The Chinese have a system of indigitation, by which
they can express on one hand all numbers less than a
a hundred thousand; the thumb nail of the right hand
touches each joint of the little finger, passing first up the
external side, then down the middle, and afterwards
up the other side of it, in order to express the nine
digits; the tens are denoted in the same way, on the
second finger; the hundreds on the third; the thou-
sands on the fourth ; and the ten thousands on the
thumb. It would be merely necessary to proceed to
the right hand, in order to be able to extend this
system of numeration much farther than could be
required for any ordinary purposes.
The common phrases ad digitos redire, in digitos
mittere, have the same meaning as computare, and
distinctly refer to digital numeration; there is also
another phrase, micare digitis, of frequent occurrence,
which alludes to a game extremely popular among the
Romans, and which was most probably the same as
the morra of modern Italy. This noisy game is played
by two persons, who stretch out a number of their
fingers at the same moment, and instantly call out a
number, and he is the winner who names a number
expressing the sum of the number of fingers thrown
out.* The same game is found amongst the Sicilians,
Spaniards, Moors, and Persians; and, under the name
tsoimoi,t is practised also in China.
There exists a species of digital Arithmetic amongst
nearly all eastern nations. The Bengaleset count as
far as fifteen by touching in succession the joints of
the fingers ; and merchants, in concluding bargains,
the particulars of which they wish to conceal from the
bystanders, put their hands beneath a cloth, and signify
the prices they offer or take by the contact of the
fingers. The same custom is prevalent also in Bar-
bary, Ś and Arabia ; }| when they conceal their hands
beneath the folds of their cloaks, and possess methods
which are probably peculiar and national, of conveying
the expression of numbers to each other.
(38.) Hin considering different systems of symbolical
Greek e te - * -
arithmeti- Arithmetic, we shall commence with that of the
cal nota- Greeks; a preference which it merits, as well from
tion. the superior developement which it received from the
hands of the people of antiquity, who cultivated the
sciences with the greatest success, as also from its
being absolutely essential to the understanding of the
ancient astronomical and other writings, in which
numbers and calculations are involved. :
* Cadell's Travels in Istria and Carniola, vol. ii. p. 118; Blunt's
Westiges of Ancient Manners and Customs in Italy and Sicily,
p. 230. When played in the night it required the utmost confi-
dence in the honour of the parties; and it is an expression of
Cicero to designate a perfectly honest man, that he is dignus,
guocum in tenebris mices. Off. lib. iii. .
+ Barrow's Travels in China.
† Halhed's Bengalee Grammar.
§ Shaw, Travels in Barbary. -
| Niebuhr's Travels in Arabia.
The Greeks expressed the natural numbers below Hiºr,
10,000, or a myriad, by means of , the twenty-four
letters of the alphabet, together with three interpolated
symbols, s, , 2, which denoted 6, 90, 900, respec-
tively. The following table exhibits the four classes
of digits and articulate numbers of the 1st, 2d, and 3d
i. into which the numerical symbols were distri-
uted : • *
(1.) a £3 y 8 e 5. § m 6
l 2 3 4 5 6 7. 8 9
(2.) & £" M. AM, lº # O 7- b
IO 20 30 40 50 6O 70, 80 90.
(3) p q t w if x \r w 2
100 200 800 400 500 600 700 800 900
(4) a, B, 7, 8, s, s, £, 7, 6,
1000 2000 3000, 4000 5000, 6OOO 7000 8000 9000,
The fourth class is a repetition of the first, each
letter having a subscribed t, or dot, by which its value
was augmented one thousand fold. -
The limit of Greek Arithmetical notation, as far First limit.
as it was dependent upon the symbols in the preceding
table, was 9999, which was expressed in symbols by
6, 2 h 9, and in words by evvea x\taðes évveakogia
evveakowra évvea. -
Their language, however, contained a term uvpias
for the next superior unit, and consequently their
numeration by words, proceeded farther than their
numeration by symbols ; by making use, however,
of the letter M or Mv subscribed or postscribed to the
symbols for any number within the limits of the pre-
ceding table, its value was augmented ten thousand
fold, in the same manner as the values of the digital
symbols were augmented one thousand fold by the
subscribed : thus
a or a . Mw = I0000.
M ..
A # or Af. Mw = 370000. .
M.
7, ºu º or n, ºp. Y. Mw = 85430000.*
MI
By this means the Arithmetical notation of the Second
Greeks was made coextensive with the powers of ex- limit.
pression of their numeral language, embracing eight
places of figures, its limit being 99999999, which was
expressed by 6, 2', 9, 6,250, or by 6.2%. 6. Mv, 6.2%. 6;
M
such is the notation which is found, with many varia-
tions, which we shall afterwards notice, in the com-
mentaries of Eutocius, and in the works of Diophantus
and Pappus. Without considering further at present
the period when this notation was introduced, or the
person by whom it was suggested, we shall assume it
as the second limit of Greek symbolical Arithmetic. 2^
The extent to which the Greeks were thus enabled -
to proceed, was sufficient for all the ordinary purposes
of life; at all events, the inconveniences which might
sometimes arise from its being confined within such
narrow limits, were greatly lessened by the very
considerable value of their primary units of length,
weight, and capacity, and particularly of money.
The speculations, however, of philosophers, which
were called forth by the progress of science in the
* Delambre, Arithmétique des Grecs; Histoire de l'Astronomie
Ancienne, vol. ii. p. 1.
A R. I. T. H. M. E. T. I C.
395
mining the number of places in this number, is very History.
Arithmetic, decline of their literary and military glory, led to the
S-N-2 consideration of greater numbers than were compre-
\-N-
Improve-
ments of
Archime-
hended in this limited symbolical Arithmetic, more
particularly when the increasing accuracy of astrono-
remarkable : he assumes the series
1, 10, 100, 1000, 10,000, &c.
des in the mical observations was giving enlarged views of the commencing by unity and proceeding by powers of
arenarius, extent of the universe; and it was a difficulty of this ten, of which the first eight terms are primary num-
nature, which suggested to the greatest of the geome- bers, the next eight secondary numbers, and so on;
ters of antiquity the necessity of inventing some ex- and the question proposed is, to determine the term in
pedient by which any numbers, however considerable, this series which is equal to the product of any two
might be brought within the compass of their lan; assigned terms, such as the (m+1)th and the (n+1)th,
guage. The work of Archimedes, in which this method or the mºh and ºth terms, omitting the first this is
is explained, is entitled *aphºrns, or Arenarius, from obviously the (m. -- n)th, or the term whose place in
the nature of the question which is primarily proposed the series, omitting the first, is the sum of the num-
to be considered : it is addressed to Gelo, King of bers which determine the corresponding places of the
Syracuse, and commences by noticing the opinion two factors. If this number be 8, the product is a
entertained by some people, that the number of the monad of secondary numbers; if 16, it is a monad of
sand is infinite it “not of that merely which is about ternary numbers; if 20, it is a myriad of monads
Syracuse and Sicily, but which is contained in the of ternary numbers ; and similarly in all other cases.
whole earth; whilst others deny that this number is Thus the product of 10 monads of secondary numbers
infinite, but greater than what can be expressed by into 100 myriads, which are the 9th and 6th terms in
any method of numeration." . In order to prove more the decuple series respectively, after the first, is a
completely the negative of both these propositions, number corresponding to the 15th term of the series
he enlarges the hypothesis, and proposes to express a under the same circumstances, and is consequently a
number which shall exceed the number of the Sand, thousand myriads of monads of secondary numbers;
even in the universe (koopos); of Aristarchus. and in this manner he proceeded to assign the suc-
#. d In order to effect this object, he forms a scale of nu- cessive products of terms in this series, to the extent
CU&CléS,
meration whose radix is a myriad of myriads,or the limit
of the ordinary Arithmetical language. All numbers
required by the conditions of his problem.
Some authors have discovered in this process of Supposed
Archimedes an anticipation of the principle and use of anticipa-
logarithms; and in one sense we may allow that such ion ºf
an opinion is not without foundation; the index of the *
comprehended in this radia are called primary numbers,
and the radix itself becomes a unit, or monad, of
secondary numbers; he thus proceeds to ternary, qua-.
ternary, and other numbers of higher orders, forming
successive classes; and the classes themselves are
called octades, or periods of eight, from their requiring
eight symbols, or in modern Arithmetic eight places
of figures, to express the numbers which are included
in each of them ; Ś and he then shows, without
actually finding or assigning the number itself, that a
number requiring for its expression not more than
eight of these octades, or, in modern notation, not
exceeding sixty-three places of figures, will exceed
the number of the sand in the sphere of Aristarchus.
The method which he has made use of for deter-
* There was another work of Archimedes on the subject of
this extended Arithmetic, entitled Apxat, or principles, addressed
to Xeuxippus, which is frequently referred to in the Wauparns.
f Tot, happiou rôv Špiðuov &repov čival rig tranóēt.
: The cooruos of the ancient Greek astronomers was the sphere,
whose centre was that of the earth, and whose radius was the distance
of the sun, and therefore also, in their opinion, of the sphere of
the fixed stars. Aristarchus, however, by a memorable anticipa-
tion of the knowledge of the true system of the universe, placed
the sun and not the earth in the centre of his mundane system,
the dimensions of which were determined by the following arbi-
trary proportion : “The sphere of the earth was to the koguos, or
universe of other astronomers, as that universe to the sphere of
the fixed stars, or kooruos of Aristarchus. Archimedes, after no-
ticing a geometrical inaccuracy in the mode of stating this pro-
portion, considers the system itself as destitute of foundation;
and assumes it an hypothesis merely which is convenient for the
more striking developement of his method of expressing very
large numbers.
§ *Earw of v šºv of uku vow sipmuévol &gtfluol és Tês uugias uupt-
dāas, trpátol kakoſſaevol. Töv 8& Trpiórov &piffuſov ai pºpuzi uupuděes,
pověs kaxeſorów Aévrépov &piðuāv kal &piðuéla'600'av Bevrépov &pið-
pºv uováðes kal &rb rôv uováðavöekáBes ral ékarovrdóss ſcal x1Aldbes
ral pupićbes és was uvpias uvgléâas. IIdaiv 8& kal ai uiptat uuptáðes
Töv Ševrépov &gt016v, uovas kaxeta.0a rgirov &ptôuðv kal &piðuéio-
600 av Tpſtav Špituſºv uováàes, kal &to Töv Plováðav Šekāāes kai éka-
Tovráčes kal XIXiădes kal uvpiáðes, és Tês pupias ºvgíadas. K. T. A.
power of ten, in any assigned term, is identical with
the number of the terms omitting the first; and the
number which determines the position of the term
which is the product of any two, is the sum of their
indices; in other words, this sum is the logarithm of
their product; and so far the method of Archimedes,
and the principles of logarithms, are identical with
each other. But there was nothing in the state of
knowledge at that time, nor in the nature of the
question which he considered, that could lead to the
invention of logarithms, properly so called, or to the
interpolation of a series of fractional or decimal
numbers between the integral indices 1, 2, 3, &c.
which should correspond respectively to the series of
natural numbers, and which should possess the funda-
mental property, that the sum of any two of these
logarithms, or interpolated numbers, should be equal to
the logarithm corresponding to the product of the
numbers to which the others corresponded. In this
ascription, therefore, of any portion of the credit of
this great invention to Archimedes, we only observe
another example of a practice which is much too
common in the history of the sciences, where the
accidental and unconscious possession of some frag-
ment of a great and general truth, or important in-
vention, is made a ground for detracting from the
honour which is due to their proper authors.
It is hardly possible to speak with certainty of the
actual state of Greek Arithmetic in the time of Archi-
medes; in his KvicAdv Me7pmots, or Treatise on the
Measure of the Circle, we find examples of the symbo-
lical representation of primary numbers exceeding a
myriad, as well as of methods of denoting fractions
whose numerators are unity: thus 14688 is denoted by
a *,x*]': 349450 by M 36, w v; the fraction # is denoted
M M -
3 F 2
396 A R IT H M ET I C.
Arithmetic.
by a peculiar symbol resembling K, the form of which,
S–S- however, varies in different manuscripts; other frac-
Notation
of Archi
medes or
Eutocius,
Improve-
ments of
Apollonius.
tions, whose numerators are unity, are denoted by simply
writing the denominator, after the monads, or whole
numbers: thus 591; is denoted by $5 a'n'; 1009; by
a,0's', 4673; by 8xoºk; 3013} + by Yºy K'6"; and the
fraction 4-3 is denoted by 8éica oa', the numerator being
expressed in words. We should not, however, be jus-
tified in asserting, that such was the notation employed
by Archimedes himself. Successive copyists of manu-
scripts appear to have altered the notation of numbers
to suit the practice which was common in their age;
and the notation of which we have just given ex-
amples, is precisely the same as that which is found in
the Commentaries of Eutocius of Ascalon upon this
Treatise, which were written six hundred years after the
death of Archimedes; and it is most probable that the
notation in the text was supplied by the commentator.
Whatever, however, was the state in which Greek
symbolical Arithmetic was left by Archimedes, it
is quite clear that the speculations contained in the
Arenarius excited the attention of succeeding geome-
ters, and particularly of the celebrated Apollonius of
Perga in Pamphylia, who flourished towards the con-
clusion of the second century before the birth of
Christ. Though the work of Apollonius has perished,
and we have no record even of its name, except in an
obscure allusion to it by Eutocius,” yet the substance
of it formed the second Book of the Mathematical Coi-
lections of Pappus ; a great part of this also has shared
the fate of the original, the unique manuscript of it,
which was left by Sir Henry Saville to the University
of Oxford, wanting the first fourteen out of the
twenty-seven propositions of which it originally.
consisted.t
The improvements introduced by Apollonius were
of various kinds; and are, many of them, of great
importance. In the first place, he appears to have
adopted the plan proposed by Archimedes, of classi-
fying numbers, only reducing his octades to tetrads, or
reducing the radix of the geometric series, by which
the units of these classes increased in value, from a
myriad of myriads to a simple myriad; the units in
each class, after the first, being severally denominated
Auvptas amºn, 6tzrM), 7putram, retpatM), and so on ; and
were denoted by M. a, M. (3, M. Y., M. 8, &c. the digital
number which designated the order of the myriad,
being written after the initial letter M ; in making
this change, Apollonius was probably as much in-
fluenced by the increased convenience of the numeral
language, which was formed by means of it, as well
as from its giving greater facilities to the symbo-
lical notation of large numbers : it will be soon seen,
from an example, to what extent he succeeded.
The chief object, however, of the work of Apollo-
mius appears to have been, the simplification of the
process of the multiplication of articulate numbers;
as the articulate numbers in Greek Arithmetic were
represented by distinct symbols and in practice, a
multiplication table was required of the different com-
* At the end of his Commentary on the Measure of the Circle.
+ This fragment of Pappus was discovered by Wallis, and
published in 1688, and afterwards in the third volume of his
works. There is no doubt of its containing the substance of the
work of Apollonius, as he frequently refers to him by name, and
quotes the specific examples which Apollonius had given,
binations not of the mine digits, but of the thirty-six History.
symbols of which their notation was composed. In S-N-7
our notation we are directed to the product of such
numbers as 50 and 70, from the very nature of the nota-
tion itself, by our knowledge of the product of 5 and 7;
but with them, the symbols v and o for 50 and 70, though
connected, have nothing immediately in common with Ae
e and g for 5 and 7, and the product of the first y, ºp
nothing in common with A. e the product of the
two last. The researches of Apollonius appear to
have been directed to the removal of this great defect,
and to make the multiplication of all numbers de-
pendent upon the combination of the nine digits
merely, with the aid of a few supplemental propo-
sitions. - -
The nine digits, a, B, Y, &c. were called by him. Theory of
Trv6aeves, or bases; and the numbers which are found bases and
in the geometrical series, whose radix is 10, and of .
which any one of these bases is the first term, are “”
called analogous to them : thus t, p, a, or 10, 100,
1000, are analogous numbers to the base a, or l;
#, x, s, or 60, 600, 6000, are analogous to the base s,
or 6 ; and similarly in other cases. In performing
multiplications, he replaces articulate or analogous
numbers by their bases, finds their product, and then,
by means of other propositions, which are in some
measure equivalent to the addition of the requisite
number of zeros, he passes to the proper result. A
few examples will furnish the best explanation of this
process : -
Example 1. To multiply together v, v, v, u,
50, 50, 50, 40, 40, 60.
w v v u pº M & Mv , Mv 60,0000,0000.
! t t t t tº p Mv 100,0000.
e e s 3 8 ºf s, Mo 6OOO. *
Example 2. To multiply together a, T, v, º, or 200,
300, 400, 500.
u, N, or
a T v Ø p : Mv. Mv 120,0000,0000.
p p p p a Mv . Mv 1,0000,0000.
£8 ºf 6 e p K. 12O.f
Example 3. To multiply together t, k, N, K, k, o, t,
v, º, or 10, 20, 30, 20, 20, 200, 300, 400, 500.
ticºkka Tvºj K,”,” Mv. Mv. Mw 28800,0000,0000,0000.
t t t t t ppp p t Mv. Mv. Mv 10,0000,0000,0000.
aſ "I B/3/378 e £,” 7 2880. - -
Example 4. Multiply together a, t, k, A, A, i, B, 7, 8,
or 200, 300, 20, 30, 40, 10, 2, 3, 4.
G. T. .
k \ u' t yºu vs Mv. Mv 3456,0000,0000.
8 y 3 - -
p p t t t t a Mv . Mv 1,0000,0000.
£3 ºf £3 ºf 8 a Bºy 8 ºw v S. 3456.5
The process appears to have been as follows ; first,
write down the numbers to be multiplied together;
secondly, the 100s or 1000s by which the bases are mul-
tiplied to produce those numbers; and, lastly, the bases
themselves. Form the product of the bases; and
afterwards of the 10s, and 100s, which would be done
by allowing 1 for every 10, and 2 for every 100: for
* Pappi Collectanea Mathematica, lib. ii. prop. 15.
+ Ibid. prop. 16.
: Ibid. prop. 18.
§ Ibid. prop. 26.
A R IT H M ET I C.
397
scarches of Archimedes and Apollonius, which could History.
naturally lead to its invention, with the exception of \-N-
Arithmetic, every four contained in the sum of them, there will be
S-S-' a corresponding myriad as a factor in the product.
Artifices of
notation.
In the first example this sum is 6, and the result
p. Mv, or 100 myriads; in the second it is 8, and the
result therefore a . Mv. Mv, or a myriad of myriads;*
in the third it is 13, and the result is t . Mv. Mv. Mv,
or ten myriads of myriads of myriads. In order to
form, therefore, the first product, it remained only to
multiply the product of the bases with the number a, t,
p, or a, which preceded the Mv in the second product;
the rules by which this was effected were contained in
those propositions which are lost; but as there were
only four cases, we may readily conceive what they
were ; thus if Yvus, as in the first example, was the
product of the bases, we should find the product of
a and Yv vs = Yvus.
* and ºvus = ºf or y Mw kat M6. 8%.
p and Yvvs = 7,4,ex or M6 Mv ca, Mo ex.
a, and Yvvs =Tue Mv kat Mo. s.
There are many of the artifices of notation em-
ployed in this work, which if pursued and properly
generalized, would have given increased symmetry as
well as extent to their symbolical Arithmetic; amongst
these we ought particularly to notice the accentuation
by the subscribed t, of the symbols of articulate num-
bers of the second and third order, increasing their
value, as in the case of the nine digits, one thousand
fold. The only reason which can easily be assigned
why this extension of their notation had not been
generally adopted for all the symbols, when once
applied to those of the nine digits, appears to have
been, that as they merely proposed by it, in the first
instance, to make their notation coextensive with the
terms of their numeral language, they paused when
that object was effected; and, however simple its ex-
tension to all the other symbols may have been, it was
not likely to be adopted when the utility of it was not
felt ; the advantages indeed of a simple and expres-
sive notation addressed to the eye, as distinct from
language, were in no respect understood by the ancient
geometers; and it is only in modern times that the
powers of symbolical language have been completely
appreciated. -
The use of the initial letters Mv in such expressions
as ºff, Mv. Mv for 12000,0000,0000, might at first
sight appear to resemble the modern zero, in a scale
of notation proceeding by myriads; but it can only
be considered in this case as the abbreviated expres-
sion of Mvpuas Mvptaëtov, and is never employed to
give value from position, without reference to its
value as a factor : thus 347900006008 is expressed by
ºvo0. Mv. Mv ca, Mo. sº, and not by Yvo9. Mv. sm.
There is nothing, in short, in Greek Arithmetical
notation, which, in the slightest degree, resembles our
own; and nothing in the object proposed in the re-
* Muglas Mupiabaov. *
+ Some Lexicographers and writers on Greek Arithmetic have
mentioned another extension of this notation, and have quoted
Herodian the Grammarian for their authority, though it is not
noticed by him; it consists in increasing the value of the first
twenty-seven symbols 1,000,000 times, by adding two accents to
them, 1000,000,000 times by adding three accents, and so on;
thus a is 1,000,000, a 1000,000,000, and so on. -
the discovery of the very important fact, that the mul-
tiplications of the articulate numbers depended upon
that of their bases. -
Pāppus, at the conclusion of this fragment, has
given from the work of Apollonius two examples, to
prove the facility of multiplying any numbers, how-
ever large, by means of the process which he had
explained in the preceding propositions. In one case
it is proposed to find the continued product of the
numbers expressed by the several letters in the
Verse :
Apréutôos k\e?re kpótos éoxov čvvéa coöpa,
and in the other, in the verse,
Măviv ćetēe 6ed Amujtepos ຠMadicapirov. . .
In the first example, and the only one which we
think it necessary to notice, he multiplies the bases
successively, and the resulting product
19,6086,8480,0000,0000 is expressed by
M3. 6 ca, My .svg kai MB. myr, where M8, My,
M6 denote myriads of the fourth, third, and second
orders respectively. If this number be multiplied into
the continued product of the decads and centuries
(écatávtaðss) which is 10 myriads of the ninth order,
the final product, or 196,0368,4800,0000,0000,0000,
O000,0000, 0000, 0000, 0000, 0000, 0000, 0000, is ex-
pressed Mº e phs FC Olt, Mið {º Tºm ka, Mia. 8,0.
Delambre has noticed forms of Greek notation,
which appear to favour the notion that the principle
of value from position was in later times in some
measure understood ; thus in Diophantus, we find Notation of
. 3069000 Ay . a, thos
the fraction : * expressed by ts. 9.” * *
331776 pr y ts. U,
where the numerator' is 7s . 6, and the denominator
My . a pos. Great and important as this simplification
of the ordinary notation certainly is, it is seldom used
by Diophantus, except in his fourth Book, and very
rarely, if ever, by later authors; in other parts of his
works, the abbreviation Mv is either prefixed or post-
fixed to the symbols which denote myriads, and the
abbreviation Mv is sometimes prefixed to monads
and sometimes omitted; thus, in one place we find
12768 denoted by Av. a. Bººn, and in the next line
the same number is denoted by Av. a. ºo. 6 ºn it
in another place 17136600 is denoted by uv a u-ty.
povaðes sp.: ; and again 163021824, the only example
in his works where a myriad of the second order is
involved, is expressed by up a. a. s.73. uo. a, wrº.f
Amidst such a total want of uniformity of notation,
we may fairly infer, that Diophantus was insensible to
the value of his own discovery. Theon, who lived at
a later period than Diophantus, and who was well
acquainted with his writings, expresses the number of
cubical stadia in the earth, or 38406364469497, by
Avptavračucuy Tpºwu ºn, Avptavtaðukwu 8tºratov 8,5),
avptačes arxat sups, kal 0.9%, after the manner of
* Diophanti Arith, lib. iv. prop. 46.
+ Ibid. lib. iii. prop. 22.
f Ibid. prop. 86.
398
A R IT H M ET I C.
Arithmetic. Apollonius, and in no case does he adopt the notation
S-N-2 in question; no notice of it is discoverable in the
Numeral
powers of
words.
Process of
imultipli-
cation of
integers.
Commentaries of Eutocius, who lived at a still later
period ; under such circumstances, we should feel
strongly inclined to ascribe this form of notation to
the omissions of the successive transcribers of the
manuscripts.
It appears to have been a favourite practice with
the Greeks of the later ages to form words in which
the sum of the numbers expressed by their component
letters should be equal to some remarkable number;
of this kind were the words appagaš and aftºpagaša ;
the letters in which express numbers, which, added
together, are equal to 365 and 366, the number of
days in the common and bissextile years respectively;
and it was also remarked that the word vet\os pos-
sessed the same property with the first of these words.”
Observations like these, however trifling, are not
without their portion of curiosity; but the same in-
dulgence cannot be shown to the absurdities of those
Pythagorean philosophers who, amongst other extra-
ordinary powers which they attributed to numbers,
maintained that of two combatants, he would conquer,
the sum of the numbers expressed by the characters of
whose name exceeded the sum of those expressed by
the other. It was upon this principle that they ex-
plained the relative prowess and fate of the Heroes in
Homer, IIatpokMos, “Ekrwp, and AxtMNevs, the sum of
the numbers in whose names are 861, 1225, and 1276
respectively.*. -
It is not very easy to give a complete account of
Greek Arithmetical operations; there is no work of
antiquity extant in which they are specifically detailed,
and it is only in the Commentaries of Eutocius on the
measure of the circle of Archimedes, that we can find
any considerable number of examples of multiplica-
tions exhibited at full length ; and even in this case
the variations which are found in different manuscripts,
in the order and form in which the different steps and
symbols in the processes are written, prevents our
speaking in a positive manner at least with respect
to them.
The following examples are taken from the Com-
mentaries of Eutocius, on the third and last proposition
of Archimedes on the measure of the circle, in which
it is chiefly required to find the squares of two num-
bers, and to assign the square root of their sum :
ExAMPLE I.
To find the square of p vºy, or 153.
p v Y
p v Y
a Me,”
e.8, ºp v
T p v 6
8Mº, v 6
In performing the operation they proceeded from the
right to the left, and the successive products are written
down separately, without any incorporation with those
which precede or follow them; they do not appear to have
adhered with much strictness to any order of magnitude
in writing down the successive results, or to have been
very solicitous about writing them underneath each
* Words in which the sums of the numbers expressed by the
letters were equal, were called äväuara ia 6ilmqa; and we have an
example in the Greek Anthology, where the Poet, wishing to ex-
press his dislike of a pestilent fellow of the name of Aguayopas,
says, that having heard that his name was equivalent in numeral
power to Aotubs, proceeded to weigh them in a balance, when the
latter was found to be the lighter of the two.
Aapayágav kai Aotubv iodºmºpov ris &Kotoras,
"Eatma' apporépov Töv Tgórov čk kavóvos'
Eus to uégos 5& ka9etaker’ &vexicva'6&v to réxavrov
Aapayápov, Aotºv 5’ eſſpew éAappérepov. -
Æſistoire de l'Académie des Inscriptions, vol. v. p. 209.
153
I53
10000 or a M.
5000 or e,
3OO Or T
5000 or e,
2500 or 8.6
150 or p v
300 Or ºr
150 or p v
9 or 6
23.409
other, as they are sometimes in the same line. They
then performed the additions much in the same way
* This very singular superstition continued in force as late as
the sixteenth century, and was transferred from the Greek to
the Roman numeral letters, I, U or V, X, L, C, D and M,
which correspond to the numbers 1, 5, 10, 50, 100, 500, and
1000 : thus the numeral power of the name of Maurice (Mauri-
tius) of Saxony was considered as an index of his success against
Charles V. It was the fashion also to select or form memorial
sentences or verses to commemorate remarkable dates. Thus the
year of the Reformation of religion in Germany (1517) was
found to be expressed by the numeral letters of the verse of the
Te Deum ; Tibi Cherubin et Seraphin incessabili voce proclamant,
in which there is one M, four Cs, two Ls, two Us or Vs, and
seven Is. In a similar manner, the defeat of Francis at Pavia.
(1525) is commemorated in both the following verses:
Regia succumbunt pugnacis lilia Galli ;
and
See Henischius, de Numeratione Multiplici ; and Hostus, de
Numeratione emendatá, veteribus Latinis et Graecis usitatá, 1632.
Captus erat Gallus, coeunt cum rure cohortes.
History.
N-N-
Arithme-
tical ope-
rations.
A R IT H M ET I C.
399
|
Arithmetic, as in the addition of concrete numbers of different
*Y* denominations in common Arithmetic, beginning with
Notation of
fractions.
the digits and advancing in succession through the
different orders of articulate numbers. The scheme
which is given of this multiplication in common
figures will render this process perfectly clear: the
product of p and p is am, or 10,000; of p and v is so or
5000; of p and y is T, or 300; of v and p is e, or
5000; of v and v is 8, º, or 2500; of , and y is p v, History.
or 150; of Y and p is t, or 300; of Y and v is p v, or
150; and of Y and Y is 9, or 9. In Greek notation it
is clearly a matter of indifference in what order these
successive products are written ; whilst in our nota-
tion the value of the digits depends on the number of
places which follow them.
ExAMPLE 2.
To find the square of ap Egº' or 1162 +.
a,p # 8'n'
a,p #3'n'
PM M M 8.pke
‘M am s, a 8 K
sM s, q, xp k & K
B, a p k 3/8"
p a e 8' 8"
—a-
p A e º M & K #
M
This example involves fractions, and the proeess
will be sufficiently explained by the scheme with
which it is accompanied. The fraction # is denoted
by the peculiar symbol K ; and the other fractions,
whose numerators are unity, by writing the numbers
in the denominator immediately after the integers, the
distinction between them being marked by an accent;
thus, in Ptolemy, we find 34 + + ºr denoted by
1162 &
1162 .
—w
1000000 or py
100000 or “M
60000 or sm
2000 or 6,
125 or p ice
100000 or “M
10000 or aw
6000 or s,
2OO Or or
12+ or v BK
60000 or sm
6000 or s,
3600 or y, x
120 or p ic
7# or g K
2000 or 6,
200 Or or
120 or p c
4 or ö
+ or 8
145; or p 1, 6
tºr or # 8'
13505.34+ ºr
A 58' 0", 8". In the following example the fraction
-ºr has its numerator and denominator written imme-
diately after the integers, thus 6' aſ: but when mixed
up with integers in a manner which might lead to
some confusion, the denominator is placed above the
numerator to the right hand, in the manner of an
index in Algebra.
400
A R IT H M ET | c.
Arithmetic. . ExAMPLE 3. . History. .
-N- . gº - \-N-"
To find the square of ā, w N 7' 6' 4 aſ or 1838 ar.
a, w X m 6', aſ 1838 ºr
a, tº N m 6', aſ 1838 ºr
p 7 ºf 7, a m’ g” IOOOOOO Or "M
M M M
g i. ; 8, s, v × v 3 s!" 800000 or 1,
ºf 3 8, 2 a p r 8's' “ 30000 or ,
M M -
m, s. v c a # 8 s' s” 8000 or m, f
w a 7/8” “x v 8' s” 818 ºr or w w n' 6”
& 3, sº *s, *, *, aſ Pº" 800000 or TM
- 640000 or § 6
t a . K Cº. - M
*** **** ºr afé 24000 or £8M 8,
or T A M a, a v 6' x tº “” 6400 or s, v
M
654+ºr or X w. 3' sº **
30000 or YM
24000 Or PM3,
900 or 2
240 or a pº
24, or k 3 s”
8000 or m,
6400 or s, v
240 or o p.
64 or # 8
6,ºr or s” s^**
81812, or w w m/8”
654 ºr or x v 8’s”
24 ºr or c 3’s' * *
6,ºr ºf r or s s^** 7 aſ P**
3381951-17F ºr
or 338,1252 +2.7F
Difficulty Eutocius, in the conclusion of his Commentary, states reason which is expressly assigned by Ptolemy for hi
of multi- that Philo, of Gadara, had brought the approxima-
plying tion to the length of the circle to greater accuracy
º than Archimedes, in consequence of extending his
. ...t multiplications and divisions to numbers involving
limit of myriads, which, he says, are difficult to follow,
Greek ... unless by a person well versed in the Logistics of
Arithmetic Magnus. The term Aoytotºxn is applied to the whole
i. science of arithmetical calculation ;
suppose that the work to which Eutocius refers,
expressly treated on these subjects to an extent which
they rarely attained in other books. The examples
which we have given, show how very difficult and
embarrassing these operations must have been, parti-
cularly when fractions were involved; and it is this
and we may
preference of sexagesimals.” *.
Eutocius has given no example of division ; and in
the repeated instances in which the square root of a
number is required, he assumes the root, and then
shows that its square coincides with the proposed
number or nearly so ; thus, in extracting the square
root of Ø a £ 3, p. A 6' 4 s, or 5472131 ºr, he assumes
M
it to be 8, T N 0. 8', or 2339 #, finds its square or
© M & £, º, K s', or 5472090 + ºr, which differs from
the number whose root is required by Mo. At a K, or 414.
* Meyaam Xuvračis, lib, i. ch. ix.
A R IT H M ET I c.
401
Arithmetic. In the Commentaries of Theon on the Almagest
S-N-2 (MeyaM) >vvrafts) of Ptolemy, we find a statement of
Sexagesi-
mal Arith-
metic.
By whom
invented,
the rule for extracting the square root, which corres-
ponds in essential points with the one in common use,
but it is not accompanied by any example exhibiting a
type of the operation. In the same author we find
also many examples of division, performed upon sex-
agesimals. Before we proceed, however, to notice
them further, it may be as well to premise a few
observations on the origin and design of this species
of Arithmetic.
(39.) The division of the circle into 360° seems to
have been pointed out to the earlier astronomers, by its
being an articulate number nearly equal to the days in
the year; and, consequently, one of these degrees
was nearly equal to the portion of the ecliptic
described by the sun in one day. Whatever, how-
ever, were the grounds upon which this division was
adopted in the first instance, it was adhered to after-
wards in the most improved periods of ancient and
modern astronomy, from a sense of the convenience
presented by the number 360 in the great number of
its divisors. The angle subtended by the side of a
hexagon inscribed in a circle was therefore 60°, and
the corresponding portions of the circumference were
called uotpat, parts or degrees; each uoupa was
divided into 60 MeTTa, or minutes, or primes, or sex-
agesimals of the first order; each minute into 60
seconds, or sexagesimals of the second order; and so
on proceeding to trines, quaternes, &c. in a descend-
ing series. But this sexagesimal division was not
confined to degrees of the circle : the side of the in-
scribed hexagon itself, which is equal to the radius,
was likewise divided into 60 poupau, and the same
series of sexagesimal subdivisions were applied to
these rectilineal degrees (uotpat evéewv) as to those
of the circumference (woupal treptºepewv ;) and as the
whole business of calculation in ancient astronomy
was reduced to the arcs and chords of circles, this
sexagesimal Arithmetic superseded every other in
works on that subject.
The invention of this species of Arithmetic is
attributed to Ptolemy by his commentator Theon,
and later authors; though, if we might judge from
the language of Ptolemy himself,” when explaining
the principles upon which his table of chords was
constructed, we might be inclined to think, that
the sexagesimal division of the degrees of the
circle was known before his time, and that he only
applied it to the division of the radius. Whoever,
however, was its author, it must be considered as the
greatest improvement in the science of calculation
which preceded the introduction of the Hindoo no-
tation; it enabled astronomers at once to get rid
of fractions, the treatment of which, in their ordinary
Arithmetic, was so extremely embarrassing-; and
enabled them to extend their approximations, particu-
larly in the construction of tables, to any required
degree of accuracy. .
The notation of sexagesimals, as it appears in
Ptolemy and his Commentator, is nearly the same as
that which is made use of in modern astronomical
writings; the degrees, or woupae, were considered as
units, and written in the ordinary manner, a stroke
being placed over the last symbol, as in a 3, or 44°.
The successive orders of sexagesimals, primes, seconds,
trines, &c. were denoted by one, two, three, &c.
accents, as in modern astronomical notation ; thus
a? n' v g” a 6” 4 s” x 9" is equivalent to
14° S. 57// 42” 26iv 397.
It is quite clear, that in , this notation all symbols
beyond #, or 60, were superfluous; and, as in many
cases a zero was necessary to signify the absence of any
one term in the series of sexagesimals, the symbol o
next in order to it was taken for this purpose, as it
could not be confounded with any of those which this
notation made significant ; thus o c 3’ t s” denotes
O° 24′ 16”; , s o' u" denotes 16° 0' 40". It is a
curious circumstance, that this symbol for zero, trans-
mitted from the Greeks to the Arabians, and from
thence to Europe, was adopted as the zero in the
Hindoo notation, having superseded the simple dot
which was generally used for that purpose amongst
the people from whom it was derived.
We shall now give a few examples of Arithmetical
operations on sexagesimal quantities, in doing which
we shall take for our guide the Commentary of Theon
on the 9th Chapter of the Almagest of Ptolemy, of
which the chief object is the exposition of the princi-
ples and practice of this Arithmetic.
ExAMPLE 1.
To find the square of x & 8' ve" or 37° 4' 55/.
ſ y e”
g
6
S’ |/ e”
&
A.
A.
a, T = } p it n' B, \ e"
p A iſ FF
t 5" a k"
8, \ e” or k'ſ/ ºf, k e////
*
A ºn // */ . . ////
a, t # 6 a 5 s o, Ts v u!” º, ke
The multiplications are performed in the same
manner as in duodecimals in our common books of
* Meyaam ºvvrafts, bibl. A. kep, 0.
WOL. I.
37° 4' 55"
37° 4/
&
2035”
16' 22O///
2035// 22O’’ 3O25”
1369° 148/
j48’
1369° 296' 4086” 440” 3025////
Arithmetic, only proceeding from the right to the
left; and it is probable that the multiplications were
rendered more easy by means of a sexagesimal table,
containing the products of all numbers with each
3 G
History.
Notation,
Symbol for
ZCTO's
A -- ~~
Example of
multipli-
cation.
402
A R. I. T H M E T I C.
Arithmetic. other as far as 60;* another question also which
S-v- Theon has considered, was to determine the order of
Of division.
the product of sexagesimals of the same or different
orders; thus, the product of two primes are seconds,
of seconds and primes are trines, of seconds and
trines are quinquines, and generally the order of
the product of any sexagesimals will be the sum
of the orders of the component factors ; a fact,
º, ke” = o of o' vſ/’ ce”/
v u// F - g" k///
// —- P //
8, it s a n s
Y
a 9 s = 3 vs’
W w
a, + £6 = a, 7 # 6
*
a, to e 3/ , sº (ſ/ ſce////
which will be very evident to any one who under- History.
p q p q
t the th -
stands eorem ºr x gº 66 m + .
It now remains to divide the successive sums of
these sexagesimals by 60, so as to reduce them within
the proper limits of the sexagesimal notation.
3O25iv = O9 O' O// 50// 25iv
440’’ = O° 0' 7” 20%
4086’ = 1° 8/ 6//
296 = 4° 56/
1369° 1369°
1375° 4' 14" lo” 25°
Additions, as well as subtractions and other arithmetical operations, appear to have been performed from
right to left; a method which was subject to considerable inconvenience, particularly in the two first cases,
from their requiring a constant reference to the numbers in the subsequent columns. “Theon has proposed
the following example of division, and detailed the process.
may, however, be easily supplied.
He gives no scheme of the operation, which
ExAMPLE 2.
To divide a, , , e iſ , º, by ic d a 3' " or 1515° 20' 15" by 25°12' 10".
\
a, q t e k! t e” ice a 3’ tº
y O f X. //
a, ‘p #" g’ A y
2 k'
y, c’
o/
/
&
/
p';
p o e”
2 t e”
7F 3//
w \, a”
a// .”
&O K. 6// 1///
w k e”
*==
a 5”
T º s///
* Such tables were in general use when operations in this Arithmetic were required amongst astronomers before the deci-
mal division of the radius, and may be found in many works, both astronomieal and arithmetical ; and, amongst others, in
Wallis's Algebra.
A R P T H M E T I C.
403
Arithmetic. - . . .
g © c (ExAMPLE 2, continued.) . History.
º \-y-f
1515° 20' 15// 25° 12/ IO/
25° x 60 1500° - g
60° 1st quotient.
15° = 900'
920.
I2' × 60 7207. \-
- 2OO/
IO’’ x 60 IO/
1907 25°12' 10"
2 O As 'A * - / t º
5° x 7 175’ 7’’ 2nd quotient.
15'-900//
915//
12” x 7” 84//
83].” f
10' x 7' I? IO///
SQ9 50/// 25° 12° 10'ſ/
O // -
25° x 33 825 33' 3d quotient.
4// 50'’ - 290///
12/ × 33" 396///
— 106
The quotient is nearly, therefore, 60° 7' 33".”
The operation requires no farther illustration than what is afforded by the preceding schemes, and accurately of extract-
resembles our processes for compound division. This example forms a natural introduction to one for ing the
extracting the square root, which Theon afterwards subjoins, referring for the proof of the operation to the **
figure and result of the fourth Proposition of Euclid's Elements.
ExAMPLE 3.
To extract the square root of 8, à, or 4500.
\
8, Ø f : 8 , ºr
6, v ºr 6 p \, 6
3 *
ºf v s! t s”
W
g. v c 6" | p \ 6 m’
& + o'
l
g” k’’
1,7// R. e//, /
Aw e// A. 6/// X. e////
* Delambre, Histoire de l'Astronomie Ancienne, tom. ii. p. 25.
3 G 2
404 A R IT H M ET I C
Arithmetic. (ExAMPLE 3, continued.)
S-N-'
4500° 67° 4' 55"
4489 134° = 2 × 67°
119 – 660'
536/ 16/
123' 44” = 7424/ 134° 8/
134° x 55 7370?
8' x 55' 7// 2O///
55" X 55” 50/// 251V
45// 49/// 35iv
Process for The process is as follows: the greatest number
extracting whose square is less than 4500 is 67; subtract the
.* square of 67 from 4500, and the remainder is 11°, or
& 660’; double 67, which makes 134 ; the next term in
the root is 4', which, multiplied into 134°4', produces
536' 16"; subtract this, and the remainder is 123'44",
or 7424"; the double of the root already obtained is
134° 8', and the next term in the root obtained by
trial is 55", which, multiplied into 134° 8' 55", and
the result subtracted, leaves a remainder 45/49' 35".
It is clear that the same process may be continued to
any required degree of accuracy. The scheme of the
operation, which we have copied from Delambre,
agrees substantially with the process given by Theon;
at its conclusion he has stated the rule with perfect
distinctness in the following manner : “Find the root
of the nearest square to the whole number ; subtract
this square, convert the remainder into primes, and
divide it by the double of the first root, and thus
determine the next term in the root; square the sum
of the terms found, subtract this square, convert the
remainder into seconds, and divide it by the double
of the root already found, and you will have the
square root very nearly.”
Reasons for (40.) The sexagesimal division of the circle has con-
continuing tinued to our times, and is likely to continue, notwith-
the º: standing the attempt made in France, at the same period
simal divi- S-> 49 49
son of the that they altered their measures of length, weight, and
capacity, to replace it by the decimal division, or
rather centesimal, and which has been sanctioned by
the authority of Laplace. If the alteration had com-
menced with the centesimal division of the degree
which should itself have remained unchanged, it would
probably have met with general adoption, as it would
have produced a considerable simplification of loga-
rithmic tables, and would have assimilated trigonome-
trical with all other processes of calculation; but, by
attempting to change the primary divisions of the
circle, they not only abandoned the advantages pre-
sented by the number of divisors of 60, 90, 360, of
which artists employed in the division of circles are
very sensible, but likewise proposed to render useless
the whole mass of existing tables, unless they had
been calculated anew.
We have likewise retained the sexagesimal division
of time, and have not merely retained the accentual
notation, but likewise the names, such as minutes and
seconds, which are connected with this division.* The
sexagesimal division of the radius continued until the
year 1464, when Regiomontanus, in his Opus Palati-
num de Triangulis, divided the radius in ten millions;
he at first proposed to divide it into sixty millions of
parts, but abandoned his intention upon farther con-
sideration, as we learn from the relation of Valentine
Otho, in his Preface to that work.
(41) In reviewing the history of Greek Arithmetic, we
find it indebted for its greatest improvements to the
same persons who contributed most to their geome-
trical and astronomical science ; to Archimedes, for
his indefinite extension of their numeral language; to
Apollonius, for his distinction of bases and analogous
numbers, and the practical methods of multiplication
which were founded upon it ; and, most of all, to
Ptolemy, for his refined invention of sexagesimals, by
which fractions and integers were brought within the
compass of a common and uniform notation, and sub-
jected to the same arithmetical operations. To this
list we might, upon the authority of the learned and
accurate Delambre, add the name of Diaphontus, for
the artifice of denoting myriads from position merely,
by interposing a dot between symbols for myriads and
monads, omitting the initial M v, or M, which are
usually attached to the former; but we have given
some reasons above for inducing us to believe, that if
this artifice was really made use of by him, he was
insensible of its advantages, as this important principle
was nearly barren in his own hands, and is never
noticed by subsequent writers. e
Delambre considers it a fact humiliating to the pride
of human genius, that the discovery of the notation by
nine digits and zero, should have escaped the sagacity
of these illustrious men, especially when engaged in
researches connected with the improvement of arithme-
tical language and notation. To us, with whom this
notation has been familiar from our boyhood, the inven-
tion of it may appear simple and easy; but with them it
ran counter to all their associations. They had been ac-
customed to the use of twenty-seven independent sym-
bols, which all appeared equally necessary for arithme-
tical notation; and it was not a very simple investigation
which showed that nine of them only were necessary
in arithmetical operations. In order to pass from this
conclusion to their use in the expression of all
History. .
circle.
* The primes were called Aerta, that is
minuta, or small
portions of the poipa, or integral part. *
Recapitu-
lation.
numbers, there was required the invention of the zero
and the device of place, both of them refinements of
a nature not easily discovered. The Greeks also were
altogether ignorant of the advantages of notation as
distinct from language; and were unacquainted both
with the powers of algebraical symbols, in exhibiting
A R L T H M E T I C.
405
Arithmetic at once to the eye and to the mind the most compli-
S-V-' cated relations of quantity, and such as language is
incapable of expressing without extreme difficulty;
they, in consequence, always appear to have considered
numerical notation as of secondary importance to
numerical language, and never attempted to make
them independent of each other.
! If Ptolemy had found the degree of the circle
divided into 10 minutes instead of 60, and similarly in
all further subdivisions, he would have been led to
the invention of the decimal instead of the sexagesi-
mal Arithmetic, with the zero, and much at least that
is most essential in the device of place; for the ac-
centual marks which distinguish the several orders of
sexagesimals, though they made the zero unnecessary,
did not supersede it ; and the order in which these
quantities were written gave their relative value with
respect to each other, and their absolute value with
respect to the primary unit. It might be objected,
indeed, that the sexagesimal division was applied in a
descending and not in an ascending scale; the units
themselves being written in the ordinary notation, and
not classified according to ascending powers of 60:*
but we must keep in mind that this notation was in-
troduced for avoiding the inconveniences of the
common notation in the treatment of fractions, and not
for the purpose of superseding it ; and that its in-
ventor and his successors naturally terminated their
innovations, when they had fully answered the purpose
for which they were introduced,
(42.) It is impossible, from any existing records or
monuments, to fix the date of the origin of Greek
arithmetical notation. We may assume it to have been
introduced subsequently to their alphabet, and that it
was unknown also at the period of the colonization of
Latium, as no traces of it are discoverable among the
Romans; t and we have before mentioned (Art. 22,); a
notation mentioned by Herodian the Grammarian, as
made use of by Solon, in writing his laws, and which
is frequently observed in ancient coins, and in monu-
mental and other inscriptions. Thus in the Arun-
delian marbles we have the inscription, (in modern
Ancient
Greek
arithmeti-
cal nota-
tion.
* We find, however, in the Commentaries of Theon on the
fourth Book of the Almagest, examples of superior sexagesimals,
though the highest order of the sexagesimals are considered, as
far as the notation is concerned, as the primary units: thus, in
reducing : v i g iſ a 6" waſ" p!", or 74.12d 10° 44' 51", 40iv to the
sexagesimal notation, he divides 7412 twice by 60, and writes
down the result after the first division under the form
grº, A 8' !" p. 67/ y a!” a'ſ/
and after the second as follows,
5 'y Ag" ," p. 5'ſ" va” aſ”
which is equivalent to
2sex sex. 3sex. 32d 10' 44' 51/// 40iv.
It is clear from these examples, that the accents had reference to
relative value from position only ; and the quantities to which
they were attached varied with the variation of the value of the
primary units.
† In the later ages of the Roman Empire, the Greek numeral
notation was sometimes made use of ; the digits were denoted
\ *
by a, b, c, d, e, f, g, h, , ;
the articulate numbers of the first order by
#, l, m, n, 0, p, q, r, s:
and those of the second order by
- t, w, w, y, z, I, V, hi, hu, ta.
Henischius, de Numeratione Multiplici.
# It is found in a short Tract repi Tây &pſguſov amongst the
Grammatici Veteres.
characters,) 'Aq} of; Kékpop. A0mvæv égagéAsvge, ka? # . History. -
xwpa Kekporta ék\#6m to Tpdrepov caxovuévn Aktuk.) STN-
âzrö Actatov Avrox06vos éry XHHHAIIIII, (1318;) and
again, 'Aſ of Quñpos & Toujins épavī0m, HHAAAAIII,
(643.)* But it by no means follows from the use of
these numerals on such occasions, that the other were
unknown ; it is sufficient that the one were more
ancient than the other, to induce engravers and others
to make use of them, whether from respect to, or
affectation of, antiquity. Such at least may be easily
imagined to have been a prevalent feeling, if we may
judge from the practice of modern times.
(43.) The Greeks derived their alphabet from the Phoe- Arithme-
nicians, and from a similar source they derived also the
tical nota-
tion of
use of their ordinary numerical notation ; for we find Semitic
the same system in use amongst the Hebrews, Syrians,
and in short amongst all Semitic nations. An enu-
meration of some of those systems, combined with
some observations on the names and positions of the
three interpolated symbols, will render their origin
perfectly clear. -
The following is the system of Hebrew numerals:
1. R Aleph. 60. D Samech.
2. - Beth. 7 O. W. Ain.
3. l Gimel. 80. S. Pe.
4. T Daleth. 90. Nº Tsadi.
5. In He. 100, p. Koph.
6. Vau. 200. - Resch.
7. Zain. 300. U Schin.
8. T Chet. 400. In Thau.
9. to Teth. 500. 1 Caph final.
IO. Jod. 600. D Mem final.
20. 5 Caph. 700. Nun final.
30. 9 Lamed. 800. In Pe final.
40. Yo Mem. 900. y Tsadi final.
50. J Nun.
The ancient Hebrew and Samaritan alphabets con-
sisted of only twenty-two letters, and the simple
numeral symbols proceeded no farther than 400; to de-
note 500, they combined the symbols for 400 and 100,
thus, ph; 600, ºn; 700, wh; 800, nh ; 900, pH n.f
The same is the case also in the Syriac characters;;
and, according to the statement of De Sacy," with
the alphabet of the ancient Arabs. It was only in
later times that they appear to have" added the five
final letters, to bring their numeral notation up to the
limits of their numeral language.
(44.) The comparison of the Hebrew numeral cnárac-
ters with those of the Greeks, will show at once their
common origin, particularly when combined with
the names which were given by the Greeks to their
interpolated symbols ; thus Alpha, . Beta, Gamma,
Delta, Epsilon, correspond with Aleph, Beth, Gimel,
* See also Rose's Inscriptiones Graeca Vetustissimae, p. 41 and
137–140.
† Professor Leslie, in his Philosophy of Arithmetic, has charac-
terised the numerical system as well as language of the ancient
Hebrews, as equally remarkable for their poverty and rudeness.
It is difficult, however, to discover upon what grounds this re-
proach is founded, in one respect at least, when we find that
system adopted, with very few changes, by the most improved
nation of antiquity; and that even under this form it was superior
to that which continued to be employed by the Romans throughout
their empire.
† Beveridge, Arithmetices Chronologicae, lib. i.
§ Grammaire Arabe, vol. i. p. 74.
nations.
Of the
Hebrew8.
Greek al-
phabet and
numeral
symbols of
Semitic
origin.
406
A R I T H M E T I C.
Arithmetic.
Daleth, He, which denote 1, 2, 3, 4, 5; and also Zeta,
S-N- Eta, Theta, with Zain, Chet, Tet, for 7, 8, 9; but in
Hebrew
Arithmetic.
ºrabic.
Russian.
the Greek there is no letter corresponding to the
Hebrew Vau, which denotes 6; and they consequently
interpolated the symbol s for this number, bearing as
much resemblance in form to the corresponding
Hebrew letter as is found amongst other letters of the
alphabet, and expressly denominated by them éréon-
pov Bao, that is, indicating Vau, to show its place in
the system from which it was taken. The other two
symbols were 5 and 2, denominated €Téamuov corrà and
&Téamuov gavri, that is, indicating Koph and Tsadi.” It
is observable also that the symbol Koph has receded one
place in its transmission to the Greek system, whilst the
other symbol, Tsadi, or oauti, may, or may not, be in
its proper place, according as it is used for the final
or initial letter of that name. Under any circum-
stances, the names as well as the positions in the
system of these interpolated symbols, are more than
sufficient to ascertain their origin, particularly when
the discordance in the second half of the second, and
the whole of the third ennead of symbols is considered,
which arises from the diversity of the alphabets ;
from the vowels in one, and the compound letters
in both. - -
(45.) In returning again to the Hebrew Arithmetic,
we find little which distinguishes it from the Greek.
Compound numbers were denoted by the combina-
tion of the symbols of the component numbers :
thus x 5 is 21; W = - is 932; the number 15 they de-
noted by to, or 9 and 6, and not by n”, Jah, one of the
names of the Deity, which could not be pronounced
without profanation. In some cases they denoted
thousands, by denoting the number of thousands by
its proper symbol, and the other numbers after it thus,
ºns, 1430, where s denotes 1000; a 5- i. 5242,
where n denotes 5000; but this is seldom done, unless
succeeded by an articulate number of the second
order; thus 1030 is hardly ever denoted by . R. It is
not our object, however, to describe all the artifices
of notation to which they resorted ; it is sufficient for
us to exemplify a system of Arithmetic made use of
in the most ancient of languages, and which has been
from thence transmitted, either directly or indirectly,
to so many nations.f
(46.) The ancient, or Cufic Arabic characters, were
derived from the Syriac, and were only twenty-two in
number. The modern characters were introduced
about the year 800 after Christ, and are twenty-eight
in number, though six of these are only different
forms of the same letters when they appear in the
middle and end of a word, like the Hebrew finals.;
The Arabians were thus possessed, not only of the
three enneads of symbols, which were used by the
Hebrews, but likewise of a simple symbol for 1000.
The same system was found also amongst the Persians,
the Copts, and every other people whose language
was in any considerable degree of a Semitic cha-
Tacter.
(47.) The Russians derived their alphabet from the
Greeks, amplified, however, so as to embrace the
* Seyffarth, de Sonis Literarum Graecarum, p. 592.
+ Beveridge, złrithmetices Chronologica”, lib, i.
# De Sacy, Grammaire Arabe, p. 74.
greater variety of sounds which their language required;
the thirty-six letters * of their alphabet gave them
numeral characters for all numbers below 10,000, and
the system of accentuation extended their notation as
far as any number less than 100,000,000. This nota-
tion continued as late as the time of Peter the Great,
who introduced the Hindoo numerals ; and in public
and formal documents is sometimes made use of even
at this time. We find it used also among Gothic and
Scandinavian, as well as Sclavonic nations ; and it
was only abandoned when the influence of the Latin
language, in the first place, made way for the Roman
numeral characters; and, lastly, by that notation
which has superseded every other.
(48.) It would be a vain and idle task to attempt to
enumerate all the conjectures which have been made to
account for the origin of the Latin numeral symbols;
it is sufficient for us to say, that they are obviously
connected with the same numeral systems which gave
rise to the more ancient Greek numeral characters,
(Art. 22.) In fig. 6, we have given from Gruter and
Beveridge a table containing the principal forms which
their numeral characters are found to assume in an-
cient inscriptions; the first five of them are subject
to very few variations. The character for 500 is IO,
or under an abbreviated form D; its value is doubled,
or becomes 1000, by prefixing a C to it, as in CIO 5
5000 is denoted by IOO, and 100,000 by CCIOO ;
and the value becomes increased in a decuple propor-
tion, by the successive addition of pairs of C, on each
side of the line I; thus 100,000 is denoted by
CCCIOOO ; 10,000,000 by CCCCIOOOO, and so on.
Though 6 is usually denoted by VI. yet in some
inscriptions we find it expressed by six lines; thus we
find IIIHIVIR for sevir, or sertumvir; 20 is mostly
denoted by XX, but sometimes by X , and 30 by 2%; but
V and L are never repeated, and X and C never more
than four times. By placing a line over these numeral
characters, their values were inereased one thousand
fold; thus I is 1000, V is 5000, X is 10,000, L, 50,000,
C, 100,000; 2000 was usually denoted by CIOCHO,
but sometimes also by IICIO, or IIM ; and in the
same manner 4000 was represented by IVCIO, 7000
by VIICIO, and similarly in other cases.
Examples without number of these notations are
every where to be found in classical authors and in
inscriptions; we shall merely give the following,
Nam ferme ante annos DCCCCL (950) floruit
Homerus, intra Co (1000) natus est.
Welleius Paterculus.
Homo qui primus factus est ante annos (ut tradunt) IIIMDC
(3600.) - Plin. Hist. Nat. Hib. xxxvi. c. 13.
Proh deſºm atque hominum ſidem 1 qui H-S CCCJOOO
CCCIOOO CCCIOOO(300,000 sestertia) questus facere noluit, (nam
History.
Roman
arithmeti-
cal nota-
tion.
Examples.
certè H-S CCCIOOO CCCIOOO CCCIOOO merere potwit et debuit,
si potest Dionysio H.S CCCIOOO CCCIOOO (200,000 sestertia)
'merere) is per summam fraudem et malitiam et perſidiam H-S
FOOO (50,000 sestertia) appetit 2
- Cicero, pro Roscio Comaedo.
In fig. 7 is given an inscription, which will illustrate
some other forms of numeral characters, which are of
frequent occurrence.
These examples will sufficiently exhibit the cum-
* Water, Grammatik der Russischen Sprache.
Fig. 7.
A. R. I T H M ET I C. 407
decimal throughout, without any intermixture of any . History.
other scale, whether quinary or vicenary. .
(51.) The existence of systems of symbolical Arith- Palpable
metic implies some considerable progress in the arts of Arithmetic.
Arithmetic. brous structure of the Roman arithmetical notation,
\-v- and will also account for the total absence of all
arithmetical operations amongst them, which were not
performed by means of the Abacus ; and it is one of
Its incon-
veniences. In this instance the simplicity of arithmetical notation people, we may refer all those systems to Egypt and
suffered from its being perfectly symbolical, and alto- Syria for their origin, however much modified in later
gether independent of language, as all numbers were times by the habits and languages of the people to
expressed by the mere apposition of the symbols for whom they were transmitted. In passing from ancient
numbers in a series commencing from unity, and to modern nations we shall find, that with the ex-
formed by successive and alternate multiplications by ception of China, possessing both a literature and in-
five and two ; a mode of forming compound numbers stitutions so different from all other nations, the
much less simple than what is followed in nearly all Hindoo Arithmetic has superseded every other species
languages, of expressing them as the sum of the of numeral symbols, both in Asia and Europe. Before
digits, and the several articulate numbers which they we proceed, however, to the notice of the gradual
contain. * advance of this Arithmetic from the East to the West,
The numeral notation of the Romans was adopted or the circumstances which accompanied its introduc-
in almost every part of their extensive empire, and tion, we shall premise a few remarks on the practice
continued to be employed wherever the Latin language of Arithmetic by means of the Abacus, which was so
was used, long after the introduction of the Arabic much used by the ancients, and which was in general
numerals, from a feeling of respect to antiquity, and a use amongst the nations of Europe until the end of
desire of conforming in every particular to the practice the XVth century.
of classical authors. The sexagesimal Arithmetic In the Theatrum Arithmeticum of Leopold, we have Roman
indeed prevailed amongst astronomers, and was used a representation of a Roman Abacus, which was pre- abacus.
in astronomical tables and calendars ; but it was served in the library of St. Geneviève, at Paris, and
clothed in Roman numerals, notwithstanding the in- which is copied in fig. 9; in this, the numbers are Fig. 9.
convenience of such a practice for the purposes of denoted by small round counters moving in parallel
calculation, and the knowledge of a better and more grooves. There are seven divisions for whole num-
commodious notation. bers, representing units, tens, hundreds, thousands,
Palmyrene (49.) We have before mentioned the extraordinary ten thousands, hundred thousands, millions; the value
and Phoeni- analogy which exists between the Roman numerals and of each superior unit being denoted by the numerical
cian nume- those which are found upon Phoenician and Palmyrene symbols which are placed between the long and the
*** coins and inscriptions,(fig.2 and 3.) In the last of these short grooves respectively. The counters in the
systems we find 2, 3, 4, denoted by the repetitions of longer grooves represent units, and in the shorter
the symbol for 1; 5 by a symbol very nearly resem- five; thus to denote 6, we put one counter in the
‘bling the Roman V in an inclined position; 6, 7, 8, 9, longer and one in the shorter groove, between which
in the same manner as in Roman numerals, the I is placed; to denote 70, we put two counters in the
symbols being written in an inverted order, conform- lenger and one in the shorter groove, between which
ably with the Eastern practice of writing; between 20 X is placed ; and similarly in other cases, the princi-
and 100, the numeration proceeds by the vicenary ple of denoting numbers by means of this instrument
scale; the symbol for 100 is the same as that for 10, being too simple to require further explanation.
with the symbol of 1 preceding it; that for 200, is Below the place of units, there is a pair of grooves
the symbol for 10, preceded by two units; the symbol appropriated to the division of the as ;” the counters in
for 300 is preceded by three units; for 500, by the the long groove denote uncia, or the twelfth part of
symbol for 5, and so on to 1000, which is formed by the pound, and those in the short groove one half of
repeating the symbol for 10 twice, and placing a unit it ; thus five counters in the long, and one in the
|before it. The Phoenician numerals generally agree short groove, would denote 11 ounces. In order to
with the Palmyrene, except that they possess no denote the divisions of an uncia, there are three short
symbol for 5; the nine digits being formed by the re- grooves added; to the first is attached the symbol 'S',
petition of the symbol for unity as often as it may be or z, which denotes semuncia, or half an ounce, which
required.* is the value of the counter placed in it ; to the second
Egyptian (50.) In the hieroglyphical symbols for numbers is attached the symbol O, which denotes sicilicum, or
hierogly- made use of by the ancient Egyptians, as ascertained sicilicus, the fourth part of an ounce; and to the last,
phical by the researches of Dr. Young,t we find the digits de- to which two counters are appropriated, belongs the
symbols. & a ſº ſº e g tº * * -
noted by the repetition of the symbol for unity, with symbol 2, designating, according to Velser, a duella,
simple symbols for 10, 100, 1000, all the interme- or third part of an ounce, but, more probably, a duode-
diate articulate numbers being denoted by the repeti- cima, or twelfth part of an ounce, a supposition which
Fig. 8. tions of those symbols, (fig. 8.) This system is would enable them to denote all the duodecimal parts
the many proofs of their extreme indifference to all
scientific improvements, that a system so incommo-
dious was not abandoned and replaced by the more
perfect and comprehensive notation of the Greeks.
*---
* A minute examination of the forms of the symbols for 5, 10,
20, 100, might, very probably, show that they were modified
forms of the letters of the alphabet which represented the same
numbers in the different and strictly alphabetical Arithmetic
Which the Greeks derived from them.
+ Discoveries in Hieroglyphical Literature.
life; and we, consequently, cannot expect that such
systems should be numerous, particularly when we
consider how few are the nations with whom civiliza-
tion has been of native growth. Amongst ancient
of an ounce, by means of the four counters in the
three grooves.f
In some cases the grooves were replaced by wires
* Leslie's Philosophy of Arithmetic. -
f Weidler and Ward, Philosophical Transactions for 1744.
408
M E T I C.
A R IT H
Arithmetic. upon which were strung perforated beads, four on the
S-N-2 longer and one on the shorter of each of them; in
Chinese
Swan Pan.
Fig. 10
Classical
allusions
to the
Tabula
Logistica.
order to represent nufmbers, the requisite number of
beads were moved on to the end of the wires, leaving
the remainder in each case, if any, on the other ex-
tremity.*
(52.) Under this form, the Roman Abacus resembled
the Swan Pan of the Chinese, which travellers have
so frequently described, and a representation of which
is given in fig. 10; it consists of ten parallel wires,
unequally divided, with four beads in the longer and
two on the shorter portions, and embraces numbers as
far as ten billions. In representing numbers upon it,
the wires are placed horizontally, the Abacus itself
being vertical, and the values of the beads increase in
descending, the greater numbers being placed under-
neath the smaller, in the same manner as in expressing
numbers by their symbols, (Art. 13.) As the decimal
division applies to their coins and to all their measures
of weight, length, and capacity, this instrument is
adapted to arithmetical operations of every kind; and
So great is their dexterity in the use of it, that they
have become celebrated throughout the Indian Archi-
pelago and the neighbouring countries for their skill
in practical Arithmetic.t
(53.) The Abacus, or Tabula Logistica, which was
generally used, was merely a rectangular tablet, strewed
with sand, in which grooves were made by the hand;
the counters, (calculi, or lapilli,) were contained in
little boxes, (loculi,) and Horace alludes to the custom
in his time, of boys marching to school with the
Abacus and its furniture suspended on their left arm :
Quo puero magnis er centurionibus orti,
Laevo suspensi loculos tabulamque lacerto.
Sat. i. vi. 75.
Persius alludes to the custom of strewing the tablet
with sand, in the following passage :
j Nec qui abaco numeros et secto in pulvere metas
Scit risisse wafer. Sat. i. 131.
This sand, according to Martianus Capella, was of a
sea-green colour:
Sic abacum perstare jubet, sic tegmine glauco
Pandere pulvereux, formosum ductibus aequor.:
Cicero makes use of the phrase eruditum attigisse
pulverem, § in a metaphorical sense to denote a person
who is skilled in the science of numbers and calcula-
tion; and Tertullian applies the terms primi nume-
rorum arenarii, to the teachers of the first rudiments
of arithmetical knowledge. -
The counters which were made use of were of
various kinds ; and in the progress of Roman luxury
were formed of the most precious materials. Thus
Juvenal alludes to the employment of counters of
ivory in the following lines,
Adeo nulla uncia nobis
Est eboris, mec Tessellae, non calculus ea hác
Materić. Sat. ii. 131.
And from a passage in Petronius Arbiter we may
History.
suppose that in later times they were sometimes made S-N-7
of silver, and even of gold : Notavi rem omnium deli-
catissimam, pro calculis albis aut migris, aureos argent-
eosque habebat denarios.” -
The familiar use of these counters gave rise to
numerous metaphorical phrases amongst classical
authors, which have reference to arithmetical opera-
rations on the Abacus; thus calculos pomere, or movere,
to state an argument ; hic calculus accedat, to signify
the addition of a proof to others which have preceded;
calculum detrahere, or subducere, to suppress a proof, or
step in an argument; calculum reducere, to change a
line of conduct or reasoning, with which you are dis-
satisfied ; and many other phrases, the proper force
of which can only be understood by a reference to the
use of this instrument.
(54.) The same instrument was likewise made use of Greek
by the Greeks, and most other ancient nations; their Abacus.
counters were called 'rmºot, and the process of calcu-
lation by their means \empoqbopla. Amongst other
distinctions which Herodotus has mentioned between
the customs of the Greeks and Egyptians, it is said,
“ that in writing and in calculating with counters,
the Greeks move the hand from the left to the right,
but the Egyptians from the right to the left.”f Some
authors have attempted to trace the derivation of the
use of this instrument from Abraham to the Egyptians,
Phoenicians, and from thence to the Greeks ; without,
however, venturing upon so minute an examination
of its history, we may certainly infer that its use was
very general amongst the nations of antiquity; and
that in almost every instance it preceded the use of
symbolical Arithmetic.
(55.) The use of counters was general throughout
SEurope as late as the end of the XVth century; about
that period they had ceased to be used in Italy and Spain,
where the early introduction of the Arabian figures,
and the number of treatises of practical Arithmetic
by means of them, had rendered them unnecessary.
They were used to a still later period in France, and
had not disappeared in England and Germany before
the middle of the XVIIth century. Shakspeare,
who may be considered as correctly representing the
customs and opinions of his times, exhibits the clown
in the play as embarrassed with an arithmetical ques-
tion, and declaring that he could not do it without
counters ::: and Iago, to express his contempt of
Michael Cassio, forsooth, a great arithmetician, terms
him a counter caster.' So general, indeed, appears to
have been the practice of this species of Arithmetic,
that its rules and principles formed an essential part
of the arithmetical treatises of that day: thus Robert
Record, in his Arithmetick, or the Ground of Arts,ſ
prefaces his second dialogue, entitled The Accounting
by Counters, by observing, “Now that you have learned
Arithmetick with the pen, you shall see the same art
in counters; which feat doth not onely serve for them
* This is the form of the Abacus, a drawing of which is given
by Velser, and which is copied by Gruter, vol. i. p. 224.
+ Philosophical Transactions for 1686, No. 180; Smethurst, in
the same Transactions for 1749; Leslie's Philosophy of Arith-
metic, p. 221 ; and Crawford's Indian Archipelago, vol. ii.
† De nuptiis Philologiae et Mercurii et de septem artibus libera-
libus, lib. vii. de Arithmetica ; Leslie's Philosophy of Arithmetic,
. 221
p § De Natura Deorum, lib, ii. 18.
| Mahudel, Académie des Inscriptions, vol. v. p. 261.
* Mahudel, Académie des Inscriptions, vol. v. p. 261,
+ Tpduuara 'ypdqoval ſcal Aoyſ; ovral phºbototy, {AAmves uév &mb
‘rāv &pia Tepāv Čiri Tà 5éâta pépovres rºw xeſpa. Alyºttiol 68 &rb Tāv
Seátóv čtri Tà apio tepē, lib. ii.
† The Winter's Tale, act iv. sc. 3. “CLowN. Let me see,
every eleven weather tods, every tod yields—pounds and odd
shillings, fifteen hundred shorn, what comes the wool to ? I can
not do it, without compters.”
§ Othello, act i. sc. l.
| First printed in 1540.
Its use to
a latc pe-
riod in
Europe.
A R IT H M ET I C.
409
Arithmetic, that cannot write and read, but also for them that can
S-N- do both ; but have not at the same time their pen or
tables with them;" and in a Treatise on Arithmetic,
published in Germany as late as 1662, we find a chapter
—j devoted to Arithmetica Calcularis, which is said to be of
Galcular
such common and general use amongst merchants, that
it might more properly be termed Arithmetica Mercatoria.”
(56.) We shall now proceed to give some account of
Arithmetic, the method of performing operations by this calcular
Notation,
Addition.
Arithmetic. They commenced by drawing seven lines
with a piece of chalk, or other substance, on a table,
board, or slate, or by a pen on paper; the counterst on
the lowest line represented units, on the next tens, and
so on as far as a million on the last and uppermost
line ; a counter placed between two lines was equiva-
lent to five counters on the lower line of the two ;
ºsmº
."
C–O—O—s
O
O
—C–0 * mammºmºmºmºmº tº ºmºmºmº
—O—G–C– tº sº-º-º-º-º-º:
—C–G–C–
O
s . —0–0–0—
thus the disposition of counters in the annexed ex-
ample represents the number 3629638; and it is
clearly very easy to increase the number of lines so as
to comprehend any number that might be required to
be expressed.
Suppose it was required to add together 788 and
383; express the numbers to be added in the two
O
O
G–C–––C–0–0–1–G-—
O 0 C
s—e-s——6–8–0–1–0–6–
G -
C–0–0 C–G–C–H–C–
first columns. The sum of the counters on the lowest
line is 6 ; write, therefore, one on that line in the
third column; carry one to the first space, which,
added to the one already there, is equal to one on the
second line; place a counter there, and add all the
counters on that line together, the sum is 7 ; leave,
therefore, two counters on that line, and pass one to
the next space; add the counters on that space toge-
ther, which are 3; leave one there, and place one also
on the next line ; add all the counters in that line
together; the sum is 6. Leave one counter, and pass
another to the next space; add all the counters in
that space together, which are 2 ; leave no counter in
the space, but pass one to the next, or fourth line; we
thus represent the sum, which is 1171.
The principle of this operation is extremely simple,
sº-
* Gasparis Schotti, Arithmetica Practica, Herbop. 1662.
+ These counters were usually of brass,
WOL. I.
º
and the process itself, after a little practice, would History.
clearly admit of being performed with great rapidity. S-N-
In giving a scheme of this operation, we have made
use of three columns; but in practice no more would
be required than are sufficient to represent the sums
to be added, the counters on each line being removed
as the addition proceeds, and being replaced by the
counters which are requisite to denote the sum.
We shall now proceed to a second example : Subtrac-
namely to subtract 682 from 1375.
() Q
—0–0–0––0 Q
C © Q
—0–0-—--&–0–0——(?–9–0–0
O
—----— — — —
Write the numbers in the first and second column.
The two counters on the last line have none corres-
ponding to them in the minuend; bring down the
counter in the first space, and suppose it replaced by
5 counters; take away 2, and 3 remain on the lowest
line of the remainder. Again, the three counters on
the second line must be subtracted from 7, (bringing
down 5,) and therefore leaving 4 on the second line of
the remainder. The counter in the second space has
now no counter corresponding to it in the minuend;
remove one counter from the next line, and replace
it by two counters in the next inferior space; there
will remain, therefore, one counter for that space in
the remainder. There is now one counter on the third
line to subtract from two in the minuend, and there
remains one for the remainder. The counter in the
next space has nothing corresponding to it, and wenust
therefore bring down the counter on the highest line
and replace it by two counters in the space below it ;
if one counter be subtracted from them, there will
remain one, and the whole remainder will be 693.
Becorde writes the smaller number in the first
column, and commences the subtraction with the
highest counters; a very little consideration will show
in what manner the operation must be performed,
with such a change in the process.
tion.


We shall now give an example of multiplication, Multiplica-
and let it be proposed to multiply 2457 by 43:
—-G--
()
(3–0–0–0 ||—
(9 () ©
e—s- —6–0-—{C}-6-0——
©
6–0–0--0 –G–C–9–0–0-——
C Q O C
—|Q–0–G–0||-0-0--|0–6–6––
C–0 s—e-s—º: Q–
tion.

~
410
A R I T H M E T I C:
Arithmetic. Write the multiplicandin the first column, and the mul-
S-N-
Division,
Manner of
using coun-
ters in
merchant's
aCCOllntS.
tiplier in the second; multiply first by 3, and write
the product in the third column, and then by 4 in a
superior place, and write the result in the fourth co-
lumn; add the numbers in these two columns together,
and the sum is the product required.
We shall conclude with an example of division,
and let it be required to divide 12,832 by 608:
Tividend. Divisor. Quotient, 1st Remain. 2d Remainder.
–G–C)
C () ()
–0–0–C–|–0 -0–
(...) C
–6–9–0– –0–0-|-0–0-|-0
()
-0–3–0–0–0–4–0-—-0–0-1---0–C–0–6—
Write the dividend in the first column, and the
divisor in the second, reserving the third for the
quotient ; then since 6 is contained twice in 12, in ,
the line above that in which 6 is written, we may put
down 2, in the last line but one in the column for the
quotient ; multiply 6 by 2, and subtract; there is no
remainder; multiply 8 by 2, and subtract 16 from the
number expressed by the counters remaining in the
dividend in the line above the last; first take one
counter from the three in the third line, and two
remain; next take 6, which is done by taking 1 from
the second line from the bottom, and bringing 1 from
the third line, replacing it by 2 in the space below,
and then subtracting one of them, thus leaving. 67
as the remainder to be denoted in the second and third
lines, and the spaces above them ; the remaining two
counters in the dividend are transferred to the corres-
ponding line in the column for the first remainder;
the operation is now repeated, the next figure in the
quotient, or 1, being written on the lowest line ; it is
now merely necessary to subtract the divisor from the
first remainder, and we get 64 for the second and last
remainder. It is evident that the same process may
be repeated to any extent that may be required; and
that the complication of the process, as exhibited in a
scheme, is much greater than in practice, where the
dividend is replaced by the first remainder, and so on
successively, until the remainder is zero, or less than
the divisor. -
(57.) Recorde” has mentioned two different ways of"
representing sums of money by means of counters, one
of which he calls the Merchant's, and the other the
Auditor's Account ; in the first, the sum of sé198.
19s. 11d. is expressed, as in the annexed scheme:
* Arithmetic, p. 213.
the lowest is the line of pence, the second of shillings,
the third of pounds, and the fourth of scores of
pounds; the single counters in the spaces denote half
of the units in the next Superior line: sixpence on the
first space; ten shillings on the second ; ten pounds
on the third; and the detached counters to the left
are equivalent to five counters to the right; the
lowest of them, therefore, representing five shillings,
the second five pounds, and the highest one hundred
pounds; and the whole sum is expressed by being
resolved into the following parts: £100. -- #380. --
3910. -- 4:5. -- 4:3. -- 10s. -- 5s. H-4s. -- 6d. + 5d.
The principle of this notation being once understood,
it is unnecessary to give examples of addition, sub-
traction, multiplication, or division, which present no
difficulty after the examples which we have given for
abstract whole numbers.
History.
~~

The mode of denoting the same sum for the ac- Auditor's
compt of auditors, is as follows:
The counters on the two lowest lines denote units of
their respective classes; on the upper line, when
placed to the left, they denote one quarter, and on the
right one-half of the next superior unit.
In both these cases, we may consider the resolution
of the number of pounds, into twenties, as a vestige of
the preference for the vicenary scale, which was so
general with our ancestors.
(58.) In ancient times, it was the custom formerchants,
bankers, or money changers, auditors of accounts,
and judges in affairs of revenue, to appear on a bank,
or bench, and before them on a board, or table, were
arranged the counters which were necessary in making
their calculations ; and the name of the Court of Ex-
chequer was derived from scaccarium, a quadrangular
table, about ten feet long and five broad, with an ele-
.vated ledge, around which the judges, tellers, and
other officers were seated ; it was covered with black
cloth, divided by white lines at right angles to each
other; they used small coins for counters, those on
the lowest bar denoting pence, the second shillings,
the third pounds, and the upper bars tens, twenties,
hundreds, thousands, and ten thousands of pounds.
The teller sat about the middle of the table ; on his
right hand, eleven pennies were heaped on the first
bar, and a pile of nineteen shillings on the second ;
while a quantity of pounds was collected opposite to
him on the third bar; for the sake of expedition, he
sometimes employed a silver penny to represent ten
shillings, and a gold penny for ten pounds.”
(59.) The term algorithmus, or algorismus, was em-
ployed universally in the XIVth and XVth centuries,
to denote the science of calculation by nine figures
and zero ; and its composition clearly shows the
source from which it was derived in our own language.
Algorism was corrupted into Augrym, or Awgrym, and
the counters which were used in calculation were
called augrym stones. Thus in Chaucer's description
of the chamber of Clerk Nicholas,
* Leslie's Philosophy of Arithmetic, p. 97.
accounts.
Bank.
Exchequer.
Algorithm,
its meaning
A R IT H M ET I c. 4II
Arithmetic. His almageste, and bokes grete and smale, have been invented for the purpose of either shorten- History.
\--~~ #. *:::::: ºn, ing arithmetical operations, or otherwise for relieving S-N-
on shelves couched at his bedies head.* - the operator from any troublesome or difficult exertion
- Millers Tale, v. 22–25. of the memory. Of this description are the virgula,
* g E g or rods of Napier, which were formerly much cele- Napier's
Modern (60.) There are not wanting in our owntimes examples brated and very generally used. The work in which rods. .
Palpable of persons who have attempted to revive the practice they were first described was published in 1617, under
Arithmetic.
of Arithmetic by counters. Professor Leslie, in his
Philosophy of Arithmetic, considers this method as better
calculated than any other to give a student a philoso-
phical knowledge º the classification of numbers,
and the theory of their notation; and with this view he
has given, in great detail, examples of the representa-
tion of numbers in different scales of notation by
counters, as well as of arithmetical operations by
means of them. With every feeling of respect for the
opinions of this very distinguished author, we shall
venture in this instance, on more grounds than one,
to dissent from them. In the first place, in this mode
of denoting numbers, the values of the counters
depend upon their position, as well as in the notation
by nine figures and zero, and as several counters cor-
respond to one digit only, the first method is, on
this account, much less simple than the other, when
viewed as a representation addressed to the eye as
well as to the mind; and, in the second place, arith-
metical operations by counters are not so easily
reducible, as in the case of figures, to rules which
are few in number, simple in principle, and rapid in
practice.
the title of Rabdologia.” In the dedication to Chan-
cellor Seton, he says, that the great object of his life
had been to shorten and simplify the business of cal-
culation ; and the invention of logarithms, which he
had just promulgated, was a noble proof that he had
not laboured in vain. These virgula, rods, or bones, as
they were often called, were thin pieces of brass,
ivory, bone, or any other substance, about two inches
in length and a quarter of an inch in breadth, distri-
buted into ten sets, generally of five each ; at the
head of each of these, in succession, was inscribed the
nine digits and zero, and underneath them in each
piece the products of the digit at the top with each of
the nine digits in succession, in a series of eight squares
divided by diagonals, in the upper part of which were
put the digits in the place of tens, and in the lower
the digits in the place of units. In order to multiply
any two numbers together, such as 3469 into 574,
those rods are to be placed in contact which are
headed by the digits 1, 3, 4, 6, 9; and the successive
products of the figures of the multiplier into the multi-
plicand are found by adding successively together the
digit on the upper half of the square to the right, so
Saunder- (61.) There are other species of Palpable Arithmetic, that in the lower half of the square to the left, in the
tº cal- some of which have been adopted especially for the line of squares which are opposite to the figure of the
º use, of blind. People; the elebrated Saundersºn, in multiplier which is employed in the operation ; thus
vented an instrument for this purpose, with which he
is said to have worked arithmetical questions with
extraordinary rapidity. His abacus, or calculating
board, was about a foot square, divided into small
squares, one hundred in each square inch, by sets of
parallel lines at right angles to each other. At every
point of intersection the board was perforated by
small holes, capable of receiving a pin, of which he
used two sorts, one with a large head, which denoted
zero, and the other with small heads ; and to every
figure was appropriated a square, consisting of four
smaller and contiguous squares. Zero was denoted
by a large pin in the centre of the square; to denote
unity this was removed and replaced by a small one;
for other digits, the large pin was placed in the centre,
and a small one either in the angle or middle of one
of the sides of the square; and the position used to
denote the several digits are given in fig. 11. The
scheme in that figure represents a portion of the board,
upon which are denoted the several sums which are
appended. It is quite evident that with such an in-
strument any arithmetical operations might be per-
formed, the sums being placed as in common figurate
Arithmetic, and the successive steps of the process
being recorded in the same manner.t
Arithmetical instruments of the kind which we
have just described, possess considerable interest and
importance from their use in lessening the privations
consequent upon one of the greatest human cala-
mities.
(62.)Many otherarithmetical instruments or machines
**-*- -*
* Leslie's Philosophy of Arithmetic, p. 221.
t Saunderson's Algebra, vol. i. p. xxi.
to multiply 3469 by 8, we take the line of squares
opposite to 8, which is represented by
i 3 4 6 9
2 3 4 7
8 24 || Z 2 || 23 Z2
and the product is 27752, being found by writing first
2, the sum of 8 and 7, 2 and 4, 4 and 3, and 2,
carrying tens when necessary, as in ordinary Arith-
metic. In the case of division, those rods are arranged
in contact which are headed by the figures of the
divisor; and from thence we are enabled to form the
products which the quotient forms with the successive
figures of the divisor.
In the case which contains these rods, which
Napier calls multiplicationis promptuarium, there are
usually found also two pieces with broader faces, one
consisting of three longitudinal divisions, and the
other of four ; one of which is adapted to the
extraction of the square, and the other of the cube
root; in the first, one column contains the nine
digits, the second their doubles, and the third their
squares; in the second, the first column contains the
digits, the second their squares, and the third and

* Rabdologiae seu Wumerationis per virgulas libri duo, authore
et inventore Joanne Nepero Baroni Merchistonii et Scoto. The
subject appears to have attracted immediate attention, and the
invention was circulated throughout Europe with extraordinary
rapidity, forming the subject of many separate publications, and
a part of almost every book on Arithmetic which was published
between 1625 and 1660,
3 H 2
412
A R I T H M E T I C.
^
Arithmetic. fourth their cubes, two columns being necessary for
S-vº-' this purpose, when the cube consists of three places:
thus the last division but one in the first is repre-
sented by
6
% is
and in the second by
5
|X| | *
a 8
In our times, when students in Arithmetic are more
perfectly acquainted with their multiplication table
than our ancestors appear to have been, we may feel
some degree of surprise at the eagerness with which
this invention was welcomed at its first publication,
when its only object was to relieve the memory from
so slight and trivial a burden ; we shall afterwards,
however, have occasion to notice examples in the
books of Arithmetic of that and the preceding age, of
the extreme anxiety of their authors to devise expe-
dients to simplify the processes of multiplication and
division ; and we shall also find, that the arrange-
ment of the half squares in Napier's rods agrees ex-
actly with the method of multiplying numbers, which
was adopted in Hindostan, Persia, and Arabia. At
the conclusion of this work of Napier is added a
Napier's short Tract, entitled Arithmetica Localis, which is
4rithmetica merely entitled to notice from its being the production
* of so great a man. It is very ill adapted, however, to
any practical use, and altogether unworthy of the
genius of its author.
Other (63.) Leibnitz invented an arithmetical machine by
arithmeti- which any numbers might be multiplied together;”
* and Leopold, in his Theatrum Arithmeticum, has re-
' corded many others, including two of his own. The
limits of this work will not allow us to enter into any
description of these inventions, which would neces-
sarily lead to great details. With respect to all of
them, however, it may be remarked, that as they
merely propose to multiply numbers together, the
importance of the object to be attained bears no
proper or reasonable proportion to the difficulty and
refinement of the means which are required to attain it.
In our own times, however, a gentleman of profound
knowledge of practical mechanics and general science,
and distinguished for the uncommon inventive powers
of his mind, is engaged in the construction of an arith-
metical machine of a very extraordinary character. It
is adapted to the performance of all arithmetical cal-
culations which depend upon differences; and, conse-
quently, to the construction of logarithmic and many
classes of astronomical tables; and is not limited to
the mere work of calculation, but distributes the types
which are required to record and register the result of
its operations without the possibility of error.
Origin, (64.) There are few subjects which have given rise
antiquity, , to more frequent controversies, than the invention of
and period the notation by nine figures and zero; whether we re-
#. * gard the country which gave it birth, the channels
... through which it was communicated to Europe, and
numerals, the period at which it was first known and generally
adopted. The total revolution which this invention
* Leibnitzii Opera, vol. iii. p. 413.
effected in the practice of arithmetical calculation, History.
whether for scientific or ordinary purposes, gives it S-N-
an uncommon degree of importance in the history
of the progress of human knowledge; and we shall
therefore make no apology for discussing its origin
and progress at considerable length. •
(65.) We have before mentioned our reasons for Antiquity
thinking that the Hindoos had possessed a very perfect of this
system of Arithmetic from great antiquity, from the in- .."
ternal evidence of their numeral language, independent #. S. €
of any external testimony. The assertion, however, of
Anquetil du Perron,” that the ancient Sanscrit alpha-
bet was distributed like the Greek into three classes
of numeral letters, would greatly invalidate such an
opinion, as such a notation must have preceded that
with nine figures and zero, it being extremely impro-
bable that a system of notation so inconvenient as the
first, could have been adopted, when the other was
already known and practised ; but the opinions of this
very fanciful and learned author have not been corro-
borated by the late researches of oriental scholars
incomparably better acquainted with the antiquities of
the Sanscrit language than himself; and we may,
therefore, venture to consider it in the light of one of
his numerous other dreams which have been found to
have no foundation in fact. It remains to consider to
what extent the antiquity of this invention may be
ascertained from the testimony of Sanscrit authors.
(66.) We have two translations of the Lildvati and Vija- Age of
ganita of Bhascara, works on Arithmetic, Mensuration, Bilascara.
and Algebra, which enjoy the highest reputation in
Hindostan ; of the first by Dr. Taylor, of the second
by Mr. Strachey, and of both by Mr. Colebroke,t
an author equally remarkable for his profound know-
ledge of oriental literature, and for his great scientific
acquirements; to the last is prefixed a dissertation on
the state of algebraic knowledge among the Hindoos,
Arabs, and the Greeks, in which the respective claims
of these people to originality in the possession and
invention of the rules of this science are discussed
with uncommon learning. He has established beyond
controversy that Bhascara, the author of the Sidd'-
hanta siromani, of which these works are a part, lived
about the middle of the XIIth century of the Chris-
tian era. He has also shown that Brahmegupta, an Brahme-
author frequently quoted by Bhascara, and portions of gupta.
whose works, containing treatises on Arithmetic and
Mensuration are extant, lived in the early part of the
VIIth century; and again, that Arya-bhatta, who is Arya.'
referred to by Brahmegupta, and considered the oldest *
of their uninspired and merely human writers, and the
subject of part of whose works was Algebra and
Arithmetic, flourished at least as early as the Vth
century, and probably at a much earlier period.
From these facts, which appear to be established Hindoo
upon very satisfactory evidence, it appears that Hindoo Arithmetic
Algebra and Arithmetic are at least as ancient as Dio- º Alger
phantus, and preceded, by four centuries, the intro- ...; º
duction of those sciences among the Arabs ; and in Diophan-
no case is the original invention of the notation by tus.
nine digits and zero referred to by any of these authors,
but is always stated to be one of the benefactions of
the Deity, which is the best proof of its possessing an
* Zendavesta, vol. i. p. 172. It is also asserted that this system
exists among some of the alphabets on the coast of Malabar.
f Algebra, with Arithmetic and Mensuration from the Sanscrit.
A R I T H M E T I C. 413 2"
Arithmetic.
Grant of
land dated
twenty-
three years
before
Christ,
Its anti-
quity
amongst
the Arabs.
Its use
general in
the Xth
century.
antiquity antecedent to all existing records. If the
royal grant of land engraved on a copper plate,
found in the ruins of Mongueer, and translated by
Dr. Wilkins,” be not a forgery, it would furnish evi-
dence of the existence of this notation at a much
earlier period than any which we have mentioned; as it
is dated in these figures in the thirty-third of Sambat,t
corresponding to the twenty-third year before the
birth of Christ ; under any circumstances, however,
whatever importance we may attach to this document,
there can be no doubt of the Hindoos possessing this
notation long before the Persians, Arabs, or any
western people. -
(67.) The testimony to the same fact derived from
the Arabs, is completely decisive of the source from
which they derived it. The first Arabian who wrote
upon Algebra and the Indian mode of computation,
is stated, with the common consent of Arabic
authors, to have been Mohammed ben Musa, the
Khuwarezmite, who flourished about the end of the
IXth century;f an author who is celebrated as having
made known to his countrymen other parts of Hindoo
science, to which he is said to have been very partial.
Before the end of the Xth century, these figures, which
are called Hindasi from their origin, were in general use
throughout Arabia; amongst others is mentioned the
celebrated Alkindi, who was nearly contemporary with
Ben Musa, and who, amongst his numerous other
works, wrote one on the Indian mode of computa-
tion, (Hisabu l'Hindi.) The same testimony is repeated,
in almost every subsequent author on Arithmetic or
Algebra, and is completely confirmed by their writing
those figures from left to right, after the manner of
the Hindoos, but which is directly contrary to the
order of their own writing.8 -
The use of this notation became general amongst
Arabic writers, not merely on Arithmetic and Algebra,
but likewise on Astronomy, about the middle of the
Xth century. We find it in the works of the astro-
nomer Ebn Younis, who died in the year 1008, and it
is found likewise in all subsequent astronomical tables.
It was, of course, communicated to all those countries
where their language and science were known. In the
XIth century, the Moors were not merely in pos-
session of the southern provinces of Spain, but had
established a flourishing kingdom, where the favourite
sciences of their eastern ancestors were cultivated with
* Asiatic Researches, vol. i. p. 127.
+ The present year (1826) is the 1882d year of the Hindoo
period Sambat.
† Colebrooke, Dissertation, p. 69. He has also mentioned an
Arabic author of the latter part of the XIIth century, who is
quoted by Casiri, in his account of the Arabic manuscripts in
the Escurial, as mentioning among other works derived from the
Hindoos, “A book on numerical computation which Abu Jäfr
Mohammed ben Musa Al Khuwärezmi amplified, and which is
a most expeditious and concise method, and testifies the acute-
ness and ingenuity of the Hindoos.” Another testimony of a
similar kind, which has been frequently quoted, is from the Com-
mentaries of Alsephadi on a poem of Tograi, who remarks that
the Hindoos have three inventions of which they boast, the game
of chess, numerical figures, and the book called Golaila Vadamna,
or the Fables of Bidpai.
§ Silvester de Sacy, Gram, Arab, vol. i. p. 76.
|| Delambre, Histoire de l'Astronomie du moyen age, p. 140.
It is stated by Dr. Bevis, in a letter to Mr. Ames, that in the
manuscript of this author in the Bodleian library, the Hindoo
numerals are used throughout ; and that when any number
is given, it is afterwards expressed in words at full length.
Selections,from Gentleman's Magazine, vol. ii. p. 162,
uncommon activity and success, and from that quarter History.
and from the Moors in Africa they chiefly appear to S-N->
have been communicated to the Spaniards and other
Europeans. - -
(68.) The learned Abbé Andres* considers that the ear- This nota-
liest example of the use of these figures which is to be tion used.
found in Spain or in Europe, is a translation of Ptolemy iºn lſº
in the year 1136; facsimiles of the forms of these figures ſº
are said to be given in the XIIth plate of the Paleo-
grafia Spagnuola of Terreros, who found them in all the
mathematical manuscripts subsequent to that period, but
in no other books or documents, nor even in accounts,
which were kept in the Castilian, which differed
little from the Roman numerals ; the calendars which
were chiefly constructed in Spain, both in that age
and until the end of the XIVth century, and were
sent from thence to other parts of Europe, continued
to be written in the old notation.
(69.) Kircher, in his Arithmologia, has advanced an Hypothesis
hypothesis which is not destitute of probability, that the of Kircher.
knowledge of these numerals was communicated to
Christian Europe by means of the celebrated astronomi-
cal tables which were formed under the direction of
Alphonso, King of Castile, and published at Toledo about Alphonsine
the year 1252. These tables were chiefly computed by Tables.
Arabian astronomers, and we should naturally expect
that they would adhere to the notation which had so
long been in general use in the writings of their
countrymen ; this question, however, cannot be de-
cided, unless by an examination of the earlier manu-
scripts of these tables.f
(70.) But we have positive evidence of the existence of Known in
a work, written expressly for the purpose of commu- Italy at the
nicating to Europe a knowledge of Arabic numerals jºins
and Algebra, at an earlier period than the publication XIij,
of the Alphonsine Tables. About the middle of the century.
last century, Targioni Tozzettif found in the Maglia-
becchian Library at Florence a manuscript, entitled
Liber Abbaci compositus a Leonardo filio Bonacci Pisano Leonardo
in anno 1202; and another work, by the same person, Pisano.
on square numbers, inserted in an anoymous Tract on
computation, (un Trattato d'Abbaco,) in the library of
the Royal Hospital in the same place. A transcript of
another Treatise of his was also found in the Maglia-
becchian library, entitled Leonardi Pisani de filiis Bonacci
Practica Geometria, composita anno 1220. The subject
of this work is the mensuration of land, and it is
mentioned by the author, in the preface to a revised
copy of the Liber Abbaci in 1228 ; Tozzetti met with
a second, though somewhat mutilated copy of the
Liber Abbaci, in the same library; a third has since
been discovered in the Riccardi collection at Florence;
and a fourth, but imperfect one, was communicated
by Nelli to Cossali. .
It appears from a short account of himself and his His life.
travels, which Leonardo has introduced into his Pre-
face to the Liber Abbaci, that he travelled into Egypt,
Barbary, Syria, Greece, and Sicily; that being in his
youth at Bugia in Barbary, where his father, by appoint-
ment of the merchants of Pisa who resorted there,
was scribe to the Custom-house, he learned the method
" * Dell' origine, dei progressi et dello stato attuale d'ogni littera-
tura, tom. iv. p. 57. -
f These tables were first printed in 1483.
+ Viaggi mella Toscana, vol. i. 11 ; Cossali, Origine e primi
progressi dell’Algebra, c. i. Sec. 5, ch. ii. sec. 1.
414
A R IT H M ET I C.
Arithmetic. of accounting by nine figures and zero ; that finding
*-N-' it much more commodious and far preferable to that
which was used in the other countries which he had
visited, he pursued the study,” and with some additions
of his own, and some propositions from Euclid, he
composed the treatise in question, that “the Latin
race might no longer be found deficient in the complete
knowledge of that method of computation.”f In the
epistle, also, which is prefixed to the revised copy of
his work, he professes to have taughtf the complete
doctrine of number according to the Indian method.
The preceding facts would refer the studies and
travels of Leonardo to the close of the XIIth century,
and the date of his first work, and consequently of
the introduction of the Arabic numerals through his
means, to the second year of the following century,
and fifty years before the publication of the Alphonsine
Tables. That this work was the first on this subject
which appeared in Italy, we know from other autho-
rity than that of Leonardo himself, as nearly all sub-
sequent Italian writers on Arithmetic and Algebra
ascribe the honour of priority to him, and particularly
Lucas Paccioli, or Lucas de Burgo Sancti Sepulchri,
whose work, entitled Summa de Arithmetica, &c. was
published in 1484, being the first work which was
printed on this subject; and a succession of writers
on Algebra, and therefore on Arithmetic, are men-
tioned by Cossali, from the beginning of the XIVth
century.
Leonardo Blancanus, in his Chronologia Mathematica, referred
considered Leonardo to the beginning of the XVth century, and
ºtº this date was adopted by Vossius, by Wallis, and by
.."...ing Montucla in the first edition of his work. Professor
flourished Leslie appears also to favour the same opinion, and
at a later founds much of his argument upon the probability
period. that the readers of the manuscripts of Leonardo
mistook the 4 for a 2, making the date 1220 instead
of 1420, those figures being easily confounded in the
older forms ; but we see no reason whatever for
doubting the judgment or authority of the numerous
persons by whom these manuscripts were examined,
and the frequent occurrence of those figures in a
work whose subject is Arithmetic and Algebra,
would appear to prevent the possibility of a mistake
of this nature; but independently of internal evi-
dence, there are other reasons which render it alto-
gether improbable that Leonardo could have published
his work at the latter period, at least if we may place
any reliance upon the testimony of Paccioli and all
the writers on these subjects of the preceding and fol-
lowing age, that he was the first person who introduced
the knowledge of Algorithm and Algebra to his
countrymen ; for, in the first place, Paccioli appears to
have taught those sciences at Venice about the year
1460; and he speaks of three persons who succes-
sively filled the professorship expressly dedicated to
their exposition, who had been his predecessors in it;
namely, Paolo della Pergola, Demetrio Bragadini, and
Antonio Cornaro, the latter of whom had been his
fellow disciple ; and in the century preceding the in-
vention of printing, innumerable Treatises de Algorithmo
Date of his
work.
/
* Quare complectens strictius ipsum modum, Yndorum, et actentius
studens in eo, ea proprio sensu quadam addens et quaedam ea sub-
tilitatibus Euclidis geometria artis apponens.
f Ut gems Latina de castero absque illa minime inveniatur.
† Plenam numerorum doctrinam edidi Yndorum, quem modum
in ipsa scientia praestantiorem elegi.
had been written, and manuscripts of them, of that age, History.’
are now found in great numbers, not merely in the S-N-'
manuscript collections of Italy, but likewise in those |
of every part of Europe. Again, Paolo de Dagomari
died about the year 1350, and obtained the surname
of Dell'Abbaco for his skill in the science of numbers,”
and Villani, the earliest Florentine historian and his
contemporary, speaks of him as a great geometer, and
most skilful Arithmetician, and who surpassed both ancients
and moderns in the knowledge of equations. Raffaello
Caracci, a Florentine Arithmetician, of the XIVth cen-
tury, also wrote a Treatise, entitled. Ragionamento di
Algebra, in which he speaks of Guglielmo di Lunis,
who before his time had translated a treatise on
Algebra from the Arabic into Italian;t and even
Professor Leslie himself refers to a date (1355,)
written in these characters in the hand-writing of
Petrarch, upon a manuscript of St. Augustin on the
Psalms, which was given him by Boccacio.f The in-
ference to be drawn from these facts is, that algebra
and algorithm, terms of contemporaneous introduction
into Europe, and the latter of which was always
applied to treatises of Arithmetic with Arabic numerals,
were perfectly well known in Italy throughout the
whole of the XIVth century, and consequently could
not have been introduced by Leonardo, if he flourished
at the beginning of the XVth ; instead, therefore, of
clearing away many difficulties by the adoption of the
latter date, we introduce others which it is impossible
to explain upon any hypothesis which is consistent
with facts, and the authority of the authors of that
a £62.
*gain, the work of Leonardo was written in Latin,
and he speaks of the Italians as the Latin race, a cir-
cumstance which makes it probable, that in his time
the Italian had not assumed the dignity of a written lan-
guage; now we know on the authority of Muratori, one of
the most profound and accurate of literary antiquaries,
that there is no authentic example of Italian prose
before the year 1264; but that after the year 1300 it
came into general use, and nearly superseded the use
of the Latin in writings on ordinary subjects; we
may consider this circumstance, therefore, as furnish-
ing a strong presumption at least, that Leonardo
wrote before the middle of the XIIIth century; and it
would likewise prove, that the translation into Italian
of the work of Mohammed ben Musa by Guglielmo di
Lunis, which some authors have considered as furnish-
ing the first source of their knowledge of Algorithm,
was made at a later period. The Tuscans generally, Early pro-
and the Florentines in particular, whose city was the ficiency of
cradle of the literature and arts of the XiiIth and *Tºgans
XIVth centuries, were celebrated for their knowledge ...”
of Arithmetic : the method of book-keeping, which e
is called especially Italian, was invented by them ; and
the operations of Arithmetic, which were so necessary
* Cossali, vol. i. p. 9.
+ This was most probably the Treatise of Mohammed ben
Musa, a translation of which was well known in Italy, as we know
from the testimony of Bombelli, who refers to it as if it were
perfectly familiar to his readers.
† Mabillon, in his noble work De re Diplomatica, has given a
fac simile of this record of Petrarch, which is as follows: Hoc
immensum opus donavit mihi vir egregius Johannes Boccacius de
Certaldo, poeta nostri temporis, quod de Florentia Mediolanum ad
me pervenit 1355, Aprilis 10. The figure of 3 is nearly the same
as in modern times, but that of the 5 is the same as is generally
found in manuscripts of the XIVth and XVth centuries.
A R. I. T H M E T I C. 415
of that system: in a letter to the Emperor Otho, he History.
Arithmetic, to the proper conduct of their extensive commerce,
styles himself extremus numerorum abaci; and in another Y-a-’
\-N-7 appear to have been cultivated and improved by them
with particular care ; to them we are indebted for our
present processes for the multiplication and division of
whole numbers, and also for the formal introduction
into books of Arithmetic, under distinct heads, of ques-
tions in the single and double rule of three, loss and
gain, fellowship, exchange, simple interest, discount,
compound interest, and so on : in short, we find in
those books, every evidence of the early maturity of
this science, and of its diligent cultivation; and all these
considerations combine to show that the Italians were
in familiar possession of Algorithm long before the
other nations of Europe.
If, therefore, we should found our decision upon the
evidence already adduced, of the question, What nations
in Europe were in the first possession of the notation by
nine figures and zero 2 we must certainly answer, Spain
in the first instance, and Italy in the second : in one
case, it was introduced in the translations of astrono-
mical works from Arabic into Latin, and appears to
have been long confined to mathematical works alone;
in the other, the algorithm itself is made the subject
of a distinct treatise, written for the purpose of making
it generally known. In the first case, it appears to
have been chiefly confined to the Moors, by whom it
was introduced, and its general propagation checked by
the contests which distracted that country, until their
final expulsion ; in the other, it passed rapidly from the
writings of arithmeticians into general use ; and in
less than a century and a half, it assumed a form much
more adapted to practice, than that which it pos-
sessed amongst the people with whom it originated.
to his friend, he says, Nam quomodo rationes abaci expli-
care contenderemus, nisi te adhortante O mi dulce solamen
laborum Constantine P In another epistle to the son of
the Bishop of Geneva, he says, De multiplicatione et
divisione numerorum Joseph sapiens sententias quasdam
edidit. Eas pater Adelbero Remorum archiepiscopus
habere cupit. This was a work, celebrated in that age,
by Joseph of Spain, which is again referred to in the
following passage, in a letter to the Abbot of Orleans:
De multiplicatione et divisione numerorum libellum a
Joseph Hispano editum Abbas Garnerius penes vos reliquit :
wt exemplar in commune sit rogamus, sc. ego et Adelbero.
If this book contained an exposition of the Hindoo
notation, it is impossible that the knowledge of it
could have been lost, when communicated to so many
persons; and in supposing that the abacus referred to
in preceding extracts meant the mensa Pythagorica,
or common multiplication table, which may or may not
have been the case, there is no reason why it should not
have been expressed in Roman numerals, as the same
is found in the works of Boethius.* Again, when in
another passage he speaks of digital, compound, and
articulate numbers : Quid cum idem numerus modo sim-
plea, modo compositus; nunc digitus, nunc constituatur ut
articulus, it must be kept in mind that these distinc-
tions originated with the Arithmeticians of the Pytha-
gorean school, and that there is no reason for us to
interpret this sentence, as was done by Wallis, as if
it was meant to assert that the same figure was some-
times employed to denote a digit, and sometimes an
articulate number, according to its position. The
Claims of (71.) A much earlier date, however, has been assigned observation which immediately follows is remarkable :
Pope by some authors to the introduction of these numerals Habes ergo (talium diligens investigator) viam rationis (sc.
Sylvester, into Europe, than those which we have mentioned. abaci ;) brevem quidem verbis sed prolia’am sententiis ; et
the Second. In the latter part of the Xth century flourished Ger-
bert, a monk of Aurillac, in Auvergne, who was after-
wards Archbishop of Rheims and of Ravenna, and
finally Pope, under the name of Sylvester II.” In
early life he travelled into Spain, and is represented
as having made himself master of all the learning of
his time, and as one consequence of these numerous ac-
quirements, was accused of dealing with the powers of
magic : he wrote largely on Arithmetic and Geometry,
and in the opinion of Wallis, f Leibnitz, f and many
subsequent writers, was the first European who ac-
quired a knowledge of the Arabic numerals from
the Saracens in Spain. This opinion is chiefly founded
upon a passage in our English historian, William of
Malmsbury; $ when speaking of Gerbert, he says,
Abacum certe primus a Saracenis rapiens, regulas dedit
qua, a sudantibus abacistis via intelliguntur. This sen-
tence, however, contains no certain intimation of the
1<nowledge of the notation by nine figures and zero,
as the rules which would be thence derived, would
tend rather to relieve than increase the labours of
the sweating ealculators.|| Other passages have been
quoted by Wallis, from his letters to his fellow disciple
Constantine, and others, which are supposed by him
and Kaestner," to give indications of his knowledge
* He died in 1003,
† Opera Mathematica, p.254.
§ He flourished about the year 1150.
| North, On the introduction of Arabic numerals, Archaeologiſt,
wol. x. p. 366. -
ºf Geschichte der Mathematik, vol. ii. p. 366.
‘f Algebra, ch. iv.
ad collectionem intervallorum et distributionem, in actua-
libus Geometrici Radii secundum inclinationem et erectio-
nem, in speculationibus et actualibus simul dimensiones Caeli
et Terra plend fide comparatam. It is difficult, how-
ever, to conceive in what manner this character, brevem
quidem verbis sed proliram sententiis, could apply to the
system in question; and the remainder of the sentence
is so obscure, that no inference respecting the method
to which it referred can properly be deduced from it.
In the third Volume of the Thesaurus Anecdotorum,
novissimus of Bernard Pez, which was printed at Augs-
burg in 1721, there is a notice of a manuscript of the
Geometry of Gerbert, which was found in the monas-
of St. Peter, at Salzburg, in which the Arabic figures
are found: the author considered the manuscript to
have been written in the year 1100, and he supposes
that the transcriber would not have inserted these
figures in his copy, if he had not found them in the
original ; it appears, however, that no practice has
been more common than alterations of this nature,
and that those figures have sometimes been inserted
* Weidler, the historian of Astronomy, discovered a manu-
script of the Geometry of Boethius in the Public Library at Altdorf,
in which the Mensa Pythagorica is given in Arabic numerals; and
in a Dissertation, de characteribus numerorum vulgaribus et eorum
aetatibus, he attempted to show that they necessarily formed a part
of the original work: it is a sufficient answer to show that they
do not appear in the most ancient manuscripts of Boethius, where
all the numbers are expressed by the Roman characters, and
that consequently, in the later manuscript which Weidler saw,
they had been inserted by the transcriber.
416
A R IT H M ET I C.
Arithmetic, at a later period than the date of the manuscript itself.
S-N-' That such has been the case with the manuscript in
of his, one in prose and the other in verse, which are History:
found together in a manuscript at Oxford; the second S-v-
Of John of
Halifax.
question, we may infer from the existence of other ma-
nuscripts of Gerbert, of nearly his own age, in which
those figures are not found, even on occasions where
they might most naturally have been looked for. Thus
William of Malmsbury mentions an Epistle Quam Adel-
bold fecit ad Gerbertum de questione diametri super Macro-
bium, which Mr. North” found in Parker's Library, in
Corpus Christi College, at the end of Macrobii Opera,
together with Gerbert's answer: in this manuscript,
which is of greater antiquity than the one of Salzburg,
the Roman numerals are constantly used by both ; a
circumstance which affords a strong presumption,
when the nature of the subject discussed is considered,
that those figures were at that time unknown.
• In the account given of Gerbert by Trithemius,t is
the following passage: -
Gerbertus docuit Fulbertum, hic etiam Fulbertus Beren-
garium, qui iterum Brunonem Remensem et alios multos
haredes Philosophia reliquit.
It would be a very extraordinary circumstancce if
Gerbert had known and taught this notation, that it
should have been lost notwithstanding this regular
chain and succession of his disciples; and it is no suffi-
cient answer to the presumption, that he did not teach
this system because he did not know it, to contend with
Raestner, that there is no necessity for a tutor to
communicate the whole of his knowledge to his pupils.
We have been more particular in our examination of
the claims of Gerbert to the knowledge of this system,
because the arguments of Wallis on the subject appears
to have convinced Mr. Colebrooke, f whose opinion and
judgment are entitled to so much respect; and though
it must certainly be allowed that it was possible
for him to have acquired the knowledge of it from the
Saracens in Spain, it is more probable, however, that
they, who were recent conquerors of that country,
amongst whom the arts of peace had hardly begun to
take root, were at that time ignorant of most of the
scientific improvements which had taken place in the
preceding century amongst some of their countrymen
in the east: at all events, it would certainly follow
from the facts which we have mentioned, that if the
system was known to Gerbert, it was barren in his
hands, as no certain traces of it are discernible, either
in his own writings or in those of his contemporaries
or successors. We may, therefore, fairly deny him
the merit of having introduced the knowledge of it
into Europe.
(72.) Wallis, who seemed to consider the figures in
every manuscript which he saw, to be of the age of
the author, and that they had never been introduced by
subsequent copyists, has given a long list of authors,
to whom this notation was known in the XIIth and
of these commences with the verses:*
Harc Algorismus ars praesens dicitur : in quá
Talibus Indorum fruimur bis quinque figuris ; -
which are there given, and are the same in form as
those which are used in the XVth century, the age to
which we should be inclined to refer this manuscript;
in short, there is no sufficient reason to attribute either
of these works to Sacro Bosco, as no notice of these
figures appear in the older manuscripts of the two
former works, where we should naturally have ex-
pected to meet with them, particularly in the latter.
(73.) The second person whom we shall mention is of Bishop
Robert Grossetête, or Grosshead, Bishop of Lincoln, and
the contemporary of John of Halifax. He also wrote De
Computo Ecclesiastico; and his calendar constructed by
means of it, under the name of Kalendarium Lincoln-
iense, appears to have long continued in repute, and
numerous copies of it of the XVth century are found
in manuscript libraries. It is impossible, however, to
judge readily of the age of the earlier manuscripts of
this work, and it is only in the later copies that the
Arabic numerals are found.
Grosshead.
(74.) There is a copy of the Calendar of the celebrated Roger
Roger Bacon in the British Museum, which Mr. Ames
considers, upon the authority of Mr. Casley, to be of
the date 1292; upon an examination of it, however,
we found it headed by the following notice: Kalen-
darium sequens extractum est tabulis Tholetanis Anno
dm. 1292, factis ad meridianum civitatis Tholeti ; or in
other words, that the calendar had been formed from
the Toledo tables published in 1292, and calculated
for the meridian of that city. In this case, as in
many others, the name of Roger Bacon had been
attached to the calendar by the fmonks who composed
it; either for the purpose of recommending it by the
authority of a name so distinguished for abstruse, and
what was in that age deemed magical learning ; or
perhaps in the arrangement and composition of it,
they had availed themselves of some of the rules of
the Treatise de Computo Ecclesiastico, of which he was
the author. o
(75.) We find it unnecessary to give other examples
of Astronomical tables, or of Treatises de Algorithmo,
which are assigned by Wallis to the XIIIth century,
either from the character of the manuscripts, or from
the names of their authors, as the instances which we
have given would show that he was too much de-
voted to his theory, to be inclined to subject his
documents to a very accurate or critical examination.
There is one hypothesis which he has made respecting
the period at which the Arabic figures were introduced
into England, which deserves a more particular ex-
amination, from the numerous discussions to which it
Bacon’s
calendar.
XIIIth centuries. Amongst others, in particular, is has given rise. He supposes that they were brought English
mentioned Johannes de Sacro Bosco, the Latinized from Spain into England about the year 1130; and to travellers.
name of John of Halifax or Holywood, who died in account for this very early introduction he has referred Spain in
1256, and who wrote a Tract de Sphaera, which for a long
time was a work of standard authority; another De Com-
puto Ecclesiastico, in the later manuscripts only of which
the Arabic figures appear; and another which is attri-
buted to him, though apparently on very insufficient
authority, De Algorithmo. Wallis speaks of two Tracts
* Archaeologia, vol. x. p. 368.
: Hindoo Algebra, Introduction, p. 54.
+ Ibid.
to several Englishmen who travelled in that country
about that period : amongst others, he mentions
Adelard, the Monk of Bath, who translated Euclid
from Arabic into Latin ; Robert of Reading, who
translated the Alcoran into Latin in 1143; Daniel
Morley, who studied Mathematics and the Arabic lan-
* There is a copy of this manuscript also in the Public Library
at Cambridge.
the XIIth
century.
A R IT H M ET I C.
417
Arithmetic. guage at Toledo about 1180. By such persons it was
natural to expect that the knowledge of these nu-
merals should not only be acquired, but likewise
communicated upon their return;* as a proof that
such was the case, he refers to the date on the Mantel-
piece in a room in the parsonage house at Helmdon
in Northamptonshire, a description of which he has
given in the Philosophical Transactions,t and afterwards
in his Algebra.f The date which is there given, is
supposed to be expressed partly in Roman and partly
in Arabic figures, and is equivalent to A° DO' Mº 133.
Dr. Ward, in the same Transactions for 1735, re-
sumed the subject ; and as he had satisfied himself,
though upon reasons which might have equally an-
swered for a much later date, that the knowledge of
these numerals had been introduced by the Crusaders,
he will allow no date to be genuine which is before
the year 1200; he therefore boldly corrects the date
to 1233. Later observers, including the celebrated
antiquary Gough, found great difficulty in making out
the A° DO", which appeared so plain to Wallis, and
still greater in identifying the other figures; and,
lastly, Mr. Denneš has shown that the fleur de lys and
dragon volant with which this rude piece of rustic
sculpture is adorned, are more appropriate to the reign
of the last than the first or third of our Henries;
and that the date, if the inscription be really meant for
one, is designed for 1533 rather than 1133 or 1233.
(75.) The Helmdon inscription, and the conclusions
founded upon it, produced in a short time a number of
others of equal or greater antiquity, which have all how-
ever yielded to a more sober and critical examination;
of this kind was the inscription at Colchester, which
was said to be 1090, but which further investigation
extended to 1490; and one at Widgel Hall, near Bun-
tingford in Hertfordshire, which appeared to be M 16,
but which was found to be M. I. G.'ſ the initial letters
\\ of a proper name having been mistaken for numerals :
on a barn belonging to Preston Hall, in Kent,” where
the date 1102 is put between two armorial shields
with the cyphers T. C. attached to each of them, but
which were shown to be the initials of Thomas Cole-
pepper, the owner of the estate, who lived about
1587, and who, most probably, commemorated in this
manner the time when his family first got possession
of the property : on a beam in a very ancient gate-
way near the great bridge in Cambridge, where the
date which Dr. Warren represented to Dr. Ward as
1332,ft in reality should be read 1552 : on a
stone found in digging up the foundation of the Black
Swan Inn, in Holborn, with the date 1144,85 though
it is difficult to make out that such rude marks repre-
sented any numerals whatsoever, as they have no
resemblance to such as were used before the end of
the XVth century; but it is needless to extend this list,
as in all the cases which have hitherto been produced,
their pretensions to uncommon antiquity have been
refuted by further investigation.]]||
Helmdon
inscription.
Different
opinions
respecting
it.
Other in-
scriptions
with dates
which have
been
mistaken,
* Algebra, ch. xi.
† No. 178, for December, 1683.
§ Archaeologia, vol. xiii. p. 142.
| Philosophical Transactions for 1735, No. 439, p. 131.
T Ibid. p. 136. -
** Hasted's History of Kent, vol. ii. p. 175.
+t Philosophical Transactions for 1744, No. 474.
†† Archaeologia, vol. x. p. 372.
§§ Ibid, vol. i. p. 149.
Hiſ Of this kind is the date 1182, said to be found on a brick
WOL. I. - -
3. Ch. ix.
(76.) The real fact appears to be, that the Roman History.
numerals were used throughout Europe long after the S-N-'
Arabic figures were in common and general use. The Monumeſ:
te 4- * tal dates in
earliest example of a monumental date in these figures ...".
in England is in the church of Ware, on a brass plate at War."
commemorating the death of Ellen Wood in 1454 ; and
the second is in the same church, and is dated 1484.”
Kaestner considers that the earliest monumental In Ger-
date in Germany is A. Di. 1497, on the wall of the *Y.
church of Grossalmerode, in Hesse.t Gatterer says
that these figures rarely ever appear in public docu-
ments during the XVth century, and only became
common for such purposes at the close of the XVIth ;
and that the earliest date which he has observed
amongst more than 1000 documents in his possession
was 1527.: Calmet, in the Mémoires de Trevoua, has
noticed a series of inscriptions, chiefly monumental,
from 1445 to 1519. He also says, that at Turkheim
there is an inscription on a stone of the date 1332,
which we might venture, however, to convert into
1552, upon the same principle that was applied to a
similar inscription noticed above. Mabillon, in the Diploma-
whole course of his inquiries, and after the examina- .
tion of more than six thousand documents, found no º
authentic date earlier than 1355, in the hand-writing
of Petrarch ;|| and the learned Benedictines, the
authors of the Nouveau Traité de Diplomatique, declare
their conviction that the appearance of a date in Arabic
numerals before the XIVth century, would be suf-
ficient to vitiate its authority." The conclusion to
be drawn from these facts unquestionably is, that it is
not in dates that we must look for the first appearance
of these numerals ; but in astronomical works and
tables, in calendars, and in Treatises of Arithmetic
and Algebra.
(77.)The earliest example of the use of these numerals The Arabic
which Montfaucon discovered, was in a chronicle in º
the Strozzi library at Rome. It is a manuscript .
written by different hands. The first and most ancient faucon in a
concludes in 1250, and is written in the Roman nume- very an-
rals; the second, and more modern, in which Arabic cient ma;
numerals are used, commences in 1268, and finishes §:
in 1317, and is therefore posterior to that date. In library.
another manuscript, there is found written the date
1245, in which the forms of the figures 2 and 5 are
different from those on the chronicle, and possibly
more ancient; but it is impossible to judge of the real
antiquity of a date which is written in such a manner,
and not embodied in the work itself.” &
(78.) The author of the Göttwich Chronicle is anxious Manu-
to secure for his countrymen the honour of an earlier scripts at
acquaintance with these numerals.ft. He speaks of a Fº and
manuscript at Fulda, in which they appear, which is “ Warsaw.
said to be 1300 years old; of the same age, therefore,
with Weidler's manuscript of Boethius; and of a
building at Shalford in Bucks, though it may be satisfactorily
proved, that there was no brick building in this country before the
end of the XIVth century.
* Archaeologia, vol. xiii. p. 148.
+ Geschichte der Mathematik, vol. i. p. 36.
# Elementa rei Diplomatica, vol. i. p. 64.
§ Mémoires pour l’Histoire des Sciences et beaua Arts & Trevolta',
pour l'an 1707, p. 1624. -
|| De re Diplomatică, p. 215, and tab. xiii. p. 373.
‘ſ Tom. iii. p. 537. -
** Calmet, Mémoires de Trevour, for 1753, p. 1692.
+t Chronicon Gottwicense, swu Annales Liberi et crempti Monas-
terii Gottwicensis ordinis S. Benedicti, p. 114. -
3 I
418
A R I T H M E T I C.
Arithmetic. calendar in the library at Warsaw for the year 1268,
\-v- which is expressed in these figures.
In order, how-
ever, to found his argument upon facts of less dispu-
table character, he asks how it is possible that the
use of these figures should be unknown in Germany
four hundred years after the Saracens had established
their dominion in Spain; more particularly after the
Alkoran and many works of Arabian physicians had been
translated in Germany in the time of Conrad the Third,
and Frederick Barbarossa in the XIIth century. It
is a sufficient answer, however, to observations like
these, to state, that the knowledge of these figures
among the Arabs in the Xth and XIth centuries
was not, properly speaking, popular, but confined to
the few who had leisure for the study of science and
Mistakes in philosophy; and that it was only in the XIIth cen-
the use of tury that those arts were much cultivated in Spain;
Arabic
numerals.
Their early
tise in
calendars.
Contents of formed.
and that in all discussions on this subject too much
stress is laid upon the facility with which the know-
ledge and practice of this notation is acquired, and
transmitted amongst a people who have been accus-
tomed to the use of a system essentially different in
its nature. The following fact will show that this
system has been in some cases introduced without
being perfectly understood.
In a manuscript of the XIVth century, an ex-
tract from which Mabillon has given a fac simile,” we
find the Roman and the Arabic numerals mixed up
together in a very curious manner; thus 10, 11, 12,
13, 14, &c. are denoted by X, X 1, X2, X3, X4, &c.;
20 by XX ; 31 by XXXI, or by 301 ; 34 by 304;
40 by XXXX ; 41 by 401; 42 by 402. It is clear
from hence, that the writer did not understand the
proper force of the zero, and had but very imperfectly
comprehended the principle of value from position.
(79.) A critical examination of the calendars which
exist in different libraries in Europe, would lead to
the most certain determination of the periods at which
these numerals were generally introduced, as they
contain within themselves the data from which the year
in which they were composed may be very nearly ascer-
tained ; and there are few inquiries which would lead
to the knowledge of more curious facts respecting the
history of the human mind, as they generally contain
all those topics of medical, astrological, and astrono-
mical science, which were most popular in their time,
and which were best adapted to the superstitions and
prejudices of the people for whose use they were
The following are the contents of a
a calendar calendar in the British Museum, consisting of eight
in the
British
Museum.
vellum leaves folded up in a portable form, and which
may serve as a specimen of others of the same date,
about 1403; the leaves are marked from A to K.
A, contains a canon for the calculation of the movable
feasts, subjoined to which is an astrological scheme.
B, C, D, E, the calendar, properly so called, three
months in each.
F. Tabula luna cum canone et imagine signorum ; the
figure of a man with the signs of the zodiac on
different parts of his body.
G.
1403 to 1469. -
H. Eclipses of the moon, with their phases, from
1398 to 1448.
J. Eclipses of the moon from 1448 to 1462; to
Eclipses of the sun, with their phases, from
* De re Diplomatică, p. 373, Plate 15.
these is subjoined a Tract, entitled Opera Apuleii, de
indiciis urinarum, with the figures of six urinals dif-
ferently coloured, with the affections of the body which
they severally indicate. -
K. Tabula ad calculandum pro futuris.
This calendar exhibits the Arabic numerals, with
the figures which they usually present before the end
of the XVth century; and the abstract of its contents
which we have given, may be taken as a sample of the
most popular scientific knowledge of those times.
Mr. Denne” has given an account of another calendar
containing a table of eclipses from 1406 to 1462,
which is nearly similar to the last ; where, instead of
enlarging on the indications of urinals, those days
are particularly marked, upon which it is expedient
to abstain from flebotomy; they amount to 133 in
the course of the year, and the enumeration of
them at the end of the year is terminated by the fol-
lowing formidable warning, Isti sunt dies mali obser-
vandi ab incisione in anno, et qui homines vel pecora
inciderint inde morientur. In the last page of this
calendar, there is the following short and very clear
account of the Arabic numerals, which appears to
have formed a common appendix to them in that and
the preceding age, when their use was becoming
general : Nota quod quaelibet figura algorismi in primo
loco signat Se ipsam, et in secundo decies se. Tertio loco
centies se ipsam. Quarto loco millesies se. Quinto loco
decies millesies se. Sexto loco centies millesies se. Sep-
timo loco mille millesies se. Et semp. incipiendum est
computare a parte sinistra more Judaico. The following
page contains the Roman and Arabic numerals from
I to 100 placed opposite each other; and in the last
page many numbers are given from 20 to 1,000,000,
being severally specified in words, Roman numerals,
and Arabic figures ; thus, Viginti, XX, 20; mille
milia, Me Ma 1,000,000.
In the manuscript library of Corpus Christi College,
Cambridge, there is a table of eclipses from 1330 to
1348, to which is also subjoined a table of the
Arabic numerals, which is extremely interesting from
its great antiquity." In fig. 12 we have given a copy of
this table arranged in three columns; the first for digits,
the second for articulate, and the third for compound
numbers; each of these columns is separated into three
divisions, in the first of which we find the Roman
numerals, in the second the Arabic, and in the third a
peculiar notation nearly identical with the Roman in
principle, though different in form. After the table is
subjoined the following explanation : Omnis numerus
vel omnis figura in algorismo primo loco se ipsum signift-
cat ; secundo loco, decies se ipsum significat ; tertio loco,
centies se ; quarto loco, milesies se ; quinto loco, decies
milesies se ; sea to loco, centies milesies se ; Septimo loco,
mille milesies se ; octavo loco, decies mille milesies se ; novo
loco, centies mille milesies se ; decimo loco, mille milesies
milesies se. Et sic multiplicando per decem centum et
mille usque in infinitum computando versus sinistram.
There is no longer any difficulty in discovering in
what manner the knowledge of Arabic notation was
propagated throughout Europe, when we find such
simple and popular expositions of its principles in
those productions which were expressly formed for
the most general circulation; and from which the
* Archaeologia, vol. xiii. p. 153.
f North, Archaeologia, vol. X, p. 373.
. History.
\-y-
Of the
calendars
which con-
tain a notice
of Algo-
rithm.
Manuscript
in the
library of
Corpus
Christi
College.
Means of
propaga-
ting the
knowledge
of Arabic
numerals.
A R I T H M ET I C. 419
Arithmetic. majority even of the better informed of our ancestors
S-N-Z derived so considerable a portion of their knowledge;
so common indeed does the use of them appear to
have been, and so frequent were the occasions of
reference to them, that they were in some cases triply
folded up in such a manner that they might be sus-
pended by a knot at the girdle, so that persons
Calendar in might peruse them without removal. Several exam-
the English ples of such calendars may be seen in the British
*ś* Museum, and, amongst others, one in ten leaves, for
perfect and matured state than in other parts of History-
Europe. S-N-"
(81.) It is not a matter of much difficulty or import- Appearance
ance, to trace the progress of these numerals after the of these mu-
commencement of the XVth century: the manuscripts merels in
: & .* in hia is g the dates of
on astronomical and arithmetical subjects which were rinted
written in that century, and which are found in such j
numbers in all manuscript libraries, show how gene- -
ral the use of these numerals had become, where
custom and respect for antiquity had not opposed
in 1431. the year 1431, and which merits particular attention,
from its being the earliest which we have discovered
in which the English language is used. Its contents
are of the usual kind, containing the calendar in Arabic
figures, with rules for the calculation of eclipses, leap-
years, indictions, &c. and the usual portion of astro-
logieal information. The author of it says that it was
formed for the use of his sovereign mistress, though
it is not easy to find a personage to whom this title
would apply during the minority of Henry VI.
If they be crafty, reckin and nombre
And tell of every thing the nombre,
Yet, shullde fail to reckin even
The wonders we met in my sweven.
If we assign the date 1375 for the writing of this
poem, the epithet new might yet be appropriate to
these figures, as distinguished from the Roman
numerals, even though they had been known from
the beginning of the century. A term of this
kind, indeed, is so indefinite in its application, that it
is impossible to found any argument upon it ; the
passage, however, is in other respects interesting, as
expressing the opinion of the author upon the power
and extent of this new Arithmetic, in every way so
superior to the old. Chaucer had been in Italy, and
might there have enjoyed opportunities of witnessing
this amongst all other arts and sciences in a more
* Denne, Archaeologia, vol. xiii. p. 123,
their introduction : it was for these reasons that they
were excluded until a late period for dates, from all
public and formal deeds and documents, from regis-
ters, inscriptions, and so on. They appear neither
in the dates, nor pages of works printed by Caxton,
but in the Myrrour, or Ymage of the World, printed
in 1480, when treating of Arsmetrike, or Algorithm,
amongst other sciences, he has given a wood-cut of
an Arithmetician sitting before a table, on which are
slates or tablets with these figures upon them.
they continued the use of the Abacus in their calcula-
tions to a still later period ; nay, even so late as the
year 1595 the use of this instrument was a subject
* These numerals appear in the Fasciculus Temporum Anti-
quorum, printed at Louvain in 1476; and in 1484 was published
the great work of Lucas de Burgo, entitled Summa de Arithme-
tieſt, &c. in which the numerals appear under very nearly their
present form, and which was adopted in all books printed in the
following century.
+ In his dedication to Sir Thomas More, he says, that the
reason which induced him to study Arithmetic was to protect him-
self from the frauds of money-changers and stewards, who availed
themselves of the ignorance of their employers; he professes to have
read, for the purpose of his work, all the books which had ever
been written, whether eruditos, ineptos, Latinos, barbaros, quorum.
calleren linguas ; and be adds, that there was hardly any nation
which did not possess books on this subject in their own lan-
guage ; and by selecting what was excellent in each, and arranging
his materials, he at last succeeded in licking them, as it were.
into shape and symmetry, as the bear does her cubs.
3 &rchaeologia, vol. xiii. p. 148. …”
We may safely infer, that when calendars contain- Amongst the earliest books printed at St. Albans, Earlypºint-
ing these figures began to be circulated, that the use is one Rhetorica nova Gulielmi de Saona, in which there º:
and advantages of this notation must have been in is the date 1478;” and the Myrrour of the World was *
some degree understood by the persons for whose use reprinted in 1506, under an enlarged and altered form
they were composed; or that, at all events, under by Laurence Andrews, in which the common opera-
such circumstances, the knowledge of it must have tions of Arsmetrike, or Algorism, are treated of with
been rapidly propagated. The addition of the Rules great clearness, and in which the figures appear under
de Algorismo, which was made to so many of them their present form ; the treatise of Cuthbert Tonstall,
between the early part of the XIVth and the beginning Bishop of Durham, de Arte Supputandi, one of the
of the XVth centuries, would seem to show that their most beautiful productions of Pynson's press, was
composers were employing a notation which was not published in 1522, and showed its author to be per-
universally known or used, and that there were some fectly well acquainted with the most improved state
of their readers for whom some explanation was of the science at that period.t
necessary. After the commencement of the XVth In the year 1537, there was printed at St. Alban's
century, however, we find few examples of such addi- “An Introduction for to lerne to reckon with the Pen
tions; and we may consider, that from such a period this and with the Counters after the true cast of Arsmetyke,
notation was known and admitted in every part of or Awgrym, in hole numbers, and also in broken ;" and
Europe, where monasteries and other establishments in 1542, was printed by Robert Recorde, Doctor in
of the Church of Rome were to be found. Physic, the first edition of “ The Grounde of Artes,
The figures (80.) In a passage which has been quoted from the teaching the Woorke and Practice of Arithmetyke, booth
%ue of Dreme of Chaucer,” these numerals are called the in whole numbers and fractions, after a more easy and
Chaucer. figures newe; and, therefore, it is presumed, that they ºactº $9 te: than any hitherto hath been set forth.” This
were not known long before his time. is a work which we have had frequent occasion to
- quote, and we do not find it necessary to notice at
THE WEDDE. present any other contemporary or subsequent publi-
, Shortly it was so full of bestes cations on this subject.
That though Argos the noble counter inv1_ _ & *
Ysate to reckin in his contour (82.) The accounts of merchants were kept in Roman Their use
For by the figures newe all ken. numerals until the middle of the XVIth century, and in mer-
chants’ ac-
counts, &c.
3 I 2
420.
A R I T H M ET I C.
Arithmetic. of popular education. In colleges, where the use of
S-N-' the learned languages prescribed by their statutes gave
a species of authority and sanction to classical nota-
tion, we find that the Roman numerals were used
'in some instances as late as the year 1600, and in
others as late as 1700; and the same feeling operates
generally even to this day, to preserve their use in
monumental and other inscriptions.
The use of (83.) A nearly contemporary author in the account of
them said to the life of Baldwin, Archbishop of Treves, states that
have been he learned the use of these figures in the University of
i.º.º. Paris, in the year 1806; a fact, which would, if well
sity of paris authenticated, show that these figures had become the
in 1306. Subject of common and popular knowledge at an
earlier period than in this country. Professor Leslie,t
mentions a Traet in the German language, dated
1390, De Algorismo, in which the notation and the
common operations of this Arithmetic are very dis-
tinctly explained. It is impossible, however, in cases
where the date of the introduction of these numerals
is to be determined by their appearance in a manu-
script, to say that other manuscripts of greater anti-
quity may not have contained them, unless the external
evidence which they afford is farther confirmed by the
contents of the manuscript itself.
The work (84.) These numerals appear to have been known about
of Planudes the middle of the XIIIth century in the Greek Empire
at Constantinople, if we may judge from the work of
Planudes, of which manuscripts are to be found in the
Bodleian, Vatican, and Royal library at Paris, of the
last of which Delambre has given an analysis;f it
is entitled Toº ºpt)\oooºwtátov uovax6v kaxovuévov
Mašuov too IIAavová-) ºr mºoftopia cat' "Ivčovs, m \eyo-
puévn ue^{d\m. Vossius, indeed, has placed Planudes
in the middle of the XIVth century; but if it be true,
that he dedicated other works to the Emperor Michael
Palaeologus, he must have flourished about the period
which we have mentioned. The forms of the digits,
which he says are Indian, bear a considerable resem-
blance to the Arabic; and, with respect to the zero,
he observes, T10éag, àe étepov tº axiſwa & caxóvat
tgºpav, cat’’Ivčo is a nuatuov oièev : # 6é Tſaghpa ſpa-
Origin and Øetal ottws o : the term Tºtºpa, or cifra, is derived from
meaning of the Arabic term tsaphara, (quod vacuum aut inane est,)
*...* and corresponds to the Sanskrit term for it, stinya,
** which signifies blank or void. The use of the zero of
cypher, so important and so essential to this species
of Arithmetic, has led to a more comprehensive
meaning of the term, all the digits being designated
by the general term of cypher, and the verb to cypher
having the same signification as to calculate or work
with these figures. To return, however, from this
digression, it is clear from an examination of this
work of Planudes, that it must have been translated
or collected from the Arabic writers, as the distribu-
tion of the subject, and the rules of operation, are
nearly the same as are found in the arithmetical and
algebraical works in that language.
Different (85.) The work which we have just noticed is
origins as- another amongst the numerous testimonies which may
signed to be brought to establish the Indian origin of those nu-
º º: merals. It must not be imagined, however, that the
”” opinions of learned men have at all times agreed in
•
* Chronicon Gottwicense, p. 114.
+ Philosophy of Arithmetic, p. 114.
: Hist, de l'Astronomie Ancienne, tom. i. p. 518.
assigning to them such an origin. as there are few
subjects concerning which a greater number of ex-ra-
vagant theories have been formed. One of the earliest
and most popular of these is the one which was first
propagated by Dasypodius, and afterwards maintained
by Huet, Bishop of Avranches,” that these figures
were formed from the Greek letters for the niue
digits. Dr. Bernard, in his Tables, f acquiescing in
this theory, has stated that these figures passed from
the Greeks to the Indians in 710, from the Indians to
the Arabians in 800, and from thence to the Spaniards
in the year 1000, for all which periods he has given
their forms, though it would be difficult to refer to
his authorities, and still more difficult to confirm
them. Dr. Ward, Rhetoric Professor at Gresham
College, adopting the same views, has traced their
course also from Greece to India, and from thence
through Arabia to the Moors in Africa and in Spain.
It is unnecessary to quote more examples of the
names even of distinguished men who have written in
favour of an hypothesis so entirely unsupported by
facts. Gatterer; imagined that he had discovered in
Egyptian manuscripts written in the enchoriac cha-
racter, that the digits were denoted by nine letters,
and that a tenth sign performed the office of zero;
and, as if not contented with assigning to the Egyp-
tians the knowledge of this Arithmetic, it must be
known likewise to Cecrops and Pythagoras, and that
it formed part of the mysterious science which was
transmitted to his followers. It was, probably, a
vestige of this mystical knowledge which showed
itself in the manuscript of Boethius, which Weidler
considered as old as the VIIIth century, and in which
the Arabic numerals were used. § Wachter, however,
found uo difficulty in tracing them to a natural origin,
as well as the Roman numerals, in the different combi-
nations of the fingers ; thus unity is expressed by the
outstretched finger, and by repeating and varying this
character, we have got = for 2, E3 Kº, or & for 4, E for
5, &c. which have degenerated from long use, and for the
greater convenience of writing, to their present forms.|
A Dutchman of the name of Rudbeck, and Brinhorne,
a Swede, with singular boldness and patriotism, have
claimed the invention for the Celts, or Scythians of the
North of Europe ; whilst a Spaniard named Antonio
Nassau has referred it to the Carthaginians in Africa,
on the authority of some Tyrian inscriptions, in which
characters somewhat resembling the Arabic figures
have been discovered." The last opinion which we
shall notice is that of Calmet, ** which was originated
by Mabillon, it that these figures were part of the signs
or abbreviations of Tiro, the freedman of Cicero, which
were so extensively used in short-hand writing by the
ancients. It is a sufficient answer to such an hypothesis
to state, that we find amongst those signs simple sym-
bols for 11, 19, 13, &c. which could not have been the
case if they had involved, in any way whatsoever, the
* Demonstratio Evangelica, p. 647.
+ See his Tables, or Plate, printed in 1689.
† Weltgeschichte bis Cyrus, p. 586 ; Elementa rei Diplo-
matica, p. 65.
§ Philosophical Transactions for 1744, No. 472, p. 81.
| Naturae et Scripturae Concardia, c. 4.
* Chronicon Gottwicense, p., 114.
* Mémoires de Trevour, for 1753, p. 1630.
tică, p. 215.
ti Nouveau Traité de Diplomatique, tom. iii. p. 527.
De re Diploma-
History.
\-N-7
Dasypo-
dius, Huet,
&c.
Gatterer,
Wachter,
Rudbeck
and
Brinhorne,
Nassau,
Malyi Hlon
and Calmet.
A R I T H M ET I C. 421
Arithmetic. principle of arithmetical notation by nine digits and
\-V-- Zero.
Explana-
tion of
figures in
Plate III.
(86.) In Plate III. we have given the principal forms
of these numerals amongst different nations of Asia, as
well as at different periods since their first introduc-
tion into Europe. The following are the explanations
of the several divisions of this Plate, to which refe-
rences are made by means of the Roman numerals.
I. The Sanskrit numerals in the Devanagari cha-
racters, or divine characters of Nagari: from the
Sanskrit Grammar of Dr. Wilkins.
II. The Bengali numerals, from Halhed's Grammar,
p. 132; the forms of these numerals are slightly varied
from the Sanskrit; the zero is a small circle, and not
a simple dot, as was usually the case in the Sanskrit:
it is probable that this symbol for zero was borrowed
from the Arabs at a later period.
III. Ancient Persian numerals, from a manuscript of
the Zenda-Vesta, in the Bodleian library.
IV. Mahratta numerals, from a manuscript of Mr.
Perry, copied by Mr. Astle in his Origin and Progress
of Writing, Plate 30.
V. The Thibet numerals, from another manuscript
of Mr. Perry, also copied by Mr. Astle; it is obvious
that all these numerals have a common origin.
VI. The Arabic numerals, which are specifically called
Indian, from De Sacy's Grammaire Arabe, Plate 8.
VII. The Arabic numerals, called Gobar, in which
there is no zero, the nine digits having a dot attached
to them to denote tens, two dots to denote hundreds,
and so on ; likewise taken from De Sacy.
VIII. Numeral characters from the manuscripts of
Maximus Planudes.
IX. Numeral characters used in manuscripts in the
XVth century.
X. The wood-cut, with numerals, in the Myrrour of
the World, by Caxton, A. D. 1480.
XI. Fac simile of the memorandum of Petrarch on
the manuscript of St. Augustin : Hoc immensum opus
donavit mihi vir egregius Johannes Boccacius de Certaldo,
poeta nostri temporis et Florentia Mediolanum ad me
pervenit 1355, Aprilis 10. -
XII. The date 1245, from the manuscript in the
Strozzi library, but which has every appearance of
having been added in the XVth century.
XIII. Anno Dom. 1292, ad meridiem civitatis Toleti.
This is the date found at the commencement of what
is called Roger Bacon's Calendar, in the British
Museum, which Ames erroneously imagined to be the
date of the manuscript.
XIV. Christi 1334 incompleto; this date is written upon
a manuscript of Postilla in Psalterium by Nicholas de
Gorron ; upon which is written, Accommodatus iste
Liber Magistro Thome Durant sacre pagine Profes-
sori, post festum S. Petri Cathedre, Anno Christi 1334
incompleto. There is every reason for supposing that
this date is perfectly genuine.*
XV. Psalterium, cum calendario auctioni, hymnis Eccle-
siaticis, Litania et vigiliis mortuorum ; in usum Johanna,
Regis Ricardi 2 matris scriptum, A. p. 1380. This very
curious calendar commences with an account of Algo-
rism, as usual in that period, and contains a more than
common quantity of purely astronomical knowledge,
the tables being calculated to the meridian of Oxford.T
* Casley, Catalogue of Manuscripts in the British Museum.
+ Ibid, Plate 16, |
XVI. Liber olim de Claustro Roffensi, per Benedictum History,
episcopum datus : to which is added, Iste liber ligatus S-a-’
erat Oxonii, in Catstrete, ad instantiam Reverendi Domini
Thome Wybarum, in sacra Theologia Baccalarii monachi
Roffensis, Anno Domini 1467.* Dates, which before
the middle of this century were very rarely expressed
in Arabic numerals, became extremely common after-
wards.
XVII. A series of dates, from 1445 to 1587, for the
purpose of exhibiting the changes which these figures
underwent during that period.t
(87.) There were two distinct species of Arithmetic Two dis-
which were cultivated by the ancients, one practical, and tinct spe-
the other purely speculative. . The history of the first
species we have already entered into with sufficient
separate notice, not merely from the great and, in some
degree, extravagant importance which was attached
to it by the Pythagorean and Platonic philosophers of
antiquity, but likewise from its influence upon the
particular speculations of arithmetical and other wri-
ters in modern times; we shall, therefore, give a very
brief abstract of this Arithmetic, as far as in the first
place regards the several species of numbers, and
their arithmetical relations with each other ; and
secondly, those properties of numbers, which were
supposed to lead to the knowledge of nature, and the
principles of the true philosophy.
(88.) Euclid has defined unity to be that according Definitions Y
to which every thing which exists is called one ; and of unity.
number to be multitude, composed of unities. The
first of these definitions is one amongst innumerable
other proofs that the ancients mistook the province of
definition in attempting to explain a term such as unity,
expressing an idea which does not admit of resolution
into others more simple than itself; the consequence of
this practice has been, that these definitions of the same
term have no accordance with each other, and are in
many cases absurd or unintelligible : thus one author
says, that the monad is the principle and element of num-
ber, which while multitude is diminished by subtrac-
tion, is deprived of all number, and remains fixed and
unchanged, since division cannot proceed beyond it. §
Another asserts, that the monad is one multitude :
some say that it is the confine of number and parts;
others that it is the form of forms, as comprehending
causally all the ratios which are in number.| Another
favourite subject also of disputation in the schools,
was, whether unity is a number, and which was treated
in many cases without reference to the definition of
number itself:" thus, according to the Euclidean
definition of numbers, the question must be answered
in the negative ; but if we define number to be
that quantity by which every thing is numbered, it
would be answered in the affirmative; since unity is a
quantity which may be numbered by itself. We do
not propose these questions as deserving of grave and
serious discussion, nor shall we attempt to reconcile
and explain definitions which are apparently so con-
* Casley, Plate 16.
+ Nouveau Traité de Diplomatique, tom. iii. Plate 60.
† Lib. vii. def. 1, 2.
§ Theonis Smyrnaei Mathematica, p. 23.
| Taylor's Theoretic Arithmetic, p. 4.
| Alstedii Encyclopædia, p. 804; Arithmétique de Simon
Stevin, where the affirmative of this question is vigorously main-
tained. -
Arithmetic
º e - ' tº a tº among the
detail ; the second, however, merits a distinct and ancients.
2.
'422
A R. I. T H M E T I C.
Arithmetic, tradictory, and which are so far removed above all
Different
species of
numbers.
Pariter
pares.
Impariter
pares.
Pariter et
inapariter
pares.
Perfect.
Amicable.
Diametral.
Triangular.
Square,
simple comprehension ; we merely mention them for
the purpose of showing how readily even the most
acute understanding may be bewildered in a labyrinth
of absurdities, unless guided in all its reasonings by
fixed and intelligible principles.
Numbers are distributed into various species, such
as odd and even, prime or compound, &c. Even
numbers are separated into such as are pariter pares,
comprehended in the series,
4, 8, 16, 32, 64,
all whose divisors are even : impariter pares, forming
the series
6, 10, 14, 18, 22,
which being divided by an even number, have an odd
number for their quotient: and lastly, pariter and im-
pariter pares, comprehending all other even numbers,
greater than 2, which are not included in the two other
classes. Again, numbers are perfect, when equal to the
sum of their divisors, deficient when less, and superabund-
ant when greater than that sum. Two numbers are
amicable,” when equal to the sum of each others divi-
sors; and imperfectly amicableſ when the sum of their
divisors is the same for both, though not equal to
either of them. Diametral numbers, the sum of the
Squares of whose two factors is a square number, the
square root of which is the diameter.f Polygonal num-
bers, which are of different species, such as triangu-
lar numbers, included in the series,
1, 3, 6, 10, 15, 21, 28,
which express the number of units which admit of
being symmetrically arranged in an equilateral triangle,
where there are 1, 2, 3, 4, 5, 6, 7, &c. units respec-
tively corresponding to each side. Square numbers,
the only species of polygonal number which Euclid has
Pentagonal,
considered, as they alone strictly correspond in their
properties, with the geometrical figures from which
they derive their denomination. Pentagonal numbers,
which only admit of symmetrical, or rather equidistant
arrangement, in the equilateral, but not equiangular
pentagon, which is formed by the junction of a square
and equilateral triangle, as in the annexed figure :
* ".
/ \
A () © (9 ".
4. 0 0 9 *
! 9 O Q !
l 0 0 9 .
! . .
!------ -
These numbers are clearly, therefore, the sum of the
r—
* Such are the numbers 284 and 220.
t The sum of the divisors of 27 and 35 respectively are equal
to 13.
† The number 12 = 3.4 is a diametral number, and 5 the
diameter, since 32 + 4* = 25 = 5*.
triangular and square numbers corresponding to bases, History.
which differ by unity, and are expressed by the series, S-N-"
1, 5, 12, 22, .35, 51, 70, 92, &c.
Since triangular numbers are formed by the addition
of the terms of the series of natural numbers
1, 2, 3, 4, 5, &c.
square numbers, by the addition of the alternate terms
of this series, or of the odd numbers,
1, 3, 5, 7, &c.
and pentagonal numbers, by the sums of every third
term of that series, or of
1, 4, 7, 10, 13, &c.
So likewise hexagonal numbers are formed by the Hexagonal.
addition of every fourth term, beginning with the
first, which are,
1, 5, 9, 13, &c.
and heptagonal numbers, by the addition of every fifth
term, and so on for the polygonal numbers of higher
orders : in those cases, therefore, there is evidently no
reference whatever to geometrical figures ; and it is
quite clear, likewise, that the polygonal numbers of
the ancients have no analogy to the figurate numbers
of modern times.
If the terms of the series of triangular nnmbers be
added together, we get the series
1, 4, 10, 20, 35, 56, &c. t
the first in the series of solid pyramidal numbers, and
which are called triangular pyramidal numbers. If
the terms of the series of squares be otherwise added
together, we get a series of square pyramidal numbers, Pyramidal.
which are
- 1, 5, 14, 30, &c.
In both cases these numbers would express the number
of equal spheres, which can be placed in complete contact
with each other, when those in the base form an equi-
lateral triangle or square; and if the highest sphere,
or the first number in the series be wanting, we get a
defective pyramid; and hence the numbers 3, 9, 19,
&c. are called defective triangular pyramids, and simi-
larly for those cases where the base is a square or
any other figure. -
If the series of squares be multiplied in succession Cubes.
by the natural numbers, we get the series of cubes.
Numbers arising from three unequal factors, as 3,
4, 5, are called parallelopipedons, and sometimes Parallelopi-
orica Mjvot aka Mºvot, or in Latin gradati, and by others pedons,
£8watakov, or little altars, from the resemblance
which they bear to their analogous solids. When
two equal numbers are multiplied into a less, as
3 × 3 × 2, the result is denominated a laterculus or Laterculi.
tile ; but if two equal numbers are multiplied into a
greater, the product is called asser, or a plank. All Asseres.
these distinctions were studiously multiplied, and *
their denominations founded upon some fancied analogy
or resemblance to objects of sense. -
Numbers were also distinguished into square, Square
eTepopºlicets, and oblong ; the second were the products numbers,
of two numbers which differed by unity, and the last, oblong, &c.
the products of any other unequal numbers; and it
was observed, that whilst square numbers are formed
by the addition of the terms of the series of odd
numbers, those of the second class are formed by
the addition of the terms of the series of even numbers.
Minute and trivial as many of the distinctions of Different
the different species of numbers may appear to be, Pºies of
they are much less so than those which refer to ratios “”
and proportions, and their different species; thus we
A R IT H M ET I C.
423
Arithmetic:
\-y-Z
have ratios of equality, of greater and less inequality;
of the second, there are five species, and as many cor-
responding to them of the third ; thus we have
multiple ratio, where the antecedent is a multiple of
the consequent; superparticular,” where the antecedent
is equal to the consequent and some part of it, which
may become therefore, sesquialteran, sesquitertian,
sesquiquintan, &c. as 3 to 2, 4 to 3, 5 to 4, &c. pro-
ceeding to infinite subdivisions ; superpartient, f where
the antecedent contains the consequent once, and
some multiple of its parts, as 5 to 3, 7 to 4, 19 to 10,
and so on ; multiple Superparticular, where the ante-
cedent contains a multiple of the consequent with
some part of it, as 5 to 2,23 to 11, &c.; and multiple
Superpartient, § where the antecedent contains a mul-
tiple of the consequent and of some part of it, as 11
to 3, 17 to 7. and so on.
science of numbers, as known to the ancients; many History.
of the absurdities even of Plato himself, and still more `Y-’
of many of his commentators and successors, being ex-
cluded by the rigid form of his demonstrations, and
the intimate dependence of his propositions on each
other; a circumstance which drew a remark from Lord?
Bacon, that nothing had been added to this science since
his time, which was worthy of the lapse of so many ages,
the truth of which it would be difficult to dispute; in
reality, the powers of the human mind were frittered
away in the minute distinctions and endless refinements
of the mystical philosophy of the schools of Pythagoras
and Plato, which Bacon considered as exspatiatio quardam.
speculationis; verifying, in short, the profound obser-
vation which he afterwards subjoins, that the human
mind is naturally disposed to indulge itself in such
speculations, when not occupied with the earnest
Different They distinguished three principal species of pro- pursuit of truth : Hoc enim habet ingenium humanum,
kinds of portion, arithmetic, geometric, and harmonic, which ut cum ad solida non sufficiat, in supervacaneis se atterat.
Proportion, were known to the more ancient philosophers; but “Arithmetic,” according to the opinion of the Pla- Platonic
later writers added three others, which were opposite
to the former. Thus, if a., 6, q, are three nnmbers,
arranged in the order of their magnitude, they will be
related to each other, according to the conditions of
tonic philosophers, “ was not to be studied with gross Arithmetic.
and vulgar views, but in such a manner as might
enable men to attain to the contemplation of the
nature of numbers; not for the purpose of dealing
the fourth analogy, or weo.orms, if with merchants and tavern-keepers, but for the im-
a : " : : 3 — "Y : a - 7 || provement of the mind, considering it as the path
.* : 7 : : Y : which leads to the knowledge of truth and reality.”
of the fifth, if & º
2 £3 : y :: 8 — y : a - 3 || In order to show how perfectly removed their arith-
and of the sixth. # gº tº ty metical speculations were from all practical objects,
5 £3 £3 g. ** it is only necessary to refer to a few examples of the
Cº., I : ; tºº Y : a, - o º
To these six, four others were added by later writers,
to complete the decad; these are as follows:
manner in which they philosophised about numbers.
Thus, perfect numbers compared with those which
are deficient or superabundant, are considered as the
a : y : : a - Y : {3 — ºf it images of the virtues, considered as equally remote
a : y : : a - Y : a - 3 #: from excess and defect, and constituting a mean point
B : " : : a - Y : B — ºf §§ between them ; thus, true courage is a mean between
B : " : : a - Y : a - 3 || audacity and cowardice, and liberality between pro-
Many im- It is certain that many curious and valuable proper- fusion and avarice: in other respects also, this analogy
portant ties of numbers were discovered by the ancients, in remarkable; as perfect numbers, like virtues, are
. their pursuit after these relations and analogies, which few in number, and generated in a constant order;
j were supposed to lead to the knowledge of the true whilst superabundant º º: º 3 Fe º:
by the philosophy; every number became, as it were, pos- Wºº, infinite in number, disposable in no regular
ańcients, sessed of a property, and all numbers possessed some *.* generated according to no certain and in-
relative analogy with each other, to which a name Vºble law. g = º g
could be given ; but it is a tedious and unprofitable The tracing of analogies, like the preceding, accom-
task to wade through their multiplied refinements, their panied as is generally the case with the most
laborious demonstrations, that all equality proceeds elºquent ºf the Philosophical writers of antiquity;
from inequality; and again, that all inequality may with moral illustrations of uncommon elegance and
be reduced to equality; that squares and cubes par- beauty, may be considered as furnishing, at least, a
take of the nature of identity or sameness, and a great pleasing, if not a useful, exercise of the understand-
variety of other questions equally trivialand unintelligi- *š3 but, in other cases, analogies are taken for proofs,
ble ; and we have been chiefly induced to enlarge upon and assumed as the bases of the most absurd and
this subject to the extent which we have done, from inconsistent theories : and it would be difficult to refer
observing the influence of these speculations, both *Y chapter in the history of the human mind which
upon the form and substance of nearly all works on affords a more degrading picture of the perversion of
Arithmetic which appeared before the end of the the reasoning faculties, than that which would be
XVIth century. furnished by the collection of the ravings of the Pytha- Pythago-
Euclidean The VIIth, VIIIth, IXth, and Xth Books of Euclid, 89*** philosophers on the subject of numbers and rean Arith-
their properties; thus Pythagoras considered “number metic.
as the ruler of forms and ideas, and the cause of Gods
and daemons :” and again, that “ to the most ancient
and all-powerful creating Deity, number was the canon, ,
Arithmetic. form an abstract of the most rational portion of the
*—
* Aoyos etruopios, Theon. Smyrn. pars ii. cap. 24.
+ Aoyos stripepſis, Ibid. cap. 25.
ſt Aoyos troAAatrâagletipoptos, Ibid. cap. 26.
§ Aoyos troXAaTAao tetriuepms, Ibid. cap. 27.
| As, for instance, the numbers 6, 5, 3.
* Aoyurtuch kal &piðueruch, ÖAkös kal &yoyos Trpès &A#0slav-
&rreov Šē Aoytotikäs, u}, idiotikós, kal, ws étrl 6&v ràs rāv &gtºuðv,
* As 2, 4, 5. ** As 1, 4, 6 ºpvorea's àpíkovt at Tă vohorel. Övös trpádeos xapu èutégou # karrºw”
++ As 6, 8, 9. †† As 6, 7, 9 &AA’ {veka juxils, 7%s éir' &Añ6etav, kal oia'tav 6500. Theon.
§§ As 4, 6, 7. ||| AS 3, 5, 8 Smyrn. cap. i.
424
A. R. I. T H M E T I C.
Arithmetic, the efficient reason, the intellect also, and the most
\-y-' undeviating balance of the composition and generation
of all things.” Again, Philolaus declared, “that num-
ber was the governing and self-begotten bond of the
eternal permanency of mundane natures.” Another
said, “ that number was the judicial instrument of the
Maker of the universe, and the first paradigm of
mundane fabrication.” And Taylor, their modern com-
mentator and advocate, in order, to add to this climax
of absurdities, asks whether it is possible that these
philosophers could have spoken thus sublimely of
number, unless they had considered it as possessing
an essence separate from sensibles, and a transcen-
dency fabricative, and at the same time paradigmatic.”
Benarius The learned Meursiust has made a collection from
Pythagori- the writings of Pythagorean philosophers, of the
i. of. names and properties which they assigned to all num-
Cllr SillS, es tº º ©
bers from 1 to 10, abounding, as might be imagined,
with all kinds of absurdities and contradictions. We
shall give a few specimens. Unity, or the monad, is
termed by Nicomachus, “ Intellect, male and female,
God, and in a certain respect, matter; the recipient of
all things (Tavöoxews,) chaos, confusion, commixture,
obscurity, darkness, a chasm, Tartarus, Styx, and
terror ; the absence of mixture, a subterranean profun-
dity, Lethe, a rigid virgin and Atlas; the axis, the
sun, and Pyralios; Morpho, the tower of Jupiter, and
the spermatic reason; Apollo, likewise, the prophet and
soothsayer.” The commentators on this passage say,
that the monad is called intellect, as being the origin
and fountain of all numbers, in the same manner as
intellect is of all ideas : it is called male and female,
as containing in itself causally, the odd and the even,
the former corresponding to the male, and the latter
to the female ; as it is the cause of multitude, there-
fore it is called God, and matter in a certain respect
only, as God is the first, and matter the last of
things, and each subsists by negation of all things,
and consequently matter is said to be dissimilarly
similar to divinity : again, the monad was considered
as the recipient of all things, from its analogy to the
divinity, as all things are comprehended in the ineffa-
ble nature of divinity: it is called chaos, as being
primaeval and first born, from which all things have
sprung, as all numbers from unity; but it is needless
to pursue this tissue of absurdities to the conclusion,
as we have given more than enough to satisfy our
readers of the nature of their idle speculations.
The work (89.) This passion for discovering the mystical
of Bungus properties of numbers descended from the ancients
on the pro- to the moderns, and numerous works have been
*...* written for the purpose of explaining them; amongst
numbers, others we may mention that of Petrus Bungus, t
whose work on this subject extends to 700 quarto
pages; illustrating all the properties of numbers,
whether mathematical, metaphysical, or theological ;
not content with collecting all the observations of the
Pythagoreans concerning them, he has referred to
every passage in the Bible in which numbers are men-
tioned, incorporating, in a certain sense, the whole
system of Christian and Pagan theology : our limits
will allow us to notice but one or two specimens of
his reasoning; thus, the number 11 being the first
* Taylor's Theoretic Arithmetic, p. 165.
t Denarius Pythagoricus, Lugd. Batavorum, 1631.
: Petri Bungi Bergamotis, Numerorum Mysteria, 1618,
which transgresses the decad, denotes the wicked who History.
transgress the Decalogue, whilst 12, the number of the S-N-
Apostles, is the proper symbol of the good and the
just ; the number, however, upon which above all
others he has dilated with peculiar industry and satis-
faction, is 666, the number of the beast in the Reve-
lations, the symbol of Antichrist; he shows that this
is the number denoted by the words, Tettav, Napºretts,
\ateuvos, and is particularly anxious to reduce the
name of the impious and perfidious Haeresiarch Mar-
tin Luther, to a form which may express the same
formidable number ; for this purpose he transfers the
numeral power of the Greek to the Latin alphabet, and
after Italianizing and mispelling his name, he finds that
M (30) A (1) R' (80) T (100) I (9) N (40)
L (20) V (200) T (100) E (5) R (80) A (1)
constitute the number 666. He seems conscious,
however, of the liberties which he had been obliged
to take in effecting his purpose, and consoles him-
self with his better success in its Hebraized form,
-nº, Lulter, which expresses the same number.”
(90.) The numbers 3 and 7 were, for very obvious Properties
reasons, the subject of particular speculation with the of the
writers of that age ; and every department of nature, hº
science, literature, and art, was ransacked for the purpose j.
of discovering ternary and septenary combinations. The
excellent old monk Pacioli, or Fra. Lucas de Burgo
Sancti Sepulchri, the author of the first printed Trea-
tise on Arithmetic, has enlarged upon the first of these
numbers in a manner which is rather amusing, from
the quaint and incongruous mixture of the objects
which he has selected for illustration. “ There are Lucas de
three principal sins,” says he, “avarice, luxury, and Burgo on
pride; three sorts of satisfaction for sin, fasting, §.
tº e º Ü T º
almsgiving, and prayer; three persons offended by sin, .
God, the sinner himself, and his neighbour ; three
witnesses in heaven, Pater, verbum, spiritus sanctus ;
three degrees of penitence, contrition, confession, and
satisfaction, which Dante has represented as the three
steps of the ladder that leads to purgatory, the first
marble, the second black and rugged stone, the third
red porphyry. There are three sacred orders in the
church militant, subdiaconati, diacomati, and presbyterati
there are three parts, not without mystery, of the most
sacred body made by the priest in the mass; and three
* It must be confessed, however, that these attacks on the
great Reformer were not altogether unprovoked, either by himself
or his followers ; he himself interpreted this number to apply
to the duration of Popery ; and his friend and disciple, Stifel, the
most acute and original of the early mathematicians of Germany,
appears to have allowed himself to be seduced by these alsurd
speculations ; he relates, in an appendix to his edition of Christo-
pher Rudolph on Algebra in 1571, that whilst a monk at Esslin-
gen in 1520, and when infected by the writings of Luther, he was
reading in the library of his convent the 13th Chapter of Přer:c-
lations, it struck his mind that the Beast must signify the Pope,
Leo X. ; he then proceeded in pious hope to make the calcula-
tion of the sum of the numeral letters in Leo decimus, which he
found to be M, D, C, L, V, I ; the sum which these formed was
too great by M, and too little by X; but he bethought him
again, that he had seen the name written Leo X., and that there
were ten letters in Leo decimus, from eithcr of which he could
obtain the deficient number, and by interpreting the M to mean
mysterium, he found the number required, a discovery which gave
him such unspeakable comfort, that he believed that his inter-
pretation must have been an immediate inspiration of God.
This is not the only instance in which this excellent person
gave way to these idle fancies; and he prophesied the downfall
of the papacy on more than one occasion, and had the misfortune
to live to see his own predictions falsified, i
A R I T H M E T I C.
425
Arithmetic, times he says Agnus Dei, and three times, Sanctus; and
Yº-Yº" if we well consider all the devout acts of Christian
Trinads of
Bungus.
Septenary
combina-
tions.
Ancient
writers on
Arithmetic.
worship, they are found in a termary combination; if
we wish rightly to partake of the holy communion,
we must three times express our contrition, Domine
non sum dignus; but who can say more of the ternary
number in a shorter compass, than what the prophet
says, tu signaculum sanctae trinitatis. There are three
Furies in the infernal regions; three Fates, Atropos,
Lachesis, and Clotho. There are three theological vir-
tues: Fides, spes, et charitas. Tria sunt pericula mundi:
Equum currere; navigare, et sub tyranno vivere. There
are three enemies of the soul : the Devil, the world,
and the flesh. There are three things which are in no
esteem : the strength of a porter, the advice of a
poor man, and the beauty of a beautiful woman.
There are three vows of the Minorite Friars: poverty,
obedience, and chastity. There are three terms in a
continued proportion. There are three ways in which
we may commit sin : corde, ore, ope. Three principal
things in Paradise: glory, riches, and justice. There
are three things which are especially displeasing to
God : an avaricious rich man, a proud poor man, and
a luxurious old man. And all things, in short, are
founded in three ; that is, in number, in weight, and
in measure.” -
The collection of trinads which Bungus has given
would alone form a little volume, embracing, as they
do, every species of knowledge, art, and science; he has
observed them in friendship, beauty, colours, eternity,
in the first letter of the alphabet, in the monad, in
music, in poetry, in a point, in a circle, in magnitude,
in time, in primitive theology, and, in short, in almost
every imaginable thing; so general, indeed, according
to him, are these ternary combinations, that they make
some approach to a general law of nature.
(91.) If the number 3 has been honoured with particu-
lar commemoration, the number 7, has received an equal,
if not greater distinction. In the year 1502 there
was printed at Leipsic a work entitled Heptalogium
Virgilii Salzburgensis,* in honour of the number 7, and
expressly composed for the use of students of the
University of Leipsic; it consists of seven parts, each
consisting of seven divisions. We think it unnecessary
to detain the reader with the enumeration of the
septads which this work contains, forming a collection,
as might indeed be expected, of the most gross
absurdities; our object being merely to show from
this instance, as might be done from a multitude of
others, how general was this passion for philosophising
about the properties of numbers; so much so indeed,
that vestiges of it may be discovered at a very late
period, when the principles of just and philosophical
reasoning were generally understood and practised.t
(92.) The history of Arithmetic, down to the period of
the introduction of the Arabic numerals, would be little
* Kaestner, Geschichte der Mathematik, vol. i. p. 204.
t As a final example, we will merely mention the following work,
the contents of which, as might be expected, are quite worthy of the
title: “ The Secrets of Numbers according to Theological, Arith-
metical, Geometrical, and Harmonical computation. Drawn, for
the better part, out of those Ancients, as well as Neoteriques.
Pleasing to read, profitable to understande, opening themselves to the
capacities of both learned and unlearned ; being no other than a
key to lead men to any doctrinal knowledge wha soever. By William
Ingpen, Gent, London, 1624.” The first chapter is entitled, “The
ercellencie of numbers ; and how far they stretch towards the
attaining of all manner of sciences.”
WOL. I.
benefitted by the analysis of the arithmetical writings
of the Platonic school, presenting as they do no great
variety in their form, and still less in their object;
the chief of those whose works we possess, was
Nicomachus of Gerasos, the author of a Treatise
entitled Isagoge Arithmetica, and who flourished pro-
bably about the Christian era, though his date is un-
certain ; of this work, the Arithmetic of Boethius is in
many respects a mere translation ; it was honoured
also with a commentary by Jamblichus, whose work,
entitled Tă 6eoMoyolaeva Tijs 'Apiðuetúcms, surpasses,
in visionary speculations on the properties of num-
bers, the most absurd and extravagant of his pre-
decessors. Martianus Capella, who flourished in
the VIth century, wrote a Poem on the seven li-
beral arts, including Arithmetic and Geometry, en-
titled De Nuptiis Mercurii et Philologiae. Theon of
Smyrna, whose work we have had occasion to quote,
and who flourished about the beginning of the IIId
century, wrote expressly on those parts of Mathema-
tics which were necessary for the understanding of
the works of Plato. Porphyry, who flourished about
the same time, and who wrote on Arithmetic and the
mysteries of numbers. Proclus, who in his Com-
mentary in the Ist Book of Euclid has furnished us
with so much information on the history of the Mathe-
matical sciences amongst the ancients. Cassiodorus,
a contemporary of Capella, Photius, Philo, Thymari-
das, and a multitude of others, whose works have
perished, but of whom we find notices in ancient
writers, and who all, in common with other mathema-
ticians of their age, appear to have written for the
purpose of expounding and amplifying the doctrines
of Plato: so universal was the reverence in which they
were held both by Christians and Pagans during the
first seven centuries after the Christian era.
(93.) In sketching the history of the progress of
Arithmetic, after the introduction of the Arabic nu-
merals, we shall follow the arrangement of subjects
and not of authors, proceeding from numeration in a
regular series through the common operations of
Arithmetic. We are well aware of the many advan-
tages regarding the history of science, which arise
from a regular analysis of the works of the principal
writers, a plan which has been adopted so successfully
by Delambre in his History of Ancient Astronomy, by
Kaestner, and by other authors ; but the extensive
details in which such a plan must necessarily lead us,
would be inconsistent with the limits within which we
are confined by the nature of this work, and which
indeed we have already very much exceeded ; we
shall include, however, in our present sketch, occa-
sional notices of the works which we shall have
occasion to refer to, without attempting any further
analysis of them than may be generally called for by
the immediate subject of discussion.
(94.) The process of numeration, as distinguished
from notation, will of course vary with different nations,
according to the genius of the language to which it is
accommodated. Amongst the Hindoos, who possess
distinct names for the first nineteen terms” of the
* Ganésa, one of the Commentators on the Lilāvati, says that
these names were carried to a greater extent by Srid'hara, and
other writers; but he does not notice them, in consequence of
the little utility of such designations, as well as of the numerous
contradictions which are found amongst them, from their not being
always appropriated to the same numbers.
3 k
History.
\-N-"
Nicoma-
chus,
Jamblichus.
Martianus
Capella.
Theon.
Porphyry,
Proclus.
Cassiodo-
rus, &c.
History of
Arithmetic
in the order
of subjects.
Numeration
in the
Lilávati.
426
A R I T H M E T I C.
Arithmetic. decuple series of numbers beginning from unity, it
S-N- assumes the form which is of all others the most
Amongst
the Italians,
A mongst
the Spa-
niards.
The term
million :
when first
used in
England.
simple in principle; it being merely necessary, in passing
from notation to numeration, to repeat in succession
the name of the digits with that of the corresponding
term of the series; the practice, however, of such
numeration is extremely tedious and embarrassing,
from its preventing that distribution of large numbers
into classes of superior units proceeding by thousands,
myriads, or millions, &c. which is so useful in relieving
the memory from the burden of so many independent
terms, and in also assisting the mind in the conception
and comparison of large numbers.
(95.) The Italians from an early period adopted the
mode of numeration which is now in universal use,
distributing the digits into periods of six, and, conse-
quently, proceeding by millions ; the units in the
several classes thus formed being millions, billions,
trillions, &c. (Art. 16.) Such is the process of nu-
meration which is given by Lucas de Burgo.
(96.) The Spaniards adopted the term cuento to denote
a million, and the following is the table of numeration
which is given in the Arithmetic of Juan de Ortega"
in 1536:
10.
100.
1000.
10000.
100000.
1000000.
10000000.
100000000.
1000000000.
Their numeration was thus limited to eleven places
of figures, as the term cuento did not admit of com-
position with the terms for two, three, four, &c. in the
same manner as the term millione in billione, trillione,
quadrillione, &c. In an earlier Spanish author, how-
ever, on the same subject,t we find the term millione
applied to designate 1000000000000, or a billion,
whilst cuento is used in its ordinary sense for 1000000;
thus the number
957 || 653 | 978 || 245 || 349 | 186 || 357 | 243
consisting of twenty-four places, and separated into
periods of three, is expressed in Latin, adapted to the
Spanish numeral-language, by nonaginti quinquaginta
septem millia | seccenti quinquaginta tres cuentos | nona-
gintiseptuaginta octo ducenti quadragintaquinquemil-
liones trecenti quadraginti movem millia | centum octa-
ginta sea cuentos trecenti quinquaginta septem millia |
ducenti quadraginta tres; the misapplication of the term
million, which is found in this case, is a very curious
example of a practice which we have already had
occasion to remark on more than one occasion.
(97.) We have no means of ascertaining the precise
period at which this term was introduced into our
Dezema.
Centena.
Millar.
Dezena de millar.
Centena de millar.
Cuento.
Dezena de cuento.
Centena de qo. (cwento.)
Centena de millar de qo.
* Tractado subtilissimo de Arismetica y de Geometria ; compuesto
y ordenado por el reverendo padre fray Juan de Ortega, de la orden
de los predicadores. This is a work of some merit, and we shall
afterwards have occasion to notice a method which it contains for
approximation to the square and cube root.
+ Arithmeticae Practicae seu Algorismi Tractatus à Petro
Sanchez Teruelo noviter compilatus earplicatusque. Impressus
Parisiis per Thomani Rees in domo rubro post Carmelitas, Anno
di 1513.
own language.
Bishop Tonstall,” who has discussed History.
at great length the Latin nomenclature of numbers, S-V-'
speaks of the term million as in common use, but he
rejects it as barbarous. Quartus locus, says he,
exhibet mille; septimus millena millia; vulgus millionem
barbare vocat. Again he says, Decimus locus capit
millies millena millia; vulgus milliones millionum vocat.
In this case, however, the combination of these terms
is erroneous, as it would designate a million of millions,
or a billion. It is not easy to say what class of persons
were meant to be designated by the term vulgus ; but,
most probably, the arithmetical writers of this and
other countries; at all events, the term appears in
Recorde’s Arithmetic, and in all subsequent writers on
this subject. . -
(98.) It appears to have been admitted into German When first
at a much later period than into English and French. used in
Kaestner says,t that he found it in no German author Germany.
on Arithmetic in the first half of the XVIth century;
and Claviusi is the first writer of that nation, who in a
Latin Treatise on Arithmetic has noticed the term ; in
the chapter on numeration he says, Si more Italorum
millena millia appellare velimus milliones, paucioribus
verbis et fortasse significantius numerum quemcumque
propositum erprimemus. He does not seem to have
carried the innovation farther, as we afterwards find
billions expressed by milliones millionum, which are
the highest numbers which he has occasion to use.
(99.) It has been from avery early period the custom of Divisions of
writers on Arithmetic to separate numbers into periods
of three and of six, as the numeration in most Euro-
pean languages must proceed by thousands and mil-
lions; these periods are called membres by Stevinus, Š
amongst whose definitions we find the following: Chas-
ques trois characteres d’un nombre s' appellent membre,
des quels le premier, sont les premiers trois characteres à
la deatre : et le second membre, les trois characteres sui-
vant, vers la sinestre : et aimsi par ordre du troisieme
membre et autres suivants, tant qu'il y en aura all mom-
bre propose. Instead of million, he says, mille mille; for
a thousand millions, he uses mille mille mille ; and for
a billion, mille mille mille mille, and so on for higher
numbers. If we might be allowed to judge from this
practice and numeral phraseology of Stevinus, as well
as from the observation of his contemporary Clavius,
we might imagine that the term million was not yet
in general use amongst mathematicians. A different
system, however, began to prevail at no very distant
period ; for we find Albert Girard, in his Commence-
7mens de l'Arithmetique, in his account of numeration
which he terms Prolation des Nombres, dividing the
places into periods of six, which he terms premiere
* De Arte Supputandi, p. 4. In numeration he divides the
places into periods of three, and calls 1000000 millena millia, or
millies millema; 1000000000, millies millena millia ; 1000000000000,
millies millena millia millies, and so on, proceeding by analogy with
the practice of classical authors in the construction of their expres-
sions for high numbers.
+ Geschichte der Mathematik, vol. i. p. 145.
† Christophori Clavii Bambergensis, e. S. J. Epitome Afrithme-
tica: Practica, Rom. 1583. -
§ Arithmetique, Livre Premier Definitiones, 1584. This work,
which bears many marks of the acuteness and originality of its author,
was published first in Flemish, and afterwards translated into bar-
harous French. The whole works of Stevinus were collected to-
gether, and published at Leyden in 1634, the year after his death,
by his friend and commentator Albert Girard.
| Invention Nouvelle en Algebre, par Albert Girard, Mathematicien,
Amsterdam, 1629. -
numbers
into periods
of three
and six.
A. R. I T H M E T I C.
427
Arithmetic.
S-N-2
Fundamen-
tal opera-
tions of
Arithmetic.
Addition.
Amongst
the Hindoos.
Subtraction.
masse, seconde masse, troisieme masse, respectively, the
first of which only is divided into two periods of three
places.
(100.) The fundamental operations of Arithmetic, as
given in the Lilāvatī, are eight in number; namely, addi-
tion, subtraction, multiplication, division, square, square
root, cube, cube root :* to the first four of these the
Arabs added two, namely, duplation and mediation or
halving, considering them as operations in some
degree distinct from multiplication and division, in
consequence of the readiness with which they were
performed ; and they appear as such in many of the
books of Arithmetic of the XVIth century.f
(101.) With respect to the two first operations, whether
in Sanscrit or other authors, we shall not find much
to remark. The rule given in the Lilávati, in the first
case, is as follows: “ The sum of the figures, ac-
cording to their places, is to be taken in the direct, or
inverse order;” which is interpreted by the Scholiast
to mean, “from the first on the right towards the left,
or from the last on the left towards the right.” In
other words, that they commenced indifferently with
the figures in the highest or lowest places, a practice
which would not lead to much inconvenience when
their mode of working addition is considered; thus
to add 2, 5, 32, 193, 18, 10, 100, they proceed as
follows:
Sum of the units. . . . 2, 5, 2, 3, 8, 0, 0 20
Sum of the tens . . . . 3, 9, 1, 1, 0 . . . . . . . . 14
Sum of the hundreds 1, 0, 0, 1. . . . . . . . . . . . 2
Sum of the sums . . . . . . . . . . 360
If they had commenced with the figures in the
highest places, the process would have stood as
follows: **
Sum of the hundreds . . . . . . . . 2
Sum of the tens . . . . . . . . . . . . 14
Sum of the units . . . . . . . . . . . . 20
Sum of the sums. . . . 360
(102.) The process of subtraction was commenced
likewise either from the right or from the left, but much
more commonly from the latter; and it is a circum-
stance sufficiently remarkable, that this practice of
* The subject of the 12th Chapter of the Brahma-sphuta-
siddhanta of Brahmegupta, written in the VIIth century, is Arith-
metic; and it commences by defining the knowledge which consti-
tutes a ganaca, or calculator: “He who distinctly and severally
knows, addition and the rest of the twenty logistics, and the eight
determinations, including measurement by shadow, is a calculator.”
The Scholiast on this passage states those rules to be the eight
fundamental operations, five pules of reduction of fractions, rule of
three terms, (direct and inverse,) of five terms, seven terms, nine terms,
eleven terms, and barter, which are twenty arithmetical operations.
Mixture, progression, plane figure, excavation, stack, saw, mound and
shadow, are eight determinations.
+ This distinction was abandoned when the processes for multi-
plication became more general and uniform, as an absurd and
unnecessary refinement; thus Gemma. Frisius, in his Arithmetica
Practica, Methodus Facilis, published in 1548, speaks of this
practice with great contempt : Solent nonnulli duplationem et
mediationem assignare species distinctas a multiplicatione et
divisione. Quid vero moverit stupidos illos nescio, cum et definitio
et operatio cadem sit. A different reason, however, is given by Lucas
de Burgo for abandoning this distribution of the parts of Algorithm.
“The ancient Philosophers,” says he, “assign nine parts of algorism;
but we will reduce them to seven, in reverence of the seven gifts of
the Holy Spirit: namely, numeration, addition, subtraction, multipli-
cation, division, progression, and extraction of roots.”
commencing subtraction from the highest place, Histºry.
which is subject to considerable inconvenience, should S-, ºr
have been so very general. It is found in Arabie
writers, in Maximus Planudes, and in many European
writers as late as the end of the XVIth century.
In Planudes, numbers to be added or subtracted Planudes.
are placed underneath each other, as in modern books
of Arithmetic ; and the sum in one case, and the
difference in the other, is placed above the whole.
When the digits in the subtrahend are greater than
those in the minuend, a unit is placed beneath them,
as in this example:
I 8 7 6 9 remainder.
5 4 6 I 2 minuend.
3 5 8 4 3 subtrahend.
I I } 1
In performing the operation, 3 is increased by the
unit in the next place to the right, and similarly for
5, 8, 4; and the digits thus increased are subtracted
from the digits above, when increased by 10, in order to
get the remainder.
In other cases the process is arranged as follows:
O 6 7 7 9 remainder.
2 9 9 1
3 O O 2 4 minuend.
2 3 2 4 5 subtrahend.
The digits 3, 0, 0, 2 in the minuend are replaced by
2, 9, 9, 1 ; and then 5 is subtracted from 4, 4 from 1,
2 from 9, 3 from 9, and 2 from 2, in order to get the
remainder. It is obvious, that when such a prepara-
tion is made, it is indifferent whether the operations
proceed from right to left, or from left to right.
(103.) Bishop Tonstall attributes the invention of the Tonstall.
modern practice of subtraction to an English Arith-
metician of the name of Garth ; a method by which
any number, however great or however intricate,
might be subtracted, mamentibus notis universis. This
method he has illustrated with great detail, and has
added for the assistance of the learner a subtraction
table, giving the successive remainders of the nine
digits when subtracted from the series of natural
numbers from 11 to 19 inclusive, the only cases which
can occur in practice. In speaking of the methods of
preceding writers he has given the following, which
will be at once explained by the example by which he Q.
has illustrated it :
2 9 10 10
3 O I O
l l l i
l 8 9 9
The digits in the minuend are replaced by the numbers,
whether digits or not, from which the subtraction must
be made.
(104.) In the Arithmetic of Ramus, which was published Ramus.
in the year 1584, though written at an earlier period,
we find the operation performed from left to right;
and the same practice is followed by some other
writers of his school.” Thus in subtracting 345 from
* Ramus was obliged to quit his country, and take refuge in
Germany, during the persecutions of the Protestants in France.
He there established a school of philosophy and mathematics,
chiefly distinguished for the introduction of more minute and
logical subdivisions of the subjects of discussion, whether mather
matical or not, than is to be found amongst preceding writers.
He had many followers and most zealous admirers, who con-
stituted his mathematical school, and who wrote upon Arithmetic
3 K 2
428
A R I T H M E T I C.
Arithmetic.
432, the sums to be subtracted, and the remainder are
-v- Written as follows:
8 7
4 3 2
3 4 5
Orontius
Fineus.
Multiplica-
tion.
Methods in
the Lilávati.
When 3 is subtracted from 4, the remainder should
be 1, but it is replaced by zero, since the next digit
in the subtrahend is greater than the one correspond-
ing to it in the minuend: in this case also the re-
mainder, which would be 9, is reduced to 8, since the
next digit, 5 in the subtrahend, is greater than 2
which is above it ; - the last remainder, 7, is not
altered. -
Ramus speaks with great respect of Orontius
Fineus,” his predecessor in the professorship of
mathematics at Paris, as having revived, and in some
measure introduced, the study of those sciences in
France. He was also the author of a work on Arith-
metic,f where the process of subtraction is taught N
under the same form in which it is found in modern tº
It is difficult to account for the wº
adoption of this very inconvenient practice by Ramus,
books of Arithmetic.
when the other method must have been familiar to
him; and we can only attribute it to that love of
singularity which led him to aspire to the foundation of
a school of his own.
(105.) The author of the Lilāvati has noticed six dif-
ferent modes of multiplying numbers, and two others are
mentioned by his commentators; these will be best
explained by their application to the following ex-
ample, which is given in that work: -
“Beautiful and dear Lilāvati,f whose eyes are like
a fawn’s ; tell me what are the numbers resulting from
one hundred and thirty-five taken into twelve 2 If
thou be skilled in multiplication, by whole or by parts,
whether by division or separation of digits, tell me,
auspicious woman, what is the quotient of the pro-
ducts, divided by the same multiplier *
Statement. Multiplicand, 135. Multiplier, 12.
(1.) Product. (Multiplying the digits of the multi-
plicand necessarily by the multiplier.)
I 3 5
12 12 12
I2 6O
3 6
} 6 20
(2.) Or subdividing the multiplier into parts, as 8
and 4, and severally multiplying the multiplicand by
them ; thus f
and Algebra ; of these was Bernard Salignac, of Bordeaux, his
companion in exile, who wrote a work on Arithmetic, as well as
Tractatus Arithmetici Partium et Alligationis, 1575; Christian
Urstis or Urstisius, whose work entitled Elementa Arithmetica,
logieis legibus explicata, was published in 1579; Joannes Freigius,
Christopher Clavius, and many others. Ramus afterwards returned
to Paris, and became one of the victims of the massacre of St. Bar-
tholomew.
* In his Preface to his Arithmetic.
+ Orontii Finei Delphinatis, Regii Mathematicarum Lutetiae :
Professoris, De Arithmetica Practica, libri quatuor, 2d edit. 1555.
: It was the daughter of Bhascara who is apostrophised in
this very affectionate manner, whom he would not allow to marry,
in consequence of having discovered, by an astrological scheme,
that such an event would be fatal to his own life. It was by way
of consolation that he dedicated this work to her, and called it by her
Jalīle.
H35 8 1080
l 35 4 540
162O
(3.) Or the multiplier 12 being divided by 3, the
quotient is 4; by which and by 3, successively multi-
plying, the last product is the result; thus
135 4 2O 540 3 190
12 15
4 *==
*==º 1620
540
(4.) Or taking the digits as parts, viz. 1 and 2, the
multiplicand being multiplied by them severally, and
the products added together, according to the places
of figures; thus ^.
135 135
l 2
27O
135
162O
(5.) Or the multiplicand being multipned by the
multiplier less 2, viz. 10, and added to twice the multi-
plicand; thus
135 iO I 350
135 2 27O
162O
(6.) Or the multiplicand being multiplied by the
multiplier increased by eight, viz. 20, and eight times
the multiplier being subtracted; thus
135 20 27.
135 8 1080
162O
The other two methods are given in the commentary
of Ganésa:
History.
(1.) Form a series of equal squares, the number of Reticulated
|Dºº
al/41/41%
l 6 2 0
which in length is the same as the number of places
in the multiplicand, and the number in depth the
same as the number of places in the multiplier;
divide these squares by diagonals, and write the mul-
tiplicand and multiplier on the adjacent sides of the
rectangle, each digit being placed opposite to a square,
and the highest place in both being reckoned from
the same angle. Multiply the several digits of the
multiplicand and multiplier together, placing the
several products in the squares which are common to
the two digits which are multiplied successively to-
gether; the digit in the unit's place being put in the
Iower half, and that in the place of tens being put in
the higher division of each square which is formed by
its diagonal. The entire product is found by adding
the digits between the same diagonals successively
together. -
II]
tion.
ultiplica-
A R. I T H M E T I C.
429
This method of multiplication, which appears to
- have been very popular in the East, was adopted by
The shaba- the Arabs, who termed it shabacah, or network, from
Arithmetic.
cah of the the reticulated appearance of the figure which it formed,
** and also by the Persians, under a slight alteration
of form. It is found likewise amongst the early
Italian writers on Algebra ; and the same principle
may be recognised in the process of multiplication by
Napier's bones.
Cross multi- The eighth and last method of multiplication
plication. is described by Ganésa in the following terms :
“After setting the multiplier under the multiplicand,
multiply unit by unit, and write the result underneath;
then, as in cross multiplication, multiply unit by ten,
and ten by unit, add together, and set down the sum
in a line with the foregoing results ; next multiply
unit by hundred, and hundred by unit, and ten by ten ;
add together and set down the result as before, and
so on with the rest of the digits; this being done, the
sum of the results is the product of the multiplica-
tion.” Thus,
- 135
$2.
IO
I F:
5.
l
16 TO
The Commentator, however, considers this method as
difficult, and not to be learnt by dull scholars without
oral instruction. -
(106.) The number and variety of these methods would
seem to show that the operation of multiplication was
considered as one of considerable difficulty ; and
it is sufficiently remarkable, that the ordinary process
of multiplying the multiplicand by the successive
digits of the multiplier, and adding together the
several results arranged in their proper places, should
not be found amongst them. We find no notice of
the multiplication table either amongst them or the
Arabs; at all events it did not form a part of their
elementary system of instruction, a circumstance
which would account for some of the expedients which
they appear to have made use of, for the purpose of
relieving the memory from the labour of forming the
products of the higher digits with each other.
(107.) The Arabs adopted most of the Hindoo methods
of multiplication, and added some others of their own.
They appear to have adopted the methods of Apollo-
nius for the multiplication of articulate numbers, as
far as the determination of the order of their product
was concerned: we find amongst them many peculiar
contrivances for the multiplication of numbers be-
tween 5 and 10, 10 and 20 ; of numbers between 1
and 10 into others between 10 and 20 ; of numbers
between 10 and 20 into others between 20 and
100; and so on. They may be considered also as the
authors of the method of quarter squares, or of finding
the product of two factors by subtracting the square
of half their difference from the square of half their
Sll III,
(108.) The Arabs were, most probably, the inventors of
Multiplica-
tion con-
sidered as
very diffi-
cult.
Hindoo
methods
adopted
generally
by the
Arabians.
The Ara-
bians the
inventors of by casting out the 9s, which is as yet unknown to
the proof of the accuracy of arithmetical operations
the Hindoos ; they called it tarazu, or the balance. History.
In general, however, they contented themselves with \-y—
the inheritance of the science transmitted to them the proof
from the Greeks, or with what they received from the by º:
East, with little or no attempt to add to them by native Out the 9s.
inventions.
(109.) It is one amongst many proofs that the work The sub-
of Planudes was chiefly collected from Arabic writers, stance of
that he was acquainted with this method of casting the work of
out the 9s. In the operation of multiplication itself, º
he has chiefly followed the method of multiplying from the
crosswise, or karū Töv xiao abv, from the figure of x, Arabians.
which is employed to connect the digits to be mul-
tiplied together; thus, in multiplying 24 into 35, the
factors are written thus,
8 4 O
3 5
X.
2 4
Multiply 4 into 5, (wovaðes,) write down 0 and His methods
retain 2 for the next place; multiply 4 into 3, and of multipli-
2 into 5, the sum is 22, which added to 2 makes 24, *.
(8ékaðes;) write down 4 and retain 2; lastly, multi-
ply 2 into 3, add 2, which makes 8, (skatov.Tabes);
we thus get the product 840.
This is not the only process of multiplication which he
has given ; there is another which he acknowledges to
be very difficult to perform with ink upon paper, (éri
xdprov Ště puéNavos,) but very commodious on a board
strewed with sand, where the digits may be readily
effaced and replaced by others; thus, taking the same
example,
8 4 O
7.
6/ 2/
3 5
2 4
we multiply 2 into 3, write 6 above 3; again, mul-
tiply 2 into 5, the result is 10; add 1 to 6, and
replace it by 7, or write 7 above it; multiply 4 into
3, the product is 12; write 2 above 5, and add 1 to
7, which is replaced by 8, or 8 written above it;
lastly, multiply 4 into 5, the result is 20; add 2 to 2,
place 4 above it, and after it the cypher; the last
figures, 840, or those which remain without accents,
will express the product required.
(110.) The Italians, who cultivated Arithmetic with so Methods of
much zeal and success, from a very early period adopted multiplica-
from their oriental masters many of their processes for the .
multiplication and division of numbers; adding, how- writers on
ever, many of their own, and particularly those which Arithmetic.
are practised at this day. In the Summa de Arith-
metica of Lucas de Burgo we find eight different
methods of multiplication, some of which are desig-
nated by names of a very quaint and fanciful nature.
We shall mention them in their order:
1. Multiplicatio bericuocoli e schacherii. The se-
cond of these names is derived from the resemblance
of the written process to the squares of a chess-board;
the first from its resemblance to the chequers on a
species of sweetmeat or cake made chiefly from the
paste of bacochi or apricots,” which were commonly
used at festivals. The process is as follows:
* Bericuocolo : spezie di comfortino; si tacevano prima que'
confortini de pasta de bacochi, com’ e da credere. -
430 A R L T H M E T I C.
Arithmetic.
9 8 7 6
8 || 8 || 8 || 8 4
To o o 8 w
6 || 9 |s
5|| 9 || 2 || 5 6
6 7 O 4 8 l 6 4
This method of multiplication, denominated schachero
at Venice, bericuocolo at Florence and Verona, and at
Verona and some other cities of Italy organetto, is exhi-
bited by Tartaglia,” and later Italian writers, without
the squares, in the appearance of which these singular
names originated. It thus became the method which is
now universally used, and which was adopted from the
beginning of the XVIth century by all writers on
Arithmetic, nearly to the exclusion of every other method.
2. Castelluccio; by the little castle. It is difficult to
discover the reason of this denomination.
9 8 7 6.
6, 7, 8 9.
6 1 1 0 1 0 O O
5 4 3 1 2 O O
4 7 5 2 3 O
4 O 7 3 4
67 o 48 I 64
This was one of the methods much practised by the
Florentines, by whom it is sometimes termed all’ indie-
tro, from the operation beginning with the highest
places, more Arabum, according to the statement of
Pacioli.
3. Columna, o per tavoletta; by the column, or by
the tablets. These were tables of multiplication com-
monly called libretti, or librettine, and at Florence
caselle ; they were arranged in columns, the first con-
taining the squares of the digits, the second the
products of 2 into all digits above 2; the third of 3,
into all digits above 3 ; and so on, extending in some
* Numerie Misure, pars 1ma Venice, 1556. This is a work in
three large volumes; the first of which contains the most elaborate
system of practical and mercantile Arithmetic that was known in that
age, and which we shall have very frequent occasion to refer to. The
other volumes are divided into five parts, the subjects of which are
geometry, mensuration, speculative arithmetic, and algebra. Its
author, Nicolas Tartaglia, justly celebrated for his important discovery
of the method of solution of cubic equations, derived his name,
according to the testimony of Tiraboschi, from the following incident.
He was born at Brescia in 1500, and at the sack of that city, by the
French in 1512, was left for dead with three sword-cuts on his head
and two on his face and lips; by the care of his mother, however, he
recovered, but in consequence of the wound on his lips he lisped or
stammered so much, that he was nicknamed by the boys Tartaglia, from
the Italian word by which this imperfection is designated. In later:
life he retained a name which was not without interest as connected
with the story of his misfortunes. In 1534 he settled at Venice, and
became Professor of Arithmetic, a situation which he filled with
extraordinary reputation for twenty-five years.
Some idea may be formed of the opinion entertained of Tartaglia
and of his work, from the following title of an abridged translation
of it: L'Arithmetique de Nicolas Tartaglia, Brescian, grand mathe-
maticien, et prince des praticiens. Recueillie et traduite de l’Italien
en Francois, par Guillaume Gosselin de Caen. Dedièe a tres illustre
et vertueuse princesse Marguerite de France, reyne de Navarre,
1578 -
cases as far as the products of all numbers less than History.
100 into each other. Pacioli says, that these tablets \--~~
were learned by the Florentines etiam a cunabulis;
and their familiarity with them was considered by him
as a principal cause of their superior dexterity in arith-
metical operations, and he consequently seizes every
opportunity of impressing on the mind of the student
in Arithmetic the necessity of a perfect acquaintance
with these tables. Tartaglia also, after giving some
examples of their utility, earnestly entreats every
amateur (dilettante) of the practice of Arithmetic to
force himself to learn, if not the whole, at least the
greater part of them, and in particular to make himself
familiar with those numbers which are used in the divi-
sion of the coins, weights, and measures of the city in
which he resides; of this kind, in the magnificent city
of Venice, are the numbers 12, 20, 24, 25, 32, and 36.
This method is used in multiplying any number,
however large, into another which is within the limits
of the table. Thus, to multiply 4685 into 13, the
digits of the multiplicand are multiplied successively
into 13, and the results formed in the ordinary manner.
4. Crocetta sive casella ; by cross multiplication. A
method which is said to require more exertion of the
understanding than any other,” particularly when
many figures are to be combined together. The fol-
lowing examples will explain it sufficiently :
3 7 4
\ - 2.
7 4 5
H– 2 0 7 9 3 6
Pacioli, who rarely omits an opportunity of moral-
izing, after expressing his admiration of this method,
as una bella et sotil cosa, but one which vol cervello a
casa e l'occhio a botega, proceeds to enlarge on the great
difficulty of attaining excellence, whether in morals or
in science, and on the species of analogy which exists
between them ; that whilst with respect to one there
is no virtue without labour, so in respect to the other
the saying of the philosophers is equally just, quod
virtus est circa difficile: that whilst the good and the
wise are few, and of a rare occurrence, the wicked and
foolish are met with everywhere, according to that
other saying, stultorum numerus est infinitus.
5. Quadrilatero ; by the square. A method which
is characterised as elegant,t and as not requiring the
operator to attend to the places of the figures when
performing the multiplications:
5 4 3 2
§ 4 3 2
1 || 0 || 8 || 6 || 4
4
1 || 6 || 2 || 9 || 6
2
2 || 1 || 7 || 2 || 8
6
2 || 7 || 1 || 6 || 0 |

2 9 5 O 6
* El quel modo vole alquanto piu fantasia e cervello che acuno
degli altri. •
tº Quale e bello e non bisogna tenere a mente le dicine.
A R I T H
M E T I C. 43]
Arithmetic. , 6. Gelosia sive graticola; latticed multiplication. “It
S-N-' is called by this name,” says Pacioli,” “because the
disposition of the operation resembles the form of a
lattice, or gelosia, a term by which we designate the
blinds or gratings which are placed in the windows
of houses inhabited by ladies, so that they may not
easily be seen, as well as by other nuns, in which the
lofty city of Venice greatly abounds; and it is not
surprising that the vulgar have found such names for
this operation, inasmuch as astronomers, even in our
days, have assumed the names and positions of many
stars from animals and terrestrial material forms.”
The method in question will be understood from the
subjoined example, and is clearly the same as one of
those noticed above as in common use amongst the
Hindoos, Arabians, and Persians.
& To multiply 987 into 987:
9 8 7
3 6 9 9
7 || 6 5 4
2 4
6
8 6
7 6 5
I 2 3
l
9 8 7 6
9 7 4
7. Ripiego; multiplication by the unfolding or reso-
lution of the multiplier into its component factors :
thus to multiply 157 by 42, resolve 42 into its ripieghi,
6 and 7, and multiply successively by them :
157
6 : 42 .
7.
8. Scapezzo; multiplication by cutting up, or sepa-
rating the multiplier into a number of parts, which
compose it by their addition: thus, multiply 2093 by
.17. • -
2 o 9 3 o
1 4 6 5 -l
35 58 I
* Or seato modo de moltiplicare e chiamato gelosia: ovvero per
graticola : e chiamasi per questi ºtomi, perche, la dispositione sua.
quando si pone in operatione torne a modo di graticola ovvero di
gelosia. Gelosia intendiamo quelli graticelli chi si costumano
mettere alle fenestre delle case dove habitano donne ; acio non si
possino facilmente vedere o altre religiose, di che molto abonda la
carcelsa cita di Venegia. E non e maravºglia che l’ vulgo habe tro-
In some cases, both multiplicand and multiplier are History.
separated into parts: thus, to multiply 15 into 12, we —y-
may separate 15 into 4, 5, 6, and 12 into 2, 4, 6, and
proceed as follows: **
4 , 5 , 6
2 , 4 , 6
8 || 16 24 30
10 || 20 ! 30 || 60
12 24 36 90
30 60 90 180
(111.) In another Italian Treatise on Arithmetic pub-
lished in 1567,” we find the same distinctions preserved,
and the same names, or nearly so, attached to them ; the
method of cross multiplication is expressly attributed
to Leonardus Pisanus, who derived it in common with
Maximus Planudes from the Hindoos, through the
Arabians; and it is not improbable that the St.
Andrew's cross, which is the sign of multiplication,
was derived from the custom of uniting the numbers
to be multiplied together by lines which crossed each
other, as in this example:
5 9
X
4 7
27 7 3
(112.) Both Lucas de Burgo and Tartaglia have men-
tioned the names of other methods of multiplication
which were made use of in their time ; such were the
methods per coppa or calice, per rombo, per triangolo,
per diamante. The first was most probably of the fol-
lowing kind, at least if we may judge from the very
imperfect description of it which Pacioli has given:
2 3 4
5 4 7 5 6
(113.) An extraordinary passion seems to have prevailed
in that age for the invention of new forms of multiplica-
tion, and every professional practitioner of Arithmetic
Origin of
the sign of
multiplica-
UlOn.

Other
methods of
multiplica-
tion.
Excessive
fondness for
novel
methods of
(and such were to be found in every mercantile city of multiplica-
Italy) considered it as an important triumph of his art
if he could produce a figure more elegant and more
refined in its composition and arrangement than those
which were used by others. They are all of them,
however, characterised by Pacioli as inconvenient, at
least compared with those which he had given; and
Tartaglia treats them as trifling and superfluous, such
as any one may invent who is acquainted with the 2d
Proposition in the IId Book of Euclid.
In performing multiplication a bocca over per testa
orally, or by the head, that is, senza penna, says Tar-
vato questi vocaboli a tali operatione; per che ancora li astronomi
hanno assumpti de molte stelle nomi e siti loro, da animali e formue
terrestri materiali.
* Le Pratiche delle due Prime Mathematiche di Pietro Catanec
Sienese. In Venetia, 1567.
tion.
432
A R I T H M ET I C.
Arithmetic. taglia, the Florentines make use of a species of indi-
gitation, working numbers by the inflections of the
Florentine figures. The methods for this purpose which he
* describes are similar, though not identical with the
methods of Bede and others which have been already
described above ; they furnish one amongst innumera-
ble other proofs of the proficiency of that extraordinary
people in Arithmetic, as well as in all the other arts of
civilized life.
Multiplica- (114.) We have beforementioned that the Hindoos had
tion *: no proper knowledge of the multiplication table; and
though the Arabs used sexagesimal tables to aid them
in their operations upon sexagesimals, they do not
appear to have made use of the table of Pythagoras as
the basis of their arithmetical education ; the credit
of introducing it, therefore, is due to the early Italian
writers on this science, who probably found it in the
writings of Boethius, and adopted it from thence.
Familiar as the use of it, even on a very extended
scale, appears to have been among the Italians, and
particularly amongst the Florentimes, yet many writers
of other countries considered it important to relieve the
memory from the labour of retaining it for the products
of all digits exceeding 5, by giving rules for the forma-
tion of them; the principal rule for this purpose, called
Sluggard's regula ignavi, or the sluggard’s rule, was adapted from
rule. the Arabians, and is found in Orontius Fineus, Recorde,
Laurenberg, Alstedius, and most other writers on
Arithmetic between the middle of the XVIth and
XVIIth centuries. It is as follows: Subtract each digit
from 10, and write down the difference; multiply these
differences together, and add as many tens to their pro-
duct as the first digit ecceeds the second difference, or the
second digit the first difference. The following are ex-
amples:
7 3 8 2 8 2 9 1 9 |
X X X X X
6 4. 7 3 8 2 8 2 9 |
*-*. *s-s-sº ſº-mºm-º.
4 2 6 4 7 2
*-mºre
5 6
*s-tº-º-º:
8 1
The principle of this rule is too obvious to require demon-
stration, and we merely mention it as an instance of the
disposition of the inferior writers of that as well as of other
ages, to adhere to trifling and particular processes, when
the same thing may be effected more rapidly by one which
is general. The Arabians, as we have seen, not only
made use of the rule in question, but likewise of others
similar in principle, which included numbers of two
places of figures; a practice which may be accounted
for, and in some measure justified, by their very general
use of sexagesimals, and the consequent importance of
being able to form the products which are found in a
sexagesimal table. .
Other (115.) Other expedients have been proposed to relieve
methods of the memory in the process of multiplication, from the
ºple. labour of carrying the tens. The following is proposed
Ile by Laurenberg, an author who endeavoured to elevate
the character of the common study of Arithmetic, by
collecting all his examples from classical authors, and by
making them illustrative of the geography, chronology,
weights and measures of antiquity, It will be readily
understood from his example: -
5142 History,
43
106
1532
108
2046
221106
(116.) “Multiplication,” says an old author in the Rule for
quaint pedantic language of his time, “observeth casting out
collocation, proceedeth to operation, and concludeth the 9s.
with probation.” This probation, or the rule for
proving the accuracy of this and other arithmetical
operations by casting out the nines, one of the few
additions which the Arabians made to the sciences
which they derived from the Greeks and the Hin-
doos, is found in the earliest European writings on
Arithmetic, beginning with Leonardus Pisanus. It is
stated also with great detail by Lucas de Burgo, who
has applied it to all the four fundamental rules ; he
has given likewise a method of proving the truth of
these operations by casting out the sevens, a process
much less rapid and commodious than the other,
though founded upon the same principle: it was re-
quisite, however, in this case, to get the remainders by
actual division by 7, and not, as in the other case, by
casting out the nines from the sum of the digits.
(117.) The extreme brevity with which the rules of ope-Process of
ration are stated in the Lilāvati, renders it difficult for division in
us to describe the Hindoo processes for division: we are theLilāvat:.
directed to abridge the dividend and divisor by an equal
number, whenever that is practicable, that is, to divide
them both by any common measure; thus, instead of
dividing 1620 by 12, we may divide 540 by 5, or 405
by 3. We find, however, in one of the commentators
on this work, a description of the process of long divi-
sion, which if exhibited in a scheme would exactly
agree with the modern rule ; taking the example just
given, “the highest places,” says Manoranjana, “ of the
proposed dividend 16 being divided by 12, the quotient
is l; and 4 over. Then 42 becomes the highest remain-
ing number, which divided by 12 gives the quotient 3,
to be placed in a line with the preceding quotient : thus 13
remains 60, which divided by 12 gives 5, and this being
carried to the same line as before, the entire quotient is
erhibited.”
(118.) We shall pass over the processes of division Italian
which are given in Arabic writers and Planudes, which methods of
exhibit nothing which merits much remark, and shall divisiºn.
proceed at once to the notice of the methods which are
given in early Italian writers. There are four different
methods given by Lucas de Burgo, which are as follow :
1. Partire a regolo
a tavaletta,
a la dritta
is the same operation, and is sometimes also termed
partire per testa, or division by the head; in this case
the divisor is a single digit, or a number of two places,
such as 12, 13, &c. included in the librettine, or Italian
tables of multiplication.
2 divisor 6 16
9876 dividend 3478 12387
4938 quotient 5793. 774+ºr
“This method of division,” says Lucas de Burgo,
“is called by the vulgar the rule, from the imitish
A. R. I. T H M E T I C.
433
Arithmetic. of the figure to the carpenter's rule, which is made use
S-N-2 of in the making of dining-tables, boxes, and other
articles, which rules are long and narrow. So likewise
is the scheme which is formed by the scholar in this
division, whose length consists of many figures, whilst
it is only one in breadth: it looks therefore like a rule,
since its themologia, or derivation, springs from thence.”
The second of these denominations arises from the
process being founded on the librettine, or multiplica-
tion tables; and the last, from the operation beginning
from the left, and proceeding to the right; it is called
by Tartaglia partire per colona over di testa, over per
discorso, over per toletta. The three first appellations
are easily understood ; the last is a popular corruption
of tavoletta.
2. Per ripiego ; by resolving the divisor into its
simple factors, or ripieghi :
63 divisor
7 250047 dividend
9 35721
3969 quotient.
3. A danda; “the third method of division is called,”
says the author, “by practicians a danda, it is thus called
for reasons which they will easily discover in the opera-
tion itself.”
Proveniens.
9876
5376
Divisor.
9876
97.33
88.884
865 lº
79008
75057
69 lº2
59256
59256
This process is evidently the same as our common
process for long division ; as the reader, however, may
not be able to find out the reason of its name from the
operation so readily as Pacioli" supposes, the following
explanation of a later author may remove the difficulty.
“The method of division called a danda,” says Cataneo,
“is most necessary to every person who wishes to be-
come an expert reckoner, and it is thus called a danda,
or by giving, since after every subtraction made in the
operation, we give or add one or more figures on the
right hand, so that the remainder after subtraction,
combined with the figure or figures given, may admit
of being divided by the divisor.”
This process, however, is, in the opinion of our
author, much less pleasant than the following, which is
termed, -
4. Galea vel galera vel batello; from the form of the
process resembling a galley, “the vessel of all others
most feared on the sea by those who have good know-
ledge of it; the most secure and swiftest; the most
rapid and lightest of the boats that pass on the water.’’t
The following is the form of the process :
* Il partire a danda e molto necessario a chi esperto ragioner esser
desidera ed e chiamato a danda el detto modo perche ogni sottrattion
fatta mel operaré se li da una o piu figure dal lato destro, talmente
che la detta sottrattione con la figura o figure date si possi partir
per il tuo partitore. Pietro Cataneo, le Pratiche delle due prime
Mathematiche.
t “El piu temuto nel mare da quelli chi ne hanno bona notitia,
WOL. I.
To divide 97.535399 by 9876,
l
2.
§
i
(9 8 7 6
ſ
;
ſ
g $ $
Such is the form of the galley, as given by Lucas de
Burgo. In order to explain its construction, we will
examine the several steps of the process: write down
first the dividend, and underneath the divisor, com-
mencing with the second figure of the dividend, as the
number formed by the first four figures of it is less than
the divisor; the result corresponding to the first figure
in the quotient will stand as follows:
8 6
g 7 5
x 33 3 1
g? § 3 ; 3.99 (9
g $ 7
The product of 9 iſ...}, is 81, which subtracted from
97 leaves 16 : cancel 97 in the dividend and 9 in the
divisor: 9 times 8 is 72, which taken from 165 leaves
93: cancel 163 in the last remainder and 8 in the
divisor ; 9 times 7 is 63, which taken from 933 leaves
870 : cancel 933 in the second remainder and 7 in the
divisor ; 9 times 6 is 54, which taken from 705 leaves
651 : cancel 705 in the third remainder and 6 in the
divisor: the remainder after the first division is
865 1399.
We now proceed to the second figure in the quotient,
which is 8, when the scheme, with the divisor in its
proper place, appears as follows:
§
%
(,
g $
9
which is equivalent (the quotient determined not being
considered) to
l
; 399 (98
6
:
Ž
º
ź
8
8651399 (
9876 -
The scheme, after the second division is completed, will
stand as follows:
/
g
i
5
r 0
3 5.
§ r 5
3 ; 2 99 (98
# 6
§ 7
el piu sicuro, el piu veloce: el piu smello, el piu leggiadro legno che
vada per aqua.” Cataneo speaks of this method as more beautiful
and rapid, though less secure than the other. “Il partir a galera,”
says he, “e molto leggiadro e speditivo, ma non tanto sicuro per un
principiante quanto il partir a danda et infra queste duo modi ci
corre gram disguaglianza mel operare, per le moltiplicatione et sot -
trattione, perche in questo di galera vanno per testa : come per
esemplo ti mostrero,”
3 L
History.
434
A R. I. T H M E T I C.
Arithmetic. The product of 8 and 9 is 72, which taken from 86
-N-' leaves 14 : cancel 86 and 9 in the divisor; 8 times 8
The galley
method of
division
superseded
all others.
JDivision
considered
very diffi-
cult.
Stifelius, Ramus, Stevinus, and Wallis;
is 64, which taken from 145 leaves 81 : cancel 145 and
8 in the divisor : 8 times 7 is 56, which taken from 81 I
leaves 755 : cancel 811 and 7 in the divisor : 8 times
6 is 48, which taken from 53 leaves 5 : cancel 53 in
the dividend and 6 in the divisor, and we get the re-
mainder 750599 after the second division.
It is very amusing to observe the enthusiastic admi-
ration of Lucas de Burgo for this method of division;
when describing the preceding method, he seems little
satisfied with it, and looks forward with a species of
impatience to the method a la galea, which possesses,
both in the description of it and in the operation itself,
a certain charm and solace. In truth, says he, it is a
noble thing to see in any species and scheme of num-
bers a galley perfectly exhibited, so as to be able
to observe in their disposition and arrangement its
4
8 8 l
0 g g 9 0
2 # 3 2 g r
$ $ $ 7 % J 0 $
g g g g g 4 3 g g g g g g r g
r 6 6 6 @ 3 g g g g g g g $ 3
$ $ $ $ $ 3 g g g g g g g $ 3
9 g g g g g g g g g g g g g
9 9 9 9 9 0 0 0 0 0 0 0 0
(119.) The analysis which we have given of this opera-
tion will be quite sufficient to enable the reader to pursue
it to its conclusion, and to understand its application to
other cases ; after making every allowance, however,
for the influence of our own practice in division, and
for the facility which long familiarity with this method
must necessarily have given, we must still feel some
degree of surprise at this preference given to it, on
account of the superior expedition with which the opera-
tion was performed. In this opinion, however, nearly
every writer on Arithmetic appears to have agreed,
as late as the end of the XVIIth century. It was
adopted by the Spaniards, French, Germans, and
English, and it is the only method which they have
generally thought it necessary to notice : it is found
almost universally in the works of Tonstall, Recorde,
and it was
only at the beginning of the XVIIIth century that this
method of division, called by English Arithmeticians
the scratch way of division, from the scratches used in
cancelling the figures, was superseded by the method
which is now in common use, which was specifically
called Italian division, from the country from which it
was derived.
(120.) The older writers on Arithmetic appear to have
considered division as an operation of considerable diffi-
culty, and one which required a very close and steady
application of the understanding. Dura cosa e la par-
tita, says Lucas de Burgo; and Ramus, after working
through an example of long division, in itself sufficiently
complicated and operose, subjoins the following re-
marks: Hoc exemplum et similia divisoris plurium
notarum mathematicum illam et Platonis uétao Tpoºjv
et Aristotelis aſpapeauv in auditorum animis imprimis
suscitabant et confirmabant. Pes bonus, oculus bonus
ait tyronibus lanista : mens boma, memoria bona, manus
bona in quotidiana divisionis experientid dicat hic
arithmeticus discipulo. Varietas enim tam multiplicis
i
stem and stern, its mast, its sail, its yards, and its History.
oars, launched in the spacious ocean of Arithmetic. S-N-
Well, then, may he exhort the student before he ven-
tures upon this last and most difficult of arithmetical
labours, to invoke Apollo in the words of Dante, at the
commencement of his Paradiso,
- O buono Apollo, ad ultimo lavaro,
to inspire him with genius and resolution which may
be equal to so important an undertaking.
It appears, according to the statement of Tartaglia,
to have been the custom at Venice for masters to pro-
pose to their pupils, as the last proof of their pro-
ficiency in this process of division, examples which
would produce the complete form of the galley, with
its masts and pendant. The last addition was supplied
by the scheme for the proof of the accuracy of the
operation by casting out the 9s. The following is the
example which he gives:
*º-º-6
0 8
* {3}
6. * { },
#} ºf
§ 2 g g g g g g g $ 3 3 & # 3
$ g g g g g g g g $ $ $ $ $ $ (8 8
g g g g g g g g g g g g g g g
9 9 0 0 0 () () () 0 O 0 9 9 9
ân und numeratione numerationis erectam mentem et
constantem memoriam, fidelemque manum marime
omnium requirunt. Ac jam memo sibi arithmeticae
discipulus vereque studiosus videatur, misi singulis arith-
metici studii diebus divisionem vel quam marimam.
potuit, effecerit. A modern school-boy might well
exult in his superior dexterity in this operation, when
the greatest masters of arithmetical knowledge of former
times felt so strongly impressed with a sense of its
difficulty, and thought it necessary to enforce its con-
stant and daily practice as the only means of acquiring
an adequate knowledge of it.
(121.) Recorde, indeed, has noticed the Italian method Italian
of division, “which,” he says, “I first learned of, and is method
practised by my ancient and especial loving friend, ...”
Master Henry Bridges, wherein not any one figure is “ e
cancelled or defaced.” The modus operandi, as given
by him, may be seen from the following example:
33) 7890 (239 sº,
66
129
99
300
297
3
(122.) The Arabians and Persians perform division by Arabian
a process which is in some degree analogous to their and Persian
methods of multiplication, as every step in the process ethods of
is put down at length. It will be best explained by an division.
example: let it be required to divide 197685 by 573:
A R IT H M ET I C.
435
Arithmetic.
\–v-”
Method of
forming
squares in
the Lilāvati.
Hindoo
process for
extracting
the square
TO0t.
3 4 5
I 9 7 6 | 8 5
l 5
4 7
2 1
2 6 6
9
2 5 7
2 0
5 7
2 8
2 9 8
l 2
2 8 6
2 5
3 6
3 5
l 5
l 5
5 7 3
5 7 3
5 7 3
(123.) The author of the Lilávati has given rules for
the formation of squares and cubes, as well as for the ex-
traction of the corresponding roots. The rule for the
formation of the square, which is very ingenious, is as
follows: “Place the square of the last digit over the
number; and the rest of the digits doubled and multi-
plied by the last are to be placed above them respec-
tively; then repeating the number with the omission of
the last digit, perform the same operation : thus to find
the square of 297,
297
88209
(124.) In performing the converse operation, every
uneven place is marked by a vertical line, and the inter-
mediate even digits by a horizontal one ; but if the
place be even it is joined with the contiguous odd digit.
If we take for an example,
§słoś
We subtract from the last uneven place 8, the square
— I — I
4, and there remains 48209. Double the root 2, and
divide by that (4,) the subsequent even digit 48;
quotient 9 a higher one cannot be taken ; for the root
of the foregoing digit would become greater than 2: History.
- e g 1– © { º Sy” º
the remainder is 12209. From the uneven place (with
I
the residue) 122, subtract the square of the quotient 9,
— I
viz. 81, the remainder is 4109. The double of the quo-
tient 18 is to be placed in a line with the former double
number 4. By this divide the even place 410; the
quotient is 7 and the remainder 49: to which uneven
digit the square of the quotient 49 answers without
residue. The double of the quotient 14 is put in a line
with the preceding double number 58, making 594 :
the half of which is the root sought, 297.
The preceding account of the Hindoo operation for
extracting the square root is taken from the Commen-
tators on the Lilávati ; and making allowances for some
little obscurity of expression, which most probably
arises from the difficulty of conveying from Sanscrit to
a modern language the full force of its idioms and
phrases, we shall find little which differs from the rule
which is given in our books of Arithmetic. The same
observation may be extended to the rule for the extrac-
tion of the cube root, which exhibits very little that is
peculiar, if we except the difference which is found in
their methods of multiplication and division, from those
which are now adopted by European authors.
(125.) The method of extracting the square root made Arabian
use of by the Arabians, resembled their method of division, process for
as far at least as the difference between the two opera- .
9 º e tº , e. he square
tions will allow ; and a little examination would serve ...
to show that they are both founded upon the Greek
methods of performing these operations upon sexagesi-
mals. An example (to extract the square root of
301401) will be quite sufficient to explain the process
which they followed:
5 4 9
3 0 | | 4 0 l
2 5
5 I
4 0
l l 4
l 6
9 8 0
9 7 2
8 I
8 I
0 8
I 0 | 1
5 4
(126.) There is not much room for variety in the rule Different
for performing this operation, and the varieties which modes 9f
are found in the form of the process itself are generally .
varieties only in the process of performing division. ...”
The earlier European writers on Arithmetic constantly
refer to the 4th Proposition of the IId Book of Euclid
3 L 2
436 <>
A R I T H M ET I.C.
Arithmetic. for the proof of the rule which they followed, with the
S-N-2 exception of the method of pointing, which is a very
obvious consequence of the principle of notation by
local value. We shall give one or two examples of the
form of the process in two or three authors, which will
be quite sufficient for our purpose. -
The first is from the Arithmetic of Pelletier,” the
first edition of which was published in 1550. It is
required to extract the square root of 92416. -
- () -
g2416
604
4 (304
Pelletier.
2416.
It may be as well to give the statement of the process
in the quaint and rude language of the author himself:
En somme, tout l'affaire des extractions quarrées se
pourra retenir par cinque mots, savoir est, chercher,
doubler, diviser, multiplier et souttraire. Premierement,
faut chercher la racine du nombre compris en la
dernier point, et d'ici lui nombre oter le nombre quarré
de telle racine. Secondement, faut doubler ce qui est la
et mettre le double entre les points. Tiercement, faut
diviser par la double, c'est adire, savoir combien de fois
il est contenu au nombre superieur. Quartement, faut
multiplier le diviseur joint avec la figure nouvelle mise
derniere le demicercle, par la figure meme. Finablement,
faut souttraire du nombre superieur, ce qui proviendra
de telle multiplication et susscrire le residu, s'aucun
on 3y a. -
The second example is from the work of Lucas de
Burgo, and is the form of the process which was most
commonly adopted: it is required to extract the square
root of 9998001.
Lucas de
Burgo.
(9 9 9 9
;
:
|
ſ
:
:
This scheme will require no explanation to one who is
acquainted with the galley form of division.
Stifelius, who usually sought to generalize the
methods of his predecessors, has considered the process
for extracting the square root in connection with those
for higher powers; by observing the formation of the
powers themselves, he discovered the following schemes
or pictures (as he calls them) for extracting the square,
cube, biquadrate, &c. roots.
mess, the terms of a binomial root a and b, we shall
have for the square root ** *
a — 20 — b
b.
Stifelius.
For the cube root,
- - a” — 300 — b
a — 30 — b”
b3
* L'Arithmétique de Jaques Pelletier de Mans departie en
quatre livres; revue et corrigée. A Lion, 1554, p. 136.
Calling, for greater clear-
For the fourth root,
a * — 4000 — 5
a * — 600 – b = &
CZ - 40 — bº
b4
For the seventh root,
a 0 – 7000000 — b
a 5 – 2100000 — bº
a * – 350000 — bº
G.3 — 35000 — bº
(Z * — 2100 — b 5
(Z - 70 — bo.
b 7
These schemes require little explanation ; a is the
greatest integer in the root of the first period; in ex-
tracting the square root it must be multiplied by 20 to
get the divisor, and from thence we determine b ; after
which the sum of the product of a, 20 and b and b°,
must be subtracted from the first remainder. We will
propose, as an example, to find the square root of
676520 !
Ž
§ 7 6 5 # 2 r (2601
2 — 20 – 6.
36. 276 .
26 – 20 – 0 . 0.
260 — 20 — | 5201
(127.) The invention of rules for approximating to the Rules for
square and other roots of numbers, where those roots approxima:
were surds, was a favourite speculation with the earlier ting to surd
writers on Arithmetic and Algebra. In order to *
state these rules with greater brevity, and to estimate
more readily their relative accuracy, we shall express
them in algebraical language.
1. The rule given by the Arabs is expressed by the By the
formula, - Arabs.
r
w/ (a + 1) = a +;
This approximation gives the root in excess; but to
increase its accuracy, we may repeat the process, making
use of the root thus obtained. Thus the first approxi-
mation to the square root of 7 is 2+ ; its square is 7.1%r.
divide ºr by twice 24, and the quotient is ºr, which
taken from 2; gives ºf for the second approximation.
This is the rule which is given by Lucas de Burgo,”
and subsequently by Tartaglia,t who derived it, in
common with the rest of his countrymen, from Leonard
of Pisa. -->
2. The rule given by Juan de Ortega i is expressed Ortega.
by the formula, • * : **
w (eit *) = a + =
tº = ..— .
- 2 a -i- 1
This approximaation is in defect, but, generally speak-
ing, more accurate than the former. - -
3. The third method of approximation was proposed Orontius
by Oronce Finé, or Orontius Fineus, Professor of Finku ~J.
Mathematics in the university of Paris, and who long e
enjoyed an uncommon reputation, in consequence of
his having introduced the knowledge of the Mathema-
* Summa de Arithmetica, &c., p. 46. -
+ Numerie misure, pars iji. - * , - '
# Tratado subtilissimo de Arithmetica, &c., 1534.

A R IT H M ET I C.
437.
Arithmeti
~!
Near ap-
proach to
decimals.
Methods
founded
upon it.
ic. tics of Italy amongst his countrymen, notwithstanding
his absurd attempts to effect the quadrature of the circle.
It consisted in adding 2, 4, 6, or any even number of
cyphers to the number whose root was required, and
then reducing the number expressed by the additional
digits of the root, which were thus introduced to sexa-
gesimal parts of an integer: thus, to extract the square
root of 10 add six cyphers, thus, , ºf
1900000 ( 3 lº
-’.--------~~ 60
6 ** | l & tº º ‘. . .
. . . t 9 720
60
43 | 200
• 60
12 || 000
The root of 10, as thus determined, when expressed
in sexagesimals, is 3.9/.43%. 12”.
The example which we have given is the mgst re-
markable approximation to the invention of decimals
which preceded the age of Stevinus. If the author
had stopped short at the first separation of the digits in
the root, it would have expressed the square root of 10
to 3 places of decimals; but the influence of the use of
sexagesimals, so familiar to the mathematicians of that
age, diverted him from this very natural extension of
the decimal notation, and retarded for more than half
a century this great improvement in the science of
calculation. * -
This very considerable improvement upon the ordi-
nary method of approximating to the values of surd
roots, as might be expected, excited the attention of
contemporary mathematicians. They did not, however,
follow the example of its author in proceeding to sexa-
gesimals, but merely subscribed as a denominator to the
whole root considered as integral, uniting with half as
many cyphers as had been added in the first instance;
3142
t = —.
hus w/ 10 1000
noticed by Tartaglia, who contends, however, that his
own method, which we have noticed above, was capable
of giving results to greater accuracy ; by Recorde, in
his Whetstone of Wit; by Buckley,” who has described
the method in the following verses:
It is under this form that it is
Quadrato numero senas praftgito ciphras,
Producti quadri radia per mille secatur.
Integra dat quotiens, et pars ita recta manebit
Radici ut vera, ne pars millesima desit.
* Buckley was a native of Litchfield, and Fellow of King's
College, Cambridge. He was also mathematical tutor to King
Edward VI. His Arithmetica Memorativa was published in 1550,
and subsequently reprinted at Cambridge at the end of Seton's Logic,
in 1631. . It consists of about 200 verses, describing, with great per-
spicuity, the principal rules of Arithmetic. He has also noticed the
second of the methods of approximation which we have mentioned
above as follows: . .
Modus collegend: minutias ea residuo
Duplo radicis numerus superadditur unus
Producto numerum mox supra scribe relictum
Lineola adjecta numeros quae separet ambos.
The practice of expressing the principles and rules of algorithm in
verse was very common before the invention of printing, and many
examples of such treatises may still be found in manuscript libraries.
They were usually confined, however, to the most simple and elemen-
tary rules of the science, and cannot be considered as exhibiting, like
the work of Buckley, its most improved state at the time they were
written. - .
Pelletier also, the pupil of Orontius Fineus, when
speaking de la manière de justifter les racines des
nombres non quarrés, after noticing the second of the
preceding methods of approximation, has described
* >
this, which he considers as more accurate and much
less tedious than any other.
(128.) We do not consider it necessary to notice in de-
History:
Methods for
tail the methods of extracting the cube root of numbers extracting
e - e * the cube
which are found amongst the Hindoos, Arabians, and root.
earlier European writers, as they present no variations
from the methods which are now in use, which may not
be inferred at once from the corresponding methods
for the extraction of the square root. It may be suf-
ficient for us to observe, that we find no trace of its
existence amongst the Greeks, though it is not very
probable that it was altogether unknown to them ; and
though it formed an essential part of all treatises on
Arithmetic, whether Sanscrit, Arabian, Persian, or
European, we may consider that their authors were
generally ignorant of the principles upon which the
rule was founded, and in some cases were incapable
even of applying it in practice.*
Methods of
(129.) Under such circumstances, it is not surprising approxima-
that mistakes should have been made in their methods tion to
of approximating to surd cube roots; that of Lucas de surd cube
Burgo may be seen from the formula,
a.
- (3 a)*
which Tartaglia says he got from Leonard of Pisa,
who had it from the Arabians; and he expresses his
surprise that he should have committed so grievous an
error, unless he had done so without consideration.
The method of Orontius Fineus errs as much in excess
as that of Pacioli in defect, as will be at once seen
from the formula which expresses it,
*w (a + rj
Tartaglia criticises the method of Cardan founded
on the formula,
- º
*... (a + i) = a + 3 & 2
with great bitterness, as might naturally be expected
from one who had been so treacherously defrauded by
him of an important discovery; his own method, more
accurate than the former, but erring in defect, whilst
the other erred in excess, is given by the formula,
tº
S S * g -
A/ (a " + o = a + ŚāIsa”
In later times, methods of approximation have
been proposed, whether founded upon rational or irra-
tional formulae, which give results much more accurate;
than any of the preceding ; as the discussion of such
formulae, however, belongs more properly to the history
of Algebra than Arithmetic, we think it unnecessary for
us to notice them in this place. ,
* V (a " + x) = a +
+ ſº
67, — .
3 a.
roots.
(130.) Fractions in the Lilávati are denoted by writing ~
the numerator above and the denominator below, with-
out any line between them. The introduction of this
line of separation is due to the Arabs, and we find it
among the earliest European manuscripts on Arith-
* Such at least was the accusation advanced by Tártaglia against
Jean Buteon, or Buteo, the author of a Treatise on Arithmetic.
f It is likewise given in the Arithmetic of Juan de Ortega.
# Of this kind are the rational and irrational formulae of Halley.
of fractions
in the
Eilávati.
2^


438 .
A R I T H M ET I C.
Arithmetic. g g o 2 ..
--~~ metic. To denote fractions of fractions, such as 3 of
B they are written consecutively, thus,
2 4
3 5
To represent a number increased by a fraction, the
fraction is written beneath the number; and when the
fraction is to be subtracted from the number, a dot is
prefixed to it; thus 24 is denoted by
2
#
and 3–4 by:
Its imper-
ſections, without verbal explanation; thus to denote “two
thirds less one-eighth, and then diminished by three-
sevenths of the residue,” the fractions are written
underneath each other, as follows:
;º:
In general, however, it may be remarked, that the
invention of a distinct, expressive, and comprehensive
notation, is the last step which is taken in the improve-
ment of analytical and other sciences; and it is only
when the complexity of the relations which are sought
to be expressed in a problem is so great as to surpass
the powers of language, that we find such expedients
of notation resorted to, or their importance properly
estimated. We, consequently, find the Hindoos,
Arabians, and earlier European writers, singularly
deficient in artifices of notation, and compelled there-
fore to express in words the relation of the numbers
which appear in their problems, or to make use of the
same notation for different relations. The following
problems, given in the Lilávati, will serve more fully
to explain our meaning. . . . -
(1.) “The quarter of a sixteenth of the fifth of three
quarters of two-thirds of a moiety of a dramma, was
given by a person from whom he asked alms: tell me
how many cowry shells* the miser gave, if thou be
conversant in Arithmetic, with the reduction called
subdivision of fractions.”
STATEMENT.
1 1 2 3 l l I
1 2 3 4 5 16 4
Reduced to homogeneousness rººrg ; in least terms
rºw, a single cowry shell. gy
(2.) “Tell me, dear woman, quickly, how much a
fifth, a quarter, a third, a half, and a sixth make, when
added together.” -
STATEMENT. . º: *] :
I I I I I 29 ‘. . . . .
5 4 3 2 6 20 2.”
(3.) “Tell me what is the residue of three, subtract.
ing these fractions.” .
** A dramma is equivalent to 1280 cowry shells.
In other cases, their notation is not intelligible
2.
*
2.
--> *
. History. •
STATEMENT.
3 l l l l l 31.
1 5 4 3 2 6, 20
In all these problems the 'statement or notation'
employed is the same, though the operations to be
performed are essentially different. \
(131.) The Lilāvati contains four rules for thereduction Rules for
and assimilation of fractions, as well as the application i. ºut-
of the eight fundamental rules of Arithmetic to them; ºn.
the rules themselves are generally sufficiently simple
and clear, and differ so little from those which are
used in modern practice, that any detailed notice of
them is unnecessary. The author, however, in the
enunciation of the following problem, would seem to
intimate that operations with fractions were not with-
out their difficulty, and that it required all the con-
fidence of long practice to avoid making mistakes.
“Tell me the result of dividing five by two and a
third, and a sixth by a third, if thy understanding,
sharpened into confidence, be competent to the division y
of fractions.” . - -
(132.) The term algorithm, which originally meant the Meaning of
notation by nine figures and zero, subsequently received º º
a much more extensive signification, and was applied algorithm.
to denote any species of notation whatever for the
purpose of expressing the assigned relations of num-
bers or quantities to each other: thus we find Stifelius
speaking of the algorithm of fractions and of fractions
of fractions, of the algorithm of proportions, of binomial
surds, of cossic numbers, &c.; and an equally extended
use of the term is sometimes made in modern times.* ,
The algorithm of fractions of fractions, if we may Algorithm
be allowed to use this term, varied with different of fractions
authors; thus with Lucas de Burgo of fractions.
jºa;
ºyd - • e
2 T 4 S. 8 uivalent t 2 f : or to * x *
s_5 was equivalent to so. 5 or to 5 × 5.
ºyd - -
where v" denoted via, or times; with Stifelius, three-
fourths of two-thirds of one-seventh was denoted by . .
3 *
4
2 -
3 - -
l
7
and the same quantity was represented by Gemma.
Frisius by .
- 3 | 2 1
4 3 7.
a notation extremely simple and convenient.
Pacioli denotes that two fractions are to be multiplied.
together by writing them thus, , -
2 - 3
3 – 4
g
k-
* It is amusing to observe the very general ignorance of the
earlier writers on the origin and meaning of the Arabic terms which
were made use of in the sciences; it was quite common with the
Italian and German writers on Algebra; to speak of Geber as its in-
ventor; and Gosselin, who in 1667, translated and abridged the work,
of Tartaglia, says that Algorithm was tierived from Algus, the inventor
of the notation by nine figures and zero. . . .
A R IT H M ET I C.
439
Arithmetic, connecting the numerators and denominators which are
S-N-2 to be multiplied together by a line. When two frac-
tions are to be added together, or subtracted from each
other, the operations to be performed are indicated
as follows,
8 9
2. 3 17 I
3 x i fa is vel. T3'
12.
where those quantities are to be multiplied together
which are connected by the lines.
(133.) The good old monk seems extremely embarrassed
by the usage and meaning of the term multiplication in
the case of fractions where the product is less than the
multiplicand, and he proposes the question, Utrum
multiplicatio fractorum augeat 2 In order to show that
this question must be answered in the affirmative, he
refers to the passage in Genesis, “Increase and mul-
tiply, and replenish the earth ;” and, again, to the
promise to Abraham, “I will multiply thy seed like the
stars of the firmament, or the sand on the sea shore,” to
show from the authority of God himself, that to multiply
means to increase; but in what manner is this to be
reconciled with the numerical result in those cases?
namely, by supposing that the units of the product are
of greater virtue and significancy than those of the
factors; thus if , and # represent adjacent sides of a
square, their product 4 will represent the area of the
square itself. '
The same difficulty appears to have occurred to
most other writers of his own and the subsequent
age, who were not all of them equally satisfied
with the correctness of his explanation. Tartaglia
2–1
Utrum anul-
tiplicatio
fractorumr,
augeat 3
says, that the meaning of the term multiplication is
different when the multiplier is an integer or a frac-
tion, denoting increase in one case and diminution in
the other. Bishop Tonstall, however, in discussing the
example 4 × # = ºr, has explained the result in this case
with singular clearness and good sense: “Curid autem
ita fiat,” says he, “si rationem poscis, illa est; quod si
numeratores in se soli ducerentur, viderentur integra
inter se multiplicari atque ita numerator mimium cres-
ceret. Veluti in exemplo dato, dum duo in tria du-
cuntur, fºunt 6, quae, si nihil praetereafteret, viderentur
integra ; capterum quia non duo integra per tria : sed
dual tertia unius integri per tres ejus quartus multipli-
candae sunt : similiter partium denominatores in se
ducuntur: ut postea divisione qua, per denominatoris
nultiplicationem, fit, (quanto enim magis denominator
crescit, tanto magis partes comminuwntur) numeratoris
augmentatio tantum corrigitur, quantum plus justo cre-
verat, atque ea ratione ad acqualitatem redigitur.”
The whole dispute furnishes a curious example of the
embarrassing effect produced by the use of a term to
which a specific and restricted meaning is attached, to
denote a general operation, the meaning and interpre-
tation of which must vary, with the nature of the
quantities to which it is applied.
(134.) There is so little difference between the opera-
tions in fractions, as they appear in ancient and modern
books of Arithmetic, that we feel it to be altogether
unnecessary to detain the reader by any further details.
In the works of Lucas de Burgo and
Tartaglia we find the number of cases and their sub-
divisions unnecessarily multiplied; and the reader upon
on the subject.
* De Arte Supputandi.
this, as well as upon other parts of Arithmetic, is fre- History.
quently more embarrassed than instructed by the Sºtº
minuteness of their explanations. The charge of pro- Prolixity of
lixity, indeed, has been made against Italian writers on Italian
this as well as other subjects of every age, and it is ".
quite impossible to deny the truth of its application
to the works of which we are speaking. It would be
unjust, however, not to attribute much of this to the
want of generality and comprehensiveness of the rules
and operations which is characteristic of the early state
of every science; and the same defect, though in a less
degree, is observable in most of the writers of other
countries who flourished at that period, with one memo-
rable exception, however, in the case of Stifelius, whose
brevity, and consequent obscurity, is as embarrassing
to the reader as the tediousness of his predecessors and
contemporaries. • * -
(135.) We have noticed above a method of approxi- Invention of
mating to the square and cube roots of numbers, which decimal
makes a near approach to the invention of decimal frac- †.
tions, though it will not be found to have in any way j
contributed to that most important improvement in lead to it.
Arithmetic, at least if we may judge from the form
under which it was first exhibited by its author.
It would seem rather tº have been suggested by the
convenience which was felt in the sexagesimal Arith-
metic in the treatment of fractions, and by observing
the connection between the series of natural numbers
and a geometrical series, whether continued upwards
or downwards. Archimedes had observed how the
order of the term, formed by the product of any two
terms of such a series, might be determined from the
sum of their exponents, or the terms in the series of
natural numbers corresponding to them ; and Stifelius
extended this remark by continuing the Arithmetic as
well as the Geometric series downwards; thus,

\
— 4 ||— 3 – 2 | – 1 || 0 l
* | * 4
. t - *
* | * | * } | 1 || 2 || 4 || 8. 16
The same distinguished author observed also, that
the proposition would be equally true if the arithmetical
series was reversed, and the positive terms made the
exponents of the descending terms of the geometric
series which were less than 1 ; thus,
0, 1, 2, 3, 4, 5, 6,
might be considered as the exponents of the sexagesi-
mal or astronomical series: tºº
I I I l I
60 3600 21.6000 12960000 777600000
It was with reference to this principle that Stifelius I -
º g tº e & . . . mproved
ventured to simplify the sexagesimal notation by writing ...tion of
the numbers 2, 3, 4, &c., accentuated, above the places sexagesi-
of the minuta secunda, tertia, quarta, &c.; thus, mals.
Grad. Min. 2 3 4
2, 3, 7, 20, 44,
means 2°, 3, 711, 2011, 44", and
Hor. Min. 2 3
6, 20, 40, 59,
means 6 hours, 20 minutes, 40 seconds, 59”; and
similarly in other cases. It is sufficiently curious that
440
A R IT H M ET I C.
Arithmetic. Stifelius, after thus viewing the theory of sexagesimals
\-v- under this very general form, should not have extended
it to decimal fractions; more particularly as the follow-
ing remarkshows that he was sensible that they depended
upon the same principle. “Facile enim vides, ut numerus
ille 60, id est, sewagenarius, limes sit totius negotii hujus-
modi fractionum, quemadmodum 10, id est denarius,
limes est calculationum vulgarium :” in other words,
that 60 in one case and 10 in the other were the roots
of the geometrical series, to which the same series of
exponents corresponded.
La Disme . (136.) Stevinus, in his Arithmétique, adopted the views
of Stevinus of Stifelius with respect to the exponents of terms in a
geometrical series, and applied them to correct the
barbarous mode of designating roots and powers of
quantities which had been prevalent before his time;
thus making a very near approach to the very impor-
tant theory of indices, as they are now used. We find
no traces, however, of decimal Arithmetic in this work;
and the first notice of decimal, properly so called, is to
be found in a short tract, which is put at the end of his
Arithmétique in the collection of his works by Albert
Girard, entitlød La Disme. ...º. was first published in
Flemish about the year 159 º' afterwards translated
into barbarous French byū, ºn of Bruges. The
ludicro-serious dedication is ºſressed Aux astrologues,
arpenteurs, mesureurs, de Tapisserie, gaviers, stereo-
metriens en general, maistres de monnoye et a tous mar-
chands; and describes in very express and ample
terms the advantages to be derived from this new arith-
metic: decimals are called nombres de disme; and those
in the first place whose sign is (1) are called primes,
those in the second place whose sign is (2) are called
secondes, and so on ; whilst all integers are charac-
terised by the sign (0), which is put after or above tre
last digit. We will subjoin a few of his examples &
arithmetical operations by means of these decimals.
1. Addition.
0 (I) (2) (3)
2 7 8 4 7
3 7 6 7 5
8 7 6, 7 8 2
9 4 1 3 0 4
Or, 941 (0) 3 (I) 0 (2) 4 (3)
2, Multiplication.
(0) (1) (2)
3 2 5 7
8 9 4 6
º —T-
I 9 5 4 2
I 3 0° 2 8
2 9 3 ] 3
2 6 0 6 6
2 9 1 3 º'7 1 2 2
OT, 29.13 (0) 7 (1) l (2) 2 (3) 2 (4)
3. Division.
(0) (l) (2) (3) (4) (5) (1) (2)
3 4 4 3 5 2 by 9 6
2'
r $ 6
(0) (1) (2) (3)
(3 5 8 7
:
º
;
;
?
9 & 3
4. Indefinite division.
40
3 T
Stevinus afterwards proceeds to enumerate the advan-
tages which would result from the decimal subdivision
of the units of length, area, and capacity, of money, and
lastly of a degree of the quadrant;' in the increased
uniformity of notation, and increased facilities in per-
forming all arithmetical operations in which fractions
of such units were involved.
(137.) Whatever advantages, however, this admirable Translation
invention, combined as it still was with the addition of of this
the exponents, possessed above the ordinary metriods of #;"
calculation in the case of abstract or concrete fractions, glish.
it does not appear that they were readily perceived or
adopted by his contemporaries. We can discover no
notice whatever of the improvement before the begin-
ning of the following century. In 1608 the tract in
question was translated into English by Richard Norton,
Gentleman, under the following title: Disme, The arte
of tenths, or decimal Arithmetike, teaching how to per-
form all computations whatsoever, by whole numbers
without fractions, by the four principles of common
Arithmetike , namely, addition, subtraction, multipli-
cation, and division, invented by the ercellent mathema-
tician, Simon Stevin.
(138.) This publication does not appear to have excited The art of
any very general or immediate notice. In the year 1619, tens by
however, we find its contents embodied in an English *Y*.
work, of which the following is the title: The art of
Tens, or decimal Arithmetike, wherein the art of Arith-
metike is taught in a more eract and perfect method,
avoyding the intricacies of fractions. Exercised by
Henry Lyte, Gentleman, and by him set forth for his
countries good. London, 1619. It is dedicated to
Charles, Prince of Wales; and in his advertisement he
says, that he had been requested for ten years to publish
his exercises in decimal Arithmetike. After enlarging
upon the advantages which attend the knowledge of
this Arithmetike to landlords and tenants, merchants
and tradesmen, surveyors, guagers, farmers, &c., and in
all men's affairs, whether by sea or land, he adds, “if
God spare me life, I will spend some time in most
cities in this land for my countries good to teach this
art. I hold the lively voice of a meane speculator
somewhat practised, furthereth tenfold more in my
judgement than the finest writer that is.” It is not
necessary to proceed further with an analysis of the
contents of this volume, as it contains nothing, either in
notation or otherwise, which is essentially different from
what was given by Stevinus.
(139.) The last and final improvement in this decimal Improve-
Arithmetic, of assimilating the notation of integers and :
decimal fractions, by placing a point, or comma, between . d
them, and omitting the exponents altogether, is un-by Napi.
questionably due to the illustrious Napier, and is not
one of the least of the many precious benefits which he
conferred on the science of calculation. No notice
whatever is taken of them in the Mirifici Logarithmorum
canonis descriptio, nor in its accompanying tables, which
was published in 1614. In a short abstract, however,
of the theory of these logarithms, with a short table of
the logarithms of natural numbers, which was published
by Wright, in London, 1616, we find a few examples
of decimals, expressed with reference to the decimal
point; but they are first distinctly noticed in the Rab- Rabdologia.
dologia, which was published in 1617. In an Admonitio
History.
0 (1) (2)
3 3 3




A. R. I T H M ET I C.
441
Arithmetic. pro decimali Arithmetica he mentions in terms of the
--~~' highest praise, the invention of Stevinus, and explains
his notation; and without noticing his own simplifica-
tion of it, he exhibits it in the following example,
in which it is required to divide 86.1094 by 432.
- 64
136
316
118,000
14l
402
429's.
86.1094,000 (1993,273
432 -
3888
3888
1296
864
3024
1296
The quotient is 1993,273, or
1993,277.3%
the form under which he afterwards writes it, in partial
conformity with the practice of Stevinus.
The same form is adopted in an example of abbre-
viated multiplication, which subsequently occurs in the
solution of the following question.
Abbreviat- If 3.1416 be the approximate value of the circum-
ed multipli ference of a circle whose diameter is 10000, what is the
“" numerical value of the circumference of a circle whose
diameter is 635.
Complete. Abbreviated.*
314 l 6 314|| 6
635 635
1884|96 1884|9 .
94.248 9.4|2 . .
15||7080 15||7 . . .
1994 9’ I’ 6” ()//// 1994 ST
}) ecimals (140.) The publication of tables of logarithms, to
not neCeS-
whatever base they might be calculated, was by no
means necessarily connected with the knowledge and use
of the decimal Arithmetic. The theory of absolute indices,
in its general form, at least, was at that time unknown;
and logarithms were not considered as the indices of
the base, but as measures of ratios merely. Under this
view of their theory, it was clearly a matter of indiffe-
rence whether we assumed the measure of the ratio of 10
to 1, to be 1, 10, 100, 100000000, or 1,00000,00000,
the number assumed by Briggs in his Arithmetica
Logarithmica. Thus the absolute logarithms of 15,
55, and 155, to ten places, are
1,1760912591
1,7403626895
2,1903316982
whilst their relative logarithms, that of 10 being
1,00000,00000, are
sary for
logarithmic
tables,
Absolute
and relative
logarithms.
* This is the "first example which we have discovered of this
abbreviated multiplication : the use of it, however, became very popular
in a short time afterwards, as furnishing some relief in the management
of the large numbers which were made use of in the construction of
tables of sines, &c. Many examples of this species of multiplication
and division may be found in the work of Kepler, on Logarithms, in
Oughtrede's Clavis, in Wallis's Algebra, &c. -
WOL. I.
1,17609,12591 History.
1,74036,26895 \-y-'
2, 19033,16982
In one case the logarithms are expressed by decimals,
in the other by whole numbers; they have the same
characteristics, and it is obvious that their use in calcu-
lations is exactly the same. It is under the latter form
that the logarithms are given in the earlier tables, such
as those of Napier, Briggs, Vlacq, Kepler and
Bartsch.
(141.)The preceding statement will sufficiently explain Noticed
the reason why no notice is taken of decimals, in the ela- § º
borate explanations which are given by Napier, Briggs, ..".}."
and Kepler, of the theory and construction of logarithms; rithmical
and indeed we find no mention of them in any Englisºarithmetike
author between 1619 and 1631. In that year thé"
Logarithmical Arithmetike was published by Gellibrand,
and other friends of Briggs, who died the year before,
with a much more detailed and popular explanation of
the doctrine of logarithms than was to be found in the
Arithmetica Logarithmica. It is there said that the
logarithms of 19695, of 1969 ºr, 1919.2% are .
4,29435,59851
3,29435,59851
1,29435,59851
differing merely in their characteristic ; and ºr, 4% ºr
are called decimal fractions. Rules are also given for
the reduction of vulgar to decimal fractions by a
simple proportion ; and lastly a table for the re-
duction of shillings, pence, and farthings, to deci-
mals of a pound sterling, of which the following is a
specimen :
S. p. f.
I9 95000 11 04.5.833 3 003] 248.
17 85000 5 020833 l 00 10416
(142.) From this period we may consider the decimal
Arithmetic as fully established, inasmuch as the explana-
tion of it began to form an essential part of all books of
practical Arithmetic. The simple method of marking the
separation of the decimals and integers by a comma,
of which Napier had given a solitary example, was not
however generally adopted. The following are different
modes of writing them, which are found amongst
English and foreign authors:
34. I’. 4//. 2”. 6////
(1) (2) (3) (4)
34 . 1 . 4. . 2 .. 6
34 . 1 4 2 6////
34 . ] 4 2 6 (4)
34 . 1 . 4. . 2 . 6
34 1426
*=ºmºsºme
34 || 1426
34°1426
34,1426
(143.) Amongst the authors who contributed most to Oughtrede's
the propagation of this Arithmetic we must mention the Clavis.
celebrated Oughtrede.” His Clavis Mathematica was
first published in 1631, in the first chapter of which, De
Different
notations of
decimals,
* William Oughtrede was a fellow of King's College, Cambridge,
and he always writes AEtonensis after his name. In those days the
members of those royal foundations had not yet begun to consider the
pursuits of literature and science as incompatible with each other.
His works enjoyed a well deserved reputation in his day, and he is
spoken of in his old age with singular reverence by Wallis. He died
in 1660, in his 87th year, from excess of joy on hearing of the restura-
tion of the monarchy.
3 M


442
A R I T H M E T I C.
stances he could not have been ignorant either of History. .
Napier's notation or of the work of Stevinus, and we \-\,-
may very reasonably doubt, therefore, the truth of his
Arithmetic. Notatione, we'find the following table, with its accom-
S--N/~' panying explanation.”
Integri. Partes, pretensions to originality, or that he should so long
* 9 || 8 7 6 5 4 3 || 2 1 0 1 2 || 3 4 5 6 7 8 9 tº: º a. [] º º such immense impor-
M M M M M M M | C x I |X C | M M M M M M M ance to the science of calculation. Q g lb
M | M M M | C X I º- I X C | MIM M | M (147.) The works of Stevinus were published in 1625 A. .
M : C X I I X C | M by his friend and pupil Albert Girard, whose own work, º n
1'.
'º';)
.* h
Other
authors.
Zogistica
Decimalis
of Beyer.
Acquainted
with the
works of
Stevinus
and Napier.
In hac tabelld numeri superiores sunt indices sive ex-
ponentes terminorum utrinque ab unitate continuo pro-
portionalium ; affirmativi in integris, negativi in par-
tibus. Estgue progressio in decuplá ratione versus simis-
tram, et in subdecupló versus deatram ; sicut litera:
*. ostendunt. Estgue igitur progressio ab unitate
ºn integris 1, 10, 100, 1000, 10000. Et in partibus,
- 1 1 D 1.
+º, ++, +9°o-w, rºo-o. Et sic in infinitum.
The integers he separates from the decimals or parts,
by a mark L_ which he calls the separatria, as in the
examples 0|56, 4815, for .56 and 48.5; and in giving
examples of the common operations of Arithmetic he
unites them under common rules.
(144.) The view of the theory of decimals which was
given by Oughtrede was generally adopted, and in some
cases his notation also, by English writers on Arithmetic
for more than thirty years after this period. Amongst
others may be mentioned Nicholas Hunt, whose Hand-
maid to Arithmetick was published in 1633 : John
Johnson, whose Arithmetic was published in 1657.
Jonas Moore, Professor of the Mathematics in the city
of Durham, whose Arithmetic was published in 1660,
with a dedication to James, Duke of York, a work
which long enjoyed a considerable reputation. Samuel
Jeake, merchant, whose Compleat Body of Arithmetick
was written in 1671, though not published before 1701;
a work of considerable learning and research on every
subject connected with practical Arithmetic, and parti-
cularly in weights and measures: besides many others,
whose works we have had no opportunity of ex-
amining.
(145.) In the year 1619 there appeared at Frankfort a
work with the following title: Logistica Decimalis,
dasist : Kunstrechnung mit Zehentheilichen Bruchen,
denen Geometris, Astronomis, Landmessern, Ingenieurm,
Wisiren, und insgemein allen Mechanicis und Arithmeticis
in unglaublicher Leichterung in rer muhsamen Rech-
mungen, Extractionen der Wurzeln, somderlich aus
Irrationalzahlem, auch zur construction einer neuen
Tabulaesinuatm, und andrer vielerhand mutzlicher camo-
mum etc, uber die maass dienstlich und mothwendig,
beschrieben durch Johann Hartman Beyern, D. Med.
The author states, that he first thought upon the sub-
ject of this decimal Arithmetic in the year 1597, but
that he was prevented from pursuing it for many years
by the little leisure afforded him from his professional
pursuits. He makes no mention of Stevinus, and
assumes throughout the invention as his own. The
decimal places are indicated by the superscription of
the Roman numerals, though the exponent correspond-
ing to every digit in the decimal places is not always
put down : thus 34.1426 is written 34°.114/121116kV, or
34°. 141*26) V, or 34°.1426 IV.
(146.) The author must have been acquainted with the
Rabdologia of Napier, as the thirty-ninth chapter of his
book is devoted to the explanation of the construction
and use of these rods, which enjoyed a most extra-
ordinary popularity at that period under such circum-
entitled Invention nouvelle en Algebre, appeared in 1629.
It contains the exposition of the principles of Arithmetic
and Algebra, and we may naturally expect to find, there-
fore, examples of the use of decimals under their most
improved form. In the solution of the equation,
I (3) esgale 3 (1) – 1
or, a * = 3 a. - 1
by a table of sines, of which method he was the author,
we find the three roots written as follows:
1,532
;)
— 1,879
“Ce qui est esprime,” says he, “en disme jusques en
trimes.” On another occasion he denotes the separa-
tion of the integers and decimals by a vertical line :
“ Divisez 3218 par 10,” says he, “il viendra 321+ºr,
le nombre est ainsi trace 321 || 8; si par 100, aimsi
32 18; et si par 1000, ainsi 3 || 218.” He does not
always, however, adhere to this simple notation, as we
afterwards find the square root of 4+ expressed by
20816 (4); and on another occasion we find similar
vestiges of the original notation of Stevinus.
(148.) Whoever has studied the history of the progress slow pro-
of the mathematical sciences must have remarked the gress of in-
extreme slowness with which improvements in nota- Provements
In the " notation.
tion have been admitted into general use.
infancy of those sciences more attention is paid to the
modus operandi, to the actual rule for performing the
operation, than to the form under which it is exhibited;
and in many cases improvements in notation, the most
important in their consequences, have originated as
much in accident as design, or at all events their
authors have had little notion of the effect of the
change which they were making. When Napier dis-
encumbered the decimal notation of the numeral ex-
ponents of Stevinus, the improvement in point of
simplicity and practical usefulness which was thus
produced, was apparently so obvious as to have at
once recommended it to universal adoption ; yet we
find it timidly proposed, and not always followed even
by its author; and though the work which contained
it was very generally circulated and read, yet the nota-
tion was not admitted in principle for fifteen years after
its first publication, even in our own country, at a
period when the discussions connected with the theory
of logarithms and the construction of tables, were calcu-
lated to bring decimal numbers and their notation into
particular notice. On the continent of Europe this
notation was not adopted generally before the middle
of the century; and even in the year 1656 we find the
Jesuit Andrew Tacquet, in his Arithmetib,” giving an
account of the theory of decimals, and uniting them
with Roman numerals as exponents, as if no improve-
ment had taken place since the original publication of
Stevinus.
* Arithmetica: Theoria et Praaris, auctore Andrea Tacquet,
Antwerpensi, e Societate Jesu Lovanii, 1656.
A R IT H M ET I C.
443
Arithmetic,
\-V-
Arithmetic
of concrete
quantities.
Observa-
tion of
Tartaglia on
the primary
units of
weights and
IllèaSu TéS,
To what
extent de-
rived from
natural
SOURTCéS.
Measures of
the Hindoos,
Of the
Hebrews.
Of the
Greeks,
(149.) We shall now proceed to the history of the
Arithmetic of concrete or denominate numbers, which
forms the second and last division of our subject, and we
shall commence with a few introductory remarks on the
divisions of the primary units of weights and measures
of different countries, and on the ultimate units in
which they are made to terminate. .
(150.) It is a remark of Tartaglia, that mankind have
generally attempted in the selection of the ultimate
units of concrete quantities to imitate the indivisible
abstract unit of number, or the mathematical point of
geometers; in other words, that those units have been
assumed to be quantities of their species so small, or
of a nature so invariable, as to be considered as in
some measure indivisible as to sense. By way of illus-
tration, he refers to examples derived from the coins,
weights, and measures, which were used in Italy.
Thus the ultimate unit of money is in Venice termed a
piccolo, or bagatino, terms used to express their ex-
treme minuteness; and in other cities of Italy a dinaro;
of the weight of medicines, gold, and precious articles,
a grain of barley; of other valuable goods, though less
precious than the former, a caratto,” equal to four
grains ; for common merchandise an onza, or ounce:
in all these cases, the minuteness of the ultimate divi-
sion being proportioned to the value of the articles
which were required to be estimated. For measures of
length, this unit was a grain of barley in breadth, and
similarly in other cases.
(151.)This observation is sufficiently curious, and quite
worthy of the very acute and philosophical genius of
its author, though we may not feel disposed to admit
its truth to the extent asserted, or in the precise terms
in which it is expressed ; it directs us, however, to an
inquiry of some interest respecting the nature of these
ultimate units, and to the extent to which they, in
common with other measures of length, weight, and
capacity, are derived from matural sources, and there-
fore generally adopted by different people independently
of each other. We shall commence with measures of
length. -
(152.) Amongst the Hindoos, 8 breadths of a barley
corn, or 3 grains of rice in length, make a ſinger;
4 times six fingers make the cubit, or fore arm ;
4 cubits make a staff, which is usually the height of
a man's body; and 20 cubits make the bambu pole,
which is used in measuring land and considerable
distances.
(153.) Amongst the Hebrews, 6 barley corns in their
greatest thickness, or 2 in length, make the etsbang, or
finger's breadth; 4 of these make the tophach, palm, or
hand; 3 of which were equal to the zereth, or span, the
distance between the ends of the thumb and the little
finger when stretched out to their greatest extent; the
double of the span made the ammah, or ordinary
cubit, the length from the elbow to the extremity of the
fingers. To these may be added the paynam, or foot,
and the tsugad, or pace, derived in common with most
of their other measures from the parts of the human
body. -
§4) There are many reasons which should make
us expect to find a resemblance between the Greek
* This term is derived from the Greek zigzroy, the carob seed, or
sweet bean, which in the Greek physical weights was considered as
equivalent to 3% grains of wheat. -
measures and those which were used by the Hebrews History.
and Phoenicians; we consequently have the Sakrv\os, -y-'
or finger, the artéaum, or span, the Tovs, or foot, the
Tnxvs, or cubit, the opyvia, or fathom, the distance
of the out-stretched hands,” with other intermediate
measures derived from the same natural source.
(155.) Amongst the Romans we find the digitus, the Of the
poller, or thumb's breadth, equal to an inch ; the palmes Romans.
7minor, or common palm of 4 digits; the palmes major,
corresponding to the artôapºn of the Greeks; the pes,
or foot; the gressus, gradus, or step ; and the passus of
5 feet, which was double of the step ; the ulna, or
ell, which corresponded to the cubit, is a term used in
later authors, and is the origin of one of the most com-
mon and most variable of the measures of modern
Europe.
(156.) Amongst the Greeks and Romans we find no
trace of the ultimateunit of length, the barley corn, either
in length or breadth, which was referred to by the Hin-
doos and Hebrews as making some approach to an
invariable standard : it reappeared, however, in modern
Europe. Thus the Venetian measures commence with Of the
the grano de orgio,f or barley corn, 4 of which make a Venetians.
dedo, (a corruption of digitus,) and 4 dedi a palmo.
Other measures are Roman, such as passo, consisting
of 5 feet, and each foot of 12 onze, or inches. In our
own country we assume 3 barley corns, taken from the English
middle of the ear, and placed end to end, as the inch.
standard of an inch. But it is not necessary to pursue
this inquiry further, as the examples which we have
already produced are sufficient to show that the ordi-
nary measures of length have been generally derived Measures
from the dimensions of the human body, or of spaces commonly
included in our ordinary motions; and iikewise that .
º º rom the
some other ultimate unit (generally a barley corn) has . lody
been assumed, as a space so small as to call for mo
further subdivision, at least in the ordinary cases
where measures of length are required, and also of a
nature so constant, or at least esteemed to be so, as to
serve as a corrective to the extreme diversity of the
other and greater measures when derived from their
natural sources.
(157.) For longer measures of length, where the parts Longer
of the human body could no longer be referred to, we measures of
must expect still less uniformity in the selection of length.
superior units. There is a general resemblance, both
in name and use, between the bambu pole of the Hin-
doos, the kaneh or reed of six cubits of the Hebrews,
the akawa of the Greeks, the decempes of the Romans,
the Spanish stadale of 11 feet, the French perche, and
* An old English author says that a pair of compasses with one leg
in the navel would graze with the other the top of the head, the sole
of the foot, and the extremities of the out-stretched arms; without
resorting to the confirmation of such an experiment, we may assume
this measure to be equal to the ordinary height of a man. The term
fathom is used in nautical measures as being the portion of the
sounding or other line which can be grasped between the hands at
one time. -
+ Tartaglia considers the grain of barley as constituting the most
correct fundarno, or basis of measures of length. It is much less
variable in breadth than the grain of wheat. He allows, however,
that it may be more corpulent in one country than another; a fact
which he ascertained from comparing the verga, or yard of England,
with the number of grains which are allowed for it, with the measures
of Italy, and which he attributes to the coldness of our climate. He
was furnished with the means of making this comparison by his friend
and pupil Richard Wentworth, to whom he dedicates the first part of
his work. - - t * - ' -
3 M 2
444
AR I.T.H M ET.1 c.
Arithmetic, the English pole, rod, or perch, whose lengths were
S-N-" taken from that of the reed, or rod which was used in
the measurement of land and large distances.
(158.) In the East, and even in modern Europe, dis-
tances were reckoned by the hour or day's journey. Thus,
in Hebrew, the cibrach haarets, or half day's journey,
was the distance which could be travelled from meal to
Chinese lih, meal. The unit of space of the Chinese is the lih, the
distance which can be attained by a man's voice, thrust
forth with all his force in a calm season, upon a clear
plain; for greater or lesser distances they proceed to
the multiplication or subdivision of this distance by 10,
presenting thus an unique example of an uniform scale
of measures of length. In the days of archery a bow
::shot presented a measure of a similar character of very
general and popular usage. The Greek a tačtov was
probably derived from the particular length of the
course of their chariots in their public games; whilst
the origin of our own furlong, a measure of nearly cor-
responding length, is sufficiently obvious in its deriva-
tion (quasi furrow long.) The parasang of ancient
Persia consisted of 30 stadia, and is of unknown origin;
and the same observation may be made of the oxotvos
of double its length, a measure of the ancient Egyp-
tians, which is mentioned by Herodotus.
(159.) The milliare, milliarium, or mille passus of the
Romans is the origin of the modern mile, varying in
different countries of Europe from its extreme length
in the German mile of 22,500 feet to the Italian of
5000; a circumstance which clearly shows that the
classical name was borrowed to designate a large dis-
tance, without any reference to its precise signification.
The term league has been supposed to be derived from
the German lugen to see, and that it originally expressed
the distance which could readily be seen by the eye on
a plain surface; and it certainly would require all the
vagueness and uncertainty which would attend the
assignation of such a space, to account for its different
lengths in the leagues of Germany, Spain, and Sweden,
in the four leagues of France under the old monarchy,
and in the common and nautical league of England.
(160.) As there are no natural, or very obvious stan-
dards, from which we can readily derive our measures of
weight, we may therefore expect to find them of a much
more arbitrary character, in their designations at least,
than the measures of length. It is very curious, how-
ever, to find how often a grain of barley has been
taken as their basis. Thus, amongst the Hindoos the
weights are derived from the barley corn and gunja, or
seed of the abrus precatorius, which is considered as
equivalent to two of them. The Greeks make two outdpua,
or grains of barley, equivalent to the XaXicos, their most
minute piece of copper money, 4 of these equal to the
kepatiov, or carob seed, and 8 to the 6eppºos, or lupine.
The Romans made their weights, however, terminate
in the siliqua, or kepartov, deriving them directly from
the Greeks, and, therefore, not proceeding lower than
such weights as were in actual use. Amongst the Ita-
lians and all other European nations the grain of barley
and the carat, which is equivalent to four of them,
have been assumed as the basis of all existing weights.
Divisions of (161.) It is not surprising that the divisions of the
the Greek. Grecian litra, or pound, which were made use of in the
litra. division of their medicines, should have been adopted in
modern Europe, when the influence of the writings of
their physicians is considered; with them,24grains made
the gramma, 3 grammata the drachm, or dram, 8
Diy's
journey.
Bow shot.
Stadium.
Furlong-
Mile.
League.
Measures of
weight.
drams the ovyyū, or ounce, and 12 ounces the litra, or . History.
pound. The Romans translated gramma into scriptu- \"Nº-
lum, scripulum, or scrupulum, which we have retained. . . .
The same divisions are continued in the Apothecaries' '
pound, ... and, therefore, in medical prescriptions in
almost every country in Europe. The Greeks had a second
pound of 16 physical ounces, called the mna, or mina,
a term derived from the Hebrew maneh, a weight of
nearly the same magnitude. The pound of Cairo” is Of the
divided in 12 ounces, each ounce into 12 dirhems, each Pêyptian
dirhem into 12 carats, and each carat into 4 grains : *
though these divisions of the pound differ from the
Grecian, there is no doubt that dirhem and drachm are
the same word, and most probably derived from some
common Phoenician root.
(162.) The Venetian libra, or lira, of weight is divided Venetian
into 12 oncie, each oncia into 6 sazzi, each sazzo into 24 lira ºf
caratti, and each caratto into 4 grani d'orgio. In this *
case, as well as in that of the Egyptian ounce, we find a
departure from the Grecian subdivisions, though in all
three of them the ounce is made to consist of the
same number (576) of grains. The modern Romans,
however, have adhered to the divisions of the pound
which prevailed amongst the ancients; it being divided
into 12 oncie, each oncia into 8 dramme, each dramma
into 3 scrupoli, each scrupolo into 2 oboli, each obolo
into 4 silique, and each siliqua into 12 grani. In this
case, the number of grains bears no relation to the
weight which they represent; a circumstance which
can only be accounted for by their being of perfectly
abitrary value. - - -
(163.) The following are the divisions of the three Divisions of
pounds which are made use of in this country: º: º
- ing 1sn
Troy. pounds.
24 grains make a pennyweight.
20 pennyweights make an ounce.
12 ounces make a pound. . .
Apothecaries.
20 grains make a scruple.
3 scruples make a dram.
8 drams make an ounce.
12 ounces make a pound.
Avoirdupois.
20 grains make a scruple.
3 scruples make a dram.
8 drams make an ounce.
16 ounces make a pound.
The two first pounds are the same weight, but diffe-
rently subdivided. The ultimate subdivisions of the
pound avoirdupois coincide with those of the Apotheca-
ries' pound, though they are never resorted to in prac-
tice. g - -
(164.)The pound troy is said to have derived its name origin of
from the town of Troyes, where a celebrated fair was for- the terms
merly held, and where this weight was used. Whatever A. and
tº º - . e * * * g © a º voirdupois.
opinion, however, may be entertained of this derivation
of the name, which is not very satisfactory, it is certain ‘.
that it was never used in any public document before
* Bishop Hooper, in his Inquiry into the state of ancient
Measures, is disposed to consider the pound of Cairo as exactly cor-
responding to our pound troy, which he supposes to have been
derived from it. - *
A R IT, H M ET I C. 445
Arithmetic, the statute of the 12th of Henry VII., where its
vº-v- subdivisions are given, and where it is said that every
gallon shall consist of 8lbs troy of wheat. The origin of
the term avoir du pois, as applied to a specific weight,
is still more difficult to trace. It is first used in this
we may consider its use, indeed, in mercantile trans- History.
actions and ordinary sales as nearly universal. S-N-2
(167.) It was one of the articles of the great charter, Laws for
that there should be one weight and one measure through- securing
out the realm; and therepeated efforts of the legislature ºy
of weights.
The pound (165.) The pound troy must be considered as the ori- plaints were made of the frauds which were practised
Troy the ginal legal and statutable weight of this kingdom, though by false and unjust measures, and particularly in the
º the libra mercatoria, corresponding nearly with the case of the purveyors in the reigns of Elizabeth, James,
wºm ... pound avoirdupois, was the weight which was in most and his unfortunate successor.
this king- common usage. In the statute of the 31st of Edward (168.) In the year 1758 a committee was appointed Committee
dom, I. it is said, that “by the consent of the whole to inquire into the original standards of weights and of weights
realm of England, the king's measure was made, so measures of this kingdom, and to examine the stan- ****
that an English penny which is called the sterling, dards which were preserved in the Exchequer, Guild- #; 1 In
round without clipping, shall weigh 32 grains of wheat, hall, and elsewhere. The report, which was drawn up *
well dryed and gathered out of the middle of the ear; by Lord Carysfort, and read on the 28th of May of the
and 20 pence make an ounce, and 12 ounces a pound, same year, is very learned and elaborate, referring to all
and 8 pounds a gallon of wine, and 8 gallons of wine the statutes which bear upon the subject, and containing
a bushel of London, which is the eighth of a quarter.” the results of the examination of most of the existing
The same division of the pound and gallon are men standards, made chiefly under the direction of the
tioned likewise in the statute of the 12th of Henry celebrated instrument-maker Bird. The standard
VII., and in all the numerous statutes which were bushel (Winchester) of 1601 was found to contain
made from time to time for securing uniformity of 2124 cubic inches, though it was defined by the statute
weights, it is the pound troy which is considered as of the 1st of William and Mary that it should contain
the standard and legal weight. 2150. The gallon, quart, and pint, of the same date,
The libra (166.) Whether this was the legitimum pondus, which contained 271, 70, 344 cubic inches respectively, and
mercatoria was recognised in the time of Henry II., it is impossi- similar and even greater variations were found to ex-
ble now to ascertain ; at all events, though this weight ist in the standards of weights and measures of length;
was the favourite of the legislature, there was another under these circumstances, it was recommended that a
pound, one-fourth greater, which was in more general new yard and a pound troy, made by Bird from a
use; it is mentioned in the Fleta, in the time of Edward mean of those which were preserved in the Exchequer,
I., in an account of the possessions of the abbey of or rather copies of those which were made with great
Bewley in Hampshire,t and also in a Tractatus de care and accuracy by Graham for the Royal Society in
Ponderibus of the same age, where the two pounds are 1742, should be the standard yard and pound troy, by
said to consist of 20 and 25 shillings respectively : in which all other weights and measures should be regula-
the statute of the 54th of Henry III., where the ted; and that the wine gallon, beer gallon, and bushel,
composition of the gallon and pound troy are given, should contain 224, 282, and 2150 cubic inches respec-
there is mentioned also una libra, pondus vigint: tively. A second report was made in the following
quinque solidorum legatium sterlingørum. On many year, chiefly consisting of recommendations for the
other occasions this libra mercatoria is referred to, and general adoption and enforcement of these standards;
- . but as the bills which were founded upon them, and
* The same statute is reenacted for the following year, but was which were proposed in 1765, never passed into a law,
repealed altogether in the 33d year of this reign, upon the petition of it is not necessary for us to particularize them further.
the butchers, who declared that they should be ruined if this custom (169.) In the year 1818, Sir Joseph Banks, P. R. S., Committee
of selling provisions by , weight, which had never been the case Sir George Clerk, Mr. Davies Gilbert, Dr. W. H. Wol- of 1818.
sense in the statute of the 24th of Henry VIII.,
which fixes the marimum prices of provisions during
a time of scarcity, and orders that carcasses, beef, pork,
victuals, &c. shall be sold by the lawful weight called
haber-de-pois.” In former times, this term appears
to have designated commodities; thus the statute of
the 9th of Edward III., made at York, speaks of
damage done to the king and his subjects by people of
cities, &c., not suffering merchants, strangers which do
bring and carry by sea or land, vins, aver-du-pois et
autres victuailles, et autres choses vendables. Again, in
the statute of staple of the 27th of the same king, it is
said, Itempur ces que mous avons entendit que auscuns
marchauntz achatent avoir de pois leymz et autres
marchandises per un pois et vendent par un autre.
The most natural inference to be drawn from these pas-
sages is, that the term which was originally made use
of to designate every description of heavy merchandise,
was afterwards transferred to the weight itself, by which
they were most commonly estimated.
before, should continue to be enforced.
+ Lord Carysfort's Report of a Committee to ascertain the origi-
mal standards of Weights and Measures of this Kingdom. 26th May,
1758.
to secure this object appear to have been thwarted by
the prevalence of the customary pound, as well as b
local variations in other measures. In the 14th of
Edward III., standard ells, bushels, gallons, and pounds,
sealed with the kings's iron seal, were sent to the
sheriffs of the different counties, and directed to be
kept and adhered to, under severe penalties. In the
27th of the same king, however, we find that the com-
plaint was general, that merchants bought by one
weight and sold by another. In the 16th of Richard II.
these standards are directed to be kept by the clerks of
the market. Enactments on the same subject were
made in almost every subsequent reign; but whether
it arose from the multitude of statutes, many of which
were inconsistent with each other, from the rapidity
with which many of them were repealed, or from the
imperfections of the standards themselves, (made by
rude artists, and tried by methods which were equally
rude,) it is certain that the uniformity at which the
legislature aimed was never attained ; repeated com-
laston, Dr. Thomas Young, and Captain Kater, were
appointed commissioners under the privy seal, for
the purpose of forming new standards of weights
446 A R I, T H M E. T.I. C.
Arithmetic. and measures, or of determining the relations of those
S-a-' already in use to some invariable standard existing in
Their report nature. Their report, which was founded partly upon
the report of a committee for the same objects in 1814,
and upon inquiries which had been conducted chiefly
by Captain Kater since that time, into the lengths of
the seconds pendulum expressed in terms of existing
standards, is of uncommon importance, from the
authority and accuracy of its determinations, and still
more so from its chief recommendations having passed
into a law. It commences by deprecating any great or
violent changes in the standards already in use, as well
Imperial bushel; and that there should be but one . History.
common gallon for corn, ale, and wine. Fº \*
A bill embodying these recommendations, drawn
up by Sir George Clerk, was passed in 1824, having
been proposed, but rejected in the preceding session
of parliament. w
(170.) The only important alteration which this bill Inconve
proposed was in our measures of capacity; and it may ºes at
very reasonably be doubted, whether this change was .
altogether consistent with one of the wisest recom- of the
mendations of the committee : that the new gallon Imperial
should contain exactly 10 pounds of distilled water, bushel.
from the great derangement which such alterations
would produce in the ordinary transactions of commerce
and trade, as from there being no peculiar advantage in
having such standards commensurable with any invaria-
ble quantity existing in nature; that it would not be ex-
pedient to alter the subdivision of those measures already
in use, proceeding as they do in most cases according to
the duodecimal scale, or by numbers admitting of two
or three successive bisections, and which were, there-
fore, better accommodated to practical uses, than if the
subdivisions had been adapted to the decimal scale.
That the parliamentary standard yard, made by Bird in
1760, should be considered as the imperial standard
yard of Great Britain ; * that the length of the pendu-
lum vibrating seconds in the latitude of London, accord-
ing to the determination of Captain Kater, was equal
to 39.13929 inches of this yard ; a relation of lengths
which would always furnish the means of recovering
this standard in case it should be lost or injured; that
though it is apparently more philosophical to determine
the measures of capacity immediately from those of
length, yet in practice they are much more easily de-
duced from measures of weight. That one-half of the
double pound troy which was made by Bird, upon the
recommendation of the committee of 1758, should be
considered as the Imperial standard pound troy, con-
taining 5760 grains, whilst the avoirdupois pound should
contain 7000 grains; that in case this standard should
be lost or injured, it might be recovered from the know–
ledge of the fact, that a cubic inch of distilled water, of
the temperature of 32° of Fahrenheit, weighs 252.724
of this pound when the barometer is at 30°. That
it was found upon examination, that the legal stan-
dards of capacity were at variance with each other,
and that the ale gallon contained 4% per cent. more
than the corn gallon, though it did not appear that
this difference was sanctioned by the legislature ; that
the Winchester gallon, according to the definition in
the statute of the 1st of William and Mary, should
contain 269 cubic inches, whilst in other acts it was
fixed at 272%. That the ale gallon of the Exchequer
contained 282 cubic inches, whilst the wine gallon was
fixed by the statute of the 5th of Queen Anne at 231 ;
that as it appeared that 10 pounds avoirdupois of dis-
tilled water at the temperature of 62°, weighed in air
when the barometer is at 30°, was equal to 277.2 cubic
inches, it was expedient to assume this capacity as the
the Imperial gallon, eight of which should make the
* The report itself recommended as the standard yard the one
which was used by General Roy, in the measurement of the base on
Hounslow Heath for the great trigonometrical survey; it was found,
however, upon further examination, before the bill was passed into a
law, that it agreed less with the average of the other standards than
that made by Bird, which was preserved in the Tower, which was then
adopted in preference.
was not a necessary condition for recovering the stam-
dard hereafter in case it should be lost, though it
might make the process for that purpose more easy,”
and the accidental coincidence of this assumed weight
with one of the standard pints of the Exchequer,
which contained exactly 20 ounces of distilled water,
was a circumstance altogether unworthy of notice. I
is undoubtedly desirable that the same term, gallon,
should indicate the same absolute measure of capacity
for whatever articles it was used; though the incon-
venience which arises from the double or triple mean-
ing of a term is trifling and speculative, whilst that
which is produced by the identification of its signifi-
cation may be serious and real. It is true, indeed,
that the alteration of the ale and wine gallon was
easily and rapidly effected, as both the measures and
the articles measured are under the simultaneous and
universal control of the excise; and it was argued as
a justification of the change of the corn gallon, that
the bushel in ordinary use was almost universally
greater than the Winchester bushel; but still it was a
legal standard which was recognised by the legisla-
ture, and which long custom had rendered familiar to
the farmers, a class of men who are generally adverse to
all changes. It formed an essential part in all leases
where the rent is regulated by the price of corn; and
the departure from this standard, which local custom
had in some cases sanctioned, was not generally very
considerable, was always understood, and was rapidly
disappearing. Under such circumstances, we may be
almost justified in characterising this act as an ex-
ample of rash and inconsiderate legislation, which
enforced a tax of £150,000. upon a class of mem
for a merely speculative object, which altered the con-
ditions of so many thousand leases, and which afforded,
by the penalties by which its adoption was enforced,
endless opportunities for fraud and litigation.
(171.) If ever an opportunity presented itself for the New French
establishment of a system of weights and measures measures.
upon perfectly philosophical principles, it undoubtedly
occurred in the early part of the French revolution,
when the entire subversion of all the old establish
ments, and the hatred of all associations connected
* It is provided in the Act, that in case any dispute should arise
concerning the accuracy of any of these measures of capacity, whether
gallon or bushel, where reference cannot readily be made to a standard,
the parties must proceed before a justice of the peace, who is re-
quired to verify the measure by weighing its content of rain water
of the temperature of 62° of Fahrenheit against the statutable.
weights. With every respect for the unpaid magistrates of this
country, we should like to know how many of them would be either
disposed or able to undertake the investigation when appealed to, and
what would be the average degree of confidence to which their
determination would be entitled; we may venture to say, that no
measures, however just and accurate, could stend the test of such an
inquiry.
A R. I. T H M E T I C.
447
Arithmetic.
Proposal of
Picart.
Of Cassini.
with them, had created a passion for universal change.
The extrºe diversity also of the old French weights
and measures in different provinces of the kingdom,
whether of the same or different denominations, was
productive of the greatest inconvenience in the trans-
actions of commerce and trade; and philosophers as
well as others had long been anxious for the intro-
duction of some more uniform system, founded, if
possible, upon some invariable quantity existing in
nature. The celebrated Picart, who first measured a
degree of the meridian in France, proposed, in accor-
dance with a suggestion of Huygens in his Horologium.
Oscillatorium, that the length of the pendulum vibrat-
ing seconds should be adopted as the unit of length, and
that it should be called le rayon astronomique. The
discovery of Richer, however, at Cayenne, in 1671, that
pendulum was not of the same length for different
latitudes, deprived it of that absolute and invariable
character which was considered essential to such a
standard. At a subsequent period, Cassini proposed
that this standard should be derived from the magni-
seconds pendulum at the equator, and at 45°, the
unit of measures, considers them in one respect as
deficient in the character of a perfect standard, inas-
much as their determination would involve the hetero-
geneous element of time ; that no such objection
applies to an unit which shall be a definite portion
of the length of a quadrant of a meridian of the earth.
They therefore propose that the 10000000th part of the
quadrant shall be called the metre, and considered as
the primary standard of measures of length, weight,
and capacity; that the quadrant shall be divided into
100 degrees, the degree into 100 minutes, and the
minute into 100 seconds; that the subdivisions of all
measures should be adapted to the decimal scale; that
in order to determine the metre, an arc of the meridian,
extending from Dunkirk to Barcelona, 6% degrees to the
north and 3 degrees to the south of the mean parallel
of 45°, should be measured; and that subsequently the
weight of a décimetre cubed of distilled water at the
temperature of melting ice should be determined, as
the unit of measures of weight. -
tude of the earth,” and that rºbºth part of a minute
of a degree should be considered as the pied géomé-
trique, and that a toise should be considered as the
u-bº-o oth part of a degree. The same idea was adopted
Of Mouton. by an astronomer of the name of Mouton, who recom-
mended that a minute of a degree should be considered
(173.) Immediate steps were now taken for the execu- Proceedings
tion of this great undertaking, under the direction of a for the de-
committee of the most celebrated men of science in ...
France. The measurement of the northern part of the . º
arc from Dunkirk to Rodez was assigned to Delambre, metricai
and of the south to Mechain. The account” given by system.
as the superior unit of length under the name of mille,
whilst the other measures, proceeding in the subdecuple
series, should be called respectively, centuria, decuria,
virga, virgula, decima, centesima, millesima, or other-
wise stadium, funiculus, virga, virgula, digitus, granum,
the former, of the difficulties which he encountered in the
course of his operations, from the jealousy and alarm of
the country-people, is extremely interesting. His first
commission ran in the name of the king; and his labours
began when the name of the king was a signal for out-
of De la punctum. In the year 1748 M. de la Condamine, rage and violence. When his work was half done, he
Condamine. who had recently returned from measuring a degree received the alarming intelligence, that his name, as well
at the equator in Peru, in a Memoir read to the Aca- as that of Borda, Laplace, Lavoisier, Coulomb, and
demy of Sciences, resumed the idea of the pendulum Brisson, had been struck out of the commission of
as the unit of length, and recommended as the best weights and measures by the committee of Public
means of quieting the feelings of national jealousy Safety,t who assigned as their reason for this proceeding,
which would attend its selection for the latitude of that they required for the public service those only who
London, Paris, or even of the parallel of 45°, which were worthy of confidence, from their republican virtues
passes through France, that it should be taken on and their hatred to kings. Fortunately, however, he
the equator: under such circumstances he felt persuaded was enabled to continue his observations, though with
that a sense of its advantages would insure its imme- great difficulty and some danger, until the termination
diate adoption by all the scientific bodies of Europe, of the reign of this sanguinary faction, when the names
and that it would speedily be received into general use. of the displaced members were restored to the commis-
Commission (172.) In the year 1788, when the ferment of the revo- sion, and the measurement of the whole arc completed.
of 1790. lution was beginning partially to show itself, the same The length of the metre which resulted was found to
subject was resumed; and in 1790 it was proposed by be 443.296 lignes less than the metre provisoire,
Talleyrand to the Constituent Assembly, that a commis- which had been adopted provisionally $ in 1794,
sion should be appointed to report on the measures
which were proper to be taken; and in consequence,
e Borda, Lagrange, Laplace, Monge, and Condorcet were . * The decree is signed by Barrere, Robespierre, Billaud Varenne,
Their report, appointed commissioners. Their report, which was Couthon, and Collot d'Herbois.
made in the following year, after noticing the proposals
which had been made to make the length of the
* It was contended by Paucton, in his Métrologie, that the side of
the great pyramid was the exact synth part of a degree of the meridian,
and that the founders of that mighty monument designed it as an im-
perishable standard of measures of length. Absurd as this notion
apparently is, it was patronized by the celebrated Bailli, with his usual
fondness for extravagant hypotheses, and who conjectured that both in
it and in the coudée nilométrique, or cubit of the nulometer, was to be
found the invariable standard of measures derived from the magnitude
of the earth : it was somewhat unfortunate for both these suppositions,
that the length of the side of the great pyramid was found to be
7163 French feet, instead of 6844, and the cubit of the nilometer 20.54
inches instead of 19.992, as it should have been.
t Base du système métrique. Discours préliminaire.
# The letter which he received in answer to an application which he
made to be allowed to complete a certain series of triangles, in order
that much of his previous labours mignt not be rendered useless, is an
admirable specimen of the style which was fashionable at that period.
Citoyen,
La commission des poids et mesures a chargé l'un de ses membres de
se rendre auprés de toi pour te remettre l'arrété du comité de salut
public quite concerne et pour concerter avec toiles moyens de clore
tes operations de manière que les signaur restent inutiles ; elle t'invite
a terminer la redaction de tes calculs et la copie de tes observations
aimsi que titles proposes.
18 Nivöse, an. 2.
§ By order of the Committee of Public Safety, who were deter-
mined to avail themselves of the impulsion révolutionnaire to effect
this change, before the conclusion of the labours of the commission.
448
A R IT H M ET I C.
1 4, 6
Arithmetic. before the completion of the operation, by rººr of a
S- ligne.*
The determination of the unity of weight, an opera-
tion of great delicacy and difficulty, was specially
confided to Lefevre Gineau, who assigned to the kilo-
gramme, or décimetre cubed of distilled water at its
greatest density, and not at the temperature of melting
ice as at first proposed, a weight of 18827.15 grains,
poids de marc.
The whole of these operations were conducted under
the general superintendence of a numerous commission
of members of the Institute, as well as of commissioners
from Italy, Spain, Holland, and Switzerland; and all
the instruments made use of, the journals of observa-
tions, and the calculations founded upon them, were
submitted to their examination.
Report of (174.) The report of the commissioners was made on
º the 10th of Prairial, 1798, and on the 4th of Messidor, the
SIOI) el'S Ill
1798. were presented, with a pompous address, to the two coun-
cils of the Legislative Body. In speaking of the metre it
is said, Cette unité, tirée du plus grand et des plus in-
variables des corps que l’homme puisse mesurer, a l’avan-
tage de me pas différer considérablement de la demi-
toise et des plusieurs autres mesures usitées dans les
différens pays ; elle me choque point l'opinion commune.
Elle offre un aspect qui n'est pgs sans interét. Il y a
quelque plaisir pour un père de famille à pouvoir se
dire: “Le champ qui fait subsister mes enfams est une
telle portion du globe. Je suis dans cette proportion
compropriétaire du monde.” After mentioning the ex-
traordinary precautions which had been taken by the
commission, and enumerating in imposing language the
names of the Savans étrangers et mationaua who com-
posed it, it is announced that these prototypes shall be
deposed amongst the national archives, to be preserved
with a religious care, from whence jamais l'ignorance et
la férocité des peuples barbares me les enleveront ; a la
vaillance, au patriotisme,aua vertus d'une mation éclairée
sur ses interéts, sur son hommeur, sur ses droits. Mais
si un tremblement de terre engloutissoit, s'il étoit pos-
sible qu'un affreuw coup de foudre mit en fusion le
zmétal conservateur de cette mesure, il m'en résulteroit
pas, citoyens legislateurs, que le fruit de tant de tra-
vaua, que le type général des mesures put átre perdu
pour la gloire mationale, mi pour l'utilité publique.
By way of provision against such a catastrophe, as un
moyen conservateur du metre, it is added, that Borda
had determined with great accuracy the length of the
seconds pendulum at Paris, and the repetition of the
experiments at any future period would furnish the
means of recovering the original relation of its length
to that of the metre, and consequently of determining
the length of the metre itself.
New no- (175.) The nomenclature of the new weights and mea-
menclature sures underwent various changes. It was proposed by the
of weights first commission, that the old names should be preserved
* as much as possible, with significations adapted to the
tº s new system. The law of the 18th of Germinal, 1794,
which established the provisional metre and kilogramme,
altered the old names entirely; whilst the law of
the 13th Brumaire, 1798, which succeeded the report
of the commission, reverted in a great measure to the
* The length of the provisional mêtre was determined from the
data furnished by Lacaille in 1758, who had assigned to a degree of
the meridian in latitude 45° a length of 57027 toises.
original metre and kilogramme (les étalons prototypes)
system of names which were first proposed. They are History.
as follows: - *... . -
Mêtre.
Measu.es of length. Synonyms. .
Lieue Myriamétre I0000
Mille Kilomètre 1000
Hectometre 100
Perche Décamétre 10
Mètre l
Palme Décimètre Tºor
Doight T-tro-
Trait rºut,
Measures of weight. Synonyms. Kilogramme.
Millier 1000
Quintal 100
Myriogramme 10
Livre Kilogramme l
Once Hectogramme Tºur
Gros Décagramme Tyruſ
Dénier Gramme T-Uuruſ
Grain Décigramme To ºut,
Measures of capacity: the unit is the metre cubed.
Muid Stëre 1.
Setier Décistère T'o'
Boisseau ++ or
Pinte Tu'uty
Verre T-U-5 Jºur
Measures of area: the unit is the metre squared.
Arpent Hectare 1000
Décare 100
Perche Are 10
Métre carré Déciare 1
(176.) The establishment of the French system of Advantages
weights and measures was an event of considerable im- and disº";
portance to the scientific world, from the imperishable ...". of
nature of its bases, and from the confidence to which their system.
determination is entitled. The power and influence of
popular and national prejudices must for ever prevent
the universal adoption of this or any other system, how-
ever perfect; but it is of comparatively little conse-
quence whether they are actually adopted by any nation
or nations, so long as they furnish a standard of refe-
rence by which those in use may be estimated, and by
that means their value become universally known.
It is only by such means that the fluctuating and
variable standards of different nations may be made
to speak the same language. The decimal sub-
division of these measures possessed many advantages
on the score of uniformity, and was calculated to sim-
plify in a very extraordinary degree the Arithmetic of
concrete quantities. It was attended, however, by the
sacrifice of all the practical advantages which attend
subdivisions by a scale admitting of more than one
bisection, which was the case with those previously in
use; and it may well be doubted, whether the loss in
this respect was not more than a compensation for
every other gain.
(177.) The centesimal division of the quadrant was not The centesi-
called for by any principle of uniformity, and it at once mal division
sacrificed all the conveniences which attend its trisection, ººl"
g ºt tº * > e g G. s. ſº . . . .” drant less
which is so important for artists in the division of cir-...t
cular instruments. It at once also made useless all than the
the trigonometrical tables which were already cal- sexagesimal.
culated, at least without previous and troublesome re-
ductions. If the change had been confined to the cen-
A R IT H M ET I C.
449
Arithmetic, tesimal division of the minute, second, &c. it might
S-N-" have bećf generally adopted, and others would have
very readily abandoned the use of sexagesimals, pro-
ceeding as they do by too high a scale to be conve-
niently used : as it was, the new division of the
quadrant was never generally used even in France, not-
withstanding the great authority of Laplace, and was
abandoned in later life, even by Delambre himself, by
whom it was once so zealously recommended.*
* (178.) The reception which the new measures experi-
* enced in France furnishes a curious proof of the extreme
jof difficulty of counteracting the prejudices, or altering the
habits of a whole people; and an instructive lesson of
the danger and inefficacy of any legislative interference
with them, unless called for by great and manifest ad-
vantages, and capable of being readily and universally
enforced. In no other nation was the grievance of
variable and uncertain weights and measures so in-
tolerable; in no other nation was the occasion for
their reformation so favourable, when the current of
popular opinions and habits had been diverted from its
ordinary channel by the violent concussion of the revo-
lution; in no other country could the change proposed
have been recommended by a greater or more imposing
authority; yet we find that the people obstinately ad-
hered to their ancient measures and their ancient names.
The metre was a new and unintelligible name, associated
in their minds with no former or natural measure, and
by no means recommended by its enabling them to
ascertain the definite portion of the earth’s surface
which their farms occupied. In other cases, the
union of old names with new measures made their intro-
duction more easy; but it required the influence of
many years, and all the authority of the government,
to effect even their partial adoption; and even at this
time, we find their metre and its third part, the foot,
with the duodecimal as well as the decimal division, in
almost universal use.
Reductions (179.) The reduction of weights, measures, and coins,
of weights from greater to lower denominations, and the contrary,
*** forms an important article in all books of Arithmetic, and
Słl reS : requires, of course, a perfect knowledge of their several
Their com- subdivisions. In Italian books of Arithmetic, these
Plexity, reductions become extremely complicated, from their
generally extending to the weights and measures of
other cities, besides those in which the authors lived,
where they varied extremely, both in denomination and
value. As an example, we shall give from Tartaglia the
measures of length and area which were used in many
of the cities of northern Italy, though he declares that
the list which he gives does not include the hundredth
part of the cities of Italy, in which such variations are
the new
system.
found.
Verona.
Measures of length. Measures of area.
Italian mea- Pertica, 6 piedi. Campo, 24 vanezze.
sures of Piede, 12 oncie. Vanezza, 30 tavole.
a ſea, Oncia, 12 ponti. Tavola, 36 piedi.
Piede, 12 oncie.
Oncia, 12 ponti,
Ponto, 12 athomi.
Athomo, 12 menicoli.
Padua. -
Pertica, 6 piedi. Campo, 4 quarteri.
* Base du système métrique, tom. iii. p. 308.
VOI, J.
Measures of area.
Quartero, 210 tavole.
Measures of length. History.
- Tavola, 36 piedi.
Treviso.
Pertica, 5 piedi. Campo, 1250 tavole.
Tavola, 25 piedi.
Milano. -
Zucata, 12 braccia. Pertica, 24 tavole.
Brazzo, 12 oncie. Tavola, 12 piedi.
The tavola is the square of the zucata, which is
divided into 12 piedi.
Bergamo.
Cavezzo, 6 braccia. Pertica, 24 tavole.
Brazzo, 12 oncie. Tavola, 12 piedi.
The tavola is the square of the double cavezzo. *
Mantua.
Cavezzo, 6 braccia. Biolco, 100 tavole.
Brazzo, 12 oncie. Tavola, 12 piedi.
The tavola is the square of the double cavezzo.
Brescia.
Cavezzo, 6 braccia. Pio, 100 tavole.
Firenza.
Brazzo, 12 oncie. Staiora, 12 panore.
Panora, 12 pugnore.
Pugnora, 12 braccia.
Brazzo, 12 oncie. -
Throughout modern Italy, oncia has the same mean- Meaning of
ing with the uncia of the ancient Romans, designating ºncº.
a twelfth part of the next superior integer, whatever
that integer may be. -
(180.) The prevalence, likewise, of the duodecimal di- General
vision in all these cases is sufficiently remarkable. The preyalence
subdivisions of the oncia never extended in practice be-, º
yond the ponto.” The Öther terms athomo and menicolo . | V | -
are introduced by Tartaglia himself, to express the more
minute terms in the duodecimal multiplication of length
into length. The misapplication of the names of mea-
sures of length to designate the area of the squares
described upon them is common in all languages ; but
in some of the cases above-mentioned the foot is taken
as the first of the duodecimal subdivisions of the tavola,
without any reference to the measures of length which
it commonly designates.
The origin of this interchange of terms, and their Interchange
misapplication to denote things essentially different of terms.
from each other, is to be ascribed in part to the poverty
of language, and partly, likewise, to the ignorance of
most men of the proper force and meaning of the terms
which they use. In the case of terms applied to de-
signate the successive subdivisions of any class of con-
crete quantities, it is a natural and easy process of the
mind to consider them as more connected with the
Yelative magnitude of the next superior unit than with
the peculiar nature of the magnitude itself. In illustra-
tion of the truth of this observation, we may refer to the
very general meaning given to the terms which were
originally confined to denote the subdivisions of the
Roman as.
* In some cases, the common people called the ponto de terra,
or any smaller subdivision of the oncia, damaro de terra, borrowing
the name of the imaginary coin so called. .
3 N
450
A H I T H M E T I C.
Arithmetic. (181.) The Arithmetic of compound or denominate
-v-/ quantities, their addition, subtraction, multiplication, and
Multiplica- division, as well as their reduction, presents not much
tion of de- room for variety, and they will be found, upon exami-
* nation of the arithmetical works of the last three centu-
numbers * - g
inj, ries, nearly under the same form. A question, how-
3ther. ever, appears to have arisen, whether it was possible to
multiply denominate numbers together, or to divide
them by each other.
It was remarked by Stifelius,” that numerus vulgariter
denominatus, non potest multiplicari per alium nume-
rum vulgariter denominatum nisi alter eorum denomi-
nationem suam depomat et fiat abstractus : but again,
that alter per alterum dividi potest, modo ambo eandem
habeant denominationem. In the latter case, the num-
bers are reduced to the same lowest denomination, and
their relation to each other is identical with the relation
of the resulting numbers, considered as abstract, and,
consequently, their quotient may be considered as an
abstract number. Tartagliaf has quoted this remark of
Stifelius with disapprobation, and seems to speak of the
possibility of multiplying money by money, and weights
by weights; but as such an operation might appear to
many people cosa nuova e forsi strania, he defers the fur-
ther discussion of such peculiarities, or, at all events, of
the exceptions to such an opinion, to another occasion.
Duodecimal (182.) The exceptions which probably suggested them-
multiplica- selves to the mind of Tartaglia were those in which the
iºn product of length into length produces area, or where the
i." product of area into length, or of length into length into
S “” length, produces capacity. It was not considered, that in
these cases the multiplication took place as if the numbers
were abstract, the inferior subdivisions forming a series
of duodecimals of the primary unit; and that the relation
between the product and the component factors (sides
or edges of the rectangle or parallelopipedon) was
merely numerical, the concrete units being essentially
different from each other: in other words, that it was
an extension of the meaning of the term multiplication
to apply it to such cases; as the analogy of which the
terms in their order are the product, the multiplicand,
the multiplier, and unity, which existed in one case, no
longer existed in the other, at least in its proper and
strict sense.
Tartaglia has given many examples of these duo-
decimal multiplications, as well as of the inverse opera-
tion of division; and we have seen before, that he
extended the nomenclature of the duodecimal sub-
divisions, so as to include all the terms which resulted
from them : beyond these cases, however, he has not
ventured to proceed, and we may consider the boast
that he would produce numerous instances in reproba-
tion of the opinion of Stifelius, as a proof of the
envious and contentious spirit with which he criticized
the writings of his contemporaries, of which he has
been accused in severe terms by Bombelli.; y
(183.) The Rule of Three, emphatically called from its
great usefulness the Golden Rule, both by ancient and
modern writers on Arithmetic, is so simple in principle,
that we can expect to find very few essential variations
In the Lilá- in the form in which it is stated. In the Lilāvati we
find the ordinary divisions of the rule into direct and
inverse, simple and compound, with statements for
Opinion of
Stifelius :
Of Tarta-
glia.
Rule of
Three :
vati.
* Arithmetica Integra, p. 81.
* - + Numeri e raisure; in fine libri tertii, pars i.
# Algebra, Preface. *
performing the requisite operations, which are suffi- History.
ciently clear and definite, due allowance being made S-
for the ordinary obscurity of Sanskrit phraseology on
scientific subjects. The terms of the proportion are
written eonsecutively, without any marks of separation
between them : the first of them is termed the measure,
or argument; the second is its fruit, or produce; the
third, which is of the same species with the first, is the
demand, requisition, desire, or question. When the
fruit increases with the increase of the requisition, as
in the direct rule, the second and third terms must be
multiplied together, and divided by the first; when the
fruit diminishes with the increase of the requisition, as
in the inverse rule, the first and second must be mul-
tiplied together, and divided by the third.
No proof of the rule is given, and no reference to
the doctrine of proportion upon which it is founded.
Proofs, indeed, are never given in the Lilávati, and on
this occasion are hardly required; the proposition is
so readily deduced by the common sense of mankind,
when its terms are once understood, that it acquires very
little additional evidence from a formal demonstration.
Under compound proportion are included the rule Compound
of five, seven, nine, or more terms. The terms are proportion.
in these cases divided into two sets, the first belonging
to the argument, and the second to the requisition : the
fruit in the first set is called the produce of the argu-
ment ; that in the second is called the divisor of the
set: they are to be transposed, or reciprocally to be
brought from one set to the other; that is, put the
fruit in the second set, and the divisor in the first ; in
other words, transpose the fruits in both sets. This
rule, which is sufficiently obscure, will be further ex-
plained in some of the examples which follow.
Erample 1. If two and a half palas” of saffron be Examples.
obtained for three-sevenths of a mishca,t say instantly,
best of merchants, how much is got for nine mishcas 2
Statement :
3 5 9
7 2 1 Answer, 52 palas and 2 carshas.
Rule of three inverse.
Erample 1. If a female slave, 16 years of age,
bring 32 mishcas, what will one aged 20 cost? If an
ox, which has been worked a second year, sell for 4
mishcas, what will one which has been worked 6 years
cost 2 -
1st question.
Statement: I 6 32 20.
2nd question.
Statement: 2 4 6. Answer,
The value of living beings is supposed to be regulated Value of
by their age, the maximum of value of female slaves slaves and
being fixed at 16 years of age, and of oxen after 2 *".
years’ work; and their relative value in the present case
being 8 to 1. So important was this traffic con-
sidered, and so fixed were the principles by which it was
regulated, that in the Arithmetic of Srid'hara it is made
the subject of a distinct chapter: this is not the only
instance in which the examples given in books of Arith-
metic will convey important information concerning
civil institutions and the trade or commerce of nations.
Answer, 25g mishcas.
14 mishcas.
* A pala = 4 carshas ; a carsha = 16 mashas ; and a masha =
5 gunjas, or 10 grains of barley.
+ A mishca = 16 drammas; a dramma =
= 4 caeinis ; and a cacini = 20 cowry shells,
16 panas ; a pana
A R I T H M
E T I C. 451
Arithmetic.
Touch of
gold.
Rule of five
ièIIIAS.
Interest of
money in
India.
Of Seven
terth S.
()f nine
terlllS.
J
º' ten may be had for one nishca of silver, what weight
Erample 2. If a gadyana of gold of the touch of
of gold of fifteen touch may be bought for the same
price?
Statement: 10 lº 15. Answer, 3.
The fineness of gold in the East is usually determined
by its colour on the touchstone, which long experience
makes a sufficiently delicate test of the quantity of alloy.
European goldsmiths, from a very early period, have
been accustomed to divide an unit of gold in 24
parts, called caratti,” or carats, and to estimate its
fineness, or degree of purity, by the number of carats
of pure gold which it contained.
Rule of five terms. - -
Example 1. If the interest of a hundred for a
month be five, what is the interest of sixteen for a year?
Statement: l 12,
100 16 -
5 or, transposing the fruit,
1 12
100 16
5
product of the larger set 960, of the lesser 100. Quo-
tient . OT : which is the answer.
Example 2. Forty is the interest of a hundred for
ten months; a hundred has been gained in eight
months: of what sum is it the interest ?
Statement: 10 8, or, transposing, 10 8
100 | 00
40 100 100 40
625 . e
whence the answer + is obtained.
The interest of money, if we may judge from the
examples in Brahmegupta and Lilāvati, varied from 3%
to 5 per cent. per month, exceeding greatly the enor-
mous interest paid in ancient Rome. The case is
similar, though not to the same degree, in modern
India, where it is not uncommon for mative merchants
or tradesmen to give 30 per cent. per annum.
Rule of seven terms.
Example. If three cloths, two wide and five long,
cost six panas, tell me how many cloths, three wide
and six long, should be had for six times six P
Statement: 2 3, or, transposing the fruits, 2 3
5, 6 5 6
3 3
6 36 36 6
The answer is 10.
Rule of nine terms.
Ewample. The price of a hundred bricks, of which the
length, the base, and breadth, are respectively sixteen,
eight, and ten, is settled at six dindras. We have re-
ceived a hundred thousand of other bricks, a quarter
less in every dimension: say what we ought to pay ?
Statement: 16 H2
8 6
10 :
. 100 100000
6 &
or, transposing the fruit and denominator, it becomes,
16 12
8 6
I0 30
100 100000
4 6
The answer is 2531}.
* Tartaglia. Numeri e misure, pars i.
*
Rule of eleven terms. - * -
Example. Two elephants, which are ten in length, J
nine in breadth, thirty-six in girt, and seven in height, Of eleven
consume one drona of grain; how much will be the terms,
rations of ten other elephants, which are a quarter more
in height and other dimensions?
History,
Statement : 2 10
10 .
9 T
36 45
35
7 4.
l .
the fruit and denominator being transposed, the answer
is #.
The principle of this very curious example would be
rather alarming, if extended to other living beings
besides elephants.
The last example which we shall give is one of Barter.
barter, included by Brahmegupta and Bhascara under
this very comprehensive rule,
If a hundred of mangoes be purchased for ten panas,
and of pomegranates for eight; how many pomegra-
mates for twenty mangoes 2 -
Statement: 10 8, or, transposing, 8 10
100 100 100 100
20 20
The answer is 25.
(184.) It was usual, according to Lucas de Burgo, for Rules used
students in Arithmetic, who wished to learn the practice in Italy. .
of la regola del tre, (or la regola delle tre cose, as it was
designated with manifest impropriety by the grossi or
ignorant,) to commit to memory one or other of the two
following rules : * :
1. La regola del tre vol che si moltiplichi la cosa che.
l’huomo vol saper per quella che non e simigliante e par-
tir per l'altra chee simigliante e quel che me vene e de
la matura de quella che non e simigliante e sia la valuta
de la cosa che volemo inquirere. -
2. La regola del tre vol che si guardi la cosa mento-
vata doi volte delle quale la prima e partitore. E la
seconda si moltiplich? per la cosa mentovata una volta.
E quella tal moltiplicatione si parta per detto partitore.
E quello che ne viene de detto partimento sia de la
anatura de la cosa mentovata, una volta ; si deve mettere
in lo mezzo quando si opera.
Tartaglia has mentioned the first of these rules
nearly in the same terms. He has given also a third
rule, differing in expression only from the preceding ;
it is as follows:
La regola dal tre sono tre cose, la prima che si mette
debbe esser sempre simile a quella chesta di drio e di drio
debbe star la cosa che si vol saper e multiplicar la
contra quella chesta de mezzo e quel produtto partirlo
per la prima e sara fatta la ragione; e nota chequella
che venira sara sempre simile alla cosa, chesta di mezzo.
This rule, expressed in popular or vernacular lam-
guage, formed part of a system of instruction of the
practice of this rule, adapted to those who had not
sufficient time to acquire, genius to comprehend, or
memory to retain the rules for the reduction and incor-
poration of fractions; a system reprobated by Tartag-
lia, and attributed by him partly to the ignorance of
the ancient teachers of Arithmetic at Venice, and
partly to the stinginess and avarice of their pupils, who
grudged the time and expense requisite for attaining a
perfect understanding of the peculiarities of fractions.
(185.) An arithmetician of Verona, named Francesco

3 N 2
452
A R I T H M E T I C.
Arithmetic. Feliciano da Lazesio, the author of a work on Arith-
S-a-' metic, entitled Scala Grimaldelli, objects to the me-
Amended morial rules of De Burgo as being too general, as it
ſºlºs of , is ve ible that the th tº a º w all be of
Feliciano da ry possible tha e three quantities may
i.e. the same or all of different species or denominations.
Thus, in the following question, If 3 ducats produce me
4, what will 6 ducats produce 2 the three quantities
are of the same species and denomination; whilst in
this, If 8 ducats purchase 4 braccia of cloth, how much
"may be got for 24 lire 2 the quantities are all of different
denominations. For these reasons, he wishes to dis-
tinguish the quantities into agents and patients, and
those again into actual or present and future. The
first term of the proportion is the agent a presente, and
its corresponding patient is the second; the third term
is formed by the agent de futuro, and its patient is the
quantity to be determined. With respect to these ob-
jections, it may be observed, that the first is true though
merely verbal ; that the second is merely imaginary,
inasmuch as the first and third terms are reducible to
the same denominations; and granting the accuracy
of the distinctions made in the proposed alterations, yet
they injure the simplicity of the old rules, by introducing
metaphysical considerations, which are not very readily
apprehended by students whose minds are not disci-
plined to the labour of systematic thought.
(186.) The following example will show the manner
in which the operation was stated and performed by De
Burgo. -
If a hundred pounds of fine sugar cost 24 ducats,
what will be the cost of 975 pounds 2
Partitore Cosa mentovata una.
2uchari. volta ºnwltiplica-
tore : ducati.
via ºyd.
I ()() 24 — 975
l X 1 — I
Example
from lucas
de Burgo:
Cosa che volento
sapere: l'altro
moltiplicatore.
975
24
3900
1950
º
:
(234 ducati
;
:
:
d
tºmºsºmºmºmºmºmº 2 g g
23400 Aſ
The quantities are exhibited under a fractional form,
for the purpose of making the process more general,
being equally applicable to fractions and whole num-
bers. It is sufficiently curious, that he should have
considered it necessary to construct the galea for the
division by 100.
(187.) The following example of the same process,
with fractions in every term, is given by Tartaglia.
If 34 pounds of rhubarb cost 2; ducats, what will be
the cost of 23# pounds?
lire. ducati. lire.
From T'ar-
taglia.
— X | --
2
partitor
da partir 1330
0 7 History.
4 g Nº-vº-'
0 g g 0
2: 3 2 0 (15 ducati.
$ 4 4.
$
0 0 0
r 3 $ 0 (20 grossi.
• * *
(188.) Different methods have been adopted by different Different
authors, for representing the terms of the proportion in mºss of
this rule. We will state a few of them with reference . ".
to the following question.
If 2 apples cost 3 soldi, what will 13 cost 2
Statement of Tartaglia :
Se pomi 2 || val soldi 3 || che valera pomi 13.
Other Italian writers write the numbers consecutively
with mere spaces, and no distinctive marks between
them ; thus,
proportion.
Pomi. Soldi. Pomi.
2 3 13
or thus,
Ima. 2 a. 3tia.
2 3 13
In Recorde and the older English writers, they are
written as follows:
Apples. Pence.
2 3
13 19% answer.
A later English author,” whose work was first published
in 1562, and afterwards in 1594, writes them thus:
2 3 13
The custom, which prevailed generally during the
XVIIth century, was to separate the numbers by a
horizontal line,f as follows :
* The Well-spring of the Sciences, which teacheth the perfect
worke and practice of Arithmetike, set forth both in whole num-
bers and in fractions; set forth by Humfrey Baker, Londoner, 1562.
In speaking of this rule, he says, “The rule of three is the chiefest,
and the most profitable, and most excellent rule of all Arithmetike.
For all other rules have neede of it, and it passeth all other; for the
which cause, it is sayde the philosophers did name it the Golden Rule
but now, in these later days, it is called by us the Rule of Three, be-
cause it requireth three numbers in the operation.”
+ Vulgar Arithmetike, explaining the secrets of that art by
Noah Bridges, 1653, p. 127: “It was the custom in that age for every
writer on the most trivial subjects to have his work recommended
by the poetical contributions of his friends. The fame of Mr.
Noah Bridges is celebrated in every form of versification, and with
every variety of extravagant eulogy. A Mr. Lovelace addresses the
incomparable Mr. Bridges on his (by his favour) No Vulgar Arith-
metic. -
Behold yon sacred bard, that doth uncharm
All your learned knots, and your strong rules disarm.
Your priest, that with delight doth sacrifice
Past errors, doubts, mistakes, unto the wise;
And with this smooth, bright, polished, easie key,
He opes a dore unto a rosie way,
So far from vexing the distracted brain,
That sculls of lead his golden rules contain.
Here he the chaste Arithmetick addresses,
Left nak’d as truth, unbrades her very tresses
With such religious skilful lust, that we
The very secrets of her secrets see.
A. R. I T H M ET I C. 453
Arithmetic. Apples. Pence. Apples - multiplying the several parts of the quantity whose value . History.
S-N-" 2 — 3 13 is demanded by the several terms of the price of its STN-
The whole algorithm of proportions appears to have Prº*Y unit, and reducing the terms of the several Te-
received the particular attention of Oughtrede,” from sults In the manner which may be required; as in the
whom the sign ::, to denote the equality of ratios, was following example:
derived. He states the rule of three as follows: What is the price of 23 braccia, 2 quarte, at 8 lire, Practica
- 13 soldi the brazzo P naturale.
2. 3 : : 13.
In still later times, the simple dot which separated the G s l. s. e ſº
terms of the ratios was replaced by two, as in the form ; º . º * * 23 braccia, 13 soldi ;
which is now used: raccia for 13 soldi
2 : 3 : : 13. 198 19 69
Classifica- (189.) Both Lucas de Burgo and Tartaglia have sought 2 quarte . . . . . . . . . . 4 5 6 piccoli. 23
tion of ques- to include, in the numerous examples which they have — wº-º-º-º-º-º-
tions in given, every possible case of mercantile practice; the Total amount . . . . 203 6 6 299
|. de a first of these authors has classified his examples with *s-as-s-s-s-sº
urgo an e e tº ll 4. 19s.
Tºglia, reference to the manner in which particular goods
are sold, whether by the hundred pounds weight, as gº *
was the case with the sugars of Palermo, Syria, Madeira, The natural practice merely requires the knowledge
or Candy, the finerspecies of wool, wax, gums, medicines, of the four fundamental rules of Arithmetic, and the
&c.; or by the thousand pounds, as was the case methods for the reduction of weights and measures
with heavier merchandise, such as metals, vitriol, galls, from higher to lower denominations, and, conversely,
rice, oils, &c., or by measures of capacity and by number. Without resorting to any of the artifices made use of in
The latter has adapted his classification partly to the the other methods of practice, which require the in-
occurrence or non-occurrence of fractions in the diffe- structions of a refined arithmetician. The rules of the
rent terms of the proportion, and partly to the peculiar first of these, as well as of the two, others, as they *
** difficulties which attend the statement or solution of appear in our author, differ very slightly from the rules P.º
the questions which are proposed. The questions them- of practice which appear in our books of Arithmetic, artificiale.
selves are in immense variety, and are stated and solved excepting only that they are not reduced to the same
with great minuteness of detail. definite and systematic form, and are worked out as
Different (190.) Amongst other abbreviations of the process for usual with a tedious particularity and diffuseness. They
species of the solution of the rule of three questions, of which the are chiefly founded upon the assignation of the aliquot
practice, Italians were the inventors, and which were adapted to the parts of their coins, weights, and measures, and more
purposes of their extensive commerce, may be mentioned especially those of the ducat and lira, or pound, of
the rules of practice. Tartaglia has divided them into which an extended list is given. The following is an
four species, la practica naturale, artificiale, Vene- example:
tiana, and Firentina ; the three first of which form, What is the value of 624 stara, 2 quarte, and 3 quº;
severally, the subjects of the IVth, Vth, and VIth Books toroli of wheat, at 9 lire, 16 soldi, 8 pizzoli, the staro 2
of the 1st part of his work. The first consists in
Stara. . . . 624 2 3
Lire . . . . 9 15 8
The reason of the disproportion between fools and wise men is very lºmºmºmºmºsºmsºmºmºmºmºs
satisfactorily explained: At 9 lire . . . . . . 5616 0 0
Why are wise few, fools num’rous in the excesse P 10 soldi. . . . . . 312 0 0
'Cause, wanting number, they are numberlesse. 6 pizzoli . . . . 15 12 0
Amongst other names in this list, we may be surprised to find those 2 pizzoli . . . . 5 4 0
of Thomas Shirley and Elias Ashmole. Another poet, whose initials *Cºmº
are T. D., is shocked and amazed at the title of his book: 6H 04 16 0
I stood amazed, when first I saw, For 2 quarte. . . . 4 17 I ()
Ev’n in thy title, such a flaw, - & © i
As made me (though engaged) withdraw. ; . h i. 5 ; IY eIl".
Why should arithmetique now be I quartorolo 2 2
Accounted vulgar? when we see
It thus ennobled by thee. 611 I 10 6 3
Alas! that title's now too poore,
Since that thv cv.phers stand for more g º [º & Up
Than all i. j. did before. The third species of practice, denominated Venetian, Practica
differs from the preceding only in the mode of solving Venetiana.
Melitides, who ne're could thrive questions, where the price is fixed at so much per hum-
*...". *::::::::. hive. dred or thousand, whether pounds or of other denomi-
Fvery student in Arithmetic may join in the poet's last wish, and in nations. The following are examples: *
thinking that the author's recompence would be well merited. If a hundred pounds of sugar of Madeira º: 9
Nay more, dear friend, free from control ducats, 18 grossi, what is the price of 3855 pounds :
I'll chant thy praise from pole to pole,
Doe but once make our fractions whole.
The reader may be somewhat surprised to be informed, that the work * The staro, like all other primary units of weights and measures,
which is thus blazoned into notice is at least a very trifling, if not a varied extremely in different cities of Italy; at Venice it weighed
very vulgar production. - 132, at Parma 110, and at Piacenza 80 lire ; and 2 stara of Mantua
* Clavis Mathematica, p. 17. were equal to 5; stara of Bergamo.
454 A. R. I T H M E T I C.
Arithmetic. 3855 - r and the following example will show that the prawis, History.
\-N-' 9 ducats, 18 grossiper hundred. which was known to or used by him, was Florentine.
&=ºmm Ulna venditur pro 15 grossis et 10 denariolis et uno
At 9 ducats 34.695 obulo; quanti vendumtur 48 ulnae de panno eodem.
12 grossi. . 1927 12 r
6 grossi. . 963 18 - ulna. fl. gro. den. ulnae.
*E=º-ºsmºs I 0 15 10; 48
Ducats. . 375.186 6 7 6 locus distractarum
Grossi. . 20/70 7 3 particularum.
Piccoli. . 22|40 16 l 1%
Answer, 375 ducats, 20 grossi, 22 piccoli. I6 -
If one thousand pounds of Spanish wool cost 27 ; : locus productorum.
ducats, 16 grossi, what is the price of 9756 pounds 2 12
9756 6
27 ducats, 16 grossi.
Facit 36 flor. 6 gro. summa producto-
68292 7"U/72. º
1951.2 In order to understand this scheme it must be ob-
At 27 ducats. . 263412 served, that 21 grossen make a florin, and, therefore,
12 grossi . . 4878 7 x 48 grossen is equal to 16 florins, the reason which
4 grossi . . 1626 -- suggested the distribution of the 15 grossen into 7, 7
mº-º-mmºmºsº and 1. The author has given six different dispositions
Ducats . . 269|916 - of the same example, to show the variety of ways in
24 which such solutions may be presented.
Grossi. . . . 21984 (192.) The great convenience of these rules for per-Rules of
- 32 forming the calculations which were continually occurring, Practice in
both in trade and commerce, made them a favourite study English
Piccoli . . 31|488 with practical arithmeticians, and they consequently ... author.
Answer, 269 ducats, 21 grossi, 31 piccoli. te sumed from time to time an increased neatness and dis-
, 41 groSS1, 51 p tinctness of form. Stevinus, indeed, speaks of them with
Practica The Florentine practice differed in no essential point some contempt, as forming “a vulgar compendium of the
Firentina. from the Venetian, adopting merely a somewhat diffe- rule of three, sufficiently commodious in countries where
rent and more artificial distribution of the aliquot parts. they reckon by livres, sous, and deniers;” but such is his
The following is an example: usual manner. Amongst the additions made to Recorde's
If one hundred pounds of mastic cost 25 lire, 12 soldi, Arithmetic by John Mellis of short Southwarke, School-
what is the cost of 18 pounds, 4} ounces? master, in 1586, is one on certain “briefe rules, called
es e g & rules of practise, of rare, pleasant and commodious effect,
25 º º fººl 18 lire, 4% oncie. j into a briefer method than hath hitherto been
I I 4 || 2 published,” where they are exhibited under a very simple
and complete form. Later works gave them still greater
360 0 0 compactness and brevity, such as that of Wingrave, as
90 0 0 edited by Kersey; and in Cocker's Arithmetic,f and
9 () 0 others printed towards the end of the XVIIth century,
I 16 0. they assumed the form which they retain at present. q
8 0 0 (193.) Amongst the questions proposed by Tartaglia, º !
0 8 0 in illustration of the rules of different species of practice, * ...”
0 2 8 are many, which he terms ragioni doppie, treppie, qua- " " `
1 1 4
Lire . . 470 8 0 - * La Pratique d'Arithmetique, p. 721.
& amsºmº f The work of Edward Cocker, late practitioner in the arts of writ-
Soldi 14 08 ing, arithmetic, and engraving, whose name for near a century en-
Pizzoli 0| +*For joyed a species of proverbial celebrity, as synonimous with the
ſº tº e & tº science of numbers and accounts, was published after his death in
In this case, the 25 lire 12 soldi are divided by 12, to iéz7, by john Hawkins, a brother writing-master in southwark, it
get the value of an ounce, and by 2, to get the value of is entitled, and very justly, “a plain and familiar method, suitable to
# an ounce. The rest of the process requires no expla- the meanest capacity, for the full understanding of that incomparable
tion art.” In his preface, he speaks with great contempt of many of the
Praxis Ila, igi Prairis ill b Italisad devolut pretended, arithmeticians of his time : “For you, the pretended
lica of (l 1.) raris uta quam go ſtatisaa nos devolutam esse numerists,” says he, “of this vapouring age, who are more disinge-
º O arbitramur, est ingeniosa quaedam inventio, quarti ter– niously witty to propound unnecessary questions, than ingeniously
*** mini regula de Tri, ez tribus terminis, mediante distrac- judicious to resolve such as are necessary; for you was this boºk
s - e ſº º, composed, if you will deny yourselves so much as to invert the
tione varia eorundum terminorum, distractarum particu streams of your ingenuity, and by studiously conferring with the
larum proportionatione atque denominationum vulgar notes, names, orders, progress, species, properties, proportions,
Tium translatione. This is the language of Stifelius,” powers, affections, and applications of numbers, delivered herein,
s - - : become such artists, indeed, as you now only seem to be.” It may be
: . worth while observing, that this modest and useful book is not honoured
* Arithmetica Integra, p. 83. . with poetical recommendations.
A R IT H M ET I C.
455
Arithmetic, druppie, &c.; questions, in short, which should properly sugar loaves, whose nett weight is 7200 lire; I pay for . History.
s - – A
Tarra.
Tret and
cloff.
Messetaria.
Other termis,
Question
require, two, three, four, or more proportions for their
solution. *#hey are chiefly those in which deductions,
whether per centage or otherwise, are to be made from
the gross weight of the articles bought or sold ; or
a tax to be deducted from the gross produce of the
sale. It may be proper to explain the terms made use
of, which have had their origin in the local or general
customs of commerce and trade, or from taxes imposed
for the particular benefit of the state or city where the
transactions took place.
Tarra, the original of our word tare, is derived from
the verb tarare, to abate or diminish, was an allowance
or deduction of so much per cent. or otherwise, upon the
gross weight of the goods sold, to make up for package,
dust, waste, or other losses; it varied, according to the
nature of the merchandise, from 2 to 10 per cent. ; the
weight netto de tarra, or clear from tare, becomes our
mett or meat weight. The term sottile, or subtle, is used
in the same sense.
The terms tret and cloff are of unknown, but probably
of Dutch, origin. The first is an allowance of 4 pound
in every 104 sold, for waste; tare, with the English
merchants, being the variable allowance for boxes,
package, &c. The term cloff has a peculiar as well as
a general sense ; in one case it denotes an allowance
of 2 pound to the citizens of London on every draught
of certain descriptions of goods which exceeded 3 cwt. ;
whilst in general it denotes a small allowance made on
goods sold in gross, to make up for deficiencies in
weight when they are sold in retail.
Messetaria, a Venetian term, sometimes expressed
by the more general word datio, or dazio, tar or im-
post, was a double tax, varying generally from 1 to 3
per cent., which was paid both by the buyer and seller
of different species of goods. It was a law of Venice,
that when one of the parties was a terrero, or inhabi-
tant of terra firma, the other might retain this tax
in his hands, as he was responsible to the datiari for
its payment. e
The following question of Lucas de Burgo introduces
other terms to our notice :
El migliaro de ramo rosso val ducati 96: el migliaro
del stagno in verga val ducate 90: el migliaro del
piombo impiastre val ducati 24: che warranno lire 9876
de bronzo, che tengano per migliaro 250 de stagno e di
rame 643 : abbattendo dono del stagno 4 per cento : e
tara del rame 10 per 1000: e callo del piombo 12 per
1000: e di gabella pesa, senseria, e bastagi in tutto 6
per cento.
Of these terms, dono may be interpreted a gift or
voluntary deduction, where no waste took place ; and
callo, a descent or allowance, for diminution of bulk and
weight which took place in the process of mixture.
Tartaglia has confined the application of this term to
denote the waste, in bulk and weight, which took place
in new oil, as distinguished from old. Of the other
terms which require interpreting, pesa was the allowance
for weighing ; and sensaria or senseria, was the fee to
the sensale, or agent, by whose means the bargain was
made.
(194.) The same author, in speaking di viaggiis mer-
propºsed by catoriis, has given in an example an account of the various
De Bu rgo,
“I buy,” says he, “for 1440 ducats at Venice 2400
charges to which a mercantile adventure was subject,
which is not without its interest, and particularly in
connection with the subject of our present discussion.
sensaria 2 per cent., to the weighers and porters (bastagi)
on the whole, 2 ducats; I afterwards spend in bowes,
cords, canvass, and in fees to the ordinary packers (le-
gatori) on the whole, 8 ducats ; for messetaria on the
first amount, 1 ducat per cent.; afterwards for duty and
taa (datio e gabella) at the office of exports, 3 ducats
per cent. ; for writing directions on the boxes and
booking their passage, 1 ducat; for the bark to Rimini,
13 ducats; in compliments to the captains, and in drink
for the crews of armed barks on several occasions, 2
ducats; in expenses for provisions for myself and ser-
vant for one month, 6 ducats; for expenses for several
short journeys or trajects over land here and there, for
barbers, for washing of linen and of boots, for myself
and servant, l ducat ; upon my arrival at Rimini, I pay
to the captain of the port for port dues, in the money of
that city, 3 lire ; for porters, disembarkation on land,
and carriage to the magazine (magazen,) 5 lire ; as a
tax upon entrance, 4 soldi per load (callo,) which are in
number 32, such being the custom ; per fontecaggia a
malatesta di marsilio, soldi 4 per callo; upon my arrival at
the fair, I find that 140 lire of weight are there equivalent
to 100 at Venice, and that 4 lire of their silver coinage
are equal to a ducat of gold. I ask, therefore, at how
much I must sell a hundred lire Rimini, in order
that I may gain 10 per cent. upon my whole adventure,
and what is the sum which I must receive in Venetian
money?” -
The author may well say, that a merchant ought to
have his wits at home (cervello a casa) who undertakes
all the reductions and calculations which an adventure
of this kind would require.
(195.) Particular species of goods appear to have been
liable to other imposts. Pepper, which was sold by the
cargo, (a weight of 400 lire,) as well as some other
articles, paid a small tax for the support of a hospital
for the poor. Cloves, (garofali,) which constituted a
most important article of a traffic at that period con-
fined to Venice, and which were of great value, (16
grossi, or 3 of a ducat per lira,) were subject to some
peculiar regulations, which appear to have been very
embarrassing to Venetian arithmeticians; they were
usually mixed up with fusti, or stalks, of much less
value than the seed itself, and 3 sazzi, out of the 72 which
every pound contained, were allowed by law. As the
quantity of them was commonly much greater, it became
a question of some complexity to determine the proper
deduction to be made. Tartaglia has mentioned, on
more than one occasion, the error which existed in the
common process for this purpose.
(196.) The IXth Book of the 1st part of the work of
Tartaglia is chiefly occupied with ordinary questions on
loss and gain per cent. ; and on the conversion of the
coins, weights, and measures, of one state or city into
those corresponding to them in another, either directly
or with certain limitations as to gain per cent. They
contain nothing which is worthy of any particular
notice.
(197.) The rule of three alla riversa, or as it is called,
in the older English writers on Arithmetic, the backer
rule of three, consists in making the third term the divi-
sor, which under the direct rule was the multiplier, and
conversely. In one case, if the second term be doubled,
the result, which is of the same species, is doubled; in
this case, if the second term be doubled, the result is
halved. This is a test of very easy application, and
\-y-
Particular
imposts on
pepper,
cloves, &c.,
Questions
on loss and
gain.
The inverse
rule of three
456
A R I T H M ET I C.
Arithmetic.
will at once ascertain in any particular case which of
S–S 2— the two rules must be used.
In what
cases used.
Rule of five
{e1 mS.
Of this kind are all questions where a rectangular
space is given, and the length and breadth are variable,
or those in which the number of measures in a given
heap of corn, or of any other quantity, is required ;
or where the time is required, in which different sums
will produce a given interest; or questions relating to
the assize of bread,” where the weight obtained for a
given sum is required, the price of the same whole
being variable.
(198.) There are different methods of solving questions
included under the rule of five or more terms, whether
by successive statements, or by the combination of all
the conditions into one. The following example is
given by Tartaglia :
If 9 porters drink in 8 days 12 casks of wine, how
many casks will serve 24 porters for 30 days 2
Let us first suppose the time the same, and state the
question as follows:
If 9 porters drink 12 casks of wine in 8 days, how
many casks will serve 24 porters for the same time 2
The answer is 32; and the second question will
stand as follows:
If 24 porters drink 32 casks of wine in 8 days,
how many casks will serve them for 30 days 2
The answer is 120; which is, likewise, clearly that
corresponding to the first question proposed.
The general principle of the other rules which are
made use of by Tartaglia, may be stated as follows:
The quantity mentioned once is of the same nature
with that which is sought, and is put in the second
place. Of the other pairs of quantities, two are put
in the first and last places, and two in the third and
fourth, in the same order in which they occur in the
question. Multiply the fifth, the fourth, and the second
together, and divide their product by the product of the
first and third : the quotient is the quantity required.
In those cases, where the inverse rule would apply
to the simple statement of three terms, omitting all the
other quantities mentioned twice, the second of the
quantities must become a factor of the divisor, and the
other a factor of the dividend.
Tartaglia usually puts the quantity mentioned once
only in the last place but one, instead of the second.
The rule may be very easily accommodated to suit this
change in the arrangement. -
The statement of the question proposed above, ac-
cording to the principle of this rule, is as follows:
9 | 2
8 30 24
Divisor 9 × 8. Dividend 12 x 30 x 24.
864
Quotient = 120.
* In all countries, the price of bread has been under the control f
the magistrates, as it was always considered necessary to protect the
people against the combinations or impositions of the bakers; for this
purpose, however, it was necessary that the persons who formed the
tariffa, or table of prices, should be good arithmeticians, as Tartag-
lia has shown that all his predecessors, including Lucas de Burgo, were
mistaken in making the price of the soldo or penny loaf vary in-
versely as the price of the staro of wheat, without taking into consi-
deration the constant expense which attended the process of making
the bread: his attention was called to this subject, when requested by
the magistrates of Verona to extend the tariff, in order that it might
meet the high prices occasioned during a period of great scarcity.
We shall subjoin a few of the most interesting ques- . History.
tions which Tartaglia has given in illustration of this X-'
rule. $ºt. Examples.
Twenty braccia of Brescia are equal to 24 braccia of
Mantua, and 28 of Mantua to 30 of Rimini; what num-
ber of braccia of Brescia corresponds to 39 of Rimini?
Rimini. Mantua. Mantua. Brescia. Rimini.
3O. 8 — 26 20 — 39
21840 780 : Answer, 28.

The lira of Pisa is equivalent to 11 oncie of that of
Florence, and the lira of Florence to 13 oncie of that
of Perugia; what is the relation between the lira of
Pisa and that of Perugia 2
143
Perugia. Fiorenza. Fiorenza. Pisa. Perugia.
— 12 12
* – º – º –
1728
13 —
Answer: they are equal to each other.
Eight soldi of Venice are equal to 13 of Ferrara, and
15 of Ferrara are equal to 9 of Bologna, and 12 of
Bologna are equal to 16 of Pisa, and 24 of Pisa are
equal to 32 of Genoa; it is required to find what num-
ber of Venetian soldi correspond to 300 of Genoa.
Gen. Ven. Pis. Pis. Bol. Bol. Fer. Fer. Gen.
32—3–24 —16 — 12 — 9 — 15 — 13 — 300
> S-2T→ ~~~~
IO368000 59904 Answer, 1734's.
Six eggs are worth 10 danari, and 12 damari are
worth 4 thrushes, and 5 thrushes are worth 3 quails,
and 8 quails are worth 4 pigeons, and 9 pigeons are
worth 2 capons, and 6 capons are worth a staro of
wheat : how many eggs are worth 4 stara of wheat?
960.
_--~~~~~~TS
1–6–10–12–4–3–3–8–4–9–3–6–3
622080. , Answer, 648.
Other questions cannot be resolved by one statement;
of this kind are the two following:
Ten excavators, (guastatori,) such as are usually em-
ployed in digging iron ore, can dig out 12 carra or
loads of earth in 16 hours, whilst 12 other common exca-
vators, less powerful than the former, dig out only 9
loads of earth in 15 hours; it is required to find in
what time they will conjointly dig out 100 loads of
earth P -
The first question is, what quantity would the second
set excavate in 16 hours, the time in which the first are
engaged, which will be found to be 9: loads; the ques-

tion is then reduced to the following:
If 22 excavators dig out 213 loads in 16 hours, in
what time will they dig out 100?
214 – 16 — 100.
A gentleman going to the wars pays 360 ducats for
12 carrette, or waggons, with a pair of oxen each, whilst
5 other waggons without oxen cost him 40; it is re-
A R IT H M ET I C.
457
Three soldiers, or adventurers, form a company for History.
Arithmetic. quired to find the sum which he must pay for 60 oxen
S-y—’ alone.
Different
species of
Italian part-
nerships and
compagnies.
Examples.
Principle
pf solution.
The first question to be solved is this, if 5
waggons gost 40 ducats, what will 24 cost? The result,
which is '96, being subtracted from 360, will give the
charge for the oxen, when the remainder of the question
is easily solved. -
(199.) There is nothing more remarkable in the ancient
commercial system of Italy, than the number, variety,
and, in some cases, complexity of their compagnies or
partnerships. The associations of different individuals
for conducting mercantile concerns, which are too ex-
tensive for the superintendence of one person, or which
require a larger capital than one individual can furnish,
must take place in all commercial countries; but in
Italy, others appear to have been formed for purposes
merely temporary, for a particular adventure, with two
or three persons, who contributed money, goods, or
labour, sometimes one, and sometimes the other, and who
divided the profits in the proportion of the capital ad-
vanced, the value of the goods furnished, or the wages
of the labour employed in conducting the concern.
Even in the most common affairs of life, they appear
to have delighted in such associations; and the partner-
ships which were formed between landlord and farmer
throughout Italy, have given a very peculiar character,
not only to their relation to each other, but likewise to
the whole of their agricultural system.
(200.) We shall mention a few only of the vast variety
of questions on this subject which are given by Lucas
de Burgo and Tartaglia, with such remarks as they
may appear to require.
Three persons form a company, the first of whom
contributes 235 ducats, the second 430, and the third
520; and at the end of a certain time, they find that
their capital and gain amount to 1732 ducats; what
portion belongs to each 2
This, and all similar questions are solved upon the
common principle, that the sum of the capitals con-
tributed by A, B, C, is to A's capital as the amount of
capital and gain together is to the sum due to A.
A person has four creditors, to the first of whom he
owes 624 ducats, to the second 546, to the third 492,
and to the fourth 368: he fails and runs away, and his
creditors find the amount of his whole property to be
only 830 ducats; in what portions ought it to be
divided amongst them 2 The principle of this question
is the same as the last. ...”
Three persons form a company, the first of whom
contributes 300 fiorini, the second 600 canne of cloth,
and the third 1200 lire of saffron; they gain 900 ſtorini,
of which the first receives 60, the second 360, and the
third 380: what was the value of the canna of cloth
and of the lira of saffrom ? -
The fiorino was the primary coin of Florence, and
under the name of florin became the general coin of
the south of Germany; a circumstance easily accounted
for, by the political connection between them.
Three companions are in a ship, one of whom has a
butt of malvasia, which holds 36 barrels, (barile,) ano-
ther one of Greek wine, which holds 24, and the third
one of wine of Romania, which holds 40. By a violent
movement of the ship the butts are upset, and the wine
is spilt in the flold. The butts are afterwards replaced
and filled with the mixture: what portion of each wine
do they severally hold? .
This question is solved on the general principle of
the regula societatis.
WOL. I.
the division of the spoil they shall gain in the wars; \*
the first, being more practised than the second, says
that he shall claim twice as much as the second, and
the second, being more expert than the third, claims
three times as much as the third, who submits to the
terms; they gain 120 ducats; what is the share of each 2
In the solution of this and similar questions, it is
convenient to take numbers, such as 6, 3, 1, in the pro-
portion of the respective shares.
In many other examples de rebus militaribus which Aventurieri.
are given both by Lucas de Burgo and Tartaglia, the
term soldier and aventuriere are used as synonimous;
the fact is, that in that age a national army was nearly
unknown in Italy, the wars being chiefly carried on by
aventurieri, who hired themselves to any party for a
limited service. Tartaglia had good reason to know
how much the horrors of war were increased, when
carried on by men who looked for their reward in the
plunder which arose from the sacking of towns, and the
wasting of a country.
Four persons, a gentleman, an artisan, a barber, and
a friar, make a pilgrimage in company, and spend 60
ducats ; the barber agrees to pay 4 times as much as
the friar, and 4 soldi more, the artisan 3 times as much
as the barber, and 16 soldi more, and the gentleman
twice as much as the artisan, and 10 soldi more ; what.
portion was paid by each 2 -
The ducats are converted into soldi, and the sum of
4, 28, and 66, are subtracted from 1200, leaving 1102;
this is divided, as in the last example, in the propor-
tion of the numbers 1, 4, 12, and 24; after which the
actual portions are easily assigned.
A man lying on his death-bed bequeathed his goods, Acelebrated
which were worth 1200 ducats, in this sort : because his . of a
Wlli,
wife was great with child, and lie yet uncertain whether
the child were a male or female, he made his bequest
conditionally, that if his wife bare a daughter, then
should his wife have two-thirds of his goods, and his
daughter one-third; but if she were delivered of a son,
then should his wife have one-third, and his son two-
thirds. Now it chanced her to bring forth both a son
and a daughter, the question is, how shall they part the
goods agreeably to the testator's will ? \
We have given Recorde’s statement, with a few
alterations, of a question which has become unusually
celebrated from the time of Lucas de Burgo. The
scholar in the dialogue is made to remark, that if some
cunning lawyers had this matter in scanning, they
would determine the testament to be void, as being in-
sufficient. The master, however, “proceeds to try the
work, not by the force of law, but by proportion geo-
metrical, seeing the testator did minde to provide for
each of them;” and as the intention was, that the son
should have double of the mother, and the mother
double of the daughter, the property must be distributed
amongst them in the order of the numbers 4, 2, and 1.
Tartaglia has proposed many other similar questions Other cases.
where the intention can be only inferred. Amongst
others, those in which a testator, from ignorance of
the nature of fractions, directs a distribution of his
property in fractional parts, the sum of which is greater
or less than unity; thus, one gives # of his property to
his son, 4 to his nephew, and 4 to his niece. In such
cases the property must be divided in the proportion of
such fractions. - gº
A person furnishes a shop with different goods by
3 O --
458
A. R. I T H M E T I C.
Arithmetic. means of a capital of 300 ducats, on the 1st of January,
^-vº-' 1549; six months afterwards, one of his friends comes
Fºllowship and offers, upon condition of being taken into partner-
with time. ship, to add 500 ducats to the capital, the division of
of sweating industry; this man employs his money in , Hºy.
merchandise, that man in trade ; and amongst other
$ g & ... Ohserva-
laudable species of industry which we every day wit- ...;
ness, we find some men who provide the means of life Lucas de
by the aid of brute animals; and this not by violence, Burgo.
the gain to be made in the conjoint proportion of the
capital and time; at the end of December, 1550, they
find the whole gain 260 ducats, what portion is due to
each 2 -
This is an example of fellowship with time.
Many other examples of compagnies are given by
Tartaglia, which, properly speaking, require the aid
of algebra for their solution ; of this kind is the fol-
lowing :
Two persons form a company, on condition that the
first should contribute 3000 lire, and the second 800
with his personal services, and that the first should
receive #, and the second # of the whole gain; the
first, however, adds 400 fiorini to his first capital, and
in consequence receives # of the gain, whilst the second
gets only ; ; what is the relative value of the florin and
the lira 2
or soccide di bestiami, which were so common, and
which lead to so many very complicated questions, that
they always formed the subject of a distinct chapter in
Italian books of Arithmetic. They arose from the
poverty of the farmers, who would not stock their farms
from their own funds, and, in many cases, could not
even buy the corn which was necessary for seed; the
consequence was, that the landlords generally, and in
some cases other persons, provided the whole or the
greatest part of the stock, and entered into an engage-
ment with the farmer to divide with him its whole
produce at the end of 3, 4, or 5 years, or to divide in
certain proportions with him the profits which occurred
in the mean time, and the whole stock which remained
at the conclusion of the sozzido. -
“Whoever wishes to support himself in this world
of misery, must govern and guide his life in the path
of barratti, or barters.
provided it be exercised in the proper mode and in
charity, according to the injunctions of the holy Scrip-
tures, which say In caritate sudate et radicate.” Such
is the preface with which Lucas de Burgo introduces
the notice of these associations of the rich and the
poor, which he says were peculiarly liable to imposition
and fraud, and that, in consequence, it was highly dam-
gerous to extend such agreements beyond 3, 4, or, at
most, 5 years; and, in every case, he recommends
them to be formed under the inspection and control
of the bishop of the diocese, since con tale consiglio sa-
lutifero raro si erra; though we might very reasonably
doubt, whether the prelates of Italy, or of any other
country, either in that age or the present,” were the
persons best calculated to regulate the terms, or to
enforce the fulfilment of such bargains. We will sub-
join a few examples of questions, which frequently
Alieged Lucas de Burgo solves this question on the principle,
error of that the consideration for personal services should be arose out of the formation of such compagnies.
# * * the same in both cases, or, in other words, that they A person gives a shepherd in sozzido 720 sheep, to Examples.
Durgo. & º e ge *
should be considered as equivalent to the same capital; keep them and their produce for 5 years, and at the
and that, consequently, the value of the florin, as deter- end of that period to divide equally with him the profit
mined from the question, should be 1; lire. Tartaglia and the capital; at the end of 3 years and 8 months
considers this principle as erroneous, and contrary to the shepherd dies, and his wife, who has no confidential
the spirit of the agreement, by which the value of the person to manage the concern, (her son not being of
personal services should increase in proportion to the sufficient age,) is compelled, with the consent of the
increase of the joint capital; if the question be solved principal, to terminate the sozzido : the number of
with this view of its meaning, the result would give the sheep is found to be 1060 ; what number will each
..florin equal to 3% lire. party receive?
Many other questions of a similar nature had been If the contract had been completed, the widow
solved by Lucas de Burgo, Piero Borgio of Venice, would have claimed 530; the number now due to her
and particularly hy Giovanni Sfortunati of Sienna, upon will be to 530 in the ratio of 33 to 5.
the first principle, and the error whether real or alleged, A person gives in Sozzido 24 cows, and the herdsman
is pointed out with great detail by Tartaglia; he seems, adds 6 to the number, to keep them for five years, and
indeed, to have experienced a peculiar satisfaction in then to divide the capital and profits equally ; at the
finding out the faults of his predecessors, and he rarely end of 3 years and 4 months they agree to terminate
omits an occasion of doing so, particularly in the case their contract, when they find 80 head of cattle; what
of his predecessor, Pacioli, who attempted the solution portion belongs to each 2
of many questions upon erroneous principles, or by A citizen gives in sozzido 18 sheep to a shepherd,
methods which were insufficient for the purpose. The who agrees to add 6 to their number, upon condition of
Freque: phrase, mai falla, which he so often uses with refe- dividing the whole equally at the end of four years; the
*...e rence to his processes, must be admitted with extreme contract being made, the shepherd returns home, and
#º. caution, being most frequently used when he is most finds that the wolves have eaten two of his sheep, and
liable to be deceived. he has, therefore, only 4 to add to the number which
soccide de (201.) There was another class of compagnies, termed he receives from the citizen; at the end of three years
pestiami. in the provincial language of the north of Italy, sozzidi, they agree to divide the Sozzido, and find that they
have 66 sheep ; what number must each receive 2
(202.) The Italians distinguished three distinct species Different
The first, simple, where goods species of
were exchanged against each other at their ready money,
or barter price; the second, compound, where the ex-
change was partly in goods and partly in ready money;
and the third, barter with time, where the barter price
is affected by the time at which the payments, whether
real or imaginary, are to be made ; in this, as well as
in every other department of their commerce, they
appear to have been fond of engagements involving the
* In modern Tuscany, the landlord furnishes stock, seed, and im-
plements of husbandry, and divides the produce equally with the
tenant: the case is somewhat different in Lombardy, where the farms
are large, and where, in consequence, the agricultural population
is in the possession of much greater wealth. -
A R IT H M ET I C. 459
Arithmetic, most complex relations, trusting to their own dexterity
S-N-' in the management of such bargains, and relying upon
Frequency
af .
Examples.
Interest
simple and
compound.
the skill of their professional Arithmeticians for the
resolution of questions, to which the majority of them
must have been altogether unequal.
(203.) So frequent were the frauds which occurred in
these transactions, either in the articles not correspond-
ing tu L11eir samples, or in fixing the durierence 1il ulie
barter price and the price a damari contadi, or for ready
money, or between the price for ready money and for
time, that it became a proverbial saying, that one of
the parties in a barratto was imbratto, cheated, or,
more literally, dirtied. It was the custom, also, accord-
ing to Lucas de Burgo, when the sensaro, or agent,
showed bad articles for barter, to ask him if he gave a
dowry with them, in allusion, says he, to the manner
in which marriages are contracted in those days; for
whilst beautiful and accomplished ladies are taken
from their fathers houses almost pennyless, the ugly and
ill-favoured are recommended by large dowries, a qua-
lity which never fails to procure a husband in this age
of avarice, in defiance of the proverb, which says, Ne
per bo me per vacca non taglia donna matta, la robba
va e vene e chi a la moglia matta se la tene.
(204.) We will add a few examples of the different
species of barter, which frequently lead to questions
of a very difficult and embarrassing nature.
Two persons wish to barter, the one wax, which sells
at 8% ducats per hundred pounds, whilst the other has
wool, of which the same quantity sells for 39% ducats;
how much wax must be given for 756 pounds of wool P
Two persons barter ginger and soap ; the hundred
pounds weight of the first is worth 16 ducats for ready
money, and 18 for barter; the second is worth 22
ducats for the thousand pounds, for ready money; if
the first pays for one-half of what he gets in ready
money, what must he give in money and ginger for
7890 pounds of soap, so that the terms of the barter
may be equal on both sides 2
Two persons barter, the one wool, the other pepper
and ginger; the hundred weight of pepper is estimated
for ready money at 30 ducats, and for barter at 35 ;
the hundred weight of ginger is estimated at 27 ducats
for ready money, and for barter at 33 ; the hundred
weight of wool is worth 10 ducats : at what price must
the wool be estimated at barter, to receive an equal
quantity of pepper and of ginger, and to gain 10 per
cent. upon the capital P
A merchant sells to another a quantity of scarlet
cloth at 6 ducats the braccio, if paid for at the end of 8
months, but the price for ready money is only 4% ducats;
afterwards the first buys of the second a quantity of
ginger for 15 ducats the hundred weight, payable in 10
months; the excess of the time above the ready money
price, in proportion to the time, being the same as in
the case of the cloth ; what is the ready money price
of the ginger?
(205.) The importance of the knowledge of the prin-
ciples of simple and compound interest, discount, annu-
ities, &c., with the proper rules for their calculation in
mercantile and other transactions, is so great, that we
may naturally expect to find the discussion of them oc-
cupy a considerable portion of all books of Arithmetic.
The rules for such calculations, however, are founded
upon algebraical formulae, and for the most part, in-
volve relations of quantities much too complicated for
any merely arithmetical investigation; under such cir-
cumstances, the questions proposed rarely extend be-, History-
yond the more common cases, such as simple interest,
discount, the ordinary cases of compound interest and
discount, and the determination of the value of tempo-
rary annuities. - -
(206.) When the excessive interest which was charged Usury.cº-
for the use of money, in those countries where commerce ºn-
lava --> --~~~lated canital. is considered, it is not very É * >
surprising that the popular indignation and prejudice
should be directed against usurers. Under the Mosaic
Law, this prejudice received a much higher sanction, and
domestic usury was not merely discouraged, but forbidden:
and in modern Europe, it was long before the same law,
which was obligatory upon Jews towards Jews, ceased
to be considered as not extending to the members of the
new covenant; at all events, religious feelings and the
denunciations of the church came partially in aid of those
which were natural and hereditary. The practice of
usury, indeed, during the middle ages was so universally
odious, that it was confined to that race of men, who by a
singular revolution had succeeded to the exclusive exer-
cise of a traffic which had been forbidden to their fore-
fathers; nor did this feeling cease to exist even in coun-
tries and cities where the conveniences of an extensive
commerce rendered it, in some measure, necessary. It
was, of course, recognised in the transactions of mer-
chants with each other; and money, time, and the con-
sideration for the delayed and anticipated payment of
money, formed an important element in all purchases
and sales: but when money was directly borrowed, not
in the course of trade, it was commonly from a Jew;
and our own Shakspeare has correctly represented the
feelings with which such transactions were regarded:
nay, even as late as in 1567, Cataneo, an arithmetican
who resided in Venice, prefaces the chapter in his work
which relates to interest and discount with the following
terms: Se quelli che alla poltronescha usura si dammo di
tal mestiero mom si vergognano, mamco mi debbo vergo-
gnare io d'insignare quanto debbi pagare quel pover
disperato, che a tali diabolichi patti s'obbliga.
(207.) Interest in Venice at the beginning of the Interest of
XVIth century varied from 5 to 12 percent per annum : . i.
in commercial transactions a much higher interest was ſh;"igi,
calculated upon, or rather a much greater considera-century.
tion paid in the difference which existed between the
ready money and time price of goods; but when money
was lent or borrowed, upon good security, whether
from Jew or Christian, it rarely exceeded the last sum
which we have mentioned; it appears to have been
estimated in very different ways; sometimes at so much
per cent. by the year or the month, sometimes at so
many damari, on each lira per mensem, and sometimes at
so many on the 100 lire per diem : it is evident that these
different customs must have materially increased the
complexity of the rules for the calculation of interest.
(208.) Simple interest is that in which no interest Definitions
springs from the interest, and compound interest, or me- of .
rito a capo d'anno,” that in which the interestis reckoned i in-
upon the arrears of interest; it was the second species terest.
only which was properly called usura, and was rarely
practised in the transactions of merchants with each
other. Stevinus terms compound interest, interest
proufttable, or celuy qu'on ajouste au capital, whilst
the corresponding discount is termed interest dom-
mageable, or celuy qu'on soubstrait du capital.
asrººm-m-m-
* Meritas: simplicemente quando da! merite “terito non wasce.
3 O 2
466)
A R I T H M E T I C.
Arithmetic. (209.) The solution of questions of simple interest and
g ° discount readily reduce themselves to the ordinary cases
Piºnt, of the rule of the three ; and there is nothing in the
'days fixed
... methods which are used for this purpose by Tartaglia,
mencement or his predecessors, which is particularly worthy of
of the mer- notice. In calculating the interest of a sum, from one day
cantile year. to another, whether of the same or different years, the de-
termination of the number of months or davs in the **-
val was in some degree embarrassing ; and Tartaglia is
proud of a rule which he has given for this purpose.
In passing from one city of Italy to another, an additional
source of embarrassment presented itself, in the diffe-
rent days on which the year was supposed to commence :
being reckoned at Venice from the 1st of March; at
Florence, from the annunciation of the Virgin; and in
most other cities of Italy, in obedience to the orders of
the church, from Christmas day.
(210.) In a running account between two merchants,
involving sums borrowed and paid at different times, upon
which simple interest, for the most part 12 per cent., was
allowed, it was important on particular days to balance
their accounts, a process which was denominated saldare
ºuma ragiome: such adjustments appear to have been very
frequently repeated, in perfect consistency with those
habits of formal punctuality for which the Italian mer-
chants were so remarkable. in such cases, the interest
upon the several sums, on the debtor and creditor side
of each account, was calculated up to the given day,
and the difference of the sum on each side, if any re-
mained, was passed in one sum to the proper side of
the ledger. Another process, also of very frequent
occurrence, was to calculate the equated time of pay-
ment of sums due at different periods, a process called
recare (li pagamenti) a un dº. It consisted in mul-
tiplying each sum by the time before it was, due, and
dividing their sum by the sum of the several payments;
this rule, which is the one commonly used at this time,
confounds interest with discount, and excludes, of
course, all consideration of compound interest. Tar-
taglia was fully aware, that the principle of this rule
was erroneous; but the principles of algebra were in
that age too imperfect to give the correct solution, or,
at all events, to give the correct interpretation to it.
(211.) Tartaglia has given some examples of cases,
chiefly of annuities, which were proposed to him pro-
fessionally ; the first, which is the following, was pro-
posed by a Jew at Venice, on the 14th of April, 1550.
A person owes me 450 ducats, payable by 9 ducats a
month for 50 months, and wishes to pay the whole at
once to another person, who undertakes to discharge
the debt; what sum must he pay, supposing interest be
allowed at the rate of 9% per cent?
He finds the equated time of payment by the ordi-
nary rule, which is 25% months, and then discounts
450 ducats, payable at the expiration of that time: the
answer is 374 ducats, 9 grossi, and 30+4++ piccoli.
A certain maestro da Barri proposed the following
question :
I lend a certain university 2814 ducats, on condition
of receiving an annuity of 618 ducats for 9 years; what
interest do I gain upon my money, the ducat being esti-
mated at 10 carlini, and the carlino at 10 grani ?
The answer, determined upon the same principles,
is 19 ducats, 5 carlini, and 3-rºr grani.
Nothing can be more unjust and erroneous in princi-
ple than this mode of calculating annuities, particularly
for a long term. Such questions were considered,
Balancing
accounts
and equa-
tion of pay-
ments.
Questions
on interest
proposed to
Tartaglia.
indeed, as peculiarly difficult and embarrassing; and History.
Tartaglia has mentioned several others of a similar \-y-
nature at the conclusion of his algebra. -
(212.) Tartaglia has noticed five methods of finding Rules for
the amount of a sum of money at compound interest. *g
Suppose the question to be, to find the amount of compound
interest.
L300. for 4 years at 10 per cent. a capo d’anno ; the
nisu Is Dy Une following Tour statements: 2 -
100 : 300 : : 110 : 330
100 : 330 : : 110 : 363
100 : 363 : : 110 : 399 fºr
100 399 fºr : 110 4394, ºr
The second merely replaces 100 and 110 by 10 and
11 in the proportion : the third, which is his own
method, multiplies 300 four times successively by 11,
and divides the last product by 10000: the fourth
consists in adding four successive tenths to the princi-
pal : the last, in calculating the amount for L100., and
then finding the amount for L300., or any other proposed
sum by a simple proportion. The last four methods
are obvious consequences of the first, and with the ex-
ception of the last, are not readily applicable, unless the
interest per cent. be an aliquot part of 100.
(213.) With the exception of discount at compound in- Discount
terest, (sconto a capo d’anno,) and its application to cor- ºld annu-
rect in part the conclusions respecting the values of an- ities.
nuities, there are few, if any, other questions of compound
interest which Tartaglia and his contemporaries can be
said to have resolved. A very natural difficulty arose A disputed
in the solution of questions of this kind ; “what is the case.
interest of 100 for 6 months, interest being reckoned
at the rate of 20 per cent. per annum.” Lucas de
Burgo, Giovanni Sfortunati, and others, made out that
this would be 10 ; in other words, they calculated that
simple interest only being allowed, it was a matter of
indifference into how many portions of time the whole
period was divided, whether into months or half years;
the conclusion, under such a view of the case, is cor-
rect, and merely proves the injustice of the very prin-
ciple of simple interest in all cases which are prospec-
tive at least, if not in those which are past.
(214.) Lucas de Burgo has an article entitled Del modo Tablesofin-
a sapere componere le tavole del merito; and he enlarges terºst used
upon the great utility of such tables for saving the " Italy.
trouble of calculation, and says, that they usually em-
braced a period of 20 years, commencing with 5 per
cent., the lowest interest which could be imagined to
be taken. This statement is sufficient to prove the
existence of such tables among the Italians, though
we are not aware of any work in which they are given.
The first compound interest tables with which we are Tables of
acquainted, are those which are given by Stevinus in Stevinus.
his Arithmetic ; they give the presentworth of 10000000
from one to thirty years, in 16 tables, the interest being
reckoned successively from 1 to 16 per cent., and in 8
other tables, where the interest is differently reckoned, *
according to the custom of Flanders, as one denier in
15, 16, 17, 18, 19, 20 (5 per cent.), 21, and 22.
There are two columns in each table, one giving
the present worths above mentioned, and the other
the values of annuities of 10000000 for the same
period, which are, therefore, the sums of the numbers
in the first column. The idea of the research of pro-
portional numbers, for the solution of questions of
interest and annuities, was suggested by the tables of
A R IT H M ET I c.
461
Arithmetic, sines, &c. commencing from a radius of 10000000, and
S-N-Z was one of many happy extensions of a common prin-
ciple, which were made by this singularly acute and
original author.
(215.) It is extremely difficult to establish from his-
torical documents the absolute antiquity of the use of
bills of exchange, or to ascertain the country where, or
* allier the places between which, they first circulated.
They are themselves documents of a very perishable
nature; and the only methods by which we are likely
to be able to trace their existence, must be from their
connection in some cases with historical transactions,
or from their appearance in legal records of disputes
which arose out of them ; of the first kind is the very
curious account given by Matthew Paris, which Mac-
pherson has quoted,” of the attempt made by the Pope,
in 1255, to depose Manfred, King of Sicily, and to
place upon his throne Edmund, the second son of our
Used in the Henry III., upon condition of being remunerated for
time of the expenses which he incurred; upon the faith of this
Bills of ex-
change,
** promise, large sums of money were advanced to the
Pope by merchants of Florence and Sienna, who were
repaid upon the failure of the enterprise by bills drawn,
at the suggestion of Henry himself, upon the prelates
of England, who were compelled to pay them with
interest, notwithstanding their protests, from apprehen-
sion of being subjected to a sentence of excommunica-
tion. -
T]ifferent (216.) This very remarkable transaction would appear
ºrigin; as to prove that the use of bills of exchange was perfectly
. ° well known to the Italian merchants of that age, though
it is probable that the date of their origin is much
earlier. Savary, in his Negociant Parfait, and in his
Dictionnaire du Commerce, says, that they were in-
vented by the Jews who were expelled from France at
different periods under Dagobert in 640, Philip the
Long in 1180, and Philip Augustus in 1316, and who
availed themselves of bills of exchange to withdraw
their property from France. At another period, also,
when the Gebelins were expelled by the Guelphs, some
Lombards took refuge in Amsterdam, and recovered
their property by the same means. These facts, how-
ever, are not supported by any very satisfactory histori-
cal evidence ; it is certain, indeed, that the Lombards,
for the purpose of the very extensive commerce of Italy,
were dispersed over every country in Europe, where
they established themselves as merchants, money-
changers, and bankers. Our own Lombard-street,
which still retains its appropriate traffic, is a proof of
their presence in our own country; and the Exchange of
Amsterdam was long known by the name of the Place
Lombarde, from similar associations.
Essential to (217.) It is not very easy, indeed, to imagine in what
* manner a very extensive international commerce could
id: be carried on without the assistance of bills of ex-
change. Though the balance of trade might disappear
in the intercourse of nations with each other, this could
rarely be the case in the transactions of individual
Their prin- merchants; we may suppose, therefore, two merchants,
ciple. A and B, at Venice, and two others, C and D, at Alex-
andria; A owes C the same sum that D owes B ; in-
stead of A sending specie to C, and D again to B, it
would save all parties both risk and expense if A should
pay the money immediately to B, and receive in return
an order, or bill of exchange, which he would transmit
to C, to enable him to receive the money from D, by
* Annals of Commerce, vol. i. p. 405.
which the accounts of the two parties would be cleared; History.
such a process as this would be pointed out by the com- -/-
mon sense of mankind, and the whole theory of ex-
changes does not require a much broader basis for its
foundation.
(218.) There is no notice of bills of exchange, or of any Not noticed
thing equivalent’ to them in the Code of Justinian, and in the Ro-
it has been inferred from thence that they were un- man law
known to the Romans, inasmuch as transactions con-
ducted by means of them, are those which of all others
require the most frequent control and regulation of the
law, and they could not, therefore, have existed, at least
to any extent, without its notice and interference. We
must allow this circumstance great force as a negative Their use
argument, notwithstanding the authority of the passage of i. lºgº.
one of the letters of Cicero to Atticus, (xii. 24,) when º ->
making inquiries concerning his son’s journey to Athens,
and the supply of money which would be requisite for
him; permutari ne possit an ipsi ferendum ? The per-
7mutatio alluded to must have been equivalent in sub .
stance at least, if not in form, with a bill or letter of
exchange, and it appears from a subsequent letter, (xii.
27,) that such was the expedient which was adopted.
(219.) Lucas de Burgo, who was duly impressed with The use of
a sense of the great importance of commerce to the . .
wealth and power of a state, complains, that in his lºy
time it was the custom of many persons to murmur usury.
against those who dealt in bills of exchange, calling,
them usurers and worse than Jews; in his opinion,
however, the inventors of them deserve a blessing with
a hundred hands, as without them the very foundation
of all that beneficial commerce would be destroyed,
which was essential to the support of the Republic.
It is true, indeed, that exchanges were sometimes
practised in a manner which was neither commendable
with God nor man; but this observation could never
be applied to their legitimate use in the general trans-
actions of commerce.
(220.) Cambio, according to the same author, might be Different
explained generally by the popular phrase to e da qua; º of
that is, togli da me questo e da me tu questo altro, take sº º
this from me and give me that in return. Four species gèS.
of exchange are noticed by Italian writers, which are
cambio menuto o commune, reale, secco e fittitio; the
first of them, minute, or ordinary exchange, is that in
which gold or silver coin is given in exchange for other
coins of different species or denominations, where the
banker or money dealer retains a small consideration
for his trouble ; an allowance, so far from being
usurious and improper, that it is approved of by the
most celebrated theologians and doctors of the church,
and amongst them by Remond Raimondo, Thomas
Aquinas, and, above all, by the most sacred doctor of
our own order, says Pacioli, Ricardus Mediavillensis.
The second, or real exchange, is of all others the most Cambio
important, being the very “water upon which the reale.
vessel of commerce floats,” and is carried on by means
of letters or bills of exchange, which have preserved
very nearly the same form for four centuries at least,
if not for a much longer period. We will give speci-
mens of such letters of exchange as were drawn in the
years 1404, 1494, and 1553.
1. Francisco da Prato et comp. a Barselona.
nome di Dio, Amen, a. d. XXIII Aprile, 1404.
Pagate per questa prima de camb. a usanza a Piero
Gilberto e Piero Olivo scuti mille a sold. X. Barselones?
Cambio
menuto.
Al Specimens
of bills of
exchange.
per scuto, e quali scut: mille sono per cambio che con
462
A. R. I. T H M E T I C.
knows that it is the highest, say 75 ducats; the bill . History.
is of course protested, returned, and A must pay B
Arithmetic. Giovanni Columbo a grossi XXII. di g. scuto: et påg.
S-N-" a nostro conto et Christo vi guardi.
Antonio Quarti Sali de Bruggias.
This is a bill of exchange which is given by Cap-
many” in his history of the town and commerce of
345 ducats, with all the charges incurred; such ex-
change was called cambio secco, and was clearly a me-
thod of avoiding the penalties and discredit of usury.
Barcelona; it was found amongst the records of a re-
e Tartaglia has illustrated this species of exchange by Practised in
ference made by magistrates of Bruges to those of Bar-
a practice which was very common in Italy, and which a certain
& ara ºr cº 4- by- *k, *
celona, respecting the practice which they followed in
the case of bills of exchange which had been protested,
and upon which undue charges had been made in its
transit from the drawer to the drawee, and which in
consequence the drawer refused to pay.
2. Domino Alphano de Alphanis e compagni in
Peroscia.
- 1494. a. d. 9 Agosto.
Pagate per questa prima mostra a Ludovico de Fran-
*cesco da Fabriano e compagni once cento d’oro Napo-
litane in su la proxima fiera di Fuligni per la valuta
d'altretante recevute qui dal magnifico homo meser
Donato da Leggi quondum meser Priamo. E. pome le
per noi. Iddio da mal vi guardi.
Wostro Paganino da Paganini da Brescia.
This is a form given by Lucas de Burgo.
3. A messer Ricardo Ventworth gentilhuomo Inglese
in Londra.
1553 a. di 4 Ottobrio in Venetia.
A uso pagareti per questa prima, a messer Giovan da
Mora delle presente latore lire vinticinque e soldi sedici
de sterlini per la valuta de altri tanti per lui medesimo
qua consignata e pometeli a vostro conto che Christo vi
conservi secondo il desiderio vostro. -
Andrea Dolphino dal bancho vostro servitor.
This form is given by Tartaglia, and is addressed to
his friend, pupil, and patron, to whom the first part of
his work is dedicated.
places the poverty of the farulers in a very suriking
light; the interval between harvest and seed-time in that
country is very considerable, and it generally happened
that the farmers were compelled by their poverty to
dispose of the whole of their produce before that period
arrived; the consequence of this forced market was,
that the price of corn was very low immediately after,
and very high immediately before the harvest; under
these circumstances they were compelled to borrow the
seed-corn upon condition of replacing an equal quan-
tity, or paying the price of it in the month of May.
The cases are clearly analogous, and show, in a very
remarkable manner, the inconveniences occasioned by
any interference with the regular trade in money, and
the extraordinary expedients which were commonly re-
sorted to for the purpose of gaining an exorbitant in-
terest, which could not have been the case had moderate
usury been sanctioned by custom or by law.
(222.) Cambio fittitio, called by the French change Cambio
feint, or adulterin, when A sells goods to B for time on fittitio.
this condition, that in case the payment is not made
when due, he shall be repaid by a bill of exchange, as
in cambio secco, reserving to himself the choice of place
and time. It is hardly necessary to observe, that such
practices were of the kind which Pacioli characterises
as commendable in the sight of neither God nor man.
The more rapid and secure communication which
takes place between different places in modern times,
Observa- With respect to the form and wording of these bills, and the many channels through which bullion may be
tº "P" very few remarks are necessary. The debtor and cre- transmitted, have materially lessened those extreme
€II1, ditor side of an account are always designated by per fluctuations in the course of exchange, which were
and a. Uso, or usanza, means the customary time in formerly so common and so certain, and in which these
different cities between the acceptance and payment of fictitious exchanges originated.
the bill, varying from ten days to three or four months, The preceding account of the terms used in ex-
according to their distance or the facility of communi- changes, which occur so frequently in Italian and other
cation ; the first of these bills is remarkable, as furnish- books of Arithmetic, is all that is requisite for our
ing an example of a bill drawn in Italian at Bruges present history; we dare not venture upon their modern
for acceptance in Spain; a proof that it had become use, history, and, still less, theory, a subject of vast
the universal language of commerce. The laws of all extent and difficulty; and we shall proceed, therefore,
commercial towns gave extraordinary power to the to the notice of another subject of purely Italian origin,
holder of a protested bill, which had been refused ac- the method of Book-keeping by Double Entry.
ceptance, or payment, by the drawee, upon the person (223.) This method of book-keeping has been ex-Italian
and goods of the drawer; and the consequence was, plained in great detail, in a distinct chapter by Lucas de book-keep-
that such bills were considered the best of all securities Burgo, and is certainly one of the most refined inventions "*
for a debt which was not real; this circumstance, and which could be devised to prevent the confusion which
the wish to evade the denunciations of the church would otherwise arise in the registering of complicated
against the practice of usury, will account for the mercantile transactions ; and though some improve-
origin of the other two species of exchange, which we ments have been introduced in later times, as far as
shall now proceed to notice. regards brevity and compactness, yet in all essential
Cambio (221.) A wishes to borrow 300 ducats of B; B selects a points the system remains unchanged. A few words
S60CO. place, Lyons for instance, where the exchange, from the will be sufficient to explain the general principle of
balance of trade at that period of the year, is very low,
say 60 ducats for a mark of gold ; B receives a
bill of exchange directed to an imaginary person at
this method, particularly as distinguished from the more
obvious method of recording accounts, which is called
Book-keeping by Single Entry.
(224.) In the latter of these methods, there is merely Principle of
required a memorial of occurrences in the order of time, book-keep-
with a ledger in which the names of all the parties between ºn-
Lyons, directing him to pay 5 marks to the holder, at
the rate of exchange at the fair of All Saints, when he
* Beckmann's History of Inventions ; the same work contains a gle entry.
custom-house tariff for 1221, and also a decree of the council of
Barcelona, dated 1394, ordering all bills of exchange to be accepted
within 24 hours of their being presented. .
whom transactions take place are entered, with an
alphabetical index of reference; the debtor and creditor
accounts of each party being arranged on the two
A R I, T H M E T I C.
463
Arithmetic. opposite pages, which afe presented at one opening,
~~ the first on the right hand and the second on the left ;
there is only one entry of each transaction, which is
either debtor or creditor; such a method enables us to
Abalance the accounts of each party, but presents no
register by which the state of the stock in trade
and the balances of capital and cash can be at once
ascertained, without a separate and independent inves-
tigation.
(225.) In book-keeping by double entry, three books
are required, the waste book or memorial, the journal,
and the ledger. This method differs from the former
chiefly in making cash, stock, goods, &c., parties as well
as persons, and in making'a debtor and creditor account
in every transaction; thus, if cloth is sold to A, A is
made debtor to cloth, and cloth creditor to A ; if cash
is received from B, cash is made debtor to B, and B
creditor to cash; and in every case the party, whether
animate or inanimate, which receives is debtor to that
which pays, and conversely. A double entry is, there-
fore, requisite in every transaction, and a balance may
at any time be struck between things as well as persons;
and in order to avoid the confusion which would arise
in a direct transfer of accounts from the memorial to
the ledger, before the proper relation of debtor and
Principle of
book-keep-
ing by dou-
ble entry.
creditor in each transaction are distinctly ascertained
and recorded, they are first entered in the order of time
in the journal, in the same form in which they must
appear in the ledger.
Sºlº (226.) Lucas de Burgo prefaces his account of Italian
... book-keeping by an enumeration of the proper qualifi-
merchant,
jin, to cations and qualities of a merchant. As he had passed
De Burgo. the greatest part of his life in a city of noble merchants,
and saw at the head of the government of his own
country a family which had risen by commerce, it is
very natural that he should have entertained the highest
respect for a character and profession which not only
led to wealth but to public honours; so high, indeed, was
the general estimation of the merchants of Italy for
honour and integrity, that the simple affirmation a la fê
d'un real mercatante, or by the faith of a true mer-
chant, was considered one of the most solemn that
could be made ; and so numerous were the accomplish-
ments which were deemed necessary for him to possess,
that it became a common and proverbial saying, “ that
it required more points to make a good merchant than
to make a doctor of laws.” Considering, indeed, the
various accidents and dangers to which he is exposed
by sea and land, in times of peace and plenty, of war
and scarcity, of pestilence and disease, and on so many
other occasions, if he possessed, like Argus, a hundred
eyes, they would not be sufficient. His proper emblem
is the cock, that watcheth by night and by day, in
summer and in winter; so watchful and so constant
ought his vigilance to be, always remembering the
maxim of the laws, vigilantibus et mon dormientibus
subveniunt jura, as well as the declarations of the holy
church and of Scriptures, that the crown is promised to
him that watcheth. He should fear no fatigue, uniting
with his labour the practice of piety and charity, trust-
ing to the truth of the adage, mec caritas opes, mec missa
7minuit iter; to all these moral qualifications, on which
the good old monk enlarges with such apparent delight,
it is requisite that he should unite others of a more
worldly nature; he must possess a sufficient capital in
money or in goods; be a ready and expert reckoner;
and possess the power of registering all his transactions
in a clear and beautiful order, so that he may at once History.
become acquainted with them by reference to his books; S-N-'
for the proverb which says, ubi non est ordo ibi est con-
fusio, which is true on all other occasions, is more par-
ticularly so in the case of mercantile affairs.
(227.) Of the books which are requisite for a merchant, Inventario,
the first is the inventario, or inventory of all his possess-
ions and goods of every description. The following is a
specimen of the mode in which it was headed : In the
name of God, on the 8th of November, 1494, at Venice.
Here follows the inventory of me M. N. of the street of
the Holy Apostle, written with my own hand, of all my
goods, moveable or immoveable, debts, credits, &c. which
I possess in the world on this present day. It then
proceeds to enumerate, with the utmost minuteness, all
his money in gold and silver, in coins of different
descriptions, lands, houses, gardens, orchards, sozzide
de bestiami, stock of all kinds, debts, credits, bills of
exchange, &c. It was sometimes usual to copy the
heads of this inventory into other books, which were
used in the conduct of mercantile affairs, which we
shall now proceed to notice.
(228.) There are three books which were necessary
for this purpose, the memoriale, giornale, and quaderno;
the first, called sometimes vacchetta, squartafoglio, Of
squartafaccia, little cow, crooked leaf, or crooked face,
from its rumpled appearance when old, corresponds to
the waste book of our merchants, and contained an Memoriale,
account of all transactions in the order of time, parti-
cularizing el chi, el che, el quando, el dove, the whom,
the what, the when, the where, in the most minute
manner, so that not an iota of the transaction may be
omitted which may be requisite to make it fully under-
stood; inasmuch as al mercante le chiarezze mai furon
troppo, a merchant cannot have too many explanations
which tend to give greater clearness.
(229.) The second book was the giornale, where the Giornale.
transactions are entered from the memoriale in the order
of time, and arranged in the form of debtor and cre-
ditor, preparatory to their being copied into the qua-
dermo 5 debtor is signified by per, and creditor by A ;
and the two entries with reference to them are separated
by two lines, thus ||. There are two terms which are
of frequent oceurrence in these entries, cassa and cave- Cassa and
dale, which it may be requisite to explain ; the first, cavedale.
which was transferred from designating the money bor
to its contents, corresponds to our own term cash, and
denotes the stock of money in hand ; the second must
be translated stock, and denotes the whole stock in
trade, (monte e corpo di faculta o di tutto il traft.co.)
The first in Italian book-keeping, properly so called,
was never made creditor, the second never debtor,
contrary to the usage of modern times.
(230.) The last and most important book was the qua- Quaderno.
dermo, or ledger, into which the entries of the giornale were
transferred in the names or designations of the several
parties, whether animate or inanimate, there being always
two entries for each transaction, one per and the other A.
It commenced with the alfabeto, repertorio or trovarello,
called in Tuscan stratto, and was ruled with as many
vertical lines as were requisite to contain the different
denominations of money or goods which were required
to be registered ; the first page contained the cash
account ; when stock was debtor, the general term
cavedale was used ; when creditor, the entry took place
under the head of the particular goods which were
concerned in the transaction; the milesimo, or date of
464.
A. R. I T H M zE T I C.
Arithmetic
A.
. the year, was put at the top of each page; the month
S-v- and day in each separate entry.
Modes of
entering
accounts of
different
classes of
transac-
‘tions.
Striking a
balance.
Other
books.
Italian
book-keep-
ing in gene-
ral use.
The same transactions were recorded in the same
memoriale, giornale, and quadermo, and to denote their
connection with each other, they were all signed with
the same letters, A, B, &c. The first set of these books,
however, were generally marked with the sign of the
cross ; that glorious sign from which all our spiritual
enemiessfly, and at the sight of which the whole host of
hell most justly trembles; and were called memoriale
croci, giornale croci, and quaderno croci. In some
places it was customary to authenticate the memoriale
before proper officers appointed for that purpose; a
most laudable and excellent practice, well calculated
to prevent disputes and frauds, as the authenticity of
the other books must be determined from it.
(231.) The author then proceeds to explain the mode
of recording and entering the accounts of different trans-
actions, whether baratti, of all their different species,
whether simple, compound or for time ; compagne,
whether personal, or what the French call en comman-
dite, where money alone or goods are contributed;
conti di botega, or accounts of traffic in detail, whether
conducted in person or intrusted to another ; accounts
with banks, which were then established in Venice,
Genoa, Bruges, Antwerp, and Barcelona; of mercantile
journeys or voyages, where separate books must be
kept, the principal ones being left at home ; of bills of
exchange and transactions connected with them, with the
notice of the expense incurred in the salaries of factors
and servants, in the ordinary maintenance of the house-
hold, as well as of extraordinary expenses incurred for
gaming, pastimes, amusements, and pleasures of dif-
ferent kinds, which are not properly included under
any kind of ordinary expenditure.
(232.) Various directions are likewise given about the
mode of striking a balance, whether general or particu-
lar, and of transferring the accounts from one ledger to
another; as also of extracting a balance sheet con-
taining the summa summarum. Every merchant is
likewise recommended to keep un libro de pagamenti,
or book of payments; un libro de recordanze, or memo-
randum book; and likewise to copy into a separate book
all letters, whether received or sent, which notice any
circumstance, the particulars of which the regular books
cannot register; the necessity also of making no change
in the books is repeatedly and strongly enforced, and if
an error is detected it must be entered as a distinct item
in the ledger; in short, no precaution is omitted which
is requisite to give perfect distinctness to the recording
of mercantile transactions, however complicated they
may be. -
(233.) If we consider the extent and influence of Italian
commerce, extending to every country in Europe, Asia,
and Africa, which was at that time known, in most of
which Italian agents, factors, bankers, and money
changers were established, it is natural to suppose, that
this system of book-keeping should be generally adopted,
recommended as it was by those whose experience and
superior progress in the arts of life gave authority to
their opinions and practice; we, consequently, find this
method described in a work written by a merchant of
Nuremberg, named Gottlieb, in 1531. In 1543, Hugh
Oldcastle, a schoolmaster of London, wrote a work on
the subject, which was afterwards published in an im-
proved form by James Peele in 1569, with the follow-
ing title: A Briefe Instruction how to keep Books of
Accounts, after the order of Debtor and Creditor, and . History.
as well for proper Accounts, Partible, &c. by three S-V-
Books, named the Memoriall, Journal, and Ledger.
(234.) Beckmann has given an account of a work of Work of
Stevinus on Italian book-keeping, written, in 1606, for his Stevinus on
patron Maurice, Prince of Qrange, and dedicated to the
great Duke de Sully, who had introduced it in the ac-
counts of the finances of France under Henry IV. It was
translated into Latin by Willebrod Snell, who has lati-
nized the modern terms with considerable elegance and
ingenuity. Book-keeping is called Apologistica, or Apo-
logismus; the book-keeper, Apologista ; the memorial,
or waste book, is liber deletitius; the ledger, coder
accepti impensique; the cash book, arcarii liber; book
of expenses, impensarum liber; the profit and loss
account, lucri damnique ratiocinium, contentio seu
comparatio sortium ; the final balance, epilogismus ;
and the counting-house, logisterium. In connection
with the subject of the names which are commonly
given to those books, we may observe, that the Italian
term quadermo is of unknown derivation; and the re-
ook-keep-
gº
mark may be extended to our own word ledger, so Different
variously written at different periods of our language, names for
though many derivations have been given; it is called ledgers.
by the French grande livre, and by the Germans
hauptbuch, or head book, to express its great im-
portance. The existence of so many independent
names proves that ledgers were used for registering
accounts in those countries long before the Italian
method was known ; as it would otherwise have been
hardly possible to have adopted the system without also
borrowing its entire momenclature.
(235.) It is not our intention to proceed farther with Modem
the notice of the books on this subject, which have been works on
written in such great numbers by merchants and others, the subject.
and by whom the method itself has been modified, from
time to time, to suit the wants and purposes of modern
commerce. Amongst the best of these we may mention
the system published by Malcolm at Edinburgh, in
1728, and by John Mair of Perth, in 1737. In the
year 1796, an accountant of Bristol, of the name of
Jones, published a work, by subscription, on book-
keeping by single entry, with double money columns,
for the purpose of showing that it might be made, by
certain modifications, equally efficient with the system
of double entry, and that it was essentially more simple.
This attempted innovation, however, was the cause of a
considerable controversy, and was closed by a pamphlet
of Mr. Mill, who showed by reducing the waste
book of Mr. Jones to a journal and ledger, according
to the old method, that his system was essentially and
unavoidably defective.
(236.) The rule for Alligation, as well as that of Po- Alligation
sition, is of eastern origin, and appears in the Lildwati, in the Lité-
though under a somewhat limited form; it is there calle
suverna-ganita, or computation of gold, and is applied
generally to the determination of the fineness or touch
of the mass resulting from the union of different masses
of gold of different degrees of fineness. The questions
mostly belong to alligation medial, and are of the fol-
lowing kind :
“Parcels of gold weighing severally ten, four, and two
mdishas, and of the fineness of thirteen, twelve, eleven,
and ten respectively, being melted together, tell me
quickly, merchant, who art conversant with the compu-
tation of gold, what is the fineness of the mass 2 If
the twenty mashas above described be reduced to six-
d vati.
A R. I. T H M ET I. C.
465,
Arithmetic, teen by refining, tell me instantly the touch of the
v-' purified mass 2 Or, if its purity when refined be six-
teen, prithee, what is the number to which the twenty
7máshas are reduced ?”
Statement:
13 12 11 10
* Touch . . . . . . . . . . . .
Weight . . . . . . . . . . . . 10 4 2 4
Products . . I30 48 22 40
The sum of the products, 240, divided by the sum of the
weights, 20, gives the fineness after melting, which
is 12.
After refining, the weight being 16, the touch is 15 ;
the touch being 16, weight is 15.
“ Eight mashas of ten, and two of eleven by the
touch, and six of unknown fineness, being mixed to-
gether, the mass of gold, my friend, became of the
fineness of twelve ; tell the degree of unknown
fineness P” -
Statement : 10 1 1
8 2 6. Fineness of the mixture, 12.
From 12 × 16, or 168, subtract 8 × 10 and 2 × 11,
the remainder, 90, divided by 6 gives 15 for the degree
of the unknown fineness.
(237.) The following is the only question given in
illustration of the rule called Alligation alternate :
“Two ingots of gold, of the touch of 16 and 10
respectively, being mixed together, the weight became
of the fineness of 12; tell me, friend, the weight of
gold in both lumps?”
The following is the rule which is given : “Subtract
the effected fineness from that of the gold of a higher
degree of touch, and that of the one of the lower de-
gree of touch from the effected fineness; tell me, friend,
the weight of gold in both lumps? The differences,
multiplied by an arbitrarily assumed number, will be
the weights of gold of the lower and higher degrees of
purity respectively.”
Statement: 16 10. Fineness resulting, 12.
If the assumed multiplier be 1, the weights are 2 and
4 mashas respectively; if 2, they are 4 and 8 ; if ,
they are 1 and 2: thus, manifold answers are obtained
by varying the assumption.
(238.) This rule, though perfectly distinct and clear, is
formed for the case of two quantities only, and there is no
appearance of its ever having been applied to a greater
number; it involves, however, the principle of the rule
which is now used, recognises the problem as unlimited,
and shows in what manner an indefinite number of
answers may be obtained. The extension of the rule to
any number of quantities, though not an easy step, in a
state of the mathematical sciences when the generali-
zation of principles and methods were little sought
after and rarely practised, was yet incomparably more
so than the invention of the rule itself, even under
its most limited form ; it is for this reason that we
feel compelled to ascribe the chief honour of this rule
to the arithmeticians of Hindostan.
(239.) It was this latter rule, under a more general
form, that was denominated Sekis by the Arabians, a term
meaning adulterous, inasmuch as it is not content with
a single, and, as it were, legitimate solution of the
question. It was sometimes called Cecca by the Ita-
liams, who appear to have known nothing further of
the word than its Arabic origin; and it constitutes the
alligation alternate of modern books of Arithmetic.
It may be as well, for greater clearness, to state alge-
braically the nature of the problems which are proposed
WOL. I.
Example in
alligation
alternate.
Rule.
Extent to
which the
rule was
known to
the Hindoos
Its Arabic
{l all]]{2.
each, so that the common quality of the compound may
for solution by means of it, and also to prove the truth History.
of the process. - -
(240.) Flet a, b, c represent the several prices, degrees Algebraical
of fineness, or other common quality of the several in-statement of
$g • 4 ; e tº g tº & the problem
gredients; it is required to find quantities a, y, and 2 of. to
be solved.
be denoted by d. -
The equations which represent the conditions of
the problem, are •
a a + b y + c 2 = m d (1)
a + y + z = m. (2)
or, eliminating m,
(a- d) a + (b — d) y + (c — d) z = 0, (3)
which is an equation of condition, which must be satis-
fied in all cases.
The value of m, therefore, makes no alteration in the
relative values of a, y, and z, which must be assigned
from equation (3); and the assignation of it can only,
therefore, in a certain sense, be said to limit the inde-
termination of the problem.
If the quantity of one of the ingredients be assigned,
if z = k, for instance, then the equation (3) becomes
(a – d) a + (b — d) y + (c – d) k = 0. (4)
In this case, the values of a and y must be determined
absolutely, so as to satisfy this equation; and those values
must satisfy another equation of condition, which is,
a + y = m — k. (4)
If m be also assigned, the determination of a and y is
complete, when there are only three ingredients.
The problem becomes more limited if a., y, and 2 are
concrete quantities, negative values of which would
admit of no meaning ; and still more so, if, in addition,
those values are likewise required to be integral; under
such circumstances there may be no answer to the ques-
tion, or at most but a limited number of them.
In alligation alternate, the only limitation is in the Different
price of the compound: in alligation total, there is a species of
imitation both of the price and quantity of the com- *
pound: in alligation partial there is a limitation of the
quantity of one of the ingredients, and of the price of
the compound: in alligation medial, the prices and
quantities of all the ingredients are given to find the
price of the compound, and the problem is, of course,
determinate. * : -
(241.)The arithmetical rulefoſſalligating the quantities Proof of
in the three first cases is the same, and the accuracy of the Arithme-
the result may be readily shown by exhibiting the pro-" "
cess and the result in algebraical symbols.
Let the prices or quality of the several ingredients
be denoted by u + a, u + b, u — a', u — b', and that
of the mixture by u : to find the quantities of each,
which are requisite to produce a compound of this
price or quality ? 4.
We will unite them in three different ways:
I. : | u + a b
at + b a/
7ſ, D
w — aſ b
w — b/ 0,
The quantities of each ingredient in their order
being b/, a', b, a, it is clear that the sum of the pro-
ducts of these quantities into their prices, ought to
be equal to the product of the quantities into their
mean price; thus,
e 3 P
466
A R IT H M E T I C. $
Arithmetic. bſ (u + a) + a' (u + b) + b (u — a') + u (u —b')
*Twº" = (a + b + a' + b/) u + a b' + aſ b – aſ b - a b'
= (a + b + a' + b') u.
2. u + a S a'
u + b W
º
u — aſ (Z
|u – V - b
In this case, also,
a/ (u + a) + b/(u + b) + a (u — a!) + b (u — bº)
= (a + b + aſ + 6') u + a a' + b bº — a a' – b bº
= (a + b + a' + bº) w
3. With double ligatures,
w -H a a' + bº
w =}. b a' + b'
7, º
u — a a + b
w — bº & + b
In this case, the several ingredients are respectively
the sums of those which were determined by the single
ligatures, and, of course, therefore answer the condi-
tions of the question; or it may be shown as follows:
(a' + bº) (w -- a) + (a' + b ) (u-H b) + (a+b) (w-a')
+ (a + b) (u — bº)
= 2. (a + b + a + b ) u + (a' + bº) (a + b)
- (a + b) (a' + bº)
= 2 { a + b + a + b) u.
In alligation total, the quantity of each ingredient
thus determined must be increased or diminished in
the proportion of the sum of the ingredients deter-
For alliga; mined to the sum required: in alligation partial, they
*P* must be altered in the proportion of the quantity of
the ingredient determined to that which is required.
In no case, does the rule attempt to determine all
the answers of the question, and in the two last cases,
it only gives as many as can arise from variation of the
ligatures.
Tor alliga-
tion total.
Meaning of (242.) The earlier Italian writers on Arithmetic, in
the term imitation of the practice of their Arabian masters, have
Accnsolare.
confined the application of this rule almost entirely to
questions connected with the mixture of gold, silver,
and other metals, with each other. This union was
designated by the term consolare, which probably ori-
ginated in the dreams of astrologers and alchemists:
Secondo che vogliono, says de Burgo, li astronomi,
dei sonnoli pianeti celestioli detti : per la virtue ordi-
7tatione che da Dio ricevano hanno li detti dei metalli
a generare e producere. Pero che la luna produce e
genera argento morto : e lo sole genera l'oro. Dell;
altri metalli se taci. It appears from hence, that it
was considered the peculiar province of the sun to
produce and generate gold; and as the process of the
alchemists in transmuting the baser metals into gold
was supposed to be under the influence of the sun,
this gradual refinement, which they in common tended History. .
to produce, was designated by the common term con- “S-
solare. In later times it was applied to silver as well -
as gold, and still more generally to the common union
of these metals with copper.
(243.) The fineness of gold was estimated by so many Mode of
carats, or parts of 24, whilst that of silver was esti- stinating
mated by so many lighe, or parts of 12. The metals º º
used in composition with them in coins were silver and ... a I) (
copper, in the case of gold; and copper only, in that of
silver: the baser metal in both cases being esteemed
of no value with reference to the other. The noble
metals were called Fired, inasmuch as they did not
waste during the process of refinement. We shall
give a few examples connected with this subject.
A person mixes 9 ounces of gold of 18 carats fine, Examples.
10 of 20 carats fine, and l l of 22 carats fine; to find
the fineness of the mixture?
A person mixes 9 marks of silver of 9 lighe of fine-
ness, 13 of 8 lighe, and 14 of 10 lighe; of what ligha
is the mixture ?
I subject 82 ounces of gold of 18 carats fine to the
fire for refinement, and draw out only 72 ounces ; of
what degree of fineness is it?
This is a common Inverse rule of Three question.
Given different species of silver of 9,8, 5 lighe, respec-
tively; in what proportions must we mix (consolaremo)
60 lbs., so that the compound may be of 64 lighe 2
The answer:
34 lbs. 3 oz. 10 gr. -- of 5 lighe.
12 lbs. 10 oz. 10 gr. -- of 8 lighe.
12 lbs. 10 oz. 10 gr. -- of 9 lighe.
A parish (communità) wish to found (gittare) a bell,
composed of 5 metals, and the hundred pounds weight
of the basis cost 16 lire, of the second 18, of the third
20, of the fourth 27, and of the fifth 31. The whole
weight of the bell is 2325 lbs., and it costs 488
lire, 5 soldi. What portions of each metal did they
use ?
The following is the form under which the ligatures
are made by Tartaglia:
16 , 18 , 20 , 27 , 31
10 3. 6 2 6 2 3 2 2
#1
The price of the mixture is 21 lire the hundred
pounds; and the quantities of each are, or may be, in
the proportion of the numbers 10, 6, 6, 3, 2.
A person has five kinds of wheat, worth 54, 58, 62,
70, 76 lire the staro respectively; what portion of each
must be taken, so that the sum may be 100 stara, and
the price of the mixture 66 lire the staro 2
The following are different solutions of this question -
1st, In the proportion of the numbers 10, 4, 10, 8,
16
A R I T H
M E T H C. 467
Arithmetic.
54 , 58 , 62 , '70 , 76
10 , 4 10 , 8 , 12
|^
66 -
2dly, In the proportion of the numbers 10, 10, 4, 4,
20.
54, 58, 62, 70, 76,
10
20
3dly, In the proportion of the numbers 14, 14, 14,
24, 24. -
54 , 58 , 62 70 , 76
10 jo ſo | 1. ' .
4. 4 4 8 8
tº-sº a sº º 4 4.
14 14 14 sºmmemº e =
24 24
Tartaglia has given two other solutions of this ex-
ample, arising from a different arrangement of the
ligatures.
macopolists to classify medicines according to their . History.
degrees of heat or coldness, moisture or dryness.
scale for this purpose was adapted to the scale of the
nine digits, the middle of which was temperature, as
follows:* -
Indices. Degrees.
9 4
8 3 tº a º
7 2 Qualities hot and dry.
6 }
5 0 Temperature.
4 l
3 2 0 2 &
2 3 Qualities cold and molst.
l 4
The solution of the question will stand therefore as
follows:
A 8 I
B 7 3
5. C 6 r 1
HD 4 3 + 1
E 2 | 2
The numbers 1, 3, 1, 4, 2, will, therefore, answer the
question. The number 5 is sometimes called the
emergent of the composition.
The S-S-
Pxamples Suppese there are five simples, A, B, C, D, E, whose The following is a more elaborate example of the
* of qualities are as followeth, viz. A is hot in 3", B is hot composition of a medicine called Dianthus, taken from
... in 2°, C is hot in 19, D is cold in 19, E is cold in 3°; Parkinson's Herbal, which is declared to be of a fine
it is required to mix four ounces of B with such quan- temperature or temperament, that is, somewhat more
tities of the rest, so that the quality of the medicine than a degree in heat, and somewhat less than a degree
may be temperate P in dryness; in this case a zero is taken as the repre-
It was the custom of the older physicians and phar- sentative of temperature.
Ingredients. Quantities. Qualities. Products.
hot, cold. moist, dry.
Rosemary flowers. . . . . . 24 × 2 — 0 – 0 — 2 48 — 0 — 0 — 48
Red roses . . . . . . . . . . . . 18 × 0 — 1 – 0 — 1 0 — 18 — 0 — 18
Violets . . . . . . . . . . . . . . l 8 × 0 — 1 — 2 -— 0 0 — 18 — 36 — 0
Licorish . . . . . . . . . . . . l8 × 1 — 0 — I — 0 18 — 0 — 18 — 0
Cloves . . . . . . . . . . . . . . 4 × 3 — 0 — 0 — 3 12 – 0 — 0 — 12
Indian spikenard . . . . . . 4 × 1 – 0 — 0 — 2 4 – 0 — 0 — 8
Nutmegs . . . . . . . . . . . . 4 × 2 — 0 — 0 – 2 8 — 0 – 0 — 8
Galonga. . . . . . . . . . . . . 4 × 3 – 0 – 0 – 3 12 — 0 – 0 — 12
Cinnamon . . . . . . . . . . . . 4 × 2 — 0 – 0 — 2 8 — 0 — 0 — 8
Ginger . . . . . . . . . . . . . . 4 × 3 — 0 — 0 — 3 12 — 0 — 0 — 12
Zedoary. . . . . . . . . . . . . . 4 × 2 — 0 — 0 — 2 8 — 0 — 0 — 8
Mace . . . . . . . . . . . . . . . . 4 × 2 — 0 — 0 — 2 8 — 0 —- 0 — 8
Wood of aloes . . . . . . . . 4 × 2 — 0 —- 0 — 2 8 — 0 – 0 — 8
Cardamoms . . . . . . . . . . 4 × 3 — 0 — 0 — 3 12 – 0 — 0 — 12
Aniseeds . . . . . . . . . . . . 4 × 2 – 0 — 0 — I 8 — 0 – 0 — 4
Dillseeds . . . . . . . . . © 4 × 2 – 0 — 0 — 3 8 — 0 – 0 — 12
126 174 36 54 178
Hot. Cold. ness, in consequence of their furnishing the solutions of
174 — 36 = +}} = 1 ºr. Hot a vast number of questions, which would otherwise have
g required the aid of Algebra. It may conduce some-
Dry. Moist. what to the clearness of some of the details which
Rules of 178 — 54 = +3+ = ##, Dry. follow, if we first state, in an algebraical form, the prin-
sºld (244.) The rules of Single and Double Position are ciples upon which these rules are foulded.
Double amongst the most celebrated in Arithmetic, and were ge- —&
Position nerally discussed by the older writers with great diffuse- * John Dee's Mathematical Preface to Euclid.



3 P 2
468
A R IT H M ET I C.
Arithmetic.
(245.) Single position includes those questions, in
which there is a result which is increased or diminished
Algebraical in the same proportion with an unknown quantity which
Statement
of their
principles.
Single
position.
Rule.
Double
Position.
Rules.
Applied to
equations
with two or
In Ore Ulrl-
known
quantities.
is proposed to be determined: of this kind are all
questions which at once resolve themselves into the
equation
(I, T = 777. (1)
The process is as follows: assume a value of a, such
as a ', and let the result corresponding to it be m', er, in
other words, let
a ar' = m/;
we from hence get,
£
ſºmeºmºmº
p
Jº
or we must multiply the first result by the position, and
divide by the new result corresponding to it.
If, however, the question is proposed in such a
manner, that the result, which is a function of the un-
known quantity, does not increase in the same propor-
tion with the increase of that quantity, or if it resolves
itself into an equation of the form,
a a + b = m, (2)
we must then make two positions, or hypotheses, for
the unknown quantity; let these be aſ and ar", and let
the corresponding errors be e' and e", or, in other words,
let
a r' + b
a " + b
we from hence get
a (r' — a = e/
a (r' — a ") = e/ — e”,
777, ſº
f
777,
ºmº
hºmºmºs
º
*
m + e
m + e”,
emº
Cºmº
which gives
P //
e – € ©
F-7 = 7-; (3)
and also
/ …?/ Aſ ºf
e' ºf ’ – € ' Tº
(p = (4)
e' – e."
The first of these results (3) being translated from
algebraical into common language, shows that the diffe-
rence of the errors is to the difference of the positions,
as the first error is to the difference of the first position
and the quantity required, a rule which is frequently
used by De Burgo and Tartaglia.
The second (4) gives the common rule, that the
product of the first error into the second position, dimin-
ished by the product of the second error into the first
position, and the result divided by the difference of the
errors, gives the quantity whose value is required.
Of course this-rule must be modified according to
the signs of the errors, whether both positive or both
negative, or one positive and the other negative, and
conversely: the sums being taken in the latter case,
where the differences are taken before.
It is not necessary that the question should at once
resolve itself into an equation of the form (2), in order
that it may come within the operation of this rule; if by
means of any simple or obvious reduction, or by the
solution of the intermediate equations, where there are
more unknown quantities than one, it can be brought
to a form in which the value of a function of the un-
kniown quantity of the form a a + b is given, it is
equally resolvable by means of it.
shall find,
In the system of equations,
a r + b y = m. (5)
a a + £3 y = p. (6)
If we assume aſ for the value of a, and determine the
value of y corresponding to it from equation (5), we
&= -º
Wºº-ºº
a w' + b y' = m
a w' + f2 y' A + e.
When the errore' is clearly the same as if we had first
solved equation (5) with respect to y, and substituted
the value thus found in equation (6): in other words,
the error is the same as if we had commenced by re-
ducing the system of equations to a single equation of
the form
gmmºns
tºmims
A a + B = u.
The same reasoning is clearly applicable to any
system of equations containing more than two un-
known quantities, where the error resulting from an
erroneous assumption of the value of one of them
necessarily shows itself in the result of one only of the
equations: of this kind are the equations,
a r + b y = m.
b’ gy + c z = n
c' z + d u = r
d' u + aſ a = s.
If we assume aſ as the value of a, determine succes-
sively the corresponding values of y, z, and u, from
the three first equations, the error of the hypothesis
appears only in the last equation, which becomes
d' w -- a' a' = s + e.
A second hypothesis gives a second error, which com-
bined with the first and the two positions, gives the true
value of a, precisely in the same manner as if we had
begun by reducing the four equations to one of the
form -
A a + B = s.
The preceding investigations include every rule
which has ever been used for the solution of such
questions in books of Arithmetic. It would be easy
to form rules for the solution of systems of equations,
by making distinct hypotheses for all the unknown
quantities: thus, in the two equations,
a a -- b y = m.
a r + £3 y = u.
If we assume aſ and y' for a and y, we shall get
a w' + b y = m + e -
a w' + 3 y' = u + f.
from whence we readily find
f gº-ºº: £3 e — b f
tº
a B — a b
f a e – a
y – y = —- af
a b — a B
It is very easy to reduce these results into Arithmeti-
cal rules; but as the rules, which are thus formed
would be less simple than those which arise from the
formulae for the direct algebraical solution of the equa-
tions, it is clearly unnecessary to notice them further,
|H istory
Other
rules
easily
formed
A R I T H M ET.I. C.
469
\
Arithmetic, particularly as they would find no application in the
S-N- illustration of the methods which are found in books
Rule of
of Arithmetic.
(246.) The rule of single position is the only one
single posi- which is found in the Irilāvati, where it is called Ishta-
tion in the
I.ilāvati.
Examples.
Derived by
the Italian
writers on
Arithmetic
from the
Arabs.
Examples
from De
Burgo.
carman, or operation with an assumed number; we
shall give a few examples from it, which, however, pre-
sent nothing very remarkable beyond the peculiarities
in the mode in which they are expressed.
Out of a heaps of pure lotus flowers, a third part,
a fifth, a sixth, were offered respectively to the gods
Siva, Vishnu, and the Sun, and a quarter was pre-
sented to Bhavani; the remaining 6 were given to
the venerable preceptor. Tell me quickly the whole
numbers of flowers ?
Statement : ++++; known, 6.
Put 1 for the assumed number; the sum of the frac-
tions +, +, +, +, subtracted from one, leaves tº ; divide
6 by this, and the result is 120, the number required.
Out of a swarm of bees, one-fifth part of them
settled on the blossom of the cadamba, and one-third
on the flower of a silimd’hri ; three times the difference
of those numbers flew to the bloom of a cutaja. One
bee, which remained, hovered and flew about in the
air, allured at the same moment by the pleasing fra-
grance of a jasmin and pandanus. Tell me, charming
woman, the number of bees 2
Statement: #, 4, ºr ; known quantity, 1 ; assumed,
30. wº
A fifth part of the assumed number is 6, a third is
10, difference 4; multiplied by 3 gives 12, and the re-
mainder is 2. Then the product of the known quan-
tity by the assumed one, being divided by the remainder,
shows the number of bees 15.
The following question is from the Manoranjana:
The third part of a necklace of pearls, broken in
amorous struggle, fell to the ground ; its fifth part
rested on the couch, the sixth part was saved by the
wench, and the tenth part was taken up by the lover;
six pearls remained strung. Say of how many pearls
the necklace was composed ? -
Statement: +, +, +, +r; remained, 6. Answer, 30.
(247.) The Italian writers on Arithmetic derived the
knowledge of these rules immediately from the Arabians,
designating them by the Arabic name El Cataym, or
Helcataym. Costumasi, says Lucas de Burgo, in
la practica de Arithmetica solversi molte e varie ques-
tioni per certa regola ditta el cataym. Quale (secondo
alcuni) e vocabulo Arabo. E in mostra lingua sona
quanto che a dire regola delle due false positioni. The
questions, proposed by him and by Tartaglia, are in
immense variety, including every case of single and
double position ; and the rules which are given for this
purpose, are such as would immediately result from
the algebraical formulae given above. A few examples
will be sufficient to illustrate the form of the process
which they followed:
A person buys a jewel for a certain number of
Jiorini, I know not how many, and sells it again for
50. Upon making his calculation, he finds that he
gains 34 soldi in each fiorino, which contains 100 soldi.
I ask what is the prime cost?
Suppose it to cost any sum you choose ; assume 30
fiorini, the gain upon which will amount to 100 soldi,
or 1 forino : I added to 30 makes 31; and you say
that it makes 50 between capital and gain ; the position
is therefore false, and the truth will be obtained by
saying, if 31 in capital and gain arises from a mere . History.
capital of 30, from what sum will b0 arise. Multi- >-N-
ply 30 by 50, the product is 1500; divide it by 31, the
result is 48##, and so much I make the prime cost of
the jewel.
The above is a translation of the account given by
De Burgo, of the first question which he has proposed
on this subject.
Three persons have coins of the same kind and
value ; the second has twice as many as the first, and
4 more : the third as many as the first and second to-
gether, and 6 more ; and the whole number is 44; how
many had the first 2
Suppose the number 8, then the second has 20, and
the third 34 ; their sum is 62, and the error is 18, which
is plus, or piu. Again, the first had 6, then the second
has 16, and the third 28 ; the sum is 50, and the
second error is 6, which is also plus or piu.
8 × 6 – 6 × 8
18 – 6
Consequently, = 5, which is the
true anSWer.
The following is the scheme which is given by De
Burgo, which will require no explanation after the pre-
ceding statement.
48
p”. p°. 8
20
veritas 5.
pmº error. dria error. 24ts error.
Onde levate, says the author, che sono le diffe-
Tentie, dice el commun proverbio, le parte stamno in pace.
Siche tu vedi per dei falsità como siamo pervenuti a la
verità. E questo è quello che diceva A. R. ex: falsis
verum : ea veris mil misi verum. Nº.
The following question admits not of translation :
Uma matta de grue volano per airi e passan sopra un
lago: dove una sta sottaqua : e sente quelle gridare
Grugru ; lei disse sete voi la su. La guida respose. Noi
siamo tante che con altre tante e con la mita de
tante e con teco in conto, siamo cento di ponto. Do-
mando quasi te erano quelli che volevano 2
This is one of a multitude of questions which were
proposed for amusement and pastime, and which were
calculated to attract notice by the singularity of the
Questions
proposed
for amuse-
ment and
terms in which they were expressed, or by presenting pastime.
something remarkable in the conditions which they in-
volved. Tartaglia says, that such questions were fre-
quently proposed as puzzles, by way of dessert at enter-
tainments, and has mixed up with his other questions
on single position a large collection of such answers
commonly proposed for this purpose. This practice,
however, does not appear to have originated in Italy,
as there are some circumstances which would make us
refer them to the Greek arithmeticians of the IVth and
Vth centuries, and probably even to an earlier period.
If the half of 5 were 3, of what number would 5 be
the quarter ? or if 4 were 6, what would 10 be 2
De Burgo has noticed other questions of this kind,

:470
A R IT H M E. T.I. C.
Arithmetic, which are only remarkable for the violation of the pro-
*-v- priety of language.
Questions
from Tar-
taglia and
De Burgo.
Indetermi-
mate pro-
blems.
The remark which he subjoins,
shows that there were bucks in Italy as well as in other
countries, and that the old monk felt a malicious
pleasure in posing them, by the proposition of such ques-
tions. In simil cosa, says he, siamo stati in dis-
putatione in piu luoghi di Italia con molti che si ten-
gono cervi e al bisogno non saltan troppo.
The following questions are taken from the same
author and Tartaglia, and will show the extent to
which these rules were applied by them.
Two persons go to a fair, and the first says to the
second, how many ducats have you? the second
answered and said, if I had 30 of yours, I should have
as many as you; and the second answered and said, if
I had 30 of yours, I should have twice as many as you:
how many ducats had each of them 2
Tartaglia has given upwards of twenty questions,
which are similar in principle to the preceding.
A schoolmaster, speaking of his scholars, says, if I
had as many more, and half as many, and one quarter
as many, and one-fifth as many, and 4 more, I should
have 240; what number had he 2
Two persons wish to buy a Turkish horse worth 120
ducats, but neither of them has sufficient money to pay
for him. If I had 3 of your money in addition to my
own, I could just pay for him ; upon which the second
answered, if I had 4 of your money besides my own, I
could also pay for the horse; how much money had
each P
A fisherman sold a sturgeon which weighed 60
pounds to three persons, the head to one, the tail to the
second, and the body to the third; the head weighed 4,
the tail ºr of the whole: what was the weight of the
body ? -
A gentleman sends his servant to the garden of a
lord, and tells him to go to the gardener and buy as
many apples, that he may bring back one to his lady:
he goes to the garden, which has four gates with four
guards; mothing is paid on entering, but on quitting
the gardens you must pay # of the whole and 1 more
at the first gate, ; of the remainder and 2 more at the
second, # of the remainder and 3 more at the third,
and # of the remainder and 4 more at the fourth : how
many must he buy, so as to bring back one to his lady,
(la qual e gravida)?
A gentleman asks a shepherd, what number of sheep
he had, who answered, that when he numbered them
2 and 2, 3 and 3, 4 and 4, 5 and 5, 6 and 6, there re-
mained 1 in each case, but if he numbered them by 7
and 7 there remained O ; what number of sheep had he?
This is an indeterminate problem, which Tartaglia
solves by finding the least common multiple of 2, 3,
4, 5, 6, which is 60, and finding amongst the multiples
of this number, increased by 1, those which were divi-
sible by 7 : of this kind are the numbers 301, 721, &c.
The following question is of a similar kind: Un
gentil huomo incontrandosi con un contadino, che con-
duceva duoi sportoni di ovi sopra una cavalla a una città
a vendere, e un cavallo di questo gentil huomo si misse
dietro a questa cavalla, talmente che gli fece rompere
tutti quelli ovi : il gentil huomo non volendo la rovina
di quel contadino per voler gli pagar li delli ovi gli
adimando, quanti erano, lui gli rispose che mom sapeva
quanti fossero, ma che sapeva ben a numerar li a 2 a 2
gli me avanzava 1 : similmente numerandoli a 3 a 3 gli
me avanzava 1 e cost a 4 a 4 gli me avanzava 1 e
cosi a 5 a 5 gli ne avanzava I : il medesimo faceva, a
6 a 6, e a 7 a 7, e a 8 a. 8, e a 9 a 9, e l'O a 10; ma
numerandoli poi a 11 a Il mi avanzava 0: si adimanda.
quanti erano li delli ovi. The least number which
answers the question is 25201.
A workman undertook to finish a piece of work in
16 days, and another workman undertook to do it in 20
days; in what time will they do it together ?
If a person ask you how many Angels there are in
Paradise, answer that there are three hierarchies, each
consisting of 3 orders, and each order of 6666 legions,
and in each Region there are 6666 Angels.
The answer is 399920004, which secondo la opinion
di theologhi sta bene. *
A person has 100 stara of wheat, and a miller has 3
mills, one of which would grind it in 10 days, the other
in 5, and the third in 4; in what time will they grind
it, all working together ? -
Four apples less a damaro are equal to 7 danari less
one apple; what is the value of one apple P
A labourer undertakes a piece of work, upon condi-
tion of receiving H0 soldi for each day that he works,
and of paying 15 soldi for every day that he is idle; at
the end of 20 days the work is finished, and he receives
only 15 soldi; how many days did he work, and how
many was he idle P
(248.) In the Greek Anthologia, we find a collectio
of arithmetical problems, the greater part of which are
attributed to one Metrodorus,” most of which are of Anthologia.
the nature of those questions which are usually resolved
by the rules of position; it is impossible, however, in
consequence of the loss of all the Greek arithmetical
writers subsequent to the age of Diophantus, to discover
any traces of the methods which they made use of for
their solution; whether these methods were merely
tentative or identical with the rules of position. We
will give a few instances :
1. Ads plot 8vo uvas, cat birMovs arod ºivopat,
Käyw Aaſºv oroo Tàs to as, orož, Terpät Movs
Other examples are given of problems which are
similar in principle.
2. The following refers to a bronze lion in a fountain,
from the mouth, eyes, and heel of which the water
flowed :
XaAlcéos etua Aéuv, kpóvvot 3'éuot du/data 8oud
Ka? otöua, a ÚV 8& 6évap befºrepoto Tööos
IPM)0e, 36 kpm Tſipa Św' juage 6eftov duaa
Kai Aatov Tptuga's kai Taipeg at 6évap :
Apktov čf weats 7Affoa, atdua, Čič’ dua Trávta
Kai ardua, ka? YMijval kal 0évap siré ºrdaaws.
3.
Eypeoff' 'Holyéveta rapéðpaue' réum Tov ćpubot,
Aetroaeums Tptagiðv oixetal oyèodtww.
4. The following possesses some interest, from its
connection with the name of Diophantus:
Otros rot Atoſhavtov exei Táºos, & p.6, a 6aoua
Kai tāºos ék Téxums uérpa Bioto Méyet :
"Ekrm v kovpigetv fliotov 6eos &mage uopmv
Aw8ékútmu ertóets uſixa ropeu XModeuv'
Tº 3’āp cºp'épôouátm To Yapºtov #yrato $6,70s
'Ek & Yáuwv Téuttw, rató' émévévaev čret.
At at 77A & Yetov Šetkov récos, jutov Tarpos
* Brunck, Anthologia, vol. ii. p. 477.
History.
n Questions
rom the
Greek
A R IT H M ET I C. #71
amongst others which ame attributed to hina, is one de History.
Arithmetic. Mérpov tº kpwégès Moºp' dºeNew Biêrov.
• mumerorum divisione, which we found to be the identi- \-y—
* * y º sº
S-N-Z IIev6os 8'āv trººpsora, trapmyopewy Śvtavroſs
:* - 5 y
Tijée Toogº ooºn Tépp' érépyge Biov.
The Epigram on the burdens of the mule and the ass,
which has been so frequently quoted by writers on Arith-
metic, we shall present to our readers in the translation
of Philip Melancthon :
Mulae asinaeque duas imponit servicius utres
Impletas vino, segmemque ut widit asellam
Pondere defessam vestigia powere tarda
Mula rogut : Quid chara parens cumctare gemisque
Uman ear utre two mensuram si mihi reddas
Duplum oneris tumc ipsa feram : sed si tibi tradam,
Unam mensurant fºunt orgwalia utrigue
Powdera: mensuras dic docte geometra istas.
(249.) Some authors” have attributed the invention of
cal treatise of Gerbert, with his prefatory letter to Con-
stantine, which we have had partieular occasion to
notice above, from its importanee in the eontroversy
about the first introduction of Arabic numerals. An
extended table of Pythagoras, which sueceeds, is clearly
the production of the same author, from its connection
with the methods mentioned in the treatise in question
for the multiplication of articulate numbers. In the
ratio cyclorum which follows, he speaks of the present
year 774, though he died in 735 ; and subsequently in
an astrological treatise, De Praecognitione copia, et paw-
pertatis futurae, he notices certain conjunctions and
configurations of the planets which threaten ruin to
Rules of
position , the rules of position to Diophantus, though it is impos- Serugaga or Seville and Corduba, and famine to the
attributed sible to discover upon what grounds. It is most probable Saracens, at least a century and a half before those
*Y* that the Greeks were in possession of some method of names were known, and the whole is merely an extract
authors to © , ſº e - * tº g ſº g -
Diophantus. analyzing and solving such questions, otherwise it is from a Spanish calendar of the XIIth or XIIIth CeIl-
hardly possible to conceive that they should have been tury. The whole treatise, De computo ecclesiastico, as
proposed in such number and variety; and when we well as those on other astronomical subjects, is clearly
consider the nature and difficulty of the problems the production of a much later age; in short, there is
solved by Diophantus, in those parts of his works which so great a part of these treatises to which he clearly has
remain to us, we should be fully justified in supposing no claim, that it is quite impossible for us not to look
that such methods were known. upon the whole as either spurious, or at least as of
Rnown to (250.) The Arabs were in possession both of the rules very doubtful authority. -
the Arabs, for double and single position, with all their applica- The fact is, that the formation of calendars, and the
.*.*.* tions, and in this instance had advanced far beyond their composition of treatises De computo ecclesiastico, was a
ºrived by I. asters: and wh ider h ll were favourite emplo t of the more learned monks in th
them. ndian masters; and when we consider now small were ploymen e
the additions which they generally made to the sciences XIIth, XIIIth, and XIVth centuries, and it was a
which passed through their hands, we might very common practice to attribute the latter to some cele-
naturally be inclined to suppose that their superior brated name: we have calendars of Roger Bacon
knowledge of these rules was derived from the Greek without number, as well as a treatise of this nature,
arithmeticians. There is, however, a vast gap in the though it is nearly certain that he had nothing to do
history of the sciences after the time of Theon, and it is with the one or the other. In that age such impos-
quite impossible to trace with certainty their trans- tures were easy, and were, indeed, considered merito-
mission to the Arabs, or to ascertain through what rious, when their object was to give additional honour
channels some portions of Greek Astronomy at least, if to a name such as that of Bede, so intimately con-
not of other sciences, were transmitted to the Hindoos : nected with the glory of the order to which he belonged.
under such circumstances we must rest contented with The first question in the collection would alone be
the rare and obscure hints which can be gathered from sufficient to throw considerable doubt upon their
the writings of authors who flourished between the VIIth authentipity.
and the XIIth centuries, who had access to many arith- Limax fuit ab hirudime invitatus ad prandium in-
metical and other writings which have perished since fra leucam unam : in die autem non potuit plusquam
that time. * wnam uneiam pedis ambulare. Dicat qui velit in quot
Arithmeti- (251.) Amongst the earliest and most remarkable of annosaut dies ad idemprandium ipse limaxperambulavit.
cal Writings these is our illustrious countryman Bede, amongst whose In the answer to the question, it is said, that the
* works there is a large collection of treatises on diffe- leuca, or league, consists of 1500 passus, and each passus
m...ba- rent arithmetical subjects, as well as many others De of 5 feet. Now it is very doubtful whether the leuca,
bly spurious computo ecclesiastico, and on several points of astrology or league, had yet become a recognised measure in
and astronomy: amongst the former is a collection of
a great number of arithmetical problems and puzzles,
which are extremely interesting under any circum-
stances, as the apparent originals of many of those
which appear in the writings of the Italian arithmeti-
cians, and which have been transmitted regularly down-
wards as stock questions to the authors of modern
times. We once felt inclined to assign them a much
earlier origin, and to suppose that they had been copied
by Bede from the works of the Greek arithmeticians,
particularly when we observe the resemblance between
many of those questions and such as are found in the
Greek Anthologia. A further examination, however,
has given us good reasons for thinking that all these
treatises are the production of a much later age;
* Gemma. Frisius, Arithmetica Practica, Methodus Facilis, 1581.
France, and it is still more doubtful that a Saxon
monk, residing in his monastery of Lindisfarn, should
have taken such a measure in preference to one which
was sanctioned by classical authority, or, at all events,
familiar to the persons to whom his writings were chiefly
addressed.
Though, for the reasons above-mentioned, we feel
compelled to deny these questions the interest and im-
portance which they would possess from the antiquity
assigned to them, yet they are not without interest, as
proving the general circulation, and even the antiquity,
of a set of very curious questions, many of which have
been familiar to us from our earlier years. We shall
mention some of them as they occur, without any
particular reference to the subject which we are im-
mediately discussing, with such remarks as may natu-
rally arise in connection with them. -
472 A R IT H M ET I C.
Arithmetic. Two men drive oxen on the same road : give me
XTX- two of yours, says the first, and I shall have as many
3. oxen as you; the other says, give me two, and I shall
have twice as many as you ; how many oxen had each 2
The same, or nearly the same question is given above
from the Anthologia. -
Quidam senior salutavit puerum cui dirit. Vivas
jili, vivas, inquit, quantum viristi et aliud tantum et
ter tantum addatgue tibi Deus unum de annis meis et
impleas annos centum. -
The same question is frequently repeated with slight
variations in its terms.
Quidam episcopus jussit 12 panes in clero dividi ;
Praecepit enim sic, ut singuli presbyteri binos accipe-
rent panes, diaconi dimidium, lector quartam partem,
ita tamen ut clericorum et panum idem sit numerus.
Tartaglia has proposed several questions which are
resolved upon the same principle as this. Of this
kind is the following: -
Eighteen persons, men, women, and children, eat 18
pigeons; the men two each, the women 1, and the
children # of one ; what number of men, women, and
children were there respectively P
A father, on his death bed, leaves his 3 sons 30
vessels, 10 of which are full of wine, 10 of them half
full, and 10 of them empty; in what manner must
they be distributed, so that each may receive an equal
quantity of wine and an equal number of vessels 2
If we reduce the conditions of this question to equa-
tions, we shall find -
a + y + z = 10 (1)
a' + y' + 2' = 10 (2)
a'ſ -- y" + 2" = 10 (3)
a + ar' + æ" = 10 (4)
y + y + y” = 10 (5)
z + 2 + 2* = 10 (6)
a + 4 = 3 (7)
• * + 4 = 5 (8)
* +4 - 5 (9)
The combination of equations (5) (6) (7) with (1)
(2) (3), gives
2 + 4 = 5
2
2 + 4 = 3
2
//
2” + 4 = 5
2
and consequently shows, that z = 1, 2' = a ', 2" = a”,
or that each must have as many empty bottles as full
ones; but it is evident, as well from the mature of the
question as from the equations themselves, that the values
ofa, a ', and a ", of y, y', and y”, and of 2, 2', and 2", are
interchangeable, and that the equations are not indepen-
dent of each other, and not sufficient therefore for the
absolute determination of the unknown quantities.
There are two sets of values which will answer the
conditions of the question,
~s’
4.
a = 5, y = 0, z = 5
lst. { a' = 1, y' = 8, 2' = I
a' = 4, y' = 2, 2" = 4
- r = 2, y = 6, 2 = 2
2d { a' = 4, y' = 2, 27 = 4
a'- 4, y' = 2, 2’, 3- 4
The following three questions, given by Tartaglia,
are of a similar character : -
A citizen dying leaves 27 vessels, 9 of which are full
of wine, 9 half full, and 9 empty, to be divided in
equal number and quantity between three monasteries;
namely, of Santa Maria dei Carmini, of Santa Maria
della Pace, and of Santa Maria della Consolatione ;
how must they be distributed ?
Two persons robbed a gentleman of a vessel of
balsam containing 8 ounces, and whilst running away
they met with a glassman, of whom they purchased
in a great hurry two vessels, one containing 5 ounces,
and the other 3 ; they at last reach a place of secu-
rity, and wish to divide their spoil; how must this
be done, so that each may have an equal portion ?
Three persons have stolen a vessel of balsam con-
taining 24 ounces, and have three vessels containing
5, 11, and 13 ounces respectively; in what manner
must they proceed to effect the distribution, so that
each may get an equal portion ?
The difficulty of questions of this kind consists in
their not being reducible to any regular analysis; the
conditions to which they are subject not being ex-
pressible in algebraical language. The following re-
presentation will show one of the sets of successive
steps which must be taken, in order to get an answer
to the question.
Vessels. . . . . . . . . . 24 i8 II 5
Successive contents 8 0 | 1 5
() 8 11 5
16 8 0 0
16 0 8 0
3 i8 8 0
3 8 8 5
8 8 8 0
The following question is taken from Tartaglia:
it is also found amongst those attributed to Bede,
brothers and sisters being substituted for husbands and
wives.
There are three men, young, handsome, and gallant,
who have three beautiful ladies for wives, who are all
jealous, as well the husbands of the wives as the wives
of the husbands : being neigbours, they go in com-
pany to visit a shrine where indulgences are granted,
and it happened that on their journey they have to
pass a broad river, with neither a bridge nor passage
boat; by good fortune, however, they find on the bank
a very small boat, which can take no more than two
at a time; in what manner must they pass, so as to
give rise to no suspicion of jealousy P
If A, B, C represent the husbands, and a, b, c their
respective wives, then a and b pass first, b returns and
takes over c, c returns and remains with C, when A and
B go over to a and b, A returns with a, and A and C
pass over to their wives, c returns and brings back a,
B returns and brings back b : they then, says Tartaghia,
attaccano il navetto alla ripa e se me vanno tutti a
braccio a braccio con le sue donne ał suo viaggia tutti
allegri e gelosi.
Historj.
N-N-"
A R I T H M ET I C. 473
Arithmetic. Tartaglia proposes the same question with 4 hus- from them, and they to devise after you are gone, History
S-N-1 bands and 4 wives, and the same method may clearly which of them shall have the keeping thereof, and that S-
be adopted for passing any number of them, without
violating the conditions, if it be allowed that the hus-
band can protect his wife, or the wife her husband.
The following are questions, similar in principle
though not in form, which appear in Bede, and which
have likewise been frequently copied by other authors,
probably from some common work. -
A person is carrying a wolf, a goat, and a bundle o
vetches, and meets with a river, which he can only pass
in a small boat, and which will only hold himself and
one of the other three; how must he contrive, so that
the wolf may be kept from the goat, and the goat from
the vetches 2
A man, his wife, each a waggon load, and their two
children, whose joint weight is equal to that of the
father or mother, have to pass a river in a boat which
can only bear the weight of a waggon load; how
must their passage be effected?
Other questions are of a very trifling kind, being
little more than a play upon words.
Bos qui tota die aratur, quot vestigia faciat in
wltima rigá 2
Of the same kind are the two following questions
from Tartaglia:
Uno cittadimoha un solo capretto ese nevuol domar uno
per uno al padre e uno al figliuolo; dimando come fară 2
Uno cittadino ha 3 fasani, li quali vorria domar a
duoi padri e duoi figliuoli e dargline uno per uno ;
dimando come lui fară 2
Other questions relate to the degrees of relationship
which result from the issue of extraordinary marriages.
Si duo homimes ad invicem alter alterius sororem in
conjugium sumpserent : dic (rogo) qua propinquitate
filii eorum sibi pertimeant 2
Si relictam vel viduam et filiam illius in conjugium.
ducant pater et filius, sic tamen ut filius accipiat matrem
et pater filiam : filii qui et his fuerint procreati dic
(quaeso) quali cognationi subjugamtur 2
Ilivinations (252.) There are many questions proposed about the
of numbers divination of a number, when the result is given, which
º ** arises from its being subjected to certain modifications,
data, tº g ſº © ºn tº a te
from additions, multiplications, &c.
Quomodo divinandum sit, quaferia septimande quilibet
homo quamlibet rem fecisset.
A is directed to double the number, to add 5 to it,
to multiply the sum by 5, and then by 10, and to give
the result: B, who is informed of the operations to
which it has been subjected, subtracts 250 from it,
and the number of hundreds which remain, is the
number required: in other words, if a be the number,
2 ar, 2 a + 5, 10 a + 25, and 100 a + 250, will de-
note the successive results of the operations performed
upon it; and, therefore, (100 a + 250) — 250 = 100 a.,
from whence the answer is obtained.
(253) Such divinations were a source of a very popular
species of pastime, and were in some measure equiva-
lent to the solution of an equation, when the connection
between the unknown quantity and the result, which
arose from certain conditions, was previously known.
The following are amongstethe most common of those
which are found in Tartagliàº'ànd later writers:
‘Game of “If in any company,” says Mellis, “you are dis-
the ring posed to make them merry by manner of divining, in
delivering a ring unto any one of them, which after you
have delivered it unto them, that you absent yourself
WOL. I. "
you, at your returne, will tell them what person hath it,
upon what hand, upon what finger, and what joint.
Which to doe, cause the persons to sit downe all on a
rowe, and to keepe likewise an order of their fingers;
now after you are gone out from them to some other
place, say unto one of the lookers on, that he double
the number of him that hath the ring, and unto the
double bid him add 5, and then cause him to multiplie
that addition by 5, and unto the product bid him add
the number of the finger of the person that hath the
ring ; and, lastly, to end the work, beyond that number
towards his right hand, let him set downe a figure, sig-
nifying upon which of the joints he hath the ring, as
if it be upon the second joint, let him put downe 2,
then demand of him what number he keepeth, from
the which you shall abate 250; and you shall have
three figures remaining at least. The first towards
your left hand shall signifie the number of the person
which hath the ring, the second, or middle number,
shall declare the number of the finger, and the last
figure towards your right hand shall betoken the num-
ber of the joint.”
If a be the number of the person, y of the finger,
and z of the ring, then the course of the process gives
successively 2 r, 2 a + 5, 10 a + 25, 10 a + 25 + y,
100 a + 250 + 10 y + 2, which, diminished by 250,
gives the number expressed by the three digits a, y, z.
Three persons play at the following game : one of Other
them must form a wish which should be chosen em- games.
peror, which king of France, and which king of Naples;
and the object of the game is, that a fourth person
should be enabled from certain data to divine upon
whom his choice had fallen. For this purpose, give to
the first (say Hannibal) the number 1, to the second
(Scipio) the number 2, and to the third (Pompey) 3,
and tell him to double the number of him whom he
wishes to be chosen emperor; add 5 to it ; multiply
the sum by 5, add to the product the number of the
person whom he wishes to be king of France, add 10
to the result, multiply by 10, and then add the number
of the person whom he wishes to be king of Naples;
if 350 be subtracted from the last sum, the remaining
digits will indicate the emperor and the two kings in
their proper order. - #
In this case, if a be the number of the emperor, y of
the king of France, and z of the king of Naples, then
the process gives, successively, aſ, 2 a., 2 a + 5, 10 a +
25, 10 a + 25 + y, 10 a + 35 + y, 100 a + 350 +
10 y, and 100 a + 350 + 10 y + z.
A similar question would be amongst three persons
who have secreted three articles, such as a glove, a
purse, and a ring, to determine by whom the first has
been taken, by whom the second, and by whom the third.
Another pastime described by Tartaglia was as fol-
lows:
Three persons seated round a table, upon which there
are 18 balls, and also a piece of gold, a piece of silver,
and a piece of copper: in the absence of a fourth, each
person takes a coin; if the first takes the piece of gold,
he also takes one ball, if the second 2 balls, and if the
third 3; if the first takes the piece of silver, he takes
2 balls, if the second 4, and if the third 6 ; if the first
takes the piece of copper, he takes also 4 balls, if the
second 8, and if the third 12 ; the absentee upon his re-
turn is required, from the number of balls which remain,
3 Q
474
A R I T H M E. T.I. C.
Arithmetic. to assign the persons who have respectively taken the
^-y-' three coins. *
If the three vowels, a, e, o, correspond to gold, silver,
and copper, respectively, the persons will be indicated
by the order of their occurrence in the following words,
according as 1, 2, 3, 4, 5, or 6 balls remain on the
table.
l 2 3 4 5 6
Absequor. Belandos. Latrones. Dochmate. Ocreas, Reportant.
1 4 12 2 2 12 l 8 6 4 2 6 4 4 3 2 8 3
(254.) The principles upon which these puzzles and
pastimes are founded, will show how easily they may be
varied ; and considering how much they were employed
for the purposes of popular amusement, and how
admirably they were calculated to excite the surprise
and admiration of those who were ignorant of the
mode in which they were formed and answered, we
may naturally expect to find them modified in a vast
variety of forms. Bachet de Meziriac, the commenta-
tor on Diophantus, was the author of a work on the
subject of such problems, * containing a collection of
all that were known in his time, accompanied by de-
monstrations and remarks, which in many cases show
uncommon ingenuity; and a still greater number of
them may be found in the Mathematical Recreations of
Ozanam, as enlarged by Montucla. Referring our
readers to their works for further information on this
very entertaining subject, we shall conclude our obser-
vations relating to it, with a notice of the problem of
the Turks and Christians, which has become unusually
celebrated.
A ship, on board of which there are 15 Turks and
15 Christians, encounters a storm, and the pilot de-
clares, that in order to save the ship one-half of the
crew must be thrown into the sea: the men are placed
in a circle, and it is agreed that every ninth man must
be cast overboard, reckoning from a certain point. In
what manner must the men be arranged, so that the lot
may fall exclusively upon the Turks 2
If the five vowels, a, e, i, o, u, represent the num-
bers 1, 2, 3, 4, 5, respectively, the rule for the arrange-
ment of the men will be expressed by the occurrence
of these vowels in the following distich or rubric:
From numbers’ aid and art
Never will fame depart.
The vowel o indicates 4 Christians.
*Problem of
the Turks
and Chris-
tians.
20 . . . . . . 5 Turks.
6 . . . . . . 2 Christians,
0 . . . . . . 1 Turk.
* . . . . . . 3 Christians.
- 0 . . . . . . 1 Turk.
(?, . . 1 Christian.
en 6 . . . . . . 2 Turks.
€ . . . . . . 2 Christians.
2. . 3 Turks.
Q. . . . . . . 1 Christian.
6 . . . . . . 2 Turks.
€ . . . . . . 2 Christians.
Q. . . . . | Turk.
Bachet de Meziriac gives the following rubric :
Mort tu me falliras pas
En me livrant le trespas.
The same purpose is answered by the Latin hexame-
ter,
Populeam virgam mater regina ferebat. History,
Tartaglia has given a series of nonsense verses, S-V-
which will answer, respectively, for the cases where the
lot falls on every third, fourth, fifth, sixth, seventh,
eighth, ninth, tenth, eleventh, or twelfth person: those
which correspond to the 9th are,
Documenta est decima perfecta,
Or,
O brunetta rizza ale ferita Elena.
Or,
O puella irata est fetida effecta ;
and for every 10th,
Rear Anglicus certo bona flamina dederat.
(255.) If any reliance could be placed upon the truth of Legend of
the following story, related by Hegesippus,” it would Josephus,
appear that the principles of such arrangements were related by
understood and practised even in ancient times; after “4”PP*
the storming of Jotapata by Vespasian, of which Flavius
Josephus, the historian, was governor, he escaped with
40 of his companions to a lake or cavern; despairing
of better fortune for their country, they determined on
destroying themselves, notwithstanding the earnest
exhortations of their commander, who was anxious
that they should commit themselves to the clemency
of Vespasian: finding all his entreaties vain, he at
last hit upon the expedient of placing himself in such
a position in the circle in which they were arranged,
that every third man, reckoning from a certain point,
being put to death, he should be one of the two which
remained. The eloquence which had failed in persuad-
ing the whole body, was successful with his sole sur-
viving companion; they agreed to live, and at once
surrendered themselves to the mercy of their conque-
TOTS. -
(256.) Stifelius has given a very elegant theory of the Theory of
steps which most probably led to the invention of the Stifelius of
rules of position, which we shall give in his own words: the inven-
& tº gº tion of the
Inventurus author regulam falsi, dissimulabat se rules of
scire numerum illum, a quo 2 subtracta relinqueret 3. position.
Recepit ergo primo 4 loco numeri illius : quem cum
examinaret subtrahendo 2, vidit (loco 3) relinqui solum-
modo 2. Itaque deficere vidit unitatem et hunc nume-
Tum, cum defectu illo, separatim annotavit. Deinde
recepit 6, quem numerum cum examinaret subtrahendo,
vidit (loco 3) relinqui 4. Itaque superfluere vidit uni-
tatem : et sic senarium cum superfluente unitate etiam
separatim annotavit. Et sic posted exploravit qua
zationeer adnotatis numeris produceretur quinarius, qui
videlicet subtractis a se 2, relinqueret 3. Facile enim
videre fuit, qua ratione hoc fieret, scilicet et aggregatione
numerorum receptorum (id est 4 et 6) fiebant 10. Et
ea aggregatione falsitatum, (id est 1 et 1) fiebant 2.
Itaque er divisione 10 per 2, proveniebatur 5, id est,
numerus qui quarebatur.
Figura positionum praedictarum
4 10 6
a
Postea recepit 4 et 7#. eos simili modo tentavit
invenire quinarium. }*bum videret figuram hujws
Minus I 1 Plus
inventionis sic stare (ut sequitur.)
* Problémes plaisans et delectables qui se font par les nombres,
1612. *
* De Bello Judaico et urbis Hierosolymitana, excidio, lib, iii.
cap, 15.
A R IT H M ET I C.
475
I
2.
3
satis videbat, quod simpler aggregatio non responderet
wtrobique inventioni priori. Tentavit igitur omnes in-
veniendi anodos possibiles, donec inveniret aggregationem
nediante multiplicatione in cruce respondere: scilicet bis
4 et semel 7 (id est 8 et 7) faciunt 15, qua divisa per 3,
faciunt 5. -
Figura unventionis praedicta.
4 I5 7
2'2^
Minus 1 3 2 Plus
Postea, ut posset concludere, tentavit ejusdem numeri
inventionem per 4 et 100: et eribat figura inventionis,
pradicto modo, hac, respondens rei.
4 480 100
D-3
Minus 1 96 95 Plus.
Conclusit ergo inventiones hujusmodi esse ratas con-
stanter, ubi falsitatum altera deficit, seu minus est, altera
superfluente, seu plus existente. Deinde convertit se ad
derteram, tentans invenire hujusmodi inventiones per
falsitates utrobique superfluentes. Recepit ergo pro
experimento 7 et 8, quibus numeris voluit invenire qui-
ºnarium, modo praedicto : unde figura inventionis sic
exibat.
7
8
>3
T'lus 2 3 Plus
Sed hic cum videret aggregationem nihil fieri, tentavit
rem per subtractionem. Et sic vidit operationem esse
bonam et respondere rei.
7.
8
N
`s
Plus 2 I 3 Plus.
Hoc est, 2 de 3 relinquunt, 1 divisorem : et 3 in 7
multiplicata faciunt 21 : et 2 in 8 faciunt 16. At 16
de 21 relinquunt 5 dividenda per 1 divisorem. -
Postea, ut de inventione a dertris etiam concluderet,
Tecepit 7 et 100, quibus numeris quinarium produceret,
7modo praedicto : ét exivit figura inventionis sic, ut
sequitur.
*
7 465 i00
Plus 2 93 35 Plus
Postea videns successum se habuisse talem a dertris,
vertit se, ut idem experiretur etiam a sinistris. Recepit
ergo 3 et 4, id est, numeros quos sciebat allaturos esse
falsitates deficientes utrinque. Per eos itaque quasivit
quinarium producere sicut prius, et invenit hanc figu-
7"Q/72,
I
Post tantos successos in questionibus ludicris, coepit
autor negotium illarum operationum transferre ad ob-
scuras quaestiones, numerorum abstractorum et contrac-
Minus 2 1 Minus.
torum. Sentiens ergo immensam latitudinem negotii
fillius, magnifice lactabatur, reputans se reperisse thesau-
Tum artis incomparabilem.
(257.) An addition was made to the Rule of False by
Gemma. Frisius, which Stifelius characterises as inven-
tum valde egregium : it consisted in applying it to the
solution of such equations as
a aº = m, or, a aº + b = m,
a ar” = m, or, a aº + b = m,
involving the squares, cubes, or higher powers of the
unknown quantity ; and the principle of it was merely
that of considering aº, aſ *, r" * (where a' and a' are the
positions) as simple quantities, such as X, X′, X",
and treating them according to the ordinary rule ; the
determination of the value of X immediately leads to
that of w. The following is an example :
To find two numbers in the proportion of the num-
bers 2 and 3, whose product shall be equal to 864.
103.68
1st Position. Square. Square. 2d Position.
g 4 16
Product 6 24 Product
3 9 19 6
Error 858 840 Error
* 18
Assume 2 and 3, their product is 6, the error 858;
again, assume 4 and 6, their product is 24, the error
840; the difference of the errors is 18: multiply 858
into 16, (the square of 4,) and from the product sub-
tract the product of 4 × 840, which is 3360; the
difference, 10368, divided by 18, gives 576, the square
of 26, the first of the two numbers required.
(258.) We shall conclude our observations on this rule
with an extract from Recorde, who, after remarking, that
in other parts of Arithmetic the numbers are taken in
just proportion, whilst in this rule they are not found
by orderly work, but taken at all adventures, proceeds
to say, “that sometimes being merie with my friends,
and talking of such questions, I have caused them
that proposed such questions, to call unto them such
children and ideots as happened to be in the place, and
History.
Extension
of the
rules of
position by
Gemma.
Frisius.
Statement
of these
rules by
Recorde.

3 Q 2
476
A R L T H M E T I C.
Arithmetic... to take their answere, declaring that I would make
S-v- them solve those questions that seemed so doubtful ;
and, indeede, I did answere to the question, and worke
the triall thereof also by those answeres which they hap-
pened at all adventures to make, which numbers seeing
that they be taken as maketh false, therefore, is this
rule for triflenesse, called the Rule of Falsehood, which
rule, for readinesse of remembrance, I have comprised
in these few verses following, in form of an obscure
riddle.
Gesse at this worke as hap doth leade,
By chance to truth you may proceed.
And first worke by the question,
Although no truth therein be done.
Such falsehoode is so goode a ground
That truth by it will soon be founde.
From many bate to many moe,
From too fewe take too fewe also.
With too much joyne too fewe againe,
To too fewe adde too many plaine.
In crosse waies multiplie contrarie kind,
All truth by falsehoode for to finde.
Whatever other merits the composition of this riddle
may possess, it is impossible to deny it the essential
one of obscurity.
(259.) The different species of Progressions, whether
Arithmetical, geometrical, or musical, as well as the sub-
ject of combinations and permutations, whether we con-
sider their theory, or a great portion of the problems
which they lead to, more properly belong to Algebra
than to Arithmetic, though they have generally been
included in books on the latter subject, as well as the
former. The great extent, however, to which this
article has proceeded, compels us to pass them over
Arithmetical
and geome-
trical pro-
gressions.
without any notice beyond a few remarks; and we feel
the less regret at the omission of more elaborate details,
however interesting they might be, as they involve the
developement of no principle which is essentially con-
nected with the progress of Arithmetical science.
(260.) The different progressions of numbers were the
object of the particular attention of the Pythagorean and
Platonic arithmeticians, who enlarged upon their most
trivial properties with the most tedious minutemess.
Their speculations, however, were directed to the eluci-
dation of the mysterious harmonies of the physical and
intellectual world, and had, therefore, no concern with
the business of real life; and they, consequently,
Particularly
noticed by
the Pytha-
gorean arith-
meticians.
passed over, as altogether unworthy of notice, the solu-
tion of those questions which maturally arise from these
progressions, and which appear in such numbers in
Hindoo, Arabic, and modern European books on Arith-
metic.
Questions (261.) Amongst the questions attributed to Bede
on Arithme- is the following: -
* There is a ladder with a hundred steps; on the first
e step is seated one pigeon, on the second 2, on the
third 3, and so on, increasing by one from each step.
Tell, who can, how many pigeons were placed upon the
ladder?
Of the two following questions, which appear in all
modern books of Arithmetic, the first originated with
the Venetian arithmeticians, as might be conjectured from
its subject; the second, of whose real origin we are
ignorant, is the subject of a very common and popular
wager.
How many strokes do the clocks of Venice strike in
24 hours ? «, - r
If a hundred stones be placed in a right line, one
yard from a basket, what length of ground must a . History.
person go who gathers them up singly, returning with
them one by one to the basket?
(262.) The extraordinary magnitude of the numbers Geometric
which result from the summation of a geometrical series, Progres-
is well calculated to excite the surprise and admiration of “”
persons who are not fully aware of the principle upon
which the increase of its terms depends ; and examples
are not wanting, where the rash and the ignorant have
in consequence been seduced into ruinous or impossible
engagements.
The most celebrated of these questions is the one Celebrated
which tradition has represented as the terms of the question.
reward demanded of an Indian prince by the inventor of
the game at chess; which was a grain of wheat for the
first square on the chess board, two for the second,
four for the third, and so on, doubling continually to
64, the whole number of squares.
Lucas de Burgo, who has solved this question, .
makes the number of grains
1844.67.440737.09551615
which he proceeds to reduce to quantities of a superior
denomination as follows:
6912 grains make a lira of Perugia.
133 lire . . . . . . . . mina.
3 mine . . . . . . SQIIla.
4 some . . . . . . corba
20 corbe . . . . . . archa
40 arche . . . . . . barca
100 barce . . . . . . magazeno.
100 magazeni ... castello.
The amount, expressed in castles of corn, would be \
209022 with a fraction; he then recommends his reader
to attend to this result, as he would then have a ready
answer to many of these babioni ignari de la Arithme-
tica, who have made wagers on such questions, and
have lost their money.
The case is similar to that of the ignorant and un-
fortunate host who undertook, on certain conditions,
to give as many dinners to 10 persons as they could
place themselves in different arrangements at the table.
In cases, indeed, of the formation of the terms of a
geometric series, or in problems on permutations,
where the result arises from the continued multiplica-
tion of the same or different factors, we speedily arrive
at numbers which surpass the powers of the imagination
to conceive ; and arithmeticians have delighted in the
proposition of questions which lead to such surprising
conclusions. The amount of a penny put out to in-
terest at five per cent. per annum, at the birth of our
Saviour, would require more than 40 places of figures
to express it; and many attempts have been made to
exhibit this result in a form which may come within
the grasp of the human mind. Political economists
have appealed to the same principle to account for the
rapidity with which population increases, when its pro-
gress is not checked by famine and disease ; whilst
the speculator on languages finds an unlimited supply
of words in those permutations of the letters of the
same or different alphabets, which form sounds within
the compass of human utterance.
(263.) We cannot conclude this history of Arithmetic Conclu-
without making some observations on the difficulty of the sion.
undertaking, and upon the many necessary defects under
which it must labour. With the exception of the very able
and interesting work of Professor Leslie on the Philoso-
A R I T H M E T I C.
477
tion of printing, at east, is very small ; and when we . History
have mentioned the great names of Lucas de Burgo, ‘---"
Arithmetic, phy of Arithmetic, to whom we are under great obliga-
S-N-'tions for having sketched an outline which we have en-
deavoured to fill up, the attempt may be considered as
altogether new. The subject is hardly noticed in the
work of Montucla, which is otherwise so admirable in
the early history of the mathematics; and the meagre
sketch which Kaestner has given of some insulated
works on the subject, generally contrives to omit
almost every particular which is essentially connected
with the history of the progress of the science; in
short, there does not exist any source of information
on this subject which can be deemed trust-worthy and
authentić, except in the original authors themselves.
In writing the history of a science, the facts are
generally distinct and positive, and the adjudication of
the honour of different inventions and improvements,
and of the just claims of different anthors to them,
may for the most part be made with certainty, from
the examination of the original works taken in the
order of time. On such subjects there is rarely any
conflicting testimony, and it is seldom necessary to
proceed to the nice weighing of probabilities, which is
so frequently requisite in the history of events ; but
there are other difficulties, almost as considerable,
which a scientific historian must encounter : he must
not only perfectly understand the subject upon which
he writes, but he must also understand it under the
form in which it appears in the work which he ex-
amines: he must not only be able fully to appreciate
the importance of a discovery or improvement, but
likewise to determine how far a hint, or partial antici-
pation of it, may have contributed to its full develope-
ment: he must weigh the relative merits of the inven-
tor and the expositor, of him who discovers a new
region in science, and of him who, by subsequent and
more minute examination, ascertains its full extent
and boundaries, and makes its productions generally
known.
In the history of Arithmetic, however, these difficul-
ties present themselves under their least formidable
aspect; the subject is easy under all its forms, and
there can be little doubt or controversy about an im-
provement when made, though some might arise on
the different steps which lead to it. Again, the num-
ber of original authors on this subject, since the inven-
Stifelius, Tartaglia, Stevinus, and Napier, the additions
made to the science by other authors are, generally
speaking, of a very trifling importance; for on all
subjects, where the difficulty of acquisition does not
necessarily limit the number of authors, the great
majority of writers are mere copiers of their predeces-
sors, and are generally contented with some little altera-
tion in form rather than in matter; and this is parti-
cularly the case with Arithmetic, a subject which so
many must learn, and so many must teach ; where the
great number of readers has a natural tendency to
make a great number of authors; and where the sim-
plicity of form under which the rules of the science are
exhibited, and the ease with which they may be learnt
and practised, must always be considered of more im-
portance than the originality of the matter.
But though the number of authors whose works must
be consulted is small, when we are in search of great and
essential improvements in this science, yet there are other
occasions where it is requisite to consult all those which
belong to a particular period. This is the case when we
wish to examine the progress of an improvement, and
to ascertain the rapidity with which it came into general
use, and the variations of form which it underwent
between its first discovery and its final developement.
Of this kind is the history of decimal fractions, from
the first publication of Stevinus to the middle of the
XVIIth century. In all cases of this kind we are sensi–
ble that this history must labour under great deficiencies,
as there are no libraries in this country which contain
all or nearly all the books which are requisite for this pur-
pose, and there are no classed catalogues by which we
can ascertain, without great labour, all the treasures
which they contain.”
* We are glad to learn, that in one case, at least, this deficiency
is speedily to be supplied, and that a classed catalogue of the library
of the British Museum, and also of the magnificent gift of the King,
is in active preparation. It is to be greatly lamented, however, that
the national bounty should be distributed in such scanty sums to the
support and increase of this great and important establishment; and
that instead of a paltry allowance of eight hundred pounds per
annum, for the purchase of books for the library, it should not be
increased to at least as many thousands.
A P P E N D I X.
(264.) SINCE the first part of this article was written
Work of guages and customs of several South American tribes;
the Abbé and printed, we have procured a copy of the work of the information which he procured was chiefly from
i. * the Abbé Hervas, entitled Aritmética di quasi tutte le personal communication with them, and his inquiries.
important
information
mazioni comosciute; it contains the numerals in 175
languages, including those of more than thirty South
were specifically directed to the construction of their
numeral language, and to their practical methods of
on South American tribes, which he obtained chiefly from the numeration. The materials which this work contains
*: ex-Jesuit missionaries who resided at Rome, after they are particularly valuable, not only from their not exist-
Illil Ileraſ.S.
had been obliged to quit their missions in South
America, upon the extinction of their order; amongst
those he particularly mentions Clavigero, the learned
historian of Mexico, his native country, Gilii, the histo-
rian of the missions on the Orinoco, Camano and
Velasco, the authors of important works on the lan-
ing in any other works, but likewise from their relating
to tribes, many of which are in the lowest state of
civilisation, amongst whom we must look for the most
certain indications of the influence of practical methods
of numeration upon the formation of their numerals.
(265.) Of the following four sets of numerals, which
478
A R H T H M E T I C.
Arithmetic. possess some points of resemblance, the first belongs to
\–N2–2 the Qquichuan, or ancient Peruvian language of the In-
Numerals eas, which was spoken anciently in Peru, and the influence
of the of which extended for more than 40 degrees of latitude
Araucani, along the western coast of America. € SeCOT1C1 IS
Aimarri,’ from the language of the Araucani, the inhabitants of
and Sapibo- Chili, who were likewise included in the great empire
CODES. of the Incas. The third is that of the Aimarri, a tribe
in the north-eastern parts of Peru; and the last, of the
Sapibocones, a neighbouring tribe.
Qquichua. Araucana. Aimarra. Sapibocona.
1. Huc, Kiſie, Mai, Pebbi.
2. Escai, Epu, Paya, Bbeta.
3. Kimsa, Kula, Kimsa, Kimisa.
4. Tahua, Meli, Pusi, Pusi.
5. Pichca, Kechu, Pisca, Pissica.
6. Socta, Kayu, Sogta, Succuta.
7. Canchis, Relghi, Pacalco, Pacalucu.
8. Passac, Pura, Kimsacalco, Kimisacalucu
9. Iscon, Ailla, Pusicalco, Pusucalucu.
10. Chunca, Mari, Tunca, Tunca.
11. Chunca Marikifie, Tuncama- Tuncapea-
huc miyoc, - yani, pebbi.
12. Chunca is-Mariepu, Tuncapayani, Tuncapeab-
cai niyoc, beta.
20. Iscaichun-Epumari, Payatunca, Bbetattunca.
Cà, * *
30. Kimsa- Kulamari, Kimsatunca, Kimisa-
X- chunca, tunca.
40. Tahua- Melimari, Pusitunca, Pusitunca.
chunca, -
100. Pachac, Pataca, Tataca, Tuncatunca.
1000. Hua- Huaranca, Huaranca, Tuncatumca-
ranca, tunca.
1000000. Hunu.
The two first systems are equally perfect, and similar
in construction, though all the terms below 100 are
essentially different from each other. The expressions
for ll and 12 in the first, mean ten one with, ten two
with, the signification of the postposition yoc being
with, the partfele ni being merely interposed for the
sake of euphony: in the second, the expressions for
the same numbers mean ten one, ten two. In the three
first systems we find the same terms for 100 and 1000,
affording an additional illustration of the truth of the
observations made in Art. 17 and 21, on the trans-
mission and adoption of the names of the higher
orders of superior units.
The second and third of these systems are curious
examples of the partial borrowing of numerals, by one
people from another more advanced in civilisation ;
the names for 1 and 2 are most probably native in
both, and that for 4 in one of them ; whilst the names
for 3, 5, and 6 are clearly Peruvian ; the names for
7, 8, and 9 are clearly compound, meaning two five,
three five, four five; calco, in one, and lucu, in the other,
meaning five, or hand; showing that the influence of
a natural method of numeration manifested itself even
in a case where part of the numerals were borrowed
from a nation who had altogether abandoned this
manual Arithmetic. The duplication and triplication of
the name for 10, in order to denote 100 and 1000, a sim-
ple and natural artifice for the expression of such num-
bers, will receive an additional illustration in the follow-
ing system of numerals of the Cayubabi, a tribe inhabiting
the banks of the Mamorè, which runs into the Marànon,
1. Carata. 10. Bururuche. History.
2. Mitia. 11. Bururuche-caratorogicné.
3. Curapa. 12. Bururuche-mitiarogicné. Numerals
4. Chadda. 19. Bururuche-chaddarirobogiené. &: bi
5. Maidaril. 20. Mitiaburuche. yūDabn.
6. Caratarirobo. 30. Curapabururuche.
7. Mitiarirobo. 100. Buruche buruche.
8. Curaparirobo. 1000. Bururuche penèbururuche.
9. Chaddarirobo. -
Hervas says, that the name for hand is arue, and
that the names 6, 7, 8, 9, respectively, mean one hand
with, two hand with, three hand with, four hand with.
The name for 10, or bururuche, is probably derived from
the reduplication of arue, quasi aruearue, or hand
hand. If this derivation be well-founded, the name
for 100 would be equivalent to hand hand hand hand, a
very remarkable result of the composition of a simple
term.
(266.) A still more remarkable example of the same Numerals
fact will be found amongst the numerals of the Coran lan- of Cora,
guage, which is spoken in New Galicia, which we now Yucatan,
subjoin, in conjunction with those of Mexico and *.
Yucatan, with which they are intimately allied.
Azteck. Yucatam. Coran.
I. Ce, Humppel, or yax, Ceåut.
2. Ome, Cappel, or ca, Hualpoa.
3. Yei, Oxppel, or yox, Huaeia.
4. Nahui, Cammpel, or cantzel, Moâcoa.
5. Macuili, Hoppel, or ho, Amxuoi.
6. Chicuace, Uacppel, or uac, Acevi.
7. Chicome, Uueppel, or uuc, Ahuapoa.
8. Chicuei, Uaxacppel, or uaxac, Ahuaeica.
9. Chicunahui, Bolomppel, or bolon, Amoacua.
10. Matlactli, Lahunppel, or lahun, Tamoãmata.
11. Matlactli-occe, Huncahumppel, Tamoãmata-
apon-ceåut.
12. Matlactli- Lahca, Tamoãmata-
On Ome, apon-hualpa.
15. Chaxtoli, Holhunte, ''.
16. Chaxtoli-occe,
20. Cempohuāli, Kal, or hunkal, Ceitevi.
30. Cempohuali-i- Ceitevi-poan-
pan-matlactli. tamoàmata.
40. Ompohuāli, Cakal, Huahcatevi.
60. Epohuali, Oxkal, Huaeicatevi.
100. Macuilpohuāli, Hokal, Anxiitevi.
200. Matlacpohuāli, Lahunkal, Tamoãmata-
tevi.
400. Cen-tzontli, Ceitevitevi.
800. Ontzontli.
8000. Ce-xikipili, Hunpic, or pic.
In the list of Mexican numerals which is given in Remarks on
Art. 28, there are both deficiencies and inaccuracies : Mexican
the name for 15 is chartoli, and the numeration re-numerals.
commences from it; the expression for 16 being fifteen
one, for 17 fifteen two, and so on, precisely in the same
manner as in the Welsh numerals, (Art. 22.) The
name for 5, macuili, is derived from maill, or hand;
and the composition of the terms for 6, 7, 8 and 9,
shows that chicu possessed a similar meaning, which
appears again in the term for 15. The name tzontli,
for 400, signifies, also, hairs of the head; and, pro-
bably, in ancient times was equivalent to innumerable,
having subsequently acquired a definite signification,
in the same manner as Avpua among the Greeks, when
A R I T H M E T I C.
479
Arithmetic, their numeration became more systematic.” Every
circumstance which tends to illustrate the composition
of the Mexican numerals possesses more than common
interest, as they constitute the most perfect example of
the vicenary scale, with the quinary and denary scales
equally subordinate to it.
Remarks on In the Yucatan, or Mayan, numerals, there are two
the unerals sets of names for the digits, which are both used, and
*** whose chief difference consists in the addition of the
final ppel. The expression for ll means one ten, for
12 two ten, for 15 five ten ; a species of composition
which might be ambiguous, if the system were denary
and not vicenary, The term pic, or humpic, eight
thousand, or one eight thousand, is the termination of
the Yucatan numerals. When the Yucatani speak of
persons, they add the final tul, instead of ppel; thus,
huntul means one person, catul, two persons, lahcatul,
twelve persons. The Coran numerals which are given
above, are those which are used for inanimate things;
for living beings they postpone the particle man. Such
instances of imperfect abstraction in the formation of
numerals are not uncommon in South American lan-
guages. -
In the last of these systems the terms for 6, 7, 8, 9
are clearly compound. The general term for hand in
the Coran language is noamati, which is clearly the
basis of the name for 10; the expression for 11 means
ten above one, that for 12, ten above two ; the name for
20 is compounded of ceòut, one, and tevit, which is
equivalent to the generic term homo, or persona, whilst
that for 400 is one twenty twenty, or more literally one
erson person,
(267.) The following numerals of the Otomiti, a tribe
allied to these above-mentioned, both in geographical
situation and language, presents an example so com-
mon amongst Celtic nations, of the vicenary scale pro-
ceeding as far as 100 and then merging in the decimal.
Remarks on
the Coran
numerals,
Numerals of
the Otomiti.
1. Na. 12. Detta-ma-yoho.
2. Yoho. 19. Detta-ma-gueto.
3. Hiu. 20. Doté.
4. Goho. 30. Doté maretta.
5. Kueta. 40. Yoté.
6. Rato. 50. Yotē maretta.
7. Yoto. 60. Hiāte.
8. Hiato. 80. Huête.
9. Gueto. 100. Nato.
10. Detta. 1000. Namao.
11. Detta-ma-na.
In this system, the names for 6, 7, 8, 9 are analo-
gous to those for 1, 2, 3, 4, a clear indication of the
quinary seale. The name doté, for 20, is probably de-
rived from yohè, man, which is its meaning in so many
South American languages.
(268.) The numeral systems given above are those
which have received the most complete developement;
those which follow are not only extremely limited in
extent, but may be considered as the expression of the
practical methods of numeration, which are required
for all numbers which exceed the radix of the natural
scales.
Numerals of the Guaranies. (See Art. 30.)
1. Petey.
2. Mocòi.
3. Mbohapi.
* The term cemiluti, in the Coran language, signifies the hairs of
zhelicad, and also innumerable. See Art. 30.
Numerals of
the Gua-
ranies,
4. Irundi. - - , History.
5. Irundi hae nirāi, four and another, or ace popetei, S-7
or the one hand, where po is hand, and ace the determi-
nate article.
6. Ace popetei hae petèi abe, the one hand and one
besides. -
9. Ace popetei hae irundi abe, the one hand and four
besides,
10. Ace pomocoi, the two hands.
20. Mbo-mbi-abe, hands feet besides.
30. Mbo-mbi hae pomocoi abe, hands feet and two
hands besides.
The missionaries never heard a Guarani count be-
yond 30.
(269.) The Omoguas, a tribe living in the kingdom of Of the
Quito, and speaking a dialect of the Guarani language, Omoguas.
notwithstanding their immense distance from each
other, have only five numerals, the last of which, upa-
pua, signifies hand. By the combination of these,
however, with the expressions for the hands and feet,
they can proceed as far as a hundred.
(270.) The following are the numerals of the Of the
Zamucoes, one of the numerous tribes of Paraguay : ***.
1. Chomara.
2. Gar.
Gaddive.
Gahagami.
Chuenayiminaete, finished hand.
Chomarahi, one of the other
. Garihi, two of the other.
10. Chuena yimanaddie, finished two hands.
11. Chomara yiritie, one of a foot,
20. Chuena yiriddie, finished feet.
The missionaries never heard a Zamuco express in Their mode
words a number greater than 20 : any number greater .
than 20 is designated by the term unaha, many : if the . beyond
number greatly exceeds 20, they say unahapuz, very 20,
many; and to express in terms of increasing intensity
their opinion of the magnitude of very large numbers,
they say unaahapuz, unaaahapuz, unaaaahapuz, re-
duplicating continually the sound of the letter a. In
common cases, however, in speaking of numbers within
the compass of their methods of numeration, they
take in their hand grains of rice, little stones, or seeds,
and count them out until they have reached the number
required, and then point to them, saying choetie, like
this. *
(271.) The numerals of the Luli, another tribe of Of the
Paraguay, present an example of a very singular con- Luli
struction, where the mere poverty of words has caused an
appearance of the quaternary scale.
1. Alapea.
2. Tamop.
3. Tamlip.
4. Lokep. -
5. Lokep moilè alapea, four with one, or is-alapea,
hand one.
6. Lokep moilè tamop, four with two.
7. Lokep moilè tamlip, four with three.
8. Lokep moilè Iokep, four with four.
9. Lokep moiſè lokep alapea, four with four one.
10. Is-yaoum, all the fingers of hand.
11. Is-yaoum moilè alapea, all the fingers of hand
with one. -
20. Is-elu-yaoum, all the fingers of hand and foot.
30. Is-elu yaoum moilè is-yaoum, all the fingers of
hand and foot with all the fingers of hand.
i
480 A R IT H M ET I C.
Arithmetic. It is a rare thing for a Lulo to attempt the expres-
sion of a number beyond 30; when driven to it by
Their mode necessity, they avail themselves of actions for the pur-
ºrs pose. Thus, to express 40 he raises his open hands to
jä0, his shoulders, and bending his head towards his feet,
he says tamop, which means twice of all that I show :
3you : with the same action, accompanied by the word
tamlip, he expresses 60, and by saying lokep moilé
alapea, he expresses 100. -
Wººls of (272.) The same expedients are made use of by the
*** Vileli, a neighbouring tribe, to express such numbers,
and it will be at once seen that their numerals, though
essentially different, are formed upon the same principle.
1. Yaaguit, or agüit.
2. Ukè.
3. Nipetuei.
4. Yepcatalèt. }
5. Isig-misle yaaguit, fingers of hand one, meaning
all the fingers of hand one.
6. Isig-teet yaaguit, hand with one.
7. Isig-teet uke, hand with two.
10. Isig-ukè-nisle, of hands two the fingers.
11. Isig-ukè-misle teet yaaguit, of hands two the fin-
gers with one.
20. Isig-ape misle cavel, fingers of hands and feet.
Of the (273.) The Mocobi are a tribe on the Paranã, in the
* neighbourhood of Buenos Ayres, the formation of whose
numerals resembles that of the Luli, but which are
still more remarkable for their extreme poverty.
1. Iniatedà.
2. Inabaca.
3. Inabacao-caini, two above.
4. Inibacao-cainiba, two above two, or natolatata.
5. Inibacao-cainiba iniatedà, two above two one, or
natolatata iniatedà, four one, -
6, Natolatatata inibaca, four two.
7. Natolata-inibacao-caini, four two above.
8. Natolata-natolata, four four.
It ought to be observed, however, that the Mocobi
possess practical methods of numeration as well as
other tribes, and that the preceding numerals are never
used, unless in cases where they wish to make an effort
to dispense with the use of their hands and feet.
of the (274.) The Mbayi, or Guaicurus,who live on the western
Guaicurus. bank of the river of Paraguay, are unable to express any
number beyond 5,without the assistance of manual action.
. Uninitegui. - -
2. Iniguata.
3. Iniguata dugani, two over.
4. Iniguata-driniguata, two two.
5. Oguidi, a word equivalent to many, and applied
equally to all numbers above four.
Uf the (275.) The Betoi are a nation who live on the banks
Betoi. of the Casanare, which runs into the Orinoco, who speak
a language whose syntax and construction is singularly
complex and artificial: their numeral language, properly
speaking however, possesses only one, or at most two,
independent names.
1. Edojojoi.
2. Edoi, another.
3. Ibutu, beyond.
4. Ibutu edojojoi, beyond one.
5. Rumocoso, hand.
It must be kept in mind, that these people, as well
as thcse last mentioned, possess practical methods of
numeration which are equally extensive with those of
other American tribes.
(276.) The Maipuri, the Tamonaki, and the Yaruroes, History.
'are considerable tribes who live on the banks of the
Orinoco, who agree in their general methods of numera- Of the .
tion, and who all give the name of man, or Indian, "Pºº
to the number 20. -
Numerals of the Maipuri :
. Papita.
. Avantime.
. Apekiva.
. Apekipaki, three one.
Papitaerri capiti, one only hand.
. Papita yana pauria capiti purena, one of the
other hand we take.
10. Apanumerri capiti, two hands.
11. Papita yana kiti purena, one of the toes we take.
20. Papita camonée, one Indian or man.
40. Avantime camonée, two men.
60. Apekiva camonée, three men.
The preceding numerals are used when counting
human beings: in speaking of other living beings,
one is termed paviata, and two avimime. In the case
of inanimate objects, one is pakiöta, and two akinime ;
and in reckoning time, the first is mapukiä, and the
second apucinume. We know of no other instance
of variations equally numerous, with the exception of
those of Japan, where the numerals are different,
according as they are applied to measures, men, animals,
inanimate things, days, nights, years, and the changes
of the moon.
(277.) Numerals of the Tamonaki: Of the
Tevinitpe. - Tamonaki.
Acchiacke. -
Acchialubve.
Acchiackemneve, or acchiackere-penè.
Amnaitone, hand entire.
. Itacono ammpona tevinitpe, of the other hand one.
10. Amna-acheponare, hands two.
11. Puitta-pona tevinitpe, of the foot one.
15. Iptaitone, foot two hands.
16. Itacono-puitta-pona tevinitpe, of the other foot
O716. |
20. Tevin-itoto, one Indian, or one mart.
21. Itatono itóto yamnar-pona tevinitpe, of the other
Indian at the hand one.
30. Itatono itóto-poma amna-ache pona, of the other
Indian hands two. :
40. Acchiake itoto, two Indians.
100. Ammaitone-itoto, hand Indians, or five Indians.
There are only two numerals tevin and acchia, for
one and two, which can properly be considered as in-
dependent, those for 3 and 4 being clearly compound.
In no case, says the Abbé Gilii, does an Indian men-
tion a number without a corresponding action : if he
asks for a fruit he raises a finger; if he mentions five, he
shows his whole hand ; if ten, both his hands; and if
twenty, he points the fingers of his hands to the toes of
his feet. The Tamonaki call the thumb the father of
the fingers; the inder is termed the finger for pointing;
and the ring finger is called the finger by the side of
the little one.
i
i
(278.) Numerals of the Yaruroes: Of the
I. Caneame, Yaruroes,
2. Noeni.
3, Tarani,
4, Kevvinë.
5, Caniicchimo, cani, one, icchi, hand, mo, alone,
10, Yoaicchibo, all the hands, -
A R. I. T H M ET I C.
481
Arithmetic. 11. Taomepe-caneame, to the foot (tao) one.
\–y—’ 12. Taonepe-noeni, to the foot two.
15. Canitaomo, one foot alone.
16. Caneamotaonepè-caneame, one foot alone one.
20. Canipume, one man. -
40. Noenipume, two men. .
In general, however, when they count beyond 20,
they take grains of sand or stones of fruit, and make
them into heaps of 20 each.
Important (279.) We have to apologize to our readersfor entering
facts esta-
e at so much length into the discussion of these South
tºº. American numerals, but we must plead as our apology,
rals the uncommon interest which they possess, as illustra-
* ting nearly all the remarks which we have had occasion
to make, in connection with this subject in the first
part of this Article ; and as proving, almost to a demon-
stration, the general truth of the propositions which
we have there stated, with respect to the origin and
universality of the natural scales of numeration. It is
extremely curious, likewise, to observe with what ex-
treme difficulty these rude children of nature abstract
words from things, and how little language, in many
cases, at least, is able to keep pace even with the ex-
pression of the most common of their wants.
(280.) Philologers have spoken with admiration of
perfection the wonderful syntax and construction of many of these
of the syn- languages, presenting so many examples of extreme
tax and the refinement and complexity; and this has been observed
rudeness of in the languages even of those tribes whose numeral
Contrast
between the
the nume systems are the most imperfect: it is in vain to attempt
rais of to account for such facts upon ordinary principles, and
many of e - e
tja- the solution of our author is, of all others, the most
guages. rational, and the most becoming a Christian philoso-
pher, who seeks for the origin of these languages, and
the laws of their construction, not in the efforts of men.
for the mutual communication of their wants, but in
the ordinance and institution of God himself.
Difficulties (281.) A person who examines minutely the analysis
in ascer-, which is given above of the grammatical construction
* * of many of these systems of numerals, will find reason
COrrett &A & g &
grammati- to suspect the existence of very considerable inaccura-
ºal con- cies in them. We have before remarked the extreme
struction of difficulty of writing down accurately the words of any
j of language where the ear is the only guide ; and the in-
º * formation which Hervas obtained from many of these
Missionaries was derived from mere recollection, twenty
years after they had been compelled to quit their sta-
tions, when old age and calamity had impaired the
activity of their memory, as well as other faculties;
besides, there were many other circumstances which
combined to diminish the value of the information de-
rived from such sources: the greater part of the
Jesuits who were sent to South America were Spaniards
possessing few of the advantages of education, which
gave such celebrity to many others of their order; who,
by living amongst savages, were compelled to adopt
many of their habits; who had no opportunities of
literary intercourse ; who saw their few books and
papers perishing from the damp and insects which in-
fest the mighty forests which characterise that vast con-
tinent ; and who were compelled to submit to privations,
of which a lively image is given in the reply of the
poor monk to Humboldt, when asked how long he had
resided in his Reduction, “On such a day I shall have
completed my twenty years of mosquitoes.”
(282.) Among the 100 systems of Asiatic numerals
which Hervas has given, we find few which suggest any
ſ WOL. I. *
Numerals of
Georgia.
particular remarks, in addition to those which we have History. .
ourselves had occasion to make, if we except the nume- ~~
rals of different dialects of Georgia, which are adapted to
the vicenary scale, and which present the only genuine
example of it in any Asiatic language. The numerals .
of Georgia Proper are as follow :
. Erti.
. Ori.
Sami.
Otchi.
. Chuti.
. Echsi.
. Sciuiti.
Rua.
Zchara.
. Athi.
. Athierti, ten one.
. Athiori, ten two.
. Athichuti, ten five.
. Athiechsi, ten sir.
. Ozierti, one twenty. -
. Samarti, three ten, or, more commonly in other
dialects, twenty ten.
. Ormazi, two twenty.
. Ormazathi, two twenty ten.
. Samotzi, three twenty.
. Samotzathi, three twenty ten.
. Otmozi, four twenty.
90. Otmozathi, four twenty ten.
100. Assi.
1000. Athachsi.
The author attempts to prove that these dialects are Basque
analogous to the Basque, and that their vicenary Arith- *
metic, as well as that of the Celtic nations, were derived
from a common source. The following is a list of
Basque numerals, which, though similar in construc-
tion, possess no other points of resemblance.
1. Bat.
2. Bi.
3. Iru.
4. Lau.
5. Bost.
6
7
8
9
. Sei.
. Zospi.
. Zortzi.
Bederatz.
Amar.
. Amaicu, ten ome.
. Amabi, ten two.
. Amairu, ten three.
. Amalau, ten four.
. Amabost, ten five.
. Amasei, ten sia.
. Amazospi, ten seven.
. Amazortzi, ten eight.
. Ameretzi, ten mine.
. Oguei.
. Oguei tabat, twenty with one.
. Oguei taamar, twenty with ten.
. Berroguei, two twenty.
. Berroguei taamar, two twenty with ten.
. Iruroguei, three twenty.
. Lauroguei, four twenty.
100. Eun.
1000. Milla,
(283.) We have examined the other parts of the work
of Hervas with considerable interest, as he has travelled
3 R
482 A R. I T H M E T I C.
of Indo Pelasgic languages, occupying a zone of History.
more than two-thirds of the circumference of the globe, S-S-
Arithmetic, over a great part of the same ground with ourselves:
S-N-' be attempts to prove, that the quinary Arithmetic, or
Observa-
rather numeration by the fingers of one hand, was
extending from the north-western extremity of Europe,
††. practised in the infancy of the world, and discovers through Persia and Hindostan, to the islands of the
ions ºf the vestiges of it in the very general resemblance of the South Sea, and which will be found to comprehend the
work of name for hand and for five, or, at least, of the roots of greatest part of the nations to whose numerals he has
Hervas. those terms. It must be confessed, however, that, in referred, in confirmation of this part of his theory.
his search after such analogies, he has ventured to
travel further into the very dangerous regions of etymo-
logy than can be considered either prudent or judi-
cious. He considers, however, the almost universal
prevalence of the decimal scale as a proof, that it had
superseded the quinary Arithmetic long before the
dispersion of nations, and appeals, in confirmation of this
opinion, to the affinity of the names for 6 and 7, which
both possess the characteristic letter s in so many
languages, and which, therefore, were most probably
derived from some common source; he possessed not,
however, the key which more modern philologists have
found out, for the classification of European and
Asiatic languages, and particularly of that great class
4a
The author has likewise discussed, with considerable
learning, the alphabetical and symbolical Arithmetic of
different nations, as well as the question so often agitated,
of the origin of the notation by nine figures and zero,
and the date and circumstances of its introduction into
Europe. The opinions which he has advanced on
these subjects are not materially different from our own,
and though some of the facts which he has collected
are new and important, we feel compelled to leave
them unnoticed, as we have already trespassed too
much upon the patience of our readers, to venture
upon the addition of any further extracts to those we
have already given.
ERRATA. *
Page. Col. Line. Error. Correction.
397, 2, 14 from top, 2004%, zougal.
do. do. 19 from bottom, Mv, Mo.
428, 1, 26 from bottom, necessarily, successively.
429, 2, 2 from bottom tacevano, facevano.
433, do. 25 from top, 97335376, 97535376. º
445, l, 3 from top, wheat, wine.
447, 2, 21 from bottom, after ligmes put a comma.
448, l, 29 from bottom, after enlöveront omit semicolor.
A R I T H M E T I C.
PART I.
Arithmetic. (284.) THERE are two great divisions of the Science
of Arithmetic, to which we shall adhere generally in the
following treatise. -
Arithmetic The first comprehends the fundamental rules, Nota-
of abstract tion, Addition, Subtraction, Multiplication, and Divi-
numbers. Sion, which will vary according to the nature of the
quantities which are considered, whether integers,
ordinary or decimal fractions, or concrete or compound
quantities; to which may likewise be added, the rules
for the extraction of the square, cube, and other roots.
The second comprehends the application of these
rules to the solution of such classes of questions as
arise in the ordinary business of life; such as questions
on the rule of three, practice, interest, and annuities,
&c.; a division of our subject which we shall treat
with great brevity, as sufficient information may be
obtained upon it in our ordinary books of Arithmetic.
Of concrete
numbers.
(285.) As the following signs are very generally used,
and contribute greatly to the distinctness of notation in
many cases, and to the abbreviation of language, it
may be expedient to premise an explanation of them.
(1.) + plus, or more, the sign of addition ; its signi-
fication in Arithmetic being, that the numbers between
which it is placed are to be added together.
Thus 7 -H 3 denotes that 7 is to be added to 3:
# + + means that is to be added to +.
(2.) — minus, or less, the sign of subtraction; its
arithmetical signification being, that the second of the
numbers between which it is placed is to be subtracted
from the other. -
Thus 7 – 3 means that 3 is to be subtracted from
7 : ; – 3 means that 3 is to be subtracted from #.
(3.) × into, the sign of multiplication, signifying
that the numbers between which it is placed are to be
multiplied together.
Thus 7 x 3 means that 7 is to be multiplied into 3.
(4.) -- by, the sign of division, signifying that the
former of the two numbers between which it is placed
is to be divided by the latter.
Thus 12 + 3 signifies that 12 is to be divided by 3.
This last sign is not very generally used, the more
common practice being to write the divisor underneath
the dividend, in the form of a fraction. Thus 12 + 3
is equivalent to *.
(5.) = equal to, signifies that the numbers between
which it is placed are equal to one another.
Thus 7 -– 3 = 10. º
There are other signs which we shall have occasion
sometimes to make use of, but their explanation may be
deferred until we come to the discussion of the opera-
tions for which they are required.
Explanation
of signs.
Numeration and Notation.
Peñnitions (286.) Arithmetical notation may be defined to be, the
expression of any number in symbols which is already Part I.
expressed in words; whilst the term numeration is \-y-.
generally applied to the converse process, of expressing
in words a number which is already expressed in sym-
bols.
We must, of course, suppose the learner to be
acquainted with the meaning of all ordinary numerical
terms, such as the names of the digits, tens, hundreds,
thousands, millions, &c., as also with the full import of
the phrases for the expression of compound numbers,
such as three hundred and sixty-five, one thousand
eight hundred and twenty-six; ten millions, three hun-
dred and ninety-five thousand, seven hundred and
eighty-four ; and so on. Unless possessed of such
elementary and fundamental knowledge, it would be
extremely difficult to make him comprehend the nota-
tion of numbers. ºr- -
(287.) The nine digits, one, two, three, four, five, six, Notation of
seven, eight, nine, are denoted by the nine figures, digits.
1, 2, 3, 4, 5, 6, 7, 8, 9.
Zero, or nothing, is denoted by 0, which is also called
a cypher.
The articulate numbers of the first order, or ten, Articulate
twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety, numbers-
are denoted by
10, 20, 30, 40, 50, 60, 70, 80, 90,
a cypher being written after the digital number, which
must be multiplied into ten, in order to produce the
corresponding articulate number.
The articulate numbers of the second order, one
hundred, two hundred, three hundred, &c., are denoted
b * —
y 100, 200, 300, 400, 500, 600, 700, 800, 900,
two cyphers being written after the respective digital
numbers.
Articulate numbers of the third, fourth, or any other
order, are denoted by writing three, four, or as many
cyphers after the digital number as may be equal to
the mumber which determines the order. Thus one
thousand is denoted by 1000, twenty thousand by 20000,
five hundred thousand by 500000, one million by
1000000, and similarly in other cases.
The zeros, or cyphers, therefore, though without value
themselves, serve to mark the values of the digits which .
they succeed ; those digits being supposed to be mul- "
tiplied into ten, a hundred, thousand, &c., according
as one, two, three, &c. cyphers or places succeed them.
(288.) An example or two will best explain the prin-
ciple of denoting compound numbers.
Let it be required to denote by figures the number Examples of
seven thousand, six hundred, and ninety-five. the notation
Write underneath each the digital and several articu- of .
tº, º e g pound num-
late numbers of which this compound number is com- fiers.
&
s
osed
P 483
3 R 2
484 A R I T H M E T I C.
Arithmetic. Five . . . . . . . . . . . . . . 5 Billion . . . . . . . . . . . • • * * * * * * * * * * 1000000000000 Part I.
\-N-' Ninety . . . . . . . . . . . . . . 90 Trillion. . . . . . . . . . . . . . . . . . I000000000000000000 º
Six hundred. . . . . . . . . . 600 Quatrillion . . . . . . . . 1000000000000000000000000
Seven thousand . . . . . . 7000
The number which is the sum of these several parts is
denoted by 7695, where 5 is in the place of units, 9 of
tens, 6 of hundreds, and 7 of thousands : in this case,
therefore, the values of the several digits, 9, 6, 7, are
determined by their position with respect to the place
of units; and the number denoted, by writing those
digits in succession in their several places, is equal to
the sum of the numbers which they would express if
written separately, with the same number of cyphers
as of places after each.
Let it be required to write down the number twenty-
three millions, sixty-nine thousands, one hundred and
SeVen.
(290.) Our language possesses no simple names for Numbers
numbers in the decuple series, 1, 10, 100, &c., except for separated
the 1st, 2d, 3d, 4th, 7th, 13th, 19th, &c.; and it has º
e * O
therefore been usual to separate numerical expressions places.
into members or periods of six, the first embracing all
numbers below a million, the second millions, the third
billions, and so on, thus affording an aid to the eye, by
which their numeration is more easily effected. Thus,
the number denoted by
2340,064039,672.107,
is two thousand three hundred and forty billions, sixty-
four thousand and thirty-nine millions, six hundred and
seventy-two thousand, one hundred and seven ; and the
number denoted by
Seven . . . . . . . . . . . . . . . . . . 7 10076,432897,158204,000621,
$.". d........' gº is ten thousand and seventy-six trillions, four hundred
sixty thousand. . . . . . . . 60000 and thirty-two thousand eight hundred and ninety-seven
i. millions. ... 3000000 billions, one hundred and fifty-eight thousand two
T º:-- . . . . hundred and four millions, six hundred, and twenty-one.
wenty millions . . 20000000
The number which is the sum of all these parts is
written 23069107 in one line; the digits being written
in succession, the zeros being written in those places
to which no digit corresponds. .
The numeration of each period is the same as for the first
six places, being only, instead of units, millions for the
second period, billions for the third, trillions for the
fourth, and so on.
(291.) As the values of the digits increase in a decuple Principle of
proportion, in passing from the place of units from the the notation
right to the left, it is a very natural extension of this of decimals.
The principle of this notation, which is sufficiently
illustrated by these examples, may be stated as follows:
Table of Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 other cases.
numeral Ten. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 (292.) Such fractions are called decimal fractions, or Decimal
terms with One hundred . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 decimals, to distinguish them from integers, though they point.
. * Thousand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1000 are all equally subject to the same decimal scale, or
g Ten thousand . . . . . . . . . . . . . . . . . . . . . . . . . . 10000 classification of values. The dot also is termed the
Hundred thousand . . . . . . . . . . . . . . . . . . . . 100000 decimal place; all the digits to the right of it being
Million. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1000000 considered as decimals, though in this respect no
Ten millions . . . . . . . . . . . . . . . . . . . . . . . . 10000000 great regard is had to propriety of language. It would
Hundred millions. . . . . . . . . . . . . . . . . . . . 100000000 be more proper to place the dot beneath the digit in
Thousand millions. . . . . . . . . . . . . . . . . . 1000000000 the place of units, which is the point of departure;
Ten thousand millions. . . . . . . . . . . . . . 10000000000 the digits to the right and to the left, possessing value
Hundred thousand millions . . . . . . . . 100000000000 from their position with respect to it; but this would
the values of the digits increase in a tenfold proportion,
in passing from the place of units from the right hand
to the left, being supposed to be multiplied by ten in
the second place, by a hundred in the third, by a
thousand in the fourth, by ten thousand in the fifth, and
so on ; and the number denoted by those digits written
in succession, is the sum of the numbers which they
severally denote, when their values are considered with
reference to the place of units. -
Thus, the number denoted by 2345 is equivalent to
the sum of 2000, 300, 40, and 5; or, if we pass from
symbols to numeral language, it is equal to two thousand,
three hundred, and forty-five. -
The number denoted by 8400018 is equivalent to
8000000, 400000, 10, and 8, or to eight millions, four
hundred thousand, and eighteen.
The number denoted by 111000111 is equivalent to
100000000 + 10000000 + 1000000 + 100 + 10 + 1,
or to one hundred and eleven millions, one hundred, and
eleven.
(289.) The following table of numeral terms, with the
expressions in figures for the equivalent numbers, will
materially assist the learner in the notation of any
number when given in words, or in its numeration when
expressed in symbols.
principle to consider digits on the right of the place of
units as decreasing in a decuple proportion from left to
right: thus
32.124
would denote 3 × 10 + 2 + -ºr + +3tr + +pºor, a
dot being placed after the place of units, to determine
its position with respect to the other digits: and again,
7634.0345 -
is equivalent to 7 × 1000 + 6 × 100 + 3 × 10 + 4
-- +3-y + +tºry + roºrov. The digits in the 1st, 2d,
3d, 4th, &c. place to the right of the place of units,
being divided by 10, 100, 1000, 10000, &c. respectively,
whilst those in corresponding positions to the left are
multiplied by the corresponding numbers in the same
series. In the numeration of such numbers, the digits
to the right of the place of units are tenths, hundredths,
thousandths, ten thousandths, &c., corresponding to the
places of tens, hundreds, thousands, ten thousands,
amongst integral numbers. Thus
3,245
is read three, two-tenths, four-hundredths, five thou-
sandths. Also,
.006934
is read six thousandths, nine ten-thousandths, three
hundred-thousandths, four millionths, and similarly in
A R IT H M ET I C.
485
Arithmetic. lead to some inconvenience, when there were no integral
\-v- numbers in the expression. Thus
75.036
denoted by
- 75036;
but there would be some degree of awkwardness, though
no ambiguity, in denoting
would be conveniently
.00062
by
00062
Decimals (293.) It has been usual in books of Arithmetic to sepa-
wº rate the rules for operation with integral numbers from
cluded those in which decimals are also involved; and though
under com- the notation of such quantities is reducible to a common
Inon rules, principle in both cases, and though it would not be
difficult to frame the rules for addition, subtraction,
multiplication, and division, so as to include them
both, we shall adhere to the common practice, as better
adapted for the purposes of elementary instruction.
A student in Arithmetic is not likely to possess much
power of generalization, and it seems expedient that
he should first be familiarized with the common opera-
tions with whole numbers only, without having addi-
tional difficulties thrown in his way, by the greater
complexity of the rules which would be necessary, in
order to embrace decimals as well as integers.
ADDITION.
Rule. (294.) To add is to collect several numbers into one
Sº???..
For this purpose, the numbers must be written under-
neath each other, so that units may stand under
units, tens under tens, hundreds under hundreds ; and
we then proceed to add the digits in each column into
one sum, and write the result underneath. Thus, if we
have to add 321 to 237, they must be written thus,
. 321
237
558 -
And the sum 558 is found by adding successively 7 to
1, 3 to 2, and 2 to 3.
Method of But if the sum of the digits in the same column ex-
carrying ceed 10, we must write down the excess, and carry 1 to
tens, the next column. Thus the sum of 27 and 56, or
27
56
83
is found, by first adding 6 and 7 together, whose sum
is 13: we write down 3, and carry 1 to the sum of the
digits of the next column, which thus becomes 8.
Let it be required to add together 303, 727, 1069,
and 35 : -
303
727
1069
35
2134
The sum of the digits in the first column is 24, write
down 4, and carry 2. The sum of 2, 3, 6, 2, in the
second column, is 13: write down 3, and carry 1: the
sum of r, 7, 3, in the third column, is 11 : write down
I, and carry 1; the sum of r and 1 is 2, which, written
down, gives the entire sum of the numbers required.
In this case, we have denoted the numbers which are Part F.
carried from one column to another with scratched -
figures, to distinguish them from those which actually
appear in the original sums to be added.
The principle of the rule for carrying the tens from
one column to another, so important in the incorpora-
tion of numbers into one sum, whether in addition or
multiplication, must be at once understood by any one
who fully comprehends the principle of notation by nine
figures and zero; and we should most probably create
a difficulty where none existed before, by any attempt
to explain it. Demonstrations become difficult and
unsatisfactory, when the relation between the premises
and conclusion is so simple that the mind at once per-
ceives it ; and in such cases, what is gained in form is
generally lost in perspicuity.
Examples:
I. 96341 2.
25784
1000]
7249
70000
209375
Examples.
12345
23456
34567
45678
56789
172835
999999
101010
1101009
3. 373.737 4.
3636.36
737373
SUBTRACTION.
(295.) To subtract one number from another, is to find
their difference, or to find a number which added to the
first will produce the second.
Place the number to be subtracted underneath the Rule.
other, in the same manner as in addition, and then
subtract the digits underneath successively from those
above. Thus, to subtract 237 from 558, write them as
follows:
- 558
237
321
Subtract 7 from 8, the remainder is l; 3 from 5, the
remainder is 2; 2 from 5, the remainder is 3: we thus
get the entire remainder, which is 321.
In this example, the digits in the subtrahend are Process of
severally less than, those above them ; in case one or borrowing
more of them are greater, we must add 10 to the upper “”
digit, and increase the lower digit in the next column
by 1; in other words, the 10 which we borrow, to in-
crease the upper digit in the first column, we must
Tepay by increasing the lower digit by 1 in the next:
thus, in the example,
32
27
2=-
5
we increase 2 by 10, which makes 12, from which we
subtract 7, which leaves 5; we increase the digit 2, in
the next column, by 1, which becomes 3, and being
subtracted from 3 leaves no remainder.
Again, in the example,
486
A R I T H M E T I C.
Arithmetic.
S-N-"
34503
1537.6
19127
we add 10 to 3 and subtract 6 from 13, which
leaves 7; we then increase 7 by 1, and 0 by 10, and,
Jherefore, subtract 8 from 10, which leaves 2; we in-
crease 3 by 1, and, therefore, subtract 4 from 5, which
leaves 1; we increase 4 by 10, and then subtract 5 from
Examples.
Definitions.
Multiplica-
tion table,
14, which leaves 9; we then increase 1 by 1, and sub-
tract 2 from 3, which leaves 1, and thus get the entire
‘remainder, which is 19127.
Examples.
1. 32104 2. 232323
7963 4.1414
24.141 190909
3. 10] 101101 4. 98765.4321
90909090 123456789
10 192011 864. 197532
MULTIPLICATION.
(296.) To multiply one number by another, is to add
the first as often as the second denotes, or conversely.
The first of these numbers is called the multiplicand,
the second the multiplier, the result of their multiplica-
tion is called the product.
The definition which we have given of multiplication
rather indicates what the operation is equivalent to, than
guides us to the mode in which it may be performed :
the product is the sum of the multiplicand repeated as
often as there are units in the multiplier, but the object
of multiplication is to enable us to find this sum, or
product, by a short and simple process, which supersedes
the necessity of these repeated additions.
(297) For this purpose, it is absolutely necessary to
commit to memory the products of all numbers as far as 10
into 10, into each other. The following table extends as
far as 12 into 12, and it is expedient and usual, though
not necessary, to learn it under this extended form.
Multiplication Table.
ºn 2 || 3 || 4 || 5 || 6 || 7- 8 || 9 || 10| 1 | 12
A TG || 3 |To T2 || || || 16'ſ Ta T20 |Tº T21
Tai TTTI, T, is a 2, 27 Taoſ as as
TTT's Tº Tº Too Tºſſos || 3 || 36|Tao Taa Tas
T. To T. T.T.TH so as Tao as Tºp 55 go
s Tº is . Tº T; T4, Tag T54 || 60'ſ 66||73
T. T.I.T. . . . Tº Tº Tºº Tº Tzoſº aſ
s is . . . . . is ſºjº. Tº Tao Tsa Tos
pººl. ... ... I gº. Tº Tº T 90'ſ 39|Tos
To ſºlº 50 TGo || 70'ſ so ſigo T100110 |Tºo
T 22 || 33 44 55 º 7 as 99 ||10|12||132
T 24 || 36 || 48 || 60 ſº 84 96 100 1901&|14.
Many of the remarks, which an examination of the Part I,
numbers in this table would suggest, however interest- X->
ing, would be useless here, as having no connection Remarks
with its immediate object and application : there are tººl
some others, however, which, though extremely simple e
and obvious, it may be worth while to point out.
The numbers included in the black squares, which Square
form the diagonal of the great square, are the squares numbers.
of the numbers from 1 to 12, or the products of 2 into
2, 3 into 3, 4 into 4, &c.
The portions of the table on each side of this diago-Two similar
nal are identical with each other, as will be imme- portions of
diately seen from an examination of the numbers in the **ble.
squares on each side, whose diagonals are in the same
straight line.
Those numbers which occur more than once on the The same
same side of the diagonal, may arise from the product product
of different combinations of numbers between 1 and 12: º diffe-
thus 18 may arise from 6 and 3, or from 9 and 2; 48 rent factors.
from 6 and 8, or from 4 and 12 ; 72 from 8 and 9, or
from 6 and 12 ; and, similarly, for the numbers 12, 20,
24, 30, 40, and 60. The only square numbers which
occur in this portion of the square are 16 and 36, which
are the squares of 4 and 6, or the products 8 and 2,
and of 9 and 4, or 12 and 3.
The series of square numbers exceed by unity the
number in the adjoining square in the same diagonal,
which are 4 and 3, 9 and 8, 16 and 15, 25 and 24, 36
and 35, and so on, as far as 121 and 120 : in other
words, the square of a number exceeds by unity the
product of the two numbers, which differ from it by 1,
one in excess and the other in defect.
Pursuing the examination of numbers in the same
diagonal, we find those in the second square from the
centre differing from the square number placed therein
by 4; those in the third by 9, in the fourth by 16, and so
on : in other words, the product of two numbers, dif-
fering in excess and defect by 1, 2, 3, 4, &c. from any
number, will be less than its square by the squares of
that difference.
Conclusions like these may be generalized, and
applied to any numbers whatsoever; but such gene-
ralizations must be made with the greatest caution and
distrust, and never admitted as proved, unless it can be
shown that the conclusion does not depend upon the
particular magnitude of the numbers which are used.
(298.) There are some cases in which it is expedient Formation
to learn by heart the products of numbers beyond the of squares
limits of this table. Of this kind, are the squares of frºm 12 to
all numbers as far as 25 or 30, and even farther, the 50.
knowledge of which is frequently useful, and particu-
larly so for enabling us to form very readily the pro-
Other
remarks.

ducts of numbers equidistant from them, by a method
founded on the preceding observations.
The square of 13, 169. 22, 484.
14, 196. 23, 529.
15, 225. 24, 576.
16, 256. 25, 625.
17, 289. 26, 676.
18, 324. 27, 729.
19, 361. 28, 784.
20, 400. 29, 841.
21, 441. 30, 900.
It is a very amusing and instructive practice to observe
the analogies which may exist amongst these, or any
other connected series of numbers, and to notice such
A R IT H M ET I C. 487
Arithmetic.
S-N-'
points of resemblance or diversity, as may serve the pur-
poses of a technical memory. Thus the squares of 13 and
14 are 169 and 196, the two last digits being the same
in each, but in an inverted order: the two last figures
in the square of 15 are the two first in that of 16: the
squares of 24 and 26, each differing from 25 by 1,
differ from each other by 100: those of 23 and 27, dif.
fering from 25 by 2, differ from each other by 200 :
those of 22 and 28, differing from 25 by 3, differ from
each other by 300: those of 21 and 29, differ by 400;
of 20 and 30, by 500. If we extend the conclusion,
the squares of 19 and 31 should differ by 600, or, in
Rule.
Formation
of squares
from 50 to
i00,
Importance
of such
observations
Rule for
multiplica-
tion where
one factor
exceeds the
limits of the
table.
other words, the square of 31 should be 961 ; whilst,
in the same manner, we should find the square of 32 to
be 1024; that of 33 to be 1089, and similarly for other
numbers as far as 50 ; the general rule being as follows:
“if two numbers are equidistant from 25, the square of
the greater exceeds the square of the less, by as many
hundreds as the number itself exceeds 25.”
(299.) If we wished to form the squares of all numbers
above 50 from those below 50, it might be easily done
by the following rule: if two numbers be equidistant
from 50, the square of the greater exceeds that of the
less by twice as many hundreds as the number itself
exceeds 50. The truth of this rule would be readily
ascertained from the actual formation and examination
of any number of the squares themselves.
(300.) Observations like these are easily made, and
save the memory from much useless labour; and it is
impossible for a student to habituate himself too soon to
the practice of such examinations as are the foundation
of them. It is true, that the rules of Arithmetic are
formed generally for the use of those who have not
arrived at an age when the reflective and reasoning facul-
ties are sufficiently exercised and strengthened to enable
them to understand fully the principles of the rules
which they follow : but it may justly be doubted, whether
the acquiescence in this principle of education, is not
much too general, and whether habits of investigation
and inquiry are not checked, at least, if not destroyed,
by teaching the student to follow merely mechanical
rules, in which the understanding takes no part.
(301.) But it is proper to return from this digression to
the immediate uses of the multiplication table, as exem-
plified in the process of multiplication of numbers, one
or both of which are beyond the limits of the table.
Let one of the numbers only be within the limits of
the multiplication table. In this case the greater number
must be made the multiplicand, and the less number
the multiplier. Multiply successively every digit of the
multiplicand by the multiplier; the several products are
known from the table, and in forming the whole pro-
duct, we must carry the tens in the product of the first
digit to the product of the second, and so on to the
end. An example will explain our meaning more
clearly.
Let it be required to multiply 237 by 9,
Write them as follows:
237
9.
gººmmºmºmº
21.33
The product of 9 and 7 is 63; write down 3 and
carry 6, or retain it in the mind as a number to be
added to the next product: the product of 9 and 3 is
27; to this add 6, which makes 33; write down 3 and
carry 3: the product of 9 and 2 is 18; to this add 3, Part I.
and the sum is 21, which, written down, gives 2132, the S-V-'
entire product required.
The same result would be obtained by the addition
of 237 nine times to itself, as follows:
237
237
237
237
237
237 N
237
237
237
*mº
2133
The multiplication of the successive digits 7, 3, 2, by
9, is equivalent to the addition of these digits 9 times
repeated, and the numbers carried in each case are
obviously the same.
Let it be required to multiply 9876 by 12.
9876
12
II85 12
The product of 12 and 6 is 72; write down 2 and
carry 7: the product of 12 and 7 is 84, add 7, and the
result is 91; write down 1, and carry 9: the product of
12 and 8 is 96, add 9, the result is 105 ; write down 5,
and carry 10: the product of 12 and 9 is 108, add 10,
the result is 118, which written down gives the entire
product.
(302.) The next case to be considered is that in which
both the numbers to be multiplied together exceed the
limits of the table.
In this case it is most convenient to make that num- Rule where
ber the multiplier which possesses the smallest nnmber both the
of digits; we then multiply the multiplicand successively . are
by the digits of the multiplier, placing the several pro- #.
ducts underneath each other, the digit in the units’ table,
place in the second under the digit in the tens’ place in
the first product, and so on throughout: we then add
these results together, in order to get the entire pro-
duct. Thus, suppose it were required to multiply 2349
by 876, the form of the process is as follows:
2349
S76
tºmsºmºsºmsºmºmº
14094
16443
I8792
2057724
We first multiply 2349 by 6, the result is 14094; we
next multiply 2349 by 7, the result is 16443, which is
written underneath the first result, so that the last
figure of one may be under the last but one of the
other; we lastly multiply 2349 by 8, and the result is
18792, which is placed in a similar manner: the digits
in the several columns are added together, and the
final product is obtained.
If the several results had been written down at full
length, the scheme of the process would have appeared
as follows:
º
488
A R I, T H M E T I C.
2349
876
sºmº-º
14094
164430
1879.200
20577 24
The fact is, that the digits of the multiplier denote
800, 70, and 6, respectively, and we, properly speaking,
multiply by 70 and 800, and not by 7 and 8. The
result, however, of the multiplication of a number by
70 differs from its product by 7, merely in having an
additional cypher after the significant digits; whilst
the product produced by multiplying by 800 differs
from that with 8, merely in having 2 additional cyphers
after it: it is quite clear, however, that the final result
Arithmetic.
\-y-
in the process of multiplication, and the first place of the Part I.
product formed by the next significant digit is removed S-2
as many places to the left as there are cyphers passed Where “y”
over. We will take the following example: phers occur
between the
f 207392 significant
504003
digits.
622 176
829568
1036960
1045261901.76
The reason of this rule will be manifest, if we should
perform the multiplications with the cyphers as well as
with the significant digits, in which case the process
would produce the following scheme:
which is obtained by following the directions of the 207392
rule, and omitting the cyphers, is the same as if they 504003
were inserted in full ; and it is an important principle 622 176
in all arithmetical rules, to dispense with the writing 000000
down of all figures which are superfluous in practice, 000000
however much they may otherwise contribute to make 829568
the operation better understood. 000000
Products of (303.) The product of 10 into 10 is 100, or, expressed 1036960
tºº. a in the abbreviated form which the use of signs enables ==ºmmº-sº
by cyphers. * to give it, 104526190 176
g IO X IO - 100. (305.) The definition which we have given of multipli- Definition
Again, cation, considering it as equivalent to the addition of the of multipli-
10 × 100 = 1000, multiplicand, repeated as often as unity is contained in cation must
I0 × 1000 = 10000, the multiplier. is stri e be modified
e multiplier, is strictly applicable to those cases only when the
I 00 x 100 = 10000, where the multiplier is an abstract whole number: in multiplier
100 × 1000 = 100000, all other cases, its meaning must be modified to suit the is not an ab-
1000 × 1000 = 1000000, particular nature of the case, and at the same time *: whole
100 × 10000 = 1000000, to coincide strictly with the preceding, which is its "*
10 × 100000 - 1000000. primitive definition, in all those points which whey
It appears from hence, and these results are an imme- possess in common. We shall have occasion to notice
diate consequence of the decimal notation, that the pro- this subject again, when we come to the discussion of
duct of two numbers, expressed by 1 and any number of the multiplication of fractions. {
cyphers after it, is the number denoted by 1 with as (306.) Examples. Examples.
many cyphers as are equal to the sum of those in the 1. 1826 2. III
two factors: and the same rule applies, as far at least 365 1 11
as the number in the product is concerned, when any wºmmº-ssº-º.
other numbers terminated by cyphers are concerned; 9 130 ll I
thus the product of 30 and 300, or 10956 I l 1
5478 I 11
30 x 300 = 9000, \- *mm-º-º-º-g
70 × 800 = 56000, 666.490 12321
I200 x 1300 = 1560000,
16000 × 16000 = 2560000000. * ..., * .
Rule. The rule, therefore, for such cases, may be stated as -º-º-º-
follows: “Multiply the significant digits as if there sº & º
were no cyphers after them, and append to their pro- 369 § sº 84
duct as many cyphers as are equal to the sum of the 24642 3693330
number of those in the multiplicand and multiplier.” 1333i
The following is an example: mºmºm-º. 40106319.538
461200 15 1807041
273000
s:" DIVISION.
92.24 (307.) To divide one number by another, is to find how Definition.
12590,7600000 often the second is contained in the first ; or, in other
(304.) In case cyphers occur between the significant
digits of the multiplier, they are, of course, passed over
4.
words, to find how often the second may be subtracted con-
tinually from the first, until nothing remains, or, at least,
wntil the number which remains is less than the second,
A R. I T H M ET I C.
489
Arithmetic.
The first of these numbers is called the dividend,
>~~' the second the divisor, and the number which results
Rule when
the divisor
is within
the limits of
the multi-
plication
table,
Examples.
Fractions—
their origin
and mean-
ing,
from the operation is called the quotient.
The quotient is perfect or complete when there is no
remainder; imperfect when there is. In the first case,
the product of the quotient and divisor produces the
dividend; in the second case, this product differs from
the dividend by the remainder. -
The operation of Division is the inverse of that of
Multiplication, and the rule is founded upon a retracing
the steps of the process of multiplication. The different
cases also depend entirely upon the divisor, in the same
manner as the cases of multiplication depend upon the
multiplier.
(308.) The first of these cases is, where the divisor is a
number within the limits of the multiplication table :
we write the divisor and dividend consecutively in the
same line, merely separating them by a small curved
line ; we then inquire, how often the divisor is con-
tained in the first one, two, or three figures of the divi-
dend; we write the quotient below, and to the remainder
we annex the next figure of the dividend; we find the
quotient of this number, and repeat the same operation
continually, until all the figures in the dividend are ex-
hausted, and the quotient, whether perfect or not, is
obtained.
(309.) The following are examples:
(1.) 7) 168
24
We find that 7 is contained twice in 16; the figure
in the quotient is 2, and the remainder 2, to which we
annex 8, which gives us 28 for the next number to be
divided, of which the quotient is 4; and there is no
remainder.
(2.) 12) 112496340
9374695
In this case, it is necessary, at first, to take three
places of the dividend, before we get a number which is
greater than the divisor.
(3.)
\
4) 315 \
78;
In this case, there is a remainder 3 after the opera-
tion, and it is usual to distinguish it from the integral
part 78 of the quotient, by writing the divisor under-
neath it, with a line between : 78 is the imperfect quo-
tient of 315 divided by 4 ; the complete quotient would
require the remainder to be appended to it in the man-
ner represented above.
(310.) The quantity represented by # is termed a frac-
tion, and originates in the process of division: it might
be termed the quotient of 3 divided by 4 ; under such a
view of its origin and meaning, it must be a quantity
of such a kind, that when multiplied by 4 the product
is 3; for the operation of division being the inverse of
that of multiplication, it follows, that the number 3
being first divided by 4, and the quotient # again mul-
tiplied by the same number 4, the final result must
coincide with the original number 3.
We are enabled, in all cases, to make the quotient
complete by appending the remainder, with the divisor
underneath it, in the form of a fraction : and it must
always be understood, when such a fraction is written after
WOL. I.
an integral number, without any sign being interposed, Partſ.
that it is to be added to the number which precedes it; -
thus 78% is equivalent to 78 + #.
It is clear, likewise, that the same notation may be
applied to denote the quotient of the division of any
number by another; thus, the quotient of 315, divided
by 4, may be denoted by *; ; for it answers the condi-
tion which the quotient must satisfy ; that is, if multi-
plied by 4, it produces 315.
The term fraction, or broken numbers, which is
generally applied to such quantities as #, originates in
a view of their origin, which is different from the pre-
ceding, though it leads to the same conclusion, as we
shall see when we come to the express discussion of
such quantities.
(311.) There are many cases where the divisor is not When the
within the limits of the table, but where it is the product divisor is f
of two or more numbers which are so, which may be ..ºt.
known from trial, or otherwise ; in such cases, the quo- in the
tient may be obtained by successive division by the limits of
factors of the divisor, as in the following examples: the multi-
(1.) To divide 20390216 by 56: ſººn
8) 20390216
7) 2548777
364III
As we obtain the same product 20390216, whether we
multiply 364111 at once by 56, or first by 7, and then by
8; so, likewise, we produce the same quotient, whether
we divide 20390216 at once by 56, or successively by
8 and 7.
(2.) To divide 70.14596 by 72:
6) 7014596
12) 1169099 – 2.
97,424 – 11
or, 9742443.
The first remainder is 2; the second is 11 ; if this The re-
be reduced to the form of a quotient, it is equivalent to mainder.
++, or ##, multiplying and dividing by the same num-
ber 6: to this must be added the first remainder 2, which
is equivalent to #3; we thus get the whole additional
part of the quotient, which is ##.
Or the same result may be obtained as follows:
every unit in the quotient of the division by 6, may be
considered as corresponding to 6 units of the dividend:
the remainder 11 of the second quotient is, therefore,
equivalent to 66 units of the dividend, to which if 2
be added, the sum is 68, which, if reduced to the form
of a quotient from the division by 72, gives the fraction
8
"(312.) When the divisor is not at once resolvable into Long divi-
factors within the limits of the table, or when its com-sion.
position is unknown, we must resort to the process
termed long division, which is applicable to all cases.
It is as follows:
Write the divisor and dividend consecutively, separa- Rule.
ting them by a curved line, as in short division ; the
quotient is written after the dividend, and separated
from it in the same way as the divisor: inquire how
often the divisor is contained in as many of the highest
places of the dividend as there are places in the divi-
sor; but if the number thus formed be less than the
divisor, an additional place of the dividend must be
taken; place the digit thus found in the quotient, and
3 s ---
496).
A R I. T. H. M. E. T.I. C.
Arithmetic, multiply the divisor by it, and place the result beneath
^-y- the assumed portion of the dividend, and subtract the
one from the other; to the remainder append the next.
figure in the dividend, and repeat the process until all
the places in the dividend are exhausted.
The process will be better understood from its appli-
cation to a few examples.
Let it be required to divide 42075 by 275:
275) 42075 (153
275 r
1457
1375
825
825
The first three places of the dividend make a mum-
ber which is greater than the divisor, which is contained
once in it: the figure in the quotientis I ; subtract 275
from 420, the remainder is 145; append to this 7, the
next figure in the dividend, when the number to be next
divided becomes 1457; the number 275 by trial is found
to be contained 5 times in it; write down 5 in the quo-
tient, and multiply 275 by 5, the product is 1375, which
subtracted from 1457 leaves 82; to this append 5, and
the next number becomes 825, which contains the divisor
thrice; write 3 in the quotient, and multiply 275 by 3, and
the result is 825, which subtracted leaves no remainder.
If the process were written at full length, it would
appear as follows:
• * 275) 42075 (100 + 50 + 3 = 153
27500 -
14575
137.50
825
825
We first multiply 275 by 100, and subtract the result,
which leaves 14575 : we next multiply 275 by 50, and
subtract the result, which leaves 825 : we then multiply
275 by 3, and subtract the result, which leaves no re-
mainder. We have thus subtracted 100 + 50 + 3 or
153 times 275 from the dividend, and there is no re-
mainder: in other words, 153 is the perfect quotient of
the division under this form of the process: the cyphers
are superfluous, and 5 is written once more than neces-
sary. The other form of it, which is a skeleton of the
complete one, is the best adapted to practice, inasmuch
as it omits all unnecessary writing.
Let it be required to divide 29137062 by 5317:
5219
5317) 29137062 (5479;
26585
25590
21268
42526
37219
53072
47853
52.19
In this example, we take 5 places of the dividend for
the first division, though there are only 4 places in the
Examples
divisor; the last remainder is. 5219, and the quotient
corresponding to it is the fraction #.
Let it be required to divide 31086917 by 71000.
When there are cyphers after the significant digits in Cyphers in
the divisor, we mark off as many places from the divi- * divisor.
dend as there are cyphers in the divisor, and then pro-
ceed to divide the remaining places of the dividend by
that divisor, which arises from the omission of the
cyphers. -
Part I.
71,000) ** (437;
gº
268.
213
556
497
599 hº
The reason of this process will be at once seen if we
write it at full length.
71000) 31086917 (400 + 30 + 7
284.00000
26869 I'7
2}30000
U 556917
497000
59917
If there are cyphers terminating both the dividend º
and divisor, we may obliterate altogether as many as and dividend
are common to each of them.
Let it be required to divide 239406000 by 12100000:
121,002.22) 2394,06322 (19;
12]
*-*
1184
1089.
9506
(313.) Other examples:
(1.) To divide 10000 by 3:
3) 10000
33333.
(2) To divide 83016572 by 240 :
8) 8301657,2
3) 103707–i
345902 — I
Other
examples.
or, 345902;.
(3) To divide 2988394.5593000 by 84050000:
8405,0202) 2938394.559,3222 (349600;
- 25215
41689
33620
80694
76645
50495.
50430
65593.
- : *---- - A. R. I “T H M E T I C.
49;
Atithmetic, (314.) On the methods of verifying the correctness of
Proof of the operations in the Addition, Subtraction, Multiplica-
rOOt O
Addition tion, and Division of whole numbers.
r l l - e g
subtraction, Different methods have been proposed for verifying
Multiplica- the correctness of the results obtained in the fundamen-
tion, and tal operations of Arithmetic. Thus, in Addition, we
* are directed to add the digits in the several columns
downwards, and see whether the result thus obtained,
agrees with that obtained by adding them in the con-
trary direction. In Subtraction, we must add the re-
7mainder to the subtrahend, and observe whether the
sum equals the minuend. In Division, we are directed
to multiply the divisor into the complete quotient, the
result P which ought to equal the dividend; but the
method, which of all others is the most popular, and we
may add, likewise, the most general, is that which is
founded upon casting out the 9's, the principle of which
we shall now proceed to explain.
(315.) If we divide the series of articulate numbers 10,
20, 30, 40, 50, 60, 70, 80, 90, by 9, the remainders are
di 1, 2, 3, 4, 5, 6, 7, 8, respectively; and the same is the
\\l C. a. Ill! IIl- tº ſº
i., § case by whatever number of cyphers these digits are
is the same succeeded: in other words, the remainder from the
The re-
mainder
from divi-
as from division of any number, such as 8 3 4 5 6 7. 23, is the
* same, whether we consider it as equivalent to 80000000
... + 3000000+400000 + 50000 + 6000 + 700 + 20
+ 3, or as simply equal to the sum of its digits, or 8 +
3 + 4 + 5 + 6 + 7 -- 2 + 3 : that is, the remainder
from dividing any number by 9, is the same as that
which arises from dividing the sum of its digits by
9. It is this theorem which is the foundation of the
rule in all cases.
(316.) In the first place, for addition, the rule
is as follows: Cast the 9's out of the digits of the
several sums to be added, and also out of the sum of the
several remainders; the last remainder thus obtained is
equal to the remainder which results from casting out
the 9's from the sum of the sums. The following is an
example:
In addition.
78.403 — 4
20465 — 8
796.39 – 7
57341 – 2
235848 – 3
The sum of 7 and 8 is 15, omit 9, there remains 6:
the sum of 6 and 4 is 10, omit 9, there remains 1: the
sum of 1 and 3 is 4, which is the remainder from
casting out the 9’s from 7, 6, 4, and 3, and also the re-
mainder from dividing 78403 by 9: the remainders ob-
tained by the same process from the other numbers, and
from the sum of the sums, are 8, 7, 2, and 3, respec-
tively; and the remainder from casting out the 9’s from
the sum of the remainders corresponding to the several
numbers, is 3, the same as that from the sum of the
numbers, as it ought to be, if the addition is correct,
for the following reasons.
The numbers 4, 8, 7, 2 are the remainders, from
dividing the several sums by 9: if we divide their sum
by 9, the remainder must be the same as that which
arises from the division of the sum of the remainders
by 9, which is 3. -
This proof, however, cannot be considered as com-
plete, inasmuch as this agreement of the remainders
may take place even when the addition is not correct :
thus, the remainder from the division of the sum would
be 3, if we should from mistake have written down Part I.'
234048 for the sum, and not 235848; but if the re-'--"
mainders are not the same, the result is eertainly wrong;
and a mistake generally so considerable, as to produce
a difference of 9 in the sum of the digits, ean hardly be
eonsidered as within the limits of ordinary errors.
(317.) The rule for proving the correctness of the
result of the multiplication of two mumbers is as follows:
Cast out the 9’s from the digits of the multiplicand, For multi-
multiplier, and product ; multiply the remainders from plication-
the two first together, and cast out the 9's from their
product: if the remainder which thus results is the
same as that from the product of the two numbers, the
operation most probably is correct; if not, it is certainly
wrong. -
The following is an example of its application:
Multiplicand. . . . 3748 — 4 5
Multiplier...... 6236–8 a
-—- 8
Product . . . . 23372528 — 5 s\
The remainders, 4 and 8, from the multiplicand and
multiplier, are placed in the opposite angles of a St.
Andrew's cross; in one of the remaining angles is
placed the remainder, 5, from their product; in the last,
is placed the remainder from the product, which is like-
wise 5, which shows that the multiplication is correctly
performed. * - -
(318.) The proof of this rule may be readily derived Proof of
from the general theorem mentioned above, and the con- the rule-
sideration of the nature of multiplication. The multi-
plicand consists of a multiple of 9, and a remainder;
and the same is the case with the multiplier. In form-
ing the product, we add the multiplicand to itself as
often as unity is contained in the multiplier: in the first
place, we add the multiplicand as often as unity is
contained in the portion of the multiplier, which is a
multiple of 9 ; the sum is clearly a multiple of 9.
Again, we add the multiplicand as often as unity is con-
tained in the remainder from the multiplier; the sum will
consist of a multiple of 9, arising from the repeated ad-
dition of the multiple of 9 in the multiplicand, and another
part, which arises from the addition of the remainder of
the multiplicand as often as unity is contained in the
remainder of the multiplier, which is clearly equiva-
lent to the product of these remainders, which is the only
part of the entire product which is not necessarily a
multiple of 9. If, therefore, we reject the 9’s from
this product of the remainders, the remainder which
results, must clearly be the same as the remainder from
casting out the 9’s from the product of the multiplicand
and the multiplier.
(319.) The process of the rule for proving the truth of For division.
division, must clearly be founded upon that for multipli-
cation; the dividend corresponding to the product, and
the divisor and quotient corresponding to the *...*
and multiplier. We must cast out the 9’s from th
divisor and quotient, and from the product of the re-
mainders, and the resulting remainder must be equal to
that which arises from casting out the 9’s from the
dividend. In case the quotient is not complete, the
remainder after the last division, must be subtracted from
the dividend, before this rule is applicable.
FRACTIONS.
(320.) Wehave before spoken of the origin of fractions, Fractions.
in connection with the process of division, where they are



3 s 2
492
A R L T H M E T I C.
Arithmetic, considered as representing the quotient of the division
of the numerator by the denominator.
Thus + represents the quotient of 1 divided by 4; #
the quotient of 3 divided by 7 ; and similarly in other
CaSeS.
We are thus lead to another view of their meaning,
which is very simple and intelligible. The denominator
is said to denote the number of parts into which unity
is divided, and the numerator denotes the number of
those parts which are taken.
Thus, if 1 represented any concrete quantity whatever,
and, therefore, divisible into parts: and if the number
of equal parts was four, one of them would be denoted
by 4, two of them by 3, three of them by 3, and the
whole by 4, or 1: if another equal portion of another
similar unit was added, the sum would be denoted by #;
if two were added, the sum would be denoted by , ; and,
in a similar manner, we should be enabled to interpret
the meaning of any fraction whose denominator is
four, whether proper or improper.
In a similar manner, & would denote 3 of the seven
equal portions into which unity was divided, and #9
would denote 10 of the same portions of which unity
contained 7.
Second in -
terpretation
of their
meaning.
In conceiving the meaning of such quantities, the
mind naturally resorts to actual objects, which are divi-
sible into parts, of whatever nature they may be, de-
priving them of that abstract quality, which in their
representation they possess equally with whole numbers.
(321.) The fractions #, 4, §, # are equivalent to each
other, it being clearly indifferent, whether we divide unity
into 2 parts, and take 1; into 4 parts, and take 2 ; into 6
parts, and take 3; or into 8 parts, and take 4: in short,
all fractions are equivalent to each other, which may be
derived from each other, by multiplying or dividing their
numerators and denominators by the same number :
thus ; and ºr are equivalent to each other, it being the
same thing whether we divide unity into 3 parts and
take 2, or divide it into 4 times as many parts, and take
4 times as many of them. The same reasoning would
apply to all other fractions which are thus related to each
other.
It is an important proposition, which is founded upon
the principle just mentioned, that fractions are not
altered in value by multiplying or dividing both their
mumerators and denominators by the same number.
Reduction (322.) It is frequently requisite, however, to reduce a
of fractions fraction to its lowest terms, when its numerator and de-
What
fractions are
equivalent
to each
other ?
Proposition.
.." nominator admit of a common divisor, or measure ;
terms. and the discovery of this common measure becomes an
inquiry of importance. In some cases, it is discover-
able by inspection: thus, 2 is a common measure of all
even numbers; and fractions, such as ºr and #3, are at
once reducible to # and 44; in other cases, the common
measures are masked in the products in such a manner
as not to be discernable, without some further knowledge
of the composition of numbers: thus, # is reducible
to Hº, from our knowledge of the multiplication table,
and the same means furnish us at once with the reduc-
g 4 & 5 5 3 6 2 5 1 gº
tions of ##, #, and ###, to 3, #, and T. The composi-
tion of the numerators and denominators of frac-
e _9_1 19.2 1 4 08 i e
tions, such as Tººf, #3, ###, is not discoverable by
such simple means, nor indeed by any methods which
are not those of successive trials. There is a general
method, however, of discovering the greatest common
measure of any two numbers, the rule for which is as
follows::
º
Divide the greater number by the less, and the last Part II.
remainder by the last divisor continually, until there is no
temainder; the last divisor is the greatest common Rule for
measure required. *
Thus, let it be required to find the greatest common É.
measure of 91 and 147: or, in other words, to reduce measure of
the fraction #7 to its lowest terms. . In Ul IIl-
erS.
91) 147 (1 Example.
91
T56) 91 (1
56
35) 56 (1
36
21) 35 (I
21
14) 21 (1
14
7) 14 (2
14
*
The greatest common measure is 7, and the reduced
fraction is, therefore, ##.
It does not require a very difficult analysis of this Proof that
operation to prove the truth of the conclusion which 7 tº º-
is thus deduced ; it being merely requisite, for this pur- . t;"
pose, to trace the steps in an inverse order: thus, 7 is " te
a divisor of 14, and, therefore, of 14 + 7, or 21 : it
is a divisor of 21 and 14, and, therefore, of their sum,
which is 35 ; it is a divisor of 35 and 21, and, therefore,
of their sum, which is 56: it is a divisor of 56 and 35,
and, therefore, of their sum, which is 91 : it is a
divisor of 91 and 56, and, therefore, of 147. It is
thus shown to be a divisor or measure, both of 91 and
147: the only principle, involved in this proof, being
the very simple one, that if a number divide two others,
it will divide their sum. -
It only remains to show that 7 is the greatest num- And also the
ber which divides 91 and 147, a conclusion which will greatest of
be established, if it be shown that every divisor of "..."
91 and 147 is necessarily a divisor of 7: for, let us IllèaSu TeS.
suppose that some number greater than 7 is a divisor
of 91 and 147; if so, it must divide their difference,
which is 56; and since it divides 91 and 56, it must
divide also their difference, which is 35 ; and if it divide
56 and 35, it must divide their difference, which is 21 ;
and if it divide 35 and 21, it must divide their diffe-
rence, which is 14; and if it divide 21 and 14, it must
divide their difference also, which is 7: but no number
greater than 7 can divide it; therefore 7 is the greatest
of all the numbers which can divide 91 and 147.
We will take another example. Let it be required Second
to reduce the fraction # to its lowest terms, example,
75) 405 (5
375
30) 75 (2
60
15) 30 (2
30
A R IT H M ET I C.
493.
Arithmetic. The reduced fraction is #. -
It is very easy to show that 15 must be a divisor of
75 and 405 : 15 is a divisor of 30, and, therefore, of
twice 30 or 60 ; it is, therefore, a divisor of 60 and
15, and, therefore, of their sum, which is 75 : it is a
divisor of 75, and, therefore, of 5 times 75, which is
375 : it is a divisor of 375 and of 30, and, therefore, of
their sum, which is 405.
It is very easy, by a reversion of these steps, to show
that every divisor of 405 and 75, is also a divisor of
15 ; and that, therefore, 15 is the greatest measure of
these numbers ; for every number which divides
405 and 75, divides also 405 and 375, and, therefore,
their difference, which is 30 ; and if it divides 75 and
30, it must also divide 75 and 60, and, therefore, their
difference, which is 15.
The principle of this proof is independent of the
particular numbers involved in the preceding examples,
and, therefore, equally applicable to every other case.
We may, therefore, consider the rule as universally true,
and that it will in all cases lead to the detection of the
greatest common measure, whenever such measures
exist which are greater than unity.
(323.) The following are other examples:
(1.) To reduce ### to its lowest terms.
2
T-7-I.
(2.) To reduce #### to its lowest terms.
(3.) To reduce gº to its lowest terms.
3
4, 5*
1 (4.) To reduce ######2 to its lowest terms. Answer,
Tsº. -
Mode of (324.) In order to compare fractions with each other,
Comparing it is requisite to reduce them to a common denominator,
tºº, when the relation between them will be that of their
other. numerators: thus, 3 and # being reduced to equivalent
fractions, with a common denominator 12, become fºr
and +3, and the relation between is that of the numbers
8 and 9. But it is not in these cases only, in which we
wish to compare the magnitudes of fractions with each
other, that such reductions are requisite, inasmuch as
they are required whenever fractions are to be added
together, or subtracted from each other. The following
is the general rule by which it is effected : ſ
Rule for Jºhen any number of fractions are to be reduced to
reducing a common denominator, each numerator must be mul-
fractions to tiplied into all the denominators ercept its own, for a
21 COIn ºn On g
den.or new numerator, and all the denominators must be mul-
tiplied together for a new common denominator.
The least consideration of this rule will show, that
the numerator and denominator of each fraction are
Proof.
Its principle
general.
Other
examples. -
Answer,
Answer, 3.
Answer,
multiplied by the same number; namely, by the pro-
duct of all the denominators except its own, and, con-
sequently, that its value is not altered. A few examples
will show this more clearly.
(1.) To reduce # and # to equivalent fractions having
a common denominator. -
Examples.
To form the new numerators, 3 × 9 = 27.
7 × 4 28.
To form the common denominator, 4 × 9 = 36.
tºm
&gamma
ſºmºs
The new fractions are #4, and #3, are formed by
multiplying the numerator and denominator of # by 9,
and the numerator and denominator of , by 4.
(2.) To reduce #, +, ++ to equivalent fractions
having a common denominator.
For the numerators, Part I-
2 × 11 × 14 = 308 \-y-
5 × 3 × 14 = 210
1 1 × 3 × 11 - 363
For the common denominator, -
3 × II X 14 = 462
The new fractions are #}}, #3, ###, which are
clearly equivalent to the original fractions, inasmuch as
the numerator and denominator of 3 are multiplied by
the same number 11 × 14, those of #1 by 3 × 14,
and those of 44 by 3 × 11.
(3.) To reduce ; and # to equivalent fractions having
a common denominator.
These fractions, determined by the rule, would be #
and ##, which are clearly reducible to two others, #%
and ##, which are equivalent to the former, but in lower
terms. r
(325.) In this case, the rule gives a common denomi- Reduction
nator, which is not the least of those which can be found, of fractions
and it is always expedient, and sometimes important, to to their lºst
exhibit the fractions under their most simple form. It .
is on this account requisite to modify the rule, so that g
the common denominator which results from it may
be the least possible. *
It is obvious, that any denominator which is a mul-
tiple of all the denominators will answer for a common
denominator, and the conditions of the question will be
fulfilled, therefore, by that denominator which is the
least common multiple of the denominators. *.
Thus, in the last example, the denominators are
and 8, or 2 × 3, and 2 × 4, where 2 is their greatest
common measure. It is clear, therefore, that 2 × 3 ×
4 is a multiple of 6 and 8, and it is their least common
multiple: the two fractions, therefore, become #. and
7 X 3 cy
#. and #.
(326.) The solution of this question requires the deter- Least com-
mination of the least common multiple of the denomi-monºlº-
nators, which may be found upon the following principle: º
Find out all the simple factors of the several numbers; numbers.
the numbers formed by the multiplication of the simple Rule.
factors, omitting one of them, as long as it occurs in any
two of the numbers, is the least common multiple required.
Thus, suppose it were required to find the least com-
mon multiple of 14 and 63; the numbers resolved into
their factors are 7 × 2, and 7 × 3 × 3. The least
common multiple is, therefore, 7 × 2 × 9, or 126
Let it be required to find the least common multiple
of the numbers 8, 12, and 18.
The numbers resolved into their simple factors are
2 × 2 × 2, 2 × 2 × 3, 2 × 3 × 3 ; and the least
common multiple is, therefore, 2 × 2 × 2 × 3 × 3, or
72. -
(327.) There is a common arithmetical rule which leads
to the same conclusion, and which is more convenient in
practice than the one just given ; it is as follows:
Write down in one line the numbers whose least com- Arithmeti-
mon multiple is required: divide those which have a cal rule.
common measure by that common measure, and repeat
these divisions as long as any common measure exists
between two or more of them : the least common multiple
is the continued product of the divisors, and of the
quotients of the several divisions. .
Thus, in the case of the example just given, we pro-
ceed as follows:
2 O
494
A R I T H M E. T. I C.
Arithmetic. 2) 8, 12, 18
~N~! 2) 4, 6, 5
3) 2, 3, 9
2, 1, 3
Examples
of the re-
duction of
fractions to
their least
common de-
nominator.
Then 2 × 2 × 3 × 2 × 1 × 3 = 72,
is the least common multiple required.
Let it be required to find the least common mul-
tiple of the nine digits:
2) 1, 2, 3, 4, 5, 6, 7, 8, 9
2) 1, 1, 3, 2, 5, 3, 7, 4, 9
3) I, 1, 3, 1, 5, 3, 7, 2, 9
1, 1, 1, 1, 5, 1, 7, 2, 3
and 2 × 2 × 3 × 5 × 7 × 2 × 3 = 5040
is the least common multiple required. * -
The least consideration of this process will show,
that by means of it, when the same common factor
occurs in two or more numbers, it is obliterated in all
of them, but is preserved singly in the divisor, which
becomes a factor of the least common multiple: it is,
therefore, clearly identical with the rule first given, but
is exhibited in a form which is better adapted to arith-
metical practice. -
(328.) We will now resume the subjectof the reduction
of fractions to their least common denominator, which
introduced the process for finding the least common
multiple; and, suppose it was required to reduce the
fraetions §, ++, and 4% to their least common deno-
minator.
The least common multiple of the denominator is
72, as we have found above; divide 72 by 8, 12 and 18
respectively, and we shall get 9, 6, and 4 for the res-
pective multipliers of the numerators; the fractions
'ſ X 9 1 1 X 6 17 × 4 -
am-ºsº ºs-m-sº g 3. 6.6
become then #, **, and *, *; or #3, #3, and
#3 respectively.
Let it be required to reduce the fractions #, 3, #,
#, and #3, to their least common denominator.
2 4 3
The least common multiple of the denominators is
7 x 8 × 9, or 506. The multipliers of the numerators
Mixed
numbers:
their reduc-
tion.
are 72, 63, 56, 21, 7.
. The fractions are
º, #3, ###, ###, #.
Let it be required to reduce the fractions
+º, +$º, Toºwo, and Toºwoo
to a common denominator.
The least common multiple is 1000000.
multipliers of the numerators are
100000, 10000, 100, and 1.
The fractions are
+%, +}}}#}o, Tožog, and Toºwww.
(329.) Mixed numbers are those which consist partly
of whole numbers, and partly of fractions, of which we
have already had examples in the quotients from the
division of a number by another, which is not con-
tained a certain number of times exactly in it; of this
kind, are 24, 7 #, 23 #, 1059 ###, &c. Such quantities
are easily reducible to a fractional form; thus 2 # is
equivalent to 2 + 4, or to 3 + 4 ; or, reducing them
to a common denominator, to 4 + 4, and incorporating
them by adding the numerators, and subscribing the
common denominator, to #. -
The
In a similar manner, 7 # = 7 -- # = } + # = # + ...Part I.
- 65 •
3. g" 1959 105.9 × 114 YºMº.
g 1 1 3 - -> & . . . . ;
Again, 1059 44; = ** + ++3 = ** + 44; =
1907.26
1208.39
Tº T'
114 +++ =
(330.) Again, improper fractions (where the numerator Improper
exceeds the denominator) are reducible to mixed num- fractions.
bers, by simply dividing the numerator by the denomi-
nator, according to the ordinary rule :
Thus, # = 2 +.
- †: = 1 +%.
+ = Il #.
# = 236 #.
* = 1209 #1.
(331.)The addition of fractions to each other is effected Addition of
by reducing them to a common denominator, adding their fractions.
numerators, and subscribing the common denominator.
Thus, let it be required to find the sum of $ and #.
Reduced to a common denominator, they become
+, and +3, and their sum, therefore, 4%.
Or, more formally, thus:
2 x 4 = 8 , *z W. ºl
3 3 - 3 3 × 4 = 12.
I'7
and the sum required 4%.
To find the sum of #, 3, #:
2 × 3 × 4 = 24.
and the sum required ## or +3.
To find the sum of 3, 3, #:
3 × 3 × 5 = 45
2 × 1 × 5 = 10 1 × 3 × 5 = 15.
7 × 1 × 3 = 21
76
and the sum is #3, or 5 #.
To find the sum of +6, +ºn, and Toºwo
3 × 10000 = 30000
7 × 100 = 700
11 x 1 - 11
307 11
and the fraction is ºw. *
(332.) Fractions are subtracted from each other by Subtraction
reducing them to a common denominator, subtracting of fractions.
their numerators, and under the remainder subscribing
the common denominator.
Let it be required to subtract # from #.
The fractions reduced, to a common denominator are
# and #3, and their difference gº.
To subtract # from #r:
7 x 13 = 91
‘8 x 11 = 88
- 3
and the remainder is +33.
º
13 x 11 = 143.
. A R IT H M ET I c. - 495
Arithmetic. To subtract +}o from +% two-thirds of three-fourths, 4 of 3 of 4, and so on. A Part I.
*N* 9 × 10 = 90 very little examination will show that the equivalent \-V-'
3 × 1 = 3 simple fractions are formed by multiplying the several
- fractions of the compound fraction together. **
87 Thus, when we say two-thirds of three-fourths, we Their mean-
mean by it two-thirds of that portion of unity which ing.
- smai is −34. º wº e tº g
and the remainder is #5 three-fourths denotes; thus, if unity be divided into 4
To subtract 2 # from 7 #: º equal parts, and three of these be taken, and if each of
The mixed numbers are reduced to the fractions : these be again divided into 3 equal parts, and 2 of
and # respectively: each of them be taken, then each of these parts will be
70 × 4 = 2; 4 × 9 = 36 one-twelfth of the original unit, and the number of
11 × 9 = 99. * Lºſ up a them taken will be 2 × 3, or 6; the result is, therefore,
181 equivalent to º, or #, or #, the product of the mul-
* , ºl. a Y- in Aar is 381 — R -1. & tiplication of $ into #. The same reasoning will apply
and the remainder is # = 5 º';. to all other cases of such compound fractions:
Multiplica- (333.) Before we proceed to state the rule for the mul- Thus, 4 of 3 of 4 = 4 × 4 x 4 = }} -
tion of tiplication of fractions, it is proper, in the first place, to A º, a # of $ ºf , "J ºr " , 9 0. Examples.
* ascertain its meaning when applied to such quantities, gain, Tāo of 12 # = +}o × 12 # = +}o x * =
and to show in what manner it is connected with the ##3 = %. - - -
definition of the term in the case of whole numbers. Also, # of # of 10 # = 4 × 4 × 10 # = 4 × 4 ×
The product of a fraction multiplied by a whole $2 – 30,
number is derived at once from that definition without r — 4.5. - e e & te º
any modification of its meaning; thus, the product of (335.) The rule for the division of fractions is founded Rule for
b 5.
3. * * * * : ~ 1.2 l- > * & A is upon that for multiplication, the operations being the in- division of
5. multip lied by 4 is , being the result of the addition verse of each other; in other words, if we multiply and fractions.
of § to itself, repeated 4 times. divide by the same fraction, the value of the multipli-
cand must remain unchanged: thus, if we multiply and
But 4 is equal to * or its value is not altered by er
divide ; by #, the first operation gives #3: ; the second
4. 7 4 X 7
being multiplied and divided by the same number 7 :
therefore the fraction 3 being multiplied by 4, the must give º otherwise the result would not be
product divided by 7, and the result again multiplied st give ºxx, -
by 7, its value is not altered. Let us take the operations equivalent to #.
in their order: When we divide, therefore, one fraction by another, Rule.
º in Ix * - 3 X 4 rºs.--> 1 > 3 X 4 we obtain the quotient by multiplying the numerator of
Mºº f Multiply § by 4, the result is ºt. Divide tº by 7, the dividend into the 㺠of the divisor for
tion, the result is #. which must be the case, inasmuch as its numerator, and the denominator of the dividend
3 X 4 º ... . 3.X 4 into the numerator of the divisor for its denominator;
à X7. being multiplied by 7, the result #; x 7 is and it is clear, that the same result would be obtained
by inverting the terms of the divisor, and then proceed-
equivalent to *::: but if we stop before this last ing as in multiplication. sº-
operation, the result #, which arises from multiply- The quotient of # divided by # is equal to 3 × 4 Examples.
*~ 3.1 -
— go-
ing by 4 and dividing by 7, may be considered as . 9 aiviaca º
equivalent to the product of the fraction ; by the The quotient of +3.5 divided by +6 = +35 × .
fraction 4. When we multiply, therefore, by a fraction, = +%. -
we mean, that we multiply by its numerator, and The quotient of 3 # divided by 9 + = ** x -3
Us tº g g tº º g e ‘r * 2 ºf X ig
divide by its denominator; the only signification which # = }.
it can admit of, so as to be consistent with the J.
definition of multiplication in the case of whole
The quotient of 3 of divided by 4 = 3 × 4 ×
numbers. # = ##.
Rule. The rule for the multiplication of fractions is The quotient of #5 of +30 divided by Tºro of +}o
founded upon this view of the meaning of the opera- 1 00 0 1 0 0 , 900 300
tion. We must multiply the multiplicand by the - To x +30 × − x * = i = **
numerator, and divide by the denominator; or, in (336.) There are some consequences of the notation of Interpreta-
other words, we must multiply the numerators of the fractions, and of the meaning attached to them, which, tion ºf the
two fractions together for a new numerator, and the two though legitimate and even necessary deductions from meaning of
e g º e G º g SOIne Oecul-
denominators together for a new denominator. them, it may be requisite to explain ; thus, let it be far .
Iºxamples, Thus, the product of +% multiplied by # is #. = required to assign the proper meaning of the frac-tional forms.
I
#. tion T.
- . 2 × 3 × 4 × 5 ſe
The product of 3, 4, 4, § is :::::::: = } = 4. This is merely the mode of denoting the quotient of
1. 's f 9 the division of 1 by 3, which, if reduced according to
The product of 9 +º, 7 Tºp, 1 Tºro , or, of #4, the general rule, is equivalent to 1 x # = 3. tº
###, 0. #, is tº
#}}, +}}}, i In the same manner, # is the quotient of 4 divided
Fractions of (334.) We frequently have occasion to make use of l
fractions, compound fractions, or fractions of fractions, such as by #1, and is therefore equivalent to # x # = }}
#
2
6
91 × 701 x 1001 688547.91 854791
1000000 = º = 63 º'
496
A R I T H M E T I C.
Arithmetic.
S-N-' Extending this conclusion, the fraction 3 is equiva-
#--
##, or ##.
lent to #; and, again, to 1 × ##,
C
#3;
1 ––
In the same manner, -i- = 4, and Ii = 3, and there
I
is, clearly, no limit to these different notations for the
same quantity. * .
(337.) Again, fractions such as the following
Continued
fractions. 1.
* 1 + #
must be reduced, by first incorporating the fractions in
the denominator, and then proceeding by the general
rule : it thus becomes
1
Tº = #.
T º
Another example is the fraction
5:#.
5 + #
which must be reduced by successive operations: we
shall thus find,
ºf 3 = ºria = ±
3+; A 1 4, 1
F 2 7
1 5 1 – 82
4 1 — iſ 5T
Such fractions are called continued fractions, and
the reductions become very complicated, when the
number of terms is great, unless simplified by rules
founded upon algebraical formulae.
Examples. (338.) The following examples will furnish instances
of most of the preceding reductions.
(1.) Reduce the fractions #4, #3}}}, and ##### to
their lowest terms. tº
(2.) Reduce the fractions #, 4}, and 43 to equiva-
lent fractions having a common denominator.
(3.) Find the least common multiple of 3, 7, 21,
27 and 63. -- -
(4.) Reduce the fractions 3, #3, ##, and 4% to equi-
valent fractions having the least common denomi-
nator. .
(5.) Reduce the mixed number 117 4 to an im-
proper fraction.
(6.) Reduce the improper fraction
number. g
(7) Add together the fractions 3 and #: 4, ##, and
#3: and 3 }, 74, and 10 +5.
(8.) Subtract ºr from 44; and 34 from 74.
(9.) Multiply §§ by ##: £ into # into #: 3 } into
74, into 10 +5.
(10.) What is the value of 4 of 4 of $ 2
(11.) Divide #6 by #5%, and 3 of # by # of #3.
(12.) Reduce the fractions
2 39
# to a mixed
O 1. 1. –2.
3. 7} 7
S 9T; 3 #
to their most simple forms.
(13.) Reduce the continued fraction
3
4 +
7
TO + 1 3
Part F.
DECIMALs. ~~
(339.) We have before explained the nature and origin Decimals.
of decimals, as connected with the notation by nine
figures and zero; the digits on the right of the place of
units being supposed to be divided by 10, 100, 1000,
10000, &c., in the same manner as those which are
respectively equidistant on the left, are multiplied by
the same numbers: thus,
783.24.2464
is equivalent to
7 x 10000 + 8 × 1000 + 3 × 100 + 2 × 10 + 4 +
2 4. 6 4.
H% + +$o + Tºo + Toºwo
and any other number involving decimals is resolvable
into its component parts in a similar manner.
The decimal
.14159
is equivalent to the sum of the fractions
l 4. 1 5 9
IOT + Too + Tooo + 10000 + 100000
which, if reduced to their least common denominator,
become
10000 4000 100
50 I tº
100000 + Iſ)0000 + 100000 + 100000 + 100000 °
and if we add them together they become
14159
100000 "
In a similar manner, the decimal expression
3.003714
is equivalent to
3
3 + 7.5 +
which, if reduced in a similar manner as the expression
last given, becomes
-º- + -t- + 4
10000 & 100000 1000000
30C37.14
1000000"
(340.) It appears from these examples (and the method Conversion
which is made use of to effect these transformations is 9f decimals
equally applicable to all cases,) that a decimal expres– º: *
sion may be converted into an equivalent fraction by G.s.
omitting the decimal point, and subscribing for a deno-
minator 1, with as many cyphers as there are decimal
places.
Thus, 90.090909
is equivalent to
- 90090909
TToooooo -
Again, .023
is equivalent to
T#85
and .0000301
is equivalent to
301
T0000000 °
(341.) Conversely, any fraction whose denominator is Converse
1, with cyphers only following it, may be converted operation.
into an equivalent decimal, by omitting the denomi-
nator, and striking off as many decimal places in the
numerator as there are cyphers in the denominator.
Thus, - +º,
is equivalent to
A R I T H M E T I C.
497
Arithmetic. .33.
S- Again,
} 41432
…” 10000
is equivalent to
14. 1432,
and
,4087
10000000 *
is equivalent to
- .0004087.
It is very important to attend to this transition from
decimals to equivalent fractions, and its converse, as it
forms the foundation of the proofs of the rules for the
multiplication and division of decimals.
Addition (342.) The rules for the addition and subtraction of
tº. f decimals are the same as those for whole numbers, care
l .." being taken to place the corresponding places under
each other. -
Let it be required to add 72.031 and 4.20123 to-
gether:
7.2.031
4.20 123
76.23223
Let it be required to add together 345.012, .02468,
7692.75, and 7.4000693 :
345,012
.02468
7692.75
7.4000693
8045. 1867 493
Let it be required to subtract3.04096 from 10.345072:
I 0.3450.72
3.04.096
7.304112
Let it be required to subtract 113.694 from 114 :
114
113.694
.306
The process in this case might, perhaps, be more
readily understood, if the decimals were written as
follows: -
114.000
113,694
It is obvious, that the addition of cyphers, after the
significant digits in decimals, makes no alteration of
their value. Thus, 114 is equivalent to 114,000, 07 is
equivalent to .070000, and similarly in all other cases.
(343.) The following is the rule for the multiplication
of decimals : * - -
Multiply the decimals as if they were whole numbers,
and strike off from the product as many decimal places
as are equal to the sum of the numbers of decimal places
in the multiplicand and the multiplier.
Let it be required to multiply together 72.037 and
3.59 : - -
VOL. I. -
Multiplica-
tion of
decimals.
72.037 Part I.
3.59
648.333
360 185
216111
258.6 1283
The sum of the numbers of decimal places in the
multiplicand and multiplier is 5, which is the number
of decimal places which must be struck off from the
product of the decimals, considered as integers.
The reason of this rule will be obvious, if we convert Proof of
the decimals into equivalent fractions: they thus the rule.
become
72037
1000
and their product is,
72037 × 359 – 25861283 .
100500 T ~ T00000 °
and if we pass from the fraction, which is the result of
the multiplication, to the equivalent decimals, it becomes
258.61283.
The same reasoning will apply in all other cases; the
numerators of the fractions equivalent to the decimals,
are the integral numbers which result from removing
the decimal point: their denominators are 1, with as
many cyphers following as there are decimal places in
each ; the product of the fractions is the product of
the numerators, which the operation performed accord-
ing to the rule always gives, divided by the product of
the denominators, which is clearly 1, with as many
cyphers as are found in the denominators of both the
fractions ; and in the transition from the fraction to
the equivalent decimal, we omit the denominator, and
strike off as many decimal places from the numerator as
there are cyphers in it.
Let it be required to multiply .00037 into .04145 :
.04145
.00037
280 15
I2435
.0000152365
In this case, it is requisite to place cyphers to the
right of the integral product, in order to get the requi-
site number of decimal places. -
Let it be required to multiply 310000 into .375.
.375
310000
375
I 125
| 16250,000
In this case the product is integral. t ---
(344.) The following is the rule for the division of Division of
decimals: - decimals.
Find the quotient in the same manner as if the deci-
mals were whole numbers ; then if the number of
decimal places in the divisor be equal to the number in
the dividend, the quotient obtained is correct: if the
number of decimal places in the divisor be less than
the number in the dividend, as many decimal places
must be struck off from the integral quotient, as is
equal to the eccess of the number in one above the num-
ber in the other; and if the number of decimal places
3 T
and **
100
498
A R. I. T H M E T I C.
Arithmetic. in the divisor be greater than the number in the dividend,
~~ as many cyphers must be written after the figures in the
quotient, (the whole being integral,) as is equal to the
ercess of the number of decimal places in the divisor
above the number in the dividend.
In the last case, it is usual, before the division is
begun, to add cyphers to the dividend, until it has as
many decimal places as the divisor. 1
Let it be required to divide 24.075 by 7.5 :
7.5) 24.075 (3.21
22.5
Examples.
157
150
75
75
The quotient of the numbers considered as integers
is 321 : but there are 3 decimal places in the dividend,
and only 1 in the divisor : we must strike off, therefore,
3 – 1, or 2 decimal places from the quotient, which
thus becomes 3.21.
(2.) If the divisor had been 75, the quotient would
have been .321.
(3.) If the divisor had been 7500, the quotient would
have been .00321.
(4.) If the divisor had been .75, the quotient would
have been 32.1.
(5.) If the divisor had been .075, the quotient would
have been 321.
(6.) If the divisor had been .00075, the quotient
would have been 32100.
The correctness of these results may be immediately
shown by passing from the decimals to their equivalent
fractions, which are # and 4; : their quotient is
$ 407 5 24 O'75
-* A A ºn-a -º - , ººgº 1. - 3.2.1 ---
1 0 () +} = H = x +}}o = 321 × Tº e ### =
3.21. *
tent is “* x -1- - -3.31 –
In case (2), the quotient is -iñº, X # = Hº's E
.321.
º ... 240 75 1 -- 3:21
In case (3), the quotient is ºn X +º, + Hºo X
sºmeº 3 2 1 *
+5 = rºw = -00321, , ,..., -
In case (4), the quotient is tº x * = ** = 32.1.
9. . 24 0.75 1 0 00
In case (5), the quotient is tº X -7E = 321.
a. g . 2 4 0 75 ... 1 00 000
In case (6), the quotient is ºf x −7: - 321 x
100 – 32 100.
The same method of proof is applicable to all other
cases, and will show very distinctly the principle upon
which the rule is founded. -
Let it be required to divide 298.89 by .1107 :
.1107) 298.8900 (2700
2214
7749
7749
In this case, the number of decimal places in the
dividend is made equal to the number of decimal places
in the divisor.
Let it be required to divide 14 by .7854
.7854) 14:0000,0000000 (17.8253119 Part I.
7854 \-N->
6146O
54978
64S2O
62832
19880
15708
41720
3997O
24500
235.62
9380
7854
1526O
7854
74060
696.86
4374
In this example, the operation does not terminate;
and in order to continue it, we have added cyphers
arbitrarily, in order to get a nearer approximation
to the true value of the quotient ; the last value ob-
tained is '###, and differs from its true value by
Tºº : and it is obvious, that by continuing the
process we may obtain a decimal value of the quotient,
differing from the true quotient by a quantity less than
any that may be assigned.
(345.) The conversion of fractions into decimals, Conversion
whether they terminate or not, is the most important of fractions
use of these quantities, as it brings them under a ºr
uniform notation. The following are examples: *
(1) + = ** = .75.
l
(2.) # = *:: * = .04
(3) + = ** = .4875.
= .088.
(4.) +º's -- **
(5.) ºr = ** = .132.
In all these cases, the factors of the denominators
are either 2 or 5, and the decimals terminate. In ex-
ample (1), the denominator is 2 × 2; in (2), it is 5
x 5; in (3), it is 2 × 2 × 2 × 2 ; in (4), it is 5 × 5
× 5 ; in (5), it is 5 × 5 × 5 × 2 ; and the number of
decimal places in each case, never exceeds the greatest
number of times that one or other of these factors are
repeated.
The fact is, that 2 and 5 are the only divisors of 10, What frac-
and, therefore, 2 × 2 and 5 × 5 are divisors of 100; tions pro-
2 × 2 × 2 and 5 × 5 × 5 are divisors of 1000; 2 × ..."
2 × 2 × 2 and 5 × 5 × 5 × 5 are divisors of 10000; “
and as the process of adding cyphers to the dividend
In the division of decimals, is equivalent, as far as the
division is concerned, to its multiplication by 10, 100,
1000, 10000, &c. respectively, it clearly follows, that
when one, two, three, four, &c. of these factors 2 and
A R. I T H M ET I C.
499
Arithmetic. 5, whether singly or conjointly, compose the divisor,
S-v- that the division must terminate after one, two, three,
four, &c. operations: it is for this reason, that the
quotient cannot involve more decimal places than the
greatest number of times that one or other of these fac-
tors is involved in the denominator.
But if the fraction in its lowest terms involves a
factor in its denominator, not resolvable into the products
of 2 or 5, such as 3, 6, 7, 9, 11, 12, &c., then the divi-
sion can never terminate, and the equivalent decimal is
interminable: for a number which is not a factor of 10,
is not a factor of 100, or of 1000, or of 10000, and, con-
quently, the continuance of the operation brings us no
nearer its termination. The following are examples:
1 - 0 00 *
(1.) # = H
The same figure is repeated continually, there being
always the same remainder; and, therefore, the same
quantity 10 to be divided. The decimal is, of course,
indefinite, and is called a circulating decimal.
(2.) # = . 16666.
The repetition begins in the second place, and the
decimal is a circulating decimal like the former.
(3) 4 = . 142857142857. . . . .
Whenever a remainder occurs, when cyphers only
are brought down, which produces a quantity to be
divided identical with any one preceding it, the same
series of quotients and remainders must occur in the
same order; the number of remainders different from
each other which can occur in succession can therefore
never exceed the divisor : in this case it is 6, and the
Tepeating period in the circulating decimal produced is
142857.
(4.) # = .11111. . . .
(5.) Tºr = .090909. . . .
(6.) +3 = .08333. . .
(7.) ºg - .076923076923. . . . .
The repeating period is 0.76923.
(8.) Tº = .06666. . . -
(9.) + = .0588235294.1352941. . .
In this case, the repeating period is 352941, and com-
mences after the first five places.
(10.) # = .05263157894736842í0526.
The repeating period consists of 18 places
(11.) # = .5925925.....
(12) #33 g = .008497133497133. . . .
(13.) #### = 4.7543543. . . .
(14.) ### = 3.14159329203...
Though in every case, when fractions are reduced to
indefinite decimals, a repeating period may be found,
yet, as the determination of it may, in an extreme case,
require a number of divisions equal to the divisor
itself, it may become too laborious to be practicable.
What frac-
tions pro-
duce inter-
minable
decimals:
Circulating
decimals.
àmº
*s
*
Rºmeºne
Conversion (346.)The preceding examples will show in what man-
ºf circula ner circulating decimals are produced: it is frequently
ting deci- important, however, to reverse the process, and to pass
mals into g & te 9
... from the circulating decimal to the equivalent fraction.
fractions. The rule for this purpose is as follows:
Multiply the circulating decimal by 1, with as many cy-
phers after it as there are decimal places before the second
repeating period ; and again multiply the circulating
decimal by 1, with as many cyphers as there are places
before the first repeating period : the products being sub-
{
tracted from each other, and the remainder divided by Part"
the difference of the multipliers, will give the fraction -v-'
which is equivalent to the circulating decimal.
Let it be required to find the fraction which produces
the circulating decimal
.0171717. ...
Multiply by 1000 : the result is
| 7.1717. . . .
Multiply by 10: the result is
.1717. . . .
Subtract these results from each other, the remainder is
17:
which divided by 1000 — 10, or 990 gives gº, the
fraction required. -
Let it be required to assign the fraction which pro-
duced the circulating decimal
34500.970097. . . .
Multiply by 10000000, the result is
3450097.0097. . . .
Multiply by 1000, the result is
345.0097. . . .
Subtract the results from each other, and the remainder
1S
34497.52 :
which, divided by 10000000–1000, or 9999000, gives
34 497 52
9 S 9, 9 00 0
for the fraction required, which, reduced to its lowest
terms, becomes
1 4 37 73
4 l 6 6 2.5
(347.) Circulating decimals present the most familiar Infinite
examines of the origin and meaning of infinite series: series.
thus
- .33333. . . .
is equivalent to
3 3 3. 3.
I, +Tº + Tº + Tº -- &c.
where the terms are supposed to be continued indefi-
nitely; the sum of the series is, likewise, the value of the
circulating decimal, and the process which determines
the one determines the other likewise.
(348.) The following examples will illustrate most of Examples.
the operations in decimals.
(1.) Add together .0345, 757.069, and 2.9168504.
(2.) Subtract 3.47965 from 5.111324.
(3.) Multiply .000395 into 27.0456.
(4.) Divide 9.6.195 by 1.21.
(5.) Divide 233.91 by .345.
1
(6.) Reduce the fractions #
decimals.
1 0 0 3
== — "º
6 9
1 x 1° 1 U U 12 and 75 33:25 to
(7.) Find the value of the circulating decimal
.003406969.
the sum of the infinite series i; + .
(8.) Find
R 3 1 3
TOOOOOO + TVNOJO) -H &c.
JEXTRACTION OF ROOTS.
Square Root.
(349.) The process for extracting the square root must Extraction
be founded upon the rule for the formation of the square, of the
in the same manner as the rules for other inverse opera- *Puare root-
3 T 2
500 A R IT H M E T I C
Arithmetic, tions are founded upon those for the direct operation:
S-TV-' the arithmetical process, however, for the formation of
the square, leaves no traces of the root which are
readily discoverable, in consequence of the incorporation
of the parts which takes place in all arithmetical pro-
cesses: we must divest the root, therefore, of its arith-
metical character, at least, as far as notation is con-
cerned, in order to detect the composition of its square.
Formation (350.) Let it be required to form the square of 74:
of the We will write it in the form
square. 70 + 4,
and consider in what manner the result arising from
multiplying this into 70 + 4 is composed.
In the first place, there is 70 times 70, which is
4900.
In the second place, there is 70 times 4, which is
- 4 × 70.
In the third place, there is 4 times 70, which is
4 × 70.
In the fourth and last place, there is 4 times 4, which
IS
4 × 4, or 16.
If all these parts be added together, so as to form
one sum, we shall get
*4900 + twice 4 × 70 + 16 ;
or the square of the number which is the sum of the
parts 70 and 4, is the square of the first part + twice
the product of the two parts + the square of the second
art.
p The same conclusion would be deduced, if the parts
were 700 and 40, 7000 and 400, or any other numbers
whatsoever.
Process for (351.) We shall now proceed to the inverse process,
ºtracting and let it be required to find the square root of
the square
TOOt. 4900 + 560 + 16 (70 + 4
4900
140 + 4) + 560 + 16
+ 560 + 16
We first find the square root of 4900, which is 70,
and subtract its square, which leaves 560 + 16: we
double 70, which gives 140, and divide 560 by it, in
order to get 4, the second part of the root: we then
add 4 to 140, and multiply the sum by 4, which gives
560 -- 16, the remaining part of the square.
We will now exhibit the same process under a some-
what more arithmetical form; let it be required to
extract the square root of 5476 :
5476 (70 + 4
4900
*mºs
140 + 4) 576
560
I6
*ºmºmºs
576
Find the greatest multiple of 10, whose square is
less than the given number ; this is 70 : subtract its
square 4900 from 5476, the remainder is 576: double
70, which is 140, and divide the remainder by it, in
order to find the second part of the root: the nearest
whole number is 4: add 4 to 140 : multiply 140 + 4
by 4: the product of 4 and 140 is 560, and that of 4
and 4 is 16: their sum is 576, which subtracted leaves
no remainder. *
It remains to give the process a purely arithmeti- Part I. .
cal form. -
5476 (74
49
*mºmºsºmºmº
144) 576
576
Divide the square into periods of two, commencing
from the place of units, by placing a dot over 6 and 4:
find the greatest number whose square is less than the
first period 54, which is 7: put 7 in the root, and
underneath the first period write its square 49, which
being subtracted, there remains 5: bring down the
next period 76, and write it after the last remain-
der: double the root 7, which is 14, and divide
57. (omitting the last digit 6) by 14: the nearest
number is 4, which must be placed after 7 in the root,
and after 14 in the divisor: multiply the divisor 144
by 4: the product is 576, which subtracted, leaves
no remainder : 74 is therefore the complete square root
of 5476.
(352.) We will proceed to another example, where Second
there are 3 places in the root: let it be required to find example.
the square root of 459684. -
459684 (600 + 70 + 8
360000
======= *-a=g
1200 + 70) 99684
84000
4900
sºme s-sm-m-m-
88900
1340 + 8) 10784
10720
64
a---
10784
Or, more arithmetically, thus:
459684 (678
36
&=====ºf
127) 996
889
*=ºmº
1348) 10784
10784
The comparison of the two schemes of the process
will show the reason of the abbreviations in the second :
the square is first divided into periods of two by mark-
ing the first, the third, and the fifth digits: the greatest
square less than the first period 45 is 36, which sub-
tracted leaves 9 : bring down the next period 96 :
double 6, the figure in the root, and divide 99 (omitting
6) by 12: the result (taken in defect) is 7: write
7 after 12, and multiply 7 into 127, and subtract the
product 889 from 996: the remainder is 107: bring
down the next period 84, and double 67, making 134:
divide 1078 (omitting 4) by 134, the result is 8: write
8 in the root, and also after 134, and multiply 8 into
1348: the result 10784 being subtracted, leaves no re-
mainder, and 678 is the complete root required.
The second scheme is the skeleton of the first, and is
founded upon the general principle of all arithmetical
rules, of avoiding all superfluous writing : the reason
of the pointing every second figure of the square,
A. R. I. T. H. M. E. T. I C.
501
Arithmetic. reckoning from the place of units, will be very obvious,
^-N-7 when we consider that the number of cyphers after the
significant digits in the square will be even, whether
the number of cyphers in the root be odd or even.
(353.) If there are decimal places in the root, there will
be double the number of them in the square, and, there-
fore, the number of decimal places in the square must
always be even. In pointing, therefore, a square,
which contains both integral and decimal places, we
must begin from the place of units, and proceed both
to the right and the left. The following is an example:
When there
are decimal
places in
the square.
1369,740i (37.01
9
*-m-msm-me
67) 469
469
7401) 7401
740 I
Indefinite (354.) When the number whose square root is required
ºpprºxima is not a complete square, we may approximate continually
º * to the true value of the root, by adding pairs of cyphers
& to the root on the right of the decimal point as often as
we choose. As an example, let it be proposed to ex-
tract the square root of 10.
10.0000. .
9
*mºsºmº
61) 100
61
626) 3900
3756
6322) 14400
12644
... (3.162
The square root of . 1 is .3162. . . . , the square root
of .01 is .1, and that of .001 is .03162.
(355.) Letit be required to extract the square root of 3.
The fraction reduced to an equivalent decimal be-
COIſleS
Square root
of a fraction.
.3756. ... (.61237
36
121) 150
I2]
1222) 2900
2444
12243) 45600
36729
122467) 887 100
85.7269
2983 l
(356.) The following are examples of the different
cases which occur in the extraction of the square root.
(1.) Extract the square root of 152399025.
(2.) Extract the square root of 119550.669.121.
(3.) Extract the square root of .0000032754.
(4.) Extract the square root of 2.
(5.) Extract the square root of 4.
(6.) Extract the square root of 7954.
Examples,
Part I.
\-N/~"
(357.) The formation of the cube, upon which the rule Formation
for the extraction of the corresponding root is founded, is of the cube.
more complicated than that of the square, and it is diffi-
cult to exhibit it clearly without the aid of algebraical
symbols. We shall assume, however, for this purpose,
74, or 70 + 4, for the root, of which the square is
4900 + twice 4 x 70 + 16;
and in order to form its cube, it is requisite to multiply
this result by 70 + 4, which being done, the several
results are as follows:
First, the product of 70 into 4900, which produces
343000, the cube of 70.
Secondly, the product of 70 into twice 4 x 70,
which is equal to twice 4 × 4900.
Thirdly, the product of 70 into 16, which produces
EXTRACTION OF THE CUBE ROOT.
16 × 70.
Fourthly, the product of 4 into 4900, which produces
4 × 4900. , -
Fifthly, the product of 4 into twice 4 x 70, which
produces twice 4 × 4 × 70, or twice 16 x 70.
Sixthly, the product of 4 into 16, which produces 64,
the cube of 4. - g
If we combine all these results together, we shall
find that the cube 70 + 4, consists of
(1.) The cube of 70, or 343000.
(2.) Three times 4 into the square of 70, or thrice
4 × 4900. -
(3.) Three times the square of 4 into 70, or thrice
16 × 70.
(4.) The cube of 4, or 64.
(358.) Assuming the sum of these expressions for the Invers:
cube, the steps in the reverse process are very obvious. process,
343000+ thrice 4 × 4900 + thrice 16 × 70+64 (70+ 4
343000 N.
Thrice 1900 thrice 4 x 4900 + thrice 16 x 70 + 64
thrice 4 × 4900 + thrice 16 x 70 + 64
We first subtract the cube of 70, (the cube of the
highest multiple of 10, which is less than the cube :)
we then take thrice the square of 70, or 3 × 4900 for
a divisor of the first term of the remainder, by which
means we determine 4, the second figure in the root:
we then subtract 3 × 4900 x 4, 3 x 70 × 16, and
64 successively, in order to take away the complete cube
of 70 + 4.
We shall now put the same example under a more
arithmetical form, and suppose that it is required to ex
tract the cube root of 405224.
405224 (70 + 4
343000
14700) 62224
58800 3 × 4900 × 4
3360 3 × 70 × 4 × 4
64 4 × 4 × 4
62224
We find the greatest multiple of 10 (70), whose
cube is less than 405224, and subtract it, leaving the
remainder 62224: we find the square of 70, which
is 4900, and multiply it by 3, which is 14700, which
we employ as a divisor of 62224, in order to find
4, the second figure in the root: we then add to-
gether thrice 4 into the square of 70, which is 58800,
502
A R L T H M E T I C.
Arithmetic, thrice 70 into the square of 4, which is 3360, and
S-' the cube of 4, which is 64, and subtracting their
sum, there is no remainder: therefore 70 + 4, or 74,
Rule.
Second
example.
is the cube root required.
It now remains to put the pro
simple form which it admits of, omitting every figure
and cypher which is not necessary in obtaining the
result.
465224 (74
343
147) 62224
588
336
64
62224
The cube is divided into periods of three places, be-
ginning from the place of units; inasmuch as there are
3 cyphers in the cube of 70, 6 in that of 700, 9 in that
of 7000, and similarly for higher orders of articulate
numbers: 7 is the greatest number whose cube is less
than the first period; the remainder is 62, to which
In the divisor we put
three times the square of seven, which is 147, and
divide 622 (omitting the two last places) to get 4, the
next figure in the root: we then form the products of
3 × 4 × 49, 3 x 16 x 7, and 4 × 4 × 4, and place
them underneath each other, so that the second may
advance one place beyond the first, and the third one
place beyond the second: they are then added together,
and their sum subtracted from the dividend, and, as
the next period is annexed.
;
X
X
X
:
6
X
X
X
there is no remainder, 74 is the cube required.
(359.) We will now proceed to a second example.
Let it be required to find the cube root of 48228544:
48228544 (300 + 60 + 4
27000000
3 × 300 × 300--- w
270000) 21228544
16200000
3240000
2I 6000
19656000
1572544
1555,200
17280
64
1572544
3 x 360 × 360 =
3888.00
Or, merely preserving the skeleton of this process,
and conforming to the arithmetical rule, the scheme
will appear as follows:
3 ×
3 ºr
300 × 300 × 60
300 × 60 × 60
4
4
60 × 60 × 60
:
.
:
º
;
48238544 (364
27
2122S
162
324
216
19656
27)
3888)
Dividend
º
3 × 3 × 3 ×
3 × 3 × 6 ×
6 × 6 × 6
Subtrahend
1572544 Dividend
!
6
6
9
7
cess under the most
*
15552 3 × 36 × 36 × 4
1728 3 × 36 × 4 × 4
64 4 × 4 × 4
1572544
(360.) Let it be required to extract the cube root Part L.
of 27054.036008 :
O & tº º Cube roots
27054.036008 (30.02 of decimals.
27
27) 54
2700) 54036
270000) 54036008
540,000 3 × 300 x 300 × 2
3600 3 x 300 × 2 × 2
8 2 × 2 × 2
54036008
(361.) Let it be required to extract the cube root of 10: Tºefinite
cube roots.
10.000000... (21.54...
8
12) 2000
12
6
I
1261
1323) 739000
6615
1575
125
677.375
138675) 61625000
554700
10320
64
55573264
605 1736
Dividend
3 × 2 × 2
3 × 2 × 1
1 × 1 × 1
Subtrahend
HDividend
3 × 21 × 21 × 5
3 × 21 × 5 × 5
5 × 5 × 5
Subtrahend
Dividend
3 x 215 ×
3 × 2.15 ×
4 × 4 × 4
Subtrahend
× I
X 1
215
4
× 4
× 4
It is quite clear that the operation can never termi-
nate, and that by continuing it we may obtain an
approximate value of the cube root of 10
required limits of accuracy.
within any
(362.) The following are other examples of the various Examples.
cases which can occur in the application of this rule:
(1.) Let it be required to extract the cube root of
343052921.70 || 0729.
(2.) Let it be required to find the cube root of
1.879080904.
(3.) Let it be required to extract the cube root of
.000000042875.
(4.) Let it be required to find the cube root of 3.
(5.) Letit be required to find the cube root of #.
(363.) The invention of rules for the extraction of the Rules for
fourth, fifth, and higher roots, depends upon the formation extraction
of the fourth, fifth, and higher powers, and is effected of higher
upon the same principles as those for the square and *
cube root, though they are not easily discovered without
the aid of algebraical formulae. The rules are also ex-
tremely complicated, and their application difficult and
embarrassing, when they extend beyond two places of
figures in the root; under such circumstances, ther e-
fore, it is expedient to defer the consideration of them
until we can avail ourselves of algebraical formulae, by
W
A R IT H M ET I C. - 503
Arithmetic, which the rules may be simplified, or other methods 5476 (74 Part I.
*~~" investigated, which may give approximate values of 49 \-y-Z
, the roots. memº --
Example of (364.) The extraction of the fourth root is equivalent to 144) 576
* º: a double extraction of the square root, and such is the 576
}OI)
arithmetical method which is most convenient to follow.
The following is an example : -
Let it be required to find the fourth root of
Consequently, 74 is the fourth root required.
(365.) There are some other subjects which might be Conclusion.
included in a Treatise on abstract Arithmetic, such as
fourth root,
29986576. the notation of numbers proceeding according to scales
different from the decimal, whether binary, quaternary,
* - Up & uinary, duodenary, &c., the formation and reduction of
-- 29986576 (5476 º: . and some of the more obvious pro-
25 perties of numbers, all of which are more properly included
104) 498 - under the Theory of Numbers: whilst the consideration
416 of others, such as arithmetical and geometric progres-
*s-sºussºmsºmº sions, combinations and permutations, which are com-
1087) 8265 monly found in treatises on this subject, may with
7609 more propriety be deferred until we are enabled to in-
tº yº” vestigate algebraically the formulae upon which the
10946) ; rules” are . % shall, º: close at this
point our Treatise on the Arithmetic of Abstract
Numbers.
A. R. I T H M E T I C.
Arithmetic. (366.) NUMBERs are concrete when the units, of which
S—— they are composed, represent magnitudes to which a
Concrete denomination is given : such as 17 shillings, 143 yards,
numbers, 74 pounds, 23 minutes, 67 gallons. t
I)ifference The arithmetic of such numbers would be nearly
in the arith- identical with that of numbers which are abstract, if
metic of , the concrete units of the same species of quantity were
*...* always of the same magnitude, not admitting of sub-
Concrete © tº e e e © g
... division into others, which are multiples or submultiples
of the first; in other words, if shillings were the only
units of money, yards of length, pounds of weight,
minutes of time, and gallons of capacity. Under such
circumstances, such numbers would be subject to all
the common operations of Arithmetic, whether of addi-
tion, subtraction, multiplication, or division, without
any reference to the particular nature of the quantities
which they denoted.
Again, supposing those subdivisions were in all cases
adapted to the decimal scale, the operations on such
quantities would be in every respect identical with
those which are required in the arithmetic of decimals.
The fact, however, is, that those subdivisions are rarely
adapted to any regular scale; the duodecimal is most
prevalent; in some cases they proceed by continued
bisections; but most commonly the successive units
are not the same submultiples or multiples of those
which precede or follow them. It is this want of
uniformity which renders it necessary for the student
in the first instance to commit to memory tables of the
subdivisions of coins, of the different units of weights,
of measures of length, area, and capacity, of time, and
of such specific quantities as are frequent subjects of
consideration, but whose subdivisions do not conform
to the general custom.
These successive units, though they neither follow
the decimal or any other scale, may be brought within
the rules of the Arithmetic of abstract numbers, by
reducing the inferior units to a vulgar or decimal
fraction of one of higher denomination. Such a mode
of proceeding is not always the most convenient or
expeditious ; but in many questions it is absolutely
necessary, and in every case it is more general than
any other process which can be followed.
We shall now put down some of the more useful of
these tables, accompanied with examples of the different
species of reductions which will be required in the
solution of questions, in which such quantities are
involved.
(367.) Table of Money.
2 farthings make I halfpenny.
4 farthings. ... 1 penny, (d.)
12 pence . . . . . . 1 shilling, (s.)
20 shillings . . . . I pound, or sovereign, ( )
21 shillings . . . . 1 guinea.
504
Table of
divisions of
English
money.
PART II.
Or, expressing each superior unit, not merely in terms Part II.
of the next below it, but also of all others which are S-N-2
inferior to it, it will stand as follows:
qrs. d.
4 = I S
48 = I2 = 1 W.
960 = 240 - 20 = 1
(368.) One of the most common species of reduction, Various re-
is to express numbers of superior denominations in ductions.
units of a lower denomination, and conversely. Thus,
suppose it was required to find how many farthings there
are in 4s. 3d. :
,-, * * S. d.
4 y 3
I 2
*
1 = 48 + 3
4 -
Answer, 204
We first reduce the shillings to pence, by multiplying
the number of shillings 4 by 12: to the product 48,
we add 3, and thus get 51, the whole number of pence
in 4s. 3d. : if this number be multiplied by 4, the last
result, 204 is obviously the number of farthings re-
quired. *
Let it be required to reduce £17. 13s. 33d. to far-
things.
This sum might be written thus,
4817. 13s. 3d. 34rs.
but it is more usual to express 3 qrs., or 3 farthings, by
the equivalent fraction #d., or three-fourths of a penny.
sº. s. d.
17, 13, 3}
20
353 -
12
4239
4
16959 = 4 x 4239 + 3 = number of farthings.
The general rule for such reductions, whether of
money or other classes of concrete units of the same
species, is to multiply the superior units by the number
which connects them with the unit next succeeding in
the table, and to add to the result whatever units of the
same order may appear in the sum to be reduced; and
the process must be continued until we arrive at the
units of the denomination required.
The following question is the converse of those just
given.
Let it be required to find how many pounds, shil-
lings and pence there are in 17347 farthings.
20 x 17 + 13 = number of shillings.
tºº
Dººmg
12 x 353 + 3 = number of pence.
A R IT H M ET I C.
505
Arithmetic.
~~~
4) 17347 - *-
* -ms-î-º-º-
12) 4336 – 3
gººm-
2,0) 36,1 — 4
18, 1, 4%
We first reduce the farthings to pence, by dividing by
4; we next reduce the pence to shillings, by dividing by 12;
and we lastly reduce the shillings to pounds, by dividing
by 20: the final result is £18. 1s. 4%d.
The steps of this process, of passing from inferior to
superior units, are clearly the inverse of those which are
followed in passing from superior to inferior units.
The following are examples of the reduction of a
compound expression to a simple fractional or mixed
inumber.
What fraction of a pound is 2s. 7d. 2
43. s. d.
I 2 y 7
20 & 12
20 31 numerator.
12
240 denominator.
The fraction is fºr.
There are 3 Id, in 2s. 7d., and 240 in a pound ; and,
consequently, if unity be divided into 240 equal parts,
and 31 of them be taken, the portion of unity, or of
13., which they denote, is #'s.
What fraction of £3. 10s. is 42. 5s. 64d. 2
£. s. 39. s. d.
3, 10 2, 5, 64
2O 20
70 45
12 12
840 546
4 4
3360 2285
The fraction is ####, or, in lower terms, ###.
The following questions are the converse of the pre-
ceding.
What is the value of 3 of a pound?
£.
2
20
7) 40 (5
35
*=ºmmºn
5
12
7) 60 (s
56
4
4
7) 16 (2
14
*
92
es/
In 482. there are 40s., and, therefore,
lent to #s. ; but if we reduce this fraction to a mixed
VOL. I.
number, it becomes 5;s. ; but 5s. are equal to 600., and, Part II.
therefore, #s. is equivalent to *d., which, reduced to a
mixed number, is 8+d. Again, 4d. are equal to 16
farthings, and, therefore, #d. is equivalent to qrs. or,
24 grs., or to #d. # qrs. : the final answer, therefore, is
5s. 8%d. # qrs, or, as it is commonly written, 5s. 8%d.}.
What is the value of ºr of £2. 12s. ?
£. s.
2, 12
20
f_*
52
17
52
&mºsºmºmºmº
34
85
113) 884 (7
79]
93
12
Ilić,
1017
tºº-º-º-º-º:
99
4
113) 396
339
º-s,
57
113) (9
(3
The answer is 7s. 9; d. ººr.
The reduction of shillings, pence, &c. to decimals
of a pound, or any other superior unit, is extremely im-
portant, being the reduction which, of all others, is
most frequently required. The following are examples:
What decimal of a pound is 2s. 6d. 2
12) 6
20) 2.5
tºº
. 125
In the first place, 6d. is equivalent to Hºrs., which re-
duced to a decimal is .5: consequently, 2s. 6d. is equi-
valent to 2.5s. ; but 2.5s. is equivalent to ###., or
.125:9.
The same result would be obtained by first reducing
2s. 6d. to a fraction of a pound, and then converting
the fraction, which is ºn, or +, to an equivalent deci-
mal.
What decimal of a pound is 19s. 11; d. 2
4) 3
12) II.75
20) 19.9891666...
amºmºmº-º-º-º-º-º-
.9994.5.833. . . .
What decimal of 13s. is 12s. 7d. 2
I2) 7
gº ºs
13) 12.5833. . .
*= ººm-º.
.9679487179487i...
*
The following questions are the converse of those
##. is equiva- just given.
What is the value of .375:9.2
3 U
A B I T H M E T I C.
Arithmetic.
29.
.375
20
7.500
12
6.00
The answer is 7s. 6d.
What is the value of .0552084.P. 2
39.
.0552084
20
1. 104 1680
12
tºº-º-º-º-º-º:
1.2500 160
4
1.0000640
Or thus, Part II.
dr. 02. \-y-Z
| 6 - l lb.
256 - 16 = l qr.
7 168 – 448 — 28 = 1 cwt.
28672 = 1792 – 1 12 = 4 = 1 ton.
573440 = 35.840 = 2240 = 80 - 20 = I.
This weight is used in weighing all heavy articles,
such as grocery goods, butter, cheese, meat, bread, corn,
&c. and all metals, except gold and silver. -
The pound avoirdupois is equal to 7000 grains Troy,
and the relation of the ounce avoirdupois to the ounce
Troy is that of 437% : 480, which is nearly that of 11
to 12; in some cases, the dram avoirdupois is sub-
divided into 3' scruples, and each scruple into 10
grains: under these circumstances, the grain Troy is
equal to 1.097 grains avoirdupois. -
(370.) The following are examples of reductions Reductions,
connected with these tables.
The answer is ls. 1; d. , bºrrºr.
What is the value of .0425 of 100:9. P
.0425
100
4.25
20
5.00
The answer is £4. 5s.
Tables of (369.) The following three tables contain the sub-
the subdivi- divisions of the weights which are used in this country.
sions of *
weights. Troy Weight.
24 grains make 1 pennyweight, dwt.
20 pennyweights 1 ounce, O2.
12 ounces. . . . . . 1 pound Troy, lb.
Or thus,
gr dwt.
24 - 1 O2.
1 lb.
480 – 20 -
57.60 = 240 - 12 = 1
This weight is used in weighing gold, silver, jewels,
and other articles of a costly nature.
Apothecaries' Weight.
20 grains make 1 scruple, sc. or 9
3 scruples. ... I dram, dr. or 3
8 drams . . . . 1 ounce, oz. or 3
12 ounces . . . . 1 pound, lb. or fö
Or thus,
gr. SC.
20 - 1 dr.
60 - 3 = 1 02.
480 = 24 = 8 = 1 lb.
5760 - 288 - 96 – 12 - 1
The apothecaries' pound is identical with the pound
Troy, differing merely in its subdivisions. It is used
by apothecaries in the composition of medicines.
Avoirdupois Weight.
16 drams make 1 ounce, O2.
16 ounces. . . . . . 1 pound, - lb.
28 pounds . . . . 1 quarter, qr
4 quarters . . . . 1 hundred-weight, cut.
20 hundred-weight 1 ton, ton.
In 3 lb. 10 oz. 7 dwt. 5 gr., how many grains?
b. oz. dwt. gr.
3
, 10 y 7, 5
12
46 oz.
20
927 dwt.
24
.22253 gr. The answer.
In 1 ton 7 cwt. 2 qr. 17 lb., how many pounds?
ton cut.gr. lb.
1, 7, 2, 17
20.
27
4
gººm-tº-yº
110
28
3097 lb. The answer. ... 1
In 27 lb. 73. 23.19, 2 gr., how many grains?
lb. 3 3 9 gr.
27 y 7, 2, 1 p 2
12
331
8
2650
3
7951
20
159022 The answer.
What fraction of a pound Troy is 3 oz. 15 dwt. 12 gr. 2.
lb. oz. dwt. gr.
l 3, 15 , 12
12 20
12 75
20 24
240 312
24 150 **
5760 1812
A R I T H M ET I C.
507
Arithmetic. The fraction is 4++# = ###.
-Y- What decimal of a ton is 7 cwt. 3 qr, 27 lb. 2
28 = 7 x 4 7) 27 -
4) 3.8571428
4 lb. = 1 qr. 4) 3.9642897
20) 7.99.1071.4
The answer, . .3995.5357
What is the value of . 12345 ib. ?
#5.
.1234
12
1.4slao
8
3.85 120
3
2.55360
20
11.07.200
The answer is 1 oz. 3 dr. 2 sc. 11 gr. ºrg; tr.
Table of (371.) Tables of Measures of Length.
measures of * e e º
length. 3 barleycorns (in length) make 1 inch, 271.
12 inches . . . . . . . . . . . . . . 1 foot, ft.
3 feet . . . . . . . . . . . . . . . . . . . . 1 yard, Syd
6 feet . . . . . . . . . . . . . . . . . . . . 1 fathom, fth
5% yards . . . . . . . . e - sº . I pole, or rod, po.
40 poles . . . . . . . . . . . 1 furlong, fur.
8 furlongs . . . . . . . . . . . . . . . . 1 mile, mi
3 miles . . . . . . . . . . . . . . . . . . 1 league, léa.
69% miles. . . . . . . . . . . . . . . . . . 1 degree, deg. or °.
Or thus,
bar. inch.
3 -: l foot.
36 - 12 -- l yard
108 = 36= 3 = 1 pole.
594 = 198 = 16} = 5} = 1 furlong
23760 = 7920 – 660 = 220 - 40 – 1 mile.
190080 = 63360 - 5280 = 1760 = 320 = 8 = 1
(372.) In 3 miles, 2 furlongs, 7 poles, 3 yards, and
2 feet, how many inches 2
7mi. furpo. Syd...ft.
3 y 2, 7, 3 p 2
Reductions,
17286;
12
The answer, 207438
What decimal of a mile is 17 yards, I foot, 6 inches 2 Part II.
12) 6 •----'
3) Tº
20) 17.5
11) sis
8) .0795454
<===w=as sº-º-º-º-º-mºme
. 0099.4318
Required the value of .67 of a league?
220 = IH X 20
.30
I2
3.60
3
1.80
The answer is 2 mi. 0 fur. 3 pol. 1 yd. 0 ft. 3 in.
Table of
measures of
à l'è8,
(373.) Table of Measures of Area.
144 square inches make 1 square foot.
9 square feet. . . . . . 1 square yard.
30+ square yards 1 square pole.
40 square poles . . . . I rood.
4 roods . . . . . . . . . . 1 acre.
Or thus,
inches. foot.
H 44 = I yard
1296 = 9 - l pole
39204 = 272} = 30} = 1 rood
1568 160 = 10890 - 1210 = 40 = 1 area.
6272640 - 43560 = 4840 - 160 = 4 = 1
The names of the inferior units of area are identical
with the names of those units of length which are the
sides of the squares; and, in general, the distinguishing
epithet (square) is altogether omitted, unless in those cases
where the meaning is not clearly defined by the context.
(374.) What decimal of an acre is 1 rood, 17 poles, 12 Reductions.
3yards 2 * -
304) 12.
or, 121) 48.
40) 17:39669
4) 1.434917
.338729.
What is the value of .12345 of an acre 2
The answer,
3 U 2
508
A R I Tº H M E T I C,
Aſſumede. .12345
\-y-' 4
.49380
40
19.752(H0
30+
22.56000
I8800
22.74800
9
6.73200
The answer is 19 poles, 22 yards, 6 feet.
Table of Measures of Capacity.
Table of (375.) (1.) For wine, ale, and other liquids,
II leaSlir CS
capacity. 2 pints make 1 quart.
4 quarts. . . . 1 gallon.
42 gallons 1 tierce.
2 tierces. ... I puncheon.
63 gallons 1 hogshead.
2 hogsheads 1 pipe, or butt.
2 pipes . . . . I tun.
(2.) Dry measure, for corn, seeds, &c.
2 pints make 1 quart, qt.
2 quarts. . . . 1 pottle, pot.
2 pottles ... I gallon, gal.
2 gallons ... 1 peck, ſpec.
4 pecks . . . . 1 bushel, bw.
4 bushels ... I coom, COO772.
2 cooms. ... 1 quarter, qr.
5 quarters ... 1 wey, or load, wey.
2 weys . 1 last, last.
Or thus,
pints. gallon.
8 = } peck.
I 6 = 2 = 1 bushel.
64 = 8 = 4 = 1 coom.
256 = 32 = 16 = 4 = 1 quarter
512 = 64 = 32 = 8 = 2 = 1 wey
2560 = 320 = 160 = 40 = 10 = 5 = 1 last.
5120 = 640 = 320 = 80 = 20 = 10 = 2 = 1
Imperial (376.) The wine gallon formerly differed from the beer
gallon.
Reductions.
gallon, and both of them from the corn gallon; the first
being 231 cubic inches, the second 282, and the third
271. In the Imperial measures of capacity, established
by act of Parliament in 1824, there is only one gallon
for wine, beer, and corn, or for liquid and dry measures,
which is equal to 277.274 cubic inches.
The Imperial gallon is nearly 4th larger than the old wine
gallon, sºoth greater than the old corn gallon, and sºoth
less than the old beer gallon. At least, these reduc-
tions are sufficiently accurate for ordinary reductions of
the ancient to the modern measures.
(377.) What number of Imperial gallons are there
in 3 pipes, 1 hogshead, 12 gallons, of the old wine
measure? --
pipe hnd. gal.
Part II.
3 , i , 12 •:
2
*ms
7
63
5) 453
90+
362; The answer.
What number of Imperial bushels are there in 7 iasts,
7 quarters of the old measure ?
last qr.
7 , 7
10
77
8
50) 616
1245.
a--as-ºs-
604; 3.
What decimal of a hogshead are 3 gallons and 3
pints?
The answer.
8) 3.
63 = 7 x 9 7) 3375
9) T.As2142857
Toš357 1428. ... The answer.
Table of the
(378.) Table of Measures of Time.
divisions of
60 seconds make 1 minute, m. or ’. time.
60 minutes . . . . 1 hour, hr.
24 hours . . . . . . 1 day, day.
7 days . . . . . . 1 week, whº,
4 weeks. . . . . . 1 month, mo.
Or thus,
seconds. minute.
6O - 1 hour.
3600 -: 60 = 1 day. -
86400 = 1440 = 24 = 1 week.
604800 = 10080 – 168 = 7 = 1 month.
24.19200 = 40320 = 672 = 28 = 4 = 1
(379.) The civil year, taking an average of four years, Different
is 365}; but if we take an average of 400 years, its years,
length is 365.2425 days: this is different from the
mean tropical year, upon which the recurrence of the
seasons depends, whose length is 365.242264 days,
differing from the former by .000136 day, or by about
11; seconds.
It is necessary, likewise, to distinguish between a And
'month, as defined by the preceding table, a calendar months,
month, which varies from 28 to 31 days, and an astro-
nomical month, which is a synodical period of the moon,
the mean length of which is 29.5505885 days. It
is the second of these which is most commonly under-
stood in arithmetical questions ; and when the particular
month is not specified, its length is assumed to be 30
days.
(380.) What decimal of a week is 1 hour, 27 minutes,
and 14 seconds P
A R IT H M ET I C.
509
Arithmetic. 60) 14 remainder 17: the number of pounds, adding 2, is 528: , Part II.
\-N- 6 0) 27.283 we thus get the entire sum, which is £528. 17s. 3}d. -
~~~~ (2.) oz. dr. sc. gr.
24) 1.45388 8, 5 , 1 v 8
*= <===== 7 º 6 ly 2 W 13
7) .0605787 ll , 7, 0 , 0
The answer, .0086541 1. y !' y 1.
g y & y & W
What is the value of .00693 of a year? 0 , 7, 1 , 19
*ś, 3 lb. , 4, 5 , 2, 19 -
—ºf The sum in the first column is 59, which, divided by
3465 20, gives a quotient 2, with a remainder 19: the sum
4.158 in the second column, adding 2, is 8, which, divided by
2079 3, gives a quotient 2, with a remainder 2: the sum in
173,25 the third column, adding 2, is 29, which, divided by 8,
253] 1825 gives a quotient 3, with a remainder 5: the sum in the
u ºf ºr 24 fourth column, adding 3, is 40, which, divided by 12,
gives a quotient 3, with a remainder 4.
101.247300 ſ (3.) Let it be required to subtract 12 ton, 7 cwt. 1 qr.
50623650 12 lb. 7 oz. from 15 ton, 11 cwt. 0 qr. I lb. 5 oz.
wºmmº-º-º-º-º-º-º-º-º-º-º-º: ton cut. qr. lb. oz.
6.074838 15 y 11 y 0 , 1 , 5
60 12 v 7 y i. p. 12, 7
4,490280 3 , 3 , 2, 16 , 14
66 In the first column, we borrow 1 lb. or 16 oz., and
n 29.4 16800 add it to 5 ; and from their sum, 21, we subtract 7,
g tº - which leaves a remainder 14: we add 1 to 12, and
The answer is 6 hours, 4 minutes, and 29} seconds. borrow 1 qr., or 28 lb., from the third column: we sub-
Addition . (381.) The reductions which We havementioned above, tract, therefore, 13 from 29, and the remainder is 16 :
* in connection with the º tables of weights and we add l to 1 in the third column, and borrow 1 cwt.,
..., measures, are those which are most commonly required or 4 qr., from the fourth column, and, therefore, subtract
quantities. in arithmetical operations with concrete quantities, and 2 from 4, which leaves a remainder 2: we add 1 to 7
particularly for bringing them within the province of
decimal arithmetic. It is not always expedient, how-
ever, to effect such reductions, and the addition and
subtraction of such quantities, and their multiplication
in the fourth column, and subtract, therefore, 8 from
11, which leaves a remainder 3: we subtract 12 from
15 in the fifth column, and the remainder is 3.
(382.) In multiplying concrete quantities of different Multiplica-
denominations by an abstract number, we multiply the tion and
terms in succession, beginning from the lowest, divide . of
the results successively by the number which connects †.
each term with the next superior, carry the quotients by abstract
successively to the next product, and leave the remain- numbers.
and division by abstract numbers, take place without
any previous preparation.
In the addition and subtraction of concrete quanti-
ties, it is requisite that they should be of the same kind,
otherwise no incorporation can take place in the results:
and in performing the operation, quantities of the
same denomination must be placed underneath each
other: under such circumstances, the numbers in the
same column are added together, or subtracted from
each other ; and when the sum exceeds the number of
units of that denomination, which constitutes an unit
of the next superior order, it must be divided by that
number, and the remainder from the division left in the
expression for the sum, and the quotient carried to the
next column. A few examples will make this rule
sufficiently clear.
ders. The following are examples:
lea, mi. fur. po. Ayd.
20 y 2, 7 , 38 y 4,
5
104 , 2, 7, 33, 3%
We multiply 5 into 4, the product is 20, which, divided
by 54, gives a quotient 3, and a remainder 3} : we mul-
tiply 5 into 38, add 3 to the product, and divide the
result, 193, by 40, which gives a quotient 4, and a re
mainder 33: we multiply 5 into 7, add 4 to the pro-
Examples, (l.) £. s. d. duct, and divide the result 39 by 8, which gives a quo-
77 , 13 , 5% tient 4, and a remainder 7: we multiply 5 into 2, and
15 , 19, 11; add 4 to the product, and divide the result 14 by 3,
107, 7, 2} which gives a quotient 4, and a remainder 2: we lastly
327 , 16 , 8 multiply 5 into 20, and add 4 to the result, which is 104.
528 , 17 , 3}, In the division of concrete quantities of different
W W 2 _*
The number of farthings is 6, which, being divided by
4, (4 far. = 14.) gives a quotient I, with a remainder
2 : the number of pence, adding 1, is 27, which, divided
by 12, (12d. = 1s.) gives a quotient 2, and a remainder
3: the number of shillings, adding 2, is 57, which, divi-
ded by 20, (20s. = £1.) gives a quotient 2, and a
denominations by abstract numbers, we commence
with the highest, and proceed to the lowest, putting
down the quotients, and carrying the remainders mul-
tiplied by the number which connects the several
denominations with each other, and adding their pro-
ducts to the corresponding terms of the dividend. The
following is an example: *.
510
A. R. I T H M E T I C.
Arithmetic. What is the fifth part of 214 quarts, 7 bushels, and
S-- 3 pecks? - -
Duodecimal
multiplica-
tion of
length into
length.
qr. bus. pee.
5) 214 , 7, 3
42 , 7, 3 , 1 gal. 2 quarts #.
The quotient of 214 is 42, and the remainder 4,
which, multiplied by 8, and the product added to 7,
makes the next number to be divided 39: the quotient
of 39 is 7, with a remainder 4, which, multiplied by 4,
and the result added to 3, makes the next number to be
divided 19: the quotient of 19 is 3, and the remainder 4,
which, multiplied by 2, is 8: the quotient of 8 (gallons)
is 1, and the remainder 3, which, multiplied by 4, the
result is 12, of which the quotient is 2, with a remain-
der 2.
In those cases in which the divisor is a mixed
number, it is necessary to multiply both the dividend
and divisor by the denominator of the fractional part,
so that the divisor may become an integral number.
The following is an example:
whº day. hrs. min.
5}) 3, 6 y 14, 53
4
21) 15, 5 , 11 , 32
5, 6, 15, 48’’;
(383.) In some cases, concrete quantities are multiplied
together, and a result is obtained which admits of in-
terpretation : thus, length being multiplied into length
produces area, and area into length produces capacity;
the units in the products are different from those in the
factors, and the meaning of the term multiplication
must be modified, so as to suit this extended applica-
tion of it: for this purpose, we must consider in what
manner the result is obtained, and also what is the
meaning of the units of which it is composed.
A rectangular area whose adjacent sides are 5 feet,
and 3 feet respectively, may be separated into 3 × 5,
or 15 equal squares, by dividing the opposite sides into
5 and 3 equal parts respectively, and drawing lines
through the points of division : in this case, the rec-
tangle is said to be the product of the two adjacent
sides, represented by numbers, whilst the units in the
numerical product are no longer lines, but squares
described upon an unit of length : it is easy to extend
this conclusion to the rectangle under two lines, which
are denoted by 5.4, and 3.7, respectively, whose pro-
duct is 19.98, which is 19 units or squares, and that
portion of one of those squares, which .98, or tºo, re-
presents.
In the same manner, the solid parallelopipedon,
whose adjacent edges are 5 feet, 3 feet, and 4 feet, re-
spectively, is equivalent to 5 × 3 × 4, or 60 equal
cubes, one whose edges is 1 foot; and it is in this
sense, that the continued product of the numbers,
whether whole or fractional, by which three lines are
denoted, gives a numerical product, of which the units
denote solids and not lines.
The subdivisions of feet proceed according to the
duodecimal scale, and artisans, in estimating rectangular
areas, or rectangular solids terminated by rectangular
surfaces, are accustomed to multiply feet and inches into
each other, for the purpose of obtaining the units of
area (squares) or of capacity (cubes), which they con-
tain: such quantities are called duodecimals, from the
scale according to which they decrease, and the pro-
cess which is made use of for this purpose is strictly S-N-
analogous to the multiplication of decimals, though re-
quiring a different notation. The following are ex-
amples:
Or thus,
:
6
6
334%;
8 8
3-#3 TTA.
37+, ++ *
The reason of the first operation will be sufficiently
obvious from the second form of the process: 5 ft. 7 in.
is equivalent to 54, feet, and 6 ft. 8 in. to 64% feet:
their product is found by multiplying these mixed num-
bers together, which is effected as follows: multiplying
first by 6, we get 6 x +, which is ##, or 34%, and
6 x 5 is 30, which, added to the former, makes 33-& :
we next multiply # into -45, the result is fºr, or
+º, ++, and, again, # into 5, which is 3%, or 3+,
which, added to the former, makes 3; +47: the sum
of these two products is 37-# Târ : if instead of re-
taining the denominators 12 and 144, we suppose their
existence understood from the position of the numerator
with respect to the place of units, we shall arrive at
the precise process which is followed in duodecimal
multiplication.
2. What is the number of cubic feet, inches, &c. in
a piece of masonry, 9 feet, 3 inches long, 11 feet, 5
inches high, and 3 feet, 2 inches thick?
9 y 3 11-#
11 , 5
101.4%
191, 9 3+} +}r
3 , 10 y 3
* ====ºmºmº-º-º-º-mºmºsº 105+ 1};
105 v 7 y 3 3-#3
3 y 2 - ***=s*
sº- 3.16% +},
316 y 9, 9 171's 134 1% a
17 y 7 y 2 p 6 - —-as-à
334.1% 1'i'ſ 1% a
334 , 4, 11 p 6
PROPORTION, THE RULE OF THREE, &c.
(384.) Before we proceed to the statement and expla-
nation of the Rule of Three, the most important of all
arithmetical rules, it appears to be requisite to give some
account of the doctrine of ratios and proportion upon
which it is founded.
Ratio exists between two numbers, or any quantities Ratios.
which are of the same kind, and admit of comparison
in respect of magnitude: thus, we speak of the ratio
A R IT H M ET I C.
5]]
Arithmetic. of 3 to 5, of 7 days to 10 days, of 11 cwt. to 14 cwt.,
S-N- and so on : but it can have no existence between quan-
tities which are dissimilar, such as #3. and 5 horses,
7 bushels and 9 feet, and so on, such quantities admit-
ting of no comparison with each other.
A ratio is denoted by placing two dots (:), one above
the other, between its terms: thus the ratio of 13 to
17 is denoted by 13 : 17: that of 3 feet to 7 feet by
ft. in.
3 : 7; and similarly in all other cases ; the first term
being called the antecedent, and the second the conse-
quent.
(385.) The term ratio, however, does not convey at
once to the mind a distinct idea of the nature of the com-
parison which is designated by it, or of the principles upon
which the magnitude of different ratios may be esti-
mated: in order to define its meaning, the antecedent
is made the numerator, and the consequent the deno-
minator of a fraction, and the magnitude of the fraction
ascertains the value of the ratio : thus the ratio of 3 to
5 is denoted by ; ; by this means ratios are brought
within the province of common arithmetic, and this
assumption respecting the mode of denoting them,
and thence of comparing them with each other, in
reality constitutes the true arithmetical definition of the
meaning of the term. -
Proportion (386.) Proportion consists in the equality of ratios:
how denoted thus the four quantities 3, 5, 9, and 15, constitute a pro-
portion, or are said to be proportionals, and are denoted
usually in the following manner:
3 : 5 : : 9 : 15 : .
The sign (::) placed between the ratios of 3:5, and
of 9 : 15, denotes the equality of the ratios; the whole
expression is equivalent to
# = +;,
the most convenient form of denoting it, inasmuch
as the equality of these fractions is the test of the pro-
portionality of the terms.
If we reduce the two fractions to a common deno-
minator, we shall find
How
denoted.
Their
meaning,
3X15 – 5 X 9
5X15 T 5x is
and, therefore,
3 × 15 = 5 × 9
Product of or, in other words, the product of the two extreme terms
the means of the proportion is equal to the product of the means,
. f a conclusion which is clearly general, inasmuch as the
#. ºne process which leads to it has no connection with the
particular numbers above given. -
It is an immediate corollary from this proposition,
that if the product of the means be divided by one of
the extremes, the quotient is the other ertreme ; or if the
product of the extremes be divided by one of the means,
the quotient is the other mean.
It will readily follow from hence, that if three terms
of a proportion are given, the fourth may be found, by
multiplying the second and third together, and dividing
by the first : thus, if it was required to find a fourth
proportional to 8,9, and 24, we should find * * * = 27,
for the number required. - -
The preceding propositions are all that are required
in the solution of questions in the Rule of Three, which
we shall now proceed to consider.
(387.) The rule itself. and the principles upon Part IF.
which it is founded, will be best understood from its \-v-
application to an example. - i. of
ree.
If 7 hats cost £9. 10s., what is the cost of 13 2
In this question, two of the three quantities are of Example.
the same kind; the third is of the same nature with
the quantity which is required to be determined.
Considering this unknown quantity as the fourth
term in a proportion, of which 7 hats, 13 hats, and
£9. 10s. are the three first terms, they will stand as
follows :
hats hats ſº. s.
7 : 13 : ; 9, 10 :
Or, reducing £9. 10s. to shillings:
hats hats S.
7 : 13 : : 190 :
13
57g
190
ºmºmºmºmº
7) 2470
352%
y
We multiply the second and third terms together,
and divide by the first, when we get 352%, or £17. 123s.,
or £17. 12s. 10%d. for the cost required.
The quantities which form the terms of the two
ratios, of which the complete proportion is composed,
are of the same kind; and these rates are, therefore,
independent of the specific denomination of their
terms: thus the ratio of 7 hats to 13 hats is identical
with that of the abstract numbers 7 and 13, whilst the
ratio of 190s. to 3525s, is the same as that of 192 to
352%; it is for this reason that we are allowed to mul-
tiply the mean terms, and divide by the extreme, pre-
cisely as in the case of whole numbers.
(388.) It is convenient in the statement of this rule, Names to
to distinguish the two known terms which are of the distinguish
same kind, by the names of the argument and the demand, the terms
and to designate the third known term as the fruit or º:
produce of the argument, the unknown term being, p º
therefore, the fruit, or produce, of the demand.
Thus, in the question proposed, the 7 hats are the
argument, the 13 hats are the demand ; and, conse-
quently, £9. 10s. is, the fruit of the argument, and
#17. 124s. is the fruit of the demand, which is the
answer to the question.
(389.) If the fruit increase with the increase of the Rule of
argument, the terms must be arranged in the follow- Three direct.
ing order;
The argument : the demand :: the fruit of the argu- Inverse.
ment : the answer.
If the fruit of the argument decrease with the in-
crease of the argument, the order of the two first terms
is inverted, and becomes as follows ; ! - *
The demand : the argument :: the fruit of the argu-
7ment : the answer. *
Questions which come under the first arrangement
belong to the direct Rule of Three; those which come
under the second arrangement, belong to the inverse
Rule of Three. br
(390.) The following are examples:
(1.) What is the value of a cwt. of sugar, at 1s. Iłd.
per lb. ?
Examples.
512
A R H T H M E T I C.
Arithmetic.
&b. lb. S. d.
1 : 112 : : 1 , 1} :
12
13;
112
336
1 12
56
12) 1512
2,0) 12,6
- £6, 6s.
In this case, 1 lb. is the argument, and ls. 1%d. its
Jruit, whilst 112 lb. is the demand, and £6.6s. is its
fruit. *
(2.) If the rents of a parish amount to £2340.17s. 6d.,
and a rate be granted of £137. 10s. 8d., what portion
of it must be paid by an estate whose rental is
#143. 9s. 10d. 2 -
39 s. d.
The answer.
4. s. d.
9, 10 :
£. s. d
20
*m-sume
46817
20
*ºmºmº ºf
2750
2340, it , G : 137, ſo, s : ; 143,
20
2869
12
m-sºad-smºs
34438
33008
27.5504
I 033 1400
103314
12
33008
12
§61810
– 12)
56181,0) 113672950,4 (2023
112362
2,0) 16,8, 7
-*—-
8, 8, 7
A31095
| 12362
187330
1685.43
isºsia
4
75 1496
56.1810
189686
The answer is £8. 8s. 74d. };}}#}.
In this case, the terms are all of the same nature,
though distinguished as the argument, its fruit, and the
demand; they involve units of different denominations,
and must all of them be reduced to the lowest. The
statement, after these reductions are effected, would be
d. d. - d.
56.1810 : 33008 11367295.04 :
The answer, or fourth term, is of the same denomi-
nation with the third, inasmuch as the two first terms
(l
might be considered as abstract whole numbers.
(3.) How many quarters of wheat can I buy for
80 guineas at 8s. 6d. per bushel?
guin. bush.
: 80 : : I :
2}
*=---
1680
2
— 8)
3360 (197
17
166
153
130
I 19
tºmmº
| 1
The answer is 24 qrs. 544 bush.
l
i
17)
24, 5
In this case, the two first terms are reduced to six-
pences, instead of pence, by which means the result is
more readily deduced.
(4.) If 12 men can reap a field of wheat in 3 days,
in what time can the same work be performed by 25
men 2
The argument is 12, and its fruit 3, and the demand
is 25: it is obvious, that the increase of the demand
must diminish the fruit, and, consequently, the state-
ment must stand as follows:
772.67?
25
777.672.
12
3
25) 36
25
11
24
ºsmºsºm-mºs
264 (10 hours.
25
14.
60
s====
840 (33 minutes.
75
T90
75
ſºmºmºz
15
The answer is 1 day, 10 hours, 33+ minutes.
days
: : 3 :
(1 day.
A very slight examination will show, that the pro-
portion is correctly assumed in this case: if the num-
ber of reapers be doubled, the work will be done in
half the time; if tripled, in one-third of the time; if
quadrupled, in one-fourth of the time; and it is pre-
sumed, and indeed implied, that in all other cases, the
time in which the same work may be done will be
diminished or increased at the same rate with which
the number of workmen is increased or diminished;
and, consequently, the argument and demand must
occupy a position in the terms of the proportion which
is the inverse of that which they occupied in the Rule of
Three Direct. -
(5.) How much in length, that is 13; poles in
breadth, must be taken to contain an acre, which is 4.
poles long and 4 poles broad 2
Part II.
A R I, T H M E T I C.
513
Arithmetic.
N-V-'
Compound
proportion.
poles
} poles
13% : :
40
4
mºmºmºsº
13%), 160
2 2
320 (11
27
wº-ºº-ºº-mº
50
27
23
5%
115
11%
27) 126,
108
18%
3
27) 55%
54
poles
: 4 :
27)
(4
(2
T;
Af 12
- 27) 18 (03.
The answer is 11 po. 4 yds. 2 ft. 03 in.
The greater the breadth, the less the length, the area
remaining the same : the demand, 13% poles, must,
therefore, be put in the first place, and the argument
40 in the second. -
(6.) If a certain number of men can throw up an
entrenchment in 10 days, when the day is 6 hours long,
in what time would they do it when the day is 8 hours
long 2 -
If the number of hours in each day be increased, the
number of days will be diminished, the number of
labourers and the work to be done remaining the same.
hours hours days
8 : 6 10
6
8) 60
- 7; days.
(391.) In many questions there are more arguments
than one, with their corresponding demands. The fol-
lowing are examples:
(1.) If a family of 9 people spend £120. in 8
months, how much will serve a family of 24 people 16
months, at the same rate of living 2
Arguments; 9 men and 8 months.
Their fruit; £120.
Demands; 24 men and 16 months.
The statement is as follows:
rº
WOL. i.,
men men
9 : 24
8 16
tºº-º-º-º-º:
144
24
mºs-s-s-s-º-º-º-º:
384
120
gº.
120
&º-º-º-º-
7;
2
es-º-º-ºm-º.
72) 46080 (6404, the answer.
432 -
&ºm===s===mºs
288
288
e====º
0
The reason of this process will be evident, if we
resolve it into two distinct statements: in the first
place, suppose the time in both cases to be 8 months;
then we should have -
771671, 7/1671, 4.
9 : 24 120
The fruit of the demand would be axis = 320.
Let us now suppose the number of men 24 in both
cases, and the time different, when ºf 320. will become
the fruit of the argument, which is 8 months: we thus get
months months
8 : 16 320 640
where the fourth term 640 = *x. - tºº.
(2.) If a barrel of beer be sufficient to last a family
of 7 persons 12 days, how many will be sufficient for a
family of 14 persons for a year?
Arguments ; 7 persons, 12 days.
Their fruit; 1 barrel.
Demands ; 14 persons, 365 days.
‘7 : 14 : : 1
12 : 365
84 1460
365
84) 51.10 (60 gº barrels. Answer.
504
70
(3.) If 248 men, in 5 days of 11 hours each, can
dig a trench 230 yards long, 3 wide, and 2 deep, in how
many days, 9 hours long, can 24 men dig a trench of
420 yards, 5 wide, and 3 deep. -
Arguments direct; 230 yds. : 3 yds. : 2 yds.
inverse ; 248 men: 11 hours.
Their fruit; 5 days.
Demands direct; 420 yds. : 5 yds.: 3 yds.
- inverse ; 24 men : 9 hours.
248 × 3 × 2 : 420 × 5 x 3 :: 5
- 24 × 9 : 248 x 11
3 x
Part II.
\-N-7
A R I T H M E T I C.
Arithmetic.
~~
'Chain rule.
Examples.
514
248 420.
3 5
744 2100
2 3
1488 6300
9 ll
13392 69300
24 248
53568 554.400
26784 277200
w 138600
32.1408 same-------
17 186400
5
321408) 85932000 (2673;{}}; days.
642816
2163040
I92S448
2365.920
2249856
116064
This question would require five successive simple
statements for its solution, three of them direct, and two
of them inverse. In combining them into one state-
ment or compound proportion, it is merely necessary to
separate the arguments and demands into direct and
inverse, and to multiply the arguments in the first into
the demands in the second, for the first term ; and the
demands in the first into the arguments in the second,
for the second term of the proportion.
(392.) The consideration of the preceding examples,
and of the modes of solving them, would lead to a rule
for their solution, in which it would be altogether un-
necessary to arrange the terms in the form of a pro-
portion: it would be as follows:
Write underneath each other the direct arguments
and the inverse demands, and, in another column, write
the direct demands and the inverse arguments, and
underneath them the fruit: divide the product of the
numbers in the second column by the product of the
numbers in the first column; the quotient is the fruit
demanded.
It is, of course, understood, that the corresponding
quantities of the same species in each column are re-
duced to units (if necessary) of the same denomina-
tion. -
It is this rule, which is denominated the Chain rule,
which is extensively used in exchange operations,
particularly by foreign merchants: the reason of its
name will be understood from a particular mode of
solving such questions of which examples may be seen
in Art. 198, as well as from the modern practice.
The following are examples of the use of this rule :
(1.) If 3 lb. of tea be worth 4 lb. of coffee, and 6 lb.
of coffee be worth 20 lb. of sugar, how many pounds
of sugar may be had for 9 lb. of tea 2
9 lb. tea.
3 lb. tea = 4 lb. coffee.
6 lb. coffee = 20 lb. sugar.
20 × 4 × 9 720
... — — = − = 40 lb.
6 x 3
GUIQ'a)",
18 tº
If the chain, connecting the corresponding quantities,
be added, it will stand as follows:
9 lb. tea.
3 lb. tea – 4 lb. coffee.
6 lb. coffee 20 lb. sugar.
(2.) Required the value of the mêtre of France in
terms of the foot of Cremona, if 48 feet of Cremona =
56 English feet, and the mêtre be = 39.371 English
inches. ."
1 foot of Cremona.
56 feet English
12 inches.
1 mêtre.
48 feet of Cremona
1 foot English
39.371 inches
tºº
*
****
tºº
*mmº
*
The result is
14
39.371 metres = 1 foot of Cremona, or
1 mêtre = 2.812 feet.
(3.) Find the value of a kilogramme of gold, weigh-
ing 15434 Troy grains, 1% fine, at £4. per ounce Troy.
+; fine.
1 kilogramme.
15434 Troy grains.
1 ounce.
9 ounces fine.
12 ounces English
standard.
gº4.
1 kilogramme
480 Troy grains
10 ounces French standard
ll ounces fine
:
1 ounce English standard
15434 × 9 × 12 × 4
480 × 10 × Il
(4.) What is the course of exchange between Lon-
don and Paris resulting from the price of gold : the
premium on the Paris mint price being 8 in the 1000,
and the price itself being 78s. per ounce, English stan-
dard, which is ++ ounce fine.
The course of exchange is expressed by the number
of francs in a pound sterling. The mint price in
France of a kilogramme of gold of 32.154 ounces, or
15434 grains Troy, being 3434.44 francs.
1 pound sterling.
20 shillings.
1 ounce standard gold.
11 ounces fine.
= £126. 5s. 6d.
1 pound sterling
78 shillings
12 ounces standard
32.154 ounces fine 3434.44 francs, mint price.
1000 francs mint pr. 1008 current.
20 × II x 3434.44 × 1008
78 × 12 × 32.154 × 1000
sterling.
In making calculations for a variable premium and
price of gold, it is usual first to determine the fixed
number
ºn
*=m,
*-
gººm
fººmſ,
lºss==
*>
fºrgº
& dº
*a-mº
= 25.3 francs per pound
20 × 11 × 3434.44
12 × 32. 154 × 1000
which, in the case before us, is multiplied by 1008, and
divided by 78.
The same rule is applicable to the solution of all
questions connected with the arbitration of exchange
and other operations of commerce ; numerous ex-
amples of which may be found in the second volume of
Kelly's Universal Cambist.
= 1.95823,
A. R. I. T. H. M. E. Tº I C.
515
Arithmetic.
Sºmnºy-ef
Practice.
Table of
aliquot
parts.
PRACTICE.
(393.) Practice is a compendious mode of solving Rule
of Three questions, when the first term, or argument,
is an unit, or 1.; in this case, it is merely requisite to
multiply the second term, considered as an abstract num-
ber, into the third term, in order to get the result.
Questions of this kind arise in the transactions of
ordinary trade, where the price is required of a certain
quantity of any species of goods generally estimated
by weight: it is the particular nature of the questions
proposed for the application of the rules of Practice,
that makes it necessary for the student to make himself
familiar with tables of the aliquot parts of a shilling and
a poundsterling, and also with those of a cwt., quarter,
and lb.
(394.) Aliquot parts of a shilling.
6d. is #, a half.
4.d. is 4, a third.
3d. is +, a fourth.
2d. is #, a sixth.
l;d. is #, an eighth.
ld. is +3, a twelfth.
#d. is +}, a sixteenth.
#d. is ºr, or # of a penny.
#d. is als, or 4 of a penny.
Aliquot parts of a pound sterling.
10s. is 4.
6s. 8d. is 4.
5s, is 4.
3s. 4d. is 4.
2s. 6d. is #.
2s. is +5.
ls. 8d. is als.
ls. is 3%.
Aliquot parts of a cwt.
2 qrs, is 4.
I qr. is #.
#d. is ; , 733 Part iſ
4.d. is # 366; S-N--
1834
12) 549;
2,0) 4,5, 9;
392, 5s, 9}d.
(2.) Find the value of 6771 at 8%d.
The answer.
6d. is 4, 6771
2d. is # 3385, 6d.
#d. is # 1128, 6d.
282, 1}d.
2,0) 479,4, 1}d.
£239, 14s, Išd. The answer.
(3.) Find the value of 969 at 19s. 11d.
10s. is #40. 969
5s. is 4:8. 484 , 10
242, 5
4s. is ##. 193, 16
6d. is 4 of 4s. 24, 4, 6
3d. is , of 6d. 12, 2, 3
2d. is of 6d. 8, 1 p 6
The answer, 964, 19, 3
It would be very easy to select other aliquot parts
which would equally make up 19s. 11d.
(4.) Find the value of 457 at £14. 17s. 9%d.
Examples of
cases of
practice.
14 lb. is #.
8 lb. is +.
7 lb. is +3.
Aliquot parts of a gr.
14 lb. is 4.
7 lb. is 4.
4 lb. is #.
34 lb. is 4.
Aliquot parts of a lb.
8 oz. is 4.
4 oz. is 4.
2 oz. is #.
I oz. is #.
i
(395.) The following examples will illustrate most of
the different the cases which can arise, and which hardly merit a
more formal classification.
(1.) Find the value of 733.(lb., oz., or units of any
other species or denomination) at #d., each.
457
14
1828
457
6398
10s. is #48. 228, 10
5s. is ; of 10s. 114, 5
2s. 6d. is , of 5s. 57 , 2, 6
3d. is-ºw of 2s. 6d. 5 & 14 ſ/ 3
#d, is # of 3d. 0, 19, 0%
The answer,
(5.) Find the value of 17 cwt. I qr. 12 lb. at £1.19s.8d.
per cwt.
1 qr. is # cwt.
7 lb. is # qr.
4 lb. is + qr.
1 lb. is + of 4 lb.
The answer,
The preceding examples include most of the cases
which are really different from each other, and are quite
sufficient to exemplify the process to be followed in all
those questions which are usually proposed for solution
aſ'6804, 10s, 9%d.
#. s. d.
1, 19 y 8
17
33, 14, 4
0, 9, 1 l
0, 2, 5%
0, 1, 5
0, 0, 4}
34, 8, 6
by the rules of Practice.
3 x 2
516
A R IT H M ET I C.
(396.) Questions, in which it is required to determine
the neat, or nett, weight, where the gross weight is to be
Arithmetic.
#. * diminished by allowances for tare, trett, &c., are com-
º monly resolved by a similar method. The following
are examples: N.
(l.) Find the nett weight where the gross weight is
173 cwt. 3 qrs. 17 lb. and the tare 16 lb. per cwt.
* cwt. qrs. 2b.
14 lb. is 4 cult. 173, 3, 17
2 lb. is 4 21, 2, 26
3, 0, 11
The answer, 149, 0, 8 -
(2.) What is the nett weight of 152 cwt. I qr, 3 lb.
gross, tare 10 lb. per cwt., and trett being, as usual, 4 lb.
in 104 lb, or ºth part of the whole?
º cwt. qrs. lb.
8 lb. is +1 cwt. 152, l y 3
2 lb. is 4 of 8 lb. 10, 3, 14
2 W 2 f/ 24
W. I3 Af 2 ty 10 Tare.
--~~" * 26) 138, 2, 21 Suttle.
- 5, 1, 9 Trett.
The answer, 133, I , 12 Nett.
(3.) What is the gross weight of 27 cwt. 3 qrs. 16 lb.,
tare being 8 lb. per cwt., trett and cloff as usual, the last
being 2 lb. in every 3 cwt. 2 e -
cwt. qrs. lb.
1, 3, 27 Tare.
26) 25, 3, 17 Suttle,
0, 3, 27 Trett.
168) 24, 3, 18 Suttle.
0, 0 , 17 Cloff.
The answer, 24, 3, 1 Nett.
INTEREST, DISCOUNT, BROKERAGE, AND
OTHER QUESTIONS CONNECTED WITH
THE PER CENTAGE RECEIVED OR PAID
ON THE LENDING, BORROWING, INVEST.
MENT, TRANSFER, AND OTHER USES OF
MONEY. - - -
Interest. (397.) Interest is the consideration due for the use of
money, whether advanced as a loan, or due as a debt:
it is generally estimated by the per centage, or sum
allowed for £100, for 1 year. - . . . .
The amount of this allowance will vary under different
circumstances, being regulated by the nature of the
security for the debt, and the abundance or scarcity of
money: in this country it is limited by the law to five
per cent, though a much higher interest.is sometimes
- } k -
paid, under different forms, by which the provisions of Part II.
the law may be evaded. - -'
(398.) The interest is usually paid at the end of each Interest
year: if the payment be forborne for a longer period, the . º
amount due will be different, according as it is esti-"
mated by simple or by compound interest. In the first
case, no interest is paid on the amount of interest due
and unpaid : in the second, the interest, when due, is
supposed to be added to the principal, and the interest
is subsequently calculated upon the whole amount.
(399.) The law allows simple interest only; in other compound
words, when the payment of the interest has been de-interest not
ferred for any number of years, such as 10, the person to legal • in
whom it is due can only demand 10 times the interest **
due in one year, without any allowance of interest upon:
the amount of interest due : as the law, however, could
enforce the payment of the interest at the end of each
year, it may always be received, added to the principal,
or otherwise invested, and thus compound interest may
be legally secured, though not légally demanded.
(400.) The rule for finding the simple interest of any
sum for any number of years is as follows: in order to
determine, in the first place, the interest for one year, we
must multiply the principal by the rate per cent, and
divide the result by 100: this quotient, multiplied by
the number of years, will give the interest required. The
following are examples: -
Required the simple interest of £237, 5s. 6d. for 3 Example.
years at 5 per cent. 2
#. s. d.
237, 5, 6
5.
Rule for
finding sim-
ple interest.
100) 11,86, 7, 6
- 20 -
--
17,27
12.
3,30
4
$mºmºmºsºmºsº
1,20
3. s. d.
11 p 17, 34.
- 3
The answer, 35, 11 , 9;
The first part of this process, for finding the interest
for one year, is identical with the following Rule of Three
statement: /
£. £. s. d. 48.
100 : 237, 5, 6 : :".5 :
Where 3100. is the argument, 395. its fruit, and
39237. 5s. 6d. the demand. . .
The whole process is equivalent to the following
Double Rule of Three statement:
£. £. s. d, e.
100 237, 5, 6 :: 5
1 : 3
Where £100, and 1 year are the arguments, £5, their
fruit, and £237. 5s. 6d. and 3 years, the demands.
(401.) Another method of solving such questions, is second
to reduce the shillings and pence to decimals of a pound method of
sterling, and to find, from the rate of interest per cent, decim”
A R I. T.H.M ETI c.
*
Arithmetic, the rate for £1.: if the number of pounds sterling be
S–y—’ multiplied by the interest of £1.. for one year, it will give,
the whole interest for 1 year; and if the interest for 1
year be multiplied by the number of years, it will give
the whole interest required. - .
Thus, in the last question, we should proceed as
follows:
100) 5
.05%
20) 5.5 -
237.275
.05
11.86375
3. Number of years.
Number of pounds.
Interest of £1.
35.59 125
20
11.82500
9.90000
4
. . . . . 3.60000
The answer is £35. 11s. 9%d. -
(2) Let it be required to find the interest of
£1229.7s. 11%d. for 7% years at 4% per cent. 2 w
By the first method :
£.
1229,
S. d. .
7, 11%
4917, 11 , 10
614, 13, 11%
100) 55,32, 5, 93
20
6.45
12
f 5.49.
- 4 -
* 1.99
- £. s. d.
Interest for 1 year 55, 6, 5%
4t 7%
387, 5, 2%
27, 13, 2}
The answer, 414, 18, 5}
By the second method: . . .
4) 2. . . .
12) 11.5
20) 79583
1229.39791
.045. = 4}.
%
00
614698955.
4917.59164 -
55.32290595 .
w 7.5 = 7%.
27661452975
38726034165
414.921794625
20 . .
18.435892.500
12
5.230910000
4
smmam-e-ºperººm-mººsºmºsº
.923640000
The answer is £414. 18s, 5}d. nearly. |
The process might be shortened considerably by
omitting all decimals after the fourth place, increasing.
the last figure by unity, when the next digit is equal to,
or greater, than 5. - -
(3.) What is the interest due upon #450. at 33 per
cent. per annum for 23 years and 67 days?
4) 3 * 450
.0375
365) 67.
.1835
2.75
100) 3.75
Ǻmºmºmº sºme
.0375
18750
1500 .
2.9335
16.8752
2.9335
84375
50625
50625
151875
33.750.
49.50282%;
20
10.0563
12
.672
4.
gºmºsºmºmºmº
2.688
The answer is £49. 10s. 03d. taking the nearest integral
values. - - - -- ©. -
(402.) The following questions are equivalentin prin-Other ques-
ciple to those in which the interest is required of a tions
sum of money for one year only. - º upon
(1.) What is the commission on £769, 3s.6d. at 2% jº
per cent. 2 . . . . . . . . . . . . Commission

5].8 A R IT H M ET I c.
Arithmetic. £. s. d. - (5.) What is the value of £2170. 5s. 6d, Bank stock Part II.
~~ 769 , 3, 6 at 2.17% per cent. 2 - S-N-4.
2; 12) 6 2170.275 -
1538, 7, 0 20);5.5 2174
384, il, 9 1519 1925
100) 19.22, 18, 9 2] 70.275 2] 70275
- 20 - 4340550
5425.6875
4.58 enºmºsºm-ºr-a
12 4714.9224|375
** 20
7.05 *=s==
- 18.4480
The answer is 319. 4s. 7d. I2
Brokerage. (2.) What is the brokerage on £7999. 11s. 4d. at # Tºszgo
per cent. 2 ”””
4) 7999, 11, 4 - 1.5040
100) 19.99, 17, 10 The answer is £4714. 18s. 5%d.
20. It is unnecessary to give other questions connected
with the purchase or sale of other species of stock,
19,97 whose value is estimated by the rate per cent. at which it
12 is saleable for ordinary money, as they are all of them
ii 7'4 solved upon the same principle, with those above given.
•'ſ 4 (403.) Discount is the deduction made in considera- Discount.
gmºmºmºsºmº - tion of the payment of money before it is due.
2.96 - The present worth of a principal sum due hereafter, Present
is the sum which, if paid immediately, will amount, at worth-
The answer is £19. 19s. 11; d. simple interest, to the principal when that principal is
Insurance, 3.) What is the insurance upon £24034. 14s.2d. at due
InSUIſàºl Ce mi º cent, P p The discount is, therefore, the difference between the
ºf - present worth and principal.
£. s. d. In questions respecting discount, the principal must Rule.
24034, 14, 2 be considered as the amount of the present worth put
11; - out to interest at a certain rate per cent. for the time
which elapses before the principal is due ; and in re-
ducing such questions to a statement, we must consider
the amount of 100 for that time as the argument, its
264381, 15, 10
12017, 7, I
100) 2763.99, 2, 11 .** interest as the fruit, and the principal as the demand.
20 (1.) What is the discount of £400, due 2 years hence
— at 5 per cent. 2
19.92 .The interest of £100. for 1 year is £5.
I2 - 2 years is £10.
II.15 29. 29. -
110 : 10 : : 400
The answer is £2763. 19s. 11d. 10
Sale of (4.) What is the value of £8334., 3 per cent. stock, at - 11,0) 400,0
stock 814 per cent. 2 sºmº
The answer is £36, 7s, 3}d.
8) 7 8334 s
.81875 The present worth is, therefore,
100) 81.875 4400. – £36. 7s. 3}d. = £363. 12s. 8:#d.
gº-º-º-º-º 327500 Or it may be found at once by the following statement,
.81875 245625
245.625 in tº wo
655000 100
assº 11,0) 4000,0
9:2500 - 4:363, 12s., 8:#d.
12 (2.) What is the present worth of £273. 4s. 6d. due
3.00 - at the end of 3 months, discounting at 4% per cent. 2
The amount of £100. in 1 year is £104. 10s.
The answer is #6823. 9s. 3d. - # year is £101, 2s. 6d.
A R. I. T H M E T I C.
519
Arithmetic. £.
~y- 101 y
20
2022
12
2427 ()
Answer, 4270. 3s. 8%d.
(3.) What ready money will discharge a debt of
#1377. 13s. 4d., due 2 years, 3 quarters, and 25 days
£. Jº. s. d.
': 273, 4, 6
20
ºmsºmºmºmºmºmº
5464
12
gººms-
2427,0) 655740,0 (270.9.
4854
17034
16989
450
20
9000 (3
'7281 -
1719
12
20628 (8
19416
1212
4
gmºmºmºmºsºms
4848 (2
4854 ,
hence, discounting at 4% per cent. per annum ?
112.33
365.) 25 8) 3
.0685 4,275
2.75 -
2.8.185
4.375
1409.25
197295
84555
| 12740
12.3309375, or 12.33%. nearly.
|00 : : 1377,6666
100
112.33) 137766.66 (1226
11233
25436
22466
297.06
22466
72406
67398
5008
20
112.33) 100 160 (8
. S9864
10296
12
112.38) iſ 3532 (In
123563
The present worth is £1226. 8s. 11d. nearly. Part II.
(404.) We have before mentioned the essential distinc- Com ºut. d
tion between simple and compound interest: it remains
to consider the principles upon which it may be calcu-
Iated. -
The most simple and obvious method is to calculate Rule.
the interest for 1 year, to add it to the principal, and
thus to find the amount at the end of the first year:
this amount becomes the principal upon which the
interest for the second year must be calculated, and
thus the whole amount at the end of it may be deter-
mined: the second amount becomes the principal for
the third year, and by the same process we may find
the amount at the end of the third year: by continuing
this process, we may find the amount during any num-
ber of years during which the interest is supposed to
accumulate: the difference between the first principal,
and the last amount, is the compound interest required.
(1.) Required to find the compound interest of £320.
for 3 years at 4 per cent. per annum ?
interest.
4 I #.
100 T 25 25) 320 Principal.
12, 16
25) 332, 16 Principal for 2d year
13, 6, 3 nearly.
25) 346 , 2, 3 Principal for 3d year.
13, 16, 10
359, 19, 1 Last amount.
320, 0, 0
The answer, 39-y 19, 1 Interest.
(2.) Required the amount of £760. 10s. forborne 3
years at 4% per cent. ”
4% 4. S.
100 = .045 760, 10 = 760.5 Principal.
1.045
38025
30420
76050
The amount of £1.. in 1
year = 1.045.
794,7225
1.045
39.736,125
31788900
794.72250
830.485|gr25 3d Principal.
1.045
2d Principal.
4152425
332]:940
8304850
867.8568.25 Final amount.
20
I 7. 1360
12
1.6320
4
*===ne-a-a-ass
2.52so
The answer is £867. 17s. 1%d.
520
A R. I T H M E T I C.
Arithmetic. In this case we determine the amount of £1.. in one
*N- year, and multiply the principal by it, in order to deter-
- mine its amount also : the same process is applied to
the several principals in succession.
It would clearly lead to the same conclusion, if we
first multiplied the decimal expressing the amount of
4:1. in one year into itself once, twice, thrice, &c., accord-
ing as the interest or amount is to be calculated for 2,
3, 4, or a greater number of years ; and, lastly, mul-
tiply the last product by the first principal.
(3.) Let it be required to find the amount of
£1057. 2s. 6d. for 5 years at 4 per cent. -
The amount of £1.. for 1 year is 1.04 (1.)
• 1.04 (2.)
gººm-º:
4 16
1040
tº-º-º-º-º-º-º-º-º-º-º
1.08 16
1.04
43264
108 160
1.124sgy
1.04
44sp2
I 12480
1.1697;
1.04
*-->
46788
I 16970
*===--- man-
I.2.165
loº
60825
24330
121 65
85 55
60825
121 650
1285.9925625
20
19.8500
12
tºmimºsºm-ºss
I 0.20
The answer is 391285. 19s. 10d.
When the number of years is considerable, the cal-
culation of compound interest becomes extremely labo-
rious : in such cases it is generally necessary to have
recourse to logarithms.
We shall not proceed to the consideration of ques-
tions on the amounts of annuities, accumulating at simple
or compound interest, the present worth of annuities,
whether perpetual or limited, equation of payments, &c.
the rules for which are founded upon algebraical formulae,
without the aid of which they admit not of explanation
or proof. -
(3.)
(4.)
(5.)
BARTER.
(405.) Questions in Barter usually resolve themselves,
with very slight modifications, into ordinary cases of
the Rule of Three. - - -
Barter,
(1.) How much sugar, at 9d. per lb., must be given Part II.
in barter for 17 cwt. of tobacco, at £3.10s. per cwt. 2 – 2–
£. s.
3, 10
17
d. 4}. s. lb.
9 : 59, 10 . : 1
20
| 190
12
9) 14:280
23) 1586
4) 56, 18
14 y 0
The answer is 14 cwt. 0 qr. 183 lb.
(2.) A merchant barters 1200 lb. of pepper, at 13a.
per lb., for equal quantities of two species of cotton at
7d. and 11d. per lb., and for 4d. in money ; how many
lbs. of each sort must he receive, and how much in
money 2
Jö.
1200
13
12) 15600
20) 130,0
3) 65
3921, 13, 4 sum paid in money.
43, 6, 8 the amount bartered
in goods.
Now it is evident, that 7 + 11, or 18d., is expended
for every lb. of each species of cotton which is given in
exchange : consequently,
d. #. s. d. b.
18 43 y 6, 8 : ; 1
20
*sº
866
I 2
10400
90
sº-º-º-º-º-
140
126
*s
140
126
14
18) (577; lb. The answer.
PROFIT AND LOSS.
(406.) Questions connected with the gain or loss per Profit and
cent. upon goods bought in gross and sold in detail, or Loss.
conversely, and, in short, under any other circumstances,
are resolved by one or more statements by the Rule of
A R L T H M E T I C. 521
(3.) If I buy tobacco at 10 guineas per cwt., at what Part H.
rate must I sell it per lb. so as to gain 12 per cent. 2
First statement :
Arithmetic. Three, combined in some cases with reductions, which
S-N-" are suggested by the nature of the questions proposed.
(1.) At 1s. 3%d. in the pound profit, how much is
gained per cent. 2
tºo p 100 :: I , 3% - lb. lb. £. s.
1; ' " I 1.2 : 1 10, 10
-ºss-mºm 20
15 gººm-w
4 112) 210 (1s.
g 112
4) 6200 cºmm-º º
98
12) 1550 12
2,0) 12,9, 2 1176 (10d.
112
- £6,9s, 20. Answer.
(2.) Bought goods at 73d. per lb., and sold them at 56
#4. 15s. per cwt., what is the gain or loss per cwt. 2 4
First statement: r
lb. lb. d. 224 (24.
1 : 112 :: 7; 224 -
t . 4 The cost price per lb. is 1s. 10%d.
T3T Second statement :
I 12 39. 29. s, d.
tºmsºmº 100 : 112 1, 10%
| 12 - 12
3 * -ºs-
36 22
4) 3472 4
12) 868 90
--- I 12
2,0) 7,2, 4 per cwt. *====
Cost price, ºf 3, 12s, 4d. 100) 10080
Second statement: 4) 100.4%;
- S. d. £. s. 39.
20 20. 2s, 1}d. The answer.
{. ; (4.) If when I sell cloth at 10s. per yard, I lose 5
*a-e Uºmº- sºm-ºs-sº per cent., how much shall I gain or lose per cent. by
868 868) 114000 (131 selling it at 12s. 6d. per yard 2-
868 Statement: f
27.20 S. S. d. aſ’.
2604. 10 : I2 # 6 ; : $9.5
12 12 150
lſº 120 150 4750
mºsºm-º-º-º-º-º: - 95
292
20 120) 14250 (11849,
sº | 20
5840 (6
225
5208
I20
632 &===
12 1050
960
7584 (8 tº-sumºus-º
6944 90
640 The gain per cent. is £183.
4 (5.) Sold goods for £75., and by so doing I lost 10
per cent., whereas in the regular course of trade I
131, 6, 8% 2560 (2 should have gained 30 per cent. : how much were they
100 , 0, 0 1736 sold under their proper value 2
---.
*º-sº-sº-sº
£31 , 6, 8%. The answer. 724
WOL. I.
3 Y
522
A. R. I T H M E T I C.
ſ
Arithmetic,
Fellowship.
Single and
Double.
Single
Fellowship,
Statement :
aff
( ; 43
90
gº £.
130 : : :
75
13
. . . 225.
75
*me
9). 975
: 108}
The goods were, therefore, sold at £1084 – £75., or
#33. 6s. 8d. under their just value.
FELLOWSHIP.
(407.) This is the rule by which the individual shares
are assigned in joint stock transactions with two or
more partners. *
The questions will be different according as the
several stocks, or their equivalents, are invested for the
same or different periods of time: questions of the first
kind belong to Single, and those of the second kind to
Double Fellowship. i
(408.) In Single Fellowship, the accumulated capital,
or the gain or loss upon it, is divided in the proportion
of the several capitals, or their equivalents, which are
invested in the concern.
(1.) A and B gain by trading £750. ; A's original
stock was £500., and B's #850. ; in what ratio must
they divide the profits?
First statement:
500
850
£1350 Joint stock,
43. 43.
1350 : 500 750
500
ºmmemºa sº-sa-a-ss tº
1350) 375000 (277
2700
10500
94.50
10500
9450
1050
2ſ)
1350) 21000 (15
1350
7500
67.50
&ºm====aa-ºº:
750
12
1350) 9000 (6
8100
tº-º-º-mºmºmºmºmºrºs-s
900
4
1350) 3600 (2
2700
*-*g
900
Second statement:
29 39
Part H.
1350 850 : ; -
43.
75ty :
850
37500
6000 -
*m-mºº smºsºmams C
1350) 637500 (47
- 5400
*
9750
94.50
*=e
3000
2700
300
20
1350) 6000 (4
5400
*===e^*
600
12
7200 (5
67.50
450
4
1800 (1
1350
$ 450
Consequently, £277.15s. 64d. #'ſ. A's portion.
- 3472. 4s. 53 d.º.º. B's portion.
the sum of which is £750.
In practice it is not necessary to work out the two
statements, inasmuch as A's portion subtracted from
4,750. will give B's portion. :
(2.) Three persons, A, B, C, invest ºf 134. 10s.
3340. 5s., and £425. 5s., respectively, in a partnership ;
at the end of 3 years they find the value of their capital
reduced by losses to £500, what portion of the loss
must they severally sustain P :
1350)
1350)
£. s. -
134, 10 A's capital.
340, 5 B's capital.
425, 5 C's capital.
900, 0
500
400 The loss. -
The following are the three statements:
º aft. £. s.
s (1.) 900 400 134, 10 A’s loss.
(2.) 900 400 340 , 5 B’s loss.
(3.) 900 400 425, 5 : C's loss.
3. s. d
59, 15, 6, #.
B’s loss = 151, 4, 543.
C’s loss 189, 0, 0
* Their sum = 400, 0, 0
(409.) In Double Fellowship we must multiply each Double
separate capital, or its equivalent, into the time of its Fellowship.
employment, and proceed with the products in the same
manner as with the simple capitals in questions in
Single Fellowship.
Consequently, A's loss
g ºº
º
gmºme
º
gº-
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A R I T H M ET I C.
523
(1.) A employs a capital of £500. in trade, and at
J- the end of 3 years takes B into partnership, who
advances a capital of £800. : at the end of 6 years from
this time, they have gained £600. : in what ratio must
the profits be divided ? * : . . . . .
500 × 9 = 4500, the product of A's capital and time.
S00 x 6 = 4800, the product of B's capital and time.
- 9300 - - º
9300 : 4500 : : 600
- r 45
93) 27000 (290
186
840
837
30
20
93) 600
558
42
12
93) 504 (5
465
39
4.
93) 156
93
63
600
48
28800
279
900
837
63
20
93) 1260 (13
93
330
279
51
12
612
558
54
4
93) 216
186
30
(6
(l
9360 4800
93) (309
93) (6
(2
£. s. d.
Consequently, A's share is 290, 6, 64 $3.
B’s share is 309, 13, 6, #}.
(2.) A ship's company take a prize of 26.4000., which
is to be divided amongst them in proportion to their
pay, and to the time they have been on board. There
are 6 officers, who have 120s. a month, and have
been on board six months; 12 midshipmen, who have
each 40s. a month, who have been on board 4 months;
and 110 sailors, who have 30s. a month, and have
been on board 3 months: what sum must each receive P
Part II.
S-N-"
We must first determine the sum due to officers, mid-
shipmen, and sailors, considered as each constituting
one body, and then divide the respective sums by the
number of officers, midshipmen, and sailors. -
s 6 × 6 × 120 = 4320
12 × 4 × 40 = 1920
Ilo x 3 x 30 = 9900,
*º-sº
16140
The following are the statements:
#.
4000
4000
4000
Officers' portion. •
Midshipmen’s portion.
: Sailors' portion.
4320
1920
9900
1614()
16140
16] 40
Consequently,
#. s. d.
The 6 officers receive 1070, 12 , 7
The 13 midshipmen
The 110 sailors. . . .
Each officer receives
Each midshipman. .
Each sailor . . . . . . 22 ,
The reader is referred to the historical notice of the
Rule of Three, Practice, Tare and Trett, Interest,
Discount, Barter, Loss and Gain, and Fellowship, for
other examples in illustration of these rules.
The ample notice which is given in the history of
Arithmetic of the rules of Alligation and of Single and
Double Position, supersedes the necessity of the more
formal statement of these rules, which is given in ordi-
nary books of Arithmetic: such rules, indeed, possess
very little practical interest or importance, as the ques-
tions to which they apply are more generally, if not
more readily, solved by algebraical processes.
2453, 10, 7
178, 8, 94.
39, 13, 0}.
6 y i.
A L G E B R A.
Algebra. (1.) ARITHMET1c and ALGEBRA are Sciences the
Relation object of which is to trace the relations and properties
i. of NUMBER. Through the medium of Number, quan-
Arithmetic tity in general is brought under their dominion; but
andAlgebra, they reject the consideration of those properties which
are peculiar to particular species of quantity, being
strictly confined to those which appertain to quantity
in the abstract. Number may be properly said to be
a means for expressing the abstract relation of one
quantity to another of the same kind; that is to say, the
relation which they have independently of the species
to which they belong. Thus, a certain length called a
foot has a relation to another length called an inch.
Again, a certain portion of time called a year has a re-
lation to another part of time called a month. Now,
although the quantities between which these relations
subsist be different in species, the one being space and
the other time, yet, notwithstanding this, the relations
are the same, and are both expressed by the number 12.
In this respect then, as being independent of any par-
ticular species of quantity, Arithmetic and Algebra
agree, and are, so far, equally abstract.
But although Arithmetic is abstract as to the species
of quantity, yet the relations which it contemplates,
and whose properties it investigates, are particular. In
other words, its objects are particular numbers, and their
properties and its notation, at least that of modern
Arithmetic, are the nine Arabic digits, and 0, or cypher.
It teaches the method of expressing, by various combi-
nation of these, all particular numbers whatever; and
... it investigates the properties of particular numbers, and
the methods of performing the various Arithmetical
operations on them, and the solutions of problems re-
specting them. t *
In the process of generalization, Algebra however
advances further than Arithmetic. The Algebraist,
not confining himself to the properties and relations of
particular numbers, takes a wider range, and investi-
gates the relations which may be considered common
to all number, and so departs one step farther from
specific quantity. While the Arithmetician is abstract
as to the quantity, but particular as to the relation, the
Algebraist is abstract as to both. An example will
illustrate this :
The problem, “To divide the number 10 into two
parts, one of which is double the other,” is Arithmetical.
It is abstract, as to the quantity expressed by the num-
ber 10 ; but the relations of the parts, into which it is
proposed that this number be divided, to each other,
and to the whole, are particular. Let the problem be
modified, so that the relations, as well as the quantities,
shall become abstract, and it ceases to be an Arithmeti-
cal question, and becomes Algebraical; in which case
it is expressed thus, “To divide any given number into
two parts which shall bear to each other a given ratio.”
The former problem is, evidently, only one individual
of an extensive class which is comprised under the
latter. When the former has been solved, the result is
merely the calculation of one particular numerical
524
question ; on the other hand, the solution of the other
furnishes a general method for the calculation of any
of the class of the questions of which the former is
only an example.
(2.) It is from this circumstance that Newton called
Algebra Universal Arithmetic. If this were the onl
respect in which the powers of Algebra exceed those
of common Arithmetic, the propriety of the title could
scarcely be disputed. But the student will not have
penetrated very deeply into the science before he will
perceive, that the title Universal Arithmetic very inade-
quately expresses the nature, objects, and extent of this
department of Analysis. -
Introduction
Algebra
called
Universal
Arithmetic.
(3.) From the more abstract nature of the objects of Difference
Algebra, it follows that the notation of Arithmetic is
insufficient for its processes. The numerical symbols
are essentially particular, and are therefore incapable
of expressing the general relations which are here
contemplated. Instead, therefore, of the Arabic digits,
of notation.
and their combinations, the letters of the alphabet have
been by universal consent adopted to express numbers
in Algebra. Thus the example already given would be
thus expressed, “To divide a given number a into two
parts, such that one should bear to the other the given
ratio of m : n.” It would be, evidently, impossible to
express this problem by the symbols of Arithmetic ;
for the moment particular numbers should be introduced
to express the different data, the problem would lose
its general character, and become an ordinary Arith-
metical question. &
The change in the nature of the symbols used to
express the numbers which are contemplated in Algebra,
renders a change in the manner also of expressing the
relations and operations on these numbers neces-
sary. In Arithmetic, the operations on numbers are
actually performed, and the results actually obtained;
but in Algebra, the operations and results are not
actually effected, but only expressed. Thus, if in Arith-
metic it be proposed to add 5 to 7, the process is
effected, and the result is 12. In Algebra, if it be re-
quired to add the number a to the number b, the pro-
cess of addition is indicated by the sign + called plus,
and the result, or the sum of the numbers a and b, is
expressed by a + b.
In examining these two processes it is remarkable,
that in the Arithmetical result no trace whatever is left
of the process by which it was obtained. The sum 12
might have been obtained by the addition of 8 and 4,
or 9 and 3, or various other numbers, for anything which
can be inferred from the mere result. But in the Alge-
braical result the process is quite apparent, and is in
effect actually expressed by a + b ; for although two
other numbers, as c and d, might have the same sum as a
and b, yet that sum would be expressed by c + d, and
not by a + b. This remark, which will be found of some
importance, is equally applicable to the result of every
Algebraical investigation, as compared with an Arith-
metical process.
(4.) It must be apparent from these observations,
Advantages
of Algebraic
notation
A L G E B R A.
525
Algebra.
that Arithmetic and Algebra are so closely connected,
\-J- that it is difficult to treat of either without, in some
For history,
see History
of Analysis,
$ymbols
wofold.
Theorem.
Problem,
degree, encroaching on the province of the other. In
the natural order of ideas in the human mind, the par-
ticulars precede the generals; and, therefore, although
all the particular properties and theorems of numbers
which form the subject of Arithmetic, are included in
the more general results of Algebra, yet we have given
the former priority in our series of Mathematical
papers.
In the following Treatise on Algebra we shall avoid,
as far as possible, a repetition of the demonstration of
principles already established in our Treatise on Arith-
metic, yet some repetition will be unavoidable, to give
that connection to the chain of reasoning without
which our investigations would be, in a great degree,
unintelligible.
(5.) We do not propose in this place to enter into
any historical account of the origin and progressive
improvement of Algebra. This and the other depart-
ments of analytical science are so intimately connected,
and, consequently, every great step in the improvement
of any one of them has produced such important effects
on the others, that it has been thought advisable, instead
of introducing each part of analysis by a historical
notice, to conclude our Mathematical papers with one
comprehensive HISTORY OF ANALYSIs. This, together
with the historical notices of Geometry and Arithmetic
already given, will, it is hoped, form a very complete
History of the Mathematical Sciences.
We shall devote the present article to an elementary
Treatise on Algebra in the most improved state to
which it has been brought in modern times. For an
account of the principal works on Algebra, we refer to
the general catalogue of Mathematical works at the
conclusion of the History of ANALysis.
SECTION I.
Notation.
(6.) IN Algebra, numbers and the operations to
which they are conceived to be submitted are repre-
sented by arbitrary symbols.
Hence there are necessarily two systems of symbols;
one to express the numbers themselves, and the other
to represent the operations to be effected on them.
Numbers are, by universal consent, expressed by the
letters of the alphabet, except in certain cases in which
particular numbers are used, in which case the symbols
and notation of common Arithmetic are preserved.
In Algebra, and indeed in Mathematical science
generally, there are two distinct species of questions:
1. A Theorem, the object of which is to establish
certain known or given properties of numbers.
2. A Problem, the object of which is to determine
certain numbers, certain other numbers being known
or given, which have, with the numbers required, known
or given relations.
In every Problem, therefore, there are two distinct
sets of numbers to be expressed, the known or given,
and the unknown or sought.
(7.) It is an universal custom to express the known
or given numbers by the first letters of the alphabet,
a, b, c, &c., and the unknown or sought numbers by
the last, r, y, &c.
WQL. I
Each of the operations to which numbers may be
submitted, is expressed by a peculiar symbol. The
four elementary operations, Addition, Subtraction, Mul-
tiplication, and Division, are expressed as follows:
(8.) 1. Addition. When two numbers are added
together, the process is signified by the sign + , called
plus, placed between the symbols which express the
numbers; and the whole combination, the symbols with
the sign between them, is understood to express the
result of the process, or the sum of the numbers.
Thus 7 -H 5 represents 12, a + b represents the sum
of the numbers represented by a and b. It may, and
frequently does happen, that more than two numbers
are to be added. This is expressed by the interposition
of the sign + between every successive pair of them.
Thus, if 7, 5 and 3 are to be added, their sum is ex-
pressed by 7 - 5 + 3, which arithmetically would be
15. If a, b, and c, are to be added, their sum is ex-
pressed by a + b + c.
In this case, it is evidently indifferent in what order
the operations may be performed. Thus, the sum will
be the same if b be first added to a, and then c added
to their sum, as if c were added to a, and b added to
their sum ; that is, a + b + c is equal to a + c + b.
And, in the same manner, it is equal to b + c + a, and
to b + a + c. In a word, the sum will be the same
in whatever order the letters may be written. …”
to-
It may happen, that the letters which are added
gether are equal to each other. Thus, if a, b, and c,
were equal, their sum would be a + a + a. It is not
usual, however, to express it in this way. The sum in
this case is expressed by the single letter a with a num-
ber prefixed to it thus, 3 a, signifying the number of
times the same letter would occur in the sum were it
expressed in the manner it would be expressed
had the letters been different. Thus, a + a + a + a
+ a is expressed by 5 a. This number is called the
coefficient of the letter; thus, in 5 a, b is the coefficient
of a. -
When a letter having a coefficient is to be added to
another, the sign of addition precedes the coefficient.
Thus, if 5 b be to be added to a, the sum is expressed
by a + 5 b.
(9.) 2. Subtraction. When one number is to be
subtracted from another, the operation is expressed by
the sign —, called minus, placed after the minuend
and before the subtrahend, and the whole combination
of symbols expresses the remainder.
Thus, if 5 be to be subtracted from 7, the process is
expressed by 7 – 5, which represents the remainder 2.
If a be the minuend, and b the subtrahend, a - b
represents the remainder. *
It may happen, that a number is to be subtracted
from the sum of several others, a + b + c. In this
case this sum may be treated as a single quantity, in
which case it is usual to enclose it in a parenthesis,
thus, (a + b + c), or to draw a line over the letters,
called a vinculum, thus, a + b + c, in which case the
remainder will be expressed thus, (a + b + c) — d,
or a + b -H c – d. These combinations of symbols
signify, that a, b, and c, are to be first added together,
and then the number d subtracted from the result.
To express the remainder in this case it is not, however,
necessary to resort to a parenthesis or vinculum. It
is evident, that the number d will be subtracted from
the sum a + b + c, if it be subtracted from any one o
3 z
Notation.
\-y-
Addition.
Coefficient.
Subtraction,
526
A L G E B R A.
the same quantity, the remainder is the same as if their Notation.
sum were at once subtracted from it. Hence we per- \-v-->
Algebra. its component parts, a, b, c. Hence it follows, that
`-- the remainder, which we have expressed above by
Positive and tractive quantity one which has the sign —.
negative
quantities.
(a + b + c) — d, may also be expressed by a + b + c
— d, without the parenthesis. The expression in this
case may be understood to mean the sum of a, b, and
the remainder c – d found by subtracting d from c.
In the same way, the remainder might be expressed
by a + b + c – d, or a – d -- b + c, or by the same
four letters placed in any order whatever, provided the
same sign + or – precede the same letters.
Here we should observe, that if the first letter a be
transposed, so as to be preceded by any other letter,
the sign + must be prefixed to it. This is obvious,
since a + b is necessarily equivalent to b + a. But
further it is necessary, that if the quantity d, to which
the sign — is prefixed, be placed first, it will not be
correct to place it without any sign prefixed, for in that
case the meaning of the whole combination would be
changed. Thus, d -- a + b + c would signify the
sum of the four numbers a, b, c, and d, instead of the
remainder when d is subtracted from the sum of a, b,
and c. If d, therefore, be placed first, it will be neces-
sary to prefix to it the sign —; indicating, that the man-
ner in which it is to be combined with the other
quantities is by subtraction. In the same sense,
therefore, when no sign is prefixed to the first quantity,
the sign + is to be understood.
These symbols + and — are called the signs of the
quantities to which they are prefixed; their true and
only meaning is, as already explained, to indicate the
manner in which the quantities which follow them are
to be united with the other quantities with which they
may happen to be combined, i. e. whether they are to
be added or subtracted. In this sense, the quantities
might with great propriety be denominated in reference
to their signs, additive and subtractive , an additive
quantity being one which has the sign +, and a sub-
But long
established usage has given to these signs the names
positive or affirmative, and negative ; that being called
a positive or affirmative quantity which has the sign +,
and that a negative quantity which has the sign —.
These terms are apt to convey wrong ideas; but the stu-
dent should endeavour to retain the notions of additive
and subtractive, and annex them to the names positive
and negative.
By generalizing the preceding results, it will be easy
to see, that if several quantities be united by different
signs, the value of the whole combination will neces-
sarily remain the same in whatever order they may be
written, provided that the same signs are always prefixed
to the same letters. Thus the following combinations,
a — b + c – d –- e – f
a + c – b – d –– e – f
a + c – d – b + e – f
a + c – d -- e – b – f
a + c – d –- e – f – b
– b + a - d - c – f –H e
– b – d – f -- a + c + e
&c. &c.
all express the same result. In effect, in all these
cases the same operations are performed with the same
quantities, but they are performed in different orders,
and this difference of orders produces no effect on the
final result.
If several quantities be successively subtracted from
ceive that the combinations
a + c + e – b – d – f
a + c + e — (b + d -- f)
are equivalent. Now if d and fare each equal to b,
we shall have the expression equivalent to
- a + c + e – 3 b.
Hence it appears, that if several negative quantities be
equal, they may be replaced by a single letter with a
coefficient, as explained in (8) with respect to positive
quantities.
It should also be observed, that if several quantities
enclosed in a parenthesis, or under a vinculum, be
positive, and that the negative sign be prefixed to the
parenthesis, the parenthesis may be removed by making
all the quantities negative. This is evident from the
preceding example.
(10.) 3. Multiplication. When two numbers are multi-
plied together, the process is represented by the sign x
placed between them, and the whole combination re-
presents their product. Thus 5 × 7 represents the
product of 5 and 7; a × b represents the product of
a and b. But when letters are used, which is generally
the case, the product is signified by a point placed
between them thus, a . b, or more usually by writing
the letters like those of one word, thus ab. This nota-
tion could not be used with particular numbers, because
there would then be no distinction between the notation
for expressing 7 times 5, and the number seventy-five.
Both would be written 75.
The terms multiplicand and multiplier as used in
Arithmetic are preserved in Algebra. There is, how-
ever, no difference between the relations which these
numbers bear to the product, and it is better to call
them by the common name factors. In other words,
the product ab will be the same, whether a be multiplied
by b or b by a, and it is indifferent whether it be
written ab or ba. In fact, the product has a relation
to its factors, which is called a symmetrical relation.
It is such, that if the values and names of the factors
be interchanged, the product remains unaltered.
It may happen, that three or more numbers are mul-
tiplied continually into one another. In this case, the
process, if the factors be particular numbers, is ex-
pressed by the interposition of the sign x between
every successive pair of factors; and if the factors be
letters, the product is expressed by writing them as in
one word. Thus, 7 × 5 × 3 signifies the product of
7 and 5 multiplied by 3, or the continued product of 7,
5 and 3, or 105. Also, abod expresses the continued
product of the numbers, a, b, c, and d.
If the several factors of a product be equal, it is
called a power, and said to be the second, third, &c.
power, according to the number of equal factors it
contains. Thus, aa is the second power of a, aaa the
third power of a, aaaa the fourth power, &c.
This, however, is not the way in which powers are
usually expressed. The number of times the same
letter occurs as a factor, is expressed by placing the
particular number above the letter, thus a”, a”, a”, &c.,
which expresses aa, aaa, aaaa, &c.; and if a. occurred
7m times as a factor, the power would be expressed a”.
The second power is usually called the square, and
the third power the cube. For the reasons of these
denominations, see GEOMETRY, pp. 330, 352, 353, also
the Definitions, pp. 314, 350.
Multiplica-
tion.
Factors.
Powers.
A L G E B F A.
527
Algebra.
\-y-'
Exponent.
Division.
Monome,
Polynome.
Likë
IlliJ Il OſſlèS
Dimensions.
Homoge-
IlêOllS
In Oſlo Iſle S.
The number which thus denotes the number of equal
factors in the power is called the exponent, and some-
times the inder of the power. f
If it be necessary to express 10 times the continued
product of the 5th power of a, the 4th power of b, the
3d power of c, the 2nd power of d and e, it is done
by this very concise notation 10aºbºcºd?e.
(11.) 4. Division. When one number is to be di-
vided by another, the process is signified by placing the
dividend above a line, and the divisor below it. If a
be the dividend and b the divisor, the quote is expressed
by #. Division is also sometimes expressed by placing
the sign : or -i- between the dividend and the divisor,
thus a : b, or a -- b, either of which signify the quote
of a divided by b.
(12.) A simple quantity is one in which the letters
of which it is composed are not connected by addition
or subtraction, or by the signs + or —. Thus, all
quantities expressed by a single letter are necessarily
simple. The quantities a b, #, &c. are simple, but
a + b, a - b, &c. are compound. Simple quanti-
ties are called monomes, and sometimes terms. Com-
pound quantities, consisting of two parts, are called
binomes, and all others polymomes.
(13.) Simple quantities are said to be like when they
are composed of the same letters combined in the same
manner. Like quantities may, therefore, differ both in
their signs and coefficients. The quantities + 3 a and
— 5 a. are like , also + 3 ab and – 10 a b. The quan-
7 6 3
tities + # and – º are like, but + 3 a b and + *:
are unlike, because although they are expressed by the
same letters, those letters are not combined in the same
In all Iſler.
(14.) The sign = interposed between two quanti-
ties, whether simple or compound, expresses their
equality. Thus,
- a + b = c + d,
means that the sum of a and b is equal to the sum of
c and d.
(15.) The sign - means greater than, and < less
than. Thus, a > b means that a is greater than b , and
a 3 b, that a is less than b.
(16.) Each of the literal factors of a monome or
term, is called a dimension of the term. The degree
of a term is its number of dimensions. Thus a b is
of the second degree, a b c of the third degree. But in
estimating the dimensions of a quote, those of the
divisor must be subtracted from those of the dividend.
a b .
Thus + is of the first degree, because there are two
dimensions in the dividend, and one in the divisor.
... a b c .
Again, + is of the second degree, &c. The reason
of this will appear hereafter.
(17.) Monomes are said to be homogeneous when
they are of the same degree, and a polynome is said to
be homogeneous when all its terms are of the same
degree. g
It should be observed, that the numeral coefficient is
not reckoned as a dimension.
3 Z 2
SECTION II.
Addition.
Addition.
(18.) SEVERAL Algebraical quantities are said to be Addition
added together, when they are arranged in a series, and defined.
connected by their proper signs. In some cases it
happens, that the operations of addition or subtraction,
indicated by the connecting signs + or —, may be
actually performed, and two or more of the quantities
may be thus incorporated, and the result so far simpli-
fied. According to what has been already observed,
when the same quantities are to be thus added together
algebraically, the result will be the same in whateve
order the operations may be performed. -
(19.) When the quantities to be added are unlike,
that is, expressed by different letters, they do not admit
of being incorporated by the operations indicated by
the signs by which they are connected. In this case,
algebraical addition consists merely in arranging them
in a series, the proper sign being prefixed to each, and
the aggregate is called their algebraical sum. When it
is considered that the numbers represented by different
letters may be referred to different units, the impossi-
bility of incorporating them will be at once perceived.
In the compound quantity a + b – c, a may represent
miles, b furlongs, and c perches ; in which case, were
the numbers represented by a and b to be actually
added, and that represented by c subtracted from the
result, the number thus obtained would neither repre-
sent the miles, the furlongs, nor the perches, in the
proposed distance. -
(20.) It is otherwise, however, if the quantities to
be added, or any of them, be like, (13.) In this case,
they are necessarily referred to the same unit, and may
always be incorporated by the actual arithmetical
operations indicated by the signs which connected them.
Thus, if the quantities to be added be + 2 a. and + 3 a,
it is evident that the sum
+ 2 a + 3 a.
is equal to 5 a.
(21.) Also, if the quantities to be added be + a,
— 2 b, and — 3 b, the result is
+ a - 2 b — 3 b,
that is, twice b is to be subtracted from a, and from the
remainder 3 b is to be subtracted. The result will
clearly be the same, if in the first instance five times b
were subtracted from a. Thus, if a be a foot and b be
an inch, two inches are first subtracted, which leave
ten inches, and again three inches are subtracted from
the remaining ten, and the remainder is seven inches.
Had five inches been subtracted at once, the remainder
would have been the same. Hence we infer the fol-
lowing equality,
a – 2 b — 3 b = a – 5 b.
So that – 2 b – 3 b is equal to — 5 b : hence negative
quantities when like are incorporated by addition in the
same manner as positive quantities.
(22.) If the quantities to be incorporated be like,
but have different signs, the process is effected by arith-
metical subtraction. Let the quantities be + 5 a. and
– 3 a. Being connected with their proper signs, the
result is
(l,
y
+ 5
528 A L G E B R A.
expediency of this change would equally apply to sub- Subtraction
Algebra. which means the actual remainder obtained by sub-
traction, multiplication, division, and numerous other S-N-
* --' tracting three times a from five times a. This is evi-
dently twice a, so that
+ 5 a - 3 a 5- -H 2 a.
In this case, therefore, the coefficient of the negative
quantity is subtracted from that of the positive quan-
tity, and the remainder is the coefficient of the result.
In the example just given, the coefficient of the posi-
tive quantity was greater than that of the negative, and
the process was sufficiently obvious. There is, however,
somewhat more difficulty in the case in which the co-
efficient of the negative quantity is greater than that of
terms. It is therefore, perhaps on the whole, better to
retain old terms in new and extended senses, than to in-
vent new ones at the risk of obscurity to students, and
to the manifest inconvenience of adepts in the science.
Algebraical addition is nothing more than the incor-
poration of a number of simple quantities by the arith-
metical processes of addition and subtraction indicated
by their signs, as far as that incorporation is rendered
possible by the nature of the quantities.
the positive quantity, and, therefore, cannot be sub- ExAMPLEs.
tracted from it. Let the quantities to be added be + æ + 5 a. 2 * – 10 ab
+ a, + 2 b and - 5 b. The result is + 2 a. + 4 a. – 27 — 4 ab
+ a + 2 b - 5 b. + 3 a. + 3 a. (, — 3 a b
By what has been proved in (21) – 5 b is equivalent + 4 + + 2 & - + — a b
to — 2 b – 3 b, and, therefore, s ====
+ a + 2 b – 5 b = + a + 2 b – 2 b — 3 b. + 10 a. + 14 a –4% – 18 a b
But it is evident that 2 b – 2 b = 0, or neutralize each b
other, and may be altogether omitted, and we infer the
following equality, – 7 :-
+ a + 2 b – 5 b = + a - 3 b, b
and hence tºm- cº-
+ 2 b – 5 b = — 3 b. + 2 a y + 7 a. (, 2 a y
Thus, when the coefficient of the negative quantity is – 5 a y — 4 y - 2 a. — 2 b y
greater than that of the positive quantity, the latter + 4 a y + ar 4 a 5 a y
must be subtracted from the former, and the remainder — 7 a. 3/ — 8 y — 5 a. — 6 b y
will be the coefficient of the result, the sign of which will — e-º-º-º-º- tº-ºººººº-º-º: -
be negative. — 6 a y + 84 – 12 y 5 a - 7 a 7 a y – 8 by
Rules for By generalizing the results of the preceding obser-
addition. Vations, we shall obtain the following rules for alge- (24.) It sometimes happens, when a compound
braical addition. quantity is to be added to a simple or another com-
pound quantity, that the operation is not actually per-
RULE I. formed but only signified. In this case, the compound
To add like quantities with like signs. quantity to be added is enclosed in a parenthesis, OI
Add their coefficients, and to the sum affix the com- placed under a vinculum, and connected by the sign +, Vinculum
mon letter orietters, and prefix the common sign with the quantity to which it is to be added. Thus, if
º p ( . a – b is to be added to 10 a, the result may be ex-
RULE II, pressed thus, -
To add like quantities with unlike signs. 10 a + (a — b)
Add the coefficients of the positive quantities, and or, 10 a + a - b.
likewise those of the negative quantities, and subtract In thi b i idered ingr! º
the lesser sum from the greater. To the remainder affix ll d i case, º i. i OllSl . º 3. . º:
the common letter or letters, and prefix the sign of those *.*.*...I.'". precedes ..". esis, or the
quantities of which the sum of the coefficients is the vinculum, does not belong to the first quantity a, but
reater to the result of the process indicated by a - b. There-
§ e fore the above complex quantity might also be ex-
RULE III. pressed thus, without any change in its meaning, or in
To add unlike quantities. its value, (9,)
Let them be arranged in a series in any order, and 10 a + (– b + a)
connected by their proper signs. or, 10 a + – b + a.
: RULE IV.
To add mired quantities, like and unlike. tºmºmºmºmº
Add the like quantities by the first and second rules,
and the results may be added by the third rule. SECTION III
Term addi- (23.) It will be observed, that the term addition in e
tion used Algebra is used in a very extended sense, the process Sub º
in an ex- being as often arithmetical subtraction as arithmetical Subtraction.
º addition. Were it not for the difficulty and inconve-
(25.) SUBTRACTION, in the popular or arithmetical
sense of the word, implies diminution. When any
quantity is said to be subtracted from another, that
other is supposed to be diminished by the quantity so
nience arising from any change in the nomenclature of
a science, it would be desirable that the algebraical
operation called addition should be otherwise denomi-
nated. But the same reasons which suggest the
A L G E B R A. 529
Algebra.
subtracted or taken away from it.
--~~' ever, the term acquires in its signification an extension
Rule.
Signs of
compound
quantities.
analogous to that already given to the term addition.
To explain the meaning of subtraction in Algebra, we
shall define it with reference to addition. By addition
we solve the problem, “Given two quantities to find
their algebraical sum.” By subtraction, then, we
solve the problem, “..Given one of two quantities, and
their algebraical sum, to find the other.” Thus, sub-
traction may be conceived to be nothing more than
wndoing, or destroying, the effect of a previous addition.
Let A represent any algebraical quantity, whether
simple or compound, from which it is proposed to sub-
tract another simple or compound quantity, which we
shall call B. The quantity A may here be conceived
to be the algebraical sum of B, and some other quan-
tity which it is proposed to discover. Let this other
quantity, whether simple or compound, be called ar, (7.)
Thus, by our hypothesis, A = a + B.
As A was obtained by annexing (18) the quantities
expressed by B to a with their proper signs, the effect
of this process will be destroyed by annexing to A
the quantities represented by B with their signs changed.
This process gives A – B = a + B — B.
But as B - - B is equal nothing, we have A — B = a.
If B were originally negative, the process would be-
come A = a – B.
A + B = a – B -- B, but – B -- B = 0
. . .” A + B = an.
(26.) Hence we may infer the following General
Rule:
“To subtract one algebraical quantity from another,
change the signs of the subtrahend, or conceive them
changed, and add the quantities by the rules of addi-
tion.”
ExAMPLEs.
From 5 a - 2 b a – 2 b + 3
Subtract 2 a + 5 b 4 a + 9 b – 5
3 a - 7 b — 3 a - 11 b + 8
From 5 a b – 18 8 & – 2 b – 5
Subtract — a b + 12 — a + 3 b + 2
6 a b – 30 9 a. - 5 b – 7
(27.) By what has been proved in the last section,
respecting the incorporation of algebraical quantities,
by actually effecting the operations indicated by the
signs which connect them, it easily appears, that the
sign of every compound quantity may be inferred from
the signs and values of its component parts.
If the simple component parts of a compound quan-
tity be all positive, it is evident that the whole quantity
is positive ; for if all the parts could be reduced to the
same denomination, and, therefore, rendered like, upon
incorporating them the result would be positive, (19.)
The same reasoning proves, that if the signs of all
the component parts be negative, the sign of the whole
is negative.
If a compound quantity be composed partly of
positive and partly of negative quantities, the sign of
the whole will be the same with that of those quantities,
positive or negative, which have the greater sum. If
the sum of the positive parts exceed the sum of the
—"
* . . . signifies, therefore
In Algebra, how-
negative parts, the whole is positive; and if the sum of Multiplica-
the negative parts exceed the sum of the positive parts,
the whole is negative.
Hence it will easily appear, that by changing the
signs of all the component parts of any compound
quantity, the sign of the whole is changed.
(28.) Hence it follows, that if the signs of the
several quantities within a parenthesis, or under a vin-
culum, be changed, and at the same time the sign
which is prefixed to the parenthesis be changed, no
real change in the compound quantity is produced,
because the two effects counteract or compensate
each other. Thus, if the quantity + (a — b) be
changed to + (– a + b) its sign is changed. But
if, again, this latter be changed to . ( a + b), its
sign being again changed, the quantity becomes what
it was before, so that
+ (a — b) = — (– a + b).
Hence we are always at liberty to change the sign of
a parenthesis, provided the signs of the quantities
enclosed be also changed.
(29.) If a complex quantity enclosed in a paren-
thesis, be connected with other quantities by the sign
+, the parenthesis may be removed, the signs of the
quantities enclosed in it being preserved. Thus, a +
(b – c) is equal to a + b – c.; and a + ( b + c) is
equal to a -- b + c.
This follows from the rules of addition ; for the
meaning of a + (b – c) is that the compound quantity
b – c is to be added to a, which is done by connecting
them by their proper signs. Addition, Rule III. (22.)
(30.) But if a compound quantity enclosed in a
parenthesis, be connected with other quantities by the
sign —, in order to remove the parenthesis it will be
necessary to change the signs of all the simple quan-
tities within it. For the sign —, which precedes the
parenthesis, indicates that the complex quantity included
within it, is to be subtracted from those quantities with
which it is connected. This subtraction, as has been
already shown, is effected by changing the signs of the
quantities within the parenthesis.
SECTION IV.
Multiplication.
tion.
(31.) MULTIPLICATION, in the original sense of the Multiplica
term, means the continual addition of the same quan-tion defined.
tity as many times as there are units in the integer,
which is called the multiplier. This term, however,
like addition and subtraction, has acquired an extended
signification, and the sense in which it was first used is
only a particular case of its present more universal
application. º
It is observed by some writers, that the multiplicand
may be any quantity, but that the multiplier must
always be an abstract number. This, however, is by
no means true. In algebraical calculations, heteroge-
neous quantities, or rather the numbers representing
them, are constantly combined by multiplication, and
it often happens, that the factors of the same product
have different significations, and are referred to units
of different species. Thus, if b represent the base of
a parallelopiped in Geometry, and a its altitude, the
product a b represents its volume. Here the factors
530
A L G E B R A.
Algebra. are not only heterogeneous, but each of them different
\–2–1 in species from the product.
Rule of
signs
(32.) The difficulty of conceiving the multiplication
of heterogeneous quantities will disappear by con-
sidering, that the letters in Algebra are not the immediate
representatives of quantities but of numbers, and these
numbers express the quantities in reference to their
specific units. Thus, in the example just given, b is the
number of superficial units in the base of the solid, a
the number of linear units in its altitude, and a b the
number of solid units in its volume. Under this view,
there is no more difficulty in conceiving the base b mul-
tiplied by the altitude a, than if the altitude were an
abstract number.
(33.) Multiplication, in the most general sense of the
term, is an operation by which a fourth proportional”
is found to the unit, and two numbers which are called
factors, and the fourth proportional so found is called
their product, (10.)
Thus, if a. and b be the factors, and a b the product,
we have 1 : a : b : a b.
Since the transposition of the means does not disturb
the proportion, it follows that there is no essential dis-
tinction between the factors, nor any grounds for giving
them different denominations, such as multiplicand and
multiplier, (10.)
When the factors of a product are considered as
having signs, as being positive or negative, a question
arises as to what sign the product should receive. The
rule commonly received is this, “When the two factors
have the same sign, the sign of the product will be +,
whether the common sign of the factors be + or – ;
and when the two factors have different signs, the sign of
the product is always —.” This rule is generally briefly
expressed thus, “In multiplication like signs produce
+, and unlike signs produce — .”
To give a general and unobjectionable demonstration
of this rule has occasioned some embarrassment with
elementary analytical writers. Before we enter upon
any investigation respecting it, let us recur to the mean-
ing of positive and negative quantities.
(34.) A positive quantity is one to which the sign +
is prefixed, and a negative quantity is one to which the
sign — is prefixed. The signs in general imply a con-
nection with other quantities, the one signifying a
connection by addition, and the other by subtraction.
In this way, therefore, we are to consider positive and
negative quantities as actually connected with some
other quantities by the arithmetical operations of addi-
tion or subtraction.
(35.) The meaning of positive and negative quantities
being thus explained, the following seems to be the
most unobjectionable of the proofs usually given for
together, the product will be a positive algebraical Multiplica.
quantitity, + a B.
2. Let it be required to multiply A + a and + b X-
together. By the last case, the product of A and + b
is + A b. Now if + b be multiplied by a greater
number than A, the product must be proportionally
greater. Therefore the product of A + a and + b
must be greater than the product of A and + b by the
product of + a and + b. Hence the product of A + a
and + b is A b + a b. Hence the product of + a and
+ b is + a b.
Hence, when the signs of both factors are +, the sign
of the product is +.
3. Let it be required to multiply A — a and + b
together. By the first case, the product of A and + b
is + Ab; but this is too great by the product of a and
b. Hence the true product is A b – a b. Hence the
product of — a and + b is — a b. When the signs of
the factors are unlike, the sign of the product is, there-
fore, — .
4. Let it be required to multiply A – a by B — b.
The product of A — a and B is A B — a B. But this
is too great by the product of b and A – a, or A b —
a b. To obtain the true product it will, therefore, be
necessary to subtract A b – a b from A B — a B, the
result of which is A B — a B — A b + a b, from which
it follows that the multiplication of — a and – b gives
the product + a b. From this and the second case
we infer that like signs produce +.
From the principles thus established, the following
consequences may be deduced without difficulty.
(36.) The sign of the continued product of several
factors is determined by the number of the negative
factors. If there be an even number of negative fac-
tors or none, the product is positive.
(37.) If there be an odd number of negative factors,
the product will be negative.
(38.) It is evident, that the same reasoning will
apply if there be no positive factor; and, therefore, if
all the factors be negative the product is positive or
negative, according as the number of factors is even or
odd.
(39.) If the signs of all the factors of a product be
changed, the sign of the product will not be changed
if the number of factors be even.
(40.) If the number of factors in the product be
odd, by changing the signs of all the factors the sign
of the product is changed.
(4.1.) When the factors of a product have numeral
coefficients, these should evidently be multiplied to-
gether, and their product taken as the coefficient of the
sought product. Thus, the product of 2 a. and 3 b is
6 a b.
the rule of signs.
1. Let it be required to multiply the number A + a
by the number B, A and B signifying absolute arith-
(42.) In all the preceding observations, the factors Multiplica-
are supposed to be single quantities. We shall now tion of
consider the case in which one of the factors is a com- complex
If A be
metical numbers independently of any signs.
multiplied by B, the product is A. B. But a greater
number than A, viz. A + a is to be multiplied by B ;
and, therefore, the product must be greater than A B
by the product a B. Hence the whole product is A B
+ a B.
Thus it follows, that if a positive algebraical quan-
tity (+ a) and an absolute number B be multiplied
* For the nature and properties of ratios and proportions, the
reader is referred to the Article GEoMETRy, p. 319
pound algebraical quantity. In this case, the product is quantities.
found by multiplying the simple factor by each term of
the complex factor, according to the rules already
established, and adding the results. The sum thus
obtained will be the true product. This may readily
be proved by the definition of multiplication already
given, and the properties of proportions established in
the Treatise on GEoMETRY in this work. But we shall
not here enter into any detail of the demonstration, the
principle being sufficiently evident.
(43.) If both factors be complex quantities, the
A L G E B R A.
53]
Algebra.
in the divisor. 3. Let each of the literal factors which Division.
product is obtained by multiplying each term of the
S-N-Z one factor by each term of the other, and adding to-
Rule of the
signs.
Division of
Ill Oſlomeş,
gether all the products by the ordinary rules of alge-
braical addition. The validity of this process may be
easily inferred from the last.
SECTION W.
Division.
(44.) DIVISIon bears the same relation to multipli-
cation that subtraction bears to addition. It is the
wndoing of what has been done, or is conceived to
have been done, by multiplication. As multiplication,
therefore, is the compounding two factors together so
as to form a product, so division, on the other hand,
consists in the decomposition of a product into its
factors. In multiplication and division there are three
quantities concerned, the two factors and the product.
Now any two of these three being given, the remaining
one may be found. When the two factors are given
to find the product, the process is called multiplication ;
and when the product and one of the factors are given
to find the other factor, the process is called division.
The product is in this case called the dividend, the
given factor the divisor, and the required factor the
uote.
q (45.) The rule for deducing the sign of the quote
from those of the dividend and divisor, is the same as
the rule in multiplication, and may be derived from it.
Let d be the divisor, D the dividend, and q the quote.
Since the product of d and q is equal to D, it follows
that when d and q have like signs, D is +, and when
d and q have unlike signs D is –. Hence it may
easily be inferred, that in division, as in multiplication,
“like signs give +, and unlike signs give – .”
(46.) If the divisor and dividend be monomes, the
division will not be capable of being executed exactly,
unless all the factors of the divisor, both numeral and
literal, be also factors of the dividend; for the dividend
is considered as the product of the divisor and the
number sought, or the quote. Let a b be the dividend,
and a the divisor, it is evident that in this case the
quote is b. -
Let the divisor be 5 a”, and the dividend 15 at b. In
15 a.º. b
5 a.3
This must be such a quantity as multiplied by 5 a” will
produce 15 at b. Let it be a, so that a × 5 a” = 15
at b. But 15 at b = 3 × 5 × a” x a b, . . . a × 5 a.3
= 3 a b x 5 a 3.
Hence it is evident that a = 3 a b.
In the same manner, if the dividend be 22 a.º. b3 c,
and the divisor ll a” b, the quote will be 2 a.2 b8 c.
By generalizing these results we shall obtain the
following rule for the division of monomes:
“The sign of the quote being determined as in
multiplication: 1. let the coefficient of the dividend
be divided by the coefficient of the divisor, and let the
result be taken as the coefficient of the quote; 2. let
each of the literal factors, which are common to the
dividend and divisor, be written after this coefficient
with an exponent equal to the excess of the exponent
in the dividend above the exponent of the same letter
the first instance the quote is expressed by (12)
occur in the dividend, but not in the divisor, be then \-,-)
written as factors of the quote.”
The following examples will illustrate this process:
75 g5 54 cºd 93 aſ as y
— = 3 a” bº —* = 3 a.º. tº
g-= 8*** a -size = 8*ev.
(47.) There are certain circumstances under which
the above process cannot be executed: 1. The co-
efficients may not be divisible one by another. 2. The
exponent of some letter in the dividend may be less
than that of the same letter in the divisor. 3. There
may be literal factors in the divisor which are not con-
tained in the dividend at all. In these cases, the divi-
sion cannot be effected, and the quote must be expressed
by the notation described in (12.)
Elet the dividend be 16 a” b c, and the divisor 10 a.”
b° c”. In this case 16 is not divisible by 10, neither
can the exponents of b and c in the divisor be subtracted
from those of the same letters in the dividend. In this
16 a.3 b c
- T0252c2
may be simplified by removing the common factors 2,
case, however, the expression for the quote
a”, b, and c.
5 b c
When complete or exact division cannot be effected,
the fractional expression for the quote may, therefore,
be simplified by the following rule :
“l. Divide the coefficients of the dividend and
divisor by their common factors. 2. If the same letter
occur in both dividend and divisor with different expo-
nents, subtract the lesser exponent from the greater,
and in place of the greater exponent place the remain-
der, omitting the letter with the lesser exponent
altogether. 3. Let the letters which are common, and
have equal exponents, be altogether omitted. 4. Let
the letters not common retain their places.”
(48.) It is evident that the value of a quote depends
on the ratio of the dividend to the divisor; and how-
ever they may be changed, provided their ratio be pre-
served, the quote will retain the same value. Since
q x d = D, '.' by (33)
d : D :: 1 : q.
The value of q must remain unchanged so long as the
unit bears to it the same ratio, that is, so long as the
divisor bears to the dividend the same ratio.
Hence it follows, that any common factors may
always be removed from the divisor and dividend with-
out affecting the quote.
The quote thus becomes
(49.) If the rule for the subtraction of the exponents value of re.
of the same letter in the divisor and dividend, when
these exponents are unequal, be applied to the case in
which they are equal, the result will assume a peculiar
form. Let the dividend be a bº and the divisor bº. The
b2
*: ; applying to this the rule for the case in
which the exponent in the dividend is greater than
in the divisor, the result is a b 2-2 = a bº. But the
quote being evidently a, we have a = a bº', '.' tº − 1.
It is usual, therefore, to say that “any quantity having
o for its exponent is = 1.” This, however, is to be
considered as a matter purely conventional, the symbol
a” having no other meaning than an expression of the
result of the division of two powers of the same letter,
having the same exponent, the process being conducted
by the rule established for the case in which the
quote is
532
A L G E B R A.
Algebra.
Division of
polynomes.
erponent of the dividend is greater than that of the
divisor.
The use of this notation is to preserve in the result
the marks of the process by which it was obtained.
(50.) If the dividend be the continued product of
several factors, it will be divided by any number by
dividing any one of its factors by that number. Thus,
72
8 × 9 = 72, and ; × 9 = 36 = 2 3 and the same,
of course, applies to algebraical quantities.
(51.) Hitherto we have supposed the divisor and
dividend to be monomes. Let us now suppose the letter
D a polynome. In this case, the quote q must be such
a polynome as multiplied by the divisor d, (supposed
a monome,) will give a product equal to D.
By (42) it appears that q is multiplied by d, by mul-
tiplying each of its terms by d, and the product
will therefore be a polynome, whose terms are the
products of d, and the several terms of q. But this
polynome must be identical with D, and, therefore, each
of the terms of D must be the product of d and the
several terms of q. Hence the several terms of q must
be the quote found by dividing the terms of D severally
by d. -
It follows from this, that in order that a polynome
should be exactly divisible by a monome, each of the
terms of the polynome must be divisible exactly by the
monome ; otherwise the quote will include terms of a
fractional form.
Let 2 a rº be the divisor, and let the dividend be
10 aſ a 5 + 20 as a 4 – 12 aſ a 3 + 6 as a 2 – 2 a. arº ;
when the several terms have been divided by 2 a aº by
the rules established for monomes, the quote will be
5 aſ a 3 + 10 a” a 2 – 6 a. a + 3 a” — 1
(52.) If the divisor be a polynome and the dividend
a monome, the exact division is impossible, and the
quote can only be expressed in the fractional form. For
the quote cannot be a monome, since the product of
a monome quote and a polynome divisor would give a
polynome dividend, contrary to hypothesis. Neither
can the quote be a polynome, since the product of the
quote and divisor, both polynomes, could not give a
monome dividend. In this case, therefore, the quote
must be expressed as in (47,) and may be simplified
if there be any factor of the dividend which is common
to all the terms of the divisor. This factor may be
removed, since both dividend and divisor may be
divided by the same quantity without affecting the
value of the quote.
The case in which the divisor is a monome, and the
dividend a polynome, admits of a similar simplification
when all the terms of the dividend contain a factor
common with the divisor.
(53.) We shall now consider the case in which both
the dividend D and the divisor d are polynomes. Each
of the terms of the dividend D being the product of a
term of the divisor d, and one of the quote q, it follows
that if we find a term of the dividend which is divisible
by a term of the divisor, this quote will be a term of
the quote q. Having thus found any one term (A) of
the quote q, this term being multiplied by the whole
divisor d, gives a product A d, which is to be considered
as that part of the dividend which has been divided by
d. This being subtracted from the whole dividend D,
the remainder is all that is now to be divided by d. As
before a term of this remainder is selected, which is
exactly divisible by some term of the divisor, and the Division.
quote being found, it is inserted with its proper sign as \-N--
another term of the quote q ; and so the process is
continued until a term of the quote q is found, which,
multiplied into the divisor d, will be equal to all that
has remained of the dividend. In this case, the division
is complete. But if in any of the remainders there is
no term which is exactly divisible by a term of the
divisor, the division cannot be effected, and we conclude
that there is no polynome q, which, multiplied by d, will
exactly give the product D.
In multiplying two polynomes together, it frequently
happens that the partial products of the several terms
of the factors destroy or modify each other, by those
which are similar being incorporated by addition or
subtraction. It may, therefore, happen that some of
the terms of the product of two polynomes are the
sum or difference of the product of two or more terms
of the factors, and not the product itself of these
terms. In the selection, therefore, of a term of the
dividend, which is to be considered as the product of a
term of the divisor, and one of the quote, it is neces-
sary that this term should be one which cannot have
proceeded from the combination of two partial products
of d and q, by the addition or subtraction of similar
terms; for if it were so, it is plain that we should not
be justified in concluding, that by dividing it by the
term of the dividend we should obtain a term of the
quote.
When the same letter occurs in two polynomes,
powers of that letter must occur in their product, and
one at least of these powers must have a higher
exponent in the product than in either of the factors.
The term containing the highest power is that which
proceeds from multiplying together the two terms of
the factors which contain the same letter with the
highest exponents. The exponent of the corresponding
term of the product will be their sum, and no other
term of the product can contain the same letter with
so high an exponent. This term, therefore, can suffer
no modification by addition or subtraction with any
other term, and must always be actually the immediate
product of the two terms of the factors which contain
the highest exponent of the same letter.
Hence it follows, that if there be a letter which occurs
with exponents in the divisor and dividend, its highest
exponent in the latter being greater than its highest
exponent in the former, that term of the dividend
which contains this letter with the highest exponent,
must be the immediate product of that term of the
divisor which contains the same letter, with the highest
exponent and a term of the quote, which term is there-
fore immediately found by dividing the one by the other.
Then, by the means already explained, a new dividend
is obtained ; and in this, likewise, a term is to be found,
in which the exponent of some letter is higher than in
the other terms, and so on. It is generally convenient
to select the highest power of the same letter in each
partial dividend, as that which is to determine the
term of the quote; this, however, is not absolutely
necessary.
In writing down the dividend and divisor preparatory
to division, it is not necessary to place the terms of
either in any one particular order rather than another.
But it is convenient to place first in each the two terms
by the division of which the first term of the quote is
to be determined. If, after the first subtraction the
A L G E B R A.
533
*
Algebra, highest power of the same letter in the next dividend be
\-N- selected, it will be also convenient that it should stand
Rule.
first in the remainder, and, therefore, that it should be
placed as the second term in the original dividend.
By continuing this reasoning we shall find, that the
terms of the dividend should be arranged according to
the descending powers of that letter, whose highest
power is selected for determining the first term of the
quote. By such an arrangement, the first term of each
remainder will be that which contains the highest power
of the same letter, and will, therefore, be that which is
proper to determine a term of the quote. Since the
first term of the quote is to be multiplied by the divi-
sor, and the result to be placed under the dividend,
preparatory to subtraction, it is evidently convenient
that the terms of the divisor should also be arranged
according to the descending powers of the same letter;
for, in that case, the corresponding powers of the terms
of the subtrahend will come under those of the dividend
preparatory to the subtraction.
Hence we obtain the following rule for the division
of polynomes:
the dividend, it follows, that the quote must consist of Of Simple
a series of terms affected by the same powers of a as Powers, and
appear in the dividend. Let A a” be any term of the
dividend, and B aſ the corresponding term of the TY"
quote. It follows, that d B a” = A a”, or dB = A.
or B = Tj } and since the same observation may be
applied to each of the terms, we deduce the following
rule for division, when the dividend contains any letter
which does not appear in the divisor : “Let the dividend
be arranged by the powers of this letter, and let each
of the multipliers of the powers be divided by the
divisor. The several quotes thus found, will be the
multipliers of the corresponding powers of the same
letter in the quote.”
SECTION VI.
Of Simple Powers and Roots.
(56.) As powers of the same quantity would be Multiplica-
multiplied by writing them down as one word, it is tion of
evident that the number of equal factors in their pro-Powers.
“Arrange the terms of the divisor and dividend
according to the descending powers of any letter which
is common to them, placing in each the term containing
this letter, with the highest exponent first, and each
succeeding term having that letter with a higher expo-
nent than that which follows it. Let the first term of
the dividend be then divided by the first term of the
divisor, and the result with its proper sign will be the
first term of the quote. Let this term be then mul-
tiplied by the whole divisor, and the product subtracted
from the dividend. Let the first term of the remain-
der be divided by the first term of the divisor,
and the result, with its proper sign, will be the second
term of the quote. Let this, in like manner, be
multiplied by the whole divisor, and the product
subtracted from the first remainder. The second re-
mainder then, constituting a new dividend, must be
treated as the former remainder, and the process must
be continued in this way until the multiplication of
some term of the quote gives a product exactly equal
to the last remainder, in which case the quote is com-
plete, and the division effected.”
(54.) It appears from what has been already proved,
that if the term of the dividend which contains the
highest power of a letter common to the dividend and
divisor, be not exactly divisible by the term of the
divisor containing the highest power of the same letter,
the exact division is impossible ; for, in this case, the
dividend cannot be the product of the divisor and any
polynome.
(55.) It is plain, that if the divisor contain any letter
which is not found in the dividend, the division is im-
possible. For a product must contain every letter
which enters either of its factors, and the division is
never possible, except when the divisor is a factor of
the dividend. On the other hand, the dividend may
contain a letter or letters which do not appear in the
divisor, because a product may contain letters which
do not appear in one of its factors, since they may be
letters of the other factor. If the dividend D contain
any letter a not contained in the divisor d, the division
may be effected by arranging the dividend by the
powers of the letter a. Since the divisor d, by hypo-
thesis, does not contain the letter a, and yet the pro-
duct of the quote q and the divisor d is identical with
WOL. I.
duct would be the sum of the numbers of equal factors
in each of the powers so multiplied. But as the expo-
nents express the number of those factors, we may im-
mediately infer, that “when powers of the same quan-
tity are multiplied together, the sum of their exponents
is the exponent of the product.” Thus
a? × a 3 = a,” a 3 a.4 = a 7
and in general am an = a m + n,
ºn and m being any positive integers.
(57.) From (46) it appears, that if a power of any Division of
quantity be divided by a power of the same quantity powers.
having a lesser exponent, the quote will be found by
subtracting the exponent of the divisor from that of
the dividend; and it has been shown (49) how this rule
has been conventionally extended to the case in which
the dividend and divisor have equal exponents. It may
also happen, that the exponent of the divisor is greater
than that of the dividend; in which case, if the division
were performed according to the rule established for
the case in which the exponent of the dividend is
greater than that of the divisor, the exponent of the Negative
Thus we should have, for exponents.
quote would be negative.
S
example, += a -2. But according to the established
0, 61 (I.
tº - a 3
rules of division (46,) we should have —- =
gº (1, (I (l, Q, Q,
l I
= — == −. Nevertheless, in order to generalize, as
a a aftº bº &
far as possible, the processes in algebraical investiga-
tions, it is found expedient to extend the rule for the
division of powers, established in (46,) to the cases in
which the exponent of the divisor is equal to, or greater
than, that of the dividend. What we have already
observed in the case of equal exponents, should, how-
ever, be carefully attended to in the case of the expo-
ment of the divisor being greater than that of the divi-
dend. The negative exponent, which the quote acquires
in this case, is to be understood only as indicating that
the power which is affected by it has been obtained by
applying the rule to a case to which it is not applicable,
otherwise than by general consent.
4 A
534
A L G E B R A.
Algebra.
Reciprocais,
Iuvolution
Evolution.
we are,
i 1
&c., and in general a-" is only another way of ex-
By the quantities a”, a *, arº, at 3, &c.,
1
therefore, to understand the quantities, 1, --,
pressing Ž, or a quantity with a negative exponent is
the reciprocal of the same quantity with the same posi-
tive exponent.
(One quantity is said to be the reciprocal of another
when their product is equal to the unit.)
(58.) The student will find no difficulty in extending
the rules for the multiplication and division of quanti-
ties with positive exponents to the case in which the
exponents of the factors are one or both negative.
a-m
E. §. a-m × a "" = a- m^*, and -T = a-m + n.
(l,
(59.) To find any required power of a quantity, or,
as it is called, to raise any quantity to a required power,
it is only necessary to form a product in which that
quantity shall be repeated as a factor as often as there
are units in the exponent of the power to which it is to
be raised. Now if the quantity to be raised be a power
of a simple quantity, as a”, it is plain that the continual
multiplication of this will give a product such as
a" + "+" *, where m is contained in the exponent as
often as there are units in the exponent of the power to
which it is to be raised. Let this exponent be m ; it is
then evident, that the m” power of a " is a "". Thus,
if it be required to find the third power of a”, we have
amama” – am + m + m – a 3m,
and this is true whether the exponent m be positive or
negative.
The general rule, therefore, to raise any simple quan-
tity to any required power, is to “Multiply the expo-
ment of the quantity by the exponent of the power to
which it is to be raised, and the product is the expo-
ment of the power sought.”
(60.) The terms power and root are correlatives; if
a be the mº" power of b, b is called the m” root of a,
and vice versä. The notation by which the root is
expressed, is the mark V called a radical, placed over
the letter, with an exponent to the left indicating the
order of the root. The quantity which is placed under
the radical, is called its suffir. Thus *va means the
third root of a, or that quantity of which the number
a is the third power. When no exponent is expressed,
the symbol means the square root, thus V a is the
square root of a, or the number whose square is a.
The processes by which powers and roots are found,
are called respectively, Involution and Evolution.
(61.) If the m” root of a simple quantity, such as
a", were required, it is evident that, if m were a multiple
of m, such as rm, the quantity a” or a” would be the
n” power of aſ, and, therefore, the nº root would be
found by dividing the exponent m or rn by the expo-
ment n of the root required. If, however, m be not a mul-
tiple of n, the mº" root cannot be algebraically extracted.
In this case, however, the same rule is extended analo-
gically, and the root is signified by assuming the quote
of the exponent m of the given quantity, by the ex-
ponent n of the required root, which is, therefore, ex-
pressed a". Thus the conventional notation derived
from the extraction of the roots of powers, gives
another method of expressing radical quantities; thus, Of Simple
*E---> * *=- y ºsmº 5 Powers and
^^a = a + 3 Ma – aft 4 vas = a +, &c. Roots,
(62.) Fractional powers of the same quantity are Mutiºn
multiplied by adding their exponents. That is, tion of
ºn m' 2n + tº * fractional
a " . a "' = a n ºn' powers.
wn. ºn.'
To prove this, let r = an aſ ~ a "'
Hence a" = a m an' = am"
Let the one be raised to the m/4" power, and the other to
the nº power, and we have
aſſin' - amºn' a/ii'n - (ſm”
These being multiplied, give
gºin' a'nn' F. am" + m’n
Taking the mn” root of these, we obtain
m n' + m, n º, wi'
ar' = a Tin WT = a n " '
Trº, m! m m'
... a7 . a7 = a n " " .
g e 777,
The same demonstration will be applicable if + and
71,
m/ g
7 are one or both negative.
, (63.) Since the product of two powers (whether frac- Division of
tional or negative, or both) is obtained by adding the ex- fractional
ponents, it follows that the quote is obtained by subtract- Powers.
ing the exponent of the divisor from that of the dividend.
For the dividend being the product of the quote and divi-
sor, its exponent must be the algebraical sum of the
exponents of the quote and divisor, (62;) therefore the
exponent of the quote must be the result of the subtrac-
tion of the exponent of the divisor from that of the
dividend. Thus the rule for the division of powers,
established in the case where the exponents are positive
integers, is general.
The rule for the multiplication of powers of the Involution
same quantity being generalized, the extension of nº evolº
that for their involution immediately follows. If"
Tº.
a *º be continually multiplied into itself, until a pro-
duct be found having a number of factors which we
may call p, the exponent +. will be added p times,
and the new exponent will be + p .#. Thus the p”
++. ºm, gº
power of a *m is a + p ..., that is, the power is obtained
by multiplying the exponent of the root by that of the
required power.
From the preceding rule is immediately derived
the extension of the rule (61) for the evolution of
powers of the same quantity. -
(64.) We shall now take the most general possible
case of the involution or evolution of powers. Let it
be required to find the (– £). power of an . The
general rule applied to this case gives
(*) $º #=
* To avoid a multiplicity of different letters, and to give symmetry
to the expressions, it is usual to express different quantities by the
same letter, distinguishing it, however, by accents thus, m, m', m”,
an", &c.
77, , ºr
an * T ~, sº
7), 7'
*
8 * (, T. &
A L G E B R A.
535
Algebra.
Sºº-y-
To prove this it is only necessary to retrace the sym-
bols to their original signification. The student will
find no difficulty in doing so.
The chief advantage which the extension of the pro-
perties established for positive integral exponents to
exponents of all kinds is, that it saves the necessity of
registering in the memory, and practising a different
Surds.
Signs of
powers and
TÜ0ts.
system of rules, and renders the results of algebraical
investigations more simple and symmetrical. Besides
this, it reduces all the operations on radicals to opera-
tions on fractions, with which every student is familiar.
(65.) The rules established for positive integral ex-
ponents being extended to those which are fractional
and negative, it may be asked how far the same rules
may be applicable to the cases where the exponents are
numbers incommensurable with the unit. Such num-
bers are called irrational mumbers or surds. As, for
example, the m” root of an integer which is not
itself the mº power of an integer. Thus V3, ºv'5,
&c. are surds. We shall see hereafter, that although
no integer or fraction can exactly express the values of
such numbers, yet we can always find a fractional
number which differs from the value of any given ir-
rational number by a quantity less than any assigned
number ; and in applying the rules already established
for exponents to an irrational exponent, it is to be
understood that they are applied to the fractional num-
ber, which represents the approximate value of the
irrational exponent. In fact, in numerical applications
we can form no distinct idea of an irrational number,
otherwise than that which we may form of the exact
fractional number which represents its value approxi-
mately.
With this limitation, therefore, and in this sense, the
properties just established may be extended to all ex-
ponents, whether rational or irrational, positive or
negative.
(66.) We have not yet pointed out how the signs
of powers and roots depend on the signs of the quan-
tities themselves. If the quantity which is involved be
positive, all its powers must be positive, for a product
is positive if all its factors be so. But in general (36)
a product is positive when it has either an even num-
ber of negative factors or none; and negative when-
ever it has an odd number of negative factors. Hence
it follows, that all powers of a positive quantity are
positive, and also all even powers of a negative quantity
are positive. Thus the only powers which can be nega-
tive are the odd powers of negative quantities. The
successive powers of + a are + a”, + a”, + at, &c. and
those of — a, are — a, + a”, - a”, + at, &c. being
alternately negative and positive.
From this it appears, that the odd powers of + a and
— a differ from each other in sign, each having the sign
of its root ; but that the even powers are the same,
being in both cases positive. Thus + a” is at the sanne
time the square of + a and of — a ; and, in like man-
ner, -- a”, + a”, &c. are the fourth, sixth, &c. powers
of + a, and also of — a.
It follows, therefore, that + a and – a have equal
claims to be considered as the square root of + a”, the
fourth root of + a”, and the same of any positive even
power of a. Thus it appears, that every positive quan-
tity must have at least two aven roots, which differ only
in their signs. Hence it is usual to prefix the double
sign + to an even radical, thus + Va, indicating
thereby that a has two square roots, one with the sign
+, and the other with the sign —, but otherwise the
Salſ]] e.
(67.) Since no quantity whether positive or negative
raised to an even power can be negative, it follows that
no negative quantity can have an even root.
theless it frequently happens in algebraical investiga-
tions, that negative quantities are found to occur under
even radicals, and although such a result is always the
consequence of a falsehood or contradiction in the
reasoning on which it is founded, yet it is found expe-
dient to preserve the mode of expressing it. The
symbol V - A, or " v - A, (where m is even) there-
fore expresses the result of an operation which cannot
be performed, yet such expressions are submitted to the
same rules, and subject to the same operations as simi-
lar radicals affecting positive quantities. They are
said, though improperly, to express impossible or
imaginary quantities. In effect, they do not represent
any quantities whatever, and are merely indicative of
an absurdity in the process from which they have been
derived. - -
(68.) With reference to such symbols, algebraical
quantities are said to be real or imaginary. An imagi-
mary quantity is the even root of a negative quantity,
and every other quantity is said to be real.
We shall reserve the further consideration of imagi-
nary expressions for a subsequent part of this section.
(69.) To obtain any proposed power of a product, it
is necessary to raise each of its factors to that power.
Thus if the third power of 2 a” b° be required, we have
(2 as bº) * = 2 as b2 × 2 as b2 × 2 as b2. But as the
order of the factors is indifferent, this may be expressed
(2 a” b2)} = 2.2.2. as . a.º. a.º. b3, b2. b3, or (2 as bº) 3
= 2* a” b°. By generalizing this result, we obtain the
following rule, “To raise a monome to any given
power, raise the numeral coefficient to that power, and
multiply each of the exponents by the exponent of the
power to which it is to be raised.”
(70.) Hence we may immediately infer the following
Of Simple
Powers and
Roots.
Impossible
or imaginary
Never- quantities.
rule for extracting any proposed root of a monome : ,
“Extract the root of its coefficient, and divide each
exponent by the exponent of the required root.”
In order, therefore, that a monome should be a com-
plete power of the m” order, it is necessary that its
coefficient should be a complete power of that order,
and that the exponent of each of its literal factors
should be a multiple of the exponent of the root re-
quired. Otherwise the result will be an algebraical
surd. -
(71.) In cases in which the roots of monomes do
not admit of absolute extraction, and are, therefore,
algebraical surds, they are nevertheless frequently
capable of considerable simplification. Some impor-
tant reductions on such quantities are founded on the
following theorem : “The mºh power of a product is
equal to a product of the mº powers of its factors, and
the m” root of a product is equal to the product of the
7m” roots of its factors, the exponent of the m being
any number whatever, integral or fractional, positive or
negative, rational or irrational.”
The first part of this theorem when m is a positive
integer has been already proved. Let it then be a
. 777.
fraction 7. The first part of the theorem announced
Simplifica-
tion of
Surds.
4. A 2
536
A L G E B R A.
Algebra. m. m. ºn m ºn
S--' algebraically is (a b c d) * = a "... b "... c.". d " . Let
a/* = a bº" = b cº = c d" = d.
-1. + —k _1
... a' = a, " bla= b " c' = c " d' = d."
17| ym. º, m.
and alm – at b'm – bºr c'm = c " d” = d."
Hence we infer, that
m. m. m. ºn
a'm b/m cºm g’m = a, "... b "... c." d "
a' ºn m. m. m.
Or (a/b/ c! d')" = a, b c” d".
Reduction
of radicals.
But also a” b” cº d" – a b c d,
... (a' b'c' d')" – a b c d
1.
... a'b' c' d' = (a b c d)"
... (a'b' cº d')" = (a b c d)"
... (a b c d) * = a "... b”. c”. d"
In this reasoning we assume the first part of the
theorem to be true, when the exponent is an integer,
whether positive or negative; but these cases may be
at once inferred from what has been already proved.
As roots are only fractional powers otherwise expressed,
the preceding demonstration establishes the second
part of the theorem. -
(72.) Since the divisor and dividend of a quote may
be multiplied or divided by the same number, (48,)
without changing the value of the quote, it follows that
the exponent of a radical, and the exponent of its
suffix, may be multiplied by the same quantity without
changing its value. For
º
"War = a"
p n †l
r m w/a p" ==a?" – d"
(73.) By this principle, two radicals may be reduced
to the same exponent. For this purpose, all that is
necessary is to multiply the exponent of each radical,
and that of its suffix, by the exponent of the other
radical, and the product of the exponent will then be
the common exponent. Let the two radicals be
" Va" and " waſ".
m Van -: m m'</ºr
” Waſ" – mm w/2777
The radicals have thus mm' as their common exponent.
It will be easily perceived that this is in effect reducing
the fractional exponents of the equivalent fractional
powers to a common denominator. (Section IX.)
The rule may be extended to several radicals. “Mul-
tiply the exponent of each radical, as well as that of its
suffix, by the product of the exponents of all the other
radicals; and the product of all the exponents will be
the common exponent.” It sometimes happens, that
the radicals can be reduced to a lower common expo-
ment than the product of their exponents. But as the
whole doctrine of the reduction of radicals may be
resolved to the reduction of the equivalent fractional
powers, we refer the student to Section IX., the results
also
then
Of Simple
Powers and
Roots.
S-N-Z
of which will be immediately applicable to the frac-
tional exponents, whose denominators are the expo-
ments of the radicals, and whose numerators are the
exponents of their suffixes. The lowest common ex-
ponent will then be the least common multiple of the
exponents of the several radicals.
(74.) The principles which have just been established Addition of
form the foundation of the rules for effecting on radi-radicals.
cals the elementary operations of addition, subtraction,
&c.
Radicals are said to be similar when they have the
same exponent and the same suffixed quantity.
Similar radicals are added or subtracted by adding
or subtracting their coefficients, and prefixing the sum
or difference as the coefficient of the result. Thus
3 3 v 5 + 2 3 Mö – 5 S v 5
3 3 Mö – 2 sw' b – 3 Mb’
3 m va. – 4 m Vas– (3 m — 4 m) w/a.
Radicals which are not similar, may sometimes become
so by reduction. Let them first be reduced to a com-
mon exponent. If then the suffixes have a common
factor, and the factors not common be complete powers
of the order expressed by the common exponent, the
radicals will be reduced to a common suffix, (the com-
mon factor,) by bringing the roots of the factors not
common outside the radical. Thus, the radicals
v48 abº and V75 a are reduced thus, V48 a 53 =
w/3.16 a bº: V75 a - v3.25 a.
factor.
3 a is a common
The factors not common are 16 bº and 25,
which are the squares of 4 b and 5. Hence V.48 a bº
= 4 b v 3a, V75 a = 5 v 3 a. Hence v 48 a 5% +
vº, a = (4b -- 5) w/3 a. Again, 3 vs as 5 -- 16 a.
= 5 v 8 a 3 (b +2a) = 2 as V5+ 2 a.
3 vöT2 a 53 = swº (5 + 2 a) = b > v 5 T2 a.
... 3 vs as 5-HT6 a 4- 3 vº-F2 a 53 = (2 a + b)
3 vº-E 2 a.
If the radicals be not similar, they do not admit of
being incorporated, and their addition or subtraction
can only be expressed by the usual signs, + or -, be-
tween them.
(75.) To multiply radicals, reduce them to the same Multiplica-
exponent, multiply their suffixes, and prefix the com- tio of
*=º - *E*-*-ºm radicals.
mon exponent. Thus "Vax "v b = "va b.
For the product of the mºh roots is equal to the m”
root of the product (71.)
In like manner, to divide radicals reduce them to a Division.
common exponent, and divide their suffixes. Thus
* = "V;
"v 5 Tº,
(76.) The rules for the involution and evolution of Involutiºn
radicals may be immediately obtained by converting and evolu-
o e ſº g tion of
them into powers with fractional exponents. If it be ...is.
required to raise "Ma" to the p" power, we have
" w/a, = a " ... ("way), = ar". Now it will be
proved in Section IX., that a fraction is multiplied by
A L G E B R A.
537
Algebra, any number, either by multiplying its numerator, or
~~~~' dividing its denominator. Hence -
p n
("Maº) * = a " - "War"
fº
Or, m + p = m + p w an
dº-º-º:
ºmmemº
Hence to raise a radical to the p" power, it is neces-
sary either to multiply the exponent of the suffix, or to
divide that of the radical by p. ~
(77.) In the same manner we may infer, that to
take the p" root of a radical it is necessary to divide
the exponent of the suffix, or multiply the exponent of
the radical by p. For the process must be exactly the
reverse of evolution.
(78.) The theorems which have been established
in the preceding articles, respecting the calculation of
radicals, are founded upon the supposition, that if the
powers of the same degree of two quantities be equal,
the quantities themselves will be equal. So long as
the theorems are applied to absolute numbers this is
strictly true, but some modification will be necessary
when applied to algebraical quantities. We have
already seen that all the even powers of + a and — a are
the same ; and we should, therefore, be wrong in con-
cluding, that if + a” be the square of two quantities,
these two quantities must be algebraically equal, since
one might be + a, and the other — a. We shall see
hereafter, that every quantity has as many roots of the
7m” order algebraically different, as there are units in m,
and, therefore, it would be wrong to conclude, that if
the m” powers of two quantities were equal, the quan-
tities themselves would be equal, since there might be
two different mº roots of the same quantity. These
observations are more especially to be attended to in
cases where imaginary expressions are concerned.
(79.) Every simple imaginary quantity may be con-
sidered as the product of a real quantity, and a power
of - 1, whose exponent has an odd numerator and
even denominator. The terms of the series 0, 2, 4, 6,
&c. are severally the doubles of those of the series 0, 1,
2, 3, &c. Hence if m represent any term of the latter
series, 2 m will represent any even integer or any term
of the former series. In like manner, the successive
terms of the series of odd numbers, 1, 3, 5, 7, &c., may
be found by adding one to each of the terms of the
first series, so that any term of the last series may be
represented by 2 m + 1, m, as before, being any term
of the second series.
A simple imaginary expression has been defined to
be a negative quantity raised to a fractional power, the
denominator of whose exponent is even. The nume-
rator must therefore be odd ; for if both were even, the
fraction might be reduced to lower terms. The nume-
rator of the exponent may, therefore, be represented by
2 m + 1, and the denominator by 2 m. The quantity,
2 m + 1
therefore may be expressed in the form (– A) *"
A being a real quantity. But by (71) we have
2 m + 1 *m-El 2 m + 1
(— A) 2 n. :- (— 1) 2 n - (+ A) 2 rº
3 m + 1
But (+ A) *" is a real quantity; let it be called B,
and we have
2 m + 1 2 m + 1
C–A) ". . = (– 1). T. B.
The results of all operations performed with imaginary On Prime
monomes are, therefore, to be determined by consider- and Com-
ing the properties of those powers of — 1 which have 1.
fractional exponents with even denominators. Jntegers,
(80.) To determine in general what powers of an Real powers
imaginary quantity are real, it is only necessary to find of imaginary
what integral multipliers will render the exponent of quantities.
(– 1) in its coefficient an integer. Let
2 m + 1 2 m + 1
(– A) *" = (– 1) *" . B.
In order that the exponent of – I should become an
integer, its numerator thust be either 2 m, or some mul-
tiple of it. Hence the real powers of such an imagi-
nary expression are 2 m, 4 n, 6 m, &c., and they are
alternately negative and positive. All other powers are
Imaginary.
The product of any two quadratic imaginary expres-
sions is real and negative.
wº- a. VT5= ( – 1)*. Va. ( – 1) + v 5
T f *-
= ( – 1) * ( – 1)* aſab
= (–1) vab = — w/abſ
The product of three such factors would be imaginary,
v'-a v- 5 M-c = — wa b. M-c = — v- i Va Sc
again, if a fourth imaginary factor be introduced, we
have
w/-a. M-5 v-2 M-d = — v- i. V-d Vab c
- (-1)*(–1). Mašod
+ V a b c of f
By continuing this reasoning it will be evident, that
a product consisting of an even number of quadratic
imaginary factors will be real, and will be positive or
negative, according as half the number of factors is
even or odd.
E-º
º
SECTION VII.
On Prime and Compound Integers.
(81.) Number is defined to be the abstract ratio of Number.
any quantity to another of the same kind, which is
called the unit. As the terms of a ratio may be either
commensurable or incommensurable, number is ac-
cordingly divided into two species.
(82.) A rational number is that which is commensu-
rable with unity.
(83.) An irrational number is that which is incom-
mensurable with unity. Irrational numbers are some-
times called surds.
(84.) Rational numbers are of two species, integral
and fractional. -
(85.) An integer is a multiple of unity.
(86.) A fraction is a submultiple of unity, or a mul-
tiple of a submultiple of unity.
Integers are divided into prime and compound.
(87.) A prime integer is one which is not measured Prime in-
by any integer greater than unity, as 3, 5, 7. 11, &c. teger.
(88.) A compound integer is one which is measured Compound
by an integer greater than unity, as 4, 6, 9, &c. integer.
538 A L G E B R A.
Algebra. (89.) An integer which measures another is called a m', 'n' being less integers than m and m, and in the TheGreatest
S-N-' divisor of it; and if it be a prime integer, it is called a same ratio. - Common
prime divisor or prime factor. It follows, also, that prime integers are the least in *: º
(90.) Every compound integer is the product of its their own ratio; for if they were not, they would have a “...
prime divisors. Thus 10 is measured by 2 and 5, and common measure, as has been already proved. Multiple.
10 = 2 × 5. It should, however, be observed, that Thus prime integers are equisubmultiples of all other \-y—
the same prime divisor may occur more than once as a integers in their own ratio, and the primes in any given
factor. Thus the only prime divisors of 12 are 2 and ratio are found by dividing any integers in that ratio
3. But 12 is not equal to 2 × 3, but = 2 × 2 × 3; by their greatest common measure.
the prime factor 2 occurring twice. In like manner, 2 (93.) If an integer a measure one of two prime
is the only prime factor of 16, but 16 = 2 × 2 × 2 integers m, it must be prime to the other n. For any
g common measure of a and n would also measure m,
(91.) Two integers are said to be prime to each which is a multiple of a, and would therefore be a com-
other, when they have no integral common factor greater mon measure of m and n, which contradicts the hypo-
than unity. Thus 7 and 9 are prime to each other, thesis. -
although 9 is not a prime integer. (94.) If an integer m measure a product a n, and be
If either or both of two integers be absolutely prime, prime to one factor n, it must measure the other a. For
it is evident that they must be relatively prime, since let it measure a n by c, so that
one or both has no factor greater than unity. . This is m, c = a m '.' m : n . . a ; c.
subject, however, to the exception of the case in which º º dº gº
one being prime the other is a multiple of it. Since ºn is prime to n it measures a.
The least (92.) The least integers in a given ratio measure all . (95.) If an integer a be prime to two others, m, n,
integers in other integers which are in the same ratio. it will be prime to their product.
a given Let m and n be the least integers, and let M, N be For if not, let c be a common measure of a and
ratio. m n. Since c measures a it is prime to m, (93;) and
any others in the same ratio.
If m measure M, it is evident from the nature of
proportion that n must measure N by the same num-
ber. Thus, if m be contained in M t times, without a
remainder, n must also be contained in N t times with-
out a remainder.
Also, from the nature of proportion it appears, that if
m be contained in M t times with a remainder m' less
than m, n must be also contained t times in N with a
remainder n' less than n. Thus we have
M = m t + m/
since it measures m n it must measure m, (94.) It, there-
fore, is a common measure of a and m, which are '.'
not prime, which contradicts the hypothesis.
The same principle being extended, shows that if an
integer be prime to any number of integers, it will be
prime to their continued product, and that if any num-
ber of integers be severally prime to any number of
others, the continued product of the former will be prime
to the continued product of the latter.
Hence, if two integers be prime to each other, every
power of the one will be prime to every power of the
Hence N = n t-i- ml other. º º &
m : n . . m t + m': m t + m' For a more complete discussion of the properties of
- * * * * s e prime and composite integers, we refer to our Treatise
but also on ARITH METIC. We have confined ourselves here
?n : n . . m f : m t strictly to what is indispensably necessary to render the
m t : n t . . m t + m' : m t + m' doctrine of fractions in Section IX. intelligible
m t : m t + m/ . . m t , n tº + n'
m, t : m." . . m t : n'
... m t : n t: : mſ : n'
m : n . . m." : n/
Hence m' and n' are integers less than m and n, and in
the same ratio with them, which is contrary to the
hypothesis. It follows, therefore, that there can be no
remainders, and that m and n must measure M and N
the same number of times, so that
SECTION VIII.
Of the Greatest Common Measure and the Least
Common Multiple.
M = m t (96.) If a quantity a measure two others, b, and c, The mea
N = n & it will also measure their sum (b + c) and their diffe-sure of any
rence (b – c.). º
It is plain that M and N are divisible by t, which is,
therefore, a common measure, and, therefore, M and N
cannot be prime.
It appears also, that t is the greatest common mea-
sure of M and N ; for if there were a greater, the quotes
found by dividing M and N by it would be less than
7m and m, and yet would be in the same ratio with them,
which contradicts the hypothesis.
Also it follows, that m and n are prime, for if they
admitted of a common factor, let it be t'. Then m =
mſ tº, m = n/t', and we should have
Tn : n : ... m': n/
For let a measure b m times, and c m times, so that
= m a, c = m a, "... b + c = m, a + n a = (m. -- n) a,
b – c = m, a - n a = (m. – m) a. Since m + m and
m – n are integers, a measures b + c and b – c. -
(97.) If the division of a greater quantity by a lesser
be partially effected, and the integral part of the quote
be obtained, any quantity which measures both the
divisor and dividend must measure the remainder, and
any quantity which measures both the divisor and the
remainder must measure the dividend. Let d be the
divisor, D the dividend, q the integral part of the quote,
and r the remainder. Hence D – q d = r, any
J A L G E B R A.
539
Algebra. quantity which measures d must measure q d, and if it
—y- measure D also, it will measure D – q d, or r, (96.)
Also, we have D = q d -H r, and any quantity which
measures d and r will also measure q d -H r or D.
(98.) To determine the greatest common measure
of two quantities. -
Let the lesser be A, and the greater B.
Let B be divided by A, and if there be no remain-
der, A is the greatest common measure, since it mea-
sures itself and B.
But if there be a remainder, let it be R. It is
necessarily - A. Let A be divided by R, and let the
remainder be Rſ. Again, let R be divided by R', and
let the remainder be Rſ', and in this manner let the
process be continued, dividing each remainder by that
which immediately succeeds it, until some remainder be
found which measures the preceding remainder. This
remainder is the greatest common measure.
First, it is a common measure ; for it measures
itself and the last divisor, and, therefore, measures the
last dividend. But this divisor and dividend were the
remainder and divisor in the preceding division, and
since it measures these, it must measure the preceding
dividend; and by the same reasoning it may be proved,
to measure every divisor and dividend until we arrive
at the given quantities A and B, which are the first
divisor and dividend. It is, therefore, a common
measure of these.
Secondly, it is the greatest common measure ; be-
cause every other common measure can be proved to
measure it. Every common measure of A and B must
measure the first remainder R. But R and A are divi-
sor and dividend in the second process of division.
Therefore the same common measure measures the
second remainder, and so on, until we arrive at the last
remainder, which it also measures. But this remainder
has been proved to be a common measure, and since
every other common measure measures it, it must be
the greatest common measure.
In this process each successive remainder is less than
that which precedes it, and the process may be con-
tinued ad infinitum, the remainders continually dimin-
ishing in magnitude, and none ever found which will
measure that which precedes it. In this case, by con-
tinuing the process, a remainder may be found which
is less than any assignable quantity. It is not difficult
to perceive, that in this case the given quantities are
incommensurable; for if they had a common measure,
however small, the process above described might be
continued, until a remainder be found smaller than this
common measure ; but this common measure would
measure every remainder, and would, therefore, mea-
sure a quantity less than itself, which is absurd. Hence
the two given quantities admit no common measure, or
are incommensurable. -
If two quantities be commensurable, all their common
measures may be found by determining their greatest
common measure. Let this be M. Every other com-
mon measure of the two given quantities measures this,
and vice versä, it is plain that every quantity which
measures M. must measure the given quantities. Now
the greatest quantity which measures M is # M. The
next in magnitude is $ M, the next + M, and so on ;
The greatest
COIn III Olſº
UneaSure.
the common measures forming the series M.-: T3 .
M M . .
--, - &
I-, -, we
If the two given quantities be integers which are TheGreatest
prime to each other, the last remainder will be unity. wººd
It is evident that the greatest common measure of “...
two quantities, A and B, is also the greatest common Common
measure of the lesser A, and the remainder resulting Multiple.
from the division of the greater B and the lesser A, and S-2
also the greatest common measure of every divisor and
remainder to the end of the process.
(99.) To determine the greatest common measure of The greatest
three quantities A, B, C, let the greatest common mea-Sommon
sure M of A and B be found, and next let the greatest tº:
common measure M' of M and C be found. This will tities.
be the greatest common measure of A, B, and C.
First, it is a common measure ; for since M' mea-
sures M, it must measure A and B, which are multiples
of M ; and it also measures C, and is, therefore, a
COIIll Il OIl II lea,SUlre.
Secondly, it is also the greatest common measure ;
for any other m, since it measures A and B, must
measure their greatest common measure M, and since
it measures M and C, must measure their greatest
common measure M’, and is, therefore, less than M'.
In the same manner, the greatest common measure
of four or more quantities may be found, viz, by finding
the greatest common measure of two, then the greatest
common measure of that and the third, and so on.
The greatest common measure of four quantities
being known, all other common measures may be found
in the same manner as for two.
(100.) To determine the least common multiple ºf The least
two quantities A, B. common
Let the common multiple sought be m A and n B, * *
so that - **
7m A = n B, g
and that m and n be integers. The question then is
to determine what are the least integral values of m
and n, which are consistent with the equality of m A
and n B. From this equality we deduce - -
m : m . . B : A. ... •
Hence m and n must be the least integers in the ratio
of B : A. Let c be the greatest common measure of
B and A. By (92) we have --
B A
771, E — ??, tº —
. C C
Hence we obtain
BA
C
The least common multiple of two quantities is,
therefore, their product divided by their greatest com-
Iſlo Il Iſlea.SUlre.
If the quantities be not numbers, this result may be
found more intelligible if announced thus, To find the
least common multiple of two quantities, let either of
them be multiplied by the number of times their
greatest common measure is contained in the other.
(101.) The least common multiple of two quantities The least
measures every other common multiple. For let m be common
the least common multiple of A and B, and let M be multiple
any other common multiple; if m do not measure M, ºne:
Iet there be a remainder r less than m. Since A and B common
measure m and M, they must also measure r, which is multiple.
therefore a common multiple of A and B, and less than
m, which is the least common multiple, which is
absurd. -
540
A L G E B R A.
Algebra.
Hence if the least common multiple m of two quan-
S—V- tities be known, every other common multiple may be
Fractions,
Denomina-
tor.
Numerator.
Ratios ex-
pressed by
fractions.
Terms of a
fraction.
found; for the least number which m measures is 2 m,
and the next is 3 m, and so on. So that the succes-
sion of common multiples is m, 2 m, 3 m, 4 m, &c.
If the two quantities be prime integers, their least
common multiple is their product. For their greatest
common measure is unity. -
We shall treat of the greatest common measure of
Algebraic quantities hereafter.
SECTION IX.
On Fractions.
(102.) ANY quantity being divided into any number
of equal parts, one, or the aggregate of several of
these parts, is called a fraction of that quantity. The
quantity which is so divided may be itself a number;
and as, in explaining the theory of fractions, it is con-
venient to suppose that all fractions arise from the
division of the same whole, we shall consider this to be
the unit.
The value of a fraction, therefore, depends on two
things, first, on the number of equal parts into which
the unit is divided, and secondly, on the number of these
parts which constitute the fraction. Two integers are,
therefore, necessary to express the value of a fraction;
that which expresses the number of parts into which
the unit is divided is called the denominator, and that
which, expresses the number of these parts in the frac-
tion is called the mumerator.
(103.) A fraction bears the same ratio to the unit as
its numerator bears to its denominator. For the for-
mer expresses the number of equal parts in the frac-
tion, and the latter expresses the number of the same
parts in the unit.
(104.) Hence it appears, that a fraction is equivalent
to the quote arising from the division of its numerator
by its denominator. For the quote bears to the unit
the same ratio as the dividend (or the numerator)
bears to the divisor (or the denominator,) (48.) Since,
then, the quote and fraction both bear the same ratio
to the unit, they are equal.
(105.) Hence the notation used to express a frac-
tion is the same as that used to express the division of
the numerator by the denominator. If a be the mu-
º . . Cl
merator, and b the denominator, the fraction is Th
(106.) If the numerators a, c of two fractions hear
the same ratio to their denominators b, d, the fractions
are equal, and vice versä, (48.) That is, if
a : b . . c : d
... a. C
* ===
b d
The latter may, therefore, be considered a more concise
way of denoting proportion.
(107.) Since the value of a fraction depends on the
relative, and not the absolute values of its terms, it fol-
lows that the same fraction may be expressed in an
infinite variety of different terms. Any change may
be made upon the terms of a fraction which does not
affect their ratio, without changing its value.
It is, however, most frequently desirable that frac-
tions should be expressed in their lowest possible
terms, and these are evidently the least quantities in
the ratio of the numerator to the denominator. Whe-
ther the fraction be arithmetical or algebraical, these
terms are found by dividing the terms of the proposed
fraction by their greatest common measure.
(108.) It is evident, also, from what has been
established in the last section, that all terms in which a
fraction can be expressed are equimultiples of its least
terms.
(109.) Also it appears, that both terms of a fraction
may be multiplied, or divided, by the same quantity
without changing its value.
(110.) It is sometimes necessary to change the
denomination of a fraction, that is, to find an equi-
valent fraction having a given denominator. It is first
to be observed, that this is only possible when the
given denominator is a multiple of the least deno-
minator of the proposed fraction; for it has been
already proved, that all terms in which a fraction can
be expressed are equimultiples of its least terms. The
numerator sought will then be the same multiple of
the least numerator, as the given denominator is of the
least denominator. The practical process for deter-
e g g Q.
mining the numerator is obvious. Let TST be the frac-
tion, d the given denominator, and a the sought
(E
(Z º
--, multi-
d b
d
plying these equal quantities by d we obtain a = 4. ;
numerator.
We have by hypothesis
in order that the problem be possible, it is necessary
that b should measure a d.
(Ill.) If several fractions be required to be reduced
to the same denomination, let them be first reduced to
their lowest terms. The common denominator to
which they are then to be reduced, must be a common
multiple of their denominators, (110 ;) and they may
be reduced to any common denominator which is a
common multiple of their denominators. The several
numerators are found by taking the same multiple of
the numerator of each fraction, as the common deno-
minator assumed is of the denominator or the fraction.
Or, what is the same, let the common denominator be
divided by each of the given denominators, and let the
quotes be severally multiplied by the respective nume-
rators. The several products will be the numerators of
the fractions sought.
The least terms in which several fractions can be
expressed, consistently with having a common denomi-
nator, is when the common denominator is the least
common multiple of their denominators.
(112.) The relative magnitudes of two fractions may
be known by reducing them to a common denominator.
For then, since the parts of unity which compose them
are the same, they are as their numerators. Let the
d {Z C
fractions be —- and d When reduced to a common
b
d b
denominator they become **. * which are as a dº be
b d b d
• T- # # . . a d : b c. Hence fractions are as the
alternate products of their numerators and denomi-
nators.
Of Frac-
tions.
S-N-
Reduction
of al frac-
tion to a
given deno-
minator.
Reduction
of fractions
to the same
denomina-
tor.
Relative
magnitudes
of fractions,
A L G. E. B R A.
541
Algebra.
Addition
and sub-
traction.
Multiplica-
tion.
By an
integer.
By a
fraction.
Four ways.
Of these the second is the most usual, because it is of Frac-
(113.) Several fractions united with the signs + or
— may be incorporated or reduced to one fraction by
reducing them to a common denominator, and adding
or subtracting their numerators according to the signs
with which the fractions are connected; taking the
result of this as the numerator, and subscribing the
v,v is a 141 via vºlva ºvkaalaavu- ~ * - Cº--- " - * J - * ~ * -- - 2 –
common denominator, the parts of the unit which they
severally contain are equal, (102;) by adding or sub-
tracting the numerators, and subscribing the common
denominator, these parts are added, and their magnitude
a c a d c b a d -- c b
preserved, thus 7, + i – T | I - T, T-
(114.) The multiplication of the numerator of a frac-
tion by any number, has the same effect on its value as
the division of the denominator by the same number.
tº (Z g g ºt tº
Let the fraction be --. If its numerator be multiplied
b
If its denominator be divided
by c, it becomes b
To prove that these results
by c, it becomes b -- c
are equal, let both numerator and denominator of
the former be divided by c, and we have
a c a c -- c Cº.,
b T b -- c T b – c
In precisely the same manner it may be proved,
that the division of the numerator of a fraction by any
number, has the same effect upon its value as the mul-
tiplication of the denominator by the same number.
(115.) A fraction is multiplied by an integer, by
multiplying its numerator by the integer. For this
multiplies the number of parts in the fraction without
changing the value of these parts. Also, the same
effect is produced by dividing the denominator by the
integer.
(116.) A fraction is divided by an integer, by divid-
ing its numerator by the integer. For this divides the
number of parts in the fraction without affecting the
value of these parts. The same effect is produced by
multiplying the denominator.
(117.) The multiplication of any quantity by a frac-
tion is an operation compounded of a multiplication
by its numerator, and division by its denominator. If
#, and the quantity be first multi-
plied by the numerator a, the product thus obtained
will be b times the true product; because the multiplier
the multiplier be
(M,
a was b times the true multiplier †. Hence, to obtain
the true product, it will be necessary to divide the pro-
duct already obtained by b. .
(118.) Hence, if one fraction + be required to be
multiplied by another #. it is necessary to multiply it
by c, and to divide it by d. As each of these opera-
tions can be performed in two different ways (115,) it
follows that there are four ways in which one fraction
may be multiplied by another; these four ways are
represented as follows:
a c -- d. Q, C (º, a + d
1.--, 2, #3 & #FF; 4. H-
WOL. I.
always possible to effect the operations. The others
are only used when the divisions indicated can be S-N-2
effected without remainders. However, when this is
the case, they are to be preferred to the second, be-
cause they give the product sought in lower terms.
**** A. Th2 divisinn nf anv quantity by a fraction Division.
(#) is an operation compounded of a division by
the numerator, and a multiplication by the denomi-
nator. If the quantity be divided by the numerator a,
the quotient will be the b% part of its true value, be-
cause the divisor a is b times the true divisor. Hence
to obtain the true quotient it will be necessary to mul-
tiply the quotient already obtained by the denomi-
nator b. It appears, therefore, that dividing by the
Gº,
fraction —-
b
to divide a quantity by a fraction, it is necessary to
multiply it by the reciprocal of that fraction.
is the same as multiplying by #. That is,
(120.) If one fraction # be required to be divided
by another #. it is the same as if required to multiply
d
it by T. Hence there are four ways (118) of effect- Four ways.
ing the object. The quote may, therefore, be expressed
in any of the following ways:
a d (Z
—;––– ; 2. --— ; 3. –F–F– ; 4.
1 b 5 b c ' b c + d 4
The second is the most usual, for the reasons assigned
in (118,) but the others, when possible, are to be pre-
ferred.
(121.) The denominator of a fraction being sup-
posed to remain the same, if the numerator be dimi-
nished the fraction itself will evidently be propor-
a —H. C.
b -- d .
a d -- c
tionately diminished. If the numerator be indefinitely Value of".
diminished, and ultimately be supposed to become
= 0, the fraction will also become = 0. Let the
fraction be
(Z - º
b
While b and a remain unvaried, let the value of a con-
tinually approach to equality with a , the fraction will
evidently be constantly diminished in value, and the
ultimate value when a = a is considered to be = 0.
Thus, when the numerator = 0, and the denominator
is finite, the fraction = 0.
If, however, the denominator = 0, the case is other-
wise. Let the fraction be
b
Cº. – º
While a and b remain unvaried, let a continually ap-
proach to equality with a The nearer a approaches to
a, the greater ratio will b bear to a - a, and, therefore,
the greater will be the value of the fraction.
proaches to equality with a, there is no limit to the
increase of the fraction. When a = a the fraction is,
therefore, considered infinitely great. -
Hence a fraction whose denominator = 0,
whose numerator is finite, is infinite.
4 B
and
As a ap-value of 4.
542
A L G E B R A.
Algebra.
Such a quantity is generally expressed by the symbol
-v- w; so that s = OO .
Infinity.
- 0
Value of; form To
If the numerator of a fraction be increased, the same
effect is produced upon its value as if its denominator
were proportionately diminished, and vice versä. Hence,
º ſº r t £a ye
*{{\{{#t ###8%jui s㺺d"thºdºši.
nator finite, the fraction is infinite. That is
OO
+ = 0 ,
and that when the denominator is infinite, and the
numerator finite, the fraction is - 0, or
b = 0.
CO
Strictly speaking, the symbols do and 0, in these
cases, should be considered as representing quantities
indefinitely increased, and indefinitely diminished; and
when they are called infinity and zero or nothing, it is
for brevity, and to avoid a circumlocutory description
of unlimited increase and diminution.
From the preceding observations, combined with the
principles established in Section VI., it follows, that
when an algebraical quantity becomes = 0, owing to
particular values or relations being assigned to the
letters of which it is composed, all its powers which
have positive exponents will be = 0, and those which
Powers of 0 have negative exponents become infinite.
Hence we
infer generally, that
0” -- 0, 0- " - op,
where m represents any positive number, integral or
fractional, rational or irrational.
In a fraction whose numerator and denominator are
algebraical quantities, it sometimes happens, that when
particular values or relations are assigned to the letters
of which these quantitities are composed, they will both
become = 0, so that the fraction will assume the form
0 .
0
To determine what the true value of the fraction is in
this case, we must examine under what circumstances
a quantity becomes = 0. If it be the product of any
number of factors, it will necessarily = 0, when any
factor having a positive exponent = 0; and if such a
factor occur with positive exponents in both numerator
and denominator, the fraction will necessarily have the
Let the fraction be #. A and B represent-
ing any algebraical quantities, which both become
= 0, when some particular values or relations are
ascribed to the letters of which they are composed.
Let the factor of A which = o be F, and let F" A" = A;
Aſ being not divisible by F, and m being a positive
number.
Also, let F" be a factor of B, and let F". B’ = B, B/
not being divisible by F; in other words, let F" and
F" be the highest powers of F which divide A and B.
Hence we have
A F". At
B T Fº. B/
If under the given conditions F = 0, the fraction
0
will become O but its true value will depend on the
relation between m and n.
1. If m = m '.'
- Of Frac-
A. rºl A! ‘tions.
T tºº Tſ’ \-,-
under which form both terms are finite, and the value
of the fraction is made evident.
2. If m > m,
A A /
*== = F' yyº - ". - .
B B!
Since F = 0, and (m. – m) is positive, '.' F *-* = 0,
f
... F" - " × Hence the fraction in this case
F O.
Tſ
tº O.
3. If m >< n '.'
and since n – m is positive, '.' F - ("-" = ap ...
A!
— (n - m) -
F * T
therefore, in this case infinite. *
Hence, in general, if an algebraical fraction assumes
= do. The value of the fraction is,
0
the form O’ by a factor common to both numerator
and denominator becoming = 0, the value of the
fraction is found by dividing both terms by the common
factor, if it have the same exponent in both. It is - 0
if the exponent be greater in the numerator than in the
denominator; and it is infinite, if it be greater in the
denominator than in the numerator.
(122.) If, however, the numerator and denominator
do not become = 0, by reason of a common factor
being = 0, but are absolutely each = 0, the value of
the fraction is indeterminate. The symbol 0 being
indicative of a quantity infinitely diminished, it will be
easily understood that a fraction whose numerator and
denominator are infinitely diminished, (except under
the circumstances already mentioned,) may have any
value whatever. Let + be any fraction, and let
a'
b/ be assumed, whose numerator and denomi-
nator are respectively the 10° parts of a and b. Then
- (7, f I
by (106) + = #. The same would be true if # Were
the 100”, or 1000%, or ten millionth parts of a and b.
In this way, both the numerator and denominator may
be infinitely diminished, and each tend to the limit 0,
and yet the value of the fraction will remain what it
originally was. And as its original value may have
another
0
been that of any number, it follows, that-i- may have
any value whatever.
The same may be illustrated geometrically: thus,
let parts A B, A C be taken on the legs of an angle,
B
C//
C/ C
and of such magnitudes that A B : A C : : a . b.
Q . o º
––, in this case the ratio of a to b
B º
A C T b
may have any value whatever; and, therefore, the
Hence


A L G E B R A.
543
Algebra.
Value of -:
Equation
defined.
Members.
Let the
base B C be moved parallel to itself towards the point
A, so as successively to assume the positions B’ Cº,
B/C/, B'C', &c. It is plain, that the ratios AB’:
A C/, A B* : A C", &c. remain the same ; and, there-
fore, r
. . C. .
fraction + may have any value whatever.
A B' – A B" . . a .
AC) - A C7, - ' ' - 5
and these ratios continue the same until the line BC
arrive at the point A, at which both terms of the frac-
g 0
tion become = 0, and it assumes the form "O’. Through-
out these changes it may have had any value whatever ;
and that value which it is supposed to have throughout
the changes, whatever it be, is its value when it be-
0
Algebraical fractions also sometimes assume the
form #-. This is always in consequence of a vanishing
factor with a negative exponent occurring in both
numerator and denominator. This may always be re-
duced to an equivalent fraction of the form TO’ by re-
moving the factor with the negative exponent from the
numerator to the denominator, or vice versä, changing
the sign of its exponent. This is equivalent to multi-
plying or dividing both terms of the fraction by the
same number.
SECTION X.
Of Equations.
(123.) WHEN a problem is to be resolved by Algebra,
the first step of the process is to translate its various
conditions from the ordinary language in which they
are usually announced, into the peculiar language of
Algebra. The result of this is always an equation, and
the resolution of this equation gives the solution of the
proposed problem. Let us suppose, for example, that
a certain number is required, such that if it be added
to a given number a, the result will be equal to double
another given number b. Now if the number sought be
called ar, when added to a the result would be a + ar,
and this by the proposed condition must be equal to 2 b,
that is, a + r = 2 b ; such is the proposed problem
when stated algebraically. -
An equation is, then, a proposition stating that the
result of certain operations performed on certain num-
bers, is equal to the result of other operations performed
on other numbers, the numbers, the operations, and
the equality being expressed by algebraical symbols.
(124.) Every equation consists, therefore, of two
parts, connected together by the sign =, the part to
the left of this sign being called the first member, and
the other the second member. Thus the first member in
the example already given is a + ar, and the second
7member is 2 b. -
(125.) Every statement of the equality of arith-
metical or algebraical quantities is not, however, called
an equation. The statements .
5 = 2 + 3
10 = 2 × 5
a — a st: 0
a – 2 b + 2 a = 3 a - 2 b |
and, in general, all equalities which are such that the Identities.
operations indicated by the signs can be performed,
and when performed render both members of the
equality identical, are called identities.
(126.) The degree of an equation is determined by Degree
the exponent of the highest power of the unknown
quantity which occurs in it. Thus, an equation in
which only the single dimension of the unknown quan-
tity occurs, is called an equation of the first degree.
Such is a + a = 2 b.
One in which the highest dimension is the square of
the unknown quantity, is called an equation of the
second degree, or quadratic. Such is
3 rº 5 + 2 a = 10 a 2.
The equation
10 a." – 2 a.” + 3 a 2- 10
is cubic, or of the third degree, and so on.
It should be observed, that in determining the degree
of an equation, it is supposed that no fractional power
of the unknown quantity occurs in it, or that the un-
known quantity is not contained under any radical, and
also that the unknown quantity does not occur in the
denominator of any fraction. A method will be here-
after explained, by which such equations may be con-
verted into equivalent ones, in which the unknown
quantity does not occur in this way. In fact, to deter-
mine the degree of an equation, it must be reduced to
a series of monomes, in each of which a power of the
unknown quantity occurs as a factor, the exponent of
which is neither negative nor fractional.
(127.) Equations, therefore, with relation to the Numerical
exponent of the unknown quantity, are classed in de- and literal.
grees. With respect to the nature of the coefficients of
the unknown quantities, they are divided into numerical
and literal. ... •
A numerical equation is one in which the coefficients
of the unknown quantity are all particular numbers.
Such are the equations
3 r + 4 + = 10
2 a. - 5 = 8.
A literal equation is one in which the coefficients of
the unknown quantity are expressed by letters, or by
letters and numbers combined. Such are
a *, +- a a = b
2 a. a + b = 3 c ar.
It will be observed, that, as applied to equations, the
term coefficient acquires an extended signification. In
this case it signifies the factor, whether literal or nume-
ral, or both, by which the power of the unknown
quantity which enters any term of the equation is
affected. Thus, the coefficients of
-g)”,
Ar", 10b r", (a + b) aº, 3 (A
are respectively
A, 10 b, a + b, 3 (A – c.).
Whenever the data of the problem are particular
numbers, the equation to which it is reduced will be
numerical. The problem in this case is always a par-
ticular one.
But if the problem be general, the data are expressed
by letters, and the equation is literal. -
(128.) The value of the unknown quantity in any
equation, whether numerical or literal, is such a num-
ber or letter, or combination of letters, as being sub-
- * 4 B 2 -
544
A L G E B R A.
Algebra. stituted for the unknown quantity would convert the
\–y— equation into an identity, (125.)
Roots.
(129.) A value of the unknown quantity, which thus
converts the equation into an identity, is said to satisfy
the equation, and such a value is called a root of the
equation. It will be seen hereafter, that an equation
may have more roots than one.
(130.) The root of an equation, or the value of an
unknown quantity, would be determined if we could
effect such changes as, without disturbing the equality
of its members, would disengage the unknown quan-
tity from those known quantities with which it is com-
bined, and so dispose the several quantities, that the
unknown quantity shall stand alone in the first member
of the equation, while the second member consists of
given quantities only combined by signs, indicating the
operations to be effected on them. The second mem-
ber will then be the value of the unknown quantity, or
the root of the equation.
(131.) In determining, therefore, the root of an
equation, it is of importance to be able to disengage
the unknown quantity from those known quantities
with which it may be combined; and the general
principle by which we are enabled to effect this is, that
“Any change may be made on the two members of an
equation which does not disturb their equality.” The
same change may always, therefore, be effected on the
two members of an equation.
Hence it follows, “That the same quantity or equal
quantities may be added to or subtracted from both
members of an equation.”
(132.) It follows from this, that any term may be
transferred from one member of an equation to the
other by changing its sign ; for this is equivalent to
adding that quantity with an opposite sign to both
members. Thus, if
a + a = b,
adding — a to both members,
a + a - a = b – a
‘. . a = b – a,
which is equivalent to transferring a to the second
member, changing its sign.
Again, if
a — a = b
a — a + a = b + a
‘. . a = b + a,
in which, as before, — a is transferred to the second
member, changing the sign.
(133.) The signs of all the terms of an equation
may be changed. For this is equivalent to transferring
all the terms of the first member to the second, and
vice versá, by (132;) it being evidently indifferent which
member is written first. w
(134.) Both members of an equation may be mul-
tiplied by the same quantity or equal quantities.
By this means, if the unknown quantity be divided
by any known quantity, whether simple or complex, it
may be disengaged from it by multiplying both mem-
bers of the equation by the divisor. Thus, if
a + b = c.
(1,
By multiplying both members by a, we obtain
- a + a b = a c.
(185.) Also, if the unknown quantity occur either
singly or in combination with known quantities as a
divisor, it may in like manner be disengaged by multi- Of Equa-
plying both members of the equation by such divisor.
This process is called “clearing the equation of STN-7
fractions.”
(136.) If several terms of an equation have different
denominators, the equation may be cleared of fractions
by multiplying both members by the least common
multiple of the denominators.
(137.) Both members of the equation may be
divided by the same quantity or equal quantities.
By these means, if the unknown quantity be affected
by a known quantity, or several known quantities, as
factors, it may be disengaged from them by dividing
both members of the equation by them. Thus, if
a w-H b = c,
by dividing both members by a we obtain
b "C
a + a T a "
(138.) Both members of an equation may be raised
to the same power, or the same root of them may be
extracted.
By this, when the unknown quantity, either singly or
in combination with known quantities, is raised to any
power, or affected by any radical, it may be disengaged.
(139.) In order to prepare an equation for solution,
it is necessary to reduce it to that state in which the
first member will be a series of monomes, each having
a power of ar, with a positive integer'as its exponent,
and the second member a known quantity or some
combination of known quantities. To this state every
algebraic equation may be reduced, by the several
means which have been just explained.
1. To clear the equation of fractions, find the least
common multiple of all the denominators which occur
in the equation, and multiply both members by this.
There will be no denominator, literal or numeral, in
the resulting equation.
2. Bring the radicals or terms affected by fractional
exponents, and involving the unknown quantity, succes-
sively to stand alone as one member, all the other
quantities being transferred to the other member, and
raise both members to that power expressed by the
exponent of the radical, or the denominator of the
fractional exponent. Each of these operations will
remove a radical, and by their successive application all
the radicals may be removed from the equation.
3. Reduce to a single term all the terms of which
the same power of the unknown quantity is a factor.
This may be done by enclosing all the coefficients of
such terms with their proper signs in a parenthesis,
incorporating by addition or subtraction such as admit
of it, and multiplying the whole parenthesis by the
power of the unknown quantity, which is the common
factor. Thus, if the several terms be
a w8 – b a' + 3 a.” – 5 a."
we have
(a – b -- 3 – 5) as
Or, (a – b – 2) a”.
4. These reductions being made, let the term in
which the highest power of the unknown quantity
occurs be placed first, and the others in the descending
order of their exponents; the terms which are indepen-
dent of the unknown quantity forming the second
member. The form to which an equation of the third
degree would be thus reduced, would be
A r" + Baº – C r = D.
A L G E B R A.
545
Algebra. , A, B, C being general representations of the coefficients,
S-v" and D of the quantities independent of w.
5. The equation may be still further simplified, by
dividing both members by any one of the coefficients.
That which is usually chosen is the coefficient of the
highest dimension. If this division were effected, an
equation of the fourth order would assume the form,
a *-i- a a " + b a' + c a' = d,
and in general an equation of the n” order would have
the form
a" + ar"---|- b a "-"+ c r"** + &c. - K,
K representing the terms which are independent of w.
(140.) We have already stated, that equations are
classed according to their degrees. It is evident that
by the process we have just explained, an equation of
the first degree would be reduced to the form a = K,
which, without further investigation, would give the
value of the unknown quantity.
We shall now proceed to the consideration of pro-
blems, the solutions of which depend on equations of
the first degree.
SECTION XI.
Of Equations of the First Degree including one unknown
quantity.
(141.) THE algebraical solution of a problem con-
sists of two very distinct parts. The first consists in
the translation of the conditions of the problem from
Examples.
the common popular language in which it is usually
proposed, into the peculiar analytical language of the
science. This is what is called “reducing the problem
to an equation.” The other part consists in discover-
ing the value of the unknown quantity from the equa-
tion, or “solving the equation.” No general rules can
be given for the reduction of a problem to an equation;
experience alone, and the study of a number of well-
selected examples, will attain this end. The following
directions will be found, however, of considerable use :
“Let the problem be considered as having been already
solved, and the known quantities being represented
either by particular numbers or by letters, and the
unknown quantity always by a letter; indicate by
algebraic signs the various relations and operations to
which these quantities would be submitted, were the
unknown quantities known.” The result of such a
process generally gives two different systems of opera-
tions on the data of the problem, and the unknown
quantity, by which some one quantity may be obtained,
and the two algebraical expressions of the results of
these operations, in general, furnish the two members
of the primary equation.
(142.) We shall now proceed to give a few examples
of the investigation of problems which are reduced to
equations of the first degree; offering such general ob-
Šervations as the peculiar circumstances of each problem
may suggest. -
(143.) A for is started at sirty of his own paces
from a hound, nine of his paces being made in the same
time as six of the hound, but three paces of the hound
being equal to seven of the foa. It is required to deter-
mine how many paces the hound will have made when
he shall have overtaken the for ?
Let H be the length of each pace of the hound.
Simple
Since three Jf the hound’s paces are equal to seven of Equations.
the fox's, if 3 H be divided by seven, the result is the S-a-
length of one pace of the fox, which is, therefore,
3 H e de
-H-. At setting out, the fox is sixty of his own
paces distant from the hound. Hence this distance is
3 H 180 H.
60 – = º
x -: 7
Let the distance sought be a, that is, the number of
paces the hound has made at the moment he overtakes
the fox. The distance the fox will, therefore, have run
will be
, 180 H
7 2
that is, the distance gone over by the hound, diminished
by the distance between them at the moment of depar-
0
The spaces a and a — 180 H
ture. being run over
in the same time, must be in the same ratio as the
speed of the two animals. It is granted that the fox
makes nine paces while the hound makes six, or, what
is the same, the fox makes three while the hound makes
tº 3 H tº ; e. - &
two. Thus, three times ~7. which is the fox's pace, is
made in the same time as 2 H. Hence, the spaces the
º 9 H
animals move through in the same time are as +:F: 2 H,
or as 9 : 14. Hence we have
180 H.
* —
7 9
tº T 14 °
Which, being cleared of fractions, becomes
14 a - 360 H = 9 a.
*...* 14 a - 9 a. = 360 H
•. • 5 a - 360 H
• a = 72 H.
The hound will, therefore, have made 72 paces when
he shall have overtaken the fox.
(144.) To divide a line of 15 inches length into two
such parts that one of them shall be three-fourths of the
other.
In this case, if one of the parts be called ar, the other
will be 15 — v. The number represented by a is here
understood to express inches. Now, by the conditions
of the question, one of the parts is three-fourths of the
3 -
other. Three-fourths of a is expressed * ; now this
and the other part 15 — a must be equal. Thus we
have the equation
3 ºr
15 — a = T4 .
It may be useful to the student to compare this pro-
cess of reduction with the observations in (141.)
Clearing this equation of fractions by multiplying
both members by 4, we obtain 60 – 4 a. = 3 a. Trans-
ferring – 4 a to the second member, changing the sign
60 = 7 a., or 7 a = 60. Dividing both members by
the coefficient 7, --
= + = 8
JC 7 4
546 A L G E B R A.
Algebra. This in inches is the length of one part, and since the – r = 15– 60 105 – 60 45 Simple
S-N- whole line is 15 inches, the other part must be 64 (Z == -7 – 7- - 7 Equations,
inches. which are equivalent to the results first obtained. S-N--
(145.) In this instance the question is particular,
and the equation numerical. It would, perhaps, be
better in every case where a particular problem is pro-
posed, to generalize it in the first instance. The re-
sult will then be a literal equation, which, when solved,
will give a general formula, by which not only the pro-
posed question may be solved, but also every question
of the same class. The preceding problem generalized
would be as follows: -
(146.) To divide a given line a into two such parts
that one shall be m times the other, (m being any
number, integral or fractional.) Af
The statement would now be thus: Let one part be
ac, and the other must be a - a. By the conditions of
the problem, a - a. and m a must be equal. Hence
7m a = a – a. Transposing — ac, and changing its
sign, we have ma' -- a = a. Collecting within a paren-
thesis the coefficients of ac, we have (m. -- 1) a = a.
Dividing by (m + 1), we obtain
(Z
a: t: — ,
m + 1 -
which is one of the parts. The other part will be
a — ar. Hence
a – a = a (Z
-- m + 1
The second member of this equation may be con-
sidered us a mixed number, and, therefore, the first
part a is to be multiplied by m + 1, and a subducted
from the result. The process will be understood from
the following steps:
_a (m+ 12– a
T m + 1 m + 1"
a (m. -- 1) — a
*mºms -------- ºt-sº-sº-ºse
m + 1 '
Q —- Jº
º
*
ſº (Z *º-º-º: 3C
gº 771 (ſ.
T m + 1
Hence we obtain the following general rule for the
solution of all such questions. To find one part, divide
the proposed line by the number which is given, ex-
pressing the proportion of the parts increased by unity,
and the quote is one part. Multiply this quote by the
same number, and the product is the other part.
It is, however, worse than useless to translate into
popular language thus, the formulae derived from
general algebraical investigation ; they are clearer and
more compendious, and much more easily retained in
the memory, wherever it is necessary to do so, when
expressed in their algebraical form. We have in the
present instance reduced the result to ordinary lan-
guage, only to show that this result is really a general
theorem or rule, and not merely the solution of a par-
ticular question or problem. In the particular instance
given, at first we have a = 15 and m =+ Hence
7
m -- 1 = T. Hence we have
- 7 4 × 15 60
ſc = 15 -- — = —— = ~~~
QC 4 7 7
obtain
(147.) A labourer is engaged for 48 days on these
conditions : for each day he works he is paid two shil-
lings, but forfeits one shilling for every idle day; at the
end of the 48 days he is entitled, under the terms of the
agreement, to 21 shillings : it is required to calculate the
number of days he worked, and the number he was idle 2
By the conditions of the problem, if the number of
days on which he worked were multiplied by 2, we
should have the wages of the entire of these days. The
number of days on which he was idle will express the
number of shillings which he forfeited. The latter
subtracted from the former will leave a remainder equal
to the sum to which he is entitled at the conclusion
of the stipulated period. This sum is, however, given
to be 21 shillings. If, therefore, an algebraical for-
mulae be adapted to represent the result of the several
operations above mentioned, and be taken as the first
member of the equation, 21 shillings will be the second
member.
Instead, however, of stating the question in the first
instance as a particular one, we shall generalize it.
Let a be the number of days for which the labourer
is engaged.
Let a be the total number of working days, and,
therefore, a - a the number of idle ones. -
Let m be the number of shillings he is paid for each
working day, and n the number which he forfeits for
each idle day.
Let S be the whole sum to which he is entitled at the
end of the period by the terms of the agreement.
The total number of shillings earned on the a work-
ing days will be ma, and the total number forfeited on
the a - a idle days will be m (a — wy.
Hence, the total sum to which he will be entitled at
the conclusion of the period a, will be m z – m (a - a).
But this, by the conditions of the question, is granted
to be equal to S. Hence we obtain the following
equation,
ma – m (a — ar) = S
Or, m a - m a + n a = S. -
Collecting within a parenthesis the coefficients of r, we
(m-H m) a - m a = S.
(m+n) r = S + m a.
Dividing both members by (m. -- m)
_ S + n a
T m + m
which gives the number of working days.
To determine the number of idle days, we have
S + m a
Transposing na,
- T m + n
which is the number of idle days.
In the particular question proposed, we have a = 48,
m = 2, n = 1, and S = 21. Hence
= ** f * = * = 28
2 + 1 .. 3 t
A L G E B R A.
547
Thus the number of working days was 23, and the idle ones
25, amounting together to the whole period of 48 days.
(148.) It is evident that the general values of a and
a – a would solve the problem with equal facility had
any other rate of payment, or any other period, been
made the subject of a similar agreement. In fact, the
result of the general algebraical investigation is not so
much an absolute solution of the problem, as an indi-
cation of a method by which similar problems may
always be solved.
If the particular numbers represented by a, m and S
be such that the product a m is less than S, it is evi-
dent that the formula expressing the number of idle days
will represent a negative number. A question then
arises, what is meant by the labourer having worked a
negative number of days?
To explain this, we must refer to the meaning of the
symbols. m. is the number of shillings paid to the
labourer for each day that he works. a is the total
period agreed upon. m. a. is, therefore, the sum which
he should receive if he worked every day of the entire
period, and spent no day idle. But, under the circum-
stances which we have supposed, he becomes entitled
to a sum S greater than the sum ma, to which he
would have been entitled had he worked every day of
the stipulated period. The inference is, that instead of
being idle on any of the stipulated days, he must have
worked as many additional days as would entitle him
to that sum by which S exceeds m a. Consequently,
the formula for a – ar, which, when positive, signifies
the excess of the stipulated period over the working
days, signifies, when it becomes negative, the excess of
the working days above the stipulated period. Such a
result as a negative number of days, considered merely
by itself, is unmeaning, but when the circumstances
which led to that result are examined, it leads to a
modification of the original question. It shows that
the conditions proposed are inconsistent with the data,
and it indicates, that to render them consistent, either
the data or the conditions must be modified ; and, fur-
ther, it points out what the necessary modifications are.
In the present instance we find that the sum S, to
which it is asserted, in the original question, that the
labourer is entitled at the end of the stipulated time,
is greater than he could have made in that time with-
out any idle days at all ; and, therefore, that if the
question be modified, and rendered consistent by
changing the data, it will be necessary to regulate the
numbers represented by a, m, and S, so that S shall
not exceed a m, which may evidently be effected by
increasing a or m, or both, or by diminishing S, or by
all these changes combined.
If, however, it be desired to modify the conditions of
the original question, so as to render them consistent
with the data, we must examine the original state-
ment. This is ma – m (a — a y = S. Now if a - a be
negative, as is supposed in the present case, that is, if
a > a the quantity – m (a – ar) is positive, and the equa-
tion being written thus, m a + n (a — a) = S, expresses
that the total sum S receivable by the labourer is com-
posed of m shillings for each of the a working days, to-
gether with n additional shillings for each of the
(a — a) days which he works over and above the sti-
pulated period of a days. Thus the n shillings, which
in the case of idle days was a forfeit, becomes a pre-
mium in the case of supernumerary working days.
The question, therefore, will be thus modified :
A labourer is engaged for a days at m shillings per
day, on condition that he shall forfeit n shillings per
day for as many days as his number of working days
shall fall short of the stipulated period a, and that, in
addition to m shillings per day, he shall receive a pre-
mium of n shillings a day for as many days as his
working days shall exceed the stipulated period a. At
the cessation of his labour he becomes entitled, under the
terms of the agreement, to a sum of S shillings. It is
required to assign the number of working days, and to
determine the number of idle or supernumerary working
days, as the case may be.
(149.) A further advantage which general alge-
braical investigations possess over particular numerical
questions is, that the same general formula may be the
means of solving other problems, besides even the ge-
neral one from which it results. In the problem just
investigated, the formula
a m + S
7m -H ºn
expresses in general a relation between the numbers
represented by ar, a, m, n, and S. Now if any one of
these five quantities be unknown, and all the others
known, the value of the unknown quantity may always
be determined.
Let us suppose, for example, that S is the unknown
quantity; the question will then be, to determine the
sum to which the labourer will be entitled at the cessa-
tion of his labour, the number of working days ar, the
daily wages m, the forfeit or premium n, and the stipu-
lated period a, being all given. To solve this problem,
it is only necessary to consider S as the unknown
quantity, and solve the equation for it. Multiplying
both members by m + n we have
(m + n) a = a m + S,
and transposing a m we have
(m. -- m) a - a m = S,
Or, S = (m + n) a - a n.
This gives the sum to which the labourer is entitled.
In the first example, m = 2, n = 1, a = 48, and
a = 23 ; hence
S = (2 + 1) . 23 – 48.
* ... • S – 21.
In this case, also, it might so happen, that the particu-
lar values assigned by the data to the quantities r, m,
m, and a, would render the value of S megative. Let
us consider the meaning of such a result.
By the equation
- S = (m. -- m) a - a m
Or, S = m, a - (a — a m,
it appears, that if S be negative we must have a - r
positive, or a > a., and m a < (a – a n, that is, the
number of working days a is less than the stipulated
period a, and the entire wages m r of the working days
is less than the sum (a — a n forfeited for the idle
days. Hence, on the whole, the labourer is a loser by
the excess of the sum forfeited (a — a n over the
wages m a., that is, by the positive value of the negative
result S.
Thus it appears, that the sum supposed in the state-
ment to be gained by the labourer becoming negative
in the result, proves that this sum is not gained, but
lost. The problem should therefore be modified, so
= 69 – 48
548 A L G E B R A.
Algebra. as that the required quantity would be the balance for
seology is, however, to be considered rather conven- simple
S-N- or against the labourer on closing the account.
tional, and derived, by analogy, from the effects of Equations.
(150.) These observations lead us to the considera-
tion of the nature of negative quantities. When posi-
tive and negative quantities are considered merely as
members of polynomes, and therefore connected by
their proper signs with other quantities, their meaning
is obvious, and they might more properly be called
additive and subtractive quantities; as has been
already explained. But we have seen that a negative
quantity is frequently the result of a calculation, and,
therefore, not considered as a member of a polynome.
What then, it may be asked, can be its meaning in
this case ?
The most simple process from which a negative quan-
tity can result is subtraction. Let the problem pro-
posed be to find a number, which, when added to a
given number b, will produce a given sum a. Thus, if
a be the number, we have
b -- a = a
*...* a = a – b.
If we suppose a = 30 and b = 20, we have
• a = 30 — 20 = 10 ;
in this case the result is positive, and is the true solu-
tion of the problem proposed. But suppose that
a = 20 and b = 30, we should have
a = 20 — 30.
Putting this expression under the form
a – 20 — 20 — 10
we have 20 — 20 = 0 °."
a = — 10,
a negative solution.
To explain the meaning of this, let us recur to the
original statement,
b -- a = a
Or, 30 + a = 20,
which expressed in ordinary language is, “To deter-
mine the number which, added to thirty, will produce a
sum equal to twenty;” a problem manifestly impos-
sible, twenty being less than thirty.
But now let us replace a by the value which the
algebraical process gives for it, and the statement
becomes 30 — 10 = 20. So that the absolute or
arithmetical value of the result obtained is a number
which, subtracted from thirty, will give a remainder
equal to twenty. -
If the original problem be considered arithmetically,
the negative solution indicates an inconsistency be-
tween the data and the conditions, and the necessity of
a modification of one or both. But if it be considered
algebraically, no such inconsistency exists; because
here the term addition is taken in a larger sense, and
includes the addition of negative quantities, which is
arithmetical subtraction.
To determine the modification which is necessary to
remove the inconsistency of a problem which gives a
negative solution, it is only necessary to change the
sign of a in the equation to which this problem is re-
duced, and then to translate the new equation into
ordinary language. The necessity of employing nega-
tive quantities in algebraic investigations, has intro-
duced a phraseology respecting them which, under-
stood literally, seems absurd. A negative quantity as
— a is said to be less than nothing ; and one negative
quantity - a being numerically greater than another
— b, is said to be less than it. Thus — 1 is said to
be less than 0, and – 3 less than — 2. This phra-
arithmetical operations on positive and absolute num-
bers. It has, however, been necessary to adopt it in
Algebra, in order to generalize the investigations and
their results. - -
It is a general principle, that when one absolute
quantity is subtracted from another, that other is dimi-
nished by the operation. Thus the operations repre-
sented by 5 – 1, 5 — 2, 5 – 3, &c. have the effect of
producing a constant diminution of the number 5.
Now let this process be continued, the successive re-
sults are 5 — 4, 5 – 5, 5 – 6, 5 — 7, 5 – 8, &c. In
an arithmetical view, all the operations represented
here after 5 – 5 cannot be performed. But in Algebra
it is necessary to perform them as far as can be done,
and to represent by a certain symbol that part which
cannot. Thus six units cannot be taken from five
units; but five of the six can, and the remaining unit
which cannot is represented by placing the negative
sign before it thus, – 1. In the same manner, 5 — 7,
5 – 8, &c. are represented by – 2, — 3, &c. Now as
in absolute numbers the remainder diminishes as the
subtrahend increases, the same property is extended
analogically to those imaginary remainders which are
the results of subtractions which cannot be executed ;
and we consider 5 – 5 to be greater than 5 – 6, and
5 — 7 greater than 5 – 8, &c.; that is, 0 is greater
than – 1, and – 2 greater than – 3, and so on.
This phraseology is not so inconsistent with the
language used in the most ordinary affairs of life as it
may at first appear. If we estimate the property of
any individual, we first compute his actual possessions
and the debts due to him ; from these we subtract the
debts which he owes, and the remainder may be con-
sidered as the value of his property. Now if it so
happen, that the amount of his debts exceed the
amount of his possessions, and the debts owing to him,
we say that he is worth less than nothing. In this case,
the result of the above-mentioned subtraction would be
a negative quantity, and one of precisely that amount
by which, in popular language, the individual in question
is said to be poorer than he who neither has, nor owes
a shilling.
In like manner, if the debts of A exceed his effects
by a, and the debts of B exceed his effects by a + b,
we say that A is richer or less poor than B. Now, in
this case, the results obtained by subtracting the debts
from the value of the effects in both cases are negative;
but the value in the case of A is numerically less than
in the case of B, although A is said to be more wealthy
than B.
From these considerations we derive a method of
expressing algebraically, that a quantity as a is posi-
tive or negative. If we wish to express that a is
positive, we write a > 0, and if it be negative, we
write a < 0.
SECTION XII.
Of Equations of the First Degree containing two or more
wnknown quantities.
(151.) IN some of the examples given in the last
section, more than one quantity was unknown, but in
A L G E B R A.
549
Algebra. all the instances which occurred, there was such an
S-N---' obvious connection between the unknown quantities,
that one unknown symbol signifying one of them, was
by proper combination with the data made to express
the other unknown quantities. Thus, in the problem
(144,) one part of the line being r, it is known that
the other part, which & priori may be considered
equally unknown, is 15 — a. But if this problem
were at once treated as one involving two unknown
quantities, we should consider the two parts as cha-
racterised by a and y, and we should have the equation
l a + y = 15
by one condition, and
y = + ,
by the other. -
The examination of the following problem will lead
us to the general principles by which questions involv-
ing two unknown quantities may be solved.
(152.) Given the sum (a) and the difference (b)
of two numbers, to find the numbers themselves.
Let a and y be the numbers, we have, by the condi-
tions of the problem, a + y = a, a - y = b. Since
equal quantities added to equal quantities give equal
results, we obtain, by adding these equations,
2 a = a + b,
an equation which is independent of the unknown
quantity y. This being divided by two, gives
H -
* = g (a + b).
In like manner, subtracting the one from the other, we
l
obtain 2 y = a – b, ..." v = T2T (a — b), and thus the
values of the two unknown quantities are found, and
we have established the following theorem.
“Of two unequal quantities the greater is equal to
half the sum of their sum and difference, and the less
is equal to half the difference of the sum and differ-
ence.”
Upon examining the preceding process it will be
found, that the contrivance by which the values of the
unknown quantities have been determined, has been
that of obtaining from the two given equations, each
containing two unknown quantities, a single equation
containing but one unknown quantity, and from this
equation obtaining the value of that. This, being
done with respect to each of the unknown quantities,
will determine their values.
(153.) By generalizing the results, we shall obtain
methods of solving all questions where two equations
containing two unknown quantities are given.
After the proposed equations are cleared of fractions
and radicals, as they cannot include any powers of the
unknown quantities higher than the simple dimensions,
they must have the forms
a a + b y = c l
a' a' + b/gy = c' [1]
a, b, c, a', b', c', being general representatives of any
numbers positive or negative, which may happen to
be the results of the reduction of the equations by the
process of clearing them of fractions and radicals, or
powers of the unknown quantities with fractional
exponents. It should, perhaps, be here observed, that
WOL. I.
if an equation of two unknown quantities contain a Simple
term of which the product (ry) of the unknown quan- *.
tities is a factor, it is accounted an equation of the STNT
second degree ; since, although it contains no term in
which the second power of either unknown quantity
occurs as a factor, yet it contains a term in which the
unknown quantities combined occur in two dimensions.
The process by which a single equation [1, contain- Elimination
ing only one unknown quantity, is obtained from the .º.
two equations, is called elimination ; and the unknown methods.
quantity which is made to disappear, is said to be eli-
minated. There are three methods by which this end
is attained.
1. The first is the method of addition or subtraction. Method of
This method consists in equalizing the coefficients of addition or
the same unknown quantity in the two equations, by *
multiplying both members of each by such a number
as will render the coefficients of the same unknown
quantity in each equal. This is done on the same
principle as that by which fractions are reduced to a
common denominator. Let the least common multi-
ple of the coefficients of the same unknown quantity
be found, and let this be divided by the coefficient of
that unknown quantity in each equation ; the quotes
will be the numbers by which it will be necessary to
multiply the two equations in order to equalize the
coefficients. Thus, if the equations be those of [1] the
least common multiple of the coefficients of y is a aſ:
consequently the multipliers sought are a' and a, and
when these are respectively multiplied into the two
members of each equation we obtain
a aſ a + b a' y = c a' 2
a a'a + b' a y = c' a [2]
in which a has the same coefficient.
Again, if the equation be
6 a. -- 8 y = 50
8 a -i- 6 y = 48.
The least common multiple of 6 and 8 is 24, which
divided by 6 and 8 gives 4 and 3. These, being mul-
tiplied by both members of each equation, give
24a –– 32 y = 200
24a –– 18 y = 144,
in which a has the same coefficient.
(154.) The same unknown quantity being by these
means reduced to the same coefficient in both equa-
tions, the next step of the process is to subtract the
one equation from the other, if this common term have
the same sign in both, and to add them together if the
common term have a different sign in the one and the
other. In either case, the result of the process will be
an equation containing but one unknown quantity.
In the first case, the two equations will be of the
form [2,] which, being subtracted, the latter from the
former, give (b a' — bºa) y = (c a' – c’a). [3]
In the second case, the equations will be of the form
a a/a, + b a' y = ca'
— a a'a + b a y = c'a,
which, being added, give
(b a' + b' a) y = (ca'+ c a). [4.]
In every case, therefore, in which the coefficients of
the same unknown quantity have been equalized, that
unknown quantity may be eliminated by addition or
subtraction, and an equation obtained, including only
4 c
550 A L G E B R A.
Algebra, the remaining unknown quantity, the value of which (b'a – aſ b) y = (c a' — aſ c) E.
S-v- may be found by the methods explained in the last * tº g quallons.
j. y P* * * * which is the same as [3, and if aſ were negative would S-N-"
Method of 2. The second method of elimination is called the be he sº [*].
*P* method of comparison, which consists in bringing the (155.) All equations whatever of the first degree
same unknown quantity to stand alone as the first between two unknown quantities can be reduced to the
member of each equation ; and thus the second member forms
of each equation would include only the remaining a r + b y = c [1..]
unknown quantity. These second members being a'a -- b' y = c' $
il l, since the first ber is common, e º e
... a. '. tWO j al, º: Hence it follows, that the solution of the equations
tion, which will therefore contain but one unknown [A] ". furnish general formulae by which the values of
quantity, and therefore the other unknown quantity is the unknown quantities in any given equations of the
by these means eliminated. first degree may be computed. By the investigations
Thus, in the equations [1,1 the first being divided by already given, it appears that the values of a and y,
a, and the second by a', we have derivable from the equations [1, are
b c in — ca' – c'a
* ++ y = + | 9 = 7.7-7,
b' c' > ! ... . of
(l, a' J a b' - a' b
* = + --" 3/ n By substituting in these formulae the particular values
Q, a 9 | of a, b, c, a', b', and c' in any proposed equations, the
c' b' values of the unknown quantities may be at once ob-
=7 – 27 y tained without further investigation.
(156.) The following example will illustrate these Examples.
The second members of this latter system being principles:
assumed as the two members of the same equation, Two couriers depart in the same direction from two
give places on the same road, the distance between which is
C b c' b' a , one A goes m miles, and the other, B, n miles per
I - a y = 7 – 7 y hour. It is required to determine at what distances from
º g e the points of departure the one will overtake the other.
which, being cleared of fractions, becomes Let a, and y be the two distances. As these dis-
ca' – b a' y = c'a - b' a y, tances are travelled in the same time, we have
and the known and unknown quantities being brought 7. a = 7m gy,
to opposite sides, we have and also
b' a y – b a' y = c'a - ca', a – 3) = d.
OI (a b' — a' b) y = c a' – c'a, Hence, by elimination, we obtain
which is the same with [3, and would, if the sign of a/ tº - 67, 777, 3/ = 0, 70, ſº
were negative, be the same as [4.] Thus these two 777, - 71, 7?? – ??,
methods lead precisely to the same results. If m × m, and therefore m — n < 0, these values for
Method of 3. The third method of elimination is called the w and y will be negative. This indicates that the
substitution
method of substitution, and in principle is the same as
the method of comparison, differing from it only in
appearance. The method of substitution consists in
bringing one of the unknown quantities in one of the
equations to stand alone as its first member. The second
member will, therefore, include only the other unknown
quantity. This member is then substituted in place of
the other unknown quantity in the second equation ;
by which substitution the second equation will contain
only one unknown quantity, and therefore the elimina-
tion will be effected.
To apply this to the equations [1, we have by the
first -
a t — — — 7/.
(l, a 9
The second member being substituted for a in the
second equation, it becomes
y) + b y = 0,
which, being cleared of fractions and reduced, becomes
C b
0. sºms º ºsmºsºms
a . (l,
courier A can never overtake the courier B in the pro-
posed direction, but that if they travel in the opposite
direction, the courier B will overtake the courier A at
the distances indicated by the values of a and y deter-
mined above.
(157.) It might happen that the values obtained for
the unknown quantities from two given equations
would be fractions, whose denominators are E 0. In
this case the roots are said to be infinite, (121.) But
the origin of such a result is always an absurdity or
inconsistency in the two given equations. It will be
easy to show this by the general formulae [1..]
The condition under which the values of w and y
derived from these equations are infinite, is
a b' — a' b = 0.
This gives -
f
b b [2.]
Now if both members of the first of the equations [1]
be divided by a, and of the second by a', they become
A L G E B R A.
551
Algebra.
b C -
•++ y = }
f f [3.]
•+% – “
27 y = 7
By the condition [2,] the first members of these equa-
tions are equal, whatever values be ascribed to a and y ;
and, therefore, unless the data be so related that the
second members are also equal, the equations are in-
consistent and contradictory.
In the same example, if m = n the results will be
infinite. In this case, the rates of travelling of the
two couriers would be the same, and consequently the
one would never overtake the other, and the con-
dition of the question would be inconsistent with the
data.
There are instances, however, in which these infinite
results do contain the true solution of the problem.
The student will find them occur frequently in our
Treatise on ANALYTIC GEoMETRY.
(158.) If the second members of these equations
were equal, as well as the first, it is evident that the
two equations would be identical. The conditions
under which this would take place would then be
b b/ c c'
— — —7,
(7, (!
from which we infer
a b'—a' b = 0, c a' – c' a = 0.
Also, by these last equations, we obtain
<=- Tº a
ſº Q,
(Ž b (l, C
a' b' ' aſ T c!’
b C
7 – 7 ‘. . c b' – c' b = 0.
It therefore follows, that under these circumstances the
()
values of a and y would assume the form -.
In this case there would be in effect but one equation
for the determination of two unknown quantities, and
the data would then be evidently insufficient for the
solution of the problem. This will be easily perceived
by substituting particular numbers for the general sym-
bols. Let the equation be
2 y -- 3 a = 50.
In this equation, any number whatever being substituted
for a, a corresponding number may always be deter-
mined, which substituted for y will satisfy the equation.
Inet y be brought to stand alone as the first member,
and we obtain
gy = 50 — # J'.
Now suppose a = 2 *.*
y = 25 – 3 = 22.
These two values, 22 and 2, being substituted for y
and a in the proposed equation, it becomes
44 -- 6 = 50
which is an identity.
Again, let any other value be substituted for a, as 5,
we find
3 a 50 – 15 35
= 25 — — . 5 = s— --
y = 25 – ; 2 2 ”
35
these two values 2. and 5 being substituted for y and
a in the original equation, give 35 + 15 = 50, which
is an identity.
In like manner, any other value being ascribed to a,
a corresponding value of y would be found, which
would satisfy the equation.
To generalize this principle, in the equation
a a + b y = C,
let any value y' be ascribed to y, so that the equation
becomes
a + b y' = c
• • c — by’
. “ — a
Substituting this value of y for y in the first equation,
it becomes
c — b y' f
a. —- + by = C,
Or c – b y' + b y' = c,
Or C F C,
which is an identity.
Thus it appears, that there may be an infinite num-
ber of systems of values of two unknown quantities,
each of which will equally satisfy the proposed equa-
tion, which, therefore, leaves the values of the unknown
quantities indeterminate.
It may, therefore, be assumed generally, that when
the values of the unknown quantities which result from
two equations assume the form To the two equations
differ only in appearance, but are really one and the
same, at least they are such that one may be inferred
from the other. In this case, therefore, there is but
one equation in reality between the two unknown
quantities, and their values are indeterminate.
In the example (156) if a = 0 and m = n, the
- 0
values of a and y would assume the form TO and
the problem would be indeterminate. In this case the
distance between the places of departure being a = 0,
they would necessarily be the same. Also, m and m
being equal, the couriers would travel at the same rate,
and since they are supposed to move in the same direction
they would necessarily keep always together. Hence, as
the object of the problem is to assign the place at
which they will be found together, every part of their
road has in this case an equal claim to be considered
as the point required. Hence the indeterminateness
0
indicated by the form of the roots TOT.
There is, however, an exception to this principle; for
it might so happen that the root assumed the form O’
from having in both its numerator and denominator a
common factor of the form a - a. The true value of
the root would then be found by dividing both nume-
rator and denominator by this common factor, (121.)
(159.) In order to determine the values of two un-
known quantities, it is therefore necessary that there
should be two independent equations between them ;
that is, two equations such that one cannot be inferred
from the other.
(160.) Three or more independent equations would
be more than sufficient data for the determination of
two unknown quantities, and the result would be, that
different and inconsistent values of the same unknown
Simple
Equations
\-y-
Two inde-
pendent
equations
necessary.
4 c 2
552
A. L G E B R A.
Algebra.
S-y-
The num-
ber of equa-
tions should
be equal to
that of the
unknown
quantities.
Consequen-
ces if not.
Examples.
quantity would be obtained from each pair of equa-
tion.S. -
(161.) It might happen that the values obtained for
the unknown quantities would be = 0. To determine
the circumstances under which this could happen, it is
only necessary to consider under what circumstances
the formulae
cb' – b c' ... a c' – c a'
9 = 77. T., a b' – b a'
shall become = 0.
necessary that
cb' – b c = 0 a c' – c a' = 0,
but that a b' – b a' should not = 0, because in that
That this should happen, it is
O
case the values of a and y would assume the form ––.
Let c be eliminated by the preceding equations, and
the result is
o!
77 (a b' *- ba') - 0.
Now since a b' – b bº cannot = 0, we must have c' = 0,
and in like manner it can be proved that c = 0. Hence
the form of the equations must be
a 4 + b y = 0
a'a' + b'3) = 0.
(162.) The principle by which one unknown quan-
tity is eliminated by two equations may be generalized.
If several equations of the first degree be given, in-
cluding several unknown quantities, any one of these
unknown quantities may be made to stand as the first
member of any one of the equations,—the other terms
being all transferred to the other member, by the methods
already explained. The second member may then be
substituted for the unknown quantity which stands
alone in the first member, in all the other equations.
One equation, therefore, has served to eliminate one
unknown quantity from all the other equations, and the
number of equations, as well as that of unknown
quantities, is thus diminished by one. The same pro-
cess may be repeated with another equation and another
unknown quantity, and the number of equations and
of unknown quantities will then be diminished by two;
and so the process may be continued. If the number
of unknown quantities be equal to the number of inde-
pendent equations, it is clear that by eliminating all
the unknown quantities but one, we shall also have
reduced the number of equations to a single one. This
single equation will determine the value of the remain-
ing unknown quantity. But if the number of equa-
tions were less than the number of unknown quantities,
after reducing the number of equations to a single one,
the number of unknown quantities remaining in it
would be two or more, and it would therefore be in-
sufficient to determine their values, and the problem
would be indeterminate. But, on the other hand, if
the number of unknown quantities be less than the
number of equations, after reducing their number by
elimination to one, more than one equation would re-
main, and the results would be contradictory if the
the given equations were independent.
(163.) We shall give one or two examples:
1. How many times do the hands of a watch coin-
cide between moon and midnight, on the supposition that
there is only an hour hand and a minute hand ; and
what are the exact moments of their coincidence. Also,
what would be the number of coincidences of three hands
moving on the same centre, an hour, minute, and second
hand, and what would be the exact moment of their
coincidence 2
2. A number is composed of three digits, of which the
sum is given. The digit in the unit's place is m times
that in the hundred's place ; and on adding a given
number consisting of three digits to the sought number,
the digits will be reversed. Investigate a general for
mula for the solution of this class of problem, and
apply it to the case where the sum of the digits is 11,
m = 2, and where the number added is 297.
3. A sum of £100,000. is placed at interest, one part
at 5 per cent., another at 4 per cent. The total interest
is £4640. ; it is required to assign the proportions which
are placed at each rate.
4. Three persons, A, B, C, have certain sums which
they place at interest. B and C have each given num-
bers of pounds more than A. The rates of interest of B
and C exceed that of A by given sums; and also the
revenues of B and C exceed that of A by given sums.
It is required to determine the capitals of A, B, and C,
and also the rates of interest they respectively receive.
Simple
Rquations.
(164.) We have already proved that all equations of Three
the first degree between two unknown quantities may
be reduced to the general forms, (153,)
a a + b y = c
a'a + b'y = c'.
The same reasoning by which this was established will
likewise prove the equations between three unknown
quantities may each be reduced to the form
a a + b y + c 2 = d,
and as in every determinate problem there must be
three of these equations, they may be represented
thus:
a r + b y + c 2 = d
a'a + b' y + c 2 = d'
a"a + b" ºy + c" z = d".
It is evident how these observations may be extended
to any number of equations between the same number
of unknown quantities.
It should be observed, that it is by no means
necessary that all the unknown quantities engaged in
the problem should occur in each equation ; and al-
though they appear to do so in the above general for-
mulae, yet, as it is supposed that any one or more of
the general coefficients a', a!... b', b. . &c. may be = 0,
they are not so restricted. These general coefficients
are, in fact, the aggregates of the coefficients of each
unknown quantity, in any particular question, after the
equations have been cleared of fractions and reduced,
as explained in (139.) g
(165.) Rules may be assigned and established by
which, when any number of equations of the first de-
gree between the same number of unknown quantities
are given, the values of these unknown quantities may
be severally obtained without the usual process of
elimination, or any other preparatory investigation.
If there be but one unknown quantity, the equation
may always be reduced to the form
a a = b,
a being the algebraic sum of all the coefficients of the
unknown quantity, and b the algebraic sum of the
unknown
quantities
A L G E B R A.
553
Algebra. terms which have no unknown factor.
S-' formula for a in this case is obviously
The general
b
tº re-
\ Q, ,
We have already shown that when there are two equa-
tions with two unknown quantities, the formula for
their values are -
c b' – b c' a c' – c aſ
* = &ºmº
T a 5'- b a' ' T a b' – b a' '
The rule by which these formulae may always be
found is as follows:
1. They have a common denominator. With the
letters a and b, which express the coefficients of a and
gy, form the two arrangements a b and b a, and place
between them the sign –, and place an accent on the
last factor of each term. Thus we first write
a b – b a,
and then placing the accents we have
a b' – b a',
which is the common denominator.
2. To determine the numerator of the value of each
unknown quantity, substitute for the letter expressing
the coefficient of that unknown quantity in the deno-
minator (already found) the absolute quantity c, and
preserve the accents as before. Thus, to determine the
numerator of the value of ar, we change a in the com-
mon denominator into c, and the result is
cb' – b c';
and to obtain the numerator of y we change b into c,
and obtain a c' – c a'.
(166.) Let us now consider the general formulae for
the values of three unknown quantities derived from
the equations of (164.)
Let z be eliminated, by multiplying the first equation
by c', and the second by c, and subtracting the one
from the other. The result is
(a c' – ca') a + (b c' – c b') y = d c' – c d'.
In like manner, eliminating z by the second and third,
we obtain
(a' c' – c' a') a + (b'c" – c' b") y = d'c" – c' d";
eliminating y by these two equations, by the usual
methods, we obtain
[(a c' – ca') (b'c" – c' b") — (a' c"—c'a") (b c'—cb')] + =
(d c' – c d') (b'c" – c' b") — (d. c." – c' d") (b c' – c l,')
Developing the several products, and dividing by the
common factor c', and arranging the factors of each
term in the order of the accents, the equation becomes
(a b'c'— a c'b" + c a' b”—b aſ c” + b c a' – c b'a") w =
d b'c"—d c'b" + c d'ò" – b d c' -- b c' d" – c 5' d".
Whence we obtain
_d b'c" – d cºb" + c d'5"— b d c” + b c d"– c 5' d"
* T a WCW— a dºſ Io.ſyſ Sº I b c'a" — a Way
and by a similar process we obtain -
a d'c' – a c' d" + c a' d"—d a' c” + d c'a"—c d'a" e
y= a b'c" — a c'b'" + ca'b'" — b a' c” + b c'a" – c b'a" 2
a b' d" — a d" b"+d a' b" – b a' d" + b d'a" – d b'a"
* = .5 cº-ac, UT-F ca'b'" – b a' c” + b c a” – c 5' a”
(167.) The last two formulae might be deduced from
Simple
the first by the symmetrical nature of the proposed Pauations.
equations. It is evident, if in the three original equa-
tions of (164) the letters a, a, a', a” were changed into
y, b, b', b", or in z, c, c', c", and vice versd, the equations
would remain unchanged. Hence we are authorized
to make similar changes in the formulae which are
deduced from these equations. I*, then, in the for
mula for a, the letters r, a, a', a” be changed into y, b,
b', b", and vice versá, we shall obtain the formula for y;
and by changing w, a, a', a' into z, c, c', c', and vice
versä, we shall obtain the formula for 2. This princi
ple will be found of very extensive use in analysis.
(168.) The preceding formulae for ar, y, and z, like
the former, have a common denominator, and may be
found by the following rule:
1. To form the common denominator, write the de
nominator (a b' , b a') in the case of two unknown
quantities without the accents, thus
a b -- b a ;
introduce the letter c in all possible positions in each
of the terms a b and b a , that is, last, middle, and
first ; and write the successive results one after another,
affecting them alternately with the signs + and —.
The result will be
a b c – a c b + c a b b a c + b c a c b a ,
accenting the second factor of each term with ', and
the third with ", the formula becomes
a b'c" — a c' b" + c a' b" — b a' c” + b c'a" – c b/a",
which is the common denominator.
2. To form the numerator of the formula for each un-
known quantity, it is only necessary to substitute for
the letter expressing its coefficient in the denominator
the absolute term d, and to preserve the accents.
Thus, to determine the numerator of the value of ar, it
is only necessary to change a into d, and the result is
d b'c' – d c'b'" -- c d' b" b d'c' + b c' d" ... c b' d",
and similarly for y and 2.
(169.) The law by which the arrangement of the
terms of these formulae is governed appears upon in-
spection, and may be extended to the cases of four or
more unknown quantities. A general demonstration
of the law has been given by LAPLACE, in the proceed-
ings of the Institute for the year 1772. It is, however,
of too complicated a nature to be properly inserted
here. -
(170.) The values of the unknown quantities de-
duced from any system of equations must be either
Q = {} o A.
positive, negative, = 0, of the form To or of the
form 0 º
()
If the value we obtain for an unknown quantity be
positive, it is generally a value which solves the problem
which was reduced to the proposed equations. It is
not, however, always so. The equation, or the system
of equations, is not always the exact transiation of the
proposed problem into the language of Algebra.
There are frequently some peculiar conditions in the
proposed problem, which the analyst is obliged to omit
from their not being of a nature to allow of being ex
pressed in an equation. The problem which is ex-
pressed by the equations is therefore more general than
the problem from which the equations are deduced;
554
A L G E B R A.
Algebra.
S-V-Z
and the roots of it, from the peculiar values of the
data, may happen to be of such a nature, that they are
inconsistent with those conditions of the problem
which are not expressed in its algebraical statement.
Thus, suppose that the problem was such that the
sought number must, from its nature, be an integer,
but that the data were such that the result of the
equation gave it a fractional value. This value is a
true and full solution of the equation, but it is not a
solution of the problem from which the equation was
deduced. The cause of which is, that the condition
that the root should be integral was not expressed in
the equation, and the result indicates that the data of
the proposed problem are inconsistent with that con-
dition. Instances of this will be seen hereafter.
(171.) If the values obtained for any of the unknown
quantities be negative, a modification of the original
problem is suggested, as has been already explained in
the case of a single unknown quantity. The modifi-
cations thus suggested may be determined by recurring
to the original equations, and changing in them the
signs of those unknown quantities which are negative.
As this is determined on the same principles as in the
case of a single unknown quantity, it will be unneces-
sary here to enter further upon the subject.
The observations already made on the other peculiar
wº A. 0 .
forms Scil., 0, -o-, and TOT, in the cases of one and
two unknown quantities, are also applicable to the
results of equations of several unknown quantities.
SECTION XIII.
Of Equations of the Second Degree.
(172.) AFTER an equation which results from the
conditions of a problem expressed algebraically has
been reduced in the manner explained in (139,) the
result, if it be an equation of the second degree,
must have the form
A a 2 + B a = C.
As the coefficients A and B are respectively the alge-
braical sums of the several coefficients of a 2 and ar,
and C the algebraical sum of those terms not affected
by a as a multiplier, it follows in general that A, B,
and C may have any values positive, negative, or = 0.
But it should be observed, that if A = 0 the equation
is no longer of the second degree; this case we shall
therefore omit in the consideration of these general
equations. If the equation be divided by A, and that
we suppose --
B L. C
TAT = p, A Q,
it becomes
a. * + p a = q,
where, as before, p and q may each be positive, nega-
tive, or = 0. %
If p = 0, the form of the equation becomes
a. * = q. -
This form is sometimes called a pure quadratic equa-
tion, and by some authors an incomplete quadratic
equation.
If p be not z 0, the equation is called a complete
or affected qacdratic equation.
The square roots of both members of the former
being taken (138) we have l
a = + V q.
If q be a number, this is done by the rules of ordinary
arithmetic. If q be a simple algebraical quantity, its
root, when it has one, may be obtained by the prin-
ciples established in Section VI. If it be a complex
algebraical quantity, the method of obtaining the root
will be explained in a subsequent section.
It may be observed, generally, that if q > 0, there
will be two values of a whose arithmetical value is the
same, but whose algebraical values have different signs,
(66.) If q >< 0, there is no arithmetical value of a,
and its algebraical values are imaginary, (68.)
(173.) The method of solving a complete equation
of the second degree is deduced from a comparison of
its first member with the form for the square of a
binomial, the first term of which is ar. Let
a + a = b :
squaring both members, we have |
a. * + 2 a a + a” = b%. [1..]
This is evidently a complete equation of the second
degree, and may be solved by taking the square roots
of both members. Upon comparing it with the form
a *-H p a = q, [2]
they are found to differ only in this, that there is an
absolute term (a”) in the first member of the former
which does not appear in the latter. This term is the
square of half the coefficient of a in the former. We
are, however, allowed to add the same known qmantity
to both members of an equation without disturbing
their equality. Hence, the first members of the two
equations will be assimilated, as to their form, by adding
to both members of the latter the square of half of the
2
coefficient p ; that is, +. By this change it becomes
2
p? p
3 -i- --- sº. —-
* + p ≤ + + I + q,
Or * + 2.4-, + + = + + q. [3.j
2 4 4
The first member here becomes identical with that of
[], by changing + into a. Hence it is easily seen
that the first member of [3] is the square of
1. –P.
* + -ā-
Quadratic
Equations.
S-N-7
Taking, then, the square roots of both members of General
[3, we obtain
–Pi— vſ.
— 4. Vº
à it + + 4.
Hence we derive a general rule by which the value of
a in an equation of the second degree may at once be
obtained.
“Let the equation be first reduced to the form [2,]
(which if it be a quadratic equation it always can ;)
the value of a will be found by taking the coefficient of
a (p), changing its sign, and dividing it by 2, and
sºmº
tºge
..". £
formulae for
solution.
A L G E B R A. 555
Algebra adding to it, or subtracting from it, the square root of
S-N-' the quantity, which is the algebraic sum of the square
2
of the half coefficient (+) and the absolute quan-
tity (q).”
Hence it will be observed, that a quadratic equation
always has two roots, inasmuch as the radical is
Susceptible of two signs.
Properties (174.) We, shall now proceed to consider some
of the roots, general properties of the roots of equations of the
second degree.
After modifying the formula
a. * + p a = q : [1]
2 . -
by the addition of + to both members, we obtained
(. + P. ) E p”. + Q
2 4 e
Let the second member of this equation be called m”,
so that
2
(* +- #) = m2,
2
Or (s + #) ---, m2 = 0,
Or (*#4 + n) ( t + - m)=0. [2.]
The first member of this equation is the product of two
factors, and the second member is 0. Now it is evi-
dent that a product will become equal 0 when either of
its factors = 0. Hence the last equation will always
be fulfilled by the condition expressed by either of the
following equations,
* + -º- + m = 0,
2
* +--- m = 0,
Or * = – 4 – m.
or, if m be replaced by its value,
— — .P. - a /4”
* = - + I- + q,
= – 4'- Pi
* = - + + V -- + q.
Since then the equation [1, or its equivalent [2,] can
only be fulfilled by one or other of the factors of [2]
being = 0, it follows, “That an equation of the second
degree admits of two roots, but not of more.”
(175.) If the equation [1] be reduced to the form
a * + pa - q = 0, [3]
its first member must be equivalent to that of [2] Let
a'a" be the roots of this equation. It is evident that
we have
a' sº sº- + * , , º, 771,
g" esº. " + + 772. 2
and by this [2] becomes
(a — a ') (a — a ") = 0;
the first member of which being equivalent to that of S-S-
[3] gives
a 2 + p a - q = (a – a ') (a — a "). [4.]
The following identity
f ~ o 2 p°
a’ + p a - q = (*-pº- * ) <--> (* + a)
is equivalent to
p \? / p?
a *-ī- pa – q = (* + +) --> (+ –– w).
From which we immediately infer -
a *-i-pa — q =
(4-4, + VºI) (++-vº)
OI’ a *-i-pa - q = (a – a ') (c. — a ").
(176.) By developing the product which forms the Product of
second member of the identity [4,] we obtain r005.S.
a *-H p a - q = w” — (a' + æ") a -- a-'a".
As this has been proved true, whatever value be ascri-
bed to r, let a be supposed = 0. Hence we obtain
Quadratic
Equations.
{
— q = x'a".
Subtracting this from the former, we obtain
a *-H p a = a," – (r' + æ") a ;
and dividing by r, and omitting the common term, we Sum of
have TO QºS.
+ p = – (r' + æ").
Hence we infer, that in a quadratic equation reduced to
the form [1, “The absolute quantity (q) with its sign
changed is equal to the product of the roots ; and the
coefficient (p), with its sign changed, is equal to their
sum.”
It will be easy to verify these results by actual addi-
tion and multiplication.
(177.) The roots of a quadratic equation are rational When
or irrational, according as the quantity under the radi- rational.
cal is an exact square or not. If it be not an exact
square, and the equation be numerical, the values of
a', a." may be obtained with any degree of approxima-
tion which may be required in rational numbers by the
arithmetical rules for the extraction of the square root.
If the equation, however, be literal, there is no other
way of signifying the root when the quantity under the
radical is not an exact square than by the radical itself,
or by the equivalent notation of fractional exponents
already explained.
(178.) If the quantity under the radical be negative, When
the radical, and therefore the roots of each of which it imaginary
is a part, will be imaginary, (68.) Of the two terms
under the radical, one is always positive, being the
square of + , a quantity supposed to be real. Hence,
in order that the suffix of the radical be negative, two
things are necessary: 1. that the absolute quantity (q)
be negative, and, 2. that it be greater than the square
2 N.
of the half coefficient ( +). It is under these con-
ditions only that the roots will be imaginary; and since
the same radical enters both roots, they must always be
both real or both imaginary together. - -
From the signs and values of the coefficient and
556 A L G E B R A.
Algebra. absolute quantity, it may, therefore, be always deter-
S-' mined whether the roots be real or imaginary.
Signs. (179.) The signs of the roots, when real, may be at
In order that the roots may be equal, it is, therefore, Quadratic
necessary that the suffix of the radical = 0, and this can Equations.
only happen when the absolute quantity (q) is negative, S-N-
determined.
1)ifference
of roots.
once deduced from the properties already established;
and from the principle that if a product of two factors be
positive, its factors will have the same sign, and if it be
negative, they will have different signs.
Hence, since the product of the roots has always a
different sign from the absolute quantity (q.) (176,) it
and equal to the square of the half coefficient. In that
case, the value of each root will be the half coefficient
with its sign changed. This may be easily verified.
(184.) If q = 0, the expressions for the roots be-
COPYle
follows, that when the absolute quantity is negative a' = — # gmm # = — p,
in [1, the roots have the same sign, and when it is
positive they have different signs. - w = – 4: * = 0 .
(180.) When two quantities have the same sign, — — . + † – " ,
their common sign is that of their algebraical sum ; and
when they have different signs, the sign of the greater
is that of their algebraical sum. Hence, when the
roots have the same sign, that sign will be different
from the sign of the coefficient (p,) and when they
have different signs, the sign of the lesser root will be
that of the coefficient (p) (176.) -
(181.) As p and q may be each positive or negative,
the general formula [1] includes under it the four follow-
ing cases: 1. a,” + p r = + q > 2. a' – p r = + q :
3. a' … p v = — q; 4. a' + p r = — q. By what has
been just established, it follows, that the roots in the
first two formulae, first, are always real; secondly, that
they have different signs, the root whose arithmetical
value is greater being negative in the first, and positive
in the second. Also, that the roots in the last two for-
2
mulae, first, are real or imaginary, according as + is
greater or less than q ; and, secondly, that when they
are real they are both positive in the third formula, and
both negative in the fourth.
(182.) When quantities have the same sign, their
algebraical sum is also their arithmetical sum, and
one of the roots being equal to the coefficient with its
sign changed, and the other being = 0. This might
also be inferred from q being the product of the roots.
If a product = 0, one of its factors must = 0, and
therefore one of the roots must = 0. The sum of the
roots (– p) will then be equal to the other root. It
will be seen, hereafter, that this is only a particular
case of a much more general principle.
(185.) In considering the case of purc, or incom-
plete, equations of the second degree, we have already
disposed of the case in which p = 0.
If p = 0, and also q = 0, both roots are = 0; for
since their product = 0, one of them at least must
= 0, but since their sum also = 0, the other must = 0.
(186.) There is a case which frequently occurs in
algebraical investigations, to explain which we must
recur to the original form in which we expressed (172)
an equation of the second degree :
a rº–H b c = c.
This equation being solved by the general rule gives
– b + v 5-HTac
* =:
2 a.
when they have different signs, their algebraical sum is “sºme *
their º j Hence i. follows, that in If we now suppose that a = 0, the values of a become
the first two of the above formulae, the coefficient p is tº sº- —b + b g
the arithmetical difference of the roots, and in the last 0
two, it is their arithmetical sum. The first two for If the upper sign be taken, we have
mulae, therefore, if interpreted in ordinary language, 2 b
become “Given the difference of two numbers, and w = —
their product, to determine the numbers themselves;” 0
and the last two, “Given the sum of two numbers, and and for the lower sign,
their product, to determine the numbers themselves.” ()
To one or other of these classes, every problem t –- ,
which produces a quadratic equation can, therefore, be ()
ultimately resolved.
(183.) To obtain the formula for the algebraic dif.
ference of the roots of an equation of the second
degree, let the values of a be subtracted from that
of a ": -
2
---, + Vº
−.
- - Vº
2
++ a
a’
a" – a ' - 2
Twice the radical is, therefore, the difference of the
roots, and is positive or negative, according to the
manner in which the subtraction is performed. -
the one being a symbol of infinity, and the other in-
determinate. &
To trace the circumstance which gave rise to these
results, it is only necessary to determine, what effect
the hypothesis a = 0 would produce upon the primitive
equation. It is evident that it would reduce it to the
form b a' = c. The division by a, which was effected
preparatory to the solution as a quadratic equation, in-
volved a distinct, though implicit, condition, that the
value of a was not = 0. The condition that a = 0,
subsequently introduced, contradicts this, and hence
, the absurdity of the results.
This process is what is called shifting the hypothesis,
and is too often used by analytical writers, who
attempt to account for the results obtained, and to give
them a meaning, notwithstanding the evident sophistry
and invalidity of the process by which they were
obtained
A L G E B R A.
557
Algebra. In the present instance it is evident, since the ori-
—y-'ginal equation becomes b a' = c when a = 0, that a has
negative. If the multiplier be positive, the signs of Inequalities
both members remain unchanged, and therefore the S-N-
but one value, and that is
C
b
If in this case b = 0, the equation becomes 0 = c,
which is absurd, if c be not := 0, and if c = 0, it
becomes an useless identity.
It is, perhaps, worth observing, that if a, b, and c
all = 0, the equation a tº + b x = c will be necessarily
true, whatever value may be ascribed to ar. The pro-
blem is in this case indeterminate, and the equation is
said to be “satisfied by its coefficients.”
& =
****
SECTION XIV.
Of Inequalities.
(187.) AN inequality is a proposition which expresses
algebraically, that one quantity is greater or less than
another. Inequalities are therefore of two kinds, and
must be expressed in either of the following forms,
A > B
A 3 B,
according as the first member is greater or less than the
second.
In an equality it is a matter of indifference on
which side of the sign = either member is placed. It
is otherwise with an inequality ; for if it be necessarily
true in one position, it will be evidently false when the
members are transposed. If, however, at the same
time that the members are transposed, the sign of in-
equality be reversed, the transposition is valid, and the
statement continues true. Thus, if A ≤ B, *.* B 3 A ;
and if A - B, "... • B > A, which is evident from the
meaning of the symbols.
(188.) Several of the changes allowable on equalities
are also allowable on inequalities. Thus, quantities
which are algebraically equal, may be added to, or
subtracted from both members of an inequality. It is
evident, that if A > B, "." A + C > B -- C, and
A – C > B – C. In executing these transformations
it should, however, be remembered, that of two negative
quantities that which is numerically less is algebraically
greater.
(189.) Hence a quantity may be transferred from
one member of an inequality to the other, provided
that its sign be changed ; for this is the same as sub-
tracting it algebraically from both members. Thus, if
A > B -- C, "." A – C > B ; and if A > B — C,
*...* A + C > B.
(190.) Hence we may infer, that if the signs of both
members of an inequality be changed, the species of
the inequality must also be changed. For if A > B,
... A – B > 0 by (189,) '... — B > – A, by (189,) or
— A 3 – B by (187.)
(191.) Both members of an inequality may be mul-
tiplied by the same positive quantity ; but if they be
multiplied by the same negative quantity, the species of
inequality will be changed.
For since products having a common factor are in
the same ratio as the factors not common, the nume.
rical inequality of both members will remain of
the same species, whether the multiplier be positive or
WOL. I. -
species of inequality remains the same; but if the
multiplier be negative, the signs of both members are
changed, and therefore the species of inequality must be
changed. Thus, if both members of A > B be multi-
plied by + C, we have A C > B C ; but if they be both
multiplied by — C, the effect is the same as if they were
first multiplied by + C, and the signs then changed.
The first result would be AC > B C ; and changing the
signs we should have by (187) — A C < – B C.
(192.) The same principles exactly, will authorize us
to divide both members of an inequality by the same
positive or negative quantity under similar restrictions.
(193.) The corresponding members of inequalities
of the same species may be added one to another.
Thus, by adding A > B, A' > B', we obtain A + A* >
B+ B',
The validity of this inference may be easily esta-
blished. It is evident, that the quantity which it is
necessary to add to the lesser member of an inequality,
in order to convert it into an equality, must be positive.
Hence m and m' will be positive quantities in the equa-
lities
A = B -- m,
Aſ = B' + m'.
These being added give
(A + Aſ) = (B -- B') + (m-H m').
Since m and m' are both positive their sum is positive,
hence A + A' > B -- B'.
(194.) A similar principle, however, is not true as
respects the subtraction of similar inequalities. It
does not follow, that if A > B, A' > B', that A — A' >
B – B'. For, as before, let A = B -- m, A' = B' + m',
‘. . (A — A') = (B — B') + (m. — m/).
The quantity m — m/ may be either positive or mega-
tive. If it be positive, we have A — A' > B — B',
and if it be negative, A — A' < B — B'. -
(195.) Both members of an inequality may be raised
to the same power, or the same roots may be extracted,
observing the condition, that if in the process of in-
volution or evolution the signs of the members be
preserved, the species of the inequality is also to be
preserved; but if the signs be changed, the species of
inequality is also to be changed.
(196.) It is evident, that the sign of the greater mem-
ber of an inequality if negative may be made positive,
and the lesser member if positive may be made nega-
tive, because by this process the former is algebraically
increased, and the latter algebraically diminished.
(197.) For the same reason any positive quantity
may be added to the greater member, or subtracted
from the lesser, and any negative quantity may be
added to the lesser member, or subtracted from the
greater.
SECTION XV.
On the changes in sign of a rational and integral for-
mula of the first or second degree, produced by changes
fin the value ascribed to the unknown or variable
quantity in it.
(198.) WHEN an algebraical formula contains a
4 D
558
A L G E B R A.
Algebra. quantity which is unknown or indeterminate, combined
by given operations with other quantities which are
given, it is said to be a rational formula when the un-
known or indeterminate quantity is not, either by itself
or in combination with other quantities, affected by a
radical or a fractional exponent. It is likewise said to
be integral when the unknown quantity, either by itself
or in combination with other quantities, is not found
in the denominator of any fraction, or affected by a
negative exponent. The degree of the formula, like
that of an equation, is decided by the highest integral
exponent. Every rational and integral formula of the
first degree must, therefore, have the form A a + B,
and every rational and integral formula of the second
degree must have the form Aaº -i- B a + C.
The general symbols A, B, C being supposed to re-
present given quantities, it follows that the values of
these formulae will entirely depend on the values which
may be ascribed to the unknown or indeterminate quan-
tity ar. We propose in this Section to determine how
the signs of the quantities represented by these formulae,
depend on the values which may be ascribed to ar,
and to distinguish what values of a will render them
positive or negative.
This may be considered as a more general investiga-
tion than the solution of equations which is the deter-
mination of the values of a, which render these for-
mulae = 0.
The formula of the first degree presents no difficulty.
B
It may obviously be expressed in the form A ( + TA
Let the value of a, which renders it = 0, be a '. We
then have a' = — B. and the original formula by
A.
this substitution becomes A (a — a'). This being the
product of two factors, its sign will be + or – , accord-
ing as its factors have like or unlike signs. Hence if
A > 0," all values of a > a.' render the formula P- 0,
and all values of r < x' render it < 0. If A 30, all
values of a > a.' render the formula 30, and all values
of a -3 v' render it - 0. Hence we find
&ºm=mºs
º
ſ A = 0 and z > – A
A r + B > 0 if {
! B
U A * 0 and a 3 T TAT.
A > 0 and a 3 — #.
At + B ~ 0 if { B
B
asºmºn
A.
Hence, if a be supposed to assume all possible values
from an unlimitedly great positive value decreasing to
0, and then to pass through all negative values from 0
to an unlimitedly great negative value, the formula
A r + B = 0 if y = —
A z + B becoming = 0, when a = — * will be posi-
tive for all values on the one side of this, and negative
* It should be carefully observed, that > and <, and the terms
greater and iess, mean algebraically greater or less, and not arith-
metically, see Sect. XIV,
for all those on the other side of it. The formula may
thus be conceived to change its sign in passing through
zero, and constantly to maintain the same sign, while
z is on the same side of the value which renders the
formula := 0, so that throughout the whole variation of
a the formula suffers but one change of sign.
This, however, is not the case with any other rational
and integral formula. In the formula
A a” -- B a + C,
let the values of a which render this = 0, be a' and a ".
We have (175)
Aa2 + B a + C = A (a – a ') (a — a ").
The quantities ac'a" are subject to all the circum-
stances incident on the roots of an equation of the
second degree : they may be, 1. real and unequal; 2.
real and equal ; 3. imaginary. We shall consider suc-
cessively these cases. -
(199.) 19. If the quantities ar' a "be real and equal,
the formula
A cº -- B a + C,
or its equivalent
A (a — ac') (c. — w!")
is the product of three factors. If two of these have
the same sign, the sign of the product will be that of the
third factor; and if two have opposite signs, the sign
of the product will be different from that of the third
factor. Of the two roots a' and a:" (being unequal) let
a' > a.". If a value be ascribed to a which is between
the values of the roots, that is, greater than the lesser
root and less than the greater root, the factors a; - a."
and a — ac" will have different signs, and therefore the
sign of the whole formula will be different from the
sign of A; but if the value ascribed to a be beyond
the limit of either root, that is, if it be greater than the
greater root or less than the lesser root, the signs of
the factors ac – ac' and a — a " will be the same, and
the sign of the whole formula will be that of A,
Thus it appears, that while continually increasing
values are ascribed to a, from negative infinity to posi-
tive infinity, the formula of the second degree suffers
two changes of sign in passing twice through zero; that
for the values of a between those which render it equal
to zero, it is > 0 when A 3 0, and < 0 when A P 0;
and that for all values of a beyond the limits of the
roots on either side, it is continually - 0 or < 0,
according as A > 0 or < 0.
(200.) 2°. If the roots a ', a "be equal, the formulae is
reduced to A (a — a ')', aſ expressing the common value
of the two roots. In this case the factor (a — a ')” is
essentially positive, whatever be the sign of a - a ',
except when a = a ', when it = 0. Hence for all
values of a whatever, except that particular value a',
which renders the formula = 0, the sign of the formula
will be that of A.
(201.) It may be observed, that in this case the for-
mula is a perfect square. For the condition on which
the equality of the roots a ', a." depends is, that the
suffix of the radical should = 0. And this gives
B? – 4 A C = 0,
... B = 2 v AC,
which being substituted in the original formula it be-
COIſleS -
Aaº + 2 M A Cr-H C,
which is equivalent to
(w/ A. r + v C)’.
Sign of an
Integral
Formula.
-N/~"
A L G E B R A. 559
(202.) It is evident also that this condition can only be
S-N-2 fulfilled when A and C have the same sign. For if they
had different signs, 4 A C would be essentially negative,
and therefore B? – 4 A C would be the sum of two
quantities essentially positive, and could not = 0.
(203.) 3°. If the roots ar', a "be imaginary, there are
no real values of a which render the formula = 0. In
this case the sign of the formula must be otherwise
determined. Any real value being ascribed to a, let
the corresponding value of the formula be y, so that
A w°-H B a + C = y,
B C
& * + -ī- — =
..". Jº ! a + ſºmºmº
which, being solved for a, gives
B VBºst y
* = - a R + V —TR-4 +,
— B E v. Bº – 4 A C + 4 A y
2 A
Since, by hypothesis, in the present case, the values a ',
a" are imaginary, it is necessary that B” – 4 AC < 0.
But also it is supposed that the values of a are real.
9.
A 2
".. ºr E
Hence B* – 4 A C – 4 A y >0;
and since B? – 4 A C ~ 0, we have
4. A y >0,
..'. A y > 0.
Hence it follows, that y must always have the sign of A,
whatever be the value of ar, provided it be real.
Thus it appears, that when the values of a which
render a rational and integral formula of the second
degree = 0 are imaginary, all real values of a whatever
will render the same formula positive when A > 0, and
negative when A 30.
It appears, as in the case where a' = a ", that in this
case A and C must have the same sign.
SECTION XVI.
Of Marima and Minima.
(204.) THE species of problems having for their ob-
ject the determination of marima and minima, belong
more properly to the Differential Calculus than to pure
Algebra. For the complete discussion of them we
therefore refer the reader to that subject. A particular
class of these questions may, however, be solved by the
aid of the theory of equations of the second degree ;
and as they frequently occur in the more elementary
parts of analysis, and particularly in the application of
Algebra to Geometry, we shall here explain the methods
of investigating them. s -
When certain operations are to be performed on
given numbers, it may so happen that the magnitude of
the result will depend on the manner in which these
operations are performed. In such a case it may be
required to determine how the proposed operations
should be performed, in order that the resulting quantity
should be of the greatest or the least values which it
could have consistently with the proposed conditions.
Such values are called marima and minima.
This will, perhaps, be better understood by an ex
ample. Let it be required to divide a given number
(2 a) into two parts, whose product is a marimum ;
that is, whose product is greater than the product of
any other two parts into which the number could be
divided.
Let y be the sought maximum value, and a one of
the sought parts, the other being 2 a - a we have
a (2 a - a) = y.
It is plain, that as there are an infinite variety of ways
in which the proposed number may be divided into two
parts, there is an infinite variety of values which may be
ascribed to the part a. In fact, a may be conceived to
express any number which is less than the given num-
ber 2 a. The value of the product y will altogether
depend on the value ascribed to a. Under these circum-
stances a is called a variable, and y is said to be a
function of a. The word function being a term im-
plying a quantity or symbol, the value of which
depends, by some given condition, on the value of
another quantity called the variable ; function and
variable being therefore correlative terms.
In order to determine the value of a, which renders
Sy a maximum, let the first member of the equality be
developed, and the result is
2 a a - a * = y,
... a 2 – 2 a. a = — y.
Let this be solved as if y were a given quantity, and
the result is
a' = a + w/ a 2 – 3/.
By the primitive equation the value of y depends on
that of ar. If such a value were ascribed to a as would
make y > a.”, that value would render the radical in
the last equation imaginary. But as this radical is a
part of the value of a by the last equation, that value
of a will itself be imaginary. Hence no real value of
a will render y >a”. The greatest value which y can
receive, consistently with the reality of a, is when
gy = a”. This therefore is the maximum value sought.
But it is still necessary to determine the parts into
which the number is divided, in order that the product
of its parts may have this value. This may be found
by substituting a” for y in the last equation, the result
of which is
- w = a + V aº – alº – a,
.*. 2 a. – a = a.
The parts into which the number is divided are there-
fore equal. From which we deduce the following
general theorem, “If a number be divided into any
two unequal parts, their product is always less than the
square of half that number.” -
This principle might also be established still more
simply, by taking half the difference of the parts as the
variable, instead of one of the parts themselves. As
before, let the number be 2 a, and let one of the parts
be a + æ. The other will be 2 a - (a + a) = a – a ;
it is evident that 2 a. is the difference of the parts, and
therefore a is half their difference. We have then
(a + ar) (a - a) = y,
a” — a * = y,
... a 2 = a” — y,
.*, ºc = aſ a Ty.
As before, if y > a”, a would be imaginary. Therefore
Maxima
and
Minima.
*
4 D 2
560
A L. G E B R A,
Algebra. 3) is a maximum when = a *, ... a = 0.
--~~ ference of the parts = 0, the parts are equal.
Since the dif.
Each step of this process involves a principle which
merits attention. By the original statement it appears,
that of two unequal quantities the greater is equal to
half their sum increased by half their difference, and
the less is equal to half their sum diminished by half
their difference. The first equation shows that the
product of two unequal quantities is equal to the
square of half their sum diminished by the square of
half their difference; and as the difference must always
be less than the sum, it is apparent, even without having
recourse to any reasoning on imaginary quantities, that
the product of the unequal parts inust be less than the
square of half the sum, or then the product of the equal
parts.
(205.) The general principle by which the property
of imaginary roots of quadratic equations becomes
instrumental in the solution of questions respecting
onarima or minima, will now be easily comprehended.
If the roots of either of the formulae
a. * – p a = — q,
a * + p a = — q,
be real, it has been already proved, that q cannot exceed
2 2
#-. Hence, if-
be supposed to be a given quan-
tity, and q to be variable, and at the same time the
values of a be supposed to be real, the greatest value
2
which q can have is £1, in which case
p
a = + = ,
+ 2
the upper sign applying to the first, and the lower to
the second formula. -
Again, if q be supposed given, and p variable, the
least value which p can have consistently with the
2
reality of the roots, is when
= q, or p = 2 aſq.
In this case p is a minimum, and the value of a is .
p
*=-º-º:
2
(206.) The principle may, however, be stated still more
generally. When the result of any problem is a qua-
dratic equation, and that a quantity whose maximum
or minimum value is to be determined, enters in
combination with given quantities under the radical in
the solution of the equation, all values of that quan-
tity which render the suffix of the radical negative
must be rejected, since they render the roots imaginary,
but that value which renders the suffix of the radical
= 0, and which stands between those which render it
positive or negative, will be the maximum or minimum
value sought. Whether this value be a maximum or
minimum, must be decided by the peculiar circum-
stances of the question.
(207.) Let it be proposed to divide a given number
(2 a) into two parts, such that the sum of the squares
of these parts shall be greater or less than the sum of
the squares of any other parts into which the same
number could be divided, or such that it shall be a
ſmarimum or minimum.
As before, let a be one of the parts, the other will be
2 a - a, and let the sum of the squares be y, so that
Lºsºl
tºmºsº
a = + + V q.
a *-ī- (2 a - a)* = y,
... 2 r * – 4 a a + 4 a * = y,
:: * – 2 a z = -4 – 2 a.º.
... a' = a + VºI.
The value of y, which renders the suffix of the radical
= 0, being found, y = 2 a” is evidently the least value
which it can have consistently with the reality of r.
The sum of the squares is therefore a minimum when
it is equal to twice the square of half the given number,
and the corresponding values of the parts are a = a,
2 a - a = a. The number is therefore divided into
equal parts. s
There is no value of y greater than a” which will
render the suffix negative; on the contrary, the value of
the suffix is continually augmented as increasing values
are ascribed to y. There is, however, notwithstanding
this, a limit. It will be remembered, that the value
of y depends on that of a ; and the mere inspection of
the original equation will show, that if a be increased
without limit, y will be also increased without limit,
and therefore no major limit to y can be inferred from
the algebraical statement of the question. In the
problem itself, however, the number 2 a is supposed to
be divided into two parts. Neither of these parts can
then be greater than the whole, consequently a cannot
exceed 2 a. If a were supposed = 2 a, which is the
extreme case, the other part 2 a - a would = 0, and y
would be greater than it could be under any other cir-
cumstances, Why, then, it may be asked, does not this
result from the algebraic investigation ? The difficulty
will be removed by examining more closely the alge-
braic statement.
The equation
a *-ī- (2 a.
means simply that the square of a number represented
by ar, added to the square of another number represented
by 2 a - a produces a result = y. Now there is
nothing here which limits the magnitude of w, or makes
it necessarily less than 2 a. The number 2 a - a may
be negative, and yet its square will be positive. In
this case 2 a will be the arithmetical difference of the
numbers a and 2 a - a, and not their arithmetical sum
as announced in the problem. So that, as frequently
happens, the algebraical statement is more general than
the original problem ; and hence it arises, that al-
though in the original problem there is a major limit
to the value of y, there is no major limit to it in the
more general algebraical statement, because the parti-
cular condition which produced the major limit is the
very condition by whose omission the problem in
generalized.
(208.) Let it be required to divide a number (2 a)
into two parts, a, 2 a - a, such that the sum of the
quotients of each part by the other shall be a maximum
O7° 77?????????/777.
Let y be the sum of the quotes.
after reduction becomes
a)* = y
The statement
4 g?
a. * ~ 2 a. a = — —-
2 + y
/ 4 as
‘.. a = a + \/ a 2 –
2 + y
Maxima
and
Minima.
A L G E B R A.
56I
•
Algebra. The suffix of the radical being equated with zero gives
2 .
* 4 a = 0,
2 + y
..'. 2 + y – 4 = 0,
..". J = 2,
a 2
w". J. E. a, 2 a. — a t a.
The number therefore must be divided into equal parts,
and the sum of the quotes is 2, each quote being 1.
In this case the sum is evidently a minimum. For
the increase of y produces a diminution in the negative
part of the suffix of the radical; and it is obvious that
no increase whatever beyond the value a will ever render
the suffix negative ; and as the diminution of y in-
creases the negative part of the suffix, no diminution
below a will ever render the suffix of the radical
positive.
SECTION XVII.
Arithmetical Progression.
(209.) A serIEs of quantities so related that each
term exceeds that which precedes it, or is exceeded by
it by the same quantity, is called an arithmetical series,
and its terms are said to be in arithmetical progression.
Thus, in the series a, , a, , as , as , &c.
if a – a – as -- a = as - a, , &c.
the quantities are in arithmetical progression.
Thus, 1, 4, 7, 10, 13, &c.
1, 3, 5, 7, 9, &c.
20, 18, 16, 14, 12, &c.
are severally arithmetical series.
The difference of every two consecutive terms in the
series being the same, is called the common difference.
The series may be conceived to be generated by the
constant addition of this common difference to the first
term ; when the series increases the common difference
being positive, and when it decreases being negative.
Thus, if a be the first term and a the common differ-
ence, the successive terms of the series will be
a, a + æ, a + 2 a., a + 3 a., &c.
The coefficient of a in any term is evidently equal to
the number of preceding terms, so that the mº term T
will be T = a + (m – 1) ar. This general formula will
determine each of the terms by substituting successively
for m the numbers 1, 2, 3, &c.
(210.) The sum of any two terms equally distant
from a given term in an arithmetical series is equal to
twice the given term. Let the given term be a + m a.
The preceding and succeeding terms are
a -- (m – 1) ar, a + (m-H 1) r,
which added are 2 (a + m a.). In like manner the
terms, two distant on each side, are
a + (m. – 2) ar, a -i- (m-H 2) ar,
which added give 2 (a + m a.), and in general the
terms distant n terms on each side are
a -- (m — n) ar, a -i- (m-H m) r,
which being added give 2 (a + m a.).
In the same manner it may be proved, that the sum
of any two adjacent terms is equai to the sum of any
two terms equally distant from them. -
(211.) Hence if any number of quantities be in
arithmetical progression, the sum of the first and last
terms is equal to the sum of the second and penulti-
mate, or of any two terms equally distant from the
extremes; and if the number of terms be odd, there
being one term equally distant from the extremes, the
sum of the extreme terms is equal to twice this middle
term.
(212.) If three quantities be in arithmetical pro-
gression, the mean is equal to half the sum of the ex-
tremes, and the common difference is equal to half the
difference of the extremes. Let the quantities be
a, b, c, Hence
2 b = a + c
H -
... b = 3 (a+c)
… a- = a- a - # * = # (a – ) = - c.
(213.) Let it be required to determine the sum of n
terms in arithmetical progression, of which the first ai
and the last a, are given. Let the common difference
be a, and the sum S ; *.."
+ (a, -ī- 2a) + (a, -ī- 3 a) + . . . .
But if we arrange the terms in the opposite direction,
beginning with a, we shall have
S = a, + (a, - a) + (a, 2 a.) + (a,
{ a, --- (n − 1) a $.
Adding these series, and observing that there are n
terms, we have .
tº tº tº e º v.
3a) + . . . .
2 S = (a, -H a,) m
71.
...' s = (a, +a.)+;
that is, the sum of the series is equal to the sum of the
first and last terms multiplied by half the number of
terms.
(214.) When an arithmetical series with a deter-
minate number of terms is given, there are five quan-
tities, viz. the first and last terms ai, a, , the common
difference ar, the number of terms m, and the sum of the
series S, between which there subsists a relation which
is expressed by the two equations,
a. = a, + (n. 1) +
2 S = (a, -í- a,) n.
Hence it follows, that if any three of these five quan-
tities be given, the remaining two may be found, and
thus there arises the ten following problems:
Given. Sought.
1. a, , a , n . . . . . . . . . . a, , S
2. a, , a , a, . . . . . . . . . . m , S
3. a, , a , S . . . . . . . . . . 7b 3 &n
4. at , 7, , a, . . . . . . . . . a , S
5. a, , n > S . . . . . . . . . . a 2 (1.
6. a, , a, , S . . . . . . . . . . a , it
7. a , 7, , a, , . . . . . . . . . a , S
8. r , n > S . . . . . . . . . . & , ºn
9. T , a, , S . . . . . . . . . . & , 72
10, r , a, , S . . . . . . . . o tº , 2.
Arithme-
tical Pro-
gression.
N-y--"
562
A L G E B R A.
Algebra.
(215.) All these problems are solved by equations
of the first degree, except those in which a and n and
a, and n are unknown. These are resolved by equa-
tions of the second degree, and it should be observed,
generally, that every value of n must be rejected, ex-
cept those which are positive integers ; for, from its
nature, n cannot be negative or fractional.
SECTION XVIII.
Geometrical Progression.
(216.) A SERIES of quantities are said to be in
geometrical progression, when they increase or decrease
in a common ratio, Thus, geometrical progression is
equivalent to continued proportion. A series in geo-
metrical progression may always be conceived to be
generated by a constant multiplier. For let the con-
stant ratio of each pair of successive terms be 1; r, and
let a be the first term. It is evident that a r will be
the second term, since a a r . . I : r, and, for a simi-
lar reason, the third, fourth, &c. terms are a rº, a rº, &c.
Thus, the n° term is a r" ~ *.
If the common multiplier r be > 1 the series in-
creases, and if it be ~ 1 it decreases.
(217.) The product of any two terms equally dis-
tant from a given term is equal to the square of the
given term. Let the given term be a r", the preceding
and following terms are a r" - ", a r"+", of which the
product is a2 rº" = (a r")*,
Those which are two terms distant on each side are
a r" - ", a r"+”
the product of which is the same, scil. = (a r")*.
And in general the terms which are n terms distant on
each side are a r" - ", a r"+", and their product is
a rº" = (a r") 2.
In like manner it may be proved, that the product of
any two successive terms is equal to the product of any
two terms equally distant from them in the series.
Let the two adjacent terms be
a r", a r"+",
and the two terms distant on each side by n terms are
a r" - ", (I, rm +n+ l,
which multiplied give
a2 r?” + “ — a r" × a r" + 1
. . . a r" - " × a r" + n + 1 - a r" X a r" + 1.
(218.) Hence if any number of quantities be in geo-
metrical progression, the product of the extreme terms
is equal to the product of any two terms equally dis-
tant from them ; and if the number be odd, this product
is equal to the square of the single term which is
equally distant from the extremes.
(219.) If three quantities be in geometrical pro-
gression, the square of the mean is equal to the pro-
duct of the extremes, and, therefore, either extreme is
found by dividing the square of the mean by the other.
If a, b, c be the three quantities
5*
a c = b% . . a = —
C
(220.) Let it be required to determine the sum (S)
Geome-
trical Pro
gression.
\-N-'
of any determinate number (n) of terms in geometrical
progression. Let the first term be ai. Hence we
have 4 -
S = a + a r + a, r"+ . . . . . . a, r*-* -- a, r" -";
multiply both members of this equality by r, and we
obtain
S r = a, r + a, r*-i- . . . . . . a r" - " -- a, r" - " -- a, r".
Subtracting this from the former we obtain
S (1 — r) = a, – a r"
-- 1 — r"
: s = a ++
ºr" — I
S = a, . tº
Or ** "TT
(221.) When the series is decreasing, it may be con-
tinued to an unlimited number of terms and yet have
a finite sum. In this case the multiplier r is 3 l, and
r" undergoes unlimited diminution, as its exponent n
is unlimitedly increased; and if n be supposed infinite,
r" will become = 0. Hence the sum of the series will be
l
1 — r
(222.) If r = 1 the formula for S assumes the form
S = a
0
To This indicates, either that the problem to deter-
mine the sum of the series is then indeterminate, or
that the formula for S has a common factor in both
numerator and denominator which becomes = 0 when
r = 1. This latter, in fact, takes place in the present
instance. For if the division indicated by the formula
1 — ?"
1 — r
1 — r"
be actually performed, we shall have
= 1 + r + r.” -- r" +
Now if in this r = 1 the second member becomes = n.
(223.) Between the five quantities ai, r, m, a, , S,
there subsist two equations, scil.
a, - a r" ~ *;
which, as in arithmetical progression, enable us when
any three of the five quantities are given to determine
the other two. But the solution of the several pro-
blems present in this case greater difficulties. The
four cases in which the unknown quantities are a, S,
r S, a, S, and a1, a, offer no particular difficulties, being
all reduced to equations of the first degree. The two
cases in which a r and a, r are sought, depend on the
solution of equations of the m” degree. By the for
mulae above mentioned we deduce
(S — a,) r" – S r" - " -- a, - 0
a, = a, r" " ',
the solution of which for r is necessary in the former
case. The degree of the problem, therefore, in this
case depends on the number of terms in the series.
In the latter case the equation is
a r" – S r + S — ai = 0.
(224.) The four other cases where n is unknown,
depend on the resolution? of an equation in which the
unknown quantity occurs as an exponent. The inves-
A L G E B R A. 563
Algebra. tigation of equations of this kind will be explained in a In this case the quantities A, B, and C may be sup- . Inde-
f g - * - ?–4. g g •s of: frmina:
J. V-' subsequent section. ppsed to be integers, since if they were fractions they º .
could be reduced to integers by multiplying the .
entire equation by any common multiple of their V
SECTION XIX.
Of the Indeterminate Analysis.-One simple equation
with two unknown quantities.
(225.) WHEN the number of equations which result
from the conditions of a problem is less than the
number of unknown quantities, the data are insufficient
for the solution, and the values of the sought quantities
cannot be determined, or rather there are an infinite
variety of values of the unknown quantities which will
equally satisfy the conditions of the problem, and all of
which, therefore, have an equal claim to be considered
as its solution.
To take a very simple instance, suppose it be required
to find two numbers which have a given ratio the one
to the other ; let the ratio be m : 1.
sought numbers we have a = m y, which expresses the
condition of the problem. Here then there are two
unknown quantities and but one equation. One of the
unknown quantities y may be supposed to have any
value whatever, and the equation will determine a cor-
responding value of the other, so that the two values
will satisfy the proposed condition. Thus the variety
of systems of values will be absolutely infinite.
(226.) The variety of values of the unknown quan-
tities in an indeterminate problem may, however, be
restricted by conditions which do not admit of being
expressed in the equation to which it is reduced. Thus,
suppose it be required that the values of the unknown
quantities be integers, all the systems of fractional
values which satisfy the equation must then be rejected,
and only the integral values retained. In the problem
5
already given, let m = Tö '.'
* = T6 3/.
Any value whatever being assigned to y, a value of a
may be found, which, together with the value so
assigned to y, will satisfy this equation. But it is re-
quired by the problem that the values of the unknown
quantities should be integers. Hence we infer, first,
that no fractional value can be assigned to y, and,
secondly, that no integral value can be assigned, ex-
cept one which is divisible by 6. For the product of
the assigned value and 5 must be exactly divisible by
6, since a must be an integer. But 6 is prime to 5,
and therefore must measure the value of y. Hence
the only values assignable to y are
6, 12, 18, 24, &c.
and the corresponding values of a are
5, 10, 15, 20, &c.
(227.) The object of the indeterminate analysis, as
applied to equations of the first degree, is to assign the
systems of positive and integral values of the unknown
quantities which satisfy them, if there be any such.
The general equation of the first degree between
two unknown quantities, is
A a + B y = C. (l.)
If w and y be the
denominators. It may also be supposed, that A, B,
and C, have no common measure; for if they had, the
entire equation might be divided by it.
These reductions having been previously performed,
if A and B be not prime, let their greatest common
measure be M, and let the whole equation be divided
by it. :
A C
† (2)
B
M. "+ ..T. & =
C
M, by hypothesis, does not measure C, therefore TVT
A B . . .
But M and M being in
tegers, since it is required that r and y should be inte-
gers, it is necessary that each term of the first member
of (2) should be an integer. And as the sum or
difference of two integers must be an integer, it is
evident that the first member is an integer, whatever be
the signs of its terms. The second member, how-
ever, is an irreducible fraction, which is absurd. Hence
there are no integers, positive or negative, which will
solve the equation (1) when the coefficients A, B are
not prime.
(228.) Let the coefficients A, B, be now supposed
prime, and let A 3 B. By solving the equation for
that unknown quantity which has the lesser coefficient
we have
is an irreducible fraction.
C B
Jº - - -
A T A 9.
If C > A the division indicated by % may be par-
II.]
tially effected. Let the integral part of the quote be
Q and the remainder R, and also let the integral part of
B us sº
the quote A. and the remainder be q and r, so that
which substitutions being made change the equation to
R. 7°
* = Q++ - a y – 3 y
sº. - R. 7-
:: * – Q+ ay = ** – ºry.
The first member of this being an integer, let it be t, so
that
t -
R.
A 3/
A.*
* R.
tº º hº sºng o 2.
... y = + t. [2.]
R. A.
Let the integral parts of the quotes +, + be Q', q',
and the remainders be R', r', and the equation becomes
R; r/
-- ſh/ — — a' f – —
y = Q-H = - 4't -- t
564
A L G E B R A.
Algebra.
S-V-->"
R! r/
gº º e- f f = --— — — t.
... y – Q-H 4 t = + --
In like manner, the first member of this being tº, we
have
, R! r)
t' = — — — ;
r
- R! r
... t = + - + t. [3.]
R!
As before, let the integral parts of the quotes →-, Trſ
be Q", q", and the remainders R", r", and we have
R. f 7."
f
t -: f/ — // f
Q ++ q" t! iſ t
R!! rºl
•." t * // ff ºf -: *g t!,
Q" + q" t -----
the first member of this being an integer, let it be t', so
that -
* = * – " ?
r/ 7./
R!! 7-'
t! == T fy t". [4]
ºr
If this process be continued, the denominator of the
fractions in the second member of some of the equations
[2,] [3, [4,] &c. must at length become = 1. For
since the numbers r, r', r", &c. are the several remain-
B A
ders from effecting the divisions indicated by TA’ T.T.
r
+, &c., the last remainder must be the greatest common
7"
measure of B and A, (98.) These numbers are by
hypothesis prime, and therefore their greatest common
measure is unity. Let us then suppose that the re-
mainder which becomes the denominator of [5] is E 1.
We have
t" = R!" — r" tº. [5]
By the equation [5] t” may be eliminated from [4,] and
by the equation thus found, t' may be eliminated from
[3, and by the equation resulting from this last process
t may be eliminated from [2,] so that we shall have y
expressed as a function of t'" alone.
By this equation y may be eliminated from [1, and
a will be obtained as a function of t'". The values of a
and y being thus obtained as functions of t, we may
obtain an unlimited number of pairs of values of a and
gy, by substituting for t in each of the values thus ob-
tained the terms of the series,
0, 1, 2, 3, 4, &c.
sy" 1, gmys 2, * 3, 4, &c.
(229.) We shall now illustrate these principles by
applying them to some examples.
Let the given equation be
13 a + 16 y = 97
97 16
se: – — —- l
1. – i. 9 [1]
- 6
* * = 7++, − y - Try.
Hence the second equation will be
f 6 3 The Inde
= -75 - -H-3) terminate
| 3 13 Analysis.
13 N-N-
Or 3y = 2 – T3T t. [2.]
The value of a may be obtained in terms of t, by
substituting this value of y in [1, which gives
97 32 16 65 16
= 15 - 15+ 4. t = 1st sº
By effecting the division [2,] becomes
gy = 2 - 4 t – $ t,
... t = – 4t,
... t = — 3 t'. [3.] -
Eliminating t between [2] and [3] we obtain.
- a = 5 – 16 t
gy = 2 + 13 t.
It is evident that the elimination of t' by these equa-
tions would give the original equations, as should be
the case, since they have been derived directly from it.
By substituting for t' successively in the above equa-
tions the values
0, 1, 2, 3, 4, &c.
we obtain the following systems of values of a and y :
a: = + 5, – 11, - 27, - 43, − 59, &c.
gy = + 2, -ī- 15, -- 28, -- 41, -ī- 53, &c.
and by substituting successively for t' the values
— 1, - 2, — 3, − 4, &c.
we obtain -
a = + 21, +37, -i- 53, -- 69, &c.
gy = — 11, - 24, - 37, − 50, &c.
Any of these systems of values substituted in the origi-
nal equation, will be found to change it to an identity.
Thus we have
13 × 5 + 16 × 2 = 65 + 32 = 97
13 x 11 + 16 × 15 = 143 –– 240 = 97
13 × 21 – 16 × 11 = 273 - 176 = 97
&c. &c. &c.
It appears that the equation admits but one solution
in positive integers, which corresponds to t = 0, and
is a = 5, y = 2.
It does not always happen, however, that the num-
ber of integral and positive solutions is limited. Let
us consider the equation
17 a - 49 y = — 8,
- 8 49
... a = - H + iy [1]
8 15
... a = − i + 2y+ Hy
t = – “. #y
17 17
8 17
. ... — ” 2
-. , – 8 2
•." * — — t
15 ' 15
A L G E B R A.
565
Algebra.
\-y-Z
15 8
... t = ·, "--,
... t = 7 t + 4 tº — 4
•.. t!! -: # t'. [4.]
By the equations [1,1 [2,] [3,1 [4,] the values of a and
gy being found in terms of t', we have
a = 49 t!" — 12
gy = 17 t” — 4.
[3.]
Substituting
1, 2, 3, 4, &c.
for t', we have
a = 37, 86, 135, 184, &c.
gy = 13, 30, 47, 64, &c.
So that the number of positive and integral solutions is
unlimited.
The number of negative integral solutions is also
unlimited, as may be proved by substituting,
- 0, - 1, -2, — 3, &c.
successively for t'.
(230.) It will be observed, that in the values of a
and y, obtained in terms of the last indeterminate
quantity which is introduced, the coefficients of the
indeterminate in the value of a, is the same with that of
3y in the original equation; and the coefficient of the in-
determinate in the value of y, is the same with that of a
in the original equation. This may be easily demon-
strated.
Let us suppose, as before, that the process stops at
equation [5.] By substituting the value of t'' obtained
in [5,.] for t' in [4,] the coefficient of t'" in the resulting
equation will evidently be ºf x r" = r.
r
being substituted in [3] the coefficient of t'" will be
f
This again
r 7" e
– ºr x → x r" = — r. The process of substitu-
r r
tion being continued to [2,] the coefficient of t'" will be
7-
* x + x # x r" = A, and in [1] it will be
B A 7' // -
A. X r X r!! X r" =
case, the coefficient of t'" in the value of a is — B, and
that of t'" in the value of y is A; the former being the
coefficient of y in the original equation, the sign being
changed, and the latter the coefficient of a.
It will be easy to generalize this demonstration. Let
the number of equations obtained before a remainder
= 1 is found, be n. The last equation will then be
# (n-3) -: R.(n -'s) mº r (n-3) * > # (n - *),
ºr
X 7 — B. Hence, in this
the numbers within the parenthesis denoting the num-
ber of accents with which each letter is affected. After
this substitution is made in the (n − 1)” equation, the
coefficient of t " - *) in it will be
(— cº-º) ×(- = -- r" - *
The substitution being continued to the (n − 2)” equa-
tion, the coefficient of t "T * in it will be
o, (n - 4) (n - 5)
e r r
* (n + 3) wº-ºº ºm-mº-º-º-ºs - sº-sº sºmeºmºse --- — 3. (n - 5)
( r )×( ...)x ( #) 7
In the (n − 3)” equation, the coefficient of the inde-
terminate t'"-" after substituting will be
VOL. I.
r (n − 2)
r(" – 3)
- (n - 4) - 5) -
7- (s * * * ,
— ºr (n - 3) — — 3r. * f –
= -- r *-*)
It appears, therefore, that after substitution the co-
efficients of the indeterminate t " - *) in each successive
equation, beginning from the last, have signs alternately
— and +, and that the values of these coefficients are
the successive remainders resulting from processes of
division. The coefficient of t!” - ?) in the third equa-
tion will be the first remainder, and in the second equa-
tion the coefficient A, and in the first the coefficient B.
One of these last will have the sign +, and the other
the sign –, according to whether the total number of
equations be odd or even. If it be odd, the second
equation will stand in an even order, counting from the
last ; and in this case the sign of A in the second equa-
tion will be +, or in general it will be the same with
that which it has in the original equation ; and that
of B in the first equation will be —, or different
from that which it holds in the original equation, and
vice versä when the number of equations is even,
SECTION XX.
On Continued Fractions. .
(231.) WHEN a fraction in its lowest terms is ex-
pressed by any high numbers, it is often desirable to
obtain a fraction nearly equivalent to it in lower num-
bers, and also in this case to determine the limit of
error to which we are subject in using this approximate
value for the true.
Let it be proposed to find an approximate value for
#3 in lower terms. To effect this, let both terms be
first divided by 159. Hence we obtain
l
1 Fººt —-
#####T.
If the fraction +º be neglected, the value 4 will be
too great, since the denominator will be too small. But
if tº be replaced by I, the value 4 will be too small,
the denominator being too great. Hence the value is
between + and 4.
A further approximation may be obtained by pro-
ceeding in the same manner with the fraction Tºº,
which gives
l
* = F-T—H-
** T 9 + 43
I
mºs-s-s-s-ºs-tº-mº-
'." Hº; =
3 + 1
9 + +;
# the fraction +} be neglected, § - +º, and '.'
3–H < ##3. But
–– = 4
3 + T **
Hence the value of the proposed fraction is less than
# and greater than #.
Now the difference between these two limits is gº,
4 E
r!" mºn 0) 3.
rtn-5)/
Continued
Fractiohs.
\ - º ſy
566
A L G E B R A.
Algebra, and, therefore, either of these two values is within # of
the true value. *
The approximation may be carried still further, by
treating the fraction 4; like the former, by which we
obtain
l -
4.32 = —
9 –– 1
1 + 1};
If the last fraction fºr be neglected here, 4 × 43.
Hence the denominator 9 + + is too great, "... the
denominator 3 + 1 is too small, and '.' the as-
9 -- 4 -
sumed value for the fraction would be too great. This
value, when reduced, is 44. Hence we infer
#}} < 4% and > *.
The difference between these limits is gºa, which is,
therefore, greater than the error to which we should be
subject in using either of these for the true value of the
fraction. -
(232.) The meaning of the expression
a -- 1
Ti
c + 1
d.Ti
&c.
must now be apparent. Such an expression is called a
continued fraction.
(233.) From the example already given, we may
derive the following rule for converting any ordinary
fraction into a continued fraction : , “Let the terms of
the fraction be submitted to the process necessary for
finding their greatest common measure, and let it be
continued until a numerator is found which exactly
measures its denominator, which, when the terms of the
fraction are prime, will always be unity ; the successive
quotes obtained in this process will be the denominators
of the fraction which constitute the successive members
of the continued fraction.”
Let – be the fraction which is to be converted into
N
a continued fraction, and let a be the integral part of
the quote § , b the integral part of the quote of N by
the first remainder, c that of the first remainder by the
second remainder, and so on. Hence we have
* = a +1
- b -- 1
c + 1
d + 1
e --
. &c. &c.
The value a is called the first approximation to
§ a -- } the second approximation, a + 1 the third
b + 1
cº
approrimation, and so on. Let these successive ap-
proximations be called æ, as, w, &c., and if they be
reduced to simple fractions, we have
Continued
Fractions,
\-y-
a t- a
_ a b + 1
r, ---
dº _ (a b + 1) c + a
* T b c + 1
_[(a b-i- 1) c + a d -- a b + 1
ar, E (b c + 1) d -- b
&c. &c.
By inspecting these values, the law by which they
may be derived from one another is very apparent. To
find the third as, the numerator of a, is multiplied by
the third quote d, and the numerator of a, added to
the result; and, in like manner, the denominator of r,
is found by multiplying the denominator of a, by the
third quote, and adding to the product the denominator
of al. Also, the numerator and denominater of a, are
found by multiplying the numerator and denominator of
a, by the fourth quote, and adding to the results the
numerator and denominators of a, And, in general,
the numerator and denominator of a, are found by mul-
tiplying the numerator and denominator of a, , by the
n” quote, and adding to the result the numerator and
denominator of an-s. .
Hence, if the numerator and denominator of re-º be
An-2, Ba-s, and those of an- be An-1, B, 1, and those of
a, be A, B, and that q be the n” quote, we have
An :- An- & Q + An-2 tº
B, E BA-1. q + B.-, .
(234.) We shall now determine the difference between
every two successive approximations. Let
An-
£, -2 = i.
An-1
ūn-l ---
n - 1
*...* @., tº A.-1. q + An-,
" B,-1. q + B,-,
Hence we find
* — ` *** An-1 An-2
n = 1 n = 2 - Ba- Ba-s
tºº An-1 Ba-, tº- An-2 B, R
hºmºnas B.-, B.-,
An- tº Q + An-2 A.-,
a'a – a n-1 E F--T-P-3 —
Ba- º Q –– B.-, Ba-
sº An-2 B,- *-* An- tº B.-,
(B. . . q + B, ,) B,-,
Hence it appears, that the numerators of the diffe-
rences between every two successive approximations
are equal, but have different signs, and that the deno-
minators are the products of the denominators of the
approximations themselves.
To determine the constant value of the numerators of
the differences, it will be sufficient to determine any one
of them. We have
a b + 1
ar, E a * = —-
_+ 1
– b :
a, - 4,
A L G E B R A.
567.
Hence, the constant value of the numerator of the
S-N->' differences is unity; and as the numerator of the dif-
ference of the first and second is + 1, that of the
second and third is – I, and so on.
(235.) Since the denominators of these differences
are essentially positive, it follows that the differences
themselves are alternately negative and positive, that is
as — a > 0
as ſc, 3 0
a', - a, P. 0
a's – a 30
&c. &c.
Hence we infer that
a', ‘ ar,
a's X as
as “3 a.
4, X as
&c.
Since the numerator of the difference between each
successive pair of approximations is constantly the
same, and the denominator constantly increasing, it
follows that this difference is constantly diminishing.
Hence we have
a', - a, X r, - rs ... a, ‘ a,
a's - as > r, - as . . 4, X 2,
ſº, - as > a. - a, . . as ‘ as
&c. &c.
Now since a, is evidently less than ar, it follows that
a', X- ar, a, 3 a., a, P. a., &c., and in general the approx-
imations of an odd order are 3 a., while those of an
even order are > a. Also, since ar, 3 a., 3 a... < z, &c.,
and all of these are 3 a., it follows that the further we
continue the approximations the nearer will those of an
odd order approach to equality with w, all, however,
being < a. And since as P. a. P are, &c., and all of
these > r, it follows that the further we proceed with
the approximations, the more nearly those of an even
order will approach to a, all being > a. The limit of
error caused by any approximation will be found by
taking the difference between it and that which is next
above it, if it be of an odd order; and that below it,
if it be of an even order.
But a still more exact limit may be determined.
Let the value of all the remaining part of the con-
tinued fraction after the (n-1)* approximation be y :
that is, if q be the m” quote, and r, s, &c. the succeed-
ing quotes, let -
y = q + 1
r —- 1
s —H l
&c.
Now we have
JC. tº: An-1 q + An-2
rt - B, - . . q + B, -g
But if we change q into y, this will become the exact
value of a .”.
* An-1 o y + An-2
– B. - 1 ſº 3/ + B.-,
g gº (An-2 B,-, <-- An- BA-2) 3/
n = * *- : •w-—a ----- —----------— »
(B, , $/ + B.-2) Ba-,
..". Jº ... dº
- A.- B-, - An-2 Ba-l
* T *-* T (B, , , y + B.) B. '
"... ſº - £, a fº - -H 3/
ſº "Tº (B, ... y + B.-:) B, , '
T 1
Jº T- ºn - 1
mº (Ba-1 • 3/ + Ba-s) Bn- º
Since y cannot be less than 1, it follows that the dif-
ference between ar, and a cannot be greater than
1 1
(Ba-l + Ba-s) Ba- Bºa-1 +- Ba- e B.-,
This gives the limit of error still more nearly than
I
B.- B.-,
before obtained. It also furnishes another
limit, scil. , though not so exact as that esta-
n - 1
blished above.
Thus we may infer that the n” approximation differs
from a by a quantity less than the fraction whose nu-
merator is unity, and whose denominator is the square
of the denominator of this approximation ; or still more
nearly by a fraction whose numerator is unity, and
whose denominator is the product of the denominator
of the n” approximation, and the sum of the denomi-
mators of the n” and (n − 1)” approximations.
(236.) We shall now investigate, by means of a
continued fraction, the value of the circumference of a
Continued
Fractions
S-y-Z
circle whose diameter is unity. This is known to be
nearly equivalent to 3,14159, or ######. Converting
this into a continued fraction, we have
a = 3 + 1
7 - 1
15 + 1
1 + 1
25 + 1
I + l
7 + +
Hence we find
a', - #, re - #, r, = ###, r. = ###, r. = }#}},
ar, - ####, r, - #####, as a ######
()
If we assume # as the true value, the error must be
less than But this is even nearer the
l - 1
7(7II) - 55.
true value, which is between * and ###, and is there-
fore nearer to it than the difference of these, which is
###. In cases, therefore, where extreme accuracy is
not required, } = 3} may be taken to represent the
circumference. This was the approximation of Archi-
medes. ! -
If the fourth approximation be taken for a, the error
must be less than the difference between the fourth and
l
fifth approximations, which in this case is II5 x 2931
3 5 5
<,00001. Thus then #} differs from the circumfe-
rence by less than the ten thousandth part of the
diameter.
568, 6.
A L G E B R A.
Algebra.
ū. SECTION XXI.
Of Exponential Equations.
(237.) AN exponential equation is one in which the
unknown quantity is an exponent, as a = b.
To explain the method of solving such an equation
as this, we shall take, in the first place, a particular
case. Let the equation be 3* = 243. Substitute suc-
cessively for a the integers 1, 2, 3, &c., and we find
31 = 3, 32 = 9, 38 = 27,
34 = 81, 35 = 243,
"... a = 5.
In this case it happens, that the second member of the
equation is an exact power of 3. But let us suppose
that the equation is 2* = 6, we have
22 = 4
23 = 8.
l
Consequently a is > 2, and < 3. Let a = 2 + Tºſ-
I ..
In this case T must be a proper fraction. We have
2*** = 6,
+ + 6 3 3 N*
•.” 22 2 º' = 6, ... ; a’ = — it T-3 •.. == *=
x 2- 2 T = g × 2 ( 2 )
Let 1, 2, 3, &c. be successively substituted for a ', and
we have
3 \| _ 3 3 Y_ 9
2 / T 2 ° 2 y - T.
3 9
Now -a < 2 and H-> 2, '...' a' > 1 and < 2. Let
3
2
Again 4 3
a ‘ a .
(# *_ 16 3
- 3-) = -g- > *g
Hence we infer that a "× 1 and < 2. Let
l
I +)* 3
! -- wºmmº-º-º-º-º: nº º *sº - —-
* = 1 ++, . (-; 2 ”
+ = (+).
3 T \ 8 J
Again, substituting 2 and 3 for a "we find
(+)= * * *
8 / T 64, 3'
9 NS 729 4 Exponen-
(+ = 513 > -a-. tial
Equations.
Hence it follows that w"> 2 and < 3. Let S-N-7
l
'Iſ – 6 tº º
40 = 2 + -a-, tº
R
9 \***_ 4 ... 81 ..)”- 4
(+) = -g, # (+ - -a-,
... 9 / 256 \ ="
** = (#.
By proceeding with this as before, we should find the
two successive integers between which the value of a "
lies, and so proceed another step; and thus the in-
vestigation might be continued as far as is desired.
We have, then,
1 l } l
* = 2 ++, w= 1 ++, "= 1 + ·, "=2++.
- l
1 + —
+ ºf 1 + l
1 + 7
= 2 ++ l
| -- l
l
+ l
2 + -ly.
Jº
º . . l l
If we omit the fractions →-, ..T. &c., we have, as a
Jº
first approximation,
l
If we include #, omitting #. #. &c., we have,
as a second approximation,
I ſº 1 5
* = 2 + —T = 2 + g = 3.
I + T2"
Again, by including +. we have
I l 3 13
1 +–T 1 ++,
l *
+ 2
and by continuing the process we should approximate
without limit to the value of a.
(238.) The general method then for resolving the
equation a * = b by approximation, is to find the
highest exact power of a which is contained in b.
Let this be a”, so that a” < b a”* > b. Hence the
value of a must be between n and n + 1. Let
1
a = n –– 7 '.'
l' I
a"+x = b . . . a”. a. -- b
b &”
Cl,
A L G E B R A.
569
s }.
* : **.
In the same manner ar' is found to be between the
Let y = n +2,
Finally we shall have
limits n' and n' + 1. and proceed
in the same manner.
l
* = n +; H
n" + 1
m" + 1
&c.
By continuing the process the value of a may thus
be obtained within any proposed degree of approxima-
tion.
By the results of Section XX. it appears, that
7. -- 7 differs from a by a quantity less than
ſº Also, that the third approximation dif.
I
(n'n" + n!-- 1)(n/n"+ 1)
fers by a quantity less than
and so on.
SECTION xxII.
Of Permutations and Combinations.
(239.) If there be any number of quantities or
things which we shall represent by letters a, b, c, &c.,
the various orders in which it is possible to arrange
these are called permutations. Thus, if there be two,
a, b, they may be arranged in either of two ways a b
or b a, and they are said to be susceptible of but two
permutations. If there be three, a, b, c, they may be
arranged in six ways, a b c, a c b, b a c, b c a, c a b,
c b a, and are said, therefore, to be susceptible of six
permutations.
(240.) Let it be required to determine in general the
number of permutations of which m letters, a, b, c, &c.
are susceptible.
Let a be the number of permutations sought, and 2
be the number of permutations of which m — 1 of the
given letters are susceptible. The remaining letter
may be placed either before the first letter in any one
of these permutations, or after the first or any succeed-
ing letter. It may, therefore, have m different places,
and for each of the z permutations of the m – 1 letters
there are m permutations of the total number. Hence
the total number of permutations is m z.
If m = 2 it is evident that 2 = 1 ... a = 2.
If m = 3 '.' z = 2, and a = 1 . 2. 3.
If m = 4 '.' z = 1 . 2. 3, and a = 1 . 2. 3. 4.
And in general we may infer, that by continuing the
process we should have
a = 1.2.3.4. . . . m — 1 . m.
(24.1.) If there be any number of quantities or things
represented, as before, by letters, a, b, c, &c., a group
consisting of any number of these, without regard to
their order, is called a combination ; and whenever two
such groups differ in a single letter, they are considered
as different combinations. Thus, if the given quantities
be a, b, c, d, a b c and a b d are different combina-
tions; but a b c, a c b, c a b are all the same combination.
&
Combinations are denominated combinations df two,
three, four, &c., according to the number of letters of C
which each group is composed. -
Each combination is susceptible of permutation.
Combinations differing in the order of their letters may
be called permuted combinations.
(242.) To determine the number of permuted com-
binations of n letters which can be formed from m let-
ters, m being supposed greater than m.
Let the number of permuted combinations of n – 1
of the m letters be 2, and the sought number be r.
Any one of the z permuted combinations of n – 1
letters being taken, and the remaining m – (n − 1)
of the m letters being successively annexed to it, will
give a corresponding number of permuted combina-
tions of m letters, and this being done with each of the
2 permuted combinations, we have a = z (m, n + 1).
If n = 2 *.’ m — 1 = 1. In this case it is evident that
z = m. Hence a = m (m – 1).
If n = 3 ... n – 1 = 2 ... z = m (m – 1)
a = m (m – 1) (m. – 2).
If n = 4 '.' n – l = 3 '.' z = m (m – 1) (m—2) '.'
a = m (m – 1) (m. 2) (m. – 3)
and in general
a = m (m – 1) (m. – 2) (m. – 3). . . . (m — n + 1).
(243.) To determine the number of combinations of
m letters which can be formed of m letters.
The number of permuted combinations was found
in the preceding article. Thus, to determine the number
of different combinations, it is only necessary to divide
the number of permuted combinations by the number
of permutations of which n letters are susceptible.
Hence the number of different combinations sought is
m (m – 1) (m. —2). . . . (m. – m + 1)
1 , 2 .. 3 e
Since this number must, from its nature, be an integer,
it appears that the continued product of all the integers
from m to m – (n − 1) inclusive, is divisible by the
continued product of all the integers from 1 to n inclu-
sive, n being less than m.
(244.) It is not difficult to prove that the number
of combinations of n letters to be made from m, is equal
to the number of combinations of m — n to be made
from m. Let m — n = 'n'. The number of combina-
tions of m' letters is
m (m – 1) (m. – 2) (m. — m/+ 1)
1 - 2 - 3. . . . . . . . . . . . . . . . m'
1 - 2 - 3. . . . . . . . . . . . [1]
First, let m – (n − 1) be greater than n + 1.
Then the preceding number may be expressed
m (m – 1) (m. – 2). . (m — (n − 1)) (m—n).. (n + 1)
1 - 2 - 3. . . . . . . . . . . . . . . . . . . . . . . . . . (m—n)
The factors from (m. – m) decreasing by unity to
n + 1 inclusive, are here common to both numerator
and denominator, and may, therefore, be omitted, and
the result is
Permuta-
tions and
ombina-
tions.
\-N-
570
A L G E B R A.
Algebra, m (m – 1) (m. – 2). . . . . .
\-y- 1 .. 2 .. 3
(m. – (n − 1) )
tº e º º ſº º º e º e º e º a º,
r". Those products in which but one letter a enters, Binomial
will have m – 1 factors of a. In these, therefore, a "- Theorem.
will be multiplied by each of the letters a, and the sum S--
which is the number of combinations of n letters to be
made from m letters.
Secondly, let m — (n − 1) be less than n + 1, and
therefore also m — n is less than n + 1. Let both
numerator and denominator of [l] be multiplied by the
successive integers from n to m – (n − 1) inclusive,
and it will become -
m (m — 1) (m. – 2). . . . . .
1 .. 2 . 3 . . . .
which, as before, is the number of combinations of m
letters to be made from m letters.
(n − 1))
(m —
. . . 7l
SECTION XXIII.
Of the Binomial Theorem.
(245.) IF the square, the third, fourth, &c. powers
of a binomial be obtained by actual multiplication, the
results will be as follows:
1. power. . . . . . a -- a.
2. power. . . . . . a" + 2 aſ a + a”.
3. power. . . . . . a' + 3 a.” a + 3 a a " + a”.
4. power. . . . . . a" + 4* a + 6 a.” a”-- 4a aº ––a4.
&c. &c.
In cases, however, where high powers are required,
the process of involution would be very laborious, and
where the exponent of the required power is expressed
by a letter, and not by a particular integer, we should
not be able to express it at all, unless the law were
known by which the exponents and coefficients of the
successive terms of the series are derived from the
exponent of the power.
The rule which determines the method of deriving
the exponents and coefficients from the exponent of the
required power in general, and independently of any
particular value which that exponent may have, is
called the binomial Theorem ; and the series thus found,
and which would also result from the continued multi-
plication by which the ordinary process of involution is
conducted, is called the developement of the power.
NEwTon first assigned the law by which the bino-
mial developement was governed, but did not give any
demonstration of it. Since his time, however, the
theorem has been submitted to rigorous proof.
(246.) We shall first consider the case in which the
exponent of the power is a positive integer. The
question then is, to obtain the developement of
(a + a)”, m being a positive integer.
If any number m of simple binomials of the forms
(a + a), (a' + a'), (a" -- a!"), &c. be multiplied so as
to form a continued product, it is evident that the de-
velopement of this product would consist of products
formed of every possible combination of m quantities,
which could be formed from the 2 m simple quantities,
r, a ', r". . . . a, a', a." If the accents be all re-
moved from letters a, and they be supposed to become
equal, the product formed of their combination will be
number of preceding terms.
of all these terms will be represented by a "-" multiplied
by the sum of all the letters, a, a', a!' . . . . Let this be
expressed by S (a). Hence the first two terms of the
developed product is a "+ +" S (a). Those terms
which have m – 2 factors of a will be multiplied by the
letters a combined in pairs; and will be equivalent to
a""" multiplied by the sum of every combination of
two of the letters, a, a', a.", &c. Let this sum be re-
presented by S (a), and the first three terms of the
product are a "+ +"-". S (a), + æ"-* S (a). And by
continuing the same reasoning, and preserving the
same notation, the continued product of m factors of
the form (x + a) (a + a') (a + a!") is, when
developed, f
a" + æ"-" S (a), +- a "-" S (a), -j- a "-" S (a), -- &c.
By the preceding section it appears, that the number of
terms in S (a), , S (a), , S (a), , &c. respectively are
, n (n − 1) . m (n − 1) (m = 2).
• T2 . . . ] ... 2
Now if the accents be removed from the letters a, and
they be supposed to become equal, we have evidently
&c.
S (a) = ma S (a), = º 2
_ m (m – 1) (m. – 2).
s(a) = *-gº-gº a
m (m – 1) m – 2) (m. – 3
S (a). = º 2 , 3 4 2 a.
&c. &c.
Hence we obtain
m (m — l) º
(a + a)” = a "-i- m a." a + 1 2 *-2 a.2
m (m – 1) (m. – 2) t”-8 as
1 .. 2 . 3
m (m – 1) (m. – 2) (m. – 3) ..., , , -
1 .. 2 . 3 . 4 a"-" at + &c. [1]
which is the binomial series.
(247.) It is plain that the coefficient of the r" term
of this series is the number of combinations of r letters
which can be formed from ºn letters, and that the ex-
ponent of a in each term is equal to the number of
preceding terms, and in the r" term it is therefore
r – l ; while the exponent of a is the given exponent
7m diminished by this number, and is therefore
m — (r – 1). Thus the r" term of the series is
m (m—1) (m-2) (m.–3). . . . (m-r-ţ-1)
1 .. 2 . 3 . 4 (r-1)
It appears that the sum of the exponents of a and a in
every term is the same, and = m + 1.
(248.) Each successive term of the series may be
conceived to be produced by multiplying the preceding
wn-r-El a’’ 1.
tº dº tº º e º 'º e
& tº . 0.
term by a fraction, one factor of which is Ta’ and the
other factor having for its numerator the given expo-
ment m, diminished by one less than the number of
preceding terms, and for its denominator the entire
Thus the third term is
A L G B B R A. 57.1
Algebra. ume .*.*, º-º Binomial
\-- found by multiplying the second by 770, 2 I & +, the (a — a)" = a " — ma"-" a + ºr D a"-8. a” †.
* . . . g 7m — 2 a. m (m – 1) (m. – 2) W.
fourth by Inultiplying the third by —a T. E. &c. In ----a--a a"-" as, &c.,
hi h * — 1 m – 2 & (252.) We have hitherto considered the binomial
this case the numeral factor 2 ” g-, &c. by series as representing the developement only where the
being multiplied into the preceding coefficient, produces
g OZ
the next coefficient, and the literal factor -, produces
&
the literal part of the term. The exponent of a is thus
continually diminished by one each step, while that of
a is increased by one. This generating fraction for the
7" term is
m – r + 2 0.
T — l Jº
The series terminates when the generating fraction be-
comes = 0. Let n be the number of terms; the gene-
rating fraction of the (n + 1)" term must = 0. Hence
by substituting n + 1 for r in the numerator of the
generating fraction, and putting it = 0, we have
7m — n – 1 + 2 = 0
‘. . m = m + 1.
Hence the total number of terms in the series exceeds
the given exponent by one.
(249.) If a be changed into a, and vice versi in the
equality [1] we shall have
m (m – 1)
(a + æ)" = a” + m a”-" a + I 2 g”-2 tº
ºn (m. - 1) (m. – 2) ...,n-, a
+ -i-,--a--a”
m (m – 1) (m. – 2) (m. – 3) ..., 4
-- 1 .. 2 . 3 - 4 a"-" r" + . . . . [2]
This series can only differ from [1] in having the terms
in an opposite order. It appears, however, that the
coefficients remain exactly the same, from whence we
infer, that in the binomial series [1] the coefficients of
every pair of terms equally distant from the extreme
terms are equal. This might also be inferred from
(244.)
(250.) The coefficients depending entirely on the
exponent m will be the same, whatever values a and a
be supposed to have. Let a = a = 1, and the series
becomes
m (m - 1)
1 .. 2
m (m – 1) (m —-2)
–– 1 .. 2 .. 3
In this case the second member of the equality is re-
duced to the sum of the coefficients. Thus it appears,
that the sum of the coefficients of the binomial series is
equal to that power of 2 whose exponent is equal to
the exponent of the binomial.
(251.) If the second member a of the binomial be
negative, it is sometimes called a residual. In this
case the odd powers of a will be negative, and as these
are factors of the alteriate terms beginning from the
second, these terms will be negative, and the binomial
will assume the form -
(1 + 1)" = 2* = 1 + m +
+ &c.
exponent m is a positive integer, and the demonstra-
tion derived from the properties of combinations; and
the continued multiplication of different binomials evi-
dently proceeds on that hypothesis. It may, however,
be proved, that the series will maintain the same form,
and be governed by the same law, when the exponent
is negative or fractional. The following demonstration,
given by EULER, extends to the cases where m is any
rational number, positive or negative.
(253.) Let (a + a) be expressed in the form
*( 1 + #) and we have
G-For-r(l –– + )
tº (Z
or if z = ---,
Jº
(a -- a)" = a " (1 + 2)”
The question is then, to show that the developement
(1 + 2)” = 1 + m -----ºr P.
m (m – 1) (m. – 2)
++====a+ =
is true whatever be the value of m.
Let the problem be converted, and let us inquire
what algebraical expression has the preceding deve-
lopement when m is a fraction. Let the sought ex-
pression be y, so that
- - n — I
y = 1 + m z + * D -
m (m – 1) (m. – 2)
++=a+== | 1.]
Let m' be another fractional exponent, and y' the
corresponding algebraical expression or equivalent for
the series, “..." .
rºl - 2
*-* + &c.
2°, + &c.
m' (m' – 1) 22
1 .. 2
m! (m' – 1) (m’ – 2) ...,
2.
++=a+–a–2 + &c. [2]
If these two equalities be multiplied, the first member
of the result will be y y'; but to ascertain by direct
multiplication the form of the second member would
be attended with some difficulty. It is evident, how-
gy' = 1 + m,' 2 +
ever, that the product of the second members of ,
[1] and [2] will necessarily be the same in form,
whatever m and m/ be supposed to represent; and,
therefore, whatever form that product will have when
7m and m' are supposed to be positive integers, will
necessarily also have when they are fractions. But in
the former case we have
(1 + 2)" = 1 +m2 +**: P × +
l .. 2
7m (m – 1) (m. – 2)
1 .. 2 .. 3 28 + &c.
572
A L G E B R A.
Algebra.
S-N-7
- ov,' ' —
(1 + 2)" = 1 + m'z ºf p 28
m' (m' – 1) (m' –
+H===
2) 2* + &c.
ºn (m. - 1) ...,
T--a-.
ºn (m. - 1) (m. – 2), e
1 .. 2 .. 3 *4 &c.)
m' (m/ 1)
... (1 + 2)"+" =(1+m2 +
+
m" (m'— I) (m' – 2)
—- 2*
Ha-‘a’ + T-a--a
e-se)
x(| + m + 2
But also
(m+m') (m+m'–1)
(1+z)”*" = 1 + (m + m) z +-
2* + &c.
Hence the second member of this last equality gives
the form of the product of the second members of [1]
and [2] when m and m' are positive integers, and the
same form must continue when they are fractions. Thus
we have
! .. 2
& 7m —— m/) (m. –- m' – 1
yy=1 + (n+m} = +& # ) ( +. )
2* + &c. [3.]
By continuing the same reasoning, if m be supposed
successively to assume the values m', m", m", &c. all
fractional, and y', y", y", &c. be the corresponding
equivalents of the series, and r = m + m' + m," +
&c., we shall have
gy y'y" y". . . . = 1 + 2 + º-P 22
r (r – 1) (r. – 2).
+ 1 a ... 3 2* + &c. [4.]
Now suppose m = m' = m." = &c., and let q be the
number of repetitions, so that r = q m, the equality [4]
hecomes
m q (m q – 1) Ö
~- 2*
1 .. 2
m q (m. q – 1) (m. 1 — 2)
++=====
gy? - 1 + m q 2 --
2* -- &c.
p
*-**
or since m is supposed to be fractional let m =
— -- p (p − 1) p (p− 1) (p−2).
y= i+p = + -ā-- * +; . 2 . a zº, &c.
But the second member of this, since p is a positive
integer, is equal to (1 + 2)” '.'
gy q = (1 + 2)?
... y = (1 + 2) + == (1 + 2)"
Hence the developement
. 7m – 1
(1 + 2)” = 1 + m z ++++
2?
+ =
holds good when m is a fraction and positive.
To extend the proof to negative exponents it is only
Binomial
Theorem.
p (ºr 2) 2* + &c.
necessary in [3] to suppose m' = — m '.' m + m' = 0
‘. . y y' = 1,
º ** I - 1
3/ 7 = y
But we have already proved that
gy' = (1 + 2).”
‘.." v = ——, ± (l -m' -- $º
v= HR = a + 2)--- (1 +3)
and, therefore,
m (m. — 1)
(1 + 2)” = 1 + m z + → z” &c.
l .. 2
is true when m is negative.
Thus, then, the binomial theorem is extended to all
cases in which the exponent is a rational number,
whether positive or negative.
(254.) This extension of the principle being made,
there will be no difficulty in applying it to the approxi-
mation to the roots of numbers.
In the series
tºmºmºry 2
a m. (m. — 1) a +&c.;
Gro-->{ + 7m . ++
TT2 T.I.,
I
Let m = -. Hence
70,
rt &==- ri 2"T" l (Z f n — 1 a?
A/ w/ 1 —— — . — — — mºss-ºs-s-s
a + a + * { ++, a m 2 m a”
1 m — 1 2 m – l aº T
— — . — — — &c.
* 7 ºn 3 m a.º. *j
Let it be proposed to apply this series to the extraction
of the cube root of 31. To effect this, it is necessary
first to find the nearest complete cube to 31, which is
27 = 38 ... a v27 = 3. Hence
31 = (a, +- a) = 27 -- 4
-resºzº"-- 1 4 l l 16
• a wººl. 4 – ; — , — — — — . ~~
27 -- *(i+. 27 3 3 729
+ * *.*, * –&e
3 3 9 19683 º
, a *-* 4 16 320
ºv'3] = 3 + -ī- — –tº– + –tº– —
* 27 T 2137 t 531441
The first three positive terms of the series expressed in
decimals, are
3 = 3,00000
4
37 = 9.44° W = 3,14875
320
53I44T T 0,00060
and the first two negative terms are
16 l
T gia; - T 0,00731 = - 0,0.0737
2560 0.00006 > = — 0,
430,4672T ~ T * J
... a v31 = 3,14138
A L G E B R A.
573
Algebra,
\-->
(255.) A series is said to converge, when the numerical
value of each term is less than that of the preceding
term ; and is said to converge more or less rapidly, as
the ratio of each term to that which succeeds it is a
greater or lesser ratio. By means of such a series we
can always approximate in rational numbers to the
value of that quantity of which it is the developement.
Such a developement may be considered to be numeri-
cally equivalent to the quantity from which it was ob-
tained. It, however, frequently happens, that the suc-
cessive terms of the developement, instead of decreasing,
increase. In this case, no numerical equality exists
between the series and the quantity whose developement.
it represents; and the sign = placed between them, is
to be understood only as indicating, that the one is
obtained from the other by a certain process which has
been instituted, and these observations are equally
applicable, whether the developement be obtained by
the binomial theorem or any other way.
If any number of terms of a converging series,
beginning from the first, be taken to represent the value
of the whole, there is a certain error introduced, the
limits of which may in some cases be assigned. Let
the series be
a – b + c – d – e – f -- &c.
the terms being supposed to decrease. Let it be re-
quired to assign the error to which we are subject in
taking a - b + c for the whole series. Let r be the
quantity to which the whole series is equal. Since
each positive quantity is greater than the negative
quantity which immediately succeeds it, it follows, that
the successive quantities (a – b), (c – d.), (e — f), &c.
are all positive, and, consequently, the sum of any
number of them, commencing from the first, will be less
than ar. Hence
a > a. — b
a > a. -- b + c : - d.
a > a - b + c – d – e – f
&c. &c.
Again, for the same reason, the successive quantities
(– b – c), (– d –- e), (— f-- g), &c. are negative,
and, therefore, when the sum of any number of them
be added to a, the result is greater than ar, so that
a 3 a.
a 3 a - b + c
a ‘ a - b + c -- d -- e
2 < a. – b + c – d –– e – f –-g
&c. &c.
Hence it follows, that the value of a is between the
values of a and a — b ; it is also between those of
a – b and a — b + c ; also between those of a - b + c
and a — b + c – d, and so on. Therefore, if a be
taken as equal to a, the error will be less than b ; if
a - b be taken for a, the error will be less than c, and
SO OIle
To apply this to the example already given, we have
*W3i = 4 16 L 320 2560
31 = 3 + 27 ăisitääriäi 4304672] --
If the first two terms be taken to represent "V3T, the
assumed value will be greater than the true value, by a
fraction less than 3+3+ ; if these terms be taken, the
WOL. I.
assumed value will be less than the true by a fraction Binomial
less than gººrt, &c. Theorem.
(256.) The general rule for the application of the \-y-Z
binomial theorem to the approximation to the roots of
numbers is as follows: Let, the n” root be sought :
find the nearest complete #h power to the proposed
number, and let this be p", and let the difference be-
tween this and the proposed number be q, so that the
proposed number being N, we shall have
N = p" -- q
when p" < N, and
= p" — q
when p" > N. In this series
- - - 1 a. I m – l a”
* = a, n - 1 -- – . — — — . *=es
(a + a) Jº {1 + 71, ſº 7, 2 m, ºr 3
I m — 1 2 m – l as Y
+ º-º: sº
substitute p" for a, and q for a. The result will be a
converging numerical series, provided a be less than r.
The several terms being reduced to decimals, and com-
bined by addition or subtraction, as indicated by the
signs, will give the value of the root with any required
degree of approximation.
(257.) Since the successive terms of the develope-
ments of (a + a)" and (r — a)” differ in nothing but
the signs of the alternate terms, beginning from the
second, it follows that if their developements be added,
the result will be twice the sum of the alternate terms,
beginning from the first ; and if they be subtracted,
their difference will be twice the sum of the alternate
terms, beginning from the second,
(258.) Also, since the alternate terms of the series,
beginning from the first, contain only even powers of
a, and the alternate terms, beginning from the second,
contain only odd powers of a, it follows that the de-
velopement of
(a + a)" -- (a — a)”
contains no odd power of a, and that of
(a + a)” – (a — a)”
contains no even power of a.
(259.) If a be a quadratic surd, such as v'5, v3, &c.
the developement of
(a + a)" -- (a — a)”
will be rational, since all the even powers of w/b, v3.
&c. are rational.
Also, if a be an imaginary quantity of the second
order, such as V–5, V–3, &c. the developement of w
(a + a)" + (a – a)"
Also, in this case, the developement of
(a + a)" — (a — a)"
w/ - T
will be real. .
(260.) The developements obtained by the ordinary
process of division, may also be obtained by the bino
mial theorem. Thus, by division we find
1 - 1 b
a + b a
The same inay be obtained by the binomial theorem,
l
a + b
will be real.
by substituting for
its equivalent (a + b)".
4 F
574 A L G E B R A.
Algebra.
-N-
SECTION XXIV.
Method of Indeterminate Coefficients.-Of Series.
(261.) IF the form of the developement of any quan-
tity be assumed, the determination of the developement
is reduced to the investigation of the values of the co-
efficients of the powers of that quantity by which the
series is supposed to be arranged. These coefficients
being supposed to be independent of the latter quantity,
will be the same, whatever value be assigned to it; and
on this fact is founded that method of developement
called the method of indeterminate coefficients,
Let the formula to be developed be
(º,
WTWE'
and let the form of the required developement be
Ao + A, a + As a " + A, a 4 + A, r + &c.
Ap, A1, A, . . . . being quantities indeterminate or un-
known, but supposed to be independent of a. Equa-
ting the series with the formula it represents, we have
(Y,
b + b'a, -
Since the values of the several coefficients in each
member of this equation are independent of a, they
will be the same whatever value a be supposed to
receive. If a = 0 the equation becomes
= A, + A, a + As a 2 + A, a " + A, a " + &c.
(N,
+ = A,
which determines the value of the first coefficient.
Making this substitution, clearing the equation of
fractions, bringing all the terms to the same side,
and arranging them by the ascending powers of ar,
we have
A. b
a b'
++
a + A, b' |aº-H A, b a. * + . . = 0
+ A, b' | + As b'
a 9-1-A, b
+ A, b'
which being divided by a becomes
A, b | + A, b a + A, b tº + A, b ré + . . = 0
b' *
+ +| + A, b + A, W +A, W
In this, if a = 0 we have
a b/ a b'
A, a' + b = 0, '." A = — b? "
This condition being observed, and the equation again
divided by ar, it becomes -
Ag b + A, aſ a + A, a' | wº +... = 0
a b/? f /
--- + A, b' | + A, b
and a being again supposed = 0, we have
a b% a b/2
A, b – b? = 0, * A = −.
and by continuing the same process we should find
a b's a b'4 a b/5
As = — h4 ° 4 = b5 ° s = – bº , &c.
In effect, each succeeding coefficient is found by multi- Methods of
plying the preceding one by ".
sº b' Coefficients.
b 2 S-N-2
and each term is found by multiplying the preceding
term by
b/
- —- ?.
b
(262.) The principle here used when generalized,
proves that if an equation of the form
A + B a + C a 3 + D a 3 + . . . . . . - 0
be fulfilled independently of a, it is necessary that each
of its coefficients severally should = 0; and it is, in
fact, equivalent to the several equations
A = 0, B = 0, C = 0, &c.
(263.) From this principle we may immediately infer,
that if an equation of the form
a + ba -- ca 3 -i- da' +... = A +Ba + Crº-H Da' +
be fulfilled independently of a, (that is, be true
whatever value be ascribed to a,) we shall have
a = A, b = B, c = C, &c.
For it may be reduced to the form
(a – A) + (b – B) a -- (c' — C) w” -- . . . . = 0.
Hence by (261) we have
a — A = 0, b – B = 0, c – C = 0, &c.
‘. . a = A, b = B, c = C, &c.
(264.) We have before stated, that in the application
of the method of indeterminate coefficients the form
of the developement is assumed. It may so happen,
that the form assumed is one in which the given quan-
tity cannot be developed. In this case the process will
lead to some manifest absurdity, indicative of the false-
hood involved in the equality which was instituted
between the given expression and the proposed form
of developement.
As an example of this, let it be required to deve-
in ascending integral and
lone the fraction
p 3 º, 2
positive powers of a, so that
3.T. = A + B a + C w” -- D a " + E a *-ī- &c.
Clearing this of fractions, and bringing all the terms
to the same side, we have
— 1 + 3 A a + 3 B a *-i- 3 C | a *-i- . . . . = 0
— A — B
Now if a = 0, we have – l = 0, which is absurd, and
shows that the expression cannot be developed in the
form required.
If, however, the original expression be resolved into
l
its factors + and , the latter may be developed
in the required form, and we find
l I *} ſp? 4, 3
à i = a- + i + -ā; + ºr +&c.
* †, = a + -ā; ++, ++, + &c.
* l q)-1 go al ºn?
5.T. = a- + -ā; + 3 + 3 + &c.
A L G E B R A.
575
Algebra.
\--'
(265.) Let the expressionot be developed be
a + aſ a
b + b/a -- b” a 2 *
and the form of the developement being as before, we
have
a + a'a,
b -H b/a + b” aſ *
which when reduced becomes
= A, + A, a + As a "+ Aaa"-- &c.
A, b | + A, b a + A, b tº + A, b |a'+A, b wº--..0
— a + A, b' + A, b/ As b/ A, b/
— aſ 1 + Ao b// A, b'ſ As b'ſ
since this is fulfilled, independently of a, it gives
Aob — a = 0
A, b + A, b' — a' = 0
A, b + A, b' + A, b" = 0
A, b + As b' + A, b" = 0
A, b + A, b' + A, b" = 0
&c. &c.
Hence we obtain
(Z
Ao = Tº
b' . . a’ — b' a + a' b
A = – ºr A, 4 + = —#—
b' b// a b/? — a'b' b — a b b%
As º “ TT Al sº Ti, o = b 3
b' b//
A. = -- A. --- A.
b/ b//
A = --- A. – H-A,
&c. &c.
and in general
b/ b'ſ
A. = --- A.-, --- A.-,
Thus we obtain a general rule for determining each
successive coefficient, viz. “ Multiply the preceding
f th
-º-, and the last but one by — T; ,
and the sum of the results is the coefficient sought.”
This rule applies to all the coefficients after the
second term. The first two terms, however, must be
determined by the formulae established for them in
particular.
Had the expression to be developed been
a + a'a, + a” a *
b -- b/w -- b" a 2 + b"a *
we should have had
A, b – a = 0
A, b + A, b' — a' = 0
A, b + A, b’ + A, b" – a”
A, b + A, b'+ A, b" + A, b"= 0
A, b + A, b' + A, b" + A, b"= 0
&c. &c.
coefficient by —
and in general
A, b + An-1. b' + An-, . b" + An-2 b" = 0.
Hence we find -
A. = -:
*=º
b
b' 2./ — a h" f Methods of
A = — — — A' -- -*— = a b' + b a Indeter-
b b b? minate
b' b” . a!! Coefficients,
As * - T, A, tºmºg -- Ao + + S-N-
a bºº — aſ b b' — a b b" + a” bº
tº- 53 ~ y
and the remaining coefficients would be determined by
b/ b'ſ bºlſ
A. = --- A.--—A, --- A.
b/ b” bºr
A = -- A. --- A. --- A.
b! - b'ſ bºlt
A. = -- A. --- A.--- A.,
&c. &c.
and in general -
b/ b'ſ b'ſ
A = --- A.----A -, --- A.-,
(266.) In the three examples which have been given,
of the application of the method of indeterminate co-
efficients, it may be observed, that in the first, each term
was derived from that which immediately preceded it
by multiplying by a constant factor; in the second, each
term was derived from the two which immediately
preceded it by multiplying each of them by a constant
factor, scil., that which immediately preceded by —
f I
+ a, and the other by — -- a *. In like manner,
in the third example each term is derived from the three
preceding terms by multiplying them respectively by
I b". a!?, b". a 3,
the constant factors — b - -,
- * –
and adding the results.
Series formed or generated in this way are called
recurring series, and the system of constant multipliers
is called the scale of relation. The order of the re-
curring series is determined by the number of constant
multipliers in the scale of relation. Thus, the first of
the preceding examples presents a recurring series of
the first order, the second gives one of the second order,
and the third one of the third order, the scales of re-
lation being respectively ~ is
(-4). (-
b ô iſ b'ſ
* Cºmº — — ºr * — — . * * *
( - *, # **, ; :)
It is evident, that by continuing the same reasoning we
should find in general that the developement of •
a + a'a' -- a” a “ —- . . . .
b'
→ *,
a ſº-1) a"-1
the scale of relation would be
b(*)
* * * * * . . . . . . - ***)
5-H bºn–E Wrº-E . . . . bº) as
would be a recurring series of the n” order, of which
b' b'ſ 5'll
— — — ar, – —- a *, - — arº
It is evident, that a recurring series of the first orde
is a geometrical progression. ... ."
4 F 2
576
A L G E B R A.
Algebra.
\-y-Z
SECTION XXV,
Of Logarithms
(267.) IN the indeterminate equation y = a for
every value, whether positive or negative, which is
assigned to w, there will result a corresponding value
of y, and vice versä, any numerical value whatever
being assigned to y, there will be a corresponding
number, which, substituted for a, will verify the equa-
tion. This, however, is on the condition that a is not
= 1, for if it were, y would also be = 1, whatever
value should be given to ar. Let N, N', N” . . . . be
any values of y, and let the corresponding values of a,
determined either exactly or approximately, be n, m',
n", &c. we have
N = a”, N' = a”, N' = a”", &c.
The value of a being arbitrarily chosen, subject to the
exception already mentioned, and being supposed to
remain the same, there will be a fixed and constant
relation between the numbers expressed by y and ar.
This peculiar relation is expressed by calling a the
logarithm of y. Thus, n is the logarithm of N, n' of
N', &c. The constant quantity a is called the base of
the logarithms.
(268.) In the theory of logarithms, therefore, all
numbers are considered as powers of some one num-
ber, which is called the base, and the exponents of the
powers are called the logarithms of the numbers. The
logarithm is usually expressed by log, or simply the
letter 1 placed before the number; thus log y, or ly,
signifies the logarithm of y.
(269.) If the base of the system be 10, we have
evidently
l (100) = 2, l (1000) = 3, l (10000) = 4, &c.
since 100 is the square of 10, 1000 the third power,
10000 the fourth power, &c.
(270.) If all numerical values whatever be supposed
to be successively ascribed to y, and written in one
column, and that the corresponding values of r in the
equation y = a” be determined, and written in another
column, the corresponding values being placed oppo-
site to one another, we shall have what is called a
table of logarithms, so that when any number is given,
its logarithm will be found registered in this table, and
vice versä, when any logarithm is given, the corres-
ponding number may also be found. The nature of
such a table, and the method of constructing it, we
shall more fully explain hereafter. We shall at present
show how such a table would be instrumental in expe-
diting several numerical operations.
Let y, y', y". . . . be several numbers, and a be the
base of the logarithms, we have
gy = aly, y' = a 'W', y' = a +", &c.
By multiplying these we obtain
W . . // – a y + y^+ (y' + . . . .
3/3/3/". . . .
? ...!?
But also $/ ?/ ?/". . .
*.* l (y y' y"..) = ly + ly' + ly" + . . . .
That is, the logarithm of the continued product of any
numbers is equal to the sum of the logarithm of the
factors. -
• == gº (yy’s”. . . .)
Hence, if it be required to multiply several numbers Logarithms.
together, it is only necessary to obtain their logarithms S-V-'
from the table; add these logarithms together, and
then obtain the number of which the sum is the
logarithm. Thus continued multiplication is reduced
to continued addition.
(271.) Let the equations y = a”, y' = aly' be
divided one by the other, and we obtain #. - aly - ly'.
S_ s
But also '. - a “º iſº ly'. That
= lu —
#)=&
is, “The logarithm of the quote is equal to the loga-
rithm of the dividend, minus the logarithm of the
divisor.”
If then it be required to divide one number by
another, let the logarithms of these numbers be taken
from the tables, and that of the divisor subtracted from
that of the dividend, and let the number be found in
the tables whose logarithm is equal to the remainder,
this number is the quote. Thus division is reduced to
subtraction.
(272.) Let both members of
y = a 9
be raised to the n” power, and it becomes
y” --- a”ly
‘... l (y") = n ly.
That is, the logarithm of any power of a number is
equal to the logarithm of the number multiplied by the
exponent of the power.
Hence, to obtain any required power of a number,
let the logarithm of the number be found in the tables,
and let the product of that and the exponent of the
power be found by the rule (270,) and this being ob-
tained, let the number be found in the tables of which
it is the logarithm. This will be the required power.
(273.) Let the n” root of both members of
y = a',
be taken, and we have
º- ly
"My = an
ly
*=-es-e.
. . . " V, -
... l "v y = 7?
That is, the logarithm of any proposed root of a number
is obtained by dividing the logarithm of the number by
the exponent of the root.
Hence, to obtain any proposed root of a number, let
its logarithm be taken from the tables, and let the
number be divided by this by the rule (271,) and then
let the number be found in the tables whose logarithm
is equal to this quote. This number is the required
root.
(274.) Thus it appears, that by the aid of a table of
logarithms we shall be able to reduce all calculations
where products, quotes, powers, or roots, are required
to simple addition and subtraction.
(275.) The number most commonly taken for the
base of a system of logarithms is 10. However, if a
system be computed with respect to any base a, it will
be easy to obtain from it a system relatively to another
base a'. Let y be any number, and let ly be its
logarithm relative to the base a, and ly relative to the
base a'. We have
3/ F aly 3/ = aſ 'y
• a '9 = aſ 'y.
A L G E B R A.
577
(279.) If a number end with any number of cyphers, Logarithms.
they may be cut off, and the logarithm of the remaining \-N-
Algebra. Taking the logarithms of these relatively to the base a,
\-v- we have
– ". . . . ... — ”
ly = 'y la' ... ly T la
(276.) Hence, when the logarithm of numbers rela-
tively to any base is known, the logarithms of numbers
relatively to any other base may be found by dividing
the given logarithms by the logarithm of the new
base in the given system.
(277.) In the computation of logarithmic tables, it
is not necessary actually to calculate the logarithms of
fractions, because they can always be found from those
of their numerators and denominators by the rule in
(271.) Neither is it necessary, in the first instance, to
compute the logarithms of any but prime integers, for
all others being products of these, their logarithms may
be derived by adding those of their factors, (270.) Thus
l6 = lº + l2.
(278.) We shall proceed first to explain the method
of using tables of logarithms, and then to show the
methods by which these tables are computed.
Let us suppose that the base of our system is 10.
The only numbers whose logarithms are rational, are
100, 1000, 10000, &c. All others must be expressed
approximately, and we shall suppose the approximation
carried to seven decimal places.
If in the equation y = 10°, a = 0, we have y = 1.
Therefore the logarithm of 1 = 0. This is common to
all systems. If c = 1, 2, 3, 4, &c., we have y = 10,
y = 100, y = 1000, y = 10000, &c. Hence the loga-
rithm of the base itself is - 1, which is also common
to all systems.
If w = — 1, - 2, — 3, &c.
R 1
9 = I0 y = 105, 9 = 7000
The logarithms of all numbers less than 1 are negative,
&c.
l
and the logarithm of 0 is — † = - 2, while the
e © 1
logarithm of an infinitely great number is + 7 =
•+ op.
The logarithm of an integer 3 10 is 3 1, and,
therefore, there is no significant digit before the deci-
mal point in the value of such a logarithm. In this
case, 0 may be conceived to precede the point, which
always happens, therefore, when the number is expressed
by a single digit.
If the number consist of two digits, it is between 10
and 100. Its logarithm is therefore × 1 and < 2, and,
therefore, the digit which precedes the point in the
logarithm is 1.
If the number consist of three digits, it is between
100 and 1000, and its logarithm is between 2 and 3.
Therefore 2 is the digit which precedes the point in
the logarithm.
In general, if the number consist of n digits, it is
between 10"- and 10", and its logarithm is between
n — 1 and n, and therefore n - 1 must be the digit
which precedes the point in the logarithm.
The digit which precedes the point in the logarithm
of a number is called the characteristic of the logarithm.
Thus the characteristic is always that integer which is
one less than the digits of the number.
part found, as many units being added to it when so
found as there were cyphers cut off. For let the value
of the number without the cyphers be N, and let m be
the number of cyphers cut off. The original number
is N × 10", the logarithm of which is lN + m.
In like manner, if a number be divided by a power
of 10, the logarithm of the quote may be found by sub-
tracting from the logarithm of the number as many
units as there are in the exponent of the power. For
let the number be N,
N
Z 10” T lN — m.
Thus, to obtain the logarithm of any number having m
decimal places, let the logarithm of the number con-
sidered as an integer be first found, and then let n be
subtracted from it. -
(280.) It appears from the uses of logarithmic
tables already explained in multiplication, division, &c.
that two processes are required in every operation:
1. to find the logarithm of a given number ; and 2.
to find the number corresponding to a given logarithm.
1. To determine the logarithm of a given number.
The given number must be either integral or frac-
tional. If it be a fraction, its logarithm is the difference
between those of its numerator and denominator. If
the fraction be expressed as a decimal, its logarithm
being found as an integer, it is only necessary to sub-
tract from the characteristic as many units as there are
decimal places. If the proposed number be composed
of an integer and a fraction, it can be reduced to a
fraction. Thus the determination of the logarithm of any
number whatever, is resolved to the determination of the
logarithms of integers.
The tables are usually constructed so as to give the
logarithms of all integers within a certain limit. If
then the integers whose logarithms are required be
within this limit, their logarithms will be immediately
found annexed to them in the tables.
If, however, it be desired to determine the logarithm
of an integer greater than any tabulated integer, let
the characteristic be first determined by the number of
places. Then let such a number of decimal places be
pointed off as will reduce the number of integral places
to the greatest number of places in the tabulated integers.
Thus, if the number of integral places in the proposed
number be 8, and the greatest tabulated integers have
but 5 places, it will be necessary to cut off three inte-
gral places by the decimal point. This will evidently
produce no other effect upon the logarithm of the
number, than to diminish its characteristic by as
many units as there are places cut off. So that if the
logarithm of the number so modified be determined,
that of the sought number may be immediately obtained,
by adding as many units to the characteristic as there
were places cut off. :
After the integral places have been thus reduced,
let the value of the number be N. Let p be the num-
ber of places cut off by the decimal point, so that the
original number is N × 10°. Find in the tables the
two integers between which the value of N lies. Let
them be m and n + 1 ; so that N > n and < (n + 1),
and let the logarithms of n and n + 1 be found. We
shall show hereafter that when numbers so high as m,
578
A L G E B R A.
Algebra.
~y-Z
N, and (n + 1), are supposed to differ by a number
less than unity, we may assume, without any consider-
able error, that the numbers are propertional to their
logarithms, so that we shall have
N — n : (n + 1) — n : , l\ – ln : l (n + 1) — lin,
or, IN — ln = {l (n + 1) - ln × (N — n)
... IN = { l (n + 1) – ln 3 × (N — m) + ln.
In fact, the error which this proportion entails upon the
value of lN, does not affect any of the first seven deci-
mal places, and beyond these we do not usually require
to extend the calculation.
The value of lN being found, we may immediately
determine that of the given number N × 10P.
l (N × 10P) = {l (n + 1) — ln (N – n) + ln + p.
By this formula, the numbers m and n + 1, and their
logarithms being known, lN may be computed
Example. Let
N × 10P = 34.735879.
If we suppose that integers as far as those consisting
of five places are tabulated, it is necessary to point off
three decimal places. Hence
N = 34735,879, p = 3, n = 34735
- m + 1 = 34.736,
lm - 4,5407673,
! (n + 1) — ln = ,0000125,
N – ??, tº ,879,
! (N × 101) = ,879 x ,0000125 + 4,5407673 + 3,
... l 34735879 = 7,5407783.
2. To determine a number when its logarithm is
given.
The given logarithm may be positive or negative.
First, Let it be positive. :
If the given logarithm, be found in the tables, the
corresponding integer will be prefixed to it.
If not, let us suppose, in the first instance, that its
characteristic is that of the highest number included in
the tables. In this case, its value will be found to be
between two successive tabulated logarithms, let these
be lin and l (n + 1), and let the sought number be N.
By the formula already established we have
N — m = lM — lin
! (n + 1) — ln
lN — lin
=Fairy-in-F".
for in this case p = 0.
Example. Let the proposed logarithm be
4,7325679,
4 being the highest characteristic in the tables.
find by the tables
72.
We
ln = 47325626,
... IN — lim = ,0000053,
* = 54021,
l (n + 1) - ln = ,0000081,
• N - # -- 54021 = 54021,65
If the characteristic be less than the greatest charac-
teristic of the tables, the formula for approximating to
N will not give sufficient accuracy. In this case,
therefore, it will be necessary to add as many units to
the characteristic, as will render it equal to the highest Logarithms.
characteristic of the tables; and to compensate for this, \-N-
it is only necessary to point off as many additional
decimal places in the result as there were units added
to the characteristic.
If the characteristic of the given logarithm be
greater than the greatest characteristic of the tables, it
is necessary to subtract as many units from it as will
render it equal to the highest tabular characteristic,
and it will be necessary to multiply the number found
by that power of 10, whose exponent is equal to the
number subtracted from the given characteristic.
Secondly, Let the given logarithm be negative.
Let as many units be added to it as will render it posi-
tive, and make its characteristic equal to the highest cha-
racteristic of the tables. This being done, let the cor-
responding number be found in the manner already
explained, and let it be divided by that power of 10
whose exponent is equal to the number of units added
to the given logarithm, or, what is the same, let the
decimal be moved as many places to the left as there
were units added.
Example. Let the given logarithm be
– 2, 4537875.
The highest characteristic of the tables being 4, let 7
be added to this, and the result is
4,4537875 = log 35173,25.
The point must now be moved 7 places to the left, and
We obtain
– 2,4537875 = log 0,003517325.
A negative logarithm is always the logarithm of a
proper fraction. If the sign be changed, it will be the
logarithm of the reciprocal of this fraction. Hence
arises another method of determining the number cor-
responding to a given negative logarithm. Let the
number corresponding to the positive value of the
given logarithm be found, and the reciprocal of this num-
ber is the number required. This method is inferior in
accuracy to the last, because two approximations are
necessary in it. First, an approximation to the num-
ber corresponding to the positive value of the given
logarithm, and, secondly, an approximation in decimals
to the value of the reciprocăl. In cases, therefore,
where much exactness is required, the former method is
to be preferred. In other cases, however, the latter has
the advantage of greater expedition.
(281.) In logarithmic calculations it frequently hap-
pens, that a number of logarithms are to be added or
subtracted. The process is somewhat abridged by the
use of what are called arithmetical complements.
The arithmetical complement of a logarithm is that
number which is found by subtracting it from 10.
Thus 10 — a is the arithmetical complement of z.
Two numbers whose sum is 10 are arithmetical com-
plements of each other. Thus, to determine the arith-
metical complement of 6,347218, we have
10,000000
6,347218
3,652782
It is easy to see that the arithmetical complement of a
logarithm may be found at once by subtracting the first
digit on the right from 10, and each of the others from 9.
To show the application of this principle, let several
A L G E B R A. 579
hitherto considered are all positive, and such are the Logarithms.
Age” logarithms be united together by the signs + and —,
only numbers whose logarithms are ever required in S-N-
-Y" thus l – l' + 1" – l'" +7" — &c. Let cl, ci", &c. be
the complements of l','!", &c., it is evident that
l’= 10 – cl’ W" = 10 – cl!//
... l – l' + 1" – l'" + "+1 = cl, -- 7"-- cl!" -- l!"—20
or in general let X () signify the sum of all the posi-
tive logarithms, and > (cl) the sum of the complements
of all the negative logarithms, and let the number of
negative logarithms be n. The whole series will then
be reduced to X (!) + X (clº) – 10 m. Thus, instead
of first adding all the positive numbers, then adding
all the negative numbers, and then subtracting the
latter sum from the former, we have only to add toge-
ther all the positive logarithms, and the complements
of the negative ones, and subtract from the result the
number of n followed by 0; a process comparatively
expeditious and simple.
(282.) Exponential equations, of which we have
given approximate methods of solution in Sect. XXI.
may be immediately solved by logarithmic tables.
Taking the logarithms of both members of the equa-
tion
a” = b
we obtain
_ !a
lb
a la = lb " . . a =
The unknown quantity may occur as the exponent
In this case let
lc .
7,
Hence ly - llc. — lla. But by taking the logarithms
of both members of b" = y, we have ly – alb'."
blo — lla
b &
(283.) The meaning of the notation llc, lla, is
obvious. The logarithm of the number c being found,
it becomes in its turn a number whose logarithm is
sought. Thus, the logarithm of le is expressed by llc.
It is, however, expressed with more elegance and bre-
vity by lºc, the number 2 not expressing an exponent,
but merely the number of ls which precede c, written as
a product or power would be. *
In like manner it may be necessary to express the
logarithm of lºc which is expressed lºc, and so on, the
meaning of lºc being sufficiently obvious.
It is evident, that l”-* c signifies the number whose
logarithm is lºc. Now if we suppose n = 1, we find
that lºc signifies the number whose logarithm is le,
and therefore lº c = c. Again, by extending the ana-
logy, let n = 0, and l'c signifies the number whose
logarithm is lºc or c.
If we call lºc the second logarithm of c, lºc the
third logarithm of c, and in general lºc the m” loga-
rithm of c, the same analogy suggests the extension of
the notation to lº"c, which signifies the number whose
n" logarithm is c.
When the student shall have advanced into the higher
departments of analysis, he will perceive the extensive
use of the principles of notation to which we have just
alluded, and of which the ordinary notation of powers
are the earliest and simplest instance.
(284.) The numbers whose logarithms we have
of the exponent, as in a' = c.
b” = y ...
a' = c '.' y =
Asº
numerical calculations.
If, however, logarithmic calculation be applied to an
algebraical formula such as
a? — b?
which gives
l (a” — b%) = 1 (a + b) + l (a – b)
it may so happen, that upon substituting the particular
values for a and b, that a ~ b may be negative. In
which case the logarithm of a negative number would
be required.
But in fact negative numbers have no logarithms.
For in a logarithmic system all numbers whatever are
considered as the powers of some one number arbi-
trarily assumed, but never changing in the same sys-
tem, and the exponents of these powers are the loga-
rithms. Now this fixed number or base is supposed to
be such, that by constantly increasing its exponent
from 0 to an unlimitedly great positive number, the value
of the power will continually increase from unity to an
unlimitedly great number; and by constantly increasing
the negative value of its exponent, it would continually
diminish to an unlimitedly small number. This would
not be the case if a negative number were assumed as
the base. On the other hand the power would some-
times be a negative quantity, (scil., when the exponent
would become an odd integer,) and sometimes an ima-
ginary quantity, (scil., when the exponent would have
an even denominator.) That continuity which consti-
tutes a part of the definition of logarithms would in
these cases be broken.
It sometimes happens, that computation by loga-
rithms is introduced into a numerical or algebraical
problem, merely as a matter of convenience to expedite
the process. If in such a case it should occur, that the
quantity to which logarithms are to be applied be ne-
gative, let its sign be changed, and after its value (con-
sidered positively) has been ascertained by logarithmic
computation, let its former sign be restored.
Thus, if the quantity a” — b” is to be computed, a
being less than b. Let b% — a” be computed, and
when determined let it be taken with a negative sign.
If, however, the question be such, that the applica-
tion of logarithms is absolutely necessary to resolve it,
the occurrence of the logarithm of a negative quantity
is a symbol of absurdity, and must be understood in the
same manner as an imaginary quantity. Suppose, for
example, a question terminated in the equation
10* = — 100
... a l 10 = l (– 100),
This is evidently an absurd equation, since there is no
power of 10, whether the exponent be positive or ne-
gative, which is - – 100.
(285.) We shall now proceed to explain methods of
computing tables of logarithms. *
The method of resolving the equation y = a”, already
explained (282,) would be attended with great labour
where the computation would be required to be extended
far, and would be absolutely impracticable in cases where
a very high degree of approximation is required. The
methods of expressing logarithms by series furnish
much more exact results, and are at the same time more
expeditious.
sº
*-ºss
580
A L G E B R A.
Algebra.
S-N-' expressed in a series.
Let y be any number whose logarithm is to be
Applying the method of inde-
terminate coefficients we have
ly = A + A y + As y” + As y” + &c.
If y = 0 the first member becomes infinite, and the
second is reduced to A, Hence it appears, that the
developement of y cannot be effected under the required
form. If, however, we assume the first member to be
l (1 + y), this difficulty will disappear, and we shall
have -
l (1 + y) = A, + A y + As y” + As y” + &c.
which when y = 0 gives
! (1) = A, - 0 :
‘. . l (1 + y) = A, y + As y” + As y” + A, y' + &c. [1]
In like manner we should have
! (1 + x) = A, a + As a 2 + A, w8 + A, tº + &c. [2]
By subtraction we have
; (1 +!
- 1 + a
= A, (y – a + A, (y” — wº) + A,(y” — wº) + &c. [3]
1 + y - 3) - a ſº
But 1 -- a = 1 + 1 + æ = 1 + u, if we suppose
1A, -3 1 -- a
And since
l (1 + u) = A, u + As w” + As u’ &c.
we have
gy — a y – a Y” y – a Y”
A. (H)+A, (H) +A. (H) + se
= A, (y – a – A, (y” — a ") + A, (y” — a ") + &c.
Dividing both members of this by y – a it becomes
l y – a (y – ar)?
A Hi-FA. iTºº F.A. iTºº F&c.
= At + As (y – a y + As (y” + y a -- a ") + &c.
As the several series are independent of any relation
between y and r, let y = a, and the preceding equality
becomes
A, H = A +2A, 4-3A.e44A, e- &c,
'.' 0 = A, + 2 A, ºr +3 A, 12 + 4A, 1.8- 5 A, a +
— All-H A, ) + 2A, + 3 A, + 4 A,
This being independent of a we shall have (261)
A, - A = 0 2A, -i- A = 0 3A, + 2 A, = 0
and in general
n An-H (n − 1) An-1 = 0.
Hence we find
As = – 4 A,
and in general
A, e - 3 A, A, a - # A,
Aa e — 1 A,
n.
Hence we find
tº º as a 4 £5
There still, however, remains one quantity A, inde- Logarithms
This might have been expected, and in- ºr--~
terminate.
deed could not be otherwise, for the question to deter-
mine the logarithm of a given number is indeterminate,
unless the base of the logarithm be given ; and we
shall find that the value of the quantity A, may be
derived from the base of the system.
(286.) The series [4] is not always sufficiently con-
grgent for the convenient determination of the loga-
itum. A series may, however, be derived from it which
will be sufficiently so. Let a be changed into - a,
and \4] becomes
* £2 gº º
! (1 – 3) = A ſ — — — — — — — — —
G-A-A (-----, -º-;
By subtracting this from [4] we obtain
1 + a tº a 3 gº &T *
H=sa (; +++++, + se) ſº
1 — a l I
Let H = 1 ++, ºr ==H.
l
i (1++
l I I
ſºmeºs 2A(H I –H 5&#TTYst 5&EI),
+se)
l (1 + 2) — l z
I l
I
=2 | * \ne 0.
A, (H+jºr 1); ' 5(2 z + 1); -- se)tº
This series is sufficiently convergent, and gives the
difference between the logarithms of two consecutive
integers. Hence, by supposing z successively equal to
1, 2, 3, &c. we have
l I l 1
• A (; ++,+,+,+*)
+se)
l l l l
2A(; + 5.7; + 5.7; +77, T &c.)
&c. &c.
(287.) Let it now be proposed to obtain the deve-
lopement of a number in terms of its logarithm, or to
develope a” in a series of powers of a. Let
a’ = A, -ī- A, a + As a " + As a " + &c.
If a = 0 we have 1 = Ao. Hence we have
a" = 1 + A, a + As a " + As a " + &c.
a' = 1 + A, y + As y”—H A, y” + &c.
By subtraction we obtain
a"—a' = A, (r-y)+ A, (wº-y”)A,--(º-y") + &c.[3]
In [1] changing a into a - y, we have
a” = 1 +-A, (w—y) + A, (z-y)* + A, (z—y)*-H, [4]
and since [3] may be written thus
a'(a"-"–1)=A(r—y)+A,(º-y)+A,("—yº)+&c.
we have
Or
I 2
l I
l 3 – l 2 B. F.;
*
l l
* A (; +;
l 4 – l 3
[1]
[2]
a” { A, (a — y) + A, (a - y)* + A, (a — y)* + &c.;
= A, (a – y) + A, (wº — y”) + A, (z" — yº) -- &c.
Dividing both members of this by a -y we have

ſº A L G E
B R A. - 581
* * ***.*.*.*.*.* tº *
If in [6] a = e, and '.' k = 1, we have Logarithms.
= A + A, (z+ y) + As (a^+ a y + y”) + &c. * , a 3 gº 3:4
- e" = 1 *E* cºmmººs t- *
Let y = a, and this becomes + l + (2) + (3) –H (4) + &c.
a” . A. = A + 2 A, r + 3A, aº-H 4A, aº-H &c. By substituting for A, or k its value H- in [5] we
and substituting for at its developement [I] we obtain . . . le
Al (1 + A, a + As a " + As a " +. . . . ) obtain — 1 (a — 1)* , (a — 1)*
= A + 2 A, a + 3 A, aº-H &c. la = le (*H mº- 3 ) + 5 ”-se)
Hence we obtain
A,” = 2 A, , A. As = 3 As , A, As = 4 A, , &c.
A.2 A.8 A.4
•. . = -: , As = + , A, - - -, &c.
A = Gy. A = [a, (5. “
where
(2) = 1 2, (3) = 1 . 2, 3, (4) = 1. 2. 3. 4, &c.
Hence
* — A, a (A, w)” (A, w)* , (A, r)*
•=1 +++ -ār---º- + -ā- + &
In this case A, still remains undetermined. To deter-
mine it, let a be the base of the system, and let a = 1
+ b, and let (1 + b)* be developed by the binomial
theorem. Hence we obtain
a (a — 1)
l + b) * = 1 + a b + —— b%
+ tº 99 Hºbº + &c.
(3)
If the multipliers of the simple dimension of a in this
series be collected, and their aggregate equated with
that of a in [1, we shall have
b 2 3 4.
A = +--- b° b' + &c.
l 2 3 4
- ---, * 2
Or Al = (a – 1) *mm (a 1)
2
(a – 1)" (a – 1)* g.,
Let the value of this series be called k. Hence
* k aſ Ac2 tº
* = 1 +++++
AES tº k+ art
+ →- + –F– + &c. 6
a; ++5–4 &c. [G]
In this series r is independent of k, but k is depen-
dent on a by [5.] Let k a = 1, or a = T." and we
have e e
–– l l I l
a * = 1 + — — — —- + → -H →- &c.
it tº) + (5 + (5 +
This is a converging series, and its value obtained to
seven decimal places is 2,7182818. Let this number
be called e, and we have
- l
Emmº,
k
(! F 6, • a = e”,
".” la = k le, '.' k = la
le
Thus the sum of the series [5] is obtained, and the
dependence of k upon a exhibited more evidently.
WOL. I.
But in the series [4] (285) if a be changed into a - 1
(a — 1)*
we have
— &c.
I 2 3 )
— I — 1 Y 2.
la = At ( – “E ºf +
therefore the indeterminate A, in that case becomes le,
so that the series [7] (285) becomes
l (1 + 2) — lz
I l l
= 2 le {# I + 3 (3. ID3 t 5(gTI); +}
The logarithms may here be related to any base.
(288.) The logarithm of the number e in any
system is called the modulus of the system.
(289.) A system of logarithms constructed with the
base e is called the Neperian logarithms, (from Neper,
the inventor of logarithms,) and sometimes hyperbolic
logarithms. -*
Hyperbolic are sometimes distinguished from other
logarithms by an accent placed over the letter thus, l'.
Thus l'a is the hyperbolic logarithm of a.
(290.) Let a be the base of a system of logarithms,
and a being any number, we have
a - al”, a = e”,
e e'r - ałº.
Taking the logarithms of both members related to the
base a, we obtain
la:
le
Hence, if the logarithm of a number in any system be
given, the Neperian, or hyperbolic logarithm of the
same number may be found by dividing the given
logarithm by the modulus.
(291.) If the hyperbolic logarithms of both mem-
bers of e” = a + be assumed, we have
l'ºr le = la', '.' l'a' =
l'a' = la l'a,
... tº 1 •. le =: l
l'r T la " ' T ||a !
Hence the modulus of any system is equal to the
reciprocal of the hyperbolic logarithm of its base.
If, therefore, the hyperbolic logarithms be given, the
modulus of any system having a given base may be
determined.
Hence, from the hyperbolic logarithms a system rela-
tive to any base may be immediately obtained by mul-
tiplying all the numbers by the hyperbolic logarithm of
the given base. -
It is evident that the modulus of hyperbolic loga-
rithms is unity.
(292.) By the equation l'a, le = lar, it follows that
the logarithms of the same number in different systems
are as their moduli. For let L denote another system,
so that l'a L e = La., '.'
4 G
Algebra.
Y t
582
lr L & la: le
le Le ' LI = T.
Hence it follows, that the logarithms of any one system
being known, those of another system having any given
base or modulus may be computed.
(293.) Let it be proposed to determine the error
produced, by assuming that the difference of the num-
bers is proportional to the difference of their logarithms
when the number of places in the numbers is 5, and
their difference not greater than 1.
! (1 + æ) — la = le {} 2 tº ' 3 as 4. a7+ +}
By the series
it appears generally, that as the number a increases the
difference of the logarithms diminishes. Also, since
I l i l
1 . º te
— is greater than the remainder of the series, we have
ū;
! (1 + æ) — lar 3 +.
If the base be 10, le = 0,4342 .... < *. Hence, in
this case,
I
1 (1 + 1) – tº < ...
If a consist of five places, its least value is 10000.
Therefore the greatest value of l (1 + x) – l a is less
than 20000 T 0,00005.
Hence we may infer, that the logarithms of every
two consecutive integers, consisting of five places,
must agree in the first four decimal places at least.
Let
A = , q + 9 = 1, = 1 ++
A-to-o-o-o-Hi
A — A' = l H-1 H-
= 1 += ( : Ha'E).
But by what has been already proved
(; ; ;&##)
l l l
**º-sº" sº-
ly (2 + y) 2 ºf (2–Ey), T 3 ya (2 + y);
l
2 y (2 + y)
If y consist of five places, its least value is 10000, and
therefore the greatest value of A — A' is less than
l l
20000 × 10002 T 200040000
to a decimal has no significant digit within the first
eight places. Hence, in tables which extend only to
te A — A' <
which when reduced
seven places, we may assume that A — A' = 0, or
A = A'.
Thus we infer, that under the circumstances which
have been supposed, the logarithms of numbers in
arithmetical progression will themselves be in arithme-
tical progression.
Let n and n + 1 be two consecutive integers, and
'm + P_ an intermediate fraction.
looked upon as three terms of an arithmetical pro-
gression whose first term is n, and whose common
difference is +. the number n + P. being the
Q.
(p + 1)* term, and n + 1 the (q + 1)" term. By
what has been already established, the logarithms of
the several terms of this series will also be in arithme-
tical progression. Let à be their common difference.
The (p + 1)" term of this series will be
ln —- p 6,
which will be the logarithm of the (p + 1)" term of
the former series, " . "
lm, + pe=t(a + #)
Also the last term of the latter series, which will be
ln —H q 6,
will be the logarithm of the last term of the former
series, " . " -
l (n + 1) = ln — q 8.
Hence we find
! (n + 1) — ln = q &
*(n++)-in-pº
1(a + +) — lº
p
! (n + 1) - ln q.
But also
m –– +) — 71.
Q = -P-
(n + 1) – m q
Hence the differences of the logarithms are as the
differences of the numbers.
ſºmeºmºmºsº
SECTION XXVI.
Of Integral Functions.
(294.) WHEN any quantity, as a , is connected with
other quantities supposed known or constant by sym-
bols indicating determinate operations to be effected
on these quantities, the formula which represents the
result of these operations is called a function of the
quantity al. The quantity a is in this case usually
called the unknown quantity, or the variable.
(295.) Functions are divided into classes, according
to the nature of the operations by which the unknown
quantity is connected with the known quantities.
If it be connected by any purely algebraical process,
that is, by addition, subtraction, multiplication, divi-
sion, involution, or evolution, the function is called an
algebraical function. Thus, a tº + b x + c, a r" — b,
#, (a + r.)", &c. are all algebraical functions of w.
If the unknown quantity enter any exponent, it is
Integral
Functions.
These may be /
A L G E B R A.
583
Algebra, , called an exponential function. Thus aº, wº, (a + wº)"
&c. are exponential functions of a.
If the logarithm of the unknown quantity, or any
function of it occur, it is called a logarithmic function.
Thus lar, l0a -- a), &c. are logarithmic functions
of ar.
(296.) Algebraical functions are divided into ra-
tional and irrational. A rational function is one in
which the unknown quantity, whether alone or in
connection with known quantities, is not affected by a
radical or fractional exponent, and an irrational func-
tion is one where it is so affected. Thus a a 4 + æ,
b
a x ++, a r" + b a "-" + c wº", (m and n being
integers, positive or negative) are rational functions;
and a Wr -- b, a + wº — va -- as, a -i- 10 w -
are irrational functions.
(Z Tº
It should be observed, that a radical or fractional
exponent does not render a function irrational unless it
affects the unknown quantity. Thus va. a + w/5. a 2
is a rational function of a, although the coefficients of
a and aº be irrational quantities. -
(297.) Rational functions are divided into integral
and fractional. An integral function is a rational
function in which the unknown quantity does not enter
any denominator, or where, being in the numerator, its
exponent is a positive integer. A fractional function
is a rational function in which the unknown quantity
occurs in some denominator, or has a negative expo-
ment in the numerator. Thus a w’ + b x + c, a aº,
a + b a
a' + b'a'
—, a a “*, &c. are fractional functions.
&
a r" — b ar", &c. are integral functions, and
a tº —
It should also be observed here, that functions are
not fractional, unless the denominator of the fraction
a + b a
C
include the unknown quantity. Thus is an
integral function of a.
(298.) Integral functions are said to be of the first,
second, or m” degree, according to the highest exponent
of the unknown quantity. Every integral function of
the first degree must come under the general form
A a + B.
Those of the second and third degrees under the form
A a 4 + B r + C
A as -- B as + C a + D,
and in general one of the n” degree under the form
Aa" + Ba”-, + C wº—a + D a "-8 .... S as-H Ta' -- V.
In these general formulae the literal coefficients A, B,
C. . . . T, V are general representatives of any number,
integral or fractional, rational or irrational. Any one
or more of the coefficients may be = 0 in particular
CàS6S.
Thus a” – l is an integral function of the second
degree, and the formula
A a 2 + B a + C
becomes identical with it by supposing A = 1, B = 0,
C = — l. It should, however, be observed, that if the
first coefficient be supposed = 0, the degree of the
Integral
function is necessarily lowered. This is not the case Functions.
with any other coefficient.
(299.) One integral function is said to divide or
measure another, when the complete quote is an integral
function of the same quantity, or, which amounts to
the same, an integral function A is said to divide or
measure another C, when there is a third integral func-
tion B of the same quantity, such that A x B shall be
identical with C.
(300.) If an integral function of a be multiplied or
divided by any quantity K independent of a, the pro-
duct or quote will be an integral function of a of the
same degree. For let the function
A r" + Ba'-i + Caº-* . . . . Ta' -- V
be multiplied and divided by K, and the results are
KAa" + KB wº" -- K Cº-º. ... KTa' + KW,
A ---. m-l + $ r- T., V
Kº" + K * K . . . . k + + K
each of which are integral functions of a of the m”
degree. *
(301.) If one integral function of a (A) divide another
(C) it will also divide it if it be multiplied or divided by
any quantity K independent of a. For let B be the
integral function of a, which multiplied by A produces
C. Hence A × B = C. Let this equality be expressed
in either of the following ways:
A
-- X
K K. B = C
B
K A x # = C.
A -
Since # and K B are integral functions of r, (300)
it follows that # measures C, and since KA and #
are integral functions of a, it follows that KA measures
(302.) Two integral functions of a are said to be
prime to one another with respect to a, when no integral
function of a measures both.
(303.) If an integral function D be prime to another
A, and measure the product of A and a third integral
function B, it will measure B.
If A be an absolute quantity independent of a, we
have, by hypothesis, , an integral function of
A × 13
D
a. If this then be divided by the quantity A, which is
independent of a, the quote T) will be an integral func-
tion of a (300,) therefore D measures B.
Let us now suppose A to be a function of a of an
higher degree than D. Let A be divided by D, and
since they are prime there will be a remainder. Let
this remainder be R, and the integral part of the quote
be Q. We have then
A = D Q -- R,
A B B R.
B = B Q --->
. A B B R.
# - B Q = pº
4 G 2
584 A L G E B R A. *
Algebra. A B , g e gº º must either measure it or be prime to it, and it cannot General
\- TDT is by hypothesis an integral function, and since the be prime to both and measure their product, (304.) Theory of
(306.) Hence every integral function (D) of the first Equations.
g ... B R . g
same is true of B Q, the quantity -B is an integral degree which divides any power of an integral function YTY-
function ; therefore D measures B. R. Now if R be
independent of a, it follows that D measures B (301,)
which was to be proved.
But if R be not independent of a, it must be an
integral function of a lower degree than D. Let D be
in this case divided by R, and let the quote and re-
mainder be Q' and R', and we have
D = R. Q' -- Rſ.
There must in this case also be a remainder, otherwise R.
would be a common measure of D and A, contrary to
hypothesis.
Multiplying the last equation by #. we have
B - BRQ' LBR'
T D D
... p BR9 B. R.
D D
B
But * has been already proved to be an integral
B R.
function of a, and therefore º | must be an integral
f
B R.
Hence - Sº must be an integral function.
D
If in this case R' be independent of a, D must measure
B (301,) which was to be proved ; and if not, the
same process must be continued. It will be observed,
that in this process the successive remainders R, R',
&c. are all integral functions of a, and each successive
remainder is of a degree lower than that which pre-
ceded it. Also, since D and A are supposed prime it
follows that no remainder can exactly measure that
which preceded it. Hence it follows, that we must at
last obtain a remainder independent of a, and since D
will necessarily measure the product of that remainder
and B, it must measure B.
In commencing this process, we supposed D a func-
tion inferior in degree to A. If A be inferior in de-
gree to D, we should commence by dividing D by A,
but in every other respect the process will be the same.
(304.) If an integral function of a divide a product,
and be prime to all its factors but one, it must measure
that one.
Let D measure A B C . . . . LM, and be prime to all
but M, it must measure M. For since D measures
function.
A × B C . . . . L. M., and is prime to A, it measures
B C . . . . L. M. Again, since it is prime to B, and mea-
sures B. × C. . . . LM, it measures C . . . . M, and ulti-
mately since it measures L. M., and is prime to L, it
measures M. -
Hence, if an integral function measure another in-
tegral function, it cannot be prime to all the integral
factors of that function.
(305.) If an integral function (D) of the first de-
gree measure the product A × B of two integral func-
tions, it must measure one of these functions. For it
A, must divide that function itself; and, also, if two inte-
gral functions be prime one to another, all their powers
will be also prime one to another.
(307.) Every integral function A, which is divided
by several integral functions D, D', D", &c. which are
prime to each other is also measured by the continued
product D, D', D", &c. of these functions. .
D
so that A = D Q. Again, D' measures A or D Q, and is
prime to D, “..' it measures Q, suppose the quote Q', so
that A = DD'Q'. Again, D" measures A or DD'Q, and
is prime to D, D', therefore it measures Q', and so on
until we obtain A = the continued product of all the
divisions D, D', D", &c. into an integral function.
(308.) Hence, if any integral functions D, D', D",
&c. prime to each other, and another integral function
A has certain powers of these D", D", D", &c. as
divisors, it is evident that any powers of these divisors,
with lower exponents than n, n', m", &c. or products of
which any combinations of these powers are factors,
will be all divisors of A.
(309.) If any integral A function be resolved into
the integral factors A', A", A", &c. every integral divi-
sor of any of these factors, and every combination of
such divisors by continued multiplication, will be
divisors of the original integral function A. Also, each
of these divisors multiplied or divided by any quantity
independent of a will be a divisor of A, and it follows,
that the original integral function A can have no other
divisors except these.
These consequences are apparent from the preceding
observations.
By hypothesis is an integral function, let it be Q,
SECTION XXVII.
The General Theory of Equations.
(310.) A compIETE equation of the mº" degree, when
cleared of fractions and radicals, and all the terms are
brought into the first member, and divided by the co-
efficient of the highest dimension of a, is of the form,
a"+Ar"---Ba'"----Crº-5. . . . Ta 4- W = 0 [1]
the coefficients A, B, C . . . . W being respectively any
quantities whatever, positive or negative, integral or
fractional, rational or irrational, or = 0.
(311.) Any quantity, whether numerical or algebrai-
cal, simple or complex, real or imaginary, which being
substituted for a will change the equation into an
identity, or make all its terms be such as necessarily to
destroy each other, so that the aggregate shall = 0, is
called a root of the equation.
(312.) If a be any root of the equation [1, the first
member of the equation is measured by (a — a.) -
For let the first member by divided by a - a, by the
ordinary process of division. The result is
A L G E B R A. 585
Algebra.
S-N-
game m -- A z*-1 —- B a "**-i- C a "T" + . . . . . . T r + V (r."- + a r"-* + as a "-a-H as r"-" + . . . .
a — a) :- + a r"-l + + - - + A +-Aa fº &
- + B —-B a
–– a a "-i-H Br"-" +C
-j- A
-- a w”-i – as w”
-- A — Aa
-- as r"-" + Ca"--
––Aa
* —H B
+ a” a "-" — as r"—a
+Aa — Aa”
—- B — Ba
-- as r"-a -- D r"-
+Aa”
+B a
—H C
&c. &c.
The coefficients of the several terms of the quote
may be observed to be integral functions of a ; that of
the second term being of the first degree, that of the
third of the second degree, and, in general, that of the
n” of the (n-1)* degree. In like manner, the suc-
cessive remainders have the same coefficients to the
highest power of a in them respectively, that of the first
remainder being an integral function of a of the first
degree, that of the second of the second degree, and,
in general, that of the n” remainder is an integral
function of a of the m” degree.
The number of terms in the original equation is
evidently m + 1, and after proceeding with the division
until the term V is brought down, the remainder with
this annexed to it will be
a”-1
+ Aa"--
+ Ba”-8
+ Ca"-
&
C.
+ T
and, therefore, the corresponding term of the quote
will be
a + V.
a”-1 + A a "-3 + B a”-* + . . . . T,
which is independent of w. This, being multiplied by
a -- a, and subtracted from the former, gives for a re-
mainder
a" + A an- + B a”-- + Ca"-a -- . . . . T a + W. [2]
But since, by hypothesis, a is a root of the equation ;
this, which is nothing more than the first member of the
given equation, changing a into a, must = 0, and,
therefore, the division is complete, and a — a is proved
to measure the first member.
(313.) The same process proves, that if a - a
measure the first member, a must be a root of the equa-
tion, for in that case the last remainder [2] must be
= 0.
(314.) This principle gives a criterion for deter-
mining whether an integral function of r of the first
degree (a – a) is a divisor of any other given integral
function of a, as A'. In A' let a be changed into a, and
if the result be identically 0, a - a is a divisor, and
otherwise not.
(315.) The law by which the successive coefficients
of the quotient in (312) are obtained, should be ob-
served. The coefficients of the several terms of the
quote may be all obtained from the formula [2;] the
coefficient of the second term of the quote is the first
two terms of [2,] (m – 1) being subtracted from each
of the exponents; the coefficient of the third term of
the quote is the first three terms of [2,] (m — 2) being
subtracted from each of the exponents; and in general
the coefficient of the n" term of the quote is the first n
terms of [2,] (m – (n − 1)) being subtracted from the
exponent.
^r, perhaps, a rule more easily impressed on the
memory would be, that the coefficient of the n” term of
the quote is an integral function of a of the (n − 1)*
degree, having the same coefficients as the original equa-
tion, and in the same order as far as the terms extend.
(316.) Every equation has as many roots as there
are units in the number which marks its degree, and
cannot have more.
We shall here take for granted, that the equation
has at least one root, whether real or imaginary. Let
the root be a. Hence, by what has been already proved,
we have
a" + Aa"-i-H B a "-2 + Ca"-s--. . . . Ta' -- W =
(a — a) (w"-" — Aſ a "-" -- B'a"-s + &c.)
where A, B', &c. express the coefficients of the suc-
cessive terms of the quote. ..
It is evident, that any number which is a root of th
equation
a"-1 4- Aſ a "-2 + B'a"-3 + . . . . = 0,
must also be a root of the original equation ; and as
this equation must at least have one root, let it be a',
so that we have
a"-" —- A'a"---|- B'a"-3 + . . . . = (r—a')
(a"-" —- A" r"-s-H . . . . )
... a " + A a "- + Ba” + . . . . (a – a) (a — a')
(r"---|- A"a"-" + . . . . )
For each simple factor thus found, the remaining factor
of the integral function in the first member is lowered
one degree, and by continuing the process through
(m – 1) steps, we should obtain an integral function of
a of the first degree, and, therefore, of the form a -a!"Tº.
We should thus have the function in the first member
resolved into m simple factors, viz. a - a, a - a',
a — a!", . . . . (a — a "T"), whose continued product is
*
*
General
Theory of
Equations.
\-º-N/~ *
586
A L G E B R A.
3. That the coefficient of the fourth term, its sign Greatest
Algebra equal to the integral function in the first member. By
factor, the equation cannot have any other root. Thus,
if there be one root there are m roots, and cannot be
II, Ore.
We are not aware of any demonstration of the prin-
ciple, that every equation must admit of one root of a
nature such as could properly be introduced here.
(317.) If any number of the quantities a, aſ, a!", . . .
be equal, the corresponding factors will be equal. In
this case the equation might be said to have a less
number of roots than is due to its degree; but in order
to generalize the principles, it is considered still to have
the full number, but the two or more of them become
equal. Thus the equation
a 2 – 2 a + 1 = 0, or (a – 1)* = 0,
is said to have two roots each equal to 1.
(318.) Since the first member of every equation of
the mº" degree admits of m divisors of the second
degree, it must admit as many divisors of the second
degree as there are combinations of two divisors of the
first degree, since the product of every two factors of
the first degree is a divisor of the second degree.
m (m – 1)
2 divisors of the second
Hence there are
m (m – 1) (m — 2)
2 . 3
divisors of the third degree, and in general there are
m (m – 1) (m. – 2) (m. – 3). . . . (m. – (n − 1))
1 - 2 - 3 . . . . . . . . . . . . . . . . . . . 70,
divisors of the n” degree, m being less than m. (243.)
These divisors of the higher degrees may become
equal, like the factors of the first degree.
(319.) If the second member of the identity
a" + A*-* + Baº-" —- . . . . Ta –– V
= (a — a) (a — a') (a — a!"). . . . (a – aft”)
be developed and arranged by the dimensions of a, it
will become (246)
a" + A*-* + B r"-* -- . . . . Ta' -- V
= r" – S (a) a "-" -- S (a), a "-" — S (a), aº-" + . . . .
. . . . S (a),— a + S (a), .
the signs being alternately + and —, because an even
number of negative factors give a positive, and an odd
number a negative, product.
The meaning of the notation S (a), &c. has been
explained in (246.)
By equating the coefficients of the corresponding
terms in both members, we have
A = – S (a), B = S (a), C = – S (a),. . . .
W = + S (a),
the sign + being used, when m is even, and – when
an is odd.
Hence we find :
1. That the coefficient of the second term, its sign
being changed, is the algebraical sum of the roots of
the equation, with their signs changed.
2. That the coefficient of the third term is the sum
of the products of every two roots, with their signs
changed.
degree, and in like manner there are
roots, with their signs changed. --
(320.) If the whole equation be divided by the last
term, and arranged by the ascending dimensions of a,
and the successive coefficients be A, B, C, &c. it as-
sumes the form
1 + A a 4- Ba” + C as -- . . . . Ma"-" —- N a "= 0,
under this form it is evident, from what has been
already proved,
1. That (A) the coefficient of the second term is the
sum of the reciprocals of the roots.
2. That the coefficient B of the third term is the sum
of the reciprocal products of every two roots.
3. That the coefficients of the fourth, fifth, and in
general of the n” term, is the sum of the reciprocal
products of every three, four, &c. and (n − 1) roots ;
and the coefficient N of the last term, is the product of
the reciprocals of all the roots.
(321.) If the last term of an equation arranged by
the descending powers of the unknown quantity be
unity, it will participate in both of the systems of pro-
perties we have just explained; for in this case it may,
without dividing by any number, be arranged in either
ascending or descending powers. In this case, the
product of all the roots is unity. And since any system
of quantities may be imagined to be the roots of an
equation, we may infer, that if the continued product of
7m quantities be unity,
1. That the sum of the reciprocals of these quantities
will be equal to the sum of the product of every com-
bination of (m – 1) of the quantities.
2. That the sum of the reciprocal products of every
two of them is equal to the sum of the products of
every (m. – 2) of them.
3. And in general, that the sum of the reciprocal
products of m of them is equal to the sum of the pro-
ducts of (m. — m) of them.
SECTION XXVIII.
On the Greatest Common Measure of Algebraical
Quantities.
(322.) ALGEBRAICAL quantities being expressed by
ſetters, their actual values are not apparent. In ap-
plying to these the principles already established re-
specting the greatest common measure of numbers, or
any quantities of the same species, it will be necessary
to explain the peculiar senses in which the terms are
applied.
When two polynomes are arranged by the dimen-
sions of the same letter, and considered as integral func-
tions of that letter, one may be considered to measure
the other exactly, if there be a third integral function of
the same letter which being multiplied by the divisor
will give a product identical with the dividend. In this
sense the exactness of the division is not considered to
be impaired, even though the coefficients of the dimen-
\-2 (311) it follows, that each of the quantities a, a', being changed, is the sum of the products of every J.
a", &c. is a root of the equation; and since the func- three roots, with their signs changed. *...
tion in the first member cannot have any other simple The last and absolute term is the product of all the .
A L G E B R A.
587
S-N-" gebraical fractions.
sions of the principal letter in the quote should be al-
Hence, when a polynome is consi-
dered to be a function of any letter as a, it will in this
sense be divisible by any other quantity, whether mo-
nome or polynome, which is independent of w.
When the different integral functions which divide or
measure such a polynome are compared together, one
is said to be greater or less than another, according as
the highest exponent of the letter by whose dimensions
they are arranged is higher or lower in the one than in
the other. --
(323.) Consequently the greatest common measure
of two integral functions of the same letter is the
highest integral function of that letter which measures
both, in the sense already explained.
Two integral functions of a are said to be prime with
respect to a, when no integral function of a measures
both. It is evident from what has been said, that
these functions may and must have many common mea-
sures, since every quantity independent of a measures
them. But provided that no integral function of a
measures them they are prime as respects ar.
(324.) The greatest common measure of two inte-
gral functions of the same letter is found by a process
exactly the same as that already established for other
quantities. It is easy to see, that the successive re-
mainders will be integral functions of a continually
decreasing in degree. If any remainder measure the
preceding one, that will be the greatest common mea-
sure, and is proved so exactly in the same manner as in
the case of numbers. If there be, finally, a remainder
independent of a, the fractions are prime with respect
to a, since all their common measures must measure
this remainder, and, therefore, none of them can be
functions of a. Y.
(325.) From the results of the last section it follows,
that every integral function can be resolved into as
many integral factors of the first degree, as there are
units in the highest power of the principal letter which
enters it. This decomposition into simple factors will
be at once effected, if the equation obtained by equat-
ing the given integral function with 0 be solved, con-
sidering the principal letter as the unknown quantity.
Each of the roots of this equation will determine a
simple factor (316) of the integral function. Thus, the
decomposition of an integral function into its factors,
is reduced to the determination of the roots of an equa-
tion.
On the other hand, if by any means the first member
of an equation, considered as an integral function of ar,
can be resolved into factors of the first or second de-
gree, the roots will be immediately obtained by putting
the factors severally = 0, and solving the equations
thus obtained. Their roots will be the several roots of
the proposed equation.
(326.) We shall now consider algebraical quantities
and their measures in another sense.
A polynome is said to be integral and rational, when
all its numeral coefficients are integers, and all its let-
ters have positive and integral exponents. In fact, it is
considered integral and rational absolutely, when it is
integral and rational with respect to all the letters and
coefficients which enter it. Thus
10 a.2 – 3 a b + b%
is integral and rational. But
vTO a” – 3 a b + 5°
10 a” — 3 + + bº
I0 a” — 3 M ab 4- b2
are not integral and rational.
(327.) It is evident, that if the product of two quan-
tities be integral and rational, and that one of the fac-
tors be integral and rational, the other factor must be
also integral and rational.
(328.) A quantity A is said to measure an integral
and rational quantity B, when there is another integral
and rational quantity C such that A C = B.
Hence it appears, (299,) that no quantity can mea-
sure an integral and rational quantity, except another
integral and rational quantity.
(329.) Two integral and rational quantities are said
to be prime to each other, when they have no common
measure in the sense just explained.
(The student should observe the difference of the
phrases “prime to each other,” and “prime to each
other with respect to a particular letter.” In the use
of the former the quantities are looked on as integral
and rational quantities; but in the other, they are only
considered integral and rational with respect to a par-
ticular letter.)
(330.) An integral and rational quantity is said to be
absolutely prime, when it is not measured by any other
integral and rational quantity.
Thus cº – b c -- a b is an absolutely prime polynome,
.lthough it be of the second degree with respect to c,
and can therefore be decomposed into two simple fac-
tors. These factors though rational with respect to c,
are irrational with respect to the other letters.
(331.) The greatest common measure of two rational
and integral polynomes, is that common measure
which has the greatest coefficients, or exponents, or
both, or that whose terms have the highest dimensions.
(332.) If two rational and integral polynomes A and
B be divided by their greatest common measure C, the
quotes A', B' will be prime to each other. For if they
have a common measure let it be c, and we have
A = Aſ x C B - B' × C
Aſ = A" × c B'-- B" X G
" . " A = A" × c × C B - B" × c X C.
Hence c x C is a common measure of A, B greater
than C, because it must have greater exponents or
coefficients, or both.
The following principles already established with
respect to other quantities may also be extended to
rational and integral polynomes.
1. All common divisors of two quantities are divisors
of their greatest common divisor.
2. The greatest common divisor of two quantities is
also the greatest common divisor of the lesser of those
quantities and the first remainder, and also of the first
and second remainders, and so on.
(333.) An example will best illustrate the method of
determining the greatest common divisor of two alge-
t
braical quantities.
Let the two quantities be
a? — a” b + 3 a bº — 3 bº
a” – 5 a b + 4 bº
Greatest
Common
Measure of
Algebraical
Quantities.
588 A L G E B R A.
Algebra. The former, according to the criterion already ex- common measure which will result from the investiga- Greatest
S-N-" plained, is the greater. Dividing it by the latter we tion. ,” - *:::
obtain the integral part of the quote, and the remainder If the former quantity be multiplied by 2, and the Algebraical
as follows: first division be effected, we have the following re- Quantities.
mainder \-,-
a”— 5 a b + 4 bº) a” — as b + 3 abº — 3 bº (a + 4b
a” — a” b – 16 abº-H 16 bº
19 abº — 19 bºa-195° (a-b)
Since the greatest common measure of the two pro-
posed quantities is also the greatest common measure
of the divisor and this remainder, and since no divisor
of the factor 19 b% measures the divisor or lesser of the
proposed quantities, it follows, that the greatest com-
mon measure of the proposed quantities must be the
greatest common measure of the lesser quantity and the
factor a -- b, and the calculation may be disembarrassed
of the simple factor. Upon the same principle, every
simple factor of each remainder which is not a factor
of the divisor may be removed; and any simple factor
of one of the proposed quantities which is not also a
simple factor of the other may be removed.
Upon nearly the same principle, any simple factor
may be introduced into one of the proposed quantities,
provided it be not a simple factor of the other. This
is sometimes necessary in order to facilitate the pro-
cess, as will be seen hereafter. --
The problem in the example under consideration, is
then resolved to the investigation of the greatest com-
mon measure of the quantities
a” 5 a b + 4 bº
a — b.
Dividing the former by the latter, we have
a b) a” -- 5 a b + 4 b% (a — 4 b
a” – 5 a b + 4 bº
33 25 * 9
There being no remainder, it follows, that a ~ b is the
greatest common measure ; and, since this is not mea-
sured by any other algebraical quantity, there is no
other common measure of the two proposed quan-
tities.
(334.) Again, let the two quantities be
15 a' + 10 a b + 4 as bº + 6 a.º. b3 — 3 a bº
12 as bº + 38 a” be + 16 a bº- 10 bº.
On examining these quantities it appears, that a is
a simple factor of the former which does not enter the
latter, and 2 b" is a simple factor of the latter which
does not enter the former. Neither of these can be
factors of the greatest common measure, and may,
therefore, be omitted in the investigation. By remov-
ing them, the quantities under consideration are re-
duced to
15 a' + 10 a” b + 4 a” b% + 6 a bº — 3 bº
6aº –– 19 as b + 8 a bº — 5 bº.
The first term of the latter will not divide that of the
former without introducing fractional coefficients. This
may, however, be avoided, by multiplying the former
by such a quantity as will render the coefficient of the
first term of the former a multiple of the coefficient of
the first term of the latter; and such a multiplier not
being a factor of the second quantity, cannot affect the
411 as bº + 274 a bº — 137 bº. e
In this remainder there is the simple factor 137 bº, and
as this does not enter the lesser of the given quan-
tities it may be omitted, and the other factor is
3 a” —- 2 a b – bº.
If the lesser of the proposed quantities be divided by
this there will be no remainder, and an exact quote will
be obtained. Hence this remainder is the greatest
common divisor.
The suppression of the simple factors which occur in
the successive remainders, without occurring in the
respective divisors, is not merely an operation effected
to expedite the process, but a matter of necessity.
For otherwise, in order to render the divisor divisible
by the remainder, it would be necessary to multiply it
by the simple factor, (for otherwise the quote would be
fractional,) in which case it would be a common factor,
and would, therefore, be also a factor of the common
measure which would result from the process, and
which would not, therefore, be a common measure of
the proposed quantities.
If, however, the proposed quantities, or any subse-
quent divisor and dividend, have any evident common
measure, whether simple or complex, it may be set apart,
and the investigation conducted as if it were suppressed.
It must, however, be finally multiplied by the common
measure which results from the investigation, in order
to find the greatest common measure of the proposed
quantities.
In general, then, it appears, that the process for the
determination of the greatest common measure of two
algebraical quantities should be conducted thus:
1. Let the two quantities be arranged according to
the dimensions of the same letter.
2. Let any simple factor which is common, or any
complex common factor which is apparent, be set apart
to be multiplied by the common measure which is the
result of the process.
3. Let any simple factors which are not common be
suppressed.
4. The quantities being thus prepared, let that which
has the higher dimensions of the letter by which they
are arranged be divided by the other, and if there be no
remainder, this other multiplied by any common fac-
tors which may have been set apart is the greatest com-
mon measure. But if there be a remainder, this re-
mainder and the divisor are to be treated in the same
manner as the original quantities, and the process is to
be continued until there be no remainder, or one which
is free of the letter by which the given quantities have
been arranged. In the former case, the last remain-
der is the greatest common measure, and in the latter
case there is no common measure.
(335.) It appears from what has been already
proved, that every common factor of two polynomes is
a factor of their greatest common measure. To in-
vestigate more particularly the composition of the
greatest common measure, let A be any rational and
integral polynome not absolutely prime; let it be sup-
posed to be arranged according to the dimensions of
the letter a. In general, such a polynome may be
A L G E B R A.
589
considered, in the first instance, as the product of
-N-' three factors: •
1. A monome factor A, common to all its terms.
This factor is the greatest common measure of all its
terms considered as simple quantities, and is formed
by finding the greatest common measure of all the
numeral coefficients, and multiplying this by the highest
dimensions of the letters which are common to all the
terms.
2. A polynome factor As independent of the letter a,
by which the proposed polynome has been previously
arranged, and which is the greatest common measure
of the several polynomes, which are the coefficients of
the several dimensions of a ; the factor Al, however,
having been previously taken out.
3. The polynome factor A, arranged by the dimen-
sions of a, which remains when the given polynome
has been divided by the two former factors. The
several coefficients of this polynome A, are evidently
prime to each other.
Hence the given polynome will be represented by the
product -
- A, X As X As.
If the coefficients of the several dimensions of a in the
given polynome happen to be prime, we shall have
Al– 1, As = 1; and if the several monomes which com-
pose the given polynome be prime, we shall have A = 1.
(336.) Let A and B be two polynomes, whose com-
mon measure is to be investigated. By what has been
just stated they may be resolved into the forms
A = A, x A, x As
B = B, x B. × Ba.
Let m, be the greatest common measure of A, and B,
ms of As and B, and me of Aa and Ba. It is evident
that m, x m, x m, is a common measure of A and B.
But it is also their greatest common measure ; for
every common measure of A and B, if it be a monome,
must measure m, ; and if it be a polynome independent
of a, must measure m, ; and if it be a polynome depen-
dent on a, the coefficients of the powers of a being
prime to each other, it must measure m, Hence,
m, x m, x m, is the greatest common measure, and
we have
A = m, x m, x m, x A'
B = m, x m, x m, x B',
A' and B' being prime to each other.
It appears, therefore, that the greatest common
measure is the continued product of the greatest com-
mon monome factor, the greatest common polynome
factor independent of the letter by which the given poly-
nomes have been arranged, and the greatest common
factor which is dependent on this letter, and, further,
that every common measure whatever of A and B must
measure this.
(337.) We shall now give a general demonstration
of the second principle announced in (332,) that the
greatest common measure of A and B is also the
greatest common measure of the lesser B, and the re-
mainder found on dividing the greater by the less.
Let us suppose that the polynomes being arranged
by the dimensions of the same letters, the coefficients
have all been divided by their greatest common factor,
and are, therefore, prime. If then A and B be the two
polynomes, let Q be the quote, and R the remainder.
Let M be the greatest common measure of A and B,
and M! of B and R. We have
WOL. I.
A = B Q -- R. |
A B - R
M = Mi Q-FM
A B R.
† = M79-F wº
By the second, since M measures A and B, M must
also measure R ; and by the third, since M' measures B
and R, it must measure A. Hence, M' being a com-
mon measure of A and B, measures their greatest com-
mon measure M ; and M being a common measure of
B and R, measures their greatest common measure Mſ.
Since M and M' measure each other, they must be equal ;
that is, the greatest common measure of two integral
polynomes is also the greatest common measure of the
lesser and remainder.
If the coefficients of the dimensions of a in the
polynomes be not prime, let their greatest common
measure be m. So that m A and m B will then be the
original polynomes. The remainder will then be m R,
the greatest common measure m M, and the greatest
common measure of m B and m R will be m M'. Now
M’ has already been proved equal to M, ... m. M is equal
to m M'. -
Hence it follows, in general, that the greatest com-
mon measure of two integral polynomes is also the
greatest common measure of the lesser and remainder.
(338.) If the greatest common measure of two inte-
gral polynomes can be determined, the greatest com-
mon measure of three or more can be found by a
process precisely similar to that explained in (99,) and
founded on the same reasoning.
(339.) Let us now investigate more particularly the
process for determining the greatest common measure
of two integral and rational polynomes, A and B.
First, let the common monome factor m, (if there be
any such) be found. This factor is composed of the
literal factors common to all the terms, and which
appear on inspection, affected by the greatest common
measure of all the numeral coefficients as a coefficient.
This last is found by the rules established in Section
VIII. This is one factor of the greatest common
measure sought, and is set apart until the others are
obtained. The monome factors common to the terms
of the one, but not of the other, may be set aside, since
they cannot enter in the common measure.
We shall now consider successively the cases in
which the remaining factors of A and B include one
letter only, two letters, and where they include three or
IIl OI’e,
FIRST CASE. To determine the greatest common
measure of two integral polynomes arranged by the
dimensions of one letter (a,) and whose coefficients are
integers which have mo common measure.
Let that of the higher degree A' be divided, if possi-
ble, by the lower, B'. This will be possible if the co-
efficient of the highest dimension of a in A' be a multiple
of the coefficient of the highest dimension of a in the
lower B'. If this be not the case, the whole polynome
Aſ must be multiplied by such an integer as will render
the coefficient of the first term of A'a multiple of that
of B'. Let m be this multiplier, so that the modified
polynomes are m A' and B'. It is easy to see that this
modification cannot affect the common measure. In
other words, that if M be the greatest common measure
4 H
Greatest
Common
Measure of
Algebraical
Quantities.
S-N-2
590 A L G E B R A.
We shall thus have obtained two polynomes, of which Transforma-
the several coefficients are prime to each other; and tion of
Algebra. of m A^ and B', it will also be the greatest common mea-
\–N2– sure of A' and B'. For since it is prime to m, and mea-
sures m A', it measures A'; therefore it is a common
measure of Aſ and B', and being so, it is evidently their
greatest common measure.
Let the division be continued in this way, rendering
the first term of each remainder when necessary a
multiple of the first term of the divisor, until a re-
mainder be obtained, in which the exponent of the
highest power of a is less than the highest exponent in
the divisor. -
Let it then be determined whether the coefficients of
this remainder have any common factor, and if so, let
it be suppressed, since it cannot be a factor of the
common measure. This done, let the divisor be
assumed as dividend, and the remainder as divisor, and
proceed as before. Continue this process, making
each remainder alternately divisor and dividend, until
a remainder is found which exactly measures the pre-
ceding remainder. This remainder is then the greatest
common divisor m, of the polynomes A', B'. If their
coefficients previously had a common measure m, the
greatest common measure would be m, x me.
(340.) SEcond CASE. To determine the greatest
common measure of two integral polynomes which in-
clude but two letters, a and b. -
Let the common monome factor m, if there be such, be
set apart, and also let any monome factors not common
be removed, since they cannot enter the greatest com-
mon measure sought. Then let the two polynomes be
arranged according to the dimensions of either of the
letters, as a.
The coefficients of the several powers of a will in
this case be integral polynomes, including no letter but
b. Let the greatest common measure of all these
coefficients in each polynome be found by the preceding
case, and the principles which regulate the determina-
tion of the greatest common measure of several poly-
nomes. Let these be Me, N. Let the greatest com-
mon measure of these be found, and it will evidently
be the factor me of the greatest common measure
sought. The remaining factors of M, and N, not
being common, cannot enter the greatest common
measure, and may, therefore, be suppressed.
The two polynomes when thus divided by M, and N,
will have their coefficients prime to each other. The
principles established in the preceding case may then be
applied to determine the common measure me, and thus
the greatest common measure of the proposed poly-
nomes m, x m, × me will be determined.
(341.) THIRD CASE. To determine the greatest com-
mon measure of two integral polynomes which include
three, a, b, c, or more letters.
Let them be arranged by the dimensions of one of
the letters a. The coefficients of the powers of this
letter will then be integral polynomes, including b and
c. Let the greatest common monome factor m, be
first found and set apart, and let any other monome
factor of either polynome be suppressed. Let the
greatest common measure M, of the several polynome
coefficients of A be then found, and the same Ne of
the coefficients of B. This may be effected by the
rules established in the second case, and in (99,) for
the greatest common measure of several quantities.
Let the greatest common measure m, of M, and N, be
then found, and let all other factors of these be sup-
pressed in A and B. •
the greatest common measure me may be found by the
principles already established. Thus the greatest com-
mon measure m, x m, x m, will be found.
By pursuing a similar method, the greatest common
measure of a polynome, including any number of letters,
may be found.
As an example of these principles, let it be required
to find the greatest common measure of the polynomes,
a? dº — cº dº — a 2 cº -H cº
4 a” d – 2 a cº -- 2 gº — 4 a c d.
There is here no common monome factor, "." m, = 1.
The monome 2 is common to all the terms of the latter
polynome, and shall therefore be suppressed. This
being done, and the polynomes being arranged by the
dimensions of d, they become
(a? – c’) dº — as cº -- c.
2 a (a — c) d – (a – c) Cº.
Since — a cº -- c = — cº (a” — cº, it is evident that
a? —cº is a factor of the coefficients of the former, and
a – c of the latter, so that M, e a” — cº, N, - a - c.
The common factor of these is a - c, "...' m, = a – c.
The factors M, and N, being suppressed, the polynomes
become
d? — Cº
2 a. d. — C*
which are evidently prime, '.' me = 1.
greatest common measure is ºn, at a - c.
The same result will be obtained if the quantities be
arranged by the dimensions of a or c.
Hence the
SECTION XXIX.
The Transformation of Equations.
(342.) THE resolution of equations of the higher
degrees presents considerable difficulties to the analyst,
and in cases where it can be effected at all requires the
aid of peculiar analytical artifices. It frequently
happens, that although the value of the unknown quan-
tity in a proposed equation cannot be immediately
determined, yet the value of some other unknown
quantity, having a given relation to the required one,
may be ascertained, and thus the required quantity
finally may be found. The process by which this end
is attained, is called the transformation of equations;
and although, properly speaking, it is a particular case
of the more general process of elimination to be
treated of hereafter, yet, in order to introduce the ab-
stract principles more gradually to the mind of the
student, we shall so far invert the order of principles,
as to investigate the principles of transformation
before we enter upon the more general field of eli-
mination.
Suppose that an equation of any degree be given,
in which w is the unknown quantity, but which cannot
immediately be solved. Suppose, also, that another
equation be given, in which y is the unknown quantity,
and which can immediately be solved. If it happen
to be known that the unknown quantity y is a number
which is greater or less than a by any given quantity,
Equations,
A L G E B R A. 59]
tion (C), and the equation which expresses the relation tº:
between the two unknown quantities ºr and y (B), E.I.s.
* , as 5, it is evident that the first equation will thus be
STVT solved by means of the second. But any other known
relation between a and y would equally attain the
desired end; as, for example, if it were known that y,
multiplied or divided by any given number, were equal
to a, or that the sum of the squares of a and y were
equal to a given number, &c.
There are here, then, in general, three things to be
considered, the equation for a, the equation for y, and
the relation between a and y. If any two of these
be given, or assumed, the third may be found. Thus,
if the equation for a (generally the proposed one) be
given, and the relation between a and y be assumed,
the equation for y may be found thus: by the assumed
relation between a and y, we know what quantity
composed of y and known quantities, or what function
of y is equivalent to a. Let this be substituted for a
in the proposed equation, and the result will be the
equation for y. In this. case the proposed equation is
said to be transformed, and the unknown quantity a
is said to be eliminated ; and the process, which in
general is called elimination, is in this particular
application of it called transformation.
(343.) In general, the object of transforming an
equation is to obtain another equation which may be
resolved with greater facility. In this process let the
proposed equation be called (A), the transformed equa-
By the explanation of the process already given, it s——
appears that the resolution of (A) depends on the
resolution of both (B) and (C). The resolution of (C)
determines the value of y. This value being substi-
tuted for y in (B), this equation (B) will determine the
value of a. Generally, therefore, the equations (B)
and (C) should be more simple and easy of solution
than the equation (A), or the process is useless.
(344.) One of the most obvious simplifications
which inay be effected on an equation, is the diminution
of the number of its terms. To investigate the means
of effecting this, let the proposed equation be
a" + A, a "-" + As a “*-i-.... An-, . a + A* = 0. (A)
Let the assumed relation between aſ and y be such that
a shall be equal to the algebraical sum of y, and
another quantity k, to which we shall assign such a
value as may be necessary to attain the end we
propose. Thus we have
a = y + k. (B) -
To determine the equation (C), let this value of a be
substituted in (A), and the result, after each of the
terms have been developed and arranged by the
dimensions of y, will be
— I - - gº ſº
yº-mk yº- + º-º-º-, + º-º-º-º-º: y-, + .... *| = 0 (c)
1 .. 2 1 .. 2 .. 3 -
— I — 2
+ A* (m. – 2) A, k + A, k"--
+ As + . . .
–– An
The law which prevails among the coefficients of y is Hence the equation (B) becomes
here easily perceived. The exponent of y in the n” term Al
is m — (n − 1), and the coefficient of this term is * = y – i.
7m (m – 1) (m. – 2) .... [m — (n − 2)]
1 - 2 - 3. . . . . . . . . . . . . . . . H; e.
(m. — 1) (m. – 2) . . . . [m — (n − 2)] n-g
+ 1 - 2 - 3. . . . . . . . . . . . . . (n − 2) Al k
(m. – 2) (m. – 3) . . . . [m – (n − 2)] fl-3
+ 1 - 2 - 3 . . . . . . . . . . . . &=; A.k
(m. – 3) (m. – 4) . . . . [m – (n − 2)] rt-4
+ 1 - 2 - 3 . . . . . . . . . . . . (n − 4) As k
+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
a s e s a º e º e s a s sº e a e e º e a e s e e º e g º e a e e s e
Since the value of k is arbitrary, let such a value be
assigned to it as will render the coefficient of the
second term of (C) = 0. The equation (C) will then
be more simple than A, as it will want the second term,
or that which corresponds to A, a "-i. To fulfil this
condition we must have * -
m k + A = 0, … k = < *.
- 777,
(345.) Hence we derive the following rule for trans-
forming an equation, so as to remove the second term :
“Substitute for the unknown quantity r, the sum of
another unknown quantity y, and the quote of the co-
efficient of the second term of the given equation by
the exponent of a in its first term, with the sign of the
coefficient being changed.”
The process already explained for the solution of a
complete quadratic equation (173) is an example of
this principle. In this case the equation (A) is
a. * + A, a -- As = 0, (A)
A
... yº -(+ * > A)= 0, (C)
: A.?
* , = + Vºs.
A,”
-i - A,
. . . . A
* * + + = +
4 H 2
592
A L G E B R A.
Algebra.
\-y-/
A A.?
--- * * Vºx.
which is the formula established in (173.)
(346.) Since the relation between k and the coeffi-
cients of the equation which is necessary to remove the
second term is a simple equation, the value of k can
always be determined, and is always real; and there-
fore this transformation can in every case be effected.
This, however, will not be the case if we attempt to
remove any of the subsequent terms of the equation.
To remove the third term, we should have the con-
dition
7m (m — l -->
*** *-(n-1)Ak+A.- 0.
If the roots k', k" of this equation be real, the third
term may be removed from the equation by substituting
3y + k', or y + k", for a in (C.)
To remove the fourth term would require the solution
of a cubic equation for k, and in general to remove the
n” term would require the solution of an equation of
the (n − 1)" degree. The removal of the last term
would require the solution of the proposed equation
itself.
In all these cases the roots may be imaginary, and
then the transformation will be impossible. It will,
however, appear hereafter, that every equation whose
degree is marked by an odd number, must have at
least one real root; but those of even degrees may
have all their roots imaginary, from whence it appears
that it is always possible to remove the second, fourth,
or any even term of an equation, but not always
possible to remove the odd terms.
tº A
It is easy to see that the substitution of y — + for
a must remove the second term. For let A,' be the
coefficient of the second term of (C), and let S (a) be
the sum of the roots of (A), and S (a') the sum of the
roots of (C.) It is plain, that since each root of (A) Transforma-
is equal to the corresponding root of (C) — -*- We
have A 770,
S (a) = S (a') — m . m = S (a') — A1,
but A = — S (a), A,' = – S (a'), '.'
— A = – As – A, , "." As = 0.
(347.) It may happen that the same condition which
removes one term will also remove some other term.
Let it be proposed to determine the relation which must
subsist between the coefficients of the equation (A) in
order that the same condition which removes the second
term shall also remove the third term. In this case it is
necessary that the same value of k shall satisfy the
conditions , -
m k + A = 0
**** * + (m – 1) k A, + As = 0.
Let the value of k derived from the first be substituted
in the second, and we obtain, after reduction,
(m – 1) A,” – 2 m. As = 0.
If then the exponent and coefficients of (A) are so
related as to satisfy this condition, the same transfor-
mation will remove the first and second terms, but
otherwise not.
In general, to determine whether the same transfor-
tion of
Equations,
mation will remove any two terms, let the correspond- .
ing coefficients in (C) be put = 0, and let k be elimi-
nated. If the resulting equation be an identity, the
effect will be produced, but otherwise not.
(348.) It is sometimes necessary to consider the
equation (C) arranged in ascending powers of y. In
this case it assumes the form
— l
k” + m k”-1 y - "º k- Ay” + . . . . m k y” + y” = 0. (D)
e |
+ A, km-l + (m *g 1) A, Żm-2 + (m H 1) *H 2) Al km-8 + Al
+A,” + on–2) A.K.- || 4 º' Tººrº A, e-
–H + . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . +A.,
+ Am-, k + Am- - - - -
+ An
The coefficient of the m” term in this case is
m (m – 1) (m. – 2) . . . . [m – (n − 2)] km- (n-1)
1 - 2 - 3. . . . . . . . . . 71 – 1
(m – 1) (m — 2) . . . . [m — (n − 1)] % = fB
+ 1 - 2 - 3 . . . . . . . . . . 77 — I Al k
(m. – 2) (m. – 3) . . . . . . . . [m — n] on – (n + 1)
+ 1 - 2 - 3 . . . . . . . . . . . . . . ; : A, k
“H. . . . . . . . . . . . ſº e º . s e s a s m e º s e º 'º s s p is e e
&c. &c.
It may be observed, that the coefficients of the succes-
an equation are some or all of them fractional.
sive powers of y may be deduced one from another,
thus, “To find any coefficient multiply the successive
terms of the preceding coefficient by the exponents of
k, and then diminish the exponents of k by unity, and
divide by the number of preceding terms.” Thus, if
any one term be known all the succeeding terms can
be found. But the first term is the first member of the
proposed equation changing a into k. Hence all the
terms may be found.
(349.) It sometimes happens that the coefficients of
If the
equation be cleared of fractions by multiplying it by
A L G E B R A.
593.
from the unit, which it is desirable to avoid.
To determine a transformation which will remove the
3/
-->
fractional coefficients, let the equation (B) be a =
Hence the equation, (C,) after multiplying by k",
will be *
y" + A, K. y” + As k°. y” +
An-1. k y” + k" A, - 0. (C)
If the coefficients A, , As . . . . or any of them be sup-
posed to be fractions in their least terms, it is necessary
that their denominators respectively should measure k,
k” . . . . in order that Al k, A, k” . . . . should be in-
tegral. This will be the case if k be an integer pro-
duced by the continued multiplication of all the prime
factors of the denominators, each factor being repeated
so as to render the powers k, k” . . . . multiples of the
several denominators. w
Let the given equation be
7 1] 25
* — — a * + — a - — = 0.
3 ** + 36 #5 = 0
The prime factors are here 2 and 3, and k = 6, there-
fore the transformed equation is
y? – 14 y” + 11 a - 75 = 0.
SECTION XXX.
Transformation continued.—First Principles of Elimi-
nation.—Equation of Differences.
(350.) BEFORE proceeding further in the theory of
equations, it will be necessary to explain the first prin-
ciples of elimination. The more complete develope-
ment of this process, however, we shall reserve for a
subsequent section.
Elimination is that process by which when two equa-
tions, (A,) (B,) each including two unknown quan-
tities, a, y, are given, a third equation (C) is deduced
from them, including but one unknown quantity, a.
In general there are certain systems of values of ac
and y which will satisfy the two equations (A,) (B.)
There are, generally, an infinite number of systems of
values (225) which will satisfy one of the equations,
but only a limited number which will satisfy both. If
it be required to determine whether any particular
number r' is a value of a., which, in conjunction with
some corresponding value of y, will satisfy thº equa-
tions, it is only necessary to substitute a' for a in the
proposed equations; and then, considering y as the
unknown quantity, if they have any common root,
such a root will be the corresponding value of y, which,
in conjunction with the proposed value of ar, will satisfy
the equations. It may even happen, that the equations
will have several common roots, in which case there
are several systems of values of a, and y, in which the
value of a is the same, and which will satisfy the pro-
posed equations. -
To determine the values of a which will satisfy the
equations (A) and (B,) it is only necessary to find the
roots of the equation (C.) This is called the final
Algebra, the least common multiple of the denominators, the
S-N-' highest dimension of a may have a coefficient different
equation, and it appears from what has been just stated, Transforma-
that if any root of this equation be substituted for a in
the proposed equations (A) and (B,) they must have
at least one common root, and therefore, considered as
polynomes, their first members must have some integral
function (Y) of y as a common factor. If this func-
tion were known, the values of y corresponding to the
proposed value of a would be the roots of the equa-
tion Y = 0.
Hence we may infer, that if such a value of a can
be found as will make the polynomes, which form the
first members of the equations (A,) (B) be divisible by
the same integral function Y of y, these values of a
being each united with the several values of y found
from the equation Y = 0, will be systems of values of
a and y, which will satisfy the proposed equations.
(351.) Before we proceed to explain the method of
obtaining the equation (C) it is necessary to observe,
that the polynomes which form the first members of
the equations (A) and (B) cannot have a common
divisor independent of particular values of the quanti-
ties a and y, unless they be supposed indeterminate,
and in fact equivalent to one equation.
For suppose that they admitted a common divisor
D, and were of the forms wº-
A' x D = 0, B' x D = 0.
1. If D be a function of a and y. In that case
the two equations are equivalent to the equation D = 0,
in which there being two unknown quantities is inde-
terminate, and therefore there are an unlimited number
of systems of values which satisfy the proposed
equation, (225.)
2. Let D be a function of orie of the unknown
quantities, as ar. The equation D = 0 determines par-
ticular values of a, which will satisfy the proposed
equations, independently of any particular value of y.
Although, therefore, the values of a are determinate,
those of y are absolutely unlimited, and therefore the
equations are indeterminate. -
3. Let D be independent of a and y. In that case
D must be a common factor of the coefficients of the
equations, and ought to be suppressed previously to
their solution.
Hence we shall consider the equations as having no
common factor, independently of particular values of
a and y.
(352.) The equations then being supposed deter-
minate, let them be both arranged according to the
descending powers of y; and let the process for finding
the greatest common measure be pursued. Since
they have no common measure by supposition, there
will be at last a remainder obtained which is indepen-
dent of y. This remainder will then be a function of
a ; and if such a value be given to a as renders it = 0,
the equations will necessarily admit a function of y as
a common divisor. Hence the function of a, which is
found in the last remainder, must be the first member
of the final equation (C.) It is plain...that the values of
a thus determined, being substituted in the preceding
remainder, render it a common measure of the first
members of (A) and (B.)
Having by the resolution of the final equation (C)
determined all the values of a, the corresponding values
of y may be found by substituting these values for a in
the preceding remainder. By this substitution, this
remainder will become the greatest common measure
tion of
Equations.
Elimination
594 A L G E B R A. O
Algebra. of the first members of (A) and (B), the same values X, - a " + A*-* + A*-* +A,a”-a+.. An-, a '+A" Transforma.
\-v- being substituted for a in them. These values of y
will, therefore, be those which correspond to the values
of a determined before.
The final equation to which we arrive in this way, is
in general of an higher degree than the second, and,
therefore, we must postpone for the present the actual
investigation of the systems of values of the unknown
quantities. It appears, however, from what has been
stated, that we can eliminate one unknown quantity
from two equations each containing two unknown
quantities, and thereby obtain a final equation includ-
ing but one unknown quantity without the resolution of
any equation whatever. -
(353.) If there be three equations including three
unknown quantities, any one of them 2 is to be elimi-
nated by the first and second equation, and also by the
first and third. We shall then have two equations be-
tween a and y which are to be treated as above, and
similar reasoning applies to four equations with four
unknown quantities, &c.
(354.) As an application of these principles, let it be
required “to find an equation whose roots have any
given relation to any two roots of a given equation.”
Let the unknown quantity in the new equation be u,
and the combination or function of the two roots a 'a",
to which u is supposed to be equal be expressed by
F (r', r") that is
w = F(t', r"). [1]
Since also the values a', a "both satisfy the given equa-
tion, we have
a" + A, a "-" — As a "-" + . . . . An-1 w!-- A, - 0, [2]
a" + A r"--- Asa" + . . . . An-1 a." + A, - 0. [3]
The question then is, to eliminate a ', a " by these three
equations, and the final equation will be the equation
sought. This elimination will of course depend on the
nature of the function in the second member of [1], or
the relation which the roots of the sought equation
are to have to those of the given equation. It should
also be observed, that the final equation being inde-
pendent of a 'and a' will be the same, whatever pair of
roots may be assumed, and therefore its roots will be
similarly related to every pair of roots of the proposed
equation, and it will in general have as many roots as
ºthere are permuted combinations of two roots of the
given equation.' Hence the degree of the final equa-
tion must be at least m (m. – 1).
Let the equation [1] be
w = a," – a '.
Hence a "= u + æ', which being substituted in [2],
and the several terms developed and arranged by the
ascending dimensions of u, give
X,+ x'. --- X. 7/2 + X'.. ** -j-u" = 0
..+x, iſ 4 x, aſ +x, aſ +...+*=
where the notation (2), (3), &c. is used to express
1. 2, 1.2.3, &c. and”
Q/,
* Those who are familiar with the differential calculus will per-
ceive, that XI, X, &c. are derived from X, by differentiation,
- d Xo & dº? Xo (in Xo -
= tºo X2 :- . . . . X == ©
X, dar’ da's ſº da’”
X,' = m a."-" + (m – 1) A, a "-a -- (m — 2) Asa"-8-|- E tion of
quations.
X, - m (m – 1) +” + (m—1) (m—2) A r"-a -- Depression.
&c. &c. * \-N-
But since a' is a root of the proposed equation X} = 0.
Observing this condition, and dividing the equation by
w, we have
X/ X,” X/ X."
2. –- ºr u + 3:... w8 + +, u% + . . . . -- w” = 0.
d) + rj “Taj "-Fai,
The equation sought is, therefore, obtained by elimi-
nating a 'by this equation, and X,' = 0.
Hence to obtain the equation of differences it is only
necessary to omit the last term of the proposed equa-
tion, to diminish each exponent of a by unity, and to
change a' into u, and the coefficients A, , As, &c. into
X′, X′, X′,
(1) " (2)” (3) '
the given equation. It is unnecessary in this process
to place the accent on ar.
(355.) For example, let the proposed equation be
acº — 6 a – 7 = 0.
Hence we have X, − arº – 6 a – 7, X = 3 a.” 6,
X, <= 6a, X, - 6, X, - 0. Hence the equations for
the elimination of a are
a:3 – 6 ºr – 7 = 0
3 as – 6 + 3 r u + w” = 0
which by elimination give
w" — 36 u" + 324 w8-|- 459 = 0,
which is the equation of differences sought.
(356.) Since m (m. - 1) must always be an even
number, the equation of differences must always be of
an even order. But it is easy to prove that its alter-
nate terms beginning with the second are wanted; in
other words, that it includes only even powers of the
unknown quantity. For since every permuted combi-
nation of the roots are to be combined by difference,
it follows, that a " — a ', and aſ — a ", are both roots
of this equation ; and in general, if a, b, c, &c. are
the numerical differences of the roots -- a and — a,
—H b and — b, &c. are roots of the equations of differ-
ences. Hence the first member of this equation is
(a – a) (a + a) ( — b) (r-i- b) (r. — c) (r.-- c) . .
Or (as — a”) (cº — bº) (a" – cº, . . . .
If the square of a be taken as the unknown quantity,
the equation will become
(2 — a”) (2 — bº) (2 – Gº) . . . . . . = 0
the degree of which will be *: )
for e." This is called the equation of the squares of the
differences. It has the advantage of being of a lower
degree than the equation of differences.
&c., and to eliminate a by this and
, 2 being put
SECTION XXXI.
Transformation continued—Depression of Equations—
Equal Roots.
(357.) WHEN particular relations are known to sub-
sist between the roots of an equation, its resolution
A L G P. B. R. A.
595
Algebra. may be reduced to that of another equation of an in-
also be divisors of X, and if they be greater than 3, Transforma-
~~' ferior degree; the process by which this reduction is they will be divisors of X, and so on. rº
effected, is called the depression of the equation. (359.) These principles being established, we are E.
If any root (a) of an equation be known, the degree prepared to determine whether an equation X, = 0
may be depressed by dividing its first member by
ir - a. In like manner, if two roots (a, b) be known,
the degree may be depressed by two units by dividing
the first member by (a — a) (c. – b), and so on.
But even when no root is absolutely known, yet if a
certain relation be known, or can be discovered to sub-
sist between the roots, the equation may be shown to
depend on the solution of an equation of a lower
degree.
(358.) One of the most simple relations which can
be imagined to subsist between two or more roots of
an equation is equality. This is the case when some of
the binome factors of the first member are of the form
(a — a)”, (a — b)", &c.
Let X, be the first member of an equation of the mº"
degree, and let a, b, c, &c. be its roots, “..."
X, + (a — a) (a — b) (a – c)
Let r be changed into a -- k and the result is
(a + k)" + A, (x + k)”- + A, (T + k)"-* + . . . .
= (a + k – a) (a + k —b). . . .
or what is the same .
(a + k)" + A, (1 + k)"- + A, (a + k)"-* + . . . .
= (k+ º- a) (k+r–b)....
By developing both members, we obtain (354) for the
first member
X, + X
k k? kº
and if the developement of the second member be ar-
ranged by the ascending powers of k, the first term or
absolute quantity will be the continued product. of
(a — a), (a — b), &c., which by the above identity
gives -
X, - (a — a) (a – b) (a – c) . . . .
a result established already. The coefficients of k give
— Xo Xo X,
X == 2++,++. ––
The equality of the coefficients of kº gives
X X X
* - l – mºsºmº, l º
(2) (TºT) + (Tajºr-5 t . . .
and so on.
If the original equation X, - 0 have equal roots, its
first member X, will have equal factors of the form
(a — a)", (a – b)", &c. Hence the several quotes to
which X, is equal, will each have the quantities
(p — a)"-", (r. — b)"-" as factors. In fact, if the com-
binations of equal factors which enter X, are
(v — a)" (a – b)"
the combinations of the same factors which enter X,
are .
tº e > s sº
(a – a)"-" (a — b)"+ . . . . . .
Hence we infer, that “if the equation X, = 0 have
equal roots, the polynomes X, and X, admit a common
divisor.” :
It appears, also, that if the exponents m, n of any of
the factors r – a, a - b be greater than 2, they will
have equal roots, and to determine these roots when it is
possible so to do. Let the exponents of the factors
a — a, a - b, a - c, &c. which occur more than once
in X, be n, n', n", &c., and let the factors which occur
but once be a - p, a - q, &c. Hence
X, = (a — a)" (r. — b)" (a – 6)" .. (r – p) (a – q).
The degree of the equation X, <= 0 being m, it is plain
from the value already found (358) for X, that it is
equal to the sum of the quotes of X, divided by each of
its simple factors. Now as the factor (a — a) occurs
n times, the sum of the quotes for this alone must be
n Xa. In like manner the sum of the quotes for a b
/
is 70, *;, and so on. So that we have
X, − m X, + m' X, + m” X, + . . . .
Jº — (1, a – b a — c
X, X,
a – p n – q
Now it is plain, that the product (a — a)"-" (a – b)”-
(a — c)”-1 .. is a common divisor of X, and Xī.
But, further, it is the greatest common divisor, because
it contains all the prime factors (a – a), (a – b),
(a – c), . . . . which are common to these polynomes.
Hence it follows, that if X, and X, have no common
divisor, the equation X, has no equal roots. But if
X, and X, have a common divisor, which can always
be determined by the principles established in Section
XXVIII., that common divisor is the product of the
equal factors of X, the exponent of each being dimi-
nished by unity.
Let D be this common divisor. If it be of the first
degree it may be reduced to the form a - h, and there-
fore (p — h)” is a factor of X, and h a root which oc-
curs twice; and it follows, that in this case there are no
other equal roots. By the division by (a — h)” the
degree of the equation is depressed by two units.
If D = 0 be of the second degree there are two cases,
either the roots of D = 0 are equal or unequal. If
they be equal, D is of the form (a – h)”; in which case
h occurs three times as a root of X, = 0, and (a – h),
is a divisor of X, which will reduce the degree of the
equation X, = 0 by three units. But if the roots of
D = 0 be unequal, D is of the form (a — h\ (a — h'),
in which h and h' each occur twice as roots, and
(a – h)? (a — h')” is a factor of X, which will depress
the degree of X, = 0 by four units. *
In general, it is necessary to resolve the equation
D = 0, in order to determine the equal roots of X, - 0,
and the number of times that each equal root occurs.
Every root which occurs once in D = 0 will occur
twice in X, = 0, every root which occurs twice in
D = 0 will occur three times in X, - 0, and so on.
(360.) When we have obtained the equation D = 0,
and that it is found to be of a degree above the second,
it may be submitted to the process already described,
to determine whether it have equal roots; and if it be
found to have them, its degree may be depressed in the
same manner as that of X, = 0, and so the process may
be continued until an equation be found which has no
596
A L G E B R A.
Algebra, equal roots.
~~ ceed the second it may be solved, and when solved its
If the degree of this equation do not ex-
roots will furnish divisors which will depress the de-
grees of all the equations from which it was deduced.
But if the equation D = 0 have not equal roots,
and that it exceed the second degree, each root will
occur twice in X, − 0; and the methods of determining
the roots will be explained hereafter. -
(361.) We shall now show that the resolution of
every equation X, - 0 which has equal roots can be
made to depend on the resolution of a system of equa-
tions, of which the first includes the roots of the given
equation which occur but once, the second those which
occur twice, the third those which occur three times,
and so on.
Let X be the product of those simple factors of X,
which occur in it but once, X" the product of those
which occur twice, and so on, so that we have
X, = X. X's. X's. X". . . . . .
and by what has been already proved
D F. X!! g X/i/2 . X/s e G
Dividing the latter by the former, we have
X = Q = X′, X′, X/. X!" . . . .
ID
which is the product of the simple factors, equal as
well as unequal, of X, -
I.et the greatest common measure D' of D and Q
be now found. It is evidently
D' st: Xſ, X!" XIm/
that is, the product of all the equal factors; each, how-
ever, being introduced but once.
If Q be divided by D', the quote is X”, which is the
product of all the factors of X, which occur but once.
The equation X, <= 0 may thus be cleared of all the
equal roots, and considerably depressed in degree. The
equation X = 0 is the first of the system to which we
proposed to reduce X = 0.
By observing the form of the quantity D, it will be
observed, that the equation D = 0, like the original
equation, includes roots which occur once, twice, thrice,
and so on. The product X" of the roots which occur
once, may be found by the same process applied to
D = 0, as we have already applied to X, - 0. Hence
we shall obtain the equation X" = 0, which is the
second of the proposed system ; and by continuing the
application of the same process, we shall obtain
X” = 0, X" = 0, &c. It may be observed, also, that
the degree of the equation X7 = 0 expresses the num-
ber of roots which occur but once in X, <= 0, and its
resolution gives the values of these roots. The de-
gree of X" = 0 represents the number of roots which
occur twice, and its resolution gives the values of these
roots, and so on.
(362.) By the principles which have been here esta-
blished, we may obtain a criterion for determining whether
a given polynome be a square, cube, or any perfect power.
For this it is only necessary to derive from it another,
in the same manner as X, was derived from X, and if
this last be an exact measure of the first, the first is a
perfect power, and otherwise not.
(363.) The results of this Section might be more
simply and expeditiously established by the differential
calculus. But as it is desirable that Algebra should
be founded on principles independent of the calculus
d X,
*mºmºmºmºrºs
we shall here merely observe, that since X = da,
(354,) and
X, <= (x – a)" (a – b)" (a – c)".... (a — p) (a – q)
we have
X
——"—
(a — a)"
&c.
from whence, and similar processes, the results may
easily be obtained.
e X
X’ = (x – a)"-1 × +(r–b)" - *(i); --
SECTION XXXII.
Depression of Equations continued.—Reciprocal Equa-
tions.
(364.) AN equation in which the last term is unity,
and of which the coefficients equidistant from the ex-
treme terms are equal, is called a reciprocal equation,
from a remarkable relation which subsists between its
roots. The most general form under which such an
equation can be expressed, is-
a" + A r"- + A, r"----.... A, ..."--A, . w—H 1 = 0.
Let a y = 1, and let each term of the equation be mul-
tiplied by that power of a y whose exponent is the num-
ber of preceding terms. Hence we obtain
a" + A, a "y + A, r" yº . . . . -- As a "y"Tº + A, a "y"
-j- a " Ay” = 0,
which being divided by r", becomes
1+A, y + A, y + . . . . A, y”-- A y”- + y” = 0,
which is the original equation, a being changed into y.
Hence it appears, that y must be a root of the equa-
tion, and since y = —, it follows that if any number
JC
be a root of this equation, the reciprocal of that num-
ber must be also a root of the equation.
Hence we may also infer, that if the degree of the
equation be expressed by an odd number, one of its
roots must be unity. For by what has been just proved,
if any number not unity be a root, its reciprocal must
also be a root; and, consequently, the number of roots
different from unity must be even ; but since the total
number is odd, there must be one root at least equal
to unity. Such an equation can, therefore, always be
reduced in degree, by dividing its first member by
a — 1.
We shall, therefore, confine ourselves to the consi-
deration of reciprocal equations of an even degree.
Let 2 m be the highest exponent, so that the equation is
a" + A, a "- + A, 4°----. . . . As a " + A, a + i = 0
Dividing the whole equation by a ", and combining the
extreme terms and those which are equally distant
from them, we shall have -
(, + #) + A, (~4. # ) + A.(“4. #
1
+ . . . . +A.(.44)= 0.
a 2.
Depression
of Equa-
tions.—
Reciprocal
Equations.
A L G E B R A.
597
Symmetri-
cal Func-
tions of the
Roots of an
Equation.
Of Symmetrical Functions of the Roots of an Equation. S-2
Algebra.
1 - -
J. t Wºmº * = •." tºº 2. -: •." :-
--~ ACT 2: a + a ’ º 2 r + 1 0, Jº
T *.
--- 1. Hence we find
#
+
SECTION XXXIII.
I |
* + + = 2 •++ = a – 2
I
— - 28 – 3 c.
tº + a 3 = 2 3 & 9 &c. -
which substitutions being made in the former, we ob-
tain an equation of the mº degree to determine 2.
For each value of z determined by this equation, we find
22
4 1.
(365.) If the extreme terms of the equation, and
those which are equally distant from them, have con-
trary signs, the equation will also have reciprocal roots
when its degree is marked by an odd number. In this
case the form of the equation is
two values of a by the formula a = + +
a" + A, a "-" -- Asa" + . . . . — A, aº – A, a - 1 = 0.
Introducing a y = 1, and its powers as before, it be-
COIſlēS
1 + A y--A, y + .... – A, y” – A, y” — y”.
& - 0.
If the negative terms of the former were the same
with the same coefficients as the positive terms, the
latter equation becomes identical with the former, by
changing y into a, and changing all the signs. This
will be necessarily the case if the number of terms
which is m + 1 be even, that is, if m be odd. And
therefore in this case the former reasoning becomes
applicable. But if m be even, there will be a middle
term, and that term will have the same sign in both
equations, while all the other terms differ in sign.
As an example of the application of these principles,
let the proposed equation be a " — 1 = 0. If this be
divided by a - 1, we have -
a"- + æ"---|- a "-" .... as + x2 + 4 + 1 = 0,
which is a reciprocal equation of an even degree when
m is odd, and of an odd degree when m is even.
Let m = 5, '.' * . , , -
a' -- as -- as + a + 1,
".. (*#)-(, ++)+1= 0.
Let r + # = s. **-* = *-2.
2” – 2 + 2 + 1 = 0,
2*-i- 2 – 1 = 0, ... z = −4 =E * V5,
* = +++ Mzº – 4
"... a = –3 +4 vºivo +2 vs. w/~l.
vol. 1.
(366.) WHEN a quantity is a function of two or more
quantities, it is called a symmetrical function when it is
similarly related to each of these quantities on which
its value depends. The test by which a symmetrical
function may be known, is that its value will not be
changed by changing any two of the quantities on
which it depends, each into the other. Some examples
will render this definition more clear. Let u be the
function, and aſ and y the quantities on which it de-
pends, and let the function be expressed by the letter F
prefixed to a, y, so that u = F (a, y). Now if the
value of u remain the same when a is changed into y,
and y into a, or u = F (y, a.), then is u a symmetrical
function of a and y. Let u = a + y ; this is evidently
a symmetrical function, since z + y = y + æ. But if
w = a – y, w is not a symmetrical function, since
ac – y is not equal to y – ar. Again, let w = c y, or
w = wº + y”; these are symmetrical functions, because
a: y = y ar, and a " + y^ = y?-- a 2. But, on the other
hand, a "y" is not a symmetrical function, because it is
not equal to y” a”, unless m = n, in which case only
it is a symmetrical function.
(367.) The most simple symmetrical function of any
number of quantities is their sum, and the most simple
class of such functions is that to which this belongs,
viz., the sum of the n” powers of those quantities.
Let al., a, as, &c. be the quantities, the class of func-
tions to which we allude, is
a, +- a, +- a, +- a, + . . . .
a H- a + a + a + . . . .
a + a + a + a + . . . .
&c. &c.
a + aſ: –H a + at . . . .
We shall express these functions severally by the nota-
tion S (a), S (a”), S (a8), &c. We shall call these
symmetrical functions of the first kind.
The class of symmetrical functions which are integral
and rational, next in simplicity to the preceding, are
symmetrical functions, each term of which is a product,
into which two different literal factors enter. A func-
tion of this class is the sum of the products of all the
letters of the function taken in permuted combinations
of two with given numbers as exponents, the same
number being the exponent of the first letter in each
permuted combination. Thus,
a; a + at a + a. a. -- a. a. -- a. a + a. a.
is a symmetrical function of a, as, as, of the second
kind. , g r . -
The general form for such a function is
aſ a + aſ a + aſ aſ . a. aſ + as aſ + . . . .
a; a + aka; +.
we shall represent such a function in general by
S (a" a”). . -
A symmetrical function of the third kind has a simi-
lar meaning, and is expressed by a similar notation
S (a" a” a”), and so on. -
(368.) If n be the number of different letters which
4 I
598
A L G E B R A.
Algebra.
w
enter a symmetrical function, the number of terms in a
symmetrical function of the first kind is evidently m.
The number of terms in a function of the second kind
is the number of permuted combinations of two letters
which are obtained from n letters, scil. n (n − 1). In
like manner, the number of terms in a symmetrical
function of the third kind is n (n − 1) (n − 2), and
SO OIle -
(N. B. There are an infinite variety of symmetrical
functions of a given number of letters, but we confine
ourselves in this place to the consideration of such as
are algebraical, rational, and integral. Those which
we have described are called elementary symmetrical
functions.) . . .
From the nature of symmetrical functions, it is evi-
dent that if any term be affected by a multiplier or
divisor, all the terms must be affected by the same mul-
tiplier or divisor; and if A be such a coefficient, the
function may be expressed A. S (a"), A S (a" a”), &c.
(369.) Having thus explained the nature of the
symmetrical functions we are about to consider, we Symmetri-
shall proceed to investigate the method of determining ºal ºne-
tºl º g tions of the
such functions of the roots of an equation. Roots cf an
Since every root of an equation must be similarly Tºai.
related to its coefficients, it follows that each of these º-y
coefficients must be a symmetrical function of the
roots of the equation. Indeed, this follows imme-
diately from the properties of the roots established in
Sect. XXXII. -
The coefficient of the second term is the sum of the
roots with their signs changed, and is, therefore, the
simplest species of symmetrical function of the first
kind. The coefficient of the second term is the sum
of the products of every two roots, and, therefore, is
the simplest species of symmetrical function of the
second kind, and so on.
Let it, however, be proposed to determine the other
symmetrical functions of the first kind of the roots.
Let the roots be a, as, as, &c. we have (312)
3 - a = r"---a, *---- a wº- + a w”- + a | *-* +.... + aſ-
l * , -
+A. --A, a +A. at + A. a. + A, . a "-"
+ A. f;"| 1: -- A, . a "T"
r ' . (2 • e s a v -
8 +A. " . . . . . . . .
-H An-1
#: = a "-" +a, w”-" + a; a”---- *. a; a”-4 + a; a” + .... -- a "-
2 - -
+ Al + A, . a, + A. . a. +A. a. -- Al aſ "º
+ As †: “I is: +&sº
t 3 - 0x - tº e º is
3 +A, . . . . . .
• + An-1
and similar developements may be obtained for a — a.’ &c.
— we w - w;
o g X X X
By adding all these developements we obtain H l + a – as –H a – as + . . . . a -a, T
m a."--- S (a) a”---|- S (a8) a"-" -- . . . . . . . . S (a"-)
-- m Al -- A, . S (a) -|- A, . S (a"-*)
- + m As + As . S (a"-8)
+. m. An-1
But by (358) it appears that the first member of this
equality is equal to X1. By equating the several co-
efficients of the powers of a in the second member of
the preceding equality with those of the same powers
of a in the value of X, found in (354,) we find after
reduction,
S (a) + A1 = 0
S (a”) + A S (a) + 2 As = 0
S (a8) + A S (a") + As S (a) + 3 As = 0
S (a"-") + A, . S (a”) + As S (a”) + . . . .
+ (m — 1) An-1 . = 0.
The first of these equations gives the value of S (a);
this being found, and substituted in the second, we may
find S (a”); this being known, the third gives S (a”),
and so on. Thus the symmetrical functions of the
first kind are determined as far as the (n − 1)"
degree. - .
To determine those of superior degrees, let al, as, as,
&c. be successively substituted for a in the given equa-
tion, and let the results be multiplied by aft, a, a., &c.
respectively, and we obtain
S (a"+") + A S (a"+"-") + A, S (a"A*-*) + . . . .
Am_i. S (a"+") + An S (a") = 0.
In this equation, let 0, 1, 2, 3, &c. be successively sub-
stituted for m, and we obtain
S (a") + A 1. S (a"-") + A, . S (a"-") + . . . .
An-1. S (a) + m, An = 0
S (a") + A. S (a") + A, . S (a"-) + . . . .
An-1. S (a") + A, . S (a) = 0 -
S (a"*) -- A. S (a"+") + A, . S (a") + . . . .
An-1. S (a”) + A, . S (a”) = 0.
&c. &c.
The first of these determines the value of S (a") where
the functions S (a"-"), S (a"-"), &c. of inferior degree
A L G E B R A. 599
The second member of this being composed of functions Symmetri-
of the second kind, has been already determined. iºi.
If n' = n”, the terms of the first member become . . .
equal in pairs, which being united, the whole will be Equation.
affected by the common factor 2. Hence we have \-y--"
Algebra, are known, and which can be found by the former pro-
S-N-' cess. The second determines S (a”) when the func-
tions of inferior degrees are known, and so on.
(370.) The symmetrical functions of the reciprocals
of the roots may be determined by making n negative
in the preceding formula. By this change it becomes
S (a"-") + A S (a"-"-") + A, S (a"-"-") + . . . .
An-1. S (a^*) -- A, . S (a") = 0.
* Substituting for n in this, 1, 2, 3, &c. we obtain
S (a"-") + A S (a""") -- .... + An-1. S (a)
+ A, . S (a-') = 0
S (a"-") + A S (a"-") + .... + A*- : S (a-)
+ A, . S (a^*) = 0 e
S (a"-") + A S (a"-") + .... + An-1. S (a-º)
-- A, . S (a-b) = 0
. &c. &c.
It is plain that S (a") = m, since a0 = 1, and the
number of terms in S (a") is m. Hence all the terms
of the first equation, except the last, have been previously
determined, and therefore S (a-') can be found. By
the second equation S (a^*) may be determined, S (a-')
being previously found, and so on. -
Hence, in general, “When any equation of any de-
gree is given, we may obtain the sum of the squares,
cubes, &c. or any similar integral powers of its roots or
the sum of the square, cubes, &c. or any similar integral
powers of the reciprocals of its roots.”
(371.) We shall now explain the method of deter-
mining the symmetrical functions of the second kind.
These we shall express by the notation S (a" a”). If
amºč
*
S (a") and S (a") be multiplied, the product will evi-
dently contain the (n + n')" powers of all the roots,
and also the product of every combination of two roots
in the n” and m” powers. Hence we have
S (a"+") -- S (a" a”) = S (a") × S (a")
... S (a" a”) = S (a"). S (a") – S (a"+")
The second member of this equation being composed
of symmetrical functions of the first kind, which have
already been determined, the first member is known.
If n = m', the second member becomes S (a")” —
S (a”). Of the m (m – 1) terms of the first member,
the permuted combinations of the same letters become
equal, and therefore the number of terms, when the
7m (m —
#9. and all
equal terms are combined, becomes
of them have 2 as a common multiplier. Hence it is
evident that the result is -
_ S (a)” – S (a”)
) *=== 2
The first member of this may be considered as a sym-
metrical function of the first kind, of the roots combined
in products of two factors. -
(372.) To determine the symmetrical functions of
the third kind, let the values of S (a" a”) and S (a") be
multiplied together. The terms of the product will be
of three forms, a "+". a”, a”. a”, and a” a” a”, and
we have evidently
S (a+”. a”) + s^a". a'4") + S (a" an a')
= S(a"a"). S (a")
... S (a' a' a') = S (a" aw). S (a") – S (an. a'4")
– S (a"+". a').
*
ºrum ºms
S(a" at a*) = + IS (a" a”). S (a") – S (a". a”)
– S (a^*. a”)]
The first member of this may be considered as a func-
tion of the second kind of the roots themselves, and
their combinations in pairs. The number of terms in
— l – 2)
it is evidently º º (m 3 2).
If n = n' = n", the terms of the first member will
be the n” powers of every permuted combination of
three roots. The terms containing the permuted com-
binations of the same letters are equal, and as the
number of such terms is 1. 2. 3. this will be a common
factor. But also in the second member since n = 'n'
... S (a" a”) = 2 S ((aa)"), and the terms S (a"+". a”)
and S (a". a”) become identical. Hence we have
S ((a a a)") = }[S ((a a)"). S (a") – S (a” a”)]
By pursuing a similar method, symmetrical functions of
all higher kinds may be determined. -
(373.) All symmetrical functions whatever, which
are integral and rational, must be combinations of those
already determined, and hence we may in general infer,
“That any integral and rational symmetrical function
whatever of the roots of an equation may be determined
when the coefficients of the equation are known.”
A symmetrical fractional function, if all its terms be
reduced to the same denominator, and added, will
become a fraction, whose numerator and denominator
are integral symmetrical functions. Hence the preced-
ing inference may be extended to all rational symme-
trical functions whatever.
If the symmetrical functions of the forms S (a),
S (a a), S (a a a), &c. be called primary symmetrical
functions, we may infer in general, without immediate
reference to equations, that if the primary symmetrical
junctions of any number of quantities be given, all ra-
tional symmetrical functions of the same quantities may
be found. For the primary symmetrical functions are
the coefficients of an equation, of which the quantities
themselves are the roots. -
(374.) Let us now apply the preceding principles to
the solution of the following problem, “to find an
equation whose roots are the sums of every pair of
roots of a given equation.” º
Let the given equation be X = 0, and a1, as, a, &c.
its roots.
Let the sought equation be Y = 0, and a + as,
a, +- a, &c. its roots. The number of roots of Y being
the number of combinations of two letters which can
be made from m letters, the degree of Y will be
m (m. — 1)
1 .. 2
The coefficient of its second term will be the sum of
the quantities a, + as, a, + as, that of the third
term will be the sum of their products in combinations
of two, that of the fourth the sum of their products in
combinations of three, and so on. These, being all sym-
metrical functions of the roots, may be determined by
the preceding principles.
4 I 2
600 gº
A L G E B R A.
And in general an equation may be found whose
\-V-2 roots are any symmetrical functions of the roots of the
given equation taken in combinations of two, or of
three, &c. For the degree of the sought equation will
be determined by the number of combinations, and its
coefficients being symmetrical functions of its roots,
which are themselves symmetrical functions of the roots
of the given equations, and these again being functions
of the coefficients of the proposed equation, it follows,
that in every case the coefficients of the sought equa-
tion may be derived from those of the given equation.
The equation of the squares of the differences deter-
mined in (356) is an example of this, since the squares
of the differences are symmetrical functions of the roots.
The coefficients of this equation may easily be deter-
mined on the principles established in the present sec-
tion.
(375.) We shall now apply the properties of sym-
metrical functions to the solution of the following im-
portant analytical problem: “To determine the degree
of the final equation resulting from the elimination of
one of the unknown quantities by two equations of
the mº and nº degrees, including two unknown quan-
tities.” *. -
A general equation of the m” degree between two
unknown quantities in which the sum of the exponents of
the unknown quantity in that term in which it is highest
is equal to m. Such an equation should include terms
containing every combination of powers of the unknown
quantities, the sum of whose exponents does not ex-
ceed m. Hence if it be arranged according to the
dimensions of a, and the coefficients be Ao, A1, As, &c.
A, a " + A, a "-" -- A, a "-" —H . . . . An-1. a + An = 0
B, a,” + B, a,”- + B, a,”-- + • * * *
Bo: a.” + B. a "- + B, a. *-* +
B. a,” + B. a "-" + B, a,” +
&c.
Since B, B, B, . . . . are rational functions of y, and
al, as, a, . . . . are in general irrational functions of y,
it follows, that those polynomes are in general irra-
tional functions of y. We shall now prove that any
value of y which renders any one of these polynomes
= 0, will, in combination with the corresponding value
of r, satisfy the proposed equations A = 0, B = 0.
Since any of the functions a1, as, a, . . . . will satisfy
A = 0 independently of y, they will also satisfy it when
Ay has such a value as renders one of the above poly-
nomes = 0. Let this value be y’. Now let y' be a
value which renders the first of the polynomes = 0.
Let y' be substituted for y in the function a, and let
the corresponding value of a, be a '. It follows then,
that y' a' are a system of values of y and a which
satisfy the equation A = 0. But they also satisfy the
equation B = 0. For since y' renders the first of the
above polynomes = 0, and this polynome is, in fact,
the first member of B = 0, a, being substituted for a,
it follows, that if y' and w' be substituted for y and a in
the first member of B = 0, it will become an identity.
Hence, in general, y'a' is a system of values of y and r,
which satisfies both of the given equations.
It is easy to perceive, also, that every value of y,
which, in combination with a value of a, will satisfy
both of the given equations, must render one of the
preceding polynomes = 0. For the value of a which
in conjunction with that of y satisfies the equations,
& © º ºs º gº e e º is
the several coefficients, the first excepted, will be inte-
Symmetri-
gral and rational functions of y, but they must be such ...cal Fune-
that their dimensions when combined with the power
of a will not exceed m. The first coefficient must,
therefore, be independent of a and y, and therefore
a known quantity; and the forms of the successive
coefficients must be respectively
a y + b
... a y” + b y + c
a y” + b y” + c y + e
&c.
a y” + b y”. ... l
. a y” + b y” + . . . k y + 1
Now let the two given equations be arranged accord-
ing to the powers of a, and let the coefficients be under
stood to be functions of y, such as those just described,
and let the equations be
A, r" + A, a "-" + A, a "-" + . . . . An-1. a + An = 0
B, a” + B, a "-" —- B, *-* + . . . . B, , . a + B, - 0
Let their first members be called A and B.
Let the former equation be imagined to be solved, as
if y was a known quantity, and let the roots be a, as,
a, &c. These will be respectively functions of y. If
any one of these functions be substituted for a in A = 0,
it will convert the equation into an identity, and it will
be true for every value whatever of y. This will not,
however, be the case if any of the same values be sub-.
stituted in B = 0. By successively substituting the
functions a, , a, as, . . . . for a in B = 0, the first mem-
ber becomes
+ B,a_1. a, + B,
-j- B, , , a, + B,
+ Bin-1. Cls + B,
&c.
must be one of the functions a1, as, a, . . . the value
of y being substituted for it ; and hence it is evident,
that the corresponding polynome becomes an identity.
Hence we may infer, that the equation whose first
member is the product of all the polynomes [1], must
contain among its roots all the values of y, which can
satisfy both the equations A = 0, B = 0. If these
several polynomes be expressed by A(i), A(2), A(2), &c.
the equation which thus gives the values of y inde-
pendently of a is *
AG). A(s). A(s)...... A(*) = 0. [2]
The first member of this equation is evidently a sym-
metrical function of the roots a, as, a, for if
any one of the roots be changed into any other in it, no
other change will be produced than a change in the
order of its factors.
Now as every symmetrical function of the roots can
be determined by the principles established in this sec-
tion, the present one may also be obtained; and hence
an equation will be established in which the unknown
quantity y alone will appear, a being eliminated ;
which is, in effect, a new process of elimination.
We shall not here go through the process for deter-
mining the form of the function in the first member of
[2]; our present object is merely to determine the
degree of the final equation [2].
The object then is to determine the highest dimen-
[1]
tions of the
Roots of an
Equation.
A L G E B R A.
60l.
Algebra, sion of y which is found among its terms. Let L a,”,
--' L'aº, L'a,”, &c. be any terms of each of the poly-
nomes of [1]. The continued product of these will be
and succeeding, sections, we propose to develope the
methods of finding the roots of numerical equations.
Let the first member of a numerical equation of the E
Limits of
the Roots of
Numerical
a term of the first member of [2] when developed, and
this term is therefore |
L. L'. L." x a,” a,” a “ . . . .
Now as products of the same combination of letters will
be permuted in every possible way in the first member
of [2], it follows, that
will necessarily be a part of this first member. The
question then is, to find what is the highest power of y
which can enter such a function as this.
The quantities L., L', L", . . . . being coefficients of
the equation B = 0, the highest dimension of y, which
enters any one of them is such that, when added to the
exponent of the power of a which it multiplies, it will
give a sum equal to n. If then p be the exponent of a,
n – p will be the highest corresponding exponent of y,
in each of the quantities L., L', L", . . . . and as the
number of these quantities is that of the roots of A = 0
or m, the highest exponent of y in the product
L L'L". . . . is m (n − p).
To determine the highest exponent of y in the func-
tion S (a,”, a ". . . . . ) we must refer to the values of
S (a), S (a8), S (a") . . . . established in (369). From
these, and from the forms of the coefficients of A = 0,
and B = 0, it appears that the dimensions of y in the
functions S (a), S (a”), S (a") . . . . are the 1st, 2d,
3d, &c. Hence, it follows, that the dimensions of y
in S (a,”. a.”. a.” . . . . ) are p +p/+ p". . . . But p,
p', p", . . . . being the exponents of a in B = 0, they must
be such, that when added to the highest exponent of y
in L L'L". . . . the sum will not exceed n. If this ex-
ponent be l, the highest value of p will be n – l ; and
since the number of these which are contained as factors
in the product S (a,” a,” a,” . . . . ) is m, the highest
dimensions of y is m (n − 1). This added to the
dimensions m (n − p) in L L'L'. . will give the highest
dimensions of y in [2] m (2 m – p-l), but p + l = n, “.."
the highest degree of y in [2] is m m.
*mºssºmsºmºmº
SECTION XXXIV.
Numerical Equations—limits of the Roots.
(376.) THE various properties of the roots of equa-
tions established in the preceding sections are appli-
cable to all equations whatever, whether their coefficients
be literal or numeral, that is, whether the equations be
algebraic or numerical. The solution of the problem
to determine the roots of a general algebraic equation
of a degree higher than the fourth has never yet been
effected. And even in the cases in which some ana-
lysts have succeeded in discovering the formulae for the
roots, the results are always complicated, and frequently
inapplicable in practice. The species of equations
which, however, most frequently occur in philosophical
investigations are numerical, and although we may be
unable to assign the general forms of the roots, yet we
can always determine their values where the numerical
values of the coefficients are known. In the present
m” degree be expressed as before, thus
a" + A, a "-" -- As a "". . . . An-1. a + A, t
the letters used here to represent the several coefficients
are to be understood as expressing particular numbers.
Any number whatever being substituted for a, let
the value of this polynome corresponding to that num-
ber be y : hence we have
9 = a "+ A. a.” + As a "T"-- . . An-1. a, + Am, [1]
the roots of the proposed equation are those numbers
which being substituted for a will render y = 0. In
general, a particular value being substituted for a must
render y either > 0, - 0, or < 0. Let two particular
values ar', a." be substituted for ac, and let the cor-
responding values of y be y', y". These values y', y''
must either have different signs or the same sign.
(377.) 1. If y' and y' have different signs, there is at
least one real root included between the numbers x' and
x', and, in general, there may be an odd number of real
roots between them.
(By the numbers included between two given num-
bers, is meant numbers greater than the lesser, and less
than the greater. It is necessary, however, to attend
to the effect of their signs (188)).
Let y' be negative, and y" positive. Let X be the
sum of the positive terms in the value of y, and X’ the
sum of the negative terms, so that we have
y = X — X'.
Since X and X’ each consist of integral powers of a
with positive numerical coefficients, it is evident that if
we suppose the value of a continually to increase from
quations.
\-v-
ac' to ac", each of the quantities X and XV must also
continually increase. But when a = a-', y = y < 0,
‘.' X 3 X', and when a = a,", y > 0. Hence, as
a continually increases from a' to a ", X and X’ both in-
crease; but X increases more rapidly than X', since it is
in the first instance less than X', and afterwards sur-
passes it. As the increase is continual, it follows, that
before X surpasses X' it must become equal to it, and
when it does, X- X' = 0, ... y = 0, and the value of a
which corresponds to this state is a real root. Hence
there is one real root at least between ac' and ac".
But it may happen, that between the values of X
and X' which correspond to aſ a:", the value of X first
increases so as to exceed X', then the rate of increase
of X becoming slower than that of X', the latter may
again surpass X, so that X X" again becomes nega-
tive, and, finally, X may again increase more rapidly
than X', and become greater than Xſ before a becomes
equal to a:". In this case, while a is increasing gra-
dually in value from aſ to ac", X first increases from
being < X to be > X', then X’ increases from being
< X to be > X, and, finally, X again increases so as
to be > X/. X must be equal to X' in three cases; and,
therefore, there will be three real roots between ac'
and ac".
By generalizing this reasoning, it appears, that the
rates of increase of X and X’ may alternately exceed
each other, while a is increasing from aſ to ac''. But that
by these changes X and X’ must be equal at least once,
and may be equal an odd number of times. From whence
it follows, that between ac'a" there must be one real
root, and may be any odd number of real roots.
= u"
602 A L G E B R A.
Algebra. This reasoning is applicable when either or both of
S—— the values aſ a '' is negative, and when either of them
= 0.
But the quantity within the parenthesis is a geometrical Exponen-
series whose first term is 1, the common multiplier w, tial
and the number of terms m. Hence the sum of the Equations.
(378.) 2. If y' and y" have the same sign, there are
either no real roots or an even number of them between
a' and ac". * . -
As before, X and X’ increase continually by the con-
tinual increase of ac.
be +, X is greater than X" at the two limiting values
corresponding to ac' and a ", and may, therefore, be
greater than it for all intermediate values. If the com-
mon sign of y' and y' be —, X is less than X" for both
the limiting values, and may, therefore, be less than it
for all intermediate values. In both cases, therefore,
there may be no real root between the limits a' and a ",
since it is not necessary that X should = X'.
But it may so happen that the rates of increase of Xand
X' may so change between the limits, that each will alter-
nately surpass the other, and in every such change they
must become equal. Now, since at the limiting values
X and X’ are similarly related to one another, it is
plain that if there be any changes of relation as to
magnitude, there must be an even number; for other-
wise the result of the whole would change their rela-
tive magnitudes contrary to hypothesis. Hence, it
follows, that between the limits ar' and ar" there must
be an even number of real roots, or none.
(379.) A value may always be assigned to x in the
second anember of [l], such that y > 0, and so that all
values greater than the assigned value will also render
> 0. -
y Let the equality, [1] be expressed in the form
A - , Am \
v=x-(1-4----. As ºr )
+ . . . . . . g"
A value may be assigned to a such as will render each
of the terms within the parenthesis, except the first,
less than any assigned value; and therefore such a
value may be given to a as will render the sum of these
terms 3 1. It is evident that one such value being
found, every greater value of a will render the sum of
these terms still less than 1. Hence, this value of ar,
and all greater values, will render the parenthesis
positive; and since a " is positive, y will be necessarily
positive.
Hence, it is evident, that no root of the equation can
be greater than such a value of w.
(380.) To determine a number, which, being substi-
tuted for x, will render the first member of an equation
positive, and such that all greater numbers will also
render it positive. - -
The number sought must be such as will render the
first term a "greater than the algebraical sum of all the
succeeding terms. Let S be this algebraical sum, and
let S' be the arithmetical sum, and let K be the greatest
numerical coefficient. It is plain that S! is generally
greater, and cannot be less than S '.' if a." > S we
must also have a " > S. Also it is plain that S' cannot
be greater than
K (*-1 + 4*-* + æ"-a a + 1).
Since K is, by hypothesis, the greatest coefficient in S',
and the several terms are affected by the same powers
of a. The problem will then be solved by any value
of a which satisfies the condition
a" > K (a"- + æ" = + æ"-" . . . . . ... a + 1).
If the common sign of y' and y'
* — I
series is ++, by which the above inequality
º Cºmº ||
becomes - a" – 1
* > K. H-.
• r"+1 — r" > Ka" – K,
a "+" > (K + 1) a” — K,
- Fº
• * > (k+1)----,
a condition which will evidently be fulfilled if a = K
+ 1. Hence we may infer that the greatest coefficient
in the equation, taken with a positive sign, and in-
creased by unity, is greater than the greatest root of
the equation. -
In obtaining this superior limit we have taken an
extreme case, scil. that in which all the terms of the
equation, except the first, are negative. This seldom
happens, and therefore the limit thus obtained is, in
general, too wide. To obtain a nearer limit, let the
exponent of the highest power of a, which has a nega-
tive coefficient, be m — n. Let S be the algebraical
sum of this and all the succeeding terms. It is evident
that any value of a which renders ar" > S will be a
superior limit. Let S' be the arithmetical sum of the
terms of S. As before, S' is generally greater and
cannot be less than S. Let K be the greatest nume-
rical coefficient of S; as before, S cannot be greater
than -
K (w"-" + æ"-"-i + . . . ... a + 1).
Hence the value of a will be a superior limit, if it fulfil
the condition
a" > K (a"-" + æ"-"-1 + . . . . . . a" + a + 1)
n-n + 1.
The sum within the parenthesis is l l Hence
the condition becomes
a. * > K.
arm-"+1 — I
a — 1
Hence it follows that the condition will be fulfilled by
the value of a determined by
Kam-n + 1
a" > — 7
ac — 1
•. a"-1 y
> a — 1
‘... a "-" (a – 1) > K.
".. a = p + 1, ..."
(p + 1)"Tº . p > K.
This inequality is evidently satisfied by p” = K, for
(p + 1)". p > p"~". p = p".
l
1
Let & – l = p,
* – 1 = K", z = K" + 1.
Hence we infer, that “ that root of the greatest nu-
merical coefficient whose exponent is the number of
terms preceding the first negative term increased by
unity, is a major limit of the roots of the equation.”
If all the terms of the equation, except the first, be
negative, this limit is equivalent to the former one.
Hence
A L G E B R A. 603
Algebra, (381.) The limits just determined are major limits Let such a value be assigned to r" as will render the º º
of the positive roots. It remains to determine their several polynomes X', , X', , X', . . . . positive, and this “ i.
minor limits, and also the major and minor limits of
value will be a major limit of the positive roots. For Equations.
the negative roots.
in that case the transformed equation cannot have any V-V-
If the equation be transformed by the substitution
a = --, and the new equation cleared of fractions, it
will be one whose greatest positive root is the least
positive root of the former; since the roots of the two
equations are reciprocals. The coefficients in the two
equations will also be the same, but occurring in an
opposite order. Hence the major limit of the roots in
the transformed equation is the minor limit of the positive
roots in the original equation.
(382.) To determine the major limit of the negative
roots, let a in the proposed equation be changed into
— y, and the positive roots of the transformed equation
are equal to the negative roots of the original equation.
Hence the major limit of the positive roots of the trans-
formed equation is the major limit of the negative roots
of the original equation.
(383.) To determine the minor limit of the negative
roots, let — — be substituted for a, and the major
limit of the positive roots of the transformed equation
will be the minor limit of the negative roots of the given
equation.
Hence it appears that the method of determining the
major limit of the positive roots being known, the other
limits may be found. *
Ea'amples. Determine the major limits of the positive
roots of the following equations:
a “ — 5 a 3 + 37 a 2 – 3 r + 39 = 0,
"v K -- 1 = 5 + 1 = 6;
a 5 + 7 a.4 — 12 w" — 49 a * + 52 a - 13 = 0,
F a/49 + 1 = 8;
a. * + ll a * — 25 a - 67 = 0,
=*V 67 + 1 = 6;
3 a 3 – 2 a.” — 11 a + 4 = 0,
| 1
= + + 1 = 5.
(384.) In particular cases it happens that transfor-
mations present themselves which expedite the process
and give nearer limits than the general method.
If the first member of the equation be such as can
be resolved into a series of products, one factor of each
being a monome, and the other a binome, whose second
term is a particular number and negative. Such is the
second of the preceding examples, which may be written
thus,
a? (r" – 49) + 7(- #)+ *(s- +)=0.
A limit will here be obtained by finding a value of a,
which will render all the binome factors positive. Such
is a = "vä9. Hence 4 is a limit nearer than 8.
(385.) There is another method of finding limits to
the roots of equations, the discovery of which is due to
NewToN. Let r + u be substituted for a, and the
transformed equation will become (354)
x. +x,44 x. +, +x. + ...wº-0.
- 0 . . "' (1) - § " (2) 3 * (3) - -
positive root, since a polynome, all whose terms are
positive, cannot = 0. Hence the real values of u.
must be essentially negative. But a = a + w, ...' a' =
a —u. Since u is essentially negative, -u is essentially
positive, '.' a' - u > a., "..' a' > a. Hence ac' is a major
limit.
(386.) If all the terms of an equation be positive, it
cannot have a real positive root, for the sum of any
number of positive monomes cannot = 0. For a
similar reason, if the terms be alternately positive and
negative, it cannot have a real negative root; for in
this case if the degree of the equation were even, all the
terms of the first member would be positive monomes;
and if the degree were odd, all the terms would be
negative monomes. In the one case, the first member
would be the sum of several positive monomes, and in
the other, it would be the sum of several negative
monomes. In neither case could it be - 0.
SECTION XXXV.
on the Real Roots of Numerical Equations.
(387.) Every equation whose degree is characterised
by an odd number, and whose coefficients are real, has
at least one real root, whose sign is different from that
of its last term.
If in the equation
y = a " + A, a "** + A, a "* + . . . . An-, . a + A,
we suppose a = 0, we have y = A, ; and, on the other
hand, a value may be assigned to a such that a "will
be numerically greater than all the succeeding terms
together; if such a value be assigned to r, with a sign
different from that of Aa, the sign of y will be different
from that of Aa. Hence, then, for a = 0, the sign of y
is the same as that of Aa, and for the other value of a
it is different. Hence one real root must be between
those values.
(388.) Every equation of an even degree in which
the least term is negative, and whose coefficients are
real, must have at least two real roots with different
Sºl S.
#. if a = 0, y = – A, ; and, on the other hand,
such a value may be assigned to a as will render a”
numerically greater than the sum of all the succeeding
terms. Whether this value of a be positive or negative,
a" will be positive, since m is even, and therefore the
value of y will be positive. Hence it follows, that
between this value of a, taken with a positive and
negative sign, and a = 0, there is in each case a real
root, the one positive and the other negative.
It is evident, that in the former case the real root
lies between K -- 1 and 0, and that in the latter case
the positive root is comprised between K+ 1 and 0,
and the negative root between — (K-H 1) and 0.
Hence, the principle assumed in (316,) that “every
equation has at least one root, is established for all
equations, except those of an even order, in which the
last term is positive.”
604 A L G E B R A.
Algebra. (389.) Imaginary roots enter equations by pairs; In the same manner it is evident, that if the number Real Roots
that is to say, there must be an even number of them, of real and positive roots be even, the last term is of Nº.
Or DOIne. positive; and if it be odd, the last term is negative. Equations.
Let the first member be divided by all the simple (392.) No equation can have a greater number ºf G –
contains one term more.
factors which correspond to the real roots; the quotient
must be rational, and its coefficients must be real. Its
roots will, by hypothesis, be all the imaginary roots of
the proposed equation, and no others. Its degree
must therefore be even, since it can have no real root
(388;) and since its degree is even, the number of its
roots is even therefore, &c.
Hence, an equation whose roots are all imaginary
must be of an even degree.
(390.) The first member of an equation whose roots
are imaginary will be positive whenever a real value is
ascribed to x. For if for any such value it were negative,
there will be another value (K -- 1) for which it will be
positive, and these values will include between them at
least one real root contrary to the hypothesis.
It is evident also, that in such an equation the last
term must be positive, (388.)
(391.) When the last term of an equation is positive,
the number of real and positive roots is even ; and when
the last term is negative the number is odd. Y"
1. Let the last term be positive.
For when a = 0, y is positive; and such a value K+1
may be assigned to a as will render y positive. Hence
there must be an even number of real and positive
roots comprised between a = 0 and a = K -- 1, or
none. ,
2. If the last term be negative. When a = 0, y is
negative ; and when a = K+ 1, y is positive. Hence,
between these limits there must be an odd number of
positive roots.
wn-F1 —— A,
— a ... •
w"- + A,
— As a
a" + A,
– A, a
This equation is one degree higher than the former, and
Each coefficient is composed
of two parts, the first part being the coefficient of the
term which holds the same order in the former equa-
tion, and the second part the coefficient of the term
which precedes that in the former equation multiplied
by — a. Thus, if A, be the coefficient of the m”
term of the former equation (An-1 — Aa-s . a) will be
the coefficient of the n” term in the latter equation.
The signs of the successive coefficients of the equation
[2] depend in some cases on the signs alone of the
successive terms of [1], and in some cases on the values
of the coefficients, and the root a.
If the (n − 1)” and nº coefficients of [1] have the
same sign, and therefore form a successive repetition,
the two parts An-, and An-s . a. of the coefficient of
the nº term of [2] will necessarily have different signs,
and in this case the sign of the whole coefficient will
depend on the particular values of the numbers An-1,
A, , , and a.
But if the (n - . 1)" and nº coefficients of [I] have
different signs, then the common sign of the parts of
the n” coefficient of [2] will be that of the n” coeffi-
cient of [1]; this common sign will then be the same
as the sign of the n” term of [2].
Thus it appears, that each successive repetition in [1]
gives a doubtful sign in [2] ; doubtful as far as it can
be determined by the signs alone of [1], and each
a"-2 + .
at least one change of sign into [2].
positive roots than there are changes of sign among its
successive terms, nor a greater number of negative roots
than there are successive repetitions of the same sign.
This rule, which was first established by Descartes,
is known by the name of Descartes rule of signs.
By “changes of sign,” and “successive repetitions of
the same sign,” is meant each successive pair of terms
which have the same sign, and each successive pair of
terms which have different signs. The number of
changes, together with the number of successive repe-
titions, must be one less than the number of terms, and
therefore must be equal to the exponent of the degree
of the equation. Thus, if the equation be
aſ + A, a " – A, aº-A, a "+ A, a "+A, a 2+A, a - A,i- 0,
there are four successive repetitions, and three changes.
We shall establish the rule of Descartes by showing,
that for every positive root which is introduced into an
equation one additional change of sign at least is also
introduced, and for every negative root which is intro-
duced dºne additional successive repetition at least is
also introduced.
Let the equation be
a" + A, a "-" — As a “*” +. . . . An-, , a -A, - 0. [1]
To introduce into this an additional positive root
(+ a) it is only necessary to multiply it by a - a, and
the result is
w°-- A. r
- An-1 a
. . . -- Am I &
- An-, a — A, a = 0. [2]
change of sign in [1] gives to the corresponding term
of [2] the sign of [1]. There will then be in [2] as
many doubtful signs as there are successive repetitions
in [1], and all the other signs will be the same with
those of the corresponding terms in [2]. The sign of
the last term of [2] will be evidently different from
that of the last term of [1].
Our object is now to prove that the number of
changes of sign in [2] must be at least one more than
in [1]. To establish this, let the doubtful signs be re-
placed in the manner least favourable to the produc-
tion of changes, which is to make every doubtful sign,
or succession of doubtful signs, the same as the sign
which immediately precedes or follows it. It should
here be observed, that when a doubtful sign is imme-
diately preceded and followed by determinate signs,
these determinate signs must be different; this neces-
sarily follows from the consideration that determinate
signs in [2] are produced by changes in [1], and
doubtful signs by repetitions. Hence it follows, that
whether a doubtful sign be replaced by the preceding
or following sign, it must be the means of introducing
In the same
manner it follows, that if several doubtful signs succeed
each other in [2], the signs which immediately precede
and follow the series must be different ; and therefore
whether the doubtful signs be replaced by the preceding
or following sign, one change at least must be introduced.
A L G E B R A.
605
Hence it follows, that if all the doubtful signs be
S-V-' replaced by determinate signs in the manner just
described, there will be the same number of changes
and of repetitions in the first m terms of [2] as there
are in [1]. But the m” and (m -- 1)" term of [2] will
necessarily give an additional change. For if it be
immediately preceded by a determinate sign the change
is manifest, since A, and – A, a must have different
signs, and the sign of A, must be that of the penulti-
mate term of [2]. But if the sign of the penultimate
term of [2] be doubtful, and that it be replaced by the
sign which precedes it, the change in the last term is also
apparent, since the term which precedes it must have a
sign different from that of the last term." If, on the other
hand, it be replaced by the sign of the last term, the
additional change will fall upon the penultimate term.
The same reasoning evidently applies, mutatis mutandis,
to the case in which the penultimate term is the last of
a succession of doubtful signs.
As an example of this, let the succession of signs in
[1] be
+ + + — — — — — — — ... — — —H.
Let the doubtful sign be expressed by , and the suc
cession of signs in [2] will be & -
+ , , – , -- – -- - , , ----
Now if each doubtful or succession of doubtful signs
be replaced by the sign which precedes it, we shall
have
+ + + — — — — — — — — — — — — — .
In this case the signs are the same as in the first, as far
as the penultimate. Between that and the last is a
change.
If the doubtful places were filled by the signs which
follow them, we should have
+ — — — — — — — — — — — — — — — — — — .
Here are seven changes, while there are but six in the
first.
Since each positive root which is introduced
necessarily adds one to the number of changes, it
follows that there cannot be more positive roots than
there are changes of sign in the equation.
By reasoning exactly similar, it is proved, that the
multiplication of [l] by the factor a + a necessarily
introduces at least one repetition more; and that,
therefore, the number of negative roots cannot exceed
the number of repetitions.
(393.) Hence it follows, that if the roots of the
equation be all real, the number of positive roots is
equal to the number of changes of sign; and the num-
ber of negative roots is equal to the number of repeti-
tions of sign.
(394.) If any power of a which is admissible in an
equation of the mº" degree be wanted, it may be con-
ceived to be supplied with a coefficient which = 0.
The term in this case may be conceived to be affected
indifferently with the sign + or —. The number of
real positive roots will be determined by the number of
repetitions of the same sign in each case when the roots
are all real. Now if this number be different when the
deficient term is supposed to have the sign + from what
it is when it has the sign –, a contradiction arises
from the supposition that all the roots are real. In
such a case, therefore, we may infer the existence of
WOL. I.
imaginary roots. But if either sign, which may be
attributed to the deficient term, satisfies the condition
established in the preceding paragraph, we cannot infer
the existence of imaginary roots.
Thus, in the equation
a *-ī- p * + q = 0,
if the deficient term be supplied thus
w? -- 0.2% + p a + q = 0,
if the upper sign be taken, we infer, that if all the roots
be real they must be all positive ; and if the lower
sign be taken, we infer, that two must be negative and
one positive, which is a contradiction. Hence we
infer, that in this case all the roots of the equation
cannot be real ; and since only an even number of
imaginary roots can occur, it follows that but one can
be real. *
But if the equation be
a" – p a + q = 0,
the deficient term being supplied, we have
a" + 0 . a' – p r + q = 0.
In this case, whichever sign be attributed to the deficient
term, the number of repetitions and changes are the
same. Hence we cannot infer the existence of imagi-
nary roots.
Hence, a test for proving the existence of imaginary
roots is this, that the change in the sign of the defi-
cient term should alter the number of repetitions and
changes.
(395.) An equation whose roots are all real has as
many positive roots, whose values are between 0 and + a,
as there are repetitions of sign in the equation obtained
by substituting x – a for x. -
All the roots of the proposed equation which are
between 0 and -- a will necessarily be negative when
a – a is substituted for a ; therefore as many changes of
sign in the original equation as are equal to the number
of roots between 0 and a will necessarily be changed
into repetitions of sign in the transformed equation.
The reverse of this may also be easily established, scil.
An equation whose roots are all real cannot have any
positive roots between 0 and + a, unless the equation
found by substituting a - a for a in the proposed equa-
tion has a greater number of repetitions than the pro-
posed equation.
SECTION XXXVI.
Method of Determining the Rational Roots of Numerical
... • Equations.
(396.) THE determination of all rational roots may
be reduced to that of integral roots. For we have
already (349) shown, that if an equation have any
fractional coefficients a transformation may be effected
which will remove them, and give an equation with
integral coefficients, that of the first term being unity.
Every rational root of such an equation must be an
Q, Q g
T; be substituted for a in
its first member, and it begomes
4 K
integer; for let a fraction
Rational
Roots of
Numerical
Equations.
~~~~
606
A L G E B R A.
Algebra.
N-y-Z
am a”-" a"-s
---. -- A. . +, + A. -i-, + . . . .
+ An-1 . ++ A. = 0.
b
Multiply the whole by b"-" and we obtain
+ + A, . a” + As . a”. b + . . . .
–– An-1. a b”** + A, . b” = 0.
The first term of this is a fraction which is irreducible.
For since b is prime to a it is prime to a”, (95;) and
all the other terms are integers, whence we should find
a fraction equal to an integer.
Hence the equation cannot have a fractional root,
and therefore every rational root must be integral.
It may be observed here, that if all the terms of an
equation but one be integral, that one must also be
integral. - -
(397.) We shall therefore consider the equation as
having only integral coefficients, and as having no
rational roots but integers. Let a be a root, and be
substituted in its first member, and the result divided
by a gives
a"-" + A, a”-2 + A, a "-a-H
A,
+ An-1. -- – t = 0.
(Z
Since A, is the continued product of all the roots
with their signs changed, a is a factor of it, and since a
is an integer.
- A
is an integral root by hypothesis
Let it be Qi, so that
a"-i –– A, a”-* + As a "rº +
An-2 a + An-1 + Q, E 0.
Let this be divided by a, and we obtain
a"-- + A, a”-a -- A, a”-- +
An-3 . a + Am-, + An-, -ī- Qi = 0.
(ſ
Since all the terms of this but the last are integers,
the last must also be an integer, and therefore a mea-
sures An-1 + Q, . By the continuance of this process
we obtain the following results,
An Am-, -}-
:- Qı } Am-, + Q, - Q, ,
Q, (!,
An-s -j- Q, Qºmºs Q Am-s-HQ, Q
(Z 3 5 (1, mºs 4 2
* † 9-2 = 9 Ali Sº- = –1
(l, m-1 5 (?, ºs-se &
(398.) Hence, to determine the integral roots it is
necessary to determine, in the first instance, the integral
factors of the last term. Among these, all the integral
roots must be found. To determine whether any one
of these be a root, it is only necessary to substitute it
in the proposed equation, and if it converts this into an
identity it is a root, and otherwise not. But this pro-
cess is generally tedious, and when the last term con-
tains several factors must be repeated for each factor.
In the cases where the factors are not roots, they may
be determined not to be so more expeditiously by the
several criterions which we have just established.
I. Let the last term be divided by the proposed factor
a, and let the quote be added to the preceding coefficient,
this sum must be divisible by a. *r
2. Let this new quote be added to the coefficient
of x*, and the sum must be divisible by a, and so on.
Now if any of these sums be not divisible by a, it
is sufficient to prove a not a root, without continuing
the process further.
But if upon continuing the process the factor a be
found to measure each sum, and if upon finally adding
to A, the quote Q, , of the preceding sum, we obtain a
result which is equal to a with a different sign, then a
is a root of the proposed equation, and not otherwise.
(399.) The practical process for obtaining the ra-
tional roots of an equation is then as follows:
1. If the equation have any fractional coefficients,
let the transformation in (349) be effected, and one
obtained which will have integral coefficients. -
2. Let the integral factors of the last term be
found.
3. Let such of these factors as are included within
the limits of the positive and negative roots be written
down in succession.
4. Let the last term be divided by each of these,
and let the quotes be written under them respectively.
5. Under these quotes let the coefficient of a be
written.
6. Let this coefficient be added respectively to the
member immediately over it, and let the sum be
placed immediately under it.
7. Let each of these sums be divided by the first
term in each column, and if the quote be an integer
let it be written under the last term of the column. If
not, the process may be stopped in that column in
which the quote is fractional ; and in this way the
process may be continued, until either every column is
stopped by fractional quotes, or until some of them
arrive at the coefficient A1.
(400.) We shall now apply this process to an
example. Let the equation be
a' — as — 13 aft + 16 a – 48 = 0.
The major limit of the positive roots is 13 + 1 = 14.
For the last two terms may be reduced to the form
16 (a – 3). The major limit of the negative roots
is — (1 + V48) or —8. The divisors of 48 are 1, 2,
3, 4, 6, 8, 12. Neither 1 nor — 1 will satisfy the
equation, since the last term alone is greater than the
numerical sum of the other coefficients. Hence we
have the following calculation:
Rationa.
ots of
Numerical
Equations,
-v->
A L G E B R A. 607

Algébrü.
\-V-'
- - - - - - ºf , s Rational
Value of A, – 48 || –48 — 48 — 48 || – 48 – 48 – 48 – 48 –48 — 48 Roots of
Factors of A, 12 || 8 6 4 3 2 | – 2 — 3 || – 4 || – 6 .
Values of Q, – 4 || – 6 || – 8 || – 12 | – 16 || –24 +24 || --16 || +12 || + 8 quon,
Values of A, , 16 16 16 | 6 16 16 | 6 | 6 16 16 -N-
An-1 + Q, 12 10 8 4 0 – 8 40 32 28 24
Q, - 1 , , , || 1 || 0 | — 4 || –20 ſy — 7 || — 4
Am-, – 13 y ty — 13 — 13 | – 13 | – 13 ſy — 13 — 13
An-, + Q. — 12 ly y — 12 — 13 | – 17 | — 33 & – 20 || – 17
& • — I ſy ty — 3 y W ſy W 5 y
Al — I W ty — l º W W ſy — 1 t/
A, —#- Q, — 2 W W — 4 W ſº By y 4 º
yº 19 p y — l ſy W £) & — 1 ſy
Hence the integral roots in this case are + 4 and
– 4. The equation is therefore divisible by (a + 4)
(a — 4) = a * – 16, which reduces it to
a *— a + 3 = 0,
the roots of which are imaginary.
(401.) It may however happen, that the equation
which is obtained by dividing the given equation by the
simple factors corresponding to the roots found by the
preceding process, may have one or more integral
roots. It is true, that the investigation already given
determines all the different integral roots which the
proposed equation can have ; but it does not indicate
whether any of these roots are more than once repeated
in the equation. If they be so, it is evident that they
will occur again as roots of the equation obtained by
dividing the given one by the simple factors. It is
proper, therefore, to submit this equation to the same
process as the first, in order to detect the existence of
these repeated roots. If the number of different inte-
gral roots of the first equation be not great, the repeti-
tion of them may be detected at once by dividing the
resulting equation again by the same simple factor.
Also it follows, that the roots cannot be repeated if
they be not factors of the last term of the new equation.
(402.) When the number of integral factors of the
last term which are included between the limits of the
positive and negative roots is considerable, the process
by which those which are not roots may be determined
may be shortened.
If a be a root, the first member is divisible by a - a,
and the quote gives
X = (a — a) (r"-" + A'a"-" + A', a "-" -- . . . . ).
The forms of the coefficients A'i, A', , &c. have been
determined in (315). This equation must be fulfilled,
whatever be the value of w. Let a = 1, and the poly-
nome X becomes equal to the algebraical sum of its
coefficients. The same is true of the polynome in the
second member. Hence we have
1 –– A. -- A. -- . . . .
+ H^+ * = 1 +A1+A, +....
By the forms of the coefficients A', , A', . . . . established
in (315) it appears that they must all be integers.
Hence it follows, that the algebraical sum of the coeffi-
cients of the proposed equation must be divisible by
1 — a, if a be a root.
In like manner, if – 1 be substituted for a, we may
prove that what the first member becomes by this sub-
stitution is divisible by – 1 — a. Hence the rule,
Substitute successively + 1 and – 1 for a in the pro-
posed equation, and let the numerical values of the
results be M and M'.
1. Every positive factor of the last term which,
being diminished by 1, does not divide M, and every
negative factor which, being increased by l, does not
divide M', must be rejected, not being roots.
2. Every negative factor whose numerical value,
increased by I, does not divide M, and every positive
factor which, diminished by 1, does not divide M',
must be rejected, not being roots of the equation.
(403.) The investigation of the real and rational
roots of equation is equivalent to the investigation of
the real and rational factors of the first degree of their
first members. After all the rational factors of the
first degree have been found, although the remaining
factors of the first degree be not rational, yet when
combined in pairs they may form rational factors of
the second degree. Before we conclude this section we
shall therefore offer some remarks on factors of this kind.
Let any rational factor of the second degree be re-
presented by a " + p a -i- q, and let p and q be con-
sidered as indeterminate quantities, whose values are
to be ascertained in rational numbers.
For this purpose let the first member X of the equa-
tion be divided by a *-H p a + q, and let the division
be continued until a remainder be found which is of a
lower degree than the divisor, and therefore of the
form M r + N. In order that X should be exactly
divisible by a *-H p r + q, it is necessary that this re-
mainder should = 0, independently of a ; and, there-
fore, that M = 0 and N = 0. But M and N are
quantities whose values depend on the numerical coef-
ficients of X, and the indeterminates p and q. These
latter, therefore, must have such values as will fulfil the
two conditions M = 0 and N = 0. In these equa-
tions, therefore, let p and q be considered as unknown
quantities; and either of them being eliminated gives
a final equation including only the other. Such roots
of this equation as are rational, being substituted in
M = 0 or N = 0, give corresponding values of the
other; and such systems of values as are rational
being substituted for p and q in a 4 + p a + q, will give
so many rational quadratic factors of the first member
Y of the proposed equation.
Since the general process here described must give
every quadratic factor, it is evident that the final equa-
tion which determines the indeterminate p or q, must
m (m. – 1)*
1 .. 2
ber of different combinations of two factors. It must
be apparent, therefore, that this process would be
attended with great difficulties in practice, and is
therefore rarely resorted to.
be of the degree, since this is the num-
4 K 2
608
A L G E B R A.
Algebra.
S-N-2
SECTION XXXVII.
On the Determanation of the Real and Irrational Roots
of Numerical Equations.
(404.) By the methods established in the preceding
section, the rational roots of an equation being deter-
mined, its first member may be divided by the several
corresponding simple factors, the result will be an
equation whose roots are severally either irrational or
imaginary. We propose to devote the present section
to explaining the methods of determining the irrational
roots, and we shall accordingly consider the equation as
having been previously cleared of its rational roots.
The general form for these roots is not known, and
can only be determined when some general method for
the solution of equations of the higher degrees shall
have been found. The want of these methods, however,
in no wise impedes the progress of practical science,
for we can always obtain the irrational roots with any
required degree of approximation, and if we had their
general forms we could do no more.
The numerical value of an irrational root, when re-
duced to decimal expression, will in general consist of
two parts, the integral part a which precedes the deci-
mal point, and the decimal part u which follows it.
To express the decimal part w exactly, would require an
infinite series of decimal places; for if the series were
finite, or even periodic, the decimal would be equiva-
lent to a rational number.” All, therefore, which can
be done in this case is to determine as many places of
w as may be necessary to give the requisite approxima-
tion, and this can always be done.
We shall, however, first consider the method of de-
termining the integral part a of the root.
(405.) Let the major limit of the positive roots be
+ L, and that of the negative roots – L'; the more
narrow these limits are determined, the more expedi-
tious will be the process. Substitute for a in the equa-
tion the successive integers from 0 to + L with positive
signs, and from 0 to — L' with negative signs. When
two successive substitutions give different signs to the
first member of the equation, one at least, and in
general an odd number of real roots must be com-
prised between the two successive integers, and the
lower of the two integers is evidently the integral part
a of the corresponding roots. If two successive sub-
stitutions give the same sign to the first member of the
proposed equation, there will either be no real root
comprised between the two integers, or there will be an
even number of them. In the latter case, the lower of
the two integers will be the integral part of all the
intermediate roots.
Before, therefore, we can determine what integers
between the limits + L and – L' belong to irrational
roots, it will be necessary to determine what number of
roots are intereepted between each pair of successive
integers. - -
We have already determined an equation which may
always be deduced from the proposed equation, and of
which the squares of the differences of the roots of the
proposed equation are the roots. Since the square of
a real quantity must always be positive, it follows, that
* See ARITHMETIc, p. 499,
roots.
the negative roots of this equation, if it have any, must
be the squares of the differences of the imaginary
Let the minor limit of the positive roots of
Real and
Irrational
Roots of
Numerical
this equation be found, and let its square root be ex- Equations.
tracted.
than it. -
If D > 1, which will be the case if the minor limit
of the positive roots of the equation of differences be
greater than unity, it follows that no two real roots of
the proposed equation can be contained between two
successive integers, and, therefore, that if two succes-
sive integers substituted for a give the first member of
the equation different signs, one, and but one, real root
will be included between them, and the integral part of
this root will be equal to the lesser of the two integers
so substituted.
If two successive integers substituted for a give the
first member the same sign, no real root can be included
between them. Thus, in this case, we determine the
number of incommensurable real roots, and the integral
part of each.
If this number be equal to the exponent of the degree
of the equation, there will be no imaginary roots. But
if it be less than that exponent, there will be a number
of imaginary roots equal to their difference.
If D > 1, several real roots of the proposed equa-
tion may be intercepted between two successive integers.
To determine if this be the case, let &
0, 0 + D, 1 + D, 2 + D, . . . . (L – 1) -- D,
0 - D, - 1 – D, - 2 – D, . . . . — (L' – 1) — D,
be successively substituted for a in the first member of
the supposed equation. Any two successive substitu-
tions which give the first member different signs, must
contain between them one, and but one real root ; and
any two successive substitutions which give the first
member the same sign, can contain between them no
real root. Henee the number of real roots is exactly
obtained, and the integer next below each real root is
known. This is the integral part of the root.
If the number of real roots in this case be equal to
the exponent of the degree of the equation, there will be
no imaginary roots; but if the number be less than that
exponent, there will be a number of imaginary roots
equal to their difference.
In this reasoning we have proceeded on the hypo-
thesis, that the equation has been cleared of its equal
roots. For if there were equal roots in the proposed
equation, one of the roots of the equation of the squares
of the differences would be = 0. Thus the minor limit
D would = 0, and the process of substitution already
explained would not be applicable. Indeed it is evi-
dent, that if there were equal roots we could not in any
case infer that the change of sign on the substitution of
two consecutive integers inferred but one intermediate
root, nor that the identity of sign inferred none.
The equation may be cleared of its equal roots by the
process explained in Sect. XXXI.
(406.) The methods which we shall explain for ob-
taining the decimal part u of the root, require that there
should not be more than one real root between two
successive integers. It will be therefore necessary in
the case in which D 3 1 to effect a transformation on
the equation, such as will render D > 1. Let the de-
nominator of D be k, and let a = #. By this substi-
Let D be this root, or any number less ºr 2-y
A L G E B R A.
609
Real and
Algebra. tution an equation will be obtained, whose roots are k -— - º
S-' times greater than the roots of the proposed equation, .* r = a + I - º
and, therefore, whose differences are k times greater. b -- } N:
For if a., w" be two roots of the proposed, we have c + Equations.
f FI l N-V-'
a' = 4. . ." = % •.” (r' — a ") k = y' — y". d–– l
k k - e - -
Hence the least difference of the roots of the trans- &c
formed equation will be k D, and as k is the denomi-
nator of D, k D cannot be less than unity. Hence, in
the transformed equation more than one real root can-
not be intercepted between two consecutive integers.
(407.) Having thus explained the methods of ascer-
taining the total number of irrational roots, the integral
part of each of them, and of so transforming the
equation that no two roots shall have the same integral
part, we shall now proceed to explain the methods of
determining the decimal part u, and in so doing we
shall suppose that this transformation has been previously
effected.
(408.) The first method of approximation which we
shall explain is that of Lagrange.
Let X be the first member of the equation, and a
the integral part of the root. Let a + 'u be substituted
for a in X = 0, and the result arranged by the dimen-
sions of u is of the form established in (354.) If in
s l º º
this w = —, and the result be cleared of fractions, it
becomes Y = 0, where Y expresses a polynome of the
form A y” —- B y” —- . . . . whose coefficients, how-
ever, are those found in (354,) a' being changed into a.
l
Since a = a + -- should determine all the values
of a when those of y are known, and no others, it
l
follows that — must have one, and but one, real value
< 1, and "... y must have one, and but one, real value
> 1 ; for were it supposed that y had more than one
real and positive value - 1, then a would have more
than one real value between a and a + 1, which is
contrary to hypothesis.
If then the successive integers 1, 2, 3,.... be seve-
By continuing this fraction we may approximate inde-
finitely to the value of a, (Sect. XX.) It is evident, that
in the process this fraction can never terminate, for if
it did, the value of a would be rational, which is con-
trary to hypothesis. None of the transformed equa-
tions Y’ = 0, Y!' = 0. . . . . can therefore have a posi-
tive and integral root.
If, however, the root were not irrational, it might be
determined exactly by this method; for in that case
some of the transformed equations would have a posi-
tive and integral root, in which case the continued frac-
tion would terminate.
(409.) There is another method of approximation
proposed by Newton, which is more expeditious than
that of Lagrange, which we have just explained.
In the method of Newton a first approximation to
within 0,1 of the value of the root is obtained by a
tentative process. The root being between the integers
a and a + 1, let a -- 0,5 be substituted for a, and
if this and a give the first member different signs,
the root is between a and a +- 0,5, but if they give it
the same sign, the root is between a -- 0,5 and a + 1.
If the root be between a -- 0,5 and a + 1, by sub-
stituting a -- 0,6, a + 0,7, a -- 0,8, &c. two results
will be found with different signs, and the root will,
therefore, be between these, and either of them will
differ from the root by a quantity less than 0,1. But it
is rarely necessary to go through all these substitutions,
as it most generally happens that the first two will
determine the root within 0,1 of its exact value.
The root being thus far determined, let the value
found be a ', so that a = a + 'u, u, being a quantity
< 0,1. By substituting this in the proposed equation,
we obtain (354)
rally substituted for y in Y = 0, it must happen that X' -- X', . u + X's . u". + . . . . = 0
some two successive substitutions will produce a change 1 (2)
of sign, and between the two integers which produce X? Y', tº X’, aſ?
this change of sign the value of y must be placed. ... u = — —#- – -º- . --— —# , º –
X's X', (2) X', (3)
l
Let these two integers be b and b + 1, and let b + y
be substituted for y in Y, and let the transformed equa-
tion be Y". This equation, as before, must have one,
and but one, real and positive root - 1. And the
integers c and c + 1, between which it lies, will be
determined as before.
o e = a a tº I
Again, substituting in Y' = 0, c + -ī; for y', we ob-
tain another transformed equation Y" = 0, which, as
before, must have one, and but one, real and positive
root - 1. And so the process may be indefinitely
continued.
Hence we have
Since u < 0, 1, " .." wº 3 0,01. The terms of this
series which succeed the first are, then, in general much
X!
less than 0,0l. 3. the
2
assumed value differs from the true by less than 0,01, and
If then we assume w = —
- ſ
therefore a' — # differs from the true valve of a by
2
less than 0,01. Let this value be r" and let a' = a,"
–H u. In this case, u < 0,01. Substituting, as before,
a"—H u for a in the proposed equation, we obtain a
result of a form exactly similar to the last; and assuming
/
l
x." the assumed value differs from the true by
2
7ſ tº —
I l I less than 0,000 l ; and in the same manner, another
a = a + -- y = b + y sy' = c + -7 approximation will differ from the exact value of a by
3/ 3/ < 0,00000001, and so on.
I To approximate to the negative roots, it is only
g" = d --
--
* * * * * * * * * *
f//
3/
necessary to change a into – a in the proposed equa-
610
A L G E B R A.
Algebra, tion, and treat them as positive roots.
This method
sometimes fails: see Lagrange, on Numerical Equa-
tions. -
(410.) After each approximation it should, therefore,
be determined whether the desired end has been
attained. This may be easily done. Let the approxi-
mate value obtained at any stage of the process be, for
example, 3,1858. Substitute this in the equation for a,
and if it give the first member a different sign from
that which it receives from the substitution of 3 for a,
the root must be ~ 3, 1858. Substitute then for a,
3,1857; and if the result have the same sign with that
which proceeds from the substitution of 3, the root is
between the numbers 3,1857 and 3,1858, and, therefore,
the requisite approximation has been obtained; but if
the results of the substitutions of 3,1857 and 3,1858
had the same sign, the requisite approximation would
not have been obtained, and it would be necessary to
diminish the last digit of the decimal.
The same observations, mutatis mutandis, apply to
the case where the substitution of 3,1858 and 4 give
the first member different signs. *
(411.) Of these two methods of approximation,
that of Lagrange has the advantage of giving a nearer
degree of approximation at each step, which Newton’s
may not ; and also Lagrange's method extends to the
exact determination of rational roots, Newton’s method,
however, is in general more expeditious. -
The method of determining rational roots, explained
in Section XXXVI., is only applicable when the
coefficients of the equation are rational numbers.
Lagrange's method may be applied when the coefficients
are irrational. It is often very advantageous to apply
both methods in the same investigation. Thus we may
employ Lagrange's method to obtain the roots to within
0, 1 or 0,01 of their exact value, and continue the
approximation by Newton’s method.
There are also other methods, but the developement
of them would lead us into details unsuitable to the
present Treatise. See Lagrange, Traité de la Résolu-
tion des Equations Numériques; Nouvelle Méthode pour
résoudre les Equations Numériques, by Budan; Théorie
des Nombres, by Legendre.
SECTION XXXVIII.
Elimination applied to two Numerical Equations between
two Unknown Quantities.
(412.) WHEN the conditions of any problem reduced
to an algebraical statement give two numerical equa-
tions of any degrees between two unknown quantities,
every pair of particular numbers which, being substi-
tuted for the unknown quantities in the equations, con-
vert these equations into identities, are to be considered
as a solution of the proposed problem.
In general, let the first members of the two equations
be A and B, and the equations being
A = 0 B = 0. g
Let a' and y' be any particular numbers which, being
substituted for a and y in A and B, render the several
terms of these polynomes such as will destroy each
other. Such a system of values we shall call conjugate
values of a and y. -
In order that any particular number should be a con- Elimination
jugate value of y, it is necessary that when it is substi-
tuted for y in the proposed equations, that they, having
applied
to two
Numerical
then no unknown quantity butt, should have a common. Equali.
root; for if not, they would be inconsistent. This
common root will be the value of a, conjugate to the
assumed value of y.
It may so happen, that when a particular number is
substituted for y, there will be more common roots, or
several values of a, which will convert both equations
into identities. In this case the same value of y will
have several different conjugate values of w.
When a conjugate value of y is substituted for it in
the given equations, their first members must admit a
common divisor which is a function of ar. If this func-
tion of a be of the first degree, there is but one value
of a conjugate to the assumed value of y : if it be of
the second degree there are two, and, in general, if it
be of the mº" degree there are n values of a conjugate
to the same value of y. -
If, however, on the substitution of a particular num-
ber for y, the first members of the proposed equations
admit of no common measure, there will be no corres-
ponding value of a, and in this case the assumed value
of y is not a conjugate value.
(413.) Elimination, properly so called, is that process
by which, from the two given equations an equation
is deduced, which includes but one of the two unknown
quantities, and whose roots are the several conjugate
values of that unknown quantity, and which has no
root which is not a conjugate value. Such an equation
is properly called the final equation.
The number of roots in this equation should be equal
to the number of systems of conjugate values which the
proposed equations admit. If a be the unknown quan-
tity which has been eliminated, the roots of the final
equation should be the several values of y. The num-
ber of unequal roots should, therefore, be the same as
the number of different conjugate values of y. But
we have observed, that it may so happen that the same
conjugate value of y may have several different conju-
gate values of ar. In this case we must consider the
several repetitions of the value of y with the different
conjugate values of a to be so many different conjugate
values of y, which have become equal, and, therefore,
in this case the value in question should be one of
several equal roots of the final equation. Hence, in
general, the degree of the final equation must be equal
to the number of different systems of conjugate values
of the unknown quantities in the proposed equations.
(414.) We shall now consider how far the final
equation, obtained by the method founded on the pro-
cess for obtaining the greatest common measure, fulfils
these conditions. It is necessary to show : 1. that
every conjugate value of y is found among its roots ;
2. that it has no root which is not a conjugate value
of y, or if it have it is necessary, 3. to show how
such roots may be distinguished, and how the equation
may be disembarrassed from them.
The first members of the proposed equations being
arranged by the dimensions of a, let the process for
determining the greatest common measure be instituted.
Let the multipliers which are successively introduced,
in order to render the first terms of the several dividends
exact multiples of those of the divisors, be a', aſ", a'",
. These will be, in general, functions of y. Let
the successive quotes be Q', Q", Q", , and the
A L G E B R A.
611
to the substitution functions of y, will now become nu-Elimination
merical, and if the component parts of these coefficients, *PPied
Algebra. remainders R', R', R/",. . . . . . By the
--~~' process we have the following identities:
nature of the
a' A = B Q'+ R! [1] .*.* not = 0, the equation Rº") = 0 will Nº.
a"B = R. Q// -- R/ [2] à - ſ w Equations.
a!!! . R^ == R// gº Q”/ —H R!// [3] A^ + B' ºr + C’ p? + ſº tº dº º tº gº - 0, ^*N-Z
F . . . . . . . . where A', B, C' . . . . are particular numbers. This
sº equation will give particular numerical values for a,
. . . . — . . . . . . . . and, therefore, the value of y thus substituted is con-
aſ”). R(n-3) = Rº-º). Q(*) -- Rº [4] jugate to aſ”) = 0, Rº-1) = 0, and is, therefore, a root
By [1] it appears, that every system of conjugate
values of a y in A = 0, B = 0, are also conjugate in
B = 0, R = 0.
By [2] it appears, that every system which is conju-
gate in B = 0 and R':- 0 is also conjugate in R' = 0
and R.' = 0.
By [3] it follows, that every system which is conjugate
in R' = 0 and R' = 0, is also conjugate in R'' – 0 and
R” = 0, and so on.
By this reasoning we infer, that every system of
values of a, y which is conjugate in A = 0, B = 0 is
also conjugate in Rº"-t) = 0 and Rºº) = 0.
Since Rºº is independent of a, all the conjugate values
ofy in A = 0, B = 0 must be roots of Rº- 0. -
But since a' is in general a function of y, it also fol-
lows from [1] that the conjugate values of a, y in
a = 0, B = 0 are also conjugate values in B = 0,
R^ = 0, and, therefore, by what has been already esta-
blished are conjugate values in Rº"-1) = 0, Rº") = 0.
But as Rºº is a function of y alone, it follows, that
every value of y which is conjugate in Rº" º = 0,
Rº") = 0 must be a root of Rſ") = 0.
In the same manner it follows, that every value of
y which is conjugate in a' = 0, R/= 0, is a root of
R") = 0, and in the same way the conjugate values of
y in a' = 0, R^ = 0, &c. are roots of Rº = 0.
Thus, in general, we may conclude, that the conju-
gate values of y in the following pairs of equations are
R! = 0
roots of R(") = 0:
. . R("−1) = 0 ) . .
a' = 0 º
A = 0 ) B = 0
B = 0 , aſ == 0
It is, however, only those values of y which are con-
jugate in the first pair which are the proposed equa-
tions which should be roots of the true final equation,
As in the succeeding pairs there may be conjugate
values of y which are not conjugate to the first pair, it
follows that all such values will be roots of Rº = 0,
and that, therefore, before Rºº) = 0 can represent the
true final equation, these roots must be determined, and
the equation Rº") cleared of them.
(415.) It will be remembered, that the functions a',
a", a'', are the multipliers which are succes-
sively introduced, in order to render the first terms
of the successive dividends A, B, R', . . . . exactly divi-
sible by the first terms of the successive divisors
B, R', R.", . . . . , and, therefore, from the nature of the
process it follows, that a', a”, a!", . . . . must be inte-
gral functions of y and independent of w. On the
other hand, the quantities B, R', R', . . . . are func-
tions of y and a, and are supposed to be arranged
according to the dimensions of w. To determine, there-
fore, the conjugate values of a, y, in any pair of the
equations already mentioned, except the first, it will be
necessary first to determine the roots of a!") = 0, and
these must be substituted for y in Rº"-1) = 0. The
coefficients of the powers of a in Rº"-19, being previous
R// = 0
a/// = 0
of R") = 0. If this value of y be not conjugate in
A = 0, B = 0, it will be neeessary to clear the equa-
tion Rº") = 0 of it before it can be considered the true
final equation. The same may be said of every value
of y determined in this way, in each pair of the equa-
tions
B - 0 R! = 0 ) . . . . . .
a' = 0 a/ - }
But it may so happen, that a value of y deduced
from a "9 = 0, when substituted in R("-1) = 0, or
A' -- B'a + Caº –– . . . .
may render the coefficients B", C/, D', . . . . each = 0,
in which case the equation Rº"-9 = 0 will not be ful.
filled, whatever be the value of a. In this case, and in
this case only, the value of y deduced from aſ”) = 0 is
not a conjugate value in aſ") = 0, R. ("-1) = 0, and,
therefore, not a root of Rſ.” = 0.
It may also happen, that a value of y deduced from
aſ") = 0, shall render all the quantities Aſ, B', C. . . . .
= 0. In this case Rº-i) will = 0, whatever be the
value of a. In this case, the quantity Rºº-tº must have
an integral function of y, independent of a as a factor,
and by the principles which have been already esta-
blished respecting the process for finding the greatest
common measure it follows, that this function of y must
be a common factor of the proposed equations A = 0,
B = 0, so that they become
A/ × Y = 0 B' × Y = 0,
if Y be the common factor. Now, both of these equa-
tions are satisfied by Y = 0, independently of a. The
equations therefore are indeterminate, since, although,
the values of y are limited in number by the equation
Y = 0, the value of a is absolutely indeterminate. To
render the equations determinate, it would be neces-
sary to disembarrass them of the common factor Y = 0.
To distinguish, therefore, the roots from which the
equation Rº") is to be cleared, in order to obtain the
true final equation, it is necessary to determine suc-
cessively the roots of the several equations a' = 0,
a ’ = 0, a' = 0, . . . . and to select such of these roots
as do not render = 0 the several coefficients of the
equations B = 0, R/ = 0, R^ = 0,. . . . ; and such of
these values as are not conjugate values in A = 0,
B = 0, should be cleared from RK") = 0, and the result
will be the true final equation. ',
In cases, however, where the equations a' = 0,
a" = 0, a” = 0, . . . . are of the higher degrees, the
determination of their roots may be attended with some
difficulty. In this case we can have recourse to a pro-
cess which will render the determination of their roots
unnecessary.
It should be observed, that the roots of a' = 0 are
always conjugate values of y in a' = 0, B = 0, except
in the particular case in which the value of y deduced
from a' = 0, renders = 0 the coefficients of all the
= 0,
612
A L G E B R A.
coefficients severally, and a', admit any function of y as
a common measure. If they do, then the values of y
found by putting this function = 0, are not conjugate
values, and are not roots of Rºº - 0. The equation
a' = O is then to be cleared of this function of y
by division; and if the quote be a function of y, its
roots will necessarily be roots of R") = 0, and such of
them as are not conjugate in A = 0, B = 0 must be
removed by division from Rº = 0. -
If it should happen that the common measure of all
the coefficients of the powers of a in B = 0 should be
also a measure of its absolute quantity, then the ori-
ginal equations will be indeterminate, for this same
function of y will be a common factor of them.
(416.) The observations just made, concerning the
equations a' = 0, B = 0, will equally apply to a' = 0,
R^ = 0, to a!// = 0, R.' = 0, . . . . . . By these means,
the equation Rº") = 0 may be successively cleared of all
the factors, or roots, which do not correspond to con-
jugate values of y in the equations A = 0, B = 0.
If the last remainder Rºº be an absolute quantity
independent of y, there is no value of a which would
render the two polynomes divisible by the same function
of y, and, therefore, there are no conjugate values of
a, y, and the given equations are inconsistent, or con-
tradictory.
If Rº = 0, independently of y, it follows, that the
value of a which satisfies the two equations is inde-
pendent of any value of y, that is, the two functions A,
B are divisible by a common function of w. Let this
be X. Both equations are satisfied by the roots of
X = 0, whatever be the value of y. Hence, in this
case, they are indeterminate.
Before we proceed further with this abstract reason-
ing, we shall illustrate it by its application to the follow-
ing examples.
Let
A = y” tº — 3 y” a - y” – 2
B = (y” — 3 y + 2) a” -- (y – 1) a - 3 y + 1,
The first multiplier a' is
... a' = y” — 3 y + 2
... R/- (— 3 y”--8 y°– 5 y”) a + 2y++ 2y?– 6 y + 4
— 3 y” + 8 y” – 5 y' = — y” (y – 1) (3 y – 5).
In this case it is necessary to take
a" = y” (y – 1) (3 y – 5)?
... R*-R") = 27 y” — 136 y” + 214 y°– 112 y?--65 y”
— 100 y” + 30 y” – 24 y” + 120 y” – 112 y +32.
This polynome includes all the conjugate values of y.
But before these can be determined, it is necessary to
determine what factors have been introduced by the
multipliers a', a”. We have sº
a' = y – 3 y + 2 = (y – 1) (y – 2)
B = (y” — 3 y + 2) + (y – 1) a - 3 y + 1 = 0.
If a' = 0, ... y = 1, or y = 2. If y = 1, B = — 2.
Hence this root does not enter Rººſ = 0. If y = 2,
= a – 5 = 0, ... a = 5. The value y = 2 is not a
conjugate value in A = 0, B = 0; for if 2 and 5 be sub-
stituted for y and a in A, we have A = 78. Hence it is
necessary to divide R' = 0 by y – 2.
Algebra, powers of w in B = 0. To determine whether this be
S-S-' the case, it will be only necessary to find whether these
- 5
If a' = 0, ...' J = 0, or y = 1, or y = 5. But
g = 0 does not satisfy R' = 0, ... y is not a factor of
R"). In like manner y = 1 does not satisfy R = 0,
‘. . y – 1, as before, is not a factor of Rºº. The same
5 --
observation applies to y = T3 . Thus it appears, that
the only factor of which R") is to be cleared is y – 2.
Being divided by this the quote becomes.
27 y” – 82 y” + 50 y? – 12 y” + 41 y” – 18 y” – 6 y”
— 36 y? -- 48 y – 16 = 0.
The roots of this equation are the conjugate values of
y, and the only ones in A = 0, B = 0. These roots
being determined, and successively substituted in
R’ = 0, will determine the conjugate values of a.
(417.) It may be observed, that in general the
Elimination
applied
to two
Numerical
Equations.
last remainder R.") being a function of y independent
of a, the preceding remainder is of the form
Ma -- N
where aſ occurs only in the first degree. The values of
y being determined by the equation R* = 0, and suc-
cessively substituted for y in the functions M and N,
the equation
Ma -- N = o
will determine all the conjugate values of a without
having recourse to the original equations at all. In
fact, any value of y which renders Rºº) = 0 must ne-
cessarily render Rºº, or Ma -- N a common mea-
sure of the first members A, B of the proposed
equations, which are therefore satisfied by
M a + N = 0.
If any root of R") = 0 renders N = 0 the conjugate
value of a = 0. If it render M = 0, a = ob, and
if it render both M = 0 and N = 0, it follows, that
since Rººt' = 0 independently of a, the preceding re-
mainder Rºº must be a common measure of A and
B. Therefore, if in this remainder we substituted the
same value of y, the roots of the equation Rº"-2) = 0
will be the conjugate values of ar. In this case
R"** = 0 will be an equation of the second degree,
and there will be two values of a conjugate to the same
value of y.
(418.) It may happen, that the value of y in ques-
tion also renders Rº-3 = 0 independently of a. In
this case the preceding remainder R. "-") will be a com-
mon measure of the quantities A, B, and the conjugate
values of a will be the roots of Rº-5) = O. This will be
an equation of the third degree, and, therefore, there
will be three values of a conjugate to the same value
of y. In the same manner, R."“ may = 0 inde-
pendently of a, in which case Rº-4) = 0 will give four
conjugate values of a, and so on.
It is evident, that whenever for the same value of y
there are several conjugate values of a, several suc-
cessive remainders must be = 0 independently of y;
for otherwise, for each value of y there would be but
one value of a determined by Rº-0 = 0, which is always
of the first degree in a.
(419.) It may be observed, that the method of eli-
mination by the greatest common divisor always gives
the true final equation, when the given equations do
not exceed the second degree. For, in this case,
A L G E B R A.
6] 3
Algebra.
A = a x2 + b a + c = 0,
B = a +2+ b/w -– c' = 0.
Here a, a must be numerical coefficients, for if they in-
cluded any dimension of y the equations would exceed
the second degree. These, being the factors first in-
troduced to render either divisible by the other, cannot
introduce any root into the final equation.
The last multiplier at") which is introduced cannot in
this case, nor in any other, be the means of introducing
a root into the final equation; any value of y deduced
from aſ") = 0 would render = 0 the coefficient of a in
R("-49, and would reduce this to a numerical quantity
which would not in general = 0.
The degree of the equation Rº") = 0 may frequently
indicate the existence in it of roots which are not conjugate
values of y. If it exceed the product of the numbers
which mark the degrees of the two equations A = 0,
B = 0, there must be at least as many roots which are
not conjugate values as the units by which the degree
of Rº = 0 exceeds that product.
(420.) We cannot, however, on the contrary infer,
that if its degree be equal to the product of the degrees
of A = 0 and B = 0, that there are, therefore, no roots
but conjugate values of y. Because, although the highest
degree the final equation can have, is the product of
the degrees of the original equations, yet, in particular
cases, it may have a lower degree.
SECTION XXXIX.
On the Imaginary Roots of Equations.
(421.) By the principles which have been already
established, we are enabled to clear an equation of its
real and rational roots. But, although we may ap-
proximate at pleasure to the irrational roots, yet unless
we could obtain them exactly, it would be impossible
to clear the equation of them by division. We shall,
therefore, in the present section consider the equation
as having irrational and imaginary roots, but no ra-
tional roots. Our object will be to determine the
imaginary roots.
We have already proved, that in an equation with
real coefficients there must always be either an even
number of imaginary roots or none. We propose now
to establish a more general theorem which includes
this, scil., Every imaginary root of an equation must
be of the form a + b v-1, and if a + b v - I be an
imaginary root of any equation, a - b v - I must be
also an imaginary root of the same equation, a and b
being real quantities.
Let
X =a.”-- A, a "-" + As a "-" —- . . . . An-1. a + A, - 0.
Let a + b v-i be substituted for w in X = 0. By
(259) it appears, that if (a + b v — 1)" be expanded
by the binomial theorem, the alternate terms begin-
ning with the first will be real, and alternately +
and —, and the alternate terms beginning from the
second will be affected with the imaginary factor
A/T 1, and alternately + and —. Observing this, it
is evident, that the substitution in X will produce
a series of real, and a series of imaginary, terms, Let
WOL. I.
the sum of the real terms be M, and that of the coeffi-
cients of V-Ti in the imaginary terms N, the result
will be -
M + N V-T = 0
•. • M = 0 N = 0.
These two equations will determine the values of a
and b.
If a - b v- I had been substituted for a, the result
would have been
M – N V - i = 0
• . . M = 0 N - 0,
which would give the same values for a and b as be-
fore. Hence, if a + b v - I be a root of X = 0,
a – b v- I will also be a root of it.
(422.) Before we proceed to show that every ima-
ginary root must have the form a + b v – I, it will be
first necessary to establish the principle, that every
algebraic function of a + b v-I may be reduced to
the form M + N v — 1.
Let w = a + b v- I
w'-a'-E b' vT
w" = a" + b'ſ v- I
&c. &c.
Let X (u), 2 (a), 2 (b), signify the algebraical sums of
u, u', uſ". . . . a, a', a” . . . . b, b', b". . . . . By addition
we have
I. X (u)= x (a) + X (b). V-T
= M + N V - I.
u w! = (a a' – b b') + (a' b + a b) v – i
uu' = M + N V-T.
u (a' + b v-I)
w! Tu (a + b v-T)
_ (a a' + b b) + (a' b – a b') w/TI
hºmºm, a” + b”
†Z
*s
b=ms
+ = M + N V- I
Q1, mº
4. w" = (a + b v – 1)".
In this case whether m be positive or negative, integral
or fractional, its developement may be reduced to the
form M + N V-1, by what has been already proved.
(423.) We shall now show that every imaginary
root of X = 0 can be reduced to the form a + b v- i.
Let a, , a, as . . . . a, be the roots of X = 0. By the
principles established in Section XXX. an equation
may be found, whose roots will be functions of each
pair of roots of X = 0, of the form
a, + a, + k a, as
k being any integer whatever. Let this equation be
Z = 0. It will, in general, have as many roots as
there are different combinations of two roots among
the m roots of the proposed equation. This is
m (m – 1)
1 .. 2
The number m being by hypothesis, even is divisible
4 L
, which is, therefore, the degree of Z = 0.
Imaginary
Roots.
\-e-V-'
614
A L G E B R A.
Algebra, by 2, and, in general, has the form 2*. m', 'm' being an
S-v- odd integer.
1. Let n = 1, " ..." T2- = m', and since this is odd,
m (m – 1)
and also m – l is odd, it follows, that is
odd, and therefore Z = 0 must at least have one real
root. Let this be 2', and let
2' = a, +- a, + k a, b,
For each integral value which is ascribed to k there
will be a different equation Z = 0, and each of these
equations will have one real root, at least, which must
be a function of some pair of roots of the proposed
equation of the form already mentioned. Since the
number of combinations of pairs is limited, it follows,
that after a certain number of substitutions for k, the
real root of the equation must be a function of some
pair, of which the real root of the equation resulting
from some former substitution was also a similar func-
tion. Let the two real roots which are functions of
a, a, corresponding to the values k k' be 2, 2', and we
have - *
2 = a, + a, + k a, a,
! — f
2 = a, +- a, + k'a, Q2
‘. . 2 – 2' = (k — k') a, as
z – 2'
"." at as E 7. Tº
2 k' – 2'k
a + a, = −.
Hence it follows, that in this case a, + a, and a, as
are real quantities, and, therefore,
(a. * a,) (a. — a.) = &º — (a, -i- a,) r + a, a,
which is a quadratic factor of X = 0, is real.
m (m — I)
s=-º-º-º-º-º-º-º-º:
I . 2
(m – 1). Hence, in this case, the equation Z = 0 is
of an even degree, but its exponent 2 m' (m – 1) if
divided by 2 gives an odd number for a quote. Hence,
by the last case, it follows, that Z = 0 must have a
real factor of the second degree. Let this be
2* + A z + B,
and let its simple factors be 2', z". These quantities
2', z" must, in general, be of the form a + b v — 1.
Let 2' = a, + as + k a, as. By the same reasoning
as in the former case, we can prove, that there is ano-
ther value of k by which another root which is a simple
factor of a real quadratic factor of Z may be found.
Let this be 2", so that we have - - . A
2' = a, +- a, + k a, as
2" = a, +- a, + k'a, a.
The values of 2’-- 2" and 2'2'', deduced from these,
being algebraical functions of 2', z” must also be of the
form a' + b v-1.
factor of the form -
* — (p + q v'-I) r + p + q' V-1.
The values of a which render this = 0 being algebraic
functions of the coefficients must be reducible to the
form p + q v' -1. We shall then have a simple fac-
tor of X of the form a - (p + q v' – 1), and, there-
2. Let m = 2, = 2 m/
... * = 2m, e
e 2 T 3 e
So that we shall have a quadratic
fore, another of the form a - (p + q w/ 1), and
hence we obtain a quadratic factor of the form
a”-- p a + q,
which will be a real quadratic factor of X.
Similar reasoning will apply when n = 3, n = 4,
&c. Hence we infer, in general, That the first member
of every equation of an even order admits, at least, one
real quadratic factor. -
This being proved, it easily follows, that the first
'member admits of being resolved into as many real qua-
dratic factors as there are units in +, or half the expo-
ment of its degree. For, since it admits of one real ſac-
tor, this may be removed by division, and an equation
of an even degree lower by 2 will be the result. This,
again, must admit of a real quadratic factor.
Hence the first member of an equation whose degree
is even, may be considered as the continued product of
as many real quadratic factors as there are units in
half the exponent. - -
And since an equation of an odd degree must always
have one real root, its first member may be considered
as the continued product of one real simple factor, and
as many real quadratic factors as there are units in
half of the earpoment of the degree diminished by unity,
on – 1 • * . . " -
O}" 2
The form of the imaginary roots being thus deter-
mined, their actual values may be found by the equation
M = 0 N = 0
in (421) which will give the values of the indeter-
minates a and b. e
Two imaginary roots, such as a + b v-1,
a – b v-1, which differ only in the sign of the ima-
ginary part, are called conjugate imaginary roots.
(424.) The equation of the squares of the differences
of the roots of an equation, has a connection with the
imaginary roots which it may be useful to trace.
The difference between any two real roots, must be
real, and either positive or negative; in either case its
square will be positive, and must, therefore, be a real
and positive root of the equation of differences.
Hence the equation of squares of differences must have,
at least, as many real and positive roots as there are
combinations of two real roots in the proposed equa-
tion. . . .
The difference of two conjugate imaginary roots
being of the form + 2 W – 1 . b, its square must in
every case be negative. Hence the equation of squares
of differences must have, at least, as many negative
roots as there are real quadratic factors, whose simple
factors are imaginary in the proposed equation.
The difference of two imaginary roots which are not
conjugate, is in general -
(a — a') + (5 – 5') V-1.
The square of this is in general imaginary, and of the
same form as each of the roots; and, therefore, there
will be an imaginary root in the equation of the
squares of differences for each pair of imaginary roots
whose rational and irrational parts are respectively
wnequal. But if the rational parts a, a’ be equal, the
difference will be (b – b') V-1, the square of which
will be negative, but always real; and if the imaginary
Imaginary
Roots.
S-N-2
A L G E B R A.
615
Algebra.
parts be equal, the difference will be a - a', the square
S-N-" of which must be positive and real.
Hence the real and positive roots of the equation of
the squares of differences must contain among them the
squares of the differences of those pairs of imaginary
roots (if there be any such) in which the imaginary
parts are equal, and the real and negative roots must
contain among them the squares of the differences of
those imaginary roots in which the real parts are equal.
The difference between a real and an imaginary root
being of the form
(a — a') + b W – 1
its square must in general be imaginary, and when so,
the corresponding root of the equation of squares of
differences will be imaginary. But if the real root be
equal to the real part of the imaginary root, then the
difference will be of the form + b v - I, the square
of which is negative, and therefore in this case the cor-
responding root of the equation of the squares of dif-
ferences will be negative.
If we suppose that the two equations have no two
imaginary roots whose real parts are equal, nor any real
root equal to the real part of an imaginary one, it fol-
lows that every negative root of the equation of the
squares of differences will be equal to — 4 b”, or four
times the square of the coefficient of V – l in the
imaginary part of one of the roots taken with a negative
sign. Let – a be a negative root of this equation, '.'
• = 4 wº, b = 2 *
2
the value of b being thus determined, let it be substi-
tuted in M = 0 or N = 0, and the corresponding values
of a will be the real part of the root.
Whether — a be a root proceeding from either of the
circumstances justmentioned, scil. the equality of the real
parts of two different roots, whether both imaginary or
one real and one imaginary, may be known by finding
whether the value of b thus determined will give the
equations M = 0, N = 0 a common root. If there be
a common value of a, which satisfies both, then the
value of b will belong to conjugate roots, and other-
wise not.
It follows from what has been inferred here, and what
has been established in (392,) that there are at least as
many changes of sign in the equation of the squares of
differences, as there are combinations of two real roots
in the proposed equation. Also it must have at least
ag many successive repetitions of sign as there are
pairs of conjugate imaginary roots in the proposed
equation, or, in other words, it cannot have a less num-
ber of successive repetitions of sign than half the
number of imaginary roots in the proposed equation.
Hence we may infer, that if the equation of differences
have its terms alternately positive and negative, and
therefore have no successive repetition of sign, there
can be no imaginary root in the proposed equation.
SECTION XL. -
On the Resolution of Algebraic Equations of the Third
and Fourth Degrees. f
(425.) THE general problem to determine the roots
of an algebraic equation of the m” degree as func-
tions of its literal coefficients, has long engaged the
attention of analysts. The converse of this problem,
scil. the determination of the coefficients as functions of
the roots, was solved in an early stage of the algebraic
analysis; but the general problem of the resolution of
literal equations has baffled the powers of the most
refined modern analysis. When it is considered, how-
ever, that all the equations which present themselves in
actual philosophical investigations, are numerical equa-
tions, the particular data of the problem furnishing the
values of the numeral coefficients, the general problem
must be considered of an interest rather speculative
than practical.
We shall, however, in the present section explain the
methods of resolving general equations of the third
and fourth degrees, which is the utmost extent, except
in very particular instances, to which the solution of
algebraical equations has been yet as carried.
By the transformation explained in (345,) it is possi-
ble in every equation to remove the second term. We
shall, therefore, consider equations of the third degree
in general represented by
a' -- a a + b = 0.
Let a = y + 2 " ." .
a” = y” + 2* + 3 y z (y –– 2)
‘. . as = y” + 2* + 3 y z a
'...' as — 3 y z a - yº – 2* = 0.
Comparing this with the proposed equation in order
to identify them, it will be necessary that
— 3 y z = a gy* -- z* = — b
as
e 8 * – —
y & = – #
gy* + 2* = — b.
Since the sum of yº and 2* is — b, and their product
8
tºº #. they must be the roots of the equation (176).
08
/2 b a' — — = 0
a" + b a 7
ſ b º, Ti-
* = -->4. a tº
b T}; a8
tº * - — — *-*º *
... y = – 3 + 4 ' 27
2° E — — — b° 0.3
- 2 T T 57
b Vº as Nºr
r = [ - - - - *-* = − ===__,
( 2 ++.)
+(– : – b” as Nº.
2 4 #)
Algebraic
Equations of
the Third
and Fourth
Degrees.
Sºº-y-Z
616 A L G E B R A.
Algebra. Let a', a” signify the arithmetical values of the third positive, and if b > 0 the first is positive and the other Algebraic
-y-' roots of the values found above for y”, zº, when the two negative. - º:
particular numbers which b and a may signify are sub- In this case it may be observed, that a must be nega- º
stituted for them, and let m', m” signify the two ima- ... . tº º © bº
gnify tive in the original equation, for otherwise — -- ;- Degrees.
ginary third roots of unity. The three values of a in
4 27 S-V->
the proposed equation will then be
a = a + a” a = m' (a' + a!") a = m." (a' + a”).
It is evident that the two roots a!, a” ought to be so
taken that their product should be — -: e
If a', a” be substituted for y, z in the equation
as — 2 y z z – yº – 28 = 0,
and the result
as – 2 a'a" a - alº – a' = 0,
divided by a - (a' + a”), the quote will be
a" + (a' -- a!') a + a” — a' a' -- a!” = 0,
which solved for a, gives
* = _a' + d" + V(HE)- *-i- a' a' — alſº
2 * = 2 -
which values may be reduced to the forms
a = — 3 (a' + a”) + š, (a' — a”) w/T 3
w = – 3 (a' + a”) — (a' – a!') V-3
The identity of these two forms with mº (a' -- a”) and
m" (a' + a”) is obvious, by attending to the values of
m/, m^,
/—- *mmemº
— 1 + V – 3 a = 1 = (−3
2 2
In considering the nature of the roots we shall
successively examine the cases in which
b? as
ſº-º-º-º-º-º - ſ), - 0, 0.
4 + 27 > <
m! =
b”
T
must be necessarily real, and, therefore, aſ + a' must
also be real. The other two roots of the proposed
equation --
w = — , (a/+a") + 3 (a' — a”) V-3
are necessarily imaginary since the coefficient of w/T3
is real, and not = 0. The sign of the real root is in
this case different from that of b.
b” a 3
2. Let — + -ā- = 0. In this case a' = a” = —
4
3 *º tº a
==-2'Vº
2
8
1. If ~7- - - # > 0. In this case the values a', a”
V; .
2
3 / 5."
a = — 3 (a' + a”) = V;
which last is the common value of the two roots, which
in the last case were imaginary, and have under the
present condition become equal. The common value
of the equal roots is, therefore, half the first root with
a contrary sign. : -
If b > 0 the first root is negative and the other two
would be composed of two terms essentially negative,
and therefore could not = 0.
b? 8
3. Let T + # < 0. In this case a must also be
negative, and the quantities aſ, a” will be imaginary.
The first root a = a + a” assumes the form'
a = (p + q V – 1)* + (p — q V – 1)*
This, although it includes imaginary terms, is essentially
real, since if its parts be developed by the binomial
theorem, the imaginary parts will mutually destroy each
other (259.) -
It follows also from the same principles (259) that
(a/ *g a”)
JH-
is real
is real, and, therefore, that (a' — a!") w/ – 3
Hence the roots
a = — , (a' + a”) + 3 (a' — a!") W – 3
are real.
Thus in this case the roots are all real.
This case of equations of the third degree is com-
monly called the irreducible case. Because, although
the formula obtained for the roots is their true algebrai-
cal expression, yet it can only be cleared of imaginary
quantities by converting it into a series, and as this
series is seldom convergent, it is useless for the actual
determination of the roots; and therefore we must
always have recourse to the methods of approximating
to the roots of numerical equations.
For other methods of solving cubic equations see
TRIGoNoMETRY.
(426.) We shall now proceed to explain the methods
of resolving equations of the fourth degree. The
second term being capable of being removed by a
transformation, we may consider all equations of this
degree included under the form
a" + p a + q r + r = 0.
Following a method of investigation analogous to that
adopted in the case of equations of the third degree,
let -
a = y + 2 + u
'...' a' = y”-H z* + wº + 2 (y 2 + y u + z u)
C e a' – 2 (y” + 2* + w”) a” + (y” + 2*-ī- u”)*
= 4 (y'2'-- yºu’-- * u") +8 y z u(y-- z +u)
‘..' a' – 2 (y?-- 2* + w”) as – 8 y z waſ -- (yº-j- 2* +u")*
– 4 (y” z*-ī-y” w? -- 22 w?) = 0.
To identify this equation with the proposed one, the
following conditions will be necessary:
1. p = – 2 (y” + 2* + w”) '.' y” + 2* + u" = — p
2. r = (y” + 2*-i- wº)” – 4 (y” z* + yºu” + 2* u”)
* —
•." y” zº-H yºu + z'u' =? 4 r
16
§
3. q = – 8 y z u : v 3 u = – 4. . . 2" us = T.
Q 3/ $/ & u 8 w = w = i
A L G E B R A. 617
This being transformed into another which will be free
of fractions by substituting i for t, the equation be-
COIſleS - - - - - -
s” + 2 p s”-- (p” – 4 r) s – q = 0.
This being an equation of the third degree, its roots
may be determined by the methods already explained.
Let them be s', s”, s". Hence
y = + , Vs, z = + , V 5" u = + , Vs".
Since a = y + 2 + u, the values of y, z, and u being
combined in every possible manner by addition, would
give eight values of a. But since y z w = — # it is
necessary that they be so combined that their product
shall have a different sign from that of q.
Hence when q is negative, either two or none of the
values of y, z, u must be negative ; hence the values
of a are in this case
* = + , Vy + k My H-] vs”
w = + , w/97 — # Ms” — ! My
r = — ; Vs – ; Vy-- Vy"
a = — Vs' -- V s” – V s”
When q is positive it is necessary that either one or
three of the values of y, z, u be negative. Hence in
this case the values of a are -
a = — VST-3 vs.” – ; Vºs’
r = – Vy H-3, Vs” + , v 37
w = + , Vis' – ; Vºsſ' -- 3 Vs”
(427.) The nature of the roots of the proposed
equation evidently depends on that of the roots sº, s”,
s". These must either be all real, or one real and the
other two imaginary.
If they be all real, they must either be all positive,
or one positive and the other two negative, since the
last term — q” of the equation of which they are the
roots is essentially negative.
If sº, s”, s” be all positive, all the values of a are
necessarily real.
If s be positive, and s”, s” negative, all the values
of r are imaginary, except in the particular cases where
two imaginary terms happen to be equal, and therefore
destroy each other when united with opposite signs. In
that case two roots will be real and two imaginary.
If one of the values sº, s”, s” be real, and the other
two imaginary, the real value is necessarily positive,
since the last term of the equation of which they are
roots is — q” essentially negative. The other two
being conjugate imaginary roots, must be of the forms
(423,)
a + b v – I, a - 5 M-1,
Algebra. By this it appears, that of the three quantities y”, 2’, and these must enter the values of a in one or other of Series for
~~ u”, the sum, the sum of the products in pairs, and the the forms . .
inued product of all three, are severally given. —ſº - —R} of Multiple
continued p 2 c y g (a + b w/º. 1)” —H (a — b w/º: 1): Arcs.
They are therefore the roots of the equation \º-,--"
e-4 e 4 ºr tº – £ = 0 (a+b /= i,” – a – b /= i\}.
+ # * + -ij- t – H = 0.
The former is a real, and the latter an imaginary quan-
tity, (259.)
Hence it easily appears, that in this case two of the
values of r must be real, and two imaginary.
SECTION XLI."
Of the Developement of the Sines and Cosines of
Multiple Arcs in Powers of the Sines and Cosines
of the Simple Arcs.
(428.) NotwitHSTANDING the elementary nature of
the trigonometrical analysis, and the attention which
has been devoted to its various details, from the time of
Euler to the present day, by the greatest mathemati-
cians, yet the analysis of angular sections remained
until a late period in a most imperfect state. Formulae
expressing relations between the sine and cosine of an
arc, and those of its multiples, were established by
Euler, and subsequently confirmed by the searching
analysis of Lagrange, which have since been proved
inaccurate, or true only under particular conditions;
and it was only within the last three years that the
complete exposition of this theory has been published,
and general formulae assigned expressing those rela-
tions. In the year 1811, Poisson detected an error in a
formula of Euler, expressing the relation between the
power of the sine or cosine of an arc, and the sines and
cosines of certain multiples of the same arc. But the
most complete discussion of this subject which has hi-
therto appeared, is contained in a Memoir read before the
Academy of Sciences at Paris by Poinsot,f an eminent
French mathematician, in the year 1823, and further
developed by him in another Memoir published in the
year 1825.
The developements respecting multiple arcs may be
divided into two distinct classes. The first includes all
series in which the sine or cosine of a multiple arc is
expressed in powers of those of the simple arc; and the
second, those in which a power of the sine or cosine of
a simple arc is expressed in a series of sines or cosines
of its multiples: to the former we shall devote the
present section, reserving the latter for the following
OIle.
The series in powers of the sine, cosine, &c. may be
either ascending or descending, and accordingly the
several problems into which our analysis resolves itself
may be enumerated as follow :
* This and the following Section are extracted, by the permission
of the Publisher and the Author, from Dr. Lardner's Treatise on the
Analysis of Angular Sections in the third part of his work on Plane
and Spherical Trigonometry.
# This mathematician has rendered himself distinguished by the
invention of the “theory of couples,” (Théorie des couples,) a most
powerful instrument of investigation in analytical mechanics, and
one which has not yet received the attention which it deserves from
mathematical writers, either here or on the continent, and which we
venture to predict it must ultimately command.
618 A L G E B R A.
Algebra. To develope | *g A, m* + 2 A, Series for
I º º :} IIl ascending powers of cos ar. + [A, (m” – 1) + 2. 3 A.] y, *:
2. sin m a . . . . . . . g - + [A, (m? – 4) + 3.4 A.] y”, - PCS,
COS 772, º in ascending powers of sin a . + [A, (m? – 9) + 4.5 A.] y”, \-N-
3. sin m :} in ascending powers of tan ar. + [A, (mº – 16) + 5.6 A.] y'.
COS 770, Jº • . . . . . . . . . . . . . . . . . . . . . . . .
4, cosm :} in descending powers of cos r. . . . . . . . . . . . . . . . . . . . . . . .
Sun m a. + { A,is [m”— (n–2)*]+ (n-1) (n) A. } y".”
5. Sin m r ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
in descending powers of sin r.
COS 772, ſº
ge e .." Since this must be fulfilled independently of y, the
(429.) To develope cos m x in a series of ascending coefficients must severally = 0. Hence we find
powers of cos x.
Let cos a = y, and let As = – º As.
2 = y + Vºyº – 1, m? – 1
emismºmºmºmºmºmºmº = — — A, ,
... + = y– v y” — 1. - As 2. 3 Tº
- - - 7m? — 4
But also (see TRIgoNoMETRY) A, E — —a 1– A.,
2 cosmº = ~ ++. * A. = on? – 9 A
s s - – -TET as ,
If then 2" be obtained in ascending powers of y,
and zº" deduced from it by changing the sign of m, . . . . . . . . . . . . . . . . . .
we shall thence obtain 2 cos m. a. in a series of the m? — (n − 2)2
required form. e' An = - -(n-1)-i-A-. 9
Let 71 - 1 J 72
z" = u = A + A y + As y” + As y” + . . . . . . F. . . . . . . . . . . . . . . . . . . . .
. . . e • v. U. Hence we obtain the following conditions :
The solution of the question will be effected if the 9.
values of the coefficients of this series can be obtained A, E – *- A.,
without introducing any condition which restricts the 2 rºo
generality of the problem. 7m? – 1
Let the series assumed to express u be twice differen- As = – -3.3- ºr ,
tiated, and the results will be t * – 4)
7m2 (m” —
- - - A, e ------—H· Ao,
# = A +2A,w-3A. v.--4A,” +... 4 2. 3. 4 ""
ty As = + (m” – 1) (mº - 9) A
dº u & 2 5 - 2. 3. 4.5 1 >
dº. T 2 A, + 2. 3 A, y + 3. 4 A, y + . . . .
3/ A. = — m° (m?–4) (m°– 16) A
Also, let a – 2. 3. 4. 5 .. 6 0 7
w = (y – aſy” – 1)" . . . . . . . . . . . . . . . . . . . . . . . . . .-
be twice successively differentiated, and the results are
d u\? The law of which is evident. These conditions,
(# (y? – 1) – m” u = 0, however, fail to determine the first two coefficients
d y Ao, A, . To find these, let y = 0 in the series for u
d u) (dºw) ,,,, d wV? **\,…, , – and du. and also in the values
(#)(;)or-o- (;) -(#)” = 0. "?"
- w = 2* = (y –- Vy? – 1)",
d e
which, divided by #. gives d u 777, 70 tº
d2 u. d u d 3/ - A/ Ay? — l 2
dy? (y? – 1) + (#) gy–mº u = 0. and equating the results, we obtain
2 amº TTY" — r — m
Let the values of u, du, dºu, derived from dif. A, F ( w/ 1)" = ( – 1)2,
d y d y” wi = 1
A = m ( V-TY"-i = m (–1) = ,
ferentiating the assumed series, be substituted in the whence wºm d ( - ) (–1)
last equation, and let the result be arranged according - 2 ſº
to the ascending powers of y. We shall thus obtain A. = 77), ( – 1)2,
the following series : , - - -;
A L G E B R A.
619.
Algebra.
m? — 12 - m-1
- - 2. 3 m (–1).
A = + “tº "...m. (- )*,
As = — *Hºº g *2 (- ),
... = +.........................
Hence we find
mº (ne – 29 (ne – 4)
l. 2. 3. 4.5 . 6
1. 2.3.4. 5. 6.7.8
(m? – 1°)
1. 2. 3
+ m (–1)*{y-
(mº – 19 (mº – 3) ...,
+-Hºi H-y
(m” – I*) (mº – 3°) (mº – 5°) ...,
a- l. 2. 3. 4.5 . 6.7 y +.... }
To find the series for 27", it is only necessary to
change the sign of m in the result which has just been
obtained. Since neither of the series in this result
contains any odd power of m, this change produces no
other effect than to change the sign of the coefficient of
the second parenthesis. Let the series in the first
parenthesis be called for brevity S, and that in the
second S', and we have
gy”
ºn m-1
2" = (–1)*. S + m ( – 1) = . S',
wn - 1
2 . S';
.
2-” – (–1) . S + m (– 1)
m – 1
2
-* - 1 *
since — m ( – 1) s = m (–1)
Hence, by addition we obtain,
wº
z" + 2* = [ ( – 1)2 + ( – 1) *]s
+ I(-1)*-(– Dººms,
•. 2 cos m r = [ ( – 1)2 + (— 1) *] S + I( – 1)*
- ºn-l
+ ( – 1) = 1 m S'.... [1],
which is the developement sought.
(430.) The form of the coefficients of this formula
may be changed. By Trigonometry we have (cos a +
v — 1 sin a Y” = cos m (2 n. 7 -- a) + v– 1 sin m
ſº n T =E 4), n being any positive integer. Let a =
2 Tr, e g - -
(-1); - cosm an ED T+V-I sin; m (4n+1)",
‘.'(V-T) "" =cosłm(4n+1)7– w/Li sin}m(4n-El)7
‘... (–1)2 + (–1) 2
*n-1 -*-l
( – 1) * + ( – 1) * = 2 cos 4 (m—1) (4 m + 1) ºr,
Hence the series for cos may becomes
cos m r = cos 4 m (4 n + 1) T. S + cos 4 (m — 1)
(4 m + 1) ºr . m S'. . . . . . [2].
In this formula n is an indeterminate integer for
each value of which the second member has two values
corresponding to the double sign +. The successive
terms of the series
0, 1, 2, 3, . . . . . .
being substituted for n in cos 4 m (4 n + 1) ºr, it will
successively assume different values until the number
substituted for n is equal to the denominator of m ; for
this value of n the value of cos 4 m (4 n + 1) ºr will
be equal to that obtained by substituting 0 for n, and
all integers greater than the denominator of m will in
like manner give a constant repetition of values before
obtained by substituting for n values less than the deno-
minator of m. It follows, therefore, that cos 4 m (4 m
+ 1) T is in general susceptible of as many different
values as there are units in the denominator of m, and
no more. In like manner, cos 4 m (4 m – 1) T is sus-
ceptible of the same number of values; and therefore
the coefficient of S is susceptible of twice as many values
as there are units in the denominator of m, and a like
observation applies to the coefficient of m S'.
Since S and S' involve no functions of a, except cos
a, the change of a into 2 n. 7 -E a makes no change in
their value ; and it follows, therefore, that for a given
value of cos a the second member of [2] is susceptible
of twice as many values as there are units in the deno-
minator of m. It is therefore necessary to show how
cos m. a can have several values corresponding to a
given value of cos ar. The angle a being changed into
2 n' T =E w, n' being an integer, makes no change in
cos a, but changes cos m.a. into cos m (2 n' T =E a),
which has twice as many values as there are units in
the denominator of m. Hence the formula [2] will be
more generally and correctly expressed thus,
cos m (2 n' T =E r) = cos 4 m (4 n + 1) T. S
+ cos 4 (m – 1) (4 n + 1) T. m. S',
where both members have the same number of values,
and where the values of the indeterminate integers
n', n are supposed to be less than the denominator
of m. &
It still remains, however, to show the values of each
member which correspond respectively to those of the
other. Since the value of each member changes by
ascribing different values to the integers n' and n, this
question only amounts to the determination of the
relation between any two corresponding values of these
integers.
Let a = 4 ºr, and therefore S = 1, S' = 0. Hence
cos m (2 n' T =E 4 t) = cos 4 m (4 n + 1) ºr,
or cos 4 m (4 n' + 1) T = cos # m (4 m + 1) T.
Since n and n' are not supposed to receive any value
greater than the denominator of m (for all the values
of the cosine after that would only be repetitions of
former values,) this last condition can only be satisfied
by
= 2 cos 4 m (4 m + 1) ºr,
m = m/.
620
A L G E B R A.
Algebra.
\-y-Z
Hence the formula becomes” -
cosm (2 m r + æ) = cos 3 m (4 m + 1) r. S
+ cos # (m – 1) (4 n + 1) r. m. S' . . . . [3].
(431.) It does not always happen that the formula
expressing the value of cosma includes both terms of
the second member ; for the angles whose cosines are
the multipliers of S and S' in [3] may one or other of
them be an odd multiple of a right angle, in which
case the multiplier will be = 0, and the term will dis-
appear.
To determine the conditions under which this can
occur, it is necessary to consider when either of the
numbers
4 m (4 m + 1) T, 4 (m – 1) (4 m + 1) T,
is an exact odd multiple of 4 T. This evidently takes
place when either of the numbers
m (4 n + 1), (m – 1) (4 m + 1),
is an odd integer.
777,
Let m = +
70,
That
the first of the above numbers be an odd integer, it is
necessary that
, and let I be any odd integer.
m" (4 m + 1) = n' I.
Since m' and n' are prime, one or other must be an
odd number ; but since 4 m + 1 and I are also odd, it
is necessary that both m' and n' should be odd.
Also, since m' is prime to n', and measures n' I, it
must measure I. Let mſ = i, which must be an odd
integer, since both I and m' are odd. Hence
4 m + 1 = n' i,
... 4 n + 1 ,
f
70,
But since n is supposed to receive no value greater
than n', i cannot be greater than 4; and since it is an
odd integer, it must be either 1 or 3. The two corres-
ponding values of n are
_ n' T 1 a 3 º'-F
- 4 - " - 4 -
The denominator n' being odd, must be either of the
form 4 t + 1, or 4 t — i.
If nº be of the form 4 t + 1, the two values of n must
be
72
3 m + 1 .
77 e- er
4 * 4 °
* In clearing the formula [1] of imaginary quantities, Lagrange
has fallen into an error which was lately detected by Poinsot, and the
difficulty explained as above. Lagrange's mistake arose from as-
suming that
w/(–1) =cos 3 m r + v — 1 sin # m or,
which is evidently erroneous, since the first member has as many dif-
ferent values as there are units in the denominator of m, and the
second member has but one value, he forgot to take into account,
that while the change of r into 2 n ºr + a produces no change on
(cos a + v — 1 sin r)",
it does produce a change on
cosm s + v - Tsin m r.
In fact, without this consideration, Moivre's formula itself is in-
volved in the absurdity of one member having a greater number of
different values than the other.
since 4 evidently would not measure n' + 1 = 4 t + 2, series for
nor 3 n' — 1 = 12 t + 3 – 1 = 12 t + 2. Sines, &c.
These values of n being substituted in [3], and m * *
/ * CS.
being changed into +, and the sign + only being used S-N-7
for the first, and — for the second, give
' /n/ – 1
2 - + )= cos 3 m' ºr . Sº
cos —T
71.
4. f ºn's,
+ cos; (m/ — n') T. - S
70, ...tº
m’ 3.n'+ 1 I tº sº. e
cos -- { --— T — a = cos 3 m" ºr . S
71. 2
| om/ !
/
+ cos 3 (m – n) = -i-S J
Since m' and n' are odd,
cos 3 m.' T = 0, cos 3 (m' — n') ºr = + 1,
cos 3 m' r = 0, cos 3 (m’ – n') T = + 1,
- | f /
7m.' A n' — I 777,
'.' cos T-ſ -— T –– a l- >E+ S'
#( 2 –– m" [5]
-- tº e e s a O
m! /3 m/–– 1 m’ - p
cº (*#) --)=+% S!
72. 2 70,
the sign + being used when 3 (m’ — m/) is even, and
— when odd.
If n' be of the form 4 t — 1, the two values of n are
n' + 1 . 3 n' – 1
4 : * ~ 4 ~ *
for it is evident that 4 would not in this case measure
n' – 1, or 3 m' + 1.
These values being substituted in [3], and m being
/
changed as before into
(*#!
777. g
++, we obtain
7,
m/
cos —T T – a j- cos 3 m" tr. ST
7. 2
f
770,
+ cos 3 (m' — n') ºr . ; s’ |
m/ (*= + - | si . . . . [6].
COS —- — 7 a j = COS # 7m.' Tr.
77' 2
m"
+ cos; (m' – n) = . ; S/ J
Hence, as before, we find
/ | I 7m'
co: (*# 7 — ſº = +. S'
m/ /3m'— I m"
coº ( 2 ---.)= ++, s
The signs –H and – being used as before.
(432.) The condition under which
' — l
(m – 1) (4 m + 1) = *-* (4n+ 1)
should be an odd integer, may be immediately derived
from those of the last case by changing m' intom' — n'.
Hence the two values of n are the same as those already
found, and n' and m' – n', must be odd integers.
Hence m' is even. Hence we have
cos 3 m.' T = + 1, cos 3 (m' — n') n = 0,
cos 3 m.' T = + 1, cos 3 (m' — n') T = 0.
A L G E B R A.
621
Algebra. Hence the formulae [4] and [6] become
\-v- ' f ' — I
co-º. (tº -+.)= +s
f 8 ſ
coº ( * -- )= –E S
m! /n/ -- I >. . . . [8]
coº ( – )= +s
7m' /3 n' – 1
coº ( 2 ++)= +s
the sign + being used when m' is even, and — when
odd.
(433.) From the preceding observations it appears,
that when the denominator of m is odd there are always
two values of an angle a whose cosine is given, of
which the cosine of the multiple m a. admits of being
expressed by a single series of ascending powers of the
given cosine;” but that for all other values of the arc
whose cosine is given, the cosine of the same multiple
requires the combination of both series S and S'.
T = +*-ºs-i:
If the denominator of m be even, there is no value Series for
whatever of the angle whose cosine is given, which Sines, &c. of
allows of cos may being expressed by a single series. Multiple
e g tº tº Arcs.
(434.) The case in which m is an integer comes , p
under the cases where the denominator of m is of the
form 4 t + I, t being in this case = 0. If m be odd,
we have by [5]
&= |*|| f
cos m. a = + m S',
the sign + being used when # (ºn’ – 1) is even, and
— when odd. If m be even, we have by the first of
[8]
cos m. a = + S,
the sign -- being used when m is even, and – when
odd.
(435.) The laws of the two series S and S' are easily
defined. Let T be the rh term of S and TV of S'; by
attending to the forms of the coefficients and exponents
we find
mºnº-2) (m-4) (mº-5').... (mº- ºr- 9'),…,
2 (r – 1)
2 r - 1
!--- (
T= + -a-, *-*.
It is evident from the forms of these terms, that the
series S can only terminate when m is an even integer,
and S' when m is an odd integer.
(436.) To determine the number of terms in each
series when it is finite, let n be the sought number.
The (n + 1)* term must therefore = 0. Substituting
m + 1 for r in T and T', and putting the results = 0,
we obtain -
m” – (2n + 2 – 4)* = 0,
777,
*...* 77, T –
2
the number of terms in S ; and
n” – (2n + 2 – 3)* = 0,
fe. 7m -- l
ºmºm? y
2
+ 1
the number of terms in S'.
(437.) To obtain the last term (z) of S, it is only
necessary to substitute the value of n in place of r in
T, and the result is
_ _ m” (m” – 2") (m” – 4*). ... (mº- (m-2)”) ...,
* = + → in yº.
Each factor of the numerator may be resolved into two,
thus
m* = m × m,
(m” – 2*) = (m. --2) × (m. – 2),
(mº – 4*) = (m + 4) × (m — 4),
* > © o º tº
(mº - on- 2)*) = &n — 2) x 2.
* Before the publication of Poinsot's Memoir, these cases were not
noticed. Lagrange expressly states, that whenever m is a fraction,
botn terms of the second member of [3] are necessary.
VO L. I.
m’ – 1) (mº-3) (*-5).... (mº- (ºr-3))
2 r – 1
The second factors of these, beginning from the lowest,
are obviously the even integers from 2 to m inclusive,
and the first factors, beginning from the highest, are the
even integers from m to 2 m — 2 inclusive. Thus the
simple factors of the numerator are all the even inte-
gers from 2 to 2 m — 2 inclusive, the factor m being
twice repeated. The numerator of z may therefore be
written thus,
2. 4.6 . . . . (2 m — 2) × m,
which is equivalent to
I . 2. 3. . . . . (m. – 1) × m x 2"-"
The factors of the denominator destroying all these
except 2"-", we have
2 = + 2*-1 º',
-- being taken when + is even, and — when odd.
7m + 1
(438.) To determine the last term 2' of S', let
be substituted for r in the general term, and we obtain
, L (n° – 1%) (m” – 3%) , ... (mº – (m – 2)”).
2= +*i-gº; ; Hy".
Each of the factors of the numerator may, as before,
be resolved into two, thus
(m” — 1*) = (m. -- 1) x (m – 1),
(m” – 3*) = (m. -- 3) × (m. – 3),
****
hºmº
-:
(m” – (m. – 2)*) = (2m – 2) × 2.
The last factors of each of these, beginning from the
lowest, are the even integers from 2 to m – 1 inclusive,
and the first, beginning from the highest, are the even
integers from m + 1 to 2 m — 2 inclusive. Hence the
factors of the numerators may be expressed thus,
4 M
e-ºl
tº-º-º:
A L G E B R A.
Aigebra.
*~~~~
(2 m – 2)
(m — i.). 2"--
I
f ºn 2m-- m.
2' = + 3/
(439.) To develope sin m x in ascending powers of
COS X.
By subtracting the value of 2-" obtained in (429)
from that of 2", and the result being disengaged from
the imaginary symbols by the method used in (430)
becomes
sin m (2n ºr + r) = sin; m (4 m + 1) T S + sin (m – 1)
(4 m + 1) T. m. S'. [9]
All the preceding observations are equally applicable
here. When the denominator of m is an odd integer
there are always two values of an angle a whose cosine
is given, which are such that sin m a will be expressed
by only one of the two series in [9].
(440.) To determine the conditions under which this
will happen, it is necessary to determine when either of
the numbers
m (4 m + 1)
is an even integer.
To find the values of n which will render m (4 m + 1)
an even integer, let
m (4 n + 1) = I
m' (4 m + 1) = I m'
".” 4 m + 1 = + ,
'm'
(m. – 1) (4 n + 1)
__ must be
I & e
Hence-F n' is an odd integer, therefore -:
7m/ 772.
I
m"
-
odd integer, therefore m' must be even. Let 2,
‘. . 4 m + 1 = i m'.
It may be proved, as in the former case, that i must
be either 1 or 3, and that when n' has the form 4 t + 1,
the values of n are
77 = n' - 1 n = ** if l
4 3. 4 2
and when m' has the form 4 t — 1, the values are
m' + 1 . 3 n' — I
— — — — — — .
(441.) In like manner, in order that (m – 1)
(4 m + 1) be an even integer, the same values of n are
obtained, and it is necessary that nº should be odd,
and m' — nº even, and therefore m' odd.
(442.) Hence if m' be even, and the values of n ob-
tained above be substituted for it in [9], we obtain
f | n
7m' /m' – 1 ſº *
sin —T- 7" + •)= SIIl # m! 7T . S Series for
70, 2 Sines, &c. of
7. ſm' "ºpe
e ! – n' S' TCS.
sin ... (m 7- . —F
+ sin 3 (m' – ") r. º. \-,-- /
f f
m! /3 m'+ 1 *
sin – **!--)- in m'..s
7/ 2
f
77?
g f f
+ sin (m! - n') : . ;-S
sº 10 g
tº on' m" + I e ł f S [ |
sin -- ( —- T - a j = sin # m 7 .
71, 2 -
m’
! e ! — m' f
+ sin # (m m') ºr . Tnſ S
... m' (3 m'— 1 º
sin – 7– wr + a – sin 4 m.' T. S
7, 2
P
77?
+ sin g (m' — n') Tr. -j- S' J
: 72.
But since m' is even, and n' odd, .
sin 3 m.' T = 0, sin 3 (m' – n') ºr s + 1,
sin 4 m.' T = 0, sin 3 (m' — n') T = + 1,
Hence
! f |
... ???, n' — 1 7??,
sin – I- — Tr + æ = + -ī- S!
70, 2 7?
. m." (3 m/+ 1 ºn' s , || ' ' ' ' [11].
sin –tº–( → * T – a j = + -ī- S
71, 2 70,
I | I
. m.' /n' + 1 771,
sin —F 7 — a = + -- S'
70, 2 71.
| 3 '— I m' * * * * * , [12].
sin + 7? -4 )= -E - S'
71, 2 7),
The two first being true when m' is of the form
4 t + 1, and the last when of the form 4 t — 1. The
sign + is used when 3 (m' — n' + 1) is odd, and —
when even.
(443.) If m' be odd,
sin 4 m.' T = + 1,
sin 4 (m' — n') T = 0,
sin 3 m" ºr = + 1, sin 3 (m' — n') T = 0.
Hence the formulae [10] become
m' /m' – 1
in (*, *, +, = + S
m" /3 n'4.
sin —7– *** !, , = + S
70, 2 X- [13]
f l l de s e º º is { } @. w
sin “ (*#!-- )= + S
72. 2
! /3 ml — 1
in 4 (“. T + a J = + S J
the sign + being used when + (m' + 1) is odd, and –
when even.
(444.) The series S and S' in [9] being the same as
those in [3], their law and properties when m is an
integer have been already determined.
It is obvious that when m is even, we have
sin m a = + m, S',
+ being used when + m is even, and — when odd.
And when m is odd
sin m w = + S,
A L G E B R A.
623
Alge” + being used when + (m -- I) is odd, and – when
\-N- even.
(445.) Another form for the developement of sin m a.
in ascending powers of the cos ar, may be established
by differentiating the series found for cos m.a. in (430.)
By this process we obtain
- d S
m sin m (2 m x + 1) = – cos; m (4 n + 1) + · ·
d S'
— cos 3 (m —1) (4 m + 1) ºr . m +,
. da:
dS m? ºn” (m” – 2°),
d y T 1 1 .. 2 .. 3
amº (m” – 2*) (m” – 4*). ºn
Hºi-E-y’ + .... = – m R.
d S' 7m2 — 13 (m” – 1%) (m” – 3*)
— = 1 — — w” 4.
d y H-y-F –Hºa–H–-y
* e s e e s e = R',
# = – in r
_, d S º d Sl / s
:-H = m R sin z. d a = — R' sin ar,
‘.’ m sin m (2 m ºr -E a) = — sin a [cos 4 m (4 m =E 1)
T. m. R — cos (m – 1) (4 m + 1) T. R.']. . . . [13].
This being deduced directly from the formula [3] is
liable to the various modifications which have been
shown to be incident to [3], on assigning particular
values to m and n. The several modifications of [13]
which correspond to these, may be deduced by differen-
tiating the several series [5], [7], [8], &c. &c.
(446.) The laws of the series R and R' are easily
defined.
Let T and T'be their r" terms respectively,
T = + m*(m°–2") (m”–4*). . . . [mº–(2r—2)"]
l. 2. 3. . . . . 2 r – 1
—I*) (m?–3°). ... [mº – (2n—3)*]
1 . 2. 3. . . . 2 (r – 1)
The number of terms in R is only finite when m is
an even integer, and in R' when it is an odd integer.
The number in R is evidently one less than in S when
2 r = 1
y-r-,
T=+"
gº r − 1)
º o 7??,
it is finite, and is therefore equal to T2 . But the num-
ber in R/ when it is finite is the same as in S', and is
7m + 1
therefore tº
The last terms of R and R' in these cases may be
obtained by differentiating those of S and S', and
dividing the one by m sin a. and the other by — sin ar.
(447.) To develope the cosine or sine of a multiple arc
in ascending powers of the sine of the simple arc.
Let y = sin r,
2 = WT- y + y WTT,
2 = (VTy 4 y WTT)";
and since
2 cos m. a = 2" sº yº Series for
*ºm 777, z" -- 2-", Sines, &c. of
2 M-Tsin m r = 2" — 2-", Multiple
Arcs.
the problem will be solved by obtaining the develope" . y
ment of 2" in ascending powers of y. tº
Let
2" = A + A y + As y” + . . . .
By proceeding exactly as in (429), we shall obtain
2 * (m” – 28
* = A, #1-#y tº ) 4.
1 . 1 - 2. 3. 4
mº (mº – 2’) (mº – 4°). }
j-a-a-i-HTH − y + . . . .
m? – 12 , , (m” — 1°) (m3 – 3°) f
+A {, -} +, + I. . 2 . 3. 4 .. 5
– ...... }
The values of A, and Ai may be determined by making,
d (2")
d — ,
d y
as in (429), y = 0 in the two values of 2" an
and equating the results, which gives*
17t ſº-º-º-º: rºt - 1
A = (1)*, A = m. Aſ — 1 (1) g .
The value of 2-" may be dedućed from that of 2" by
changing the sign of m. Hence, if the series which
enter these values be Q, Q', we obtain
2" = (1)* Q + V-I (1) m Q',
m. — – “tº
2-" = (1)" a Q — W – 1 (1) * m Q',
... 2 cos m r = [(1)* + (1) *] Q + y = i
m – 1 rºl – 1
[(1) * + (1) = } m Q',
2 v- I sin m r = [(1)* — (1) Fl Q + v = i
m - I _m-l
[(1) * + (1) * } m Q'.
It will be observed, that by changing a into 2 m r + ar,
no change is made on the series Q and Q'; but there
is a change made upon the first member of each equa-
tion. The coefficients of Q and Q' have exactly as
many different v.lues as the first members of the equa-
tion. This is a circumstance which has been hitherto
overlooked.t
The above formula can be cleared of imaginary
quantities by the usual method, s
(1) * = cos n m r + V – I shi n m ºr,
in - 1
(1) * = cos n (m. — 1) T + w/ — 1 sin n (m – 1) ºr,
the number m being an indeterminate integer. All the
arcs which have the same sine may be included under
the formula n T =E ar, a being taken with the sign +
when m is even, and — when n is odd. Hence the
formulae become
* Lagrange, and all mathematicians after him, have fallen into
an error in the determination of these coefficients. Poinsot has lately
corrected it.
+ Poinsot, 1825.
4 M 2
624 A L G E B R A.
cos m 'm T. Q — sin m (m – 1)
emº
ſºmº
Algebra. cosm (n ºr + æ) (450.) To develope the sine and cosine of a multiple Series for
.*
T. m. Q' [14] are in a series of ascending powers of the tangent of the Sºº of
• . . . e ſº gº tº º simple arc. - *
sin m (n T + æ) = sin n m ºr . Q -- cos n (m – 1) TCS.
By developing the formula -V-
T. m. Q' . . . . [15].
(448.) There are certain values of n, for which each
of the coefficients of these formulae = 0. To determine
770, ge
these, let m = -r-, and let it be remembered that no
71,
value is supposed to be assigned to n greater than m'.
We have thence the following conditions:
m! 3 mſ
cos n m r = 0, m = —, or m = .
2 2
f f
7? 3 m.
cos m (m – 1) 7 = 0, * 7 = —, or n = -,
2 2
sin n m 7T = 0, m = 0, or n = m',
sin m (m. – 1) ºr = 0, m = 0, or n = 'n'.
The first two conditions can only be satisfied when
the denominator (n!) of m is even. Hence it follows,
that of all the arcs whose sines have any given value,
there are always two (X) for which the formulae [14],
[15], are reduced to a single series. These two arcs
m' . . 3 m' n'
are of the forms a 7 + æ, T2- 7r — ar, or-g- 7ſ — ar,
3 n'
T2- + æ. For these two values of m we have
cos m. X = -E m Q', sin m X = + Q.
The last two conditions can be fulfilled, whatever be
the value of n', and the formulae [14], [15], become
cos m X = + Q, sin m X = + m Q';
where X is an arc of the form a or m' ºr + æ when n' is
even, and a or n' T — a when n' is odd.
It appears, therefore, that among the values of an
are whose sine is given there are always two, the
cosines and the sines of whose multiples admit of
being expressed by a single series. In this respect,
the developements by the powers of the sine differ
from those by the powers of the cosine, in which,
when the denominator of m is even, there is no value
of the simple arc, the cosine or sine of whose multi-
ple can be developed in a single series.
(449.) If m be an integer, one of the coefficients
of each of the formulae [14], [15], must necessarily
= 0.
This comes within the case in which m has an odd
denominator, since the denominator is unity, and since
no value is supposed to be given to m greater than m',
it is in this case necessarily = 1. Hence in this case
cos m. a = + Q, sin m a = + m Q'.
The double sign applies to the two values of r, scil.
ºr and T — ar, which have the same sine. The value
of cosm a with the sign + is used when m is even,
and that with the sign — when m is odd; and in the
value of sin m a the sign + is used when m is odd, and
— when m is even.
When m is even, the series Q is finite and Q' infinite,
and when m is odd, Q' is finite and Q infinite. The
form of these series being the same as the series S,
S', in (429) the law, the number of terms when finite,
and the last term is determined in the same manner.
cos m r + V- I sin m r = (cos r + V- I sin r)";
by the binomial theorem we shall obtain
cos m r + V- I sin m r = R + v-TI R' . . . . [16],
where R represents the sum of the odd, and R' of the
even terms of the developement, and therefore
R = cos" a - A, cos"-* a sin” a + A, cos"-" a sin" v — . .
R! = A, cos"- a sin a – A, cos"-8 a sinº a
+ As cos"Tº a sin” a - . . . .
where A, A, A, . . . . represent the coefficients of the
second and succeeding terms of the expanded bino-
mial, whose exponent is m.
As each side of the equation [16] consists partly of
real, and partly of imaginary quantities, it is equivalent
to two distinct equations, between each separately. If
we consider R composed exclusively of real, and
^^ - I R' of imaginary quantities, we should therefore
have
cos m r = R \
sin m r = R! j [17].
These formulae, which were first published by John
Bernouilli in the Leipsic Acts, 1701, have been, even
to the present day, considered as exact and general.
This, however, is not the case.
To explain this, let
T = 1 — tan A, tan” a + A, tan” a - . . . .
T = A, tana – A, tan” a + A, tan” a - . . . .
•. R = cos” ar. T,
R! — cos” aſ . T'.
By changing a into 2 m r + c, the factors T, T' of the
second members of
cos m. a = cos” a . T,
sin m r = cos” aſ . T",
undergo no change, since these arcs have the same
tangent, and since T, T' include no powers except in-
tegral powers of tan ar, they can have each but one
value for an arc, whose sine and cosine are given. The
first factor cos” a has, however, as many different va-
lues as there are units in the denominator of m, of
which two, at most, can be real, and all the others
must be imaginary. On the other hand, for an arc
whose sine and cosine are given, and which is of the
form 2 m n + ar, m being any integer, the first members
of thºse equations have as many different values as
there are units in the denominator of m, and all these
values are real. Thus the two members of the equa-
tions are inconsistent.
It is not difficult to perceive, that this absurdity has
arisen from the false assumption, that the real and ima-
ginary parts of the second member of [16] were R and
w = I R'. We shall find, upon consideration, that
neither of these quantities are altogether real, or alto-
gether imaginary, but that each of them is composed
partly of real and partly of imaginary quantities, and is
of the form a + v — 1. b.
A L G E B R A.
625
Algebra.
In the formula
cosm a + V-1 sin m r = cos" a (T + WTi T'),
let the absolute, real, or arithmetical, value of cosm ar,
cos a being considered merely as a number, be P. It
is plain that its several algebraical values will be ex-
pressed by the formula P(+ 1)". And since
(+ 1)" = cos m n T + v = i sin m n ºr,
'.' cos" a = P (cos m n tr–H v — 1 sin m n ir),
the indeterminate integer n being even when cos w is
positive, and odd when it is negative. *.
Making this substitution in the former equation,
and in place of a, substituting the general formula
n' T-E a for all arcs having the same cosine, in which
the sign + is used when n' is even, and — when it is
odd, we obtain
cosm (n' ºr -E w} + V- i sin m (n't + 1)
= P (T cosm m T – T' sin m n ar)
+ V - i.P (Tsin m n + -i-T' cosm nºr).
Here the real and imaginary parts are separated on
each side, and equating them, we have
cosm (n' T =E a) = P (T cos m n T — T' sin m n tr),
sin m (n' ºr + 1) = P (Tsin m n T + T' cos m n t).
Each member of these equations is susceptible of as
many different values as there are units in the denomi-
nator of m. But it remains still to be determined,
which of the values of the second members correspond
or are equal to those of the first severally. In other
words, it is necessary to determine what relation sub-
sists between the indeterminate integers n' and n, nei-
ther of which are supposed to exceed twice the deno-
minator of m. To determine this, let a = 0, ..." P = I,
T = 1, T^ = 0. Hence
cos m n' T = cos m n ºr,
sin m n' T = sin m n ºr,
‘. . m = m. -
These integers are, therefore, always equal, and the
formulae become
cosm (n ºr H- a) = P (T cos m n tr-T' sin m n 7) [18],
sin m (n ºr + 1) = P (Tsin m n ºr + TV cosmºn 7) [19].
Whether the odd or even integers are to be substi-
tuted for m in these formulae, and whether a is to be
taken with + or —, is to be determined by the signs of
sin a and cos r, which are supposed to be given. If
cos a be positive, the values of n are to be selected
from the series
0, 2, 4, 6, . . . . ;
if it be negative, they are to be selected from
1, 3, 5. . . . .
If sin a be positive, a is to be taken with +, and if
negative with —. In all cases, however, the coeffi-
cient P in the second members is to be considered as
an abstract number independent of any sign.
If m be an integer, the formulae are reduced to the
forms
cos m. a = cos” a T, sin ºn a = cos" ar. T",
which have hitherto been taken to be general for all
values of m.
There are, however, particular values of n' even when Series for
m is a fraction, for which one or other of the series by Siº."
which cos m. a. and sin m a. are expressed will dis-
appear. In order that cos m n T should = 0, it is ,
necessary that m n should be a fraction whose deno-
minator is 2, and, therefore, whose numerator is an
odd number. This can only happen when m is a frac-
tion with an even denominator, and therefore an odd
numerator, and when n is equal to half the denomi-
nator. Also in this case, if half the denominator of m
be an even number, it is necessary that cos a should
be positive, (otherwise n should be odd,) and if half
the denominator be an odd number, it is necessary that
cos a should be negative, for otherwise n should be
even. Hence we may conclude, that if m be a fraction
with an even denominator, there is always one arc,
whose cosine has any given positive value when half the
denominator of m is even, and whose cosine has any
given negative value when half the denominator is odd,
which is such, that each of the formulae [18], [19], are
reduced to a single series, since under the conditions
just stated,
cos m. m. T = 0, sin m n T = + 1.
In order that sin m n T = 0, it is necessary that m n
should be an integer, and therefore that m should be
equal, either to the denominator of m, or to twice the
denominator.
m m r = + 1. If cos a be positive, n must be even,
and in this case, if the denominator of m be even, there
are two values of n, which will reduce the formulae
[18], [19], to a single series; but if it be odd, since n
must be even, there is but one value will satisfy this
condition. If cos a be negative, n must be odd, and,
therefore, when the denominator of m is odd, there is
but one value of m, which will reduce the formulae to a
single series, and when m is even, there is no value of
7, will effect this.
It appears, therefore, that when m is an integer,
cos m ar, and sin m r, can always be expressed in a
single series of powers of the tangent; but that when
m is a fraction, there are only certain values of an arc
of a given sine and cosine, which admit of a develope-
ment without both the series of [9], [10], and that in
some cases there is no arc which admits it.
If the two formulae [18], [19], be divided one by the
other, we shall obtain -
T cos m. m. T – T' sin m n it
Tsin m n T + T cos m. m. T’
tan m (nºr =E r) =
gº-ºº: T – T'tan m n ir
TT tan m n + -i-. TV
which, when m is an integer, and in the particular
cases already mentioned when m is a fraction, becomes
tall 777 aſ E T.
T
Or tan 771 a E T7
(451.) To develope the cosine or sine of a multiple
arc in descending powers of the cosine of the simple arc.
This problem was investigated by Euler, and subse-
quently by Lagrange, and both obtained the same
result, although they proceeded on different principles
Multiple
Arcs.
--~~
In each case sin m n 7 = 0, and cos
626
A L G E B R A.
Algebra.
\-y-
and by different methods. The series which were the
results of their investigations, and which have, even to
the present time, been received as general and exact,
are the following,
2 cosm w = (2 y)”—m (2 y)*-* + º (2 y)" --
m(m–4) (m.–5) 7m - 6 m(m — 5) (m. – 6)(m —7)
– Hºgºg (2)”--------,
(2 y)” . . . . [21],
T - . . . . . . . . . . . . . . . . . . . + . . . . . . . . . . . . . . . . . . .
+ (2)-4-m (29)------tº (2)--.
ºn(m+4)(m+5)/2.0-,-,-, m(m+5)(m+6)(m+7)
+++++(2) “-Hºº-º-º:
(2 y)-"--
& º e º s a s s º a ſº e º e s e º e º s a a e g º e º 'º e s s º a m e º ſº a
where y = cosa. The series for sin m a was deduced
from this by differentiation.
In the memoir already cited, Poinsot has examined
the analysis by which these results were obtained, and
shown that it is fallacious, and that the results them-
selves are false. To render this refutation intelligible,
it would be necessary to detail the process by which
Euler and Lagrange established the formulae, which
would lead to investigations unsuited to the purposes of
the present treatise. As, however, the results of La-
grange have been hitherto universally received as cor-
rect, it is proper to make the student aware of the fact
of their having been proved erroneous; and if he be
desirous to examine the details of the process, he is
referred to the memoir itself.
We shall confine ourselves here to that part of the
memoir in which the true developement of cos m. a. and
sin m r is investigated.
Let p = cosa and q = sin a . We have
7m, q? q' q”
cosm r = p" (1 -A, ºr--Air-A. 77 + . . . . )
where 1, A1, Aa, . . . . are the coefficients of the bino-
mial series, m being the exponent. We have
q* = | — p", q‘ = l — 2 pº -- p", tº 6 m e
Let these values be substituted for q*, q+, &c., and
let the results be arranged according to the descending
powers of p, and we have
COS 771, º st Ap" – B pº- ++ cº-
I
* =s=== mº - 6 ſh
Ira D p"+&c.
where -
A = 1 + As –H A, + A, . . . .
B = A, -ī- 2 A, + 3 A, + 4A, -ī- 5 A, . . . .
#C = A, H–3A, -i- 6 As H- 10 Alo -- . . .
1.
Tala P=A. H-4A, -- 10 A, -ī- 20 A,
The law by which these coefficients are formed is Series for
evident, but it is necessary to obtain finite expressions Sines, &c, of
for them as functions of m. For this purpose, let us **ple
suppose that the successive terms of the first coefficient, CS.
A were multiplied by the successive powers of an
arbitrary quantity y, so that it becomes
1 + A, y + A, y” + As y” + . . . .
m(m – 1) m(m—1)(m—2)(m—3) 2
or 1+++y++,-,-,--Hº-
But this last is equivalent to
G F (y)" f ( – ’) = U ,
2 2
so that U becomes equal to A when y = 1. It is not
difficult to perceive, that the other coefficients are what
the successive differential coefficients of U taken with
respect to y as a variable become when y = 1. We
have
U = 1 + A, y + A, y + A, v' + . . . .
d U -
#;=A, + 2 A, y + 3 A, y” + 4 As y”. . . .
d2 U - o
d y” = A, + 3 Aegy -- 6 As y” +
I
2
When y = 1, the second members of these equations
become equal severally to A, B, , C, H,. D. . . . Let
1.2.3'
the values of the function U and its successive differen-
tial coefficients when y = 1 be called Y, Y′, Y′, Y”,
&c.; we have hence
A = Y = +{2 + o-3,
B = Yeº.4 m (27- - 0--0
* *=sº 22 770, y
C = Y!' – * {m (m – 1) (2*-* -- 0"-2)
— ???, (2n-1 * on-1) }.
I • * * g)7?? - ??? - ºl Y -
D =Y///= # (m (m – 1) (m. – 2) (2"-" — 0"-")
– 3 m (m – 1) (2"---|- 0"-2) + 3 m (2"-1 – 0"-) }.
E = Y!!!! - |-}n (m—1) (m—2) (m—3) (2"---|-0"-4)
– 6 m (m – 1) (m. – 2) (2"-" — 0"-") + 15m (m – 1)
(2*-a-Lor-º) – 15 m (27- - 0"-) }. &c. &c.
In these analytical expressions for the coefficients of
the sought series, it is necessary to preserve the terms
0”, 0°-1, 0"-*, &c. because each of these powers of 0
become either unity, 0, or infinite, according as the
exponent of the power is - 0, positive or negative.
The true developement, therefore, of cos m. a. in de-
scending powers of cos aſ or p, the angle a being sub-
A L G E B R A.
627
Algebra. posed less than a right angle, and only considering a
J- single value of cos m. a relative to the arc a, is
To determine the last term 2, let the values of n Series for
already found be substituted for r in this formula. Sines, &c. of
- Multiple
Arcs.
cos m r = Y p"—Y' p"-" + 4 Y"p"-" — saxº~ ––.
If m be a positive integer, this series will be finite,
since all the terms beyond a certain term will = 0, and
it will thus give the exact value of cos m.a. Thus
when m = 0, or m = 1, we find that the first coeffi-
cient only has a finite value, and all the others = 0.
For m = 2 and m = 3, the first two coefficients are
finite, and all the rest = 0. For m = 4, m = 5, there
are three terms finite, and all the rest equal nothing ;
and in general, if m be an even integer, the number of
m + 1
finite terms is + + 1, and if it be odd,
But if m be a fraction, the series never terminates,
and the coefficients only continue finite as long as
the exponent of 0 which occurs in them is not nega-
tive. After this happens, all the succeeding coefficients
are infinite. Thus, if m be a fraction between 0 and 1,
the first coefficient alone is finite, and all the rest infi-
mite. If m be between 1 and 2, the first two coeffi-
cients are finite, and all the rest infinite, and so on.
If m be a fraction between m – l and m, the first m
terms are finite, and all the rest infinite. The series,
therefore, in these cases is useless and absurd, and the
same happens when m is negative. From whence we
may conclude, that the developement of the cosine of a
multiple arc in descending powers of that of the simple
arc is never possible, except when the coefficient of the
multiple is a positive integer; and in this case, since
the number of terms is finite, the series is nothing more
than the series already obtained in ascending powers,
the order of the terms being reversed. So that in effect,
the only case in which the developement by descending
powers is possible, it is useless.
It is worthy of remark, that in the analytical expres-
sion for the coefficients A, B, # C, &c. if the powers
0", 0"--, 0”, &c. be neglected, the coefficients will be
exactly those of the series [21], which has been hither-
to considered exact. Whence may be seen the reason
why this series gives false values for cos m a., and also
why, in the particular case in which m is an integer,
the value resulting from it will be exact, if we retain
in it only the positive powers of p, for that is, in effect,
rejecting all that part of the true developement which
becomes = O.
(452.) The series for cos m. a. in descending powers
of cos a or p, m, being supposed to be an integer, is
therefore
(m. – 3)
2 cos m. a = (2p)"—m (2 p)"-s + 777, 1 .. 2 (2 p)"--
wºmma 4 -
- m (m tºº. 2–G py-
m. – 5 — 6) (ºn – 7
| m (m **** ) (2 p)"-" — . . . . [22].
(453.) To define the law of this series, let the rº
term be T,
T a m(m-r)(m-ſ-1)... (m-2r-4)(m-2-3)
*=w 1 . 2. 3. . . . . . . . . . . . ... (r-1)
(2 p)"-sº-1).
If m be even, let + + 1 be substituted for r, and
the result is
m (; – )(; -2)....s.º.
2 = + - 777, 7??,
. 2. 3. . . . ſ - – 1 || –
1, 2 .. 3 2 );
All the factors of the numerator, except the first,
destroy all the factors of the denominator, except the
last, and therefore
(2 y)"-"
2 = + 2,
+ being taken when # + 1 is odd, and — when
: –– 1 is even
1
If m be odd, let *† be substituted for m, and
the result is
m – 1 7m — 3
777, (**) (**) tº tº º ſº tº e g e 3 ſº 2 o l
2 = + 2 v).
777 – 3 m – 1 (2 y)
2 2
The factors of the denominator destroying those of
the numerator, except the first, we obtain
z = + 2 m y,
I
–H being taken when 7??, is odd, and — when it is
€ Ven.
(454.) To develope sin m x in descending powers of
COS X. -
To effect this, it is only necessary to differentiate the
series [22]. This being done, and the result divided
by 2 m, and observing that d y = d cos a = — sin
a day, we obtain
sin m a. m - I L tºº m = 3
sin. F (2 y) (m. – 2) (2 y)
(m. – 3) (m. – 4) (2 y)"-" — . . . . [23].
–– I . 2
This developement, like the last, is only possible
when m is an integer.
When m is an even integer, the number of terms in
the series for 2 cos m a being + + 1, and the last
2 = + 2,
it follows, since d z = 0, that in the present case the
term
7??,
number of terms must be T2 .
The rºh term in the present case is evidently
-E (m-r) (m—r—1) ... (m- 2 r + 3) (m –2 r + 2)
1 - 2 - 3 . . . . . . . . . . . . . .
(2 y)"-(2-1).
Hence the last term, m being an even integer, may
7 – 1
be found by substituting 4; for r in this formula,
628
A L G E B R A
Algebra. which gives
\-N-7 #(; – )........ 3.2
z = -E 770, (2 y).
2. 3. . . . . . . . tº – – 1
+-)
“. . 2 = + 7m gy.
If m be odd, the last term in the series for 2 cos m a.
being + m (2 y), that of the series for *:::: is
z = + 1,
m –– 1
the number of terms being
, and + being taken
when this is odd, and — when it is even.
(455.) To develope the cosine and sine of a multiple
arc in descending powers of the sine of the simple arc.
In [22] and [23] let a be changed into # — ar, and
the two series being expressed by M and M', and p
being understood to express sin a instead of cos ar, we
shall have
* cosm (; -- = M,
7.
SIIl 111. (; — a = cos a . M'.
In this case, as in the former, m must be an integer.
If m be even,
7-
cosm (; — a = + cos m a.,
7- e
in m(;- a ) = HE sin m a.,
+ being taken when 3 m is even, and — when odd.
Hence, in these cases,
2 cos m. a = + M,
2 sin m w = -F cos a . M'.
If m be odd
COS m(; — a 1 = -E sin m a.,
7- -
in m (; – )= + cos m r,
m – 1
-- being used if
Hence
be even, and — if odd.
sin m a = + M,
cos ma = + cosa. M'.
SECTION XLII
Of the Developement of a Power of the Sine or Cosine
of an Arc in a Series of Sines or Cosines of its
Multiples.
(456.) To develope cos" x in a series of cosines or
sines of multiples of x.
We have (Trigonometry)
2 cos r = e^* + e--",
??! ~ ,-, ...??! vi. –V-TR m
... 2" cos" w = (e' '-i –– e’ )".
If this be developed by the binomial theorem,
obtain
2" cos" r = e^*Y-i-H. A. &m-s)*V=ſ+ B gn-ov- + . . . .
1, A, B, C, . . . .
are the coefficients of the binomial series.
Eliminating e by the general formula,
We
where
cos m. a + v - I sin m r = ensv-1,
we obtain
2” cos” a = cos m.a. -- A cos (m — 2) r
+ B cos (m — 4) a + . . . .
+ V - I [sin m r + A sin (m – 2) r
+ B sin (m. – 4) a + . . . . ].
Let the first series be P., and the second Q, , and we
have
(2 cosa)" = P. -- v — 1 Q..
Let cos aſ be first supposed to be positive, and in
that case (2 cos ar)" must have at least one real value.
Let this be X, and all its other values will be found by
multiplying X by the values of (1)". They are, there-
fore, all expressed by the formula
X (cos 2 m n ºr + V - I sin 2 m n tr),
m being an integer not exceeding the denominator
of m.
Also, in
(2 cos r)" = P. -- v-i Q.,
no change is made in the first member by changing a
into 2 m t + æ, and therefore
(2 cos a)" = Pan-E. H. w/TT Q, n-4-
Hence
x cos 2 m n r + V – 1 . x sin 2 m n r = P, n.4,
+ V-I Q...+..... [1].
Equating the real and imaginary parts of this equa-
tion, we find
I * = ºl I
cos 2 m n ºr ******
sin 2 m m 19.42. . [2]
Hence it appears that the real and positive value X
of (2 cosa)” can be indifferently expressed, either in a
series of powers of the cosines or sines of the mul-
tiples of a, and that the two series differ from one
another only in the constant coefficients.
Between the two series thus found, there subsists a
constant relation,
cos 2 m 'm 7t Pan-4-
3.
sin 2 m n ir 93.4.
by which it appears that these series have a constant
ratio, whatever be the value ascribed to a, for 0 to
7-
TET
If n = 0, we obtain by [2]
X = P, , ,
Series for
Sines, &c. of
Multiple
Arcs.
S-N-2
A L G E B R A. 629
Algebra. which is therefore perfectly general, provided a be
supposed less than #, and X confined to the real and
positive value of (2 cos r)".
The second formula of [2] gives sin 2 m n n = 0, . . .
0
X = +.
This fails in giving any value of X, but shows that
Q. - 0 for all values of a from 0 to + # &
(457.) If the cos a be negative, let (2 cos w)" be
expressed thus,
(–2 cos ar)” = (2 cos r)” ( – 1)" – X ( – 1)”.
But since
(–1)" = cosm (2n + 1) ºr + v - I sin m (2n + 1) ºr.
Hence
X eos m (2 m + 1) ºr + V – 1 X sin m (2 m + 1) ºr
* r an r-i.a. + w/ — l Q, n. 42.
By equating the real and imaginary parts, we find
l
X = Pan. ... . . . . [3],
cosm (2 m III) ºr ***** [3]
l
-: . . . . [4].
sin m (2 m + D. Q.--4- [4]
In which the integer n is susceptible of any value from
0 to the denominator of m.
If m = 0, we have
l l
X = — P, , X = ---- Q.,
COS 777, 7. sin m ºr
which give developements of the real value of (2 cosa.)"
when cos a is negative.
From this it appears, that Q, is not = 0, as in the
former case, where cos a was supposed positive.
But although Qama. 4., may not = 0 when m = 0,
yet there may be some other value of n, which will
render this series = 0. To discover this, let it be de-
termined what value of m will satisfy the condition,
sin m (2 m + 1) T = 0,
‘. . cosm (2 m + 1) ºr = + 1.
That these conditions be fulfilled, it is necessary that
f
m (2 m + 1) be an integer. Let m = #, and let l
be any integer, '.'
m! (2 m + 1) = I m';
but m' being prime to m' measures I. Let 7 = i, ‘I’
2 m + 1 = i n'.
Since 2 m + 1 is odd, both i and n' must be odd.
But since n is supposed not to exceed n', 7 must
be = 1.
Hence
7 n' – 1
l, E —,
2 -
which is therefore the only value of n which can satisfy
the proposed condition. *- - .
WOL. I.
Hence, if m be a fraction with an odd denominator Series for
f Sines, &c. of
m"), we have ; :
( ), Multiple
X = -E PGº-)-4--, Qo'-1)-4-> = 0, Arcs.
S-N-'
-j- being used when m' is even, and – when odd.
But if m be a fraction with an even denominator,
there is no arc (2 m + 1) ºr which can render cos m.
(2 m + 1) m = + 1 ; and, consequently, no arc 2 m
m ºr -j- a for which the series.P. can become equal to the
real value of (2 cos r)". -
By the formulae [3], [4], it follows that when cos w is
negative, the real and positive value of (2 cos a)" may
be expressed either in a series of sines or cosines of the
multiples of a, and that the two developements differ
only in the coefficients; and, finally, that their ratio is
3 ºr
the same for all values of a between # and ~5-.
(458.) If m be a positive integer Q. = 0, and we
have -
(2 cosa)” = P.
The number of terms in P, is m + 1, being those of
the binomial series. Hence the last term must be
cos (m — 2 m) a = cos m ar,
which is equal to the first. And, in like manner, the
penultimate term is equal to the second, and every pair
of terms equidistant from the extremes are equal.
It follows, therefore, that when m is odd, and . m + 1
even, the first half of the series S is equal to 4 (2" cos"
ac) = 2*T* cos" a ; and when m is even, and therefore
m + 1 odd, the first + terms together with half the
t
+ –– 1) h. term is equal to 2"-" cos" a.
Hence we conclude,
1. When m is odd,
2*** cos" a = cos m. a. -- A cos (m. – 2) r
+ B cos (m – 4) a + . . . . . .
m + 1
2
term S.
continued to
The last term of this series is
M co-ſn-2(++- 1)] = M coºr M being
2
7??,
th.
the coefficient of the ( ) term of an expanded
1
2
binomial. From the law of the binomial series we
have
7m — 3
7m . 7m — 1 . 7m — 2 . . . m — –5–
*=T. m – l
• Wºº e s sº e e º s m s e e s is e 2
This may, however, be reduced to a somewhat
simpler form. Let both terms of the fraction be mul-
ºn-l
tiplied by 23 , the operation being effected on the
denominator by doubling each of its factors; the
result is
4 N
630
A L G E B F A.
Algebra.
S-N--
Again, multiplying both numerator and denominator
by the odd integers from 1 to m inclusive, in order to
complete the series of factors in the denominator,
7m — 3
m.m.-1. n-2.... (m "n-1
2 * (13.5..m.)
772. -
*
Expunging from both numerator and denominator
the descending factors from m to m — inclusive,
we obtain
?m – 1
M = * : * * s & e º 'º E *H*.
777,
1. 2. 3. . . . . .
2
Hence the last term z is
, - 1.3.5 . . . . . . 772. gºo.
1.2. 3...... m + 1
2
2. If m be even,
2*** cos” a = cos m a -i- A cos (m — 2) a + . . . . . .
77? º
continued to T2- + 1 terms, the coefficient of the last
th
term being half that of the (+ + 1) * term of the
expanded binomial. Let z be the last term,
2 = 4 M cos (m-m) w = }; M,
m . m – 1 . m — 2 . . . . (m. – + + 1)
M =
1.2.3 ºn
e ºf e º a sº as a s s e º e º e º e s ∈ e º º ºs 2
Multiplying both numerator and denominator by 2”
in the same manner as in the last case, and intro-
ducing the deficient factors 1.3.5 . . . . m – 1, we
obtain
#72,
m . 7m — 1. ºn —2 ... (m-g + 1)
º
M = 2*(1.3.5.m.–1)
Expunging from the numerator and denominator the
descending factors from m to (m — # + 1) inclu-
sive, we obtain
a l. 3. 5. . . . (m-l) oft
tº- 22
1. 2. 3. . . . . . . . m .
2
Arºn 1.3.5 .... (n-1) gº-
777, 2
I . 2, 3 as-s-s
2
which is the value of the last term.
(459.) The developement which has been thus ob- Series for
tained, gives the value of the mºº power of the cosine Sines, &c of
of an arc in a series of cosines or sines of its multiples. Multiple
Similar series for the mº power of the sine may be *
obtained in a similar way.
By expanding
(2 sing)" (VTI)" = (e."-" — e--"F)",
and eliminating e by the formula,

cos m r + V - I sin m r = et"****
we obtain
(2 sin ry" (v- 1)" = cos m. a. — A cos (m — 2) a
+ B cos (m — 4) a - . . . . -- W - I
[sin m r - A sin (m. – 2) a + B sin (m. – 4) r — . . ].
If the series be called P, and Q., we have
(2 sin a Y” ( — 1); = P + V-1 Q..
This formula being treated in a manner similar to that
for (2 cosa)”, will give similar results.
(460.) If m be a positive integer, the number of terms
in each of the series P, and Q, will be m + 1, and
one or other of them will = 0. We shall consider
successively the cases in which m is even and odd.
1. Let m be even.
The number of terms in Q, being (m. -- 1) and '.'
odd, the sign of the last term is by the law of the
series +, and it is therefore -
+ sin (ºn – 2 m) a = — sin m a.
The penultimate term is
— A sin [m –2 (m-1)] a = — A sin (– m + 2) r
= -- A sin (m–2) ar,
and by continuing the process, it appears that the
extreme terms, and those equally distant from them,
destroy each other. Hence Q. = 0, and therefore
2” (w/ – 1)" sin" a = P.
But since m is even,
(— 1)* :- + 1,
& 771, .
+ being taken when g- is even, and — when odd.
Therefore
+ 2" sin" a = P. .
In the same manner as in the former case, it follows
that in the series P, the extreme terms, and those
which are equidistant from them, are equal, and have
the same sign, and hence, as before, we find
+ 2*-* sin” w = P.,
the number of terms being 4. + 1, and the last term
being the same as for 2"-1 cos” a when m is even.
2. Let m be an odd integer.
In this case the number of terms being m + 1, the
sign of the last term of P, is by the law of the series –,
and it is therefore
— cos (m – 2 m) r = — cos m a.
A L G E B R A. 631
Algebra, and the penultimate term is
-- A cos [m-2 (m-1)] + = + A cos (m-2) w,
and by continuing the process, it appears that the ex-
treme terms, and those which are equidistant from
them, are equal with different signs, and therefore
destroy each other. Hence P. = 0, and
2” (w/ - i)" sin" w = V – I Q.
... 2" (w/ - i)"— sin” a = Q.
But since m – l is even,
(M-T)"- = + 1,
•." + Q"sin" r = Q,
g 7m — 1 Series for
+ being taken when −5– is even, and — when it Sines, &c. of
º 2 Multiple
is odd. Arcs.
In the same manner as before, it may be shown that `
the extreme terms of Q, are equal and have the same
sign. Hence we find
2"-1 sin” w = Q,
º 7m –– 1 g
continued to +++ terms, the last term being
3.5 . . . . * .
2 = ** *H2" in r.
1, 2.3 . . . . –3–
4 N 2
Geome-
trical
Analysis.
Introduc-
tion. -
Ancient
analysis
compared
with the
modern.
GEO METRICAL AN A LY SIS.
- sº ºn-
SECTION I.
(1.) ANALYSIs, or resolution, is a process by which,
commencing with what is sought as if it were given, a
chain of relations is pursued which terminates in what
is given, (or may be obtained,) as if it were sought.
SYNTHESIs, or composition, is a process the very reverse
of this ; being one in which the series of relations ex-
hibited commences with what is given, and ends with
what is sought. Consequently analysis is the instru-
ment of invention, and synthesis that of instruction.
The analysis of the ancients is distinguished from
that of the moderns by being conducted without the aid
of any calculus, or the use of any principles except
those of Geometry, the latter being conducted entirely
by the language and principles of Algebra. The an-
cient is, therefore, called the Geometrical Analysis.
For its origin and history, the reader is referred to our
HISTORY OF ANALYSIS.
The interest which the Geometrical Analysis derives
from its antiquity, and from having been the instrument
by which the splendid results of the ancient Geometry
were obtained, would alone be sufficient to render it an
object of attention even after the discovery of the more
powerful agency of Algebra. But this is not its only
nor its principal claim upon our notice. Its inferiority,
compared with the modern analysis, in power and
facility, is balanced by its extreme purity and rigour;
and though its value as an instrument of discovery be
lost, yet it must ever be considered as a most useful
exercise for the mind of a student ; and it may be
fairly questioned, whether it may not be more conducive
to the improvement of the mental faculties than the
modern analysis, unless the latter be pursued much
farther than it usually is in the common course of
academical education, in which the student acquires
little more than a knowledge of its notation. Newton
was fully aware of the advantages attending the culti-
vation of this branch of mathematical science, and in
many parts of his works laments that the study of it
has been so much abandoned. He considered, that,
however inferior in power and despatch the ancient
method might be, it had greatly the advantage in
rigour and purity; and he feared, that by the premature
and too frequent use of the modern analysis the mind
would become debilitated and the taste vitiated. We
must however confess, that the pretensions of the
ancient method to superior rigour do not seem to us to
be as well founded as they are sometimes considered.
It would be no very difficult matter to expunge the
algebraical symbols from a modern investigation, and
substitute for them their meaning expressed in the
language used in geometrical investigations; but
would such a change confer upon them greater rigour,
or would it give to the conclusions greater validity?
And yet this is precisely what Newton himself has done
632
in many parts of his great work, the Principia. His
theorems are, evidently, investigated algebraically ; but
in demonstrating them, the process is disguised by the
substitution of lines and geometrical figures for the
algebraical species and formulae. It cannot but excite
astonishment, that a man of his extraordinary sagacity
could so far deceive himself, as to suppose that by such
a proceeding his reasoning acquired greater rigour.
But, without reference to the modern analysis, we
conceive that the ancient method has sufficient claims
to our attention on the score of its own intrinsic
beauty. It has this further advantage, that we can
enter at once upon its most interesting discussions
without the repelling task of learning any new lan-
guage or system of notation.
In the application of the Geometrical Analysis to the
solution of problems, or the demonstration of theorems,
no general rules nor invariable directions can be given
which will apply in all cases. The previous construc.
tion to be used, and the preparatory steps to be taken,
depend on the particular circumstances of the question,
and must be determined by the sagacity of the analyst;
and his skill and taste will be evinced in the selection
of the properties or affections of the given or sought
quantities on which he founds his analysis; for the same
question may frequently be investigated in many dif-
ferent ways.
In submitting a problem to analysis, its solution, in
the first instance, is assumed ; and from this assump-
tion a series of consequences are drawn, until at
length something is found which may be done by esta-
blished principles, and which if done will necessarily
lead to the execution of what is required in the problem.
Such is the analysis. In the synthesis, then, or the
solution, we retrace our steps; beginning by the execu-
tion of the construction indicated by the final result
of the analysis, and ending with the performance of
what is required in the problem, and which constituted
the first step of the analysis.
When a theorem is submitted to analysis, the thing
to be determined is, whether the statement expressed by
it be true or not. In the analysis this statement is, in
the first instance, assumed to be true; and a series of
consequences are deduced from it until some result is
obtained, which either is an established or admitted
truth, or contradicts an established or admitted truth.
If the final result be an established truth, the theorem
proposed may be proved by retracing the steps of the
investigation, commencing with that final result, and
concluding with the proposed theorem. But if the
final result contradict an established truth, the proposed
theorem must be false, since it leads to a false con-
clusion.
These general observations on the nature of the
Geometrical Analysis, and the methods of proceeding
in it, will be more clearly apprehended after the inves-
Section I.
S-N-2
No general
rules in
Geometrical
Analysis.
Analysis of
a problem.
Of a
theoretn.
G E O M ET. R. I C A L A N A. L. Y S I S. 633
Geome- tigations contained in the subjoined treatise have been Produce the line B P beyond P, until P E is equal Section II.
trical examined. to PA, and join AE. The angles B P D and E PC S-
Analysis are equal; but also (hyp.) B P D and A PC are also
--~~~~ equal, therefore the angle A PC is equal to the angle
E PC. But also the sides PA and PE are equal, and
SECTION II. the side PF is common to the triangles A PF and
g EP F. Therefore the angles A F P and E F P are
Miscellaneous Problems. equal, and therefore are right angles, and also the
(2.) Definition. A point is said to be given when its A F is equal tº PF: . g
situation º either gº or may be jºi. But since A and CD are given, the perpendicular
(3.) Definition. A right line is said to be given in A F is given, and hence the solution of the problem
position when it is either actually exhibited and drawn, "*. be derived. tº e s
or may be exhibited and drawn by previously established From either of the given points. A draw a perpendi.
principles. cular A F to the given right line CD, and produce it
through F, until FE is equal to A F, and draw the
PRGPosition. right line E B meeting the line C D in P. Draw AP,
and the lines A P and B P are those which are required.
(4.) To draw from a given point a right line inter- For since A F and FE are equal, and P F common to
secting two right lines given in position, so that the the triangles A FP and E FP, and the angles A FP
segments between the point and the right lines shall have and E FP are equal, the angles A PF and EP F are
a given ratio. equal. But B P D and E PF are also equal, therefore
fior tº g g the angles A PF and BPD are equal.
Fig, 1. hº i..." º . A ºr...ish Scholium. If the given points lie at different sides of
Let PM : P N :: m : m. If any other line as PL the given right line, the problem IS solved by merely
be drawn intersecting AB and CD, and a parallel to joining the Points. -
C D be drawn from N, that parallel will divide PL
similarly to PM, and therefore in the required ratio. PRoPos ITION.
This parallel may, or may not, coincide with the line (8.) To inscribe a square in a triangle.
N. K. First, let us suppose that it does. In that case Let A B C be the triangle, and D F E the required Fig. 3
the two lines given in position will be parallel, and the square. Draw the perpendicular B G, and draw. A E § - d -
line PL, or any other line, drawn intersecting, them, to meet a parallel B H to A C at H. It is easy to
will be cut similarly to PM, and therefore all such lines see that D'F. FET, . G. B. B. H.; for the triangles
will be cut in the required ratio. Hence it appears, AFI) and A Bo AFE and A B'H are respectively
that in this case the problem is indeterminate, since similar each to each. Hence, since D F is equal to
every line which can be drawn intersecting the given FE, G B is also equal to B H. But GB is given in
lines will equally solve it. magnitude and position, and therefore B H is given in
Secondly,if the given lines A.B. 92.9° nºt Parallel, magnitude and position. To solve the problem there.
let the parallel to CD from N meet P L in O, so that fjit is only necessary to draw B H and join A H, and
PL: PO :: n : n. But PL may be drawn, and the the point E where A H meets B C will be the vertex of
point o therefore may be determined ; and since the the angle of the square.
direction of CD is given, the direction of ON is (9.) Cor. 1. It is evident that the same analysis will
determined, and therefore the point N may be found. solve the more general problem, “To inscribe in a
Hence, the solution is as follows: let any line P L be triangle a rectangle given in species.” For in this case
drawn. If PL : P K . . m : n, the problem ls solved. the ratio B H : B G is given, and therefore B H is as
If not, let P L be cut at O, so that P L : P o . . .”.; ſº before given in position and magnitude.
and from O draw O N parallel to CD, meeting A B in (10.) Schol. If B H be drawn equal to B G and on Fig. 4.
N, and through N draw PN M. Then PM : P N : : the same side of the vertex with A, then it will be
Pº PO : : an : m. e º º ... necessary to produce A H and CB, in order to obtain
A ſººn will apply if the line their point of intersection E. In this case, however,
© D FE will still be a square, for the corresponding
(6.) Cor. 2. If the parallel to C D through O do not triangles will be similar † to F D A ..". B Å
meet the line A B, the solution is impossible. If A B to Eis A." Hence G B E H . D F : FE.
be a right line, this happens when it is parallel to C D. (11.) Cor. 2. In the same manner the more general
And therefore we conclude in general, that when the problem, “To inscribe a rectangle given in species 5 y
two right lines A B and C D are parallel, the problem may be extended 2.
is either indeterminate or impossible. o
PROPos ITION.
PRoPosition. -
sº *g * (12.) To draw a line from the verter of a given
(7.) From two given points to draw to the same point triangle to the base, so that it will be a mean propor-
in a right line given in position, two lines equally tional between the segments. -
inclined to it. Let A B C be the triangle, and let B D be a mean Fig. 5.
Fig. 2 Let the given points be A and B, and let C D be the proportional between A D and D C. Produce B D to
line given in position. Let P be the sought point, so
that the angle A P C shall be equal to the angle
B P D.
E, so that D E shall be equal to B D, and join C E.
Since
A D : B D : : E D : D C,
634
G E O M ET. R. I C A L A N A L Y S I S.
Geome-
trical
Analysis.
S-N-a-’
Fig. 6.
and the angles B D A and E D C are equal, the trian-
gles B D A and C D E are similar. Therefore the
angles E and A are equal, and are in the same segment
of a circle described on C B. If from the centre of this
circle F D be drawn, the angle F D B will be a right
angle, and the point F will therefore be in a circle de-
scribed on F B as diameter. But the point F is given,
since it is the centre of a circle circumscribed about
the given triangle, and the line F B is therefore given,
and the circle on it is as diameter is given, and therefore
the point D is given. The solution of the problem is
therefore effected by circumscribing a circle about the
given triangle, and drawing from its centre to the angle
B a radius. On that radius, as diameter, describe a
circle ; and to a point D, where this circle meets the
base, draw the line B D, and it will be a mean propor-
tional between the segments. For the angle B D F
in a semicircle is right, therefore B D = DE ; and
therefore the square of B D is equal to the rectangle
under A.D and D C.
If the circle on B F intersect A C, there will be two
points in the base to which a line may be drawn, which
will be a mean proportional between the segments. If
this circle touch the base there will be but one such
line, and it may happen that the circle may not meet
the base at all, in which case the solution is im-
possible.
If the centre F be upon the base AC, the angle ABC
will be right, and the point F itself is one of the points
which solve the problem ; for in that case A F, B F,
and C F are equal. The other point D is the foot of a
perpendicular B D from the vertex on the base.
(13.) Cor. Hence, in a right angled triangle, the per-
pendicular on the hypothemuse is a mean proportional
between the segments ; and it is the only line which
can be drawn from the right angle to the hypothemuse
which is a mean, except the bisector of the hypo-
thenuse.
Schol. It has been observed, that the solution of the
problem to draw a line to the base which shall be a
mean proportional between the segments is impossible
when the vertical angle is acute. That this is erro-
neous, must be evident from the preceding analysis. For
let one circle be described upon the radius of another
as diameter. Let any line, as A C, be drawn not
passing through F, but intersecting the inner circle;
and so that the point of contact B and the centre F
shall lie at the same side of it. Draw. A B and C B,
and also B D. It is evident that B D is a mean pro-
portional between A D and CD, and yet the angle
A B C is acute, being in a segment greater than a
semicircle.
The possibility of the solution of this problem does
not at all depend on the magnitude of the vertical
angle. It may be obtuse, right, or acute, and may be
equal in fact to any given angle, and yet the solution
be possible.
Let it be required to determine the conditions on
which the solution is possible. If the circle on B F
meet the base, the perpendicular distance of its centre.
from the base must be less than its radius ; that is,
less than half the radius of the circle which circum-
scribes the given triangle. From F and B draw per-
pendiculars FI and B H on A C, and from the centre
of the lesser circle G draw the perpendicular G. K.
Since G F is equal to G B, G. K is equal to half the
sum of FI and B H. Hence it follows, that the solu-
*
l
tion will only be possible when half the sum of F I Section II.
and B H is not greater than B G, or when the sum of S-v-
FI and B H is not greater than B F ; that is, when the
sum of the perpendiculars on the base from the vertex
and the centre of the circumscribed circle is not greater
than the radius of that circle.
PROPOSITION.
(14.) Right lines being drawn bisecting the internal
and external angles of a triangle, and being produced
to meet the base, and the production of the base to deter-
mine the conditions on which the rectangle under the
sides of the triangle will be a geometric, arithmetic,
or harmonic mean between the rectangle under the
segments of the base by the internal bisector, and the
rectangle under the segments of the base by the external
bisector.
Let A B C be the triangle, B D the bisector of the Fig. 8.
internal angle, and B E the bisector of the external
angle. By the principles of Geometry we have
A E : C E : : A B : B C,
also A D : D C : : A B : B C.
Hence it follows, that the three rectangles A E x C E,
A B × B C, A D X D C are similar.
1. Let the rectangle under A B and B C be a geo
metric mean between the other two. If three similar
figures be in geometrical progression, their homologous
sides must also be in geometrical progression ; hence
C E : C B : C D. But since the angle D B E is equal
to A B D and E B F together, it is a right angle, and
therefore since B C is a mean proportional between
D C and C E, B C A must be a right angle, (12.)
Hence the rectangle under the sides is a geometric
mean, when either of the base angles is right.
2. Let the rectangle A B × B C be an arithmetic
mean between the other two. In that case the rectan-
gle A E × E. C should exceed A B × B C by as much
as this last exceeds A D x DC. But by Geometry
the excess of AEx EC above A B × B C is the square
of B E, and the excess of A B x B C above A D x B.C
is the square of B. D. Hence in the present instance
the squares of B E and B D are equal, and therefore
the lines themselves are equal. Hence the angles
B D C and B E C are equal, and since D B E is a right
angle, B D C must be half a right angle, and therefore
the difference between B D C and B D A is a right
angle. But since by adding to each of the base angles
BAD and B C D the equal halves of the vertical angles,
we obtain sums equal to the angles B D C and B D A,
it follows that the difference between the base angles
B C D and B A D is a right angle. Hence when the
difference of the base angles is right, the rectangle
A B × B C is an arithmetic mean between the other
two rectangles.
3. Let the rectangle A B × B C be an harmonic
mean. In that case, by the nature of harnionic pro-
portion, we have
A. E. × E. C : A D x D C . . A E x E C – A B x B C
: A B x B C — A D x DC;
that is, the first rectangle is to the third as the difference
between the first and second is to the difference
between the second and third. But these differences
are the squares of the lines B E and B D, and therefore
G E O M ET. R. I C A L A N A L Y S I S.
635
Geome-
trical
Analysis.
-V--"
Fig 10.
we have the rectangle A E x EC to the rectangle
A D x D C as the square of B E is to the square of
B D. Since then the similar rectangles of which CE
and C D are homologous sides, are proportional to the
squares of B E and B D, these lines themselves are
proportional. Therefore
B E : B D : : C E : C D.
Hence the line B C bisects the angle D B E ; but
since D B E is right, C B D is half a right angle, and
therefore A B C is a right angle. Hence if the bisected
angle be right, the rectangle A B x B C is an harmonic
imean between the other two rectangles.
PROPosſTION.
(15.) To draw a right line from the verter of a
triangle to the base, or to the base produced, so that its
square shall be equal to the difference between the rec-
tangle under the sides, and the rectangle under the seg-
ments into which it divides the base.
Let the triangle be A B C, and let the required line
be B D, and let a circle be circumscribed about the
triangle. *.
1. Let the line be drawn to the base itself, and let
it be produced to meet the opposite circumference at E,
and draw C E. By hypothesis, the square of B D,
together with the rectangle A D x DC, is equal to the
rectangle A B × B C. But the rectangle A D x D C is
equal to the rectangle B D × D E. Add to both the
square of B D ; and the rectangle A D x DC, together
with the square of BD, is equal to the rectangle B D x
D E, together with the square of B. D. But the
former rectangle and square are together equal to the
rectangle A B x B C, and the latter rectangle and square
are together equal to the rectangle BE x B. D. Hence
the rectangle A B × B C is equal to the rectangle B E
× B D. Hence we have
AB : B D : : E B : B C ;
and the angles A and E are equal. Therefore in the
triangles A B D and E B C the sides A B, B D are pro-
portional to E B, B C, and the angles opposite to one
pair of homologous sides B D and B C are equal, and
therefore the angles opposite the other pair of homo-
logous sides must be either equal or supplemental. If
they be equal, the triangles A B D and E B C are
similar, and therefore the line B D bisects the angle
A.B.C. If the angles B D A and B C E be supple-
mental, the sum of the arcs which they subtend must
be equal to the whole circumference. Hence the arcs
BA, A E, BA, and C E are together equal to the
circumference. But B A, A E, B C, and C E are also
together equal to the circumference. Take away from
both the arcs B.A, A E, and CE, and the remaining
arcs B C and BA are equal; and therefore their chords
are equal, and therefore the triangle is isosceles.
Hence we infer, that “ if a line be drawn from the
vertex of a triangle to the base, so that its square,
together with the rectangle under the segments, shall
be equal to the rectangle under the sides, that line will
bisect the vertical angle, except when the triangle is
isosceles, in which case any line drawn from the vertex
to the base will have the required property.”
2. Let the line B D meet the base produced. By
hypothesis, the rectangle A D x D C is equal to the
rectangle A B × B C, together with the square of B D.
But the rectangle A D x D C is equal to the rectangle
ED × B D, which is equal to the rectangle E.B. × B D,
together with the square of B. D. From these equals
take away the square of B D and the remainders, the
rectangles E B × B D and A. B. × B C are equal.
Hence -
E B : B C : : A B : B D.
Draw CE, and the angles E and A are equal. Hence
in the triangles E B C and A B D there are two sides
E B and B C proportional to two A B, BD, and the
angles opposite one pair of homologous sides equal,
and therefore the angles opposite to the other homolo-
gous sides must be either equal or supplemental. If
they be equal, take A B C from both, and the re-
mainders E B A and C B D are equal ; but E B A and
F B D are also equal, and therefore B D bisects the
external angle C B F of the given triangle.
If the angles A B D and E B C be supplemental.
Since the angles A B D and FBD are also supplemental,
we should have the angles FB D and E B C equal ; and
therefore E B A and E B C equal; and therefore the
point B cannot in this case lie between E and D. It
must therefore be placed as in fig. 11.
square of B D is manifestly greater than the rectangle
CD x D A, and therefore the proposed condition must
be that the rectangle C D x D A, together with the
rectangle A B x B C, is equal to the square of B.D.
But the rectangle C D x D A is equal to the rectangle
B D x D E ; and taking these equals from the former,
the remainders, viz. the rectangles A B × B C and
B D x B E are equal. Hence
E B : B C : : A B : B D.
Draw CE, and in the triangles E B C and A B D the
two sides E B, B C are proportional to two A B, BD,
and the angles B E C and B A D opposite to one pair
of homologous sides are supplemental, (for B A C
and B E C are equal,) and therefore the angles B C E
and D opposite the other pair of homologous sides are
equal. Hence the difference of the arcs subtended by
D is equal to the are subtended by B C E, that is, the
difference between the arcs B C and A E is equal to the
arc B E ; or the arcs B E and A E together, that is, the
arc A E B is equal to the arc AB, and therefore their
chords are equal, but their chords are the sides A B, BC
of the triangle, which is therefore isosceles. k
Hence it follows, that “if a line be drawn from the
vertex of a triangle to the produced base, so that its
square, together with the rectangle under the sides,
shall equal the rectangle under the segments of the
base, that line will bisect the vertical angle, except when
the given triangle is isosceles, in which case there is
no line which has the required property. In this case,
however, the square of every line drawn from the vertex
to the produced base is equal to the sum of the rectan-
gles under the sides and segments.”
SECTION III.
Of the Contact of Right Lines and Circles.
(16.) PROBLEMs of contact of right lines and circles
furnished the ancients with an extensive subject for
the exercise of the Geometrical Analysis. In general
Section II.
Section III.
-N2-’
Here the Fig. 11.
636
G E O M ET. R. I. C. A. L. A. N. A. L. Y S I S.
Geome- three conditions are necessary to determine a circle. In
Analysis.
trical
'ig. 13.
the class of problems to which we allude, one at least
of these conditions is, that it should touch a given right
line or a given circle. The other data may be, that it
should pass through one or two given points, or that it
should have a given radius or centre, or that the locus
of its centre should be a given right line or circle. It
would not be easy to enumerate all the problems of this
class; but by combining the following data for the de-
termination of a circle, a considerable number of them
may be found.
To describe a circle
I. Passing through a given point.
2. Passing through two given points.
3. Passing through three given points.
4. Touching a given right line.
5. Touching two given right lines.
6. Touching three given right lines.
7. Touching a given circle. -
8. Touching two given circles.
9. Touching three given circles.
10. Having a radius given in magnitude.
11. Having its centre on a given right line.
12. Having its centre on a given circle.
13. Having a given centre.
Every combination of three which can be formed
from these data, may be taken as the limiting circum-
stances in problems for the determination of a circle.
In the invention of such problems it should however
be observed, that 2, 5, 8, and 13 are each to be counted
as two data, and 3, 6, 9 are each to be counted as
three data. Each of the latter is, therefore, itself
sufficient to determine the circle, but each of the for-
mer ought to be combined with some one of the data
1, 4, 7, 10, 11, 12.
We cannot here enter at large on this class of pro-
blems, we shall therefore confine ourselves to a few
examples.
PRoPosition.
(17.) To describe a circle passing through two given
points, and touching a right line given in position.
If the given points be at different sides of the given
line, the solution is manifestly impossible.
Let them then be A, B at the same side of the
given right line C D. Let the required circle be A B C,
and let A B be produced to meet the right line at D.
The square of C D is equal to the rectangle A D
× D B. But this rectangle is given, therefore the
square of C D is given, and therefore C D itself is
given in magnitude and position, and hence the point
C is given. But also the points A, B being given, the
circle through these points A, B, C is given.
The solution, therefore, is effected by producing A B
to D, and taking D C equal to a mean proportional
between A D and D B, and then describing a circle
through A, B, C.
But it may happen, that the line A B is parallel to
CD, and will not meet it when produced.
In this case draw A C and B C. The angle B CD
is equal to the angle A in the alternate segment, and
also equal to the alternate angle B. Hence the angles
A and B are equal, and therefore the sides A C and
B C are equal. Draw C E perpendicular to A B, and
A E and B E are equal. The point E is, therefore,
given, and the perpendicular EC is given in position,
and therefore the point C is given.
To solve the problem in this case therefore, bisect
A B at E, and draw the perpendicular through E, in-
tersecting CD in C. A circle passing through A, B, C
will be that which is required.
PROPOSITION.
(18.) To describe a circle passing through a given
point, and touching two right lines given in position.
1. Let the given right lines be parallel. In this case
it is necessary that the point should be between them,
for otherwise the solution would be impossible.
Let the lines be A B, CD, and the point be P. Let
A PC be the required circle, and draw A P and the
diameter A. C. Through P draw P P' parallel to the
given right lines, and describe any circle B P D, touch-
ing the right lines at B, D, and intersecting the parallel
at P', and draw P'B. Since the circle B P D may be
drawn, the point P' is given, and therefore the line
P' B is given in magnitude and position. But the
triangles B P/D and A PC are similar, and since
B D and A C are parallel, BP' and A P are parallel.
Therefore the line PA is given in direction, and since
the point P is given, it is also given in position. Hence
the given points A and C are given, and therefore the
circle A PC is given. -
To solve the problem therefore, describe any circle
touching the two lines, and draw the parallel through
P to meet it at P'. From P' draw P’ B, and draw PA
parallel to it. Draw A C perpendicular to A B, and it
will be the diameter of the required circle.
2. Let the given lines A B, C D intersect at E.
As before, describe any circle B P D touching the
right lines, and from E draw E P intersecting this
circle at P'. Draw the radii G A, G P, F B, and FP7.
Since G A is parallel to F B, we have
G. A : F B : : G E : F E.
Therefore G P : FPl: : G E : F. E.
Therefore G P : G E :: F P': F. E.
Hence the lines GP and FP' are parallel. But FP
is given in position, and therefore G P is given in
direction, but P is given, and therefore G P is given in
position. But the line E G bisects the angle A E C
under the given lines, and is therefore given in posi-
tion, and therefore the point G where it intersects PG
is given. Hence the centre G and the radius G P of
the required circle are given, and therefore the circle
itself is given.
To solve the problem, draw EP, and also E G,
bisecting the angle E. Describe any circle B P D
touching the given right lines, and draw P' F. Through
P draw P G parallel to P' F, meeting the bisector E G
in G. With G as centre and GP as radius, let a circle
be described. This circle will touch the right lines.
The demonstration is obvious.
It is evident, that in each of the preceding cases
there may be two circles drawn, which will solve the
problem. This circumstance arises from the line PP'
meeting the circle B P D in two points. The principle
used in the solution of both cases is the same. The
parallel in the first case corresponds to the bisector of
the angle in the second.
Section III.
S-N/~
Fig. 14.
Fig. 15.
G E O M ETR I C A L A N A L Y S I S.
637
Geome-
trical
Analysis.
Fig. 16
and 17.
Trisection
of an angle.
Fig. 18.
therefore F D is equal to twice A E, or to twice A. B. Section IV.
But AB is given, and therefore D F is given in mag- > --"
PRoPosition.
(19.) To describe a circle passing through two given
points, and touching a given circle.
Let A and B be the given points, and let C be the
centre, and C D the radius of the given circle. Let D
be the point of contact sought. Draw A DE, B D F,
and F.E. Also, let a tangent FG at F be drawn, and
from B draw B C I. -
By the properties of the circle it appears that A B
and FE are parallel, and therefore the angles A and E
are equal. But also the angle E is equal to the angle
G F B, and therefore G F B is equal to the angle A,
and therefore the triangles ABD and GFB are similar.
Hence we have
A B : B D : ; F B : B G.
Therefore the rectangle A B × B G is equal to the rec-
tangle B D x B. F. But also the rectangle B D × B F
is equal to the rectangle B I × B H. Hence the rec-
tangle A B × B G is equal to the rectangle B I × B H.
But since the point B and the circle C are given, the
rectangle B I × B H is given, and therefore the rec-
tangle A B × B G is given in magnitude.
But one side A B is given, and therefore also the
other side B G is given, hence the point G is given.
Hence the line G B is given in magnitude and posi-
tion, and the point of contact D where it intersects the
given circle is given. The circle through this point D,
and the given points A, B is therefore given.
The problem is therefore solved by taking B G, a
fourth proportion to A B, B I, and B H ; and from G
drawing the tangent GF, and from F the point of
contact drawing the line F.B. The point D where this
line intersects the given circle is the point where the
sought circle through A, B touches it.
ºmºmº
SECTION IV.
Trisection of the Angle.—Investigation of Two Mean
Proportionals.-Delian Problem.
PROPOSITION.
(20.) To trisect a given angle,
Let A R C be the given angle, and from any point
A in one leg draw a perpendicular A C to the other,
and from the same point A draw a parallel A D to the
other leg B C. Let B D be the line which cuts off the
angle C B D one third of the given angle A B C.
Hence the angle A D B, which is equal to D B C, is
one third of the given angle A B C, and the angle
A B D is two thirds of A B C, and therefore is double
the angle A D B •
Draw A E, making E A D equal to E D A, an
therefore A E is equal to DE, and the angle A E B is
equal to twice the angle A D B. Hence the angle
A E B is equal to the angle A B E, and A B is equal
to A E. But also since A FE together with A DE is
equal to a right angle, and also FAD is a right angle;
if from these equals the equal angles FDA and DAE
be taken, the remaining angles FA E and A FE will
be equal, and therefore A E is equal to E F, and
WOL. I.
nitude.
The problem to trisect an angle is therefore reduced
to the inflection of a line of given magnitude between
the legs of a right angle, and passing through a given
point. This is a problem not capable of solution by
the right line and circle.
The condition may also be reduced to the inflection
of a right line from a given point in the circumference
of a circle, so that the part intercepted between the
circle and a diameter produced passing through an-
other point shall have a given magnitude. For if with
the centre A and the radius A B or A E a circle be
described, it will be sufficient if from B a line B D be
inflected on A D, so that the external part D E shall
be equal to the radius. This condition is, in effect, the
same as the former.
PRoPositroN.
(21.) To trisect a given ratio, or to find two continued
mean proportionals between two lines. -
This, like the last, is a problem the solution of Trisection
We of a ratio.
which surpasses the powers of Plane Geometry.
can, however, investigate the conditions on which its
solution depends.
Let the terms of the ratio, expressed by lines, be Fig. 19.
placed at right angles, and the rectangle A C B D com-
pleted, let C E and B F on the produced sides of this
rectangle be the two means, so that
A C : C E : B F : A B. 1,8 %. &
By the similar triangles formed by the sides of the rec-
tangle we have --- - -
F D : D E : : A C : C E,
F D : D E : : C E : B F.
Hence the rectangle F D x B F is equal to DE x C E.
Let a circle be circumscribed round the rectangle inter-
secting FE in G. The rectangle D F x FB is equal
to iiie rectangle A F x FG, and the rectangle D E x
C E is equal to the rectangle E A x GE. But the rec-
tangles D F x FB and DE x C E have been proved
$3
therefore
equal, and therefore the rectangles A F x FG and ... .
GE x A E are also equal. But G A the difference
of the sides of these rectangles is common, and there-
fore the sides are respectively equal, viz. G E is equal
to A F, and FG is equal to A. E. -
Hence it follows that two mean proportionals will be
found, if through the point A a line can be drawn, so
that the parts FG and A E intercepted between the
circle and the produced sides of the rectangle be
equal. 4 -
The same problem leads also to a different con-
dition.
Let the former construction remain, and on B D con- Fig. 20
struct an isosceles triangle whose side KB or K D is
equal to half of D C. Bisect D C at N, and draw
K F. The square of K F is equal to the square of
KB, together with the rectangle D F x FB. But
also the square of N E is equal to the rectangle DE
× E. C, together with the square of N C. Since N C
is equal to KB, (Constr.) and the rectangle D Ex CE
has been already proved to be equal to the rectangle
D F x FB, it follows that the square of N E is equal
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638
G E O M ET. R. I. C. A. L. A. N. A. L. Y S I S.
Geome-
trical
Analysis.
S-N-
Duplication
of the cube.
Geometric
loci.
to the square of K F, and therefore these lines thern-
selves are equal. Since -
D E : C E : : D F : D B,
therefore
D E + C E : D C : : D F + D B : B F.
But D E + C E is equal to twice N E, or to twice
KF; and if D L be produced equal to B D, D F --
D B is equal to LF, and D C is equal to twice K.B.
Hence the preceding proportion becomes
2 K F : 2 KB : : L. F : B F.
Draw L K and B M parallel to it through B. Hence
we have
K F : M F : ; L F : B F,
2 K F : 2 M F : ; L F : B F.
Therefore twice MF is equal to twice KB, and there-
fore MF is equal to K B, and therefore to half of
D C.
Hence it follows that the insertion of two means
between A B and A C depends on the inflection of a
line across the sides of the angle FB M, so that it
shall pass through the given point K ; and the part
MF intercepted by the sides of the angle shall be of a
given magnitude, viz. equal to half of A B, one of the
given lines.
This condition is similar to that required for the
trisection of an angle, so that if one of these problems
could be solved the other would also be solved.
The insertion of two mean proportionals is necessary
to solve the celebrated problem of “the duplication of
the cube,” or to find a cube which doubles a given cube.
The general proposition, of which this is a particular
case, is to construct a solid of a given species, and
bearing a given ratio to a given solid of that species.
This problem is thus solved. Find a line to which
any edge of the given solid has the given ratio. Be-
tween this line and that edge find two mean propor-
tionals, and with the first of these means as an edge
construct a solid similar to the given one. This will
be that which is required. For similar solids are in
the triplicate ratio of their homologous edges; and
therefore the given solid is to the constructed one as
its edge is to the fourth continued proportional, that is,
in the given ratio. -
Thus on this principle depends the change of the
scale of solids in any required proportion.
The problem of the “ duplication of the cube” is
called the Delian problem. See History of GEOME-
TRY ; also HISTORY OF ANALysis.
Or
SECTION V.
Geometric Loci.
(22.) WHEN a point is required to be determined in
a problem with data which are insufficient for its
solution, the problem is said to be indeterminate, be-
cause the position of the point cannot be found from it.
But although the position cannot be absolutely deter-
mined, yet it may be so restricted by the conditions
which are prescribed in the problem, that it may be
known to be on some line, the nature of which may
frequently be determined. This line is called the locus Section v.
This will easily be understood by the S-V-
of the point.
following examples: suppose that the base of a triangle
were given in magnitude and position, and that its area
were given in magnitude, to determine its vertex. In
this case, it is evident, that the problem is indetermi-
nate, since innumerable triangles may be constructed
on each side of the given base having equal areas.
But since the area is equal to the rectangle under the
perpendicular and half the base, it follows that the per-
pendiculars from the vertices of all these triangles on
the base must be equal, and therefore these vertices
must all lie on parallels to the base at such a perpen-
dicular distance that the rectangle under it, and half
the base shall be equal to the given magnitude.
The locus of the vertex is therefore in this case two
right lines parallel to the base, and at equal perpendi-
cular distances at opposite sides of it.
If the base of a triangle be given in magnitude and
position, and the vertical angle be given in magnitude,
to determine the vertex, the problem is evidently inde-
terminate ; for an unlimited number of different
triangles may be constructed on the same base whose
vertical angles are equal. But the vertices of all the
triangles on the same side of the base will in this case
be placed on the arc of a circle containing an angle
equal to the given angle. Hence the locus will be two
segments of circles containing an angle equal to the
given angle, and constructed on opposite sides of the
given base.
The investigation of loci is of very extensive use in
the solution of determinate problems. In cases where
the determination of a point is required from certain
data, by omitting any one of the data the point will
have a locus which may be found by the remaining
data. This being successively applied to two of the
data, two loci will be found, the intersection of which
will determine the point.
This may be illustrated by the examples already
given. Let the base of a triangle be given in magni.
tude and position, and the area and vertical angle in
magnitude, to determine the vertex. If we omit the
vertical angle, the locus is the parallels already des-
cribed. If we omit the area, the locus is the segments
of the circle. The vertex being then at the same time
on both loci must be at the intersection of the two loci,
and will therefore be at the points where the parallels
meet the circle. In general there will be in the present
case four such points, and consequently four triangles,
but these triangles will differ only in position, being
equal as to their sides and angles.
The following propositions will illustrate the theory
of Geometric loci.
PRoPosition.
(23.) Given in magnitude and position the base of a
triangle, and the difference of the squares of its sides, to
Jind the locus of the vertex.
Let A B be the given base, and C be a point of the Fig. 21,
sought locus. Draw A C, B C, and from C draw the
perpendicular CD. The difference of the squares of
the sides A C, B C is equal to the difference of the
squares of the segments A D, DB, which is therefore
given. The points at which the perpendicular meets
the base are therefore given, and therefore the perpen-
G E O M ET H I C A L A N A L Y S I S.
639
Geome-
trical
Analysis.
\-->
Fig. 22
Fig. 23
dicular itself is given in position; and since the vertex
must be on the perpendicular, the locus is determined.
To construct the locus, it is therefore only necessary to
cut the base at D, so that the difference of the squares
of the segments shall be equal to the given difference
of the squares of the sides, and the perpendicular CD
through the point of section will be the locus sought.
It is evident that there are in general four points D at
which the line may be cut as required, two on the line
itself and two in its production, and that these points
are respectively equally distant from the middle point.
PRoPosition.
(24.) Given the base of a triangle in magnitude and
position, and the sum of the squares of the sides, to find
the locus of the verter.
Let the base be A B, and let C be any point of the
locus. Draw CD to the middle point of the base, and
draw C A and C B. The sum of the squares of CA
and C B is equal to twice the sum of the squares of
CB and D B. But the sum of the squares of C A and
C B is given, and therefore also twice the sum of the
squares of CD and DB is given, and therefore the sum
of the squares of C D and D B is given. But the
square of D B (half the given base A B) is given ;
therefore the square of C D and C D itself are given.
The point C, whose locus is sought, is therefore at a
given distance from the middle point D of the base,
and its locus is therefore a circle whose centre is the
middle point of the base, and whose radius is the given
distance. This radius is evidently a line whose square
is half the difference between the given sum of the
squares of the sides, and double the square of half the
base. - -
This proposition is only a particular case of the
following more general one : “Any number of points
being given, to find the locus of a point such that the
sum of the squares of its distances from the several
given points shall be given.”.” If the given and
sought points be in the same plane, the locus will be a
circle; but if they be not limited to the same plane, the
locus will be the surface of a sphere. In this case the
centre of the sphere will be the centre of gravity of
equal masses placed at the several points, or that point
which is mathematically denominated the centre of mean
distances.
PROPos ITION.
(25.) Given in magnitude and position the base of a
triangle, and the ratio of the sides, to determine the locus
of the verter.
Let A B be the base of the given triangle, and let
C be a point of the sought locus, and let the given
ratio be m : m. Draw A C, B C. Also draw C D,
C D', bisecting the internal and external angles at C.
Hence -
A D : D B : : A C : B C : : m : m
A D': D’B : : A C : B C : ; m : 71.
The ratio of the segments into which the line A B is
cut at D and D' is therefore given, and therefore the
* See Lardner's Algebraic Geometry, p. 113.
points D and D' are given.
C must be placed upon a circle whose diameter is
D D', and therefore this circle is the locus of the
vertex of the triangle sought.
As there are two points D at which the line may be
divided in the given ratio, and as it may be produced
through either end, the locus, strictly speaking, is two
circles.
PROPoSITION.
(26.) Given the base of a triangle, the sum of the
squares of the sides and the vertical angle, to construct
the triangle.
If the base and the sum of the squares of the sides
be given, the locus of the vertex is found by (24,)
and if the base and vertical angle be given, the locus
of the vertex is found by (22.) The intersection of
these loci will determine the vertex.
It may happen, that the loci do not intersect. In
this case the solution is impossible, and the data are
inconsistent. -
It may also happen, that the two loci are identical,
in which case the problem is indeterminate, and the
data are not distinct. This happens in the present
instance, when the sum of the squares of the sides is
equal to the square of the base, and the vertical angle
is right. Either of these data follows necessarily from
the other, and the two loci are the same circle. º
(27.) These observations, however, apply to all de-
terminate problems solved by two loci, viz. when the
loci do not meet, the problem is impossible, and the
data contradictory; and when they become identical,
the problem is indeterminate, and the data not inde-
pendent.
PROPOSITION.
(28.) Given the base of a triangle, the ratio of the
sides, and the difference of their squares, to determine the
triangle.
This problem is solved by the intersection of the loci
determined in (25) and (23), and is subject to the obser-
vations in (26.)
PROPosition.
(29.) A circle is given in magnitude and position,
and a chord passes through a given point, to find the
locus of the intersection of tangents through the ex-
tremities of the chord.
Let C B A be the circle, P the given point, A B any Fig.24 and
chord through it, and D the corresponding point of the 25.
locus. Draw CD, which will evidently bisect B A at
right angles, and we have by the known properties of
the circle C E : C F : C D. Hence the rectangle DC
x C E is equal to the square of the radius C F. Draw
D G perpendicular to C P produced, and the angles G
and E being right, the quadrilateral D EP G may be
circumscribed by a circle; therefore the rectangle DC
x C E is equal to the rectangle G C x C P, and there-
fore the rectangle G C x C P is equal to the square of
the radius. Hence the point G is independent of the
point D, and a perpendicular from any point of the
locus will meet C P produced at the same point D.
Hence to construct the locus, find a third proportional
The bisectors CD and Section v.
C D' form a right angle at C, and therefore the point \-/-
4 c 2
64()
G E O M ETR ICAL AN A L Y S I s
Geome-
trical
Analysis.
*
to CP and the radius, and take C G equal to this third
proportional, and through G draw a perpendicular to
C G. This perpendicular will be the locus sought.
The nearer the given point P is to the centre, the
more remote will be the locus G. D, and when P coin-
cides with the centre, C G will become infinite, so that
in this case the locus may be considered a right line at
an infinite distance.
There will be no difficulty in establishing the con-
verse of this principle, viz. “if tangents be drawn from
each point in a given right line to a given circle, the
chords joining the points of contact will all pass through
a certain given point.”
SECTION VI.
Porisms.
(30.) THE term porism f has been variously defined
by Geometers. Pappus states, that Euclid wrote
three books on porisms, (which have been lost,) but is
so obscure and indistinct on the subject, that it is im-
possible merely from what he has stated to determine
to what species of Geometrical proposition the Ancients
applied this term. It is certain, that it was sometimes
used synonimously with corollary ; thus Euclid, in his
Elements, calls the corollaries of his propositions
zroptopata. In an elaborate dissertation on the subject
of porisms, in the Transactions of the Royal Society of
Edinburgh, Playfair has, however, succeeded in giving
the word a meaning more worthy of the importance
which is evidently attached to this class of propositions.
The porisms of Euclid are said to be “collectio artift-
ciosissima multarum rerum qua, spectant ad analysin
difficiliorum et generalium problematum.”
According to Playfair, a porism is “a problem in
which the data are so related to each other that it be-
comes indeterminate, and admits of numberless solu-
tions.” -
It is easily conceived that a problem which in general
is determinate will, when its data are submitted to
certain conditions, become indeterminate.
cases it becomes a porism ; and it may be proposed in
a porism to determine what condition or restriction will
render a determinate problem indeterminate.
Thus, if it be required to draw a right line through
a given point, subject to some given condition, the
problem may be in general determinate; and it may be
possible to draw but one such right line. But, on the
other hand, such a position may be selected for the given
point, as that every line passing through it will fulfil
the given condition. When this position is assigned to
the point, the problem becomes a porism. The follow-
ing examples will render these observations more in-
telligible.
* A numerous collection of Local problems will be seen in Lard-
ner's Algebraic Geometry. The solutions there given are, however,
by the Algebraical Analysis.
+ From regſºo, I establish; or, according to some, from ºréeos, a
transition.
† Pappus defines a porism to be something between a theorem and
problem, or that in which something is proposed to be investigated.
Simson follows Pappus, and says, that a porism is a theorem or
problem in which it is proposed to investigate or demonstrate some-
thing. -
In such
PROPos ITION.
(31.) To draw a line passing through a given point,
and crossing a given triangle, in such a manner that the
sum of the perpendiculars on it from the two vertices on
one side of it shall be equal to the perpendicular on it
from the other verter placed on the other side of it.
Let D be the given point, and A B C the given
Section VI
\-N-
triangle, and let D E be the required line, so that AE Fig. 26.
and B G taken together are equal to C F. Draw C H
from C to the middle point H of A B, and draw HK
perpendicular to D. E gº
In the trapezium A E G B, the parallels A E, H K,
and B G are in arithmetical progression; therefore the
sum of A E and B G is equal to twice H.K.; but this
sum is also equal to C F. Therefore C F is equal to
twice H K.
triangles H L K and CFL are similar, and therefore
C F : H K : : C L : L H.
But C F is equal to twice H K, and therefore C L is
equal to twice L H, or L H is one third of C H. Since
C H is given in magnitude and position, the point L is
given. Hence the problem is solved by drawing a line
from any angle C of the triangle, bisecting the opposite
side A B, and taking on this one third of it H. L. The
line drawn from the given point D through the point L
will be that which is required.
If the given point happen to be the point L itself,
any line whatever passing through it will have the pro-
posed property, and hence we have the following porism :
“A triangle being given in position, a point may be
determined, such that any line being drawn through it,
the sum of the perpendiculars from two angles of the
triangle placed on one side of it, shall be equal to the
perpendicular from the remaining angle and the other
side.”
The point L is evidently the centre of gravity of
equal masses placed at the three vertices, or, considered
mathematically, it is the centre of the mean distances
of the three points A B C.
This porism is only a particular case of a much more
general one; “any number of points being given in the
same plane, a point may be found through which any
line whatever being drawn, it will pass amongst the
points in such a manner, that if perpendiculars be
drawn from them upon the line the sum of the perpen-
diculars at the one side will be equal to the sum of
the perpendiculars on the other side.” In this case, as
in the former, the sought point is the centre of mean
distances. • * ,
The same porism may receive another modification
which generalizes it further. “Any number of points
being given in the same plane, to determine the condi-
tion under which a right line may be drawn amongst
them, so that the sum of the perpendiculars from the
points on one side shall exceed the sum of the per-
pendiculars from the points on the other side by a
given line.”
In this case, it may be proved that the line must be
a tangent to a circle, whose centre is the centre of mean
distances, and whose radius is equal to the given line
divided by the number of given points. . . .
If the given points be not in the same plane, the
* See Lardner's Algebraic Geometry, p. 34.
Since C F and H K are parallel, the
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G E O M ET R I C A L A N A L Y S I S.
641
Geome-
trical
porism may be made still more general : “Given any
number of points in space, to determine a plane passing
Analys”, among them, so that the sum of the perpendiculars from
Fig. 27
the points on one side shall exceed the sum of the per-
pendiculars from the points on the other side by a given
line.”
In this case the plane must touch a sphere whose
centre is the centre of mean distances, and whose radius
is the given line divided by the number of points.
If the sum of the perpendiculars on one side be
equal to those on the other, the given line and the
radius of the sphere vanish, and the sphere is reduced
to its centre, i. e. the centre of mean distances. Hence,
“if a plane be drawn through the centre of mean dis-
tances, the sum of the perpendicular from the points
on the one side is equal to the sum of the perpendicu-
lars from the points on the other side.”
PRoPosition.
(32.) A circle and a straight line being given in posi-
tion, a point may be found such that any right line
drawn from it to the given line shall be a mean propor-
tional between the parts of the same line, intercepted
between the given right line and the circumference of the
given circle.
Let A B be the given right line, H K F the given
circle, and D the sought point. Draw GDI perpen-
dicular to A B through D, and also any other line C D F
Also join CI and draw H. K.
The square of CD is equal to the rectangle CE
x C F; but it is also equal to the squares of C G and
G D, and the rectangle C E x C F is equal to the
rectangle C K x C I. Hence the rectangle C K x C I
is equal to the sum of the squares of C G and G. D.
The square of G D is equal to the rectangle G. H.
× G. I; therefore the rectangle G H x G I, together
with the square of CG, is equal to the rectangle C K
x C I. Also the square of C I is equal to the sum of
the squares of C G and G. I. But the square of C I is
equal to the rectangle C K x C I, together with C I
x K I, and the sum of the squares of C G and GI is
equal to the square of C G, together with the rectangles
G H x GI and G. I x H I. Taking away from these
equals the rectangle C K x C I, and its equivalent the
rectangle G H x G I, together with the square of G C
the remainders, the rectangles C I x IK and G. I. x IH
are equal. Hence, we have -
G I : I C : ; IK : I H.
Therefore, in the triangles CIG and H IR the angle
I is common, and the sides which include it are propor-
tional, and therefore the triangles are similar ; but G
is a right angle, and therefore H K I is a right angle,
and therefore H I is a diameter. Since, then, H I
passes through the centre of the given circle, and is
porism cannot be converted into a local theorem.
fixed by the given conditions.
perpendicular to A B, the given right line, it is given in Section VI
Also G H and G. I are given in magnitude, \-N--
position.
and therefore G D, which is a mean proportional between
them, is given in magnitude, and therefore the point D
is given in position.
(33.) There is between local theorems and porisms a
close analogy. In fact, every local theorem may be
converted into a porism; but, on the contrary, every
In
local propositions the indeterminate is always a point,
the position of which is restricted, but not absolutely
Such may always be
expressed as a porism. But this class of propositions
is more general than geometric loci ; the indeterminate
may be a line, the direction of which is not restricted
by the conditions, but which is otherwise limited, as,
for example, to pass through a given point, or to touch
a given circle. It may also be a plane similarly re-
stricted to pass through a given point, or to touch a
given sphere. Instances of these have been given in
(31.)
Porisms, in common with geometric loci, take their
rise from the conditions of a problem becoming inde-
terminate. This may happen in two ways. The num-
ber of conditions may not be sufficient, or among the
given conditions there may exist some particular rela-
tion, by which some one or more of them may be de-
duced from the others. Thus, for the determination of
a triangle three conditions are necessary, and such a
problem becomes manifestly indeterminate if only two
conditions be given. But even though three be given,
the problem will still be indeterminate, if any one of the
three can be inferred from the other two. For example,
suppose the base of a triangle, the point where the
perpendicular intersects it, and the difference of the
squares of the sides be given, the problem to determine
the triangle is indeterminate, because the difference of
the squares of the sides is equal to the difference of the
squares of the segments of the base, and may, there-
fore, be inferred from the base and the point of section.
The geometrical circumstances by which determinate
problems in Geometry are converted into porismatic and
local problems, are precisely similar to those under
which the solution of an algebraical question becomes
indeterminate. In such a question there should be as
many equations as unknown quantities, and the problem
is indeterminate evidently if there be less. But it may
also be indeterminate, even if the number of equations
be equal to that of the unknown quantities, and will be
so when any one of the equations can be deduced from
the others. It may in general be observed, both in
geometrical and algebraical problems, that the number
of independent conditions should be equal to the num-
ber of quantities sought, and should neither be more
nor less. If they be more, the results may be incon-
sistent, and if they be less, the solution will be indeter-
minate. -
T H E O R. Y O F N U M B E R S.
Theory of THE Theory of Numbers is a branch of Analysis by
Numbers, which we investigate the properties, dependencies, and
relations of integral numbers, as by Geometry we in-
ing to any other measure or modulus, as 4 n + 1, Introduc-
6 m + 1, &c. tion.
10. Numbers of the same form with respect to any -->
quire into the dimensions, position, and relations of
lines; and as in the latter science a combination of
lines, or a certain disposition of them, receives particular
denominations, so in this branch of Analysis, numbers
are distinguished into classes, according to the nature
and dependence of the integral parts of which they are
composed. It will be convenient, therefore, to pro-
ceed in this case, as in the other, by definitions and
propositions.
I. Introduction, showing the forms, properties, and rela-
tions of simple Integral Numbers,
I) EFINITIONs.
1. An integer, or integral number, is an unit, or any
number of units.
2. The factors of a number, are those numbers by
the multiplication of which the former number is pro-
duced; and the number thus formed, is called the pro-
duct of those factors.
3. The multiple of a number is the product of that
number by some integral factor.
4. Even numbers are those which can be divided into
two equal parts; and uneven, or odd, numbers are those
which cannot be so divided.
5. A composite number is any number produced by
the multiplication of integral factors.
6. A prime number is that which cannot be pro-
duced by the multiplication of any integra 'actors, or
that cannot be divided into any equal integral parts
greater than unity.
7. Commensurable numbers are any two or more
numbers having a common integral divisor; and in-
commensurable mumbers are those which have not a
common divisor. The latter numbers are also said to
be prime to each other.
8. A square, or second power, is the product of two
equal factors. A cube, or third power, the product of
three equal factors; and, generally, the n” power of a
number is the continued product of n equal integral
factors; and the number from the multiplication of
which any power is produced, is called the root of that
power.
9. The forms of numbers, or formulae, are certain
algebraical expressions under which those numbers are
contained.
Thus, every even number is of the form 2 m, and
every odd number of the form 2 n + 1 ; because an
even number may be divided by 2, and will produce
an integral quotient which may be represented by n,
and, consequently, the number itself by 2 n ; and
an even number increased or diminished by unity is an
odd number ; therefore all odd numbers may be ex-
pressed by, or are of the form, 2 m + 1.
In a similar manner, numbers may be classed accord-
642
modulus, are all those which can be represented by the
same formula. Thus, 13, 17, 21, &c. are all of the
form 4 m + 1 ; and 19, 25, 31, &c. of the form 6 n + 1 ;
4 and 6 being the moduli.
The forms and relations of Integral Numbers, and of
their sums, differences, and products.
1. The sum or difference of any two even numbers
is an even number.
For let A = 2 m and B = 2n be any two even
numbers: then
A -EB = 2n + 2 m = 2 (n + n) = 2n",
which being of the form 2 m is an even number.
2. The sum or difference of two odd numbers is even,
but the sum of three odd numbers is odd.
Let A = 2 m + 1, B = 2 m' + 1, and C = 2n” + 1
be three odd numbers: then
A + B = 2 m + 2 m' + 2 = 2 m.",
and A-i-B -- C = 2n + 2 n' + 2 n' + 3 = 2 n." -- l;
the former being the form of an even, and the latter
of an odd number.
In a similar way it may be shown:
(1.) That the sum of any number of even numbers is
eVCI).
(2.) That any even number of odd numbers is even,
but that any odd number of odd numbers is an odd
number.
(3.) That the sum of an even and odd number is an
odd number.
(4.) That the product of any number of factors, one of
which is even, will be an even number, but the product
of any number of odd numbers is odd; and hence,
again,
(5.) Every power of an even number is even, and
every power of an odd number is an odd number.
(6.) Hence the sum and difference of any power and
its root is an even number.
For the power and root will be either both even or
both odd, and the sum or difference in either case is
an even number.
3. If an odd number divide an even number, it will
also divide the half of it.
Let A = 2 m, B = 2 m'—H 1 be any even and odd
number, such that B is a divisor of A; let the division
be made, and call the quotient p, then we have
2 m = p (2 n' + 1),
consequently p is even, or of the form 2n", hence
2 m = 2 m." (2 m' + 1),
72. – c.”
gºſ II = **
that is, n = }; A is divisible by B, if A itself be so.
and
T H E O R Y O F N U M B E R. S.
643
Theory of
Numbers.
\-->
4. If a number p divide each of two numbers a and
b, it will divide their sum and difference, or the sum
and difference of any multiples of them.
h
Let * = q and — = q', then
p p
a + b
= q + q = q",
p
which is an integer, because q and q' are both in-
tegers.
In like manner, n a, mb being multiples of a and b,
we have -
n a + m, b
p = n. q + m, q'an integer.
DEDUCTIONs.
It follows from the preceding propositions:
(1.) That if a number divide the whole of another
number, and a part of it, it will also divide the other
part.
(2.) It follows, also, that if a number consist of
many parts, and each of these parts be divisible by
another number, that the whole number, or the parts
taken collectively, will be divisible by the same
number.
5. If a and b be any two numbers prime to each
other, their sum a + b is prime to each of them.
For if (a + b) and a had a common divisor, their dif-
ference (a + b) — a = b would have the same divisor;
that is, a and b would have a common measure, which
is contrary to the supposition; and, in the same way, it
may be shown that a + b, and b, cannot have a com-
II) OIl Iſlea,SUI"e,
DEDUCTIONs.
(1.) In like manner, it may be demonstrated, that if
a and b be prime to each other, their difference a - b
will also be prime to each of them, if a - b > 1.
(2.) Conversely, if a number consist of two parts,
and be prime to one of those parts, it will be prime to
the other.
(3.) And if a number consist of many parts, and
each of those parts but one be divisible by another
number p, then the whole number taken collectively
is not divisible by p.
6. If a and b be two numbers prime to each other,
their sum and difference will be prime to each other, or
they can have only the common measure 2.
For if a + b and a — b have a common measure,
their sum and difference 2 a. and 2 b will have the same;
but a and b are prime to each other, therefore 2 a. and
2 b can only have the common measure 2; therefore
a + b and a — b can only have the common measure
2; and if one of these numbers a or b be even and the
other odd, then a + b and a — b are both odd ; in
this case, therefore, they are prime to each other; but
if a. and b are both odd, then their sum and difference
will have the common measure 2, but no other
7. If a and p be any numbers prime to each other,
a being the greater, then may a be always represented
by the formula a = m p + r, in which r shall be less
than p and prime to it.
Let a be divided by p, and give a quotient m, and
remainder r, which makes
a = m p + r,
where r is obviously less than p, m being supposed the
greatest quotient.
And r is prime to p, because if p and r had a com-
mon measure n p and r, as also m p + r, and r would
have the same common measure, but a = n p +-r; there-
fore a and p would have the same, which is contrary to
the supposition, these being prime to each other.
The same expression may be employed if a be less
than p, but in this case n = 0 and a = r.
8. The same conditions being made with respect to
a and p, it is always possible to express a by the formula
a = n p + r.
in which r shall be less than 3, p. For if in the formula
a = n p + r,
r, which is less than p, be less than 4 p, the formula
agrees with the enunciation of this proposition; and if r
should be greater than ; p, then we may make
a = (n + 1) p – (p — r)
or making n + 1 = n' and p - r = r",
a = n' p – r",
and here since r > #p, r = p – r' < , p. The same
formula applies to all numbers whatever, except that
r and p in this case are not necessarily prime to each
other.
9. If a and p be any two numbers prime to each
other, there cannot be another number b prime to a
which will render the product a b divisible by p. Or if
a number p be prime to two other numbers a and b,
it will be prime to their product a b.
First, if there be such a number b as will render a b
divisible by p, let us suppose it to be the least of all
those that will make a b divisible by p ; and since p is
prime to b, let
p = m b + b',
so that b' shall be less than b, and also prime both to p
and b. Then, multiplying both sides by a, we have
a p = a n b -- a b', or
a p – a m b = a b'.
If therefore a b' be divisible by p, a n b, and conse-
quently a p – a n b, as also its equal a b' will be so
likewise.
But b is by the supposition the least number that
renders a b divisible by p, whereas we have now found
a less b', which is absurd. There cannot, therefore, be
a number which is the least that renders a b divisible
by p, but if there were any such numbers one of them
must be the least; therefore there is no such number;
that is, if p be prime both to a and b it is prime to
their product a b.
DEDUCTIONs.
(1.) From this it follows, that if a number p be prime
to any number of factors a, b, c, d, &c., it is also prime
to their product a . b. c. d, and if p be prime to any
number whatever, it is prime to all its factors.
(2.) If those factors are all equal, then the product
becomes a power; if therefore p be prime to a, it is
prime to any power of a, as a ".
(3.) Hence again, conversely, a power can only have
the same prime divisors as its root.
(4.) Consequently if p divide the product a b, but is
prime to one of its factors, it must be a divisor of the
other; and if p be a divisor of a continued product
Introduc-
tion.
\-v- '
644 T H E O R Y O F N U M B E R S.
Theory of a . b. c. d, &c., and is prime to one of the factors a, it one of its factors, as cº, and so on. Then ultimately we Introduc-
Numbers. must be a divisor of the other factors b, c, d, &c. shall obtain ... tiſ)n.
S-N-
\-v-sº
(5.) If a be prime to p, and b less than p, then, whe-
ther b be prime or not, the product a b is not divisible
by p. -
(6.) If there be any number of factors a, b, c, &c.
respectively prime to any other factors p, q, r, then
will the products a . b. c. p. q . r, be prime to each
other. -
(7.) If a product a b be divisible by p, and one of
those factors as a be prime to p, then will the quotient
be divisible by a.
10. Neither the sum nor the difference of two frac-
tions which are in their lowest terms, and of which
the denominator of the one contains a factor not com-
mon with the other, can be equal to an integer.
Cº. - g
Let —– and be any two fractions in their
b
A B #
lowest terms, and of which the denominator of the
One, aS , contains a factor t not contained in A,
b
H #
then the equation
C. b :
TAT -E Hiſ = e an integer
is impossible. •
(M, b a B. t =E A. b
F Gºmºsºme — = —— 2
Or A + E. A B t
which cannot be an integer unless A 5 be divisible by
t; but A and b are each prime to t; their product
A b is therefore also prime to t. Consequently,
B t + A. b
*A* cannot be an integer: that is,
(l, -E b Cºmsº integer
A. HP = e an integ
is impossible under the conditions of the proposition.
DEDUCTIONS
(1.) The same is also true if the first fraction be not
b
in its lowest terms, if t be prime to A and By * frac-
tion in its lowest terms.
(2.) The sum or difference of two fractions each in
its lowest terms is also in its lowest terms, provided
the denominators be prime to each other: that is, if
a and # be in their lowest terms, and A prime to
A
a B =E b A
B, then will TA BT
(3.) If two fractions are each in its lowest terms,
their product is in its lowest terms.
11. Every integral number may be represented by
the formula a” b” cº &c.
First, if p be a prime, then b = 1, c = 1, &c. and
m, m, q, &c. may also be supposed = 1, and we shall
have p = a.
Secondly, if p be not a prime, divide it first by the
highest power a” of one of its prime factors contained
in it; and the quotient again by the highest power of
b, as b", and the new quotient by the highest power of
be also in its lowest terms.
p = a b” cº, &c.
where a, b, c, &c. are all prime numbers.
DEDUCTIONs.
(1.) Since every number is of the above form, the root
of any square number is of that form, and therefore
every square number is of the form
p3 = a”. bº. cº, &c.
(2) If p = a” b” cº, and any one of the exponent
n, m, q, be an odd number, p is not a square number.
And if n, m, q, &c. be not each divisible by 3, p is not
a cube, and so on in the higher powers.
(3.) Hence a square multiplied by a square will pro-
duce a product which is a square; but a square multi-
plied by a factor which is not a square, will give a pro-
duct which is not a square, and so on with the higher
powers.
12. If any square p” can be divided once by some
other number p', and after that, neither by p' nor by
any factor of p', then is p' also a square.
For let p be resolved into the form p = a”. b”. Cº.
then p' = a” ba" cº,
and since pº is divisible by p', this last must contain
some of the prime factors of p, that is, p' must have the
form
p' = a b', &c.
p” a 2" bºn Cºq
p' T aſ bº
which latter quotient will still be divisible by a, b, &c.,
unless r = 2 m, s = 2 m, &c.; and since, by the Sup-
position, this quotient is not again divisible either by p'
or by any factor of p", it follows, that p' = a”. b”, &c.
and – ag"-". 52*-*, &c.
DEDUCTIONs.
(1.) In the same manner, if any power p" be divisible
once by some other number p’, and after that neither
by p' nor by any factor of it, then will p' itself be a com-
plete m” power.
(2.) It follows from this, that no product arising
from any number of different prime numbers can be a
square ; for let p' be one of those prime numbers;
then the product may be divided once by p', and only
once, therefore that product is not a square.
(3.) The same is true of any two or more numbers
prime to each other, unless they be all squares.
(4.) Therefore, conversely, the product of the square
roots of non-quadrate numbers prime to each other
cannot produce an integer.
For if p and q be two such numbers, and
w/ p x v q = r, then p q = r",
which we have seen is impossible. -
13. The square root of an integer that is not a
complete square cannot be expressed by a fraction.
If it be possible, let w/ a = + ; + being sup-
posed in its lowest terms, so that m is prime to n, then
7773 e
a = −; and consequently mº must be divisible by
71,
n”, which is impossible, because m and n are prime to
each other.
T H E O R Y O F N U M B E R. S. 645
Theory of DEDUCTIONs. them can be the same, is truly expressed by the for- sect. I.
Numbers. g g mula Divisors
2–0 From the two preceding propositions it follows: (m + 1) (n + 1) (p + 1) (q + 1) & S-2-
\->/- 7??, 7?, p Q C.
(1.) That any root of a number which cannot be
expressed by an integer, cannot be expressed by a
rational fraction.
(2.) The product of the square roots of any two or
more non-quadrate numbers, cannot be expressed by
any rational fraction.
(3.) And, generally, if "Va and "V b be neither of
them expressible in integers, and if a be prime to b,
then can "w/ a × "V b be neither expressed in integers
nor in rational fractions.
14. Neither the sum nor the difference of the square
roots of two numbers which are not both squares, can
be expressed by any rational quantity.
Let p and q be two such numbers, and if possible,
let
w/p + V q = c,
then p + q + 2 v p q = cº,
c" — p – q * º
and W p q = —g— a rational fraction,
which is impossible.
PEDUCTIONs.
(1.) In the same way it may be shown, that
M p + W q = W c is impossible.
For then w p q = *=}=1,
which is impossible.
(2.) If p and q be prime to each other, then
W p + V q = W r + W. s is impossible.
For squaring both sides and reducing we obtain
r —H s — p —
+ V p q + V r s = .*.*,
which is impossible, whether V r s be rational or irra-
tional.
I. On the divisors of Composite Numbers.
15. To find the number of divisors of any given
number.
Let N be the given mumber, let N be resolved into
the form N = a” b" cº dº, &c. then will the number of
its divisors be expressed by the formula
(m. -- 1) (n + 1) (p + 1) (q + 1) &c.
For it is evident that N will be divisible by a, and by
every power of a to a”, that is, by every term in the
series
a, a”, a”, a”, &c. a”,
and also by b, and by every power of b to b", that is, by
every term in the series
* b, b”, bº, bº, &c. b”,
and in the same manner by c, and every power of c to
c”; by d and every power of d to dº, &c.
And also by every possible combination of the terms
of the above series; that is, by every term in the con-
tinued product
(1 + a + a”. . . . . . a”) × (1 + b + bº + &c. b”)
(1 + c + c + &c. op) × (1 + d -- dº + &c. d")
but the numbers of terms in this series, since no two of
WOL. I.
which is, therefore, the number of the divisors sought,
unity and N being both included as divisors.
Thus 360 = 28, 3”. 51.
Has (3 + 1) (2 + 1) (1 + 1) = 24 divisors. &
And 1000 = 28. 53. w
Has (3 + 1) (3 + 1) = 16 divisors.
DEDUCTIONs.
(1.) As the number N = a” b" cº dº has
(m + 1) (n + 1) (p + 1) (q -- 1) divisors,
it is obvious that the number of ways in which it can
be divided into two factors will be expressed by
# (m. -- 1) (n + 1) (p + 1) (q + 1) &c.
being equal to half the number of its divisors.
(2.) If it be required, in how many ways a number,
N = a” b” cº, &c., may be resolved into two factors
prime to each other, it is evident, that this number no
longer depends upon the value of the exponents
7m, n, p, &c., but will be the same as if N was simply
resolved into the factors a, b, c, &c.; and is, therefore,
equal to
(1 + 1). (1 + 1). (1 + 1), &c.
2 5
hence, if k represents the number of prime factors,
a, b, c, d, &c., then will 2*Tº be the number of ways
in which N. may be resolved into two factors prime to
each other. Thus, for example, 360 has twenty-four
divisors (example 1,) and, consequently, may be re-
solved into factors twelve different ways; but it has
only three prime factors, 2, 3, and 5, and can, therefore,
be resolved into factors prime to each other only,
2* = 4, different ways.
16. To find a number that shall have any given
number of divisors.
Let w represent the given number of divisors, and
resolve w into factors, as w = a x y x 2. Take
7m = a, – I, n = y – 1, p = 2 – 1, &c.; so shall
a" b” cſ, &c.
be the number required, as is evident from the fore-
going proposition, where a, b, c, &c. may be taken any
prime numbers whatever. -
Thus, to find a number that shall have 30 divisors,
we have 30 = 2 × 3 × 5. Wherefore r = 2, y = 3,
2 = 5, and m = a – 1 = 1, m = y – l = 2, p = 2 – l = 4,
and a” b” c” is the number sought, a, b, c being any
prime numbers whatever.
If a = 5, b = 3, c = 2 we have
5.3°. 2* = 720, the number sought,
and this is the least of all numbers having 30 divisors,
because a, b, c are the least three prime numbers, and
that which is involved to the highest power is the
least.
17. To find the sum of all the divisors of any given
number.
Let N be the number, and make N = a” b” c”, &c.,
then the sum of all the divisors of N is expressed by
the formula
4 P
646
T H E O R Y O F N U M B E R. S.
Theory of
Numbers.
w For
a"+1 — I b"+1 — I cºp-ti — I
==========" ºmºmºmºm —— &c.
(***) (º) (ºil)
we have seen that the formulas
(1 + a + a” &c. a”) (1 + b + bº + &c. b.")
(1 + c + c &c. cy) (1 + d -- dº -- &c. dº)
include all the divisors of N, and by the laws of arith-
metical series,
& a"+1 — 1
2 tre -——
1 -- a + a” + &c. a a – 1
bº-Fi — I
1 + b + bº + &c. b” = &===tº
a — l
&c. &c.
Consequently, this product is equal to
a”-h1 – 1 b"t — I cP+1 — I &
(*#): (*#): (**)se.
which, therefore, expresses the sum of all the divisors
of N.
In this expression, N is considered as a divisor of
itself; because, from the developement of the above
product, the last term will evidently be a” b" cº, &c.;
X
N × −1 × , &c.,
that is, the last term of the product will be the number
N itself.
Required the sum of all the divisors of 360.
First, 360 = 2*. 3”. 5; therefore,
24 – 1 38 – 1 X 5? – 1
2 — 1 3 – 1 5 – 1
= 15. 13.6 = 1170;
which is the sum of all the divisors of 360, itself being
considered as one of them.
18. If N = a” b” cF &c. represent any number,
a, b, c, &c. being its prime factors, then will
b – 1 X c — 1
b C
express the number of integers that are less than N,
and also prime to it.
First, if N be a prime number, or N = a, then we
know, that all numbers less than a are also prime to it;
a - 1
= a – l is the real
and, consequently, N ×
expression for the number of them in this case.
And if N be any power of a prime number, or
N = a”, then, in the series of numbers
1, 2, 3, 4, 5, &c., a”,
every ath term is a multiple of a, these forming of
themselves the series
a, 2a, 3a, 4a, 5a, &c., a”,
and therefore, from the a” terms in the first series, we
must deduct the a”-' terms in the last, and the remain-
der will be the number of those terms in the first, that
are prime to N, or to a”; that is, a” — a” are the
number of integers prime to N ; but since N = a” we
have
a — 1
l
(Z N × ——,
(º,
*
* -º
–- - -
ºn,
a" – a”-1 - a” x
for the number of those integers; which is likewise the
form in question.
Again, if N = a” b", it is evident, from the same
consideration as before, that we shall have
a"~ b", terms divisible by a ;
a" b"-", terms divisible by b :
a"** b."-", terms divisible by a b.
But as the first expression includes all numbers divi-
sible by a, and the second all those divisible by b, it
follows, that the latter expression is included in each of
the former; and therefore we have
a"-" b" – a”-i b"--, terms divisible by a only;
a"b"-" — a”- b"-", terms divisible by b only;
a"-" b"-", terms divisible by a b ;
and these, together, include all those terms of the
series
1, 2, 3, 4, 5, &c., a” b",
that have any common divisor with a” b", or with N ;
and, consequently, their sum, taken from N, will be
the number of those that are prime to it: hence, then,
we have
a" b” * --> g”-1 b” sºmy a" bn-l –– a”-1 bn-1 --
(a" &m a"-1) b" sºm, (a” ſºme a"-1) bº-1 -:
(a" – a”-1) × (b" – bº) =
( *#) ( b — l
a" X × { blº X
(Z b
a – 1 b – 1
:- N X 3.
X 0. b
which is again the formula in question.
Let, now, N = a” b” cº, then, on the same principles
as above, we shall have
P = a”-1 bº cºp,
Q = a” b”-1 cºp,
R – a” b” cp-1,
S = a”-1 b”-1 cºp,
terms divisible by a ;
terms divisible by b :
terms divisible by c,
terms divisible by a b,
T = a”-1 b” cº-, terms divisible by a c,
V = a” b”-1 cº-, terms divisible by b c ;
W = a”-1 bººi cº-, terms divisible by a b c.
But since all the terms W are necessarily included in
those of S, T, and V, and these last again in P, Q,
and R, we shall have, by subtraction,
S — W, divisible by a b only ;
T – W, divisible by a c only;
W – W, divisible by b c only:
and then, again,
P – S – T + 2 W – W.; or,
P – S – T – W, divisible by a only;
Q – S – V + W, divisible by b only;
R – T – V + W, divisible by c only;
W, divisible by a b c only.
And, consequently, the sum of all these expressions
will be the number of terms that have a common divi-
sor with a "b" cº, or with N ; and, therefore, N minus
this sum will be the number of integers prime to N,
and less than itself; which, by addition and subtrac-
tion, will be expressed as follows:
N – P – Q – R + S + T -- W - W.
And by reestablishing again the values of P, Q, R, &c.,
it becomes
Sect. I.
Divisors.
S-N-Z
T H E O R. Y O F N U M B E R. S. 647
§: (a" b" cº - on-l b” cP) — (a" b”-1 cºp — a "-1 b”-l cº) * m (n + 1) (rt + 2) (m –H 3) &c. n. —— 772. #:
til #TS, tº
(a" b" cº-1 — am-1 bn cP-1) + 1 .. 2 3 4 &c. m + 1 2 Numbers.
(a" bº-1 cº-i – am-1 bº-1 cP-1) =- Sºº-y-'
(a" gº m—) (b." CP — bn-l CP — b” cp-1 + bº-1 cº-) -:
(a" *º m-) X (b" tº b”-1) gº (cp tº- cº-) -:
N × a — 1 b — I X c — I
b C
X
the same form as before.
Amd, exactly in the same manner, if N were the pro-
duct of a greater number of factors, we should still
find, that the number of integers less than, and prime
to N, would be represented by
b – 1 c – 1 d – 1
N × - x -: d
Where it is only necessary to observe, that unity is
included as one of those integers.
Find how many numbers there are under 100 that
are prime to it.
First, 100 = 2* 5°; therefore,
a — 1
X
, &c.
2 – l 5 – 1
100 × 2 X 5 - 40,
the number sought: these being as follows; viz.
1 13 27 39 51 63 77 89
3 17 29 41 53 67 79 91
7 19 31 43 57 69 81 93
9 21 33 47 59 71 83 97
| 1 23 37 49 61 73 87 99
Again, how many numbers are there less than 360
which are also prime to it.
360 = 2*. 3*. 5*, therefore
2 – l 3 — I
360 × —
X 2 X 2
the number sought.
* = 1 = 96
2 *mm 3.
II. Of Figurate Numbers, &c.
19. The theory of figurate, amicable, and polygonal
numbers must be admitted to be rather a subject of
curiosity than of utility, we shall confine ourselves,
therefore, almost entirely to a definition of them, and
to a statement of some of their properties, and for the
investigations we shall be content to refer to Barlow's
Theory of Numbers.
DEFINITIONs.
20. A Perfect Number is that which is equal to the
sum of all its aliquot parts, or of all its divisors.
6 6 6
Thus 6 - 2 +. 3 + 6 and is, therefore, a per-
fect number.
21. Amicable Numbers are those pairs of numbers
each of which is equal to all the aliquot parts of the
other: thus 284, and 220, are a pair of amicable num-
bers; for it will be found, that all the aliquot parts of
284 are equal to 220; and all the aliquot parts of
220 are equal to 284.
22. Figurate Numbers are all those which fall under
the general expression
and they are said to be of the 1st, 2d, 3d, &c. order,
according as m = 1, 2, 3, &c.
23. Polygonal Numhers are the sums of different and
independent arithmetical series, and are termed lineal or
natural, triangular, quadrangular or square, pen-
tagonal, &c., according to the series from which they
are generated.
24. Natural Numbers are formed from a series of
units; thus,
Units, 1 1 1 1 1, &c.
Nat. Numbers, 1 2 3 4 5, &c.
25. Triangular Numbers are the successive sums of
an arithmetical series, beginning with unity, the com-
mon difference of which is l; thus,
Arith. Series, 1 2 3 4 5, &c.
Trian. Num. 1 3 6 10 15, &c.
26. Quadrangular, or Square, Numbers are the sums
of an arithmetical series, beginning with unity, the
common difference of which is 2 ; thus,
Arith. Series, I 3 5 7 9
Quadrang. or
Square Num.
27. Pentagonal Numbers are the sums of an arith-
metical series, beginning with unity, the common dif-
ference of which is 3; thus,
Arith. Series, 1 4 7 10 13 16, &c.
Pentagonal g
º 1 5 12 22 35 5.1, &c.
II, &c.
I 4 9 16 25 36, &c.
And universally, the m-gonal Series of Numbers
is formed from the successive sums of an arithmetica)
progression, beginning with unity, the common differ.
ence of which is m — 2.
28. Perfect Numbers are expressed, or determined,
as follows:
Find 2" – l a prime number, then will N = 2*-*
(2” – 1) be a perfect number. For from what has
been demonstrated in the preceding section, the sum
of all the divisors of this formula will be represented by
2” – I (2” – 1)” – I
2 – 1 (2" — I) — I
because 2" — 1 is a prime by hypothesis. But in this
expression l is included as a divisor, which must be
excluded in the case of perfect numbers; exclusive of
this, therefore, the formula will be
2” – 1 (2” – 1)4 – 1
2 – 1 (2” – 1) – 1
(2” – 1) x (2” – 1 + 1) — 2"-" (2” – 1) =
2 (2” – 1). 2” – 2'-' (2” – 1) = 2*-*. (2” – 1) = N,
that is, the sum of all the aliquot parts of N exclusive
of itself, or of 1 as a divisor, is equal to N, and is,
therefore, by the definition a perfect number.
The only perfect numbers known are the following
eight:
– 2"-" — 1 (2” – 1) =
6 335,50336
28 85898.69056
496 13743869 1328
81.28 230584,300.81399.52128.
4 P 2
648 T H E O R Y O F N U M B E R S.
Theory of 29. To find a pair of amicable numbers N and M, 4 m + 1, or 4 m -- 3; but 4 m + 3 = 4 (n + 1) – 1; = Sect. III.
****, or such a pair that each shall be respectively equal to 4 n' – 1, therefore all prime numbers are included in N:
all the divisors of the other. the general formula 4 n + 1. º
Make N = a” b” cº, &c., and M = a” Bº Y", then, Mr-
according to the definition, and from what has been
demonstrated in the last section, we must have
a"+4 – 1 bº-Fi – 1 cp-Fi – I e
- = N + M,
a — I b – 1 X c – 1 +
cº-Fl mºme l g-H — 1 y-Fi yms I
= M –– M.
a – 1 £3 – 1 ºf — 1 —-
Firid, therefore, such a power of 2 as 2", that
3. 2" — 1, 6. 2" — 1, and 18. 2” – 1
may be all prime numbers, then will
N = 2^++. d, and M = 2^+ b c
be the pair of amicable numbers sought.
The least three pair of amicable numbers are,
284 220
I 7296 184 16
93635.83 94.37056.
III. Of the forms and properties of Prime Numbers.
30. If a number cannot be divided by some other
number, which is equal to, or less than, the square
root of itself, that number is a prime.
For every number p, that is not a prime, may be re-
presented by p = a b. Now if a = b, then a and b are
each equal to w/ p ; and, consequently, p, which is not
a prime, is divisible by w/ p. Again, if a P V p,
then will b → y p ; for otherwise, we should have
a × b = a b > p, which is contrary to the supposi-
tion ; therefore, if a P. V p, then will b > v p ; and
if b > y p, then will a >< x/ p , and, consequently,
since p is divisible both by a and b, it is divisible by a
number less than the square root of itself: and this is
evidently true of all numbers that can be resolved into
the form p = a b ; that is, of all numbers that are not
primes: therefore, if a number cannot be so divided,
that number is a prime.
Hence, in order to ascertain whether a given number
be a prime number or not, we must attempt the divi-
sion of it by all the prime numbers less than the
square root of itself; and if it be not divisible by any
of them, it is a prime. It is obvious, that we need
only essay the division by prime numbers; for if it be
divisible by a composite number, it is evidently also
divisible by the prime factors of that divisor. This
method, however, although it admits of some contrac-
tions, is, notwithstanding, extremely laborious for large
numbers; nor has any easy, practical rule been yet
discovered, for ascertaining whether a given number be
a prime or not.
31. Of the different linear forms of prime numbers.
Every prime number greater than 2, is of one of the
forms 4 m + 1, or 4 m — 1. For every number divided
by 4 will leave a remainder 1, 2 or 3; that is, every num-
ber whatever is included in one of the four forms
4 m, 4 n + 1, 4 n + 2, 4 m + 3 ;
but the first and third of these are not primes, being
both even or divisible by 2, therefore all prime num-
bers must fall under one of the other two, viz.
**i.
DEDUCTIONs.
(1.) In a similar way it may be shown, that all prime
numbers are included in the forms
8 m =E 1 8 m -- 3
6 m =E 1
12 m + 1 12 m + 5,
&c. &c.
(2.) It may be proper just to observe, that although
all prime numbers are included in these sets of for-
mula, the prime number 2 only excepted, yet the con-
verse is not true, viz. that all numbers contained in
these forms are prime numbers; indeed no algebraical
formula whatever can be found that includes prime
numbers only. This is demonstrated in the following
proposition.
32. No algebraical formula can contain prime num,
bers only.
Let p + q a + r a* + sa" –- &c.
represent any general algebraical formula. It is to be
demonstrated that such values may be given to a, that
the formula in question shall not with that value pro-
duce a prime number, whatever values are given to
p, q, r, &c. -X-
For suppose, in the first place, that by making
a = m, the formula
P = p + q m + r m” + sm” +, &c.,
is a prime number.
And, if now we assume a = m -- ºf P, we have
p = • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.
q r = . . . . . . . . . . . . . . . . . . . . . . . . . . . . q m + q j, P
* r = . . . . . . . . . . . . . . r m2 + 2 r m ºf P + r p? P.
S v8 e- sm” + 3 s m2 ºff P + 3 s m ºff? P2 + sqº Pº
&c. &c.
Or,
p + qi + ra’--st" = (p + qm-i-rm?-- sm” +, &c.) +
P(q p + 2 r m 'ft + 3 s m° ºp) + P2 (r p2 + 3 s m 'pº) +
s pº Pº = P + P(q ft + 2 r m p + 3 s m? q}) +
Bº (r pº -- 3 s m p2) + s $3 P3.
But this last quantity is divisible by P; and, conse-
Quently, the equal quantity
p + q a + r aº –– s a " +, &c.
is also divisible by P, and cannot, therefore, be a prime
number. Hence, then, it appears, that in any alge-
braical formula, such a value may be given to the in-
determinate quantity, as will render it divisible by some
other number ; and, therefore, no algebraical formula
can be found that contains prime numbers only.
But although no algebraical formula can be found
that contains prime numbers only, there are several
remarkable ones that contain a great many; thus
a” + a + 41, by making successively a = 0, 1, 2, 3, 4,
&c., will give a series 41, 43, 47, 53, 61, 71, &c., the
first forty terms of which are prime numbers. The above
formula is mentioned by Euler in the Memoirs of Ber-
lin, (1772, p. 36.)
T H E O R Y O F N U M B E R S,
649
Theory of
Numbers.
To the above we may add the following a 2–1 r + 17,
and 2 a.” -- 29, the former has seventeen of its first
~~~' terms primes, and the latter twenty-nine.
Fermat asserted that the formula 2" + 1 was always
a prime, while m was taken any term in the series
1, 2, 4, 8, 16, &c.; but Euler found that 29% -- 1 =
641 × 6700417 was not a prime.
33. The number of prime numbers is infinite.
For if not, let the number of them be represented
by n, and the greatest of all those primes by p, then
it is evident that the continued product of all the prime
numbers not exceeding p, as
2. 3. 4. 5, &c. p
will be divisible by each of those numbers, and, there-
fore, if 1 be added to the product, the sum will be divi-
sible by no one of them ; consequently, if the formula
(2.3.4. 5, &c. p) -- 1
be divisible by any prime number, it must be by some
one greater than p, and if not it will be itself a prime,
and, consequently, greater than p. Hence there must
be a prime number greater than p, and, consequently,
a greater number of prime numbers than m, and the
same may be shown, however great m and p may be,
therefore the number of prime numbers is infinite.
34. If a and b be any two numbers prime to each
other, and each of the terms of the series
b, 2 b, 3 b, 4 b, &c. (a – 1) b
be divided by a, they will each leave a different position
remainder.
For if any two of these terms when divided by a
leave the same remainder, let them be represented
by a b, y b : then it is obvious, that a b – y b would be
divisible by a, or (a — y) b would be divisible by a.
But this is impossible, because a is prime to b, and
a — y is less than a, (art. 9,-5 :) therefore b (a — y)
is not divisible by a ; but it would be so divisible if
the terms a b, y b left the same remainder ; these do
not, therefore, leave the same remainder, conse-
quently every term of the series
b, 2 b, 3 b, &c. (a – 1) b
divided by a, will leave a different remainder.
DEDUCTIONs.
(1.) Since the remainders arising from the division
of each term in the series
b, 2 b, 3 b, &c. (a – 1) b
by a, are different from each other, and a — 1 in num-
ber, and each of them necessarily less than a, it follows
that these remainders include all numbers from
l to a - 1.
(2.) Hence, again, it appears, that some one of the
above terms will leave a remainder l ; and that, there-
fore, if b and a be any two numbers prime to each
other, a number a 3 a may be found that will render
b c – l divisible by a ; or, the equation b a - a y = 1
is always possible if a. and b are numbers prime to each
other.
And it is always impossible if a. and b have any com-
mon measure, as is evident; because one side of the
equation b a - a y = 1 would be divisible by this
common measure ; but the other side, l, would not be
so : therefore, in this case, the equation is impossible.
(3.) We have seen, in the foregoing deduction, that
the equation b a' — a y = 1 is always possible, when a
and b are prime to each other ; and the same is evi-
dently true of the equation b a' - a y = — I, for a – 1
is one of the remainders in the above series, so that a
value r < a. may be found, that renders b a' — (a – 1)
divisible by a ; or the equation b a' – a y = a – l is
always possible; but this is the same as b a - a
(y – 1) = – 1; or, making y – 1 = y, b a' – a y'—
— 1 is always possible ; and, consequently, the equa-
tion a a - b y = + 1 is always possible, when a and b
are prime to each other.
35. If a be any prime number, then will the for-
mula
1. 2. 3. 4. 5, &c. (a – 1) + 1
be divisible by a.
For it is demonstrated in our preceding second de-
duction, that, if a. and b be any two numbers prime to
each other, another number a may be found < a., that
renders the product b c – l -º a ; or, which is the
same thing, b a = y a + 1 ; and that there is only one
such value of a 3 a may be shown as follows:
The foregoing equation gives, by transposition,
b a' — a y = 1 ;
and, if it be possible, let also
b r'– a y' = 1;
and make a' = a + m, and y' = y + m, where m is
necessarily less than a, because both a and a' are so
by the supposition. Now, by this substitution, we
have
(b r + b m) — (a y + a n) = 1; but
ba, — a y
therefore + b m = HE a n, or b m -- a ; but this is im-
possible, since b is prime to a, and m × a, (art. 9,-5.)
There cannot, therefore, be two values of a less than
a, that renders the equation ba – a y = 1 possible.
But in the series of integers
1, 2, 3, 4, 5, . . . . . .
every term is prime to a, except the first, a being itself
a prime; if, therefore, we write successively, b = 2,
b' = 3, b" = 4. &c., a corresponding term ºr, in the
same series, may be found for each distinct value of b,
that renders the product a b = a y + 1, a 'b' ºr, a y' + 1,
a" b" HR a y + 1, &c.; and it is evident, that no one of
these values of a can be equal either to 1, or a – l ; for,
in the first case, we should have 1 × b = a y + 1,
which is impossible, because b >< a ; and the second
would give (a – 1) b = a y + 1, or a (b – y) =
b -- 1 ; that is, b -- 1 -- a ; which can only be when
= a – I, or when b = a, which case is excepted,
because we suppose two different terms of the series.
In fact, since (a — 1)* = a y + 1, there can be no other
term, in the same series, that is of this form ; for if
a” ºf a y' + 1, then (a – 1)* — a 2 would be divisible
by a, or (a — 1 -- a) x (a – 1 – ar) -- a, which is
impossible, since each of these factors is prime to
a, as is evident, because a < a., and a is a prime
number.
= 1 ;
a — 1,
* To save the repetition of the words divisible by, which fre-
quently occur, the sign -- is used to express them ; and for the
same reason the symbol HE is introduced, to express the words of the
form of, which are also of frequent occurrence.
Seet. III.
Prime
Numbers.
S-V-”
650 T H E O R Y OF N U M B E R. S.
Theory of Hence, then, our product we may derive a great many others; as Sect. IV.
Numbers. 1. 2. 3. 4.5 (a – 1), becomes Square
a £º e º & tºº e º Aº e s sº a º a 3. 1*. 2%. 3, 4, 5, &c., – 3 sº 1 •
\ / I . b ar. b/a'. 5"a".... a = 1; C º ) (a — 1) + = e, an integer; Number.
but each of these products, bar, b'a', b"a", &c., is, as
we have seen, of the form a y + 1 ; therefore, their con-
tinued product will have the same form, and the whole
product, including I and a — 1, will be
tº (a y + 1) x (a – 1) = a” y + a y -- a - 1,
to which, if unity be added, the result will be evidently
divisible by a, that is, the formula
(a – 1) + 1
is always divisible by a, when a is a prime number.
DEDUCTIONs.
(1.) The product,
1. 2. 3. 4.5 . . . . . .
is the same as
1 (a – 1) 2 (a – 2) 3 (a – 3), se, (+)
and this product, with regard to its remainder, when
divided by a, is the same as
+ 12. 22. 32.42 . . . . . . (*, !).
2
the ambiguous sign being plus (+) when a – 1 is
even, and minus (–) when a – 1 is odd ; that is,
+ when a is a prime number of the form 4 m + 1,
and — when a is a prime number of the form 4 m – l;
also this product,
— i \, 2
+ 12. 22. 32.42 . . . . . . ( 2 !)
is the same as
+(..*.*.* e is a ºn tº º
and, consequently, from what is said above relating to
the ambiguous sign, we shall have
{(i. e.g., & s tº a s ſº *#) +1}-a.
}-a.
when a tº 4 m — 1.
Whence it follows, that every prime number of the
form 4 m + 1 is a divisor of the sum of two squares.
Again, the latter form may be resolved into the two
factors
{(i.e. 3........ **) + 1}x
{(1,2,3,4 tº s º º tº º **)-1 }
which product, being divisible by a, it follows, that a
is a divisor of one or other of these factors, when it is
a prime number of the form 4 m — 1.
(2.) From the first product, which we have demon-
strated to be divisible by a, viz.
1. 2. 3. 4, &c., (a — 1) + 1
&
= e, an integer,
1*. 2°. 39.4. 5, &c., (a — 4) (a – 1) + 1
= e, an integer,
(M,
and so on, till we arrive at the same form as that in the
first deduction.
The theorem above demonstrated was first proposed
by Sir John Wilson, as we are informed by Waring, in
his Meditationes Algebraica, p. 380; but, notwith-
standing the simple principles on which its demonstra-
tion is founded, it escaped the observation of these two
celebrated mathematicians; the latter of whom speaks
of it, at the place above quoted, as an extremely dif-
ficult proposition to demonstrate, on account of our
having no formula for expressing prime numbers.
Lagrange was the first who demonstrated this theorem,
in the New Memoirs of the Academy of Berlin, 1771,
(which demonstration is, as might be expected from
the celebrity of its author, very ingenious ;) and, after-
wards, Euler gave a different demonstration of the
same proposition, in his Opusc. Analyt, tom. i. p. 329,
which is upon a similar principle to the foregoing ; and,
finally, Gauss, in his Disquisitiones Arithmetica, ex-
tended the theorem by demonstrating, that “The pro-
duct of all those numbers less than, and prime to a given
number a + 1 is divisible by a ;” the ambiguous sign
being —, when a is of the form p", or 2 p", p being any
prime number greater than 2; and, also, when a = 4;
but positive in all other cases, (Recherches Arithmé-
tiques, p. 57.)
The theorem of Sir John Wilson furnishes us with
an infallible rule, in abstracto, for ascertaining whether
a given number be a prime or not; for it evidently be-
longs exclusively to those numbers, as it fails in all
other cases, but is of no use in a practical point of
view, on account of the great magnitude of the product
even for a few terms.
IV. On the forms of Square Numbers.
36. Every square number is of one of the forms 4 n,
or 4 m + 1. -
Every number is either even or odd, that is, every
number is of one of the forms 2 n, or 2 m + 1 ; and,
consequently, every square is of one of the forms
4 m? --, 4 m.
4 m? -- 4 n + 1 + 4 n + 1.
DEDUCTIONs.
(1.) Every even square number is divisible by 4.
(2.) Since every odd square by the above is of the
form 4 (n2 + n) – l ; and since at” + n is necessarily
even, it follows, that every odd square is of the form
8 m + 1. And, consequently, no number of the
forms 8 m + 3, 8 m + 5, 8 m + 7, can be a square
number.
(3.) The sum of two odd squares cannot be a square;
for
(8 n + 1) + (8 n + 1) + 4 m + 2,
which is an impossible form.
T H E O R Y O F N U M B E R S. 651
Theory of 37. Every square number is of one of the forms 5 m, 0°, 1*, 2*, 3°, &c (; a)2, a being even, Sect. IV.
Numbers. or 5 m + 1. and as Square
S-N-' For all numbers compared by the modulus 5, are of (Z - Numbers.
one of the forms >~~~
5 m, 5 m + 1, 5 m + 2,
and all squares, therefore, are of one of the forms
25 m? tº 5 m.
25 n° -E 10 m + 1 + 5 m + 1
25 m” + 20 n + 4 + 5 m + 4, or 5 m — 1.
Therefore all squares are of one of the forms 5 n, or
5 m + 1. -
DEDUCTIONs.
(1.) If a square number be divisible by 5, it is also
divisible by 25; and, if a number be divisible by 5,
and not by 25, it is not a square.
(2.) No number of the form 5 m + 2, or 5 m + 3, is
a square number.
(3.) If the sum of two squares be a square, one of
the three is divisible by 5, and,. consequently, also by
25. For all the possible combinations of the three
forms 5 m, 5 m + 1, and 5 m – 1, are as follows :
(5 m + 1) + (5 n' + 1) = 5 m +2,
(5 n – 1) + (5 n' – 1) + 5 n – 2 = 5 m + 3,
5 m + 5 m' HR 5 m,
5 m + (5 m' + 1) = 5 m + 1,
5 m. + (5 n' – 1) + 5 m — 1,
(5 m + 1) + (5 n' – 1) == 5 m.
Now of these six forms, the latter four have one of the
squares divisible by 5, and, therefore, also by 25. And
the two first are each impossible forms for square num-
bers ; that is, neither of these two combinations can
produce squares: therefore, if the sum of two squares
be a square, one of the three squares is divisible
by 25.
(4.) In a similar way it may be shown, that all
square numbers compared by modulus 10, are of one
of the forms
10 m, 10 m + 5, 10 m + 1, 10 m -- 6,
10 n + 4, or 10 n +9.
Therefore all square numbers terminate with one of
the digits 0, 1, 4, 5, 6, or 9; and hence, again, no
number terminating with 2, 3, 7, or 8, can be a square
number.
(5.) By examining, in like manner, the forms of
squares to modulus 100, we may deduce the following
properties.
(6.) A square number cannot terminate with an odd
number of cyphers.
(7.) If a square number terminate with a 4, the last
figure but one must be even.
(8.) If a square number terminate with a 5, it must
terminate with 25.
(9.) If the last digit of a square be odd, the last
digit but one must be even ; and if it terminate with
any even digit except 4, the last but one must be odd.
(10.) A square number cannot terminate with more
than three equal digits, unless they are 0's; nor can it
terminate with three, unless they are 4's.
38. All square numbers are of the same form with
regard to any modulus a, as the squares
*- J l -
2 T 2 92 ºº e
o, ºr se ( 2 ), a being odd.
For every number may be represented by the formula
a n + r, in which r shall never exceed 4 a., (Art. 8.)
Now (a n + r.)2 = a” n°-E 2 a r n + rº,
where it is obvious that r* and (a n + r.)2 will leave the
same remainder, when divided by a ; therefore (a n + r.)”
and r* will be of the same form compared by modulus a ;
but r never exceeds , a, therefore all numbers com-
pared by modulus a are of the same forms as
02, 12, 22, 33, &c. re,
or, as the squares
0°, 1*, 2*, 3°, &c. (4 a)”, when a is even,
and as
0°, 12, 2’, 32, &c. (: g
IN2
) when a is odd.
DEDUCTIONs.
(1.) When a is even, the general formula
a? n° -E 2 a m r -- rº becomes
4 a” m2 + 4 a'n r + ra
== 4 a' (aſ n2 =E m r) + re.
Therefore all square numbers are of the same form to
modulus 4 a, as the squares
02, 12, 22, 3°, &c. aº ;
and hence we see immediately, that all square num-
bers to modulus 8, must be of the same forms as the
squares
02, 12, 22 ;
that is, they are all of the form
8 m, 8 m + 1, 8 m + 4,
as we have already demonstrated.
(2.) The following tables exhibit the possible and
impossible forms of square numbers for all moduli
from 2 to 10.
Possible formula.
2 m, 2 m + 1,
3 m, 3 n + 1,
4 m, 4 m + 1,
5 m, 5 m + 1,
6 m, 6 m + 1, 6 m + 3, 6 n + 4,
7 m, 7 n + 1, 7 n + 2, 7 m + 4,
8 m, 8 m + 1, 8 n + 4, -
9 m, 9 m + 1, 9 m + 4, 9 m + 7,
10 m, 10 m + 1, 10 n + 4, 10 m + 5.
Impossible formula.
3 m,
4 m, 4 m + 3,
5 m, 5 m + 3,
6 m, 6 m + 5,
7 m, 7 m + 5, 7 m + 6,
8 n, 8 n + 3, 8 m + 7,
9 m, 9 m + 3, 9 n + 5, 9 m + 8,
10 m, 10 m =E 3.
652 T H E O R. Y O F N U M B E R S.
Theory of V. Of the possible and impossible forms of Indeter- being an integer. Now this equation is the same as Sect. W.
Numbers. minate Equations of the first and second degree. (y –- 1) b — r Indeter-
\-/~ a = b – l — ; Inlinate
(Z Equations.
39. If a and b be any two numbers prime to each
other, the equation
a a - b y = + c
is always possible ; and an infinite number of different
values may be given to a and y, that answer the con-
dition of the equation in integers.
By (Art. 34,-2) it appears the equation
a al – b y = 1
is always possible while a and b are prime to each
other, and, consequently,
a c a' – b c y = + c, or a w' – b y' = + c,
by making ca' = a-', and c y = y';
and we have, evidently, the same result if we write
a (w' + m b) for a w'
b (y' + m a) for b y',
for these still give
a (r' + m b) – b (y' + m a) = -E c.
Or, again, making
a' + m, b = r
Sy’ =E m a = y,
our equation becomes
a 4 – b y = + c,
which is therefore always possible when a and b are
prime to each other.
And it is evident, that by means of the indeterminate
sign +, and indeterminate quantity m, the formulas
a' + m, b = a
y’ + m a = y,
will furnish an indefinite number of values of a and y,
which will answer the conditions of the problem.
It is also obvious, that m may be so assumed that a
shall be less than b, and y less than a.
EXEDUCTIONs.
(1.) In any of our future investigations we may, there-
fore, when the state of the question requires such an
artifice, substitute ta' — w y = c ; t and u being num-
bers prime to each other, and c any number whatever
prime to each of them, without inquiring about the
particular values of a and y, it being sufficient for our
purpose, in many cases, to know that the equation is
possible.
(2.) But if t and u have any common measure, them
such a substitution cannot be made, unless c has the
same common measure.
40. The equation a r + b y = c is always possible,
if a. and b be prime to each other, and
c > (a b – a – b).
For let c = (a b – a – b) + r, then the equation
becomes
a r + b y = (a b – a – b) + r ;
the possibility of which depends upon
a b – a – b – b y + r
Q,
flººm,
and, therefore, it depends upon the possibility of
(y-H 1) b - r = ap' being an integer;
O.
= y',
= 7° 3
or, which is stil the same, by calling y + 1
upon the possibility of the equation y' b − a ſt'
which we have seen may always be established, so
that y' < a., or y + 1 < a ; by the foregoing pro-
position.
Since, then, in the equation
(V-E 1)" - " - aſ
(M,
gy + 1 is less than a, aſ must necessarily be less than 5,
and, consequently,
2
* = b – 1–9 tº *-* = b - 1 - 2,
and since a 3 b, therefore r = b – 1 – aſ E 0, or
some integer number: whence the equation
a m + b y = c
is always possible when a and b are prime to each
other. s
41. Investigation relative to indeterminate integral
equations of the form
a tº + b u% = w”.
First, in an equation of this form, we may always
consider a and b as quantities that have no square fac-
tor, or divisor; for, if a = a' $2, and b = b'6°, our
equation becomes a' (pº tº + b 0°u% = w”; or, making
q t = tº, and 0 w = w', we have aſ tº + b' uſ? = wº;
and, consequently, if the above equation obtain when
the quantities a and b, or either of them, have a square
divisor, it may always be put in another form,
a' tº + b' u" - wº, in which the similar quantities
aſ and bº have not a square divisor; and, therefore, in
what follows, with regard to the possibility or impos-
sibility of equations of the form a tº + b wº, we may
always consider a and b as not having a square
divisor.
Again, if the equation a t” + b u% = w” be pos-
sible, when tº, u%, and wº, have a common square divisor
Ø9, it is also possible when divided by it ; thus, if
a @” tº + b º uſ" = @” w” be possible, so also is
a tº + b w? = w”,
which is a similar equation to the first, and in which
t”, uſ”, and w”, have now no common square divisor.
And it is evident, that no two of these squares can
have a common divisor, unless the third square has
the same. For, if it be possible, let tº - tº º, and
w” = w” p"; then, a tº q} + b w!” pº = wº, where the
first side of the equation is divisible by غ, but the
second is not, by the supposition, and yet it is equal
to the first, which is absurd : and the same may be
demonstrated if any other two of those squares are
supposed to contain a square divisor, not common with
the third ; a and b having no square divisor, as is
shown above. --
Hence, then, we may draw this conclusion, in any
case where we are investigating the possibility of an
equation of the form a tº + b w” = wº, the quantities
{e
T H E O R Y O F N U M B E R S.
653
Theory of a and b may be considered as not containing a square
Numbers, divisor; and also the three quantities t, u, and w, as
being prime to each other: for if the equation be pos-
sible under these conditions, it is possible when those
quantities have a common measure; and if it be im-
possible under the former case, it is also impossible
under the latter.
And it may be farther observed, that if any equation
of the form a tº + b w” = w be impossible in integers,
it is so likewise in fractions; for make
r 3/
tº; *
t = —, w = + , and w = —, then it becomes
S Q) 2
º 2 q2 e º tº
a – HE u * = −; which reduces it to this
Sº v2 22
s” v2 r2
a rºvº -- b sº y? = ; or, making
2 os? ~2
sº tº r" ..s
= MUT,
r" v = tº, sº y = wº, and —
which last must evidently be an integral square, we
have again a tº + b u” = w”; so that the possibility of
any fractional equation of this kind depends upon
a similar integral equation, and if, therefore, an equa-
tion be impossible, in integers, with any specified value
of a and b, it is also impossible in fractions.
DEDUCTION.
All that has been proved of the equation a tº +
b w” = w” is also true of the equation a tº + b w” = w8,
and generally of the equations a t” + b u" = w”, it
being always understood, that neither a nor b contain
any factor that is a complete m” power. -
42. The equation (3 p + 2) tº + 3 q u" = wº is always
impossible either in integers or fractions. We have
seen in the foregoing article, that it will be sufficient to
consider t and u as integers, and that we always suppose
tº, wº, wº to be prime to each other. Now since 3 q w? is
always of the form 3 m, whatever may be the form of wº,
and since tº must be one of the forms 3 m or 3 m + 1,
(Art. 38,) we shall either have -
First (3 p + 2) 3 m + 3 quº = w”, or
Second (3 p + 2) (3 m + 1) + 3 q u" = wº.
But in the first equation, where we suppose tº t+, 3 m,
we have the first side of the equation divisible by 3, and,
consequently, the other side w” is also divisible by 3 ;
that is, both tº and wº are divisible by 3, which cannot
be, because they are prime to each; therefore the equa-
tion, when tº is of the form 3 m, is impossible.
Again, in the second equation, in which we suppose
tº tº 3 m + 1, we have
(3 p + 2) × (3 m + 1) + 3 q wº e w”, or
9 p n + 6 m + 3 p + 2 + 3 q u" = wº, or
3 (3 p n + 2 m + p + q w”) + 2 = w”, or
w° tº 3 m + 2,
which is impossible, (Art. 38;) therefore the equa-
tion (3 p + 2) tº + 3 quº = w” is impossible, under the
limitations of the problem.
DEDUCTIONs.
(1.) By means of this general form we may derive
many particular cases of impossible equations, by
WOL. I.
giving different values to p and q : thus is q = 1, and sect. V
p = 0, then 2tº + 3 us - wº, º:
p = 1 . . . . 5 tº + 3 w? = wº, Equations.
p = 2 . . . . 8 tº + 3 w? = w”, -—
are all impossible equations.
And if q = 2, then
p = 0 gives 2 p” + 6 w” = wº,
p = 1 . . . . 5 p” -E 6 w” = wº,
p = 2 .... 8 p” + 6 w” = wº,
which are all impossible equations.
(2.) In a similar manner it may be demonstrated,
that the general equations
(5 p + 2) tº at 5 q u" = w”,
(7 p + 3) tº + 7 q u” = w”,
(7 p + 5) tº + 7 q u” = w”,
(7 p + 6) tº + 7 quº = w”,
are all impossible equations, either in integers or frac-
tions, under the same limitations as before.
And from these general forms we readily deduce the
following particular cases,
2 tº 4, 5 wº = w”, 2 tº ºt, 10 w” = w”,
3 tº +, 5 wº = w”, 3 tº ºt, 10 w” = w”,
7 tº at 5 w” = w”, 7 tº #, 10 wº = w”,
8 tº at 5 wº = w”, 8 tº ºt, 10 wº = w”,
&c. &c.
3 p” + 7 w” = w”,
5 p" + 7 w” = w”,
6 p" + 7 w” = w”,
3 p" + 14 w” = w”,
5 p" -E 14 u" = w”,
6 p" -- 14 u" = w”,
=
w”,
10 p" + 7 w” = wº, 10 p" + 14 w”
12 p” + 7 w” = wº, 12p* + 14 wº = w”,
13 p” + 7 wº = w”, 13 pº -- 14 wº = w”,
&c. &c.
which are all impossible equations.
(3.) By examining the above impossible forms it will
be seen, that the multipliers of tº are all impossible
forms with regard to that particular prime modulus to
which they are referred, thus
3 p + 2 to modulus 3,
5 p =E 2 to modulus 5,
7 p + 3 -
7 p + º to modulus 7 ;
7 p + 6
and we are hence led to an inference, that the same is
true for any other prime modulus: that is, the
Equations (11 p + 2) tº -E 11 q u” = wº,
(11 p + 6) tº + 11 quº = wº,
(11 p + 7) tº + 11 g wº = wº,
(11 p + 8) tº + 1 I q tº − w”,
(11 p + 10) tº + 11 quº = wº,
are all impossible, while q is taken prime to 11.
Also, (13 p + 2) tº at 13 q wº = wº,
(13 p + 5) tº at 13 q u" = w”,
(13 p + 6) tº at 13 q w” = wº,
when q is taken prime to the modulus 13.
4 Q
654
T H E O R Y OF N U M B E R S.
Theory of And (17 p + 3) tº at 17 quº = wº,
Numbers. 17 p + 5) tº at 17 q w? = wº,
* p (17 p Q
(17 p + 6) tº # 17 g wº = w”,
(17 p +7) tº at 17 q w? = w”,
when q is taken prime to the modulus 17.
. Likewise (19 p + 2) tº + 19 q u% = w”,
(19 p + 3) tº + 19 q u% = wº,
(19 p + 8) tº + 19 q u” = wº,
(19 p + 10) tº + 19 quº = wº,
(19 p + 12) tº + 19 q w? = wº,
(19 p + 13) tº + 19 q w? = wº,
(19 p + 14) tº + 19 q w? = w?,
(19 p + 15) tº + 19 qu% = wº,
(19 p + 18) tº + 19 g wº = wº,
when q is prime to 19; are all impossible forms of
equations in rational numbers.
These latter forms are only deduced from observa-
tion, upon the supposition that the product of a pos-
sible and impossible form is also of an impossible form;
which property may be satisfactorily demonstrated; we
shall not, however, enter upon the inquiry in this place,
but refer the reader who is desirous of following out
this proposition, to Barlow's Theory of Numbers,
(Art. 51 and 52.) We shall here content ourselves
with the induction, and proceed to a practical applica-
tion of the theorem in question.
43. To ascertain the possibility or impossibility of
any equation of the form
a tº + b y” = c 2°.
First, since a possible and impossible form multi-
plied together always produce an impossible form, it
follows, that a aº is always of the same form as a, with
regard to possible or impossible; and, in the same
manner, b y” is of the same form as b, and c 2° of the
same form as c. Now a aº ºf n a, therefore c z* – b y”
must be also of the form n a 5 and, consequently, c 2"
must leave the same remainder, when divided by a, as
by” does when divided by the same: it is evident,
therefore, that these remainders must be both of the
class of possible remainders, or both impossible, for
otherwise they could not be equal; but these remain-
ders will be of the same classes as c and b are ; and
hence it follows, that, if c and b are both found among
the remainders to modulus a, or neither of them are
found there, the equation may be possible ; but if one
of them is found there, and the other mot, the equation
is certainly impossible. And, in the same manner, if
a and c be both found among the remainders to modul
lus b, or if neither of them be found there, the equation
may be possible; but if one is found there, and the
other not, the equation is certainly impossible. And,
for the same reason, a and — b, or, which is equiva-
lent, a and c – b, must be either both found among
the remainders of modulus c, or neither of them, if
the equation be possible. Having thus shown the
principle of the rule, it may be delivered more briefly
thus:
Find the forms of all squares to modulus a, or, which
is the same, the remainders arising from dividing the
squares,
1*, 2*, 3°, 4°, &c. (; a)”, by a ;
and if b and c are both found in this series of remain-
ders, or if neither of them be found there, the equation
may obtain; but if one of them be found there, and the
other not, the equation is certainly impossible, and it
will be needless to proceed any farther in the investi-
gation. But if one of the two first conditions have
place, then find the remainders of
1*, 2*, 3°, 4°, &c. (; b)”, divided by b :
and these remainders must be submitted to the same
test, with regard to a and c; and if one of them be
found there, and the other not, the equation is im-
possible, and we need proceed no farther in the inves-
tigation. But if this be not the case, find the re-
mainders of -
1*, 2*, 3°, 4°, &c. (; c)*, divided by c ;
and if a. and (c — b) be both found in this series, or if
neither of them be found there, the equation is pos-
sible, supposing the same to have had place in the
other two series; but otherwise the equation is cer-
tainly impossible.
It is to be observed, that when any one of those
three quantities is greater than the modulus, with the
remainders of which it is compared, it must be divided
by the modulus and the remainder used, instead of the
quantity itself. It may be also farther observed, that
if any one of the three quantities, a, b, or c, be unity,
only two trials will be necessary, and if two of them be
unity, but one.
These operations will be considerably abridged by
means of the following table, which exhibits the re-
mainders to every modulus, from 2 to 51, excepting
only those numbers that contain square factors, because
a, b, and c, contain no square factors (by Art. 41 ;)
and hence the possibility or impossibility of any equa-
tion, in which the coefficients do not exceed 50, may be
ascertained by inspection.
Table of the Remainders of Squares to every Modulus,
from 2 to 51.
Moduli. Remainders.
/- _\- Y
* | }
3
5 1 4
6 1 3 4
7 1 2 4
10 1 4 5 6 9
ll 1 3 4 5 9
13 1 3 4 9 10 12
14 1 2 4 7 8 9 11
15 1 4 6 9 || 0
17 1 2 4 8 9 13 15 16
19 1 4 5 6 7 9 ll 16 17
21 1 4 7 9 15 16 18
22 1 3 4 7 9 11 12 14 15 16 20
23 1 2 3 4 6 8 9 12 13 16 18
26 1 3 4 9 10 12 14 16 17 22 23 25
29 1 4 5 6 7 9 13 16 20 22 23 24 25 28
30 1 4 6 9 10 15 16 19 21 24 25
31 1 2 4 5 7 8 9 10 14 16 18 19 20 25 28
33 1 3 4 9 12 15 16 22 25 27 31
34 | 1 2 4 8 9 13 15 16 17 18 19 21 25 26 30
l 32 33
35 1 4 9 11 14 15 16 21 25 29 30
37 | 1 3 4 7 9 10 ll 12 16 21 25 26 27 28 30
i 33 34 36
38 1 4 5 6 7 9 ll 16 17 19 20 23 24 25 26
{ 28 30 35 36
39 1 3 4 9 10 12 13 16 22 25 27 30 36
41 1 2 4 5 8 9 10 16 18 20 21 23 25 31 32
{ * 33 3: 3; 3, 4.
42 1 4 7 9 15 16 18 21 22 25 28 30 36 37 39
43 ſ l 4 6 9 10 11 13 14 15 16 17 21 23 24 25
l 31 35 36 38 40 41 .
46 f 1 2 3 4 6 S 9 12 13 16 18 23 24 25 26
l 27 29 31 32 35 36 39 41
47 1 2 3 4 6 7 8 9 12 14 16 17 18 21 24
- { 25 27 28 32 34 36 37 42
51 i l 4 9 13 15 16 18 19 21 23 25 30 34 36 49
l 43 49 -
Sect, V.
Indeter-
minate
Equations.
T H E O R Y O F
N U M B E R S. 655
Theory of
Numbers.
Example 1.
equation 7 a.” -E 11 y” = 132° be possible or impos-
sible. . -
11 := 7 m + 4, and 13 tº 7 m + 6.
Now 4 is found in the table to belong to modulus 7,
but 6 is not found there, whence the equation is im-
possible.
Example 2. Find whether the equation
7 *-i- 11 y2 = 23:2° be possible or impossible.
Il == 7 m + 4, and 23 = 7m + 2.
And 4 and 2 being both found to belong to modulus 7,
the equation may be possible.
Again,
7 = 11 m + 7, and 23 = 11 m + 1.
Now one of these remainders, 1, belongs to modulus
11, but 7 does not, therefore the equation is impos-
sible.
Example 3. Find whether the equation
14 a 2–1–6 y” = 17 2° be possible or impossible.
6 tº 14 m + 6, and 17 tº 14 n + 3.
And neither 6 nor 3 belongs to modulus 14, therefore
the equation may be possible.
Again,
14 Ha 6 m + 2, and 17 ± 6 m + 5.
And neither 2 nor 5 belongs to modulus 6, the equa-
tion therefore may still be possible.
Also,
14 = 7 m + 14, and 17 – 6 = 17 m + 11.
And neither 11 nor 14 belongs to modulus 17, there-
fore the equation is possible. In fact,
14. 119 + 6. 12 = 17. 102.
These examples will be quite sufficient for explaining
our operation; it may not, however, be superfluous to
add, that, when an equation appears under the form
a wº – b y? = c 22, it is immediately transformed to the
sort of equation we have been investigating, by writing
it c 22 + b y” = a a”. The cases in which one or two
of the coefficients become unity, are evidently involved
in the general form above given, and, therefore, need
no examples.
44. The equation aº — y? = a 2° is always possible
in integers.
For, if we resolve wº — y? into its factors a + y, and
r — y, (which are the only two literal factors that the
formula admits of,) and also a 2° into any two factors
a m t”, and m w”, we have, by comparison,
:#y-º! Or {i+y=::
a — y = m u%, a — y = a m tº,
which, by multiplication, becomes aº — y? = a m” tº wº
or aº — y” = a 2°, by making z = m tw.
Now these equations give,
a m tº -i- m w?
2 2
m w? -- a m tº
2
es
_ a m tº gamºs m w” .
and y =
1st, a =
2 2
on atº — a m tº
2 ©
On making m = 2, in order to clear the expressions
of fractions, they become,
1st, * = at + wº, and y = a tº — u";
2d, a = w” + atº, and y = wº — at’;
2d, a = , and y =
It is required to ascertain, whether the therefore the equation is always possible in
in-
tegers.
We may also take m = 1, or any odd number, only s
observing, that if a be odd, we must have t and u both
odd; for otherwise a and y would not be integers.
And if a be even, then u must be even likewise.
DEDUCTIONs.
(1.) If a be a prime number, the solution above
given is the only one the equation admits of in integers,
for a + y and a — y are the only literal factors of
— y”; and a m tº and mu” are the only factors of a 2°,
with regard to form ; and, consequently, one of the
two equalities must obtain : but the quantities t and u
being indeterminate, they will furnish an infinite num-
ber of numerical solutions. But if a be a composite
number, then the equation may have, beside the two
solutions given above, as many different literal solu-
tions as there are different ways of producing a by two
factors; thus, if a = b c, we may have
*= 2 **** 2
lst, {:tº Or {iji.
a — y = m w”, a — y = a mt”; and,
£ = b m, #2 tº = c m w°,
2d, { + y ..} OT { + y
a – 3) = cºm u%, a — y = b m, tº.
(2.) The equation aº – y? = a 2% includes the two
forms a' — a 2° = y”, and a " + a z* = y”; for, by
transposition, the first of these becomes aº — y” = a 2°,
and the latter y” — a = a 2°, which are evidently both
of the same form.
Therefore, if it be required to make a* + a 2* = y” a
square, we may have a = a tº — u", or = u” — a tº, and
2 = 2 t w ; whence aº – a z* = (a tº + wº)”; or we may
a tº — wº
2
a tº —- wº
t? a z* = ( —— l.
+ a 2-(**)
And to make a* — a 2* = y” a square, we may
assume w = a tº + u”, and z = 2 tºu, which give
w” – a z* = (a t” — wº)", or = (wº — a tº)";
or we may take
have a = , and z = t u, which give
a #4 + wº
*=dº * * and z = t u.
(3.) But if a = 1, and the equation become
a” + 2* = y”, then we may have indifferently a = t” – wº,
and 2 = 2 t u, or a = 2 tºu, and z = tº — w”, unless
there be any thing in the nature of the equation which
limits these forms : as, for example, if it be necessary
that one of the quantities, a or 2, be even ; then it is
obvious, that the even quantity must have the form 2 tw.
With regard to the equation a’ – 2* = y”, it gives
either a = tº + w”, and z = 2 t u, or z = tº — wº, both
of which values of 2 answer the required conditions of
the equation.
Example. Find the values of a, y, and 2, in the
equation aº – y” = 30 z*.
Here the following substitutions may be made,
l a -- y = a + y = 30 m tº,
e {T}Enº. r {{ſ- 7m tº.
2 {{ſ- 3 m tº, a + y = 10 m tº,
Ur – y = 10 m wº, “ {{I 3 m w?.
*ms
m tº,
Sect. W.
Indeter-
minate
quations.
\-/-
4 Q 2
656
T H E O R Y O F N U M B E R S.
Theory of
Numbers.
S-N-"
3 {:::= 2 m tº,
Ua — y = 15 m w”,
4 {...tº 5 m tº,
a — y = 6 m w”,
r {{ſ- 15 m tº,
a — y = 2 in u”
r {: tº 6 m tº,
a – y = 5 m w”.
And making, in each of these, m = 2, in order to
avoid fractions, we have the following general integral
values of a and y :
an — #2 2
1. {I t--80 u, ,
3/ = t” – 30 w”,
a = 3 tº + 10 wº, {i}...t.
2 Or 2
{;I.T. gy = 10 tº — 3 w”.
r {..I 15 tº + 2 w",
y = 15 tº — 2 w”.
4 {...it 6 w”, O {..I 6 tº + 5 wº,
Uy = 5 tº — 6 u”, r 3) = 6 tº — 5 w”.
t and u may be any integer num-
'll,”,
{j 30 tº —- wº,
g = 30 tº — wº
w = 2 t”—H 15 wº,
{; = 2 * – 15 w”,
In which formulae,
bers whatever.
45. The two indeterminate equations,
a” — y” = z*, and a “ — gy* = w”,
cannot both obtain, with the same values of a and y.
For, in the first place, a and y may be considered
prime to each other, (art. 41,) and therefore a and y
odd, or one even and one odd; and we see, imme-
diately, that it is y that must be even : for if
a” ºr 4 m + 1, and y = 4 m + 1, then a "+y^ = 4 m + 2,
which cannot be a square; and if a.” <= 4 n, and
gy* = 4 n + 1, then a” — y” tº 4 m + 3, which is also
an impossible form ; therefore a is odd, and y even.
Hence, then (art. 44, -3) we must have,
- * — sº - ?? 2
1st, {I, ST, 2d, {, t” —- w”,
y = 2 r s. gy = 2 t u.
Which furnish the following equations:
{" — s” = f° + w”,
7° S t u.
**
ſºme=s*
Now, in these equations, ris prime to s, and t prime
to u ; for otherwise a and y would have a common
measure, which is contrary to the supposition ; and,
farther, as a = r" – s” is odd, one of these quantities,
r or s, is even, and the other odd; and the same is
also true of t and u, because tº + wº = a is an odd
number.
º 7' S g g g
Again, since + = u is an integer, either r or s,
or both, must contain the factors of t ; for otherwise
the quotient would not be an integer: we may, there-
fore, make t = a, b, supposing a, b, to be its two fac-
tors, which may always be dome, because, in the case of
# being a prime, we have only to make one of these
two factors equal to unity: and, since these factors are
also contained in rs, we may write r = ar', and s = b s',
whence u = r's'; and now, substituting these values
for r, s, t, and u, the above equation becomes
a” r" — bºs” – a” b” + r" s”.
And here, since r is prime to s, and t to u; r", sº, a,
and b, are all prime among themselves, as is evident;
for if we suppose any two of the quantities to have a
common measure, as, for example, a and b, then, since
a and b enter, either separately or connectedly, into
three of the above quantities, the fourth, r's', must
have the same common measure, that is, t = a b, and
w = r" sº, would have a common measure, whereas we
have seen that they are prime to each other; and, con-
sequently, r", s', a, and b, are all prime to one another.
Now, by transposition, this equation becomes
* rºº — sº fº = gº b” + s” b”, or
Sect. V.
Indeter-
minate
Equations.
\-N-
(L" ºr
(a” – s”) r" = (a” + s”) b”, or
a” + s^ r^*
a” — s” 52
And here, since a” is prime to s”, a”-- s” is prime to
a” – s”, or they have only the common measure 2 ;
and we have, therefore, these two cases to consider
separately. First, suppose a” —- s” and a” — s” to be
a”-- s”
is in its
a” – s”
prime to each other, then the fraction
r’s
Tº
and hence, the two fractions being equal to each other,
and in their lowest terms, we must have, as resulting
from the first supposition,
{. + s? -- r”,
lowest terms, as is also because r" is prime to b;
a” – s” - b”.
Again, let a* + s” and a” — s” have a common mea-
sure 2, then t
* -- *
*=======
* *m-g
# (a” – s”) -
the first and last of which fractions are in their lowest
terms, and, consequently,
# (a” + s”) = ...} Or {. + s^ = 2 r",
4 (a” – s”) = bº, a” – s” = 2 b";
the last of which gives
gº -: r? + b°,
s” = r" – bº.
Now these two results in both cases are exactly simi-
lar to the original equations, only here the quantities
are much smaller than in that, at least r", s' and b, a,
are less than y, because y = r s a b.
Hence, then, it follows, that if the equations
* + y” -- z”,
a” — y” = 20°
were both possible, with the same values of a and y, it
would also be possible to find similar equations,
r” + y” - 2”,
{.. — y” = w”;
which would also be possible, and in which y' < y.
And, in the same manner, if these last were possible,
we might still find others,
f/2 '^2 –
{ + y” =
r!!” f/2
gºme y” = wº,
where y" < y, and so on of others, ad infinitum.
But it is impossible for a series of positive integers,
9°, Ay”, gy”, y”, &c.,
to go on decreasing to infinity, without becoming zero;
in which case our equations are
a’ = 2*,
gº - wº.
T H E O R Y O F N U M B E R S.
657
Theory of And, consequently, the two proposed equations can
N*, never obtain, with the same values of r and y, except
when y = 0; that is, the double equality
a" + y” = 2*,
{ a" — y” :- w”,
is impossible.
DEDUCTIONS.
(1.) Hence, also, it appears, that the two equations,
{. –– gy” - 2 z”,
a’ — y” = 2 w",
are impossible, with the same values of r and y, for
these may be reduced to
a’ = 2* + 2*,
{ y” = z* – wº;
and the two last being impossible, the former are im-
possible also.
(2.) The two equations
2 *-ī-y” = 2*,
2 a.” — y” = wº,
are both impossible, with the same values of a and y.
For we may consider a and y as prime to each
other; and therefore both odd, or one even and one
odd ; but they cannot be both odd, for then
2 * + y^ = 2 (4 m + 1) + (4 n' -- 1) + 4 m + 3,
which cannot be a square. Neither can a be even and
gy odd, for then
2 a.” — y” = 2 (4 m) — (4 n' + 1) = 4 n + 3,
which is an impossible form. And if y were even and
a odd, then
2 a.” + y” = 2 (4 m + 1) + 4 nſ = 4 m -- 2,
which is also impossible; and therefore the two given
equations cannot both obtain.
(3.) And this, again, shows the impossibility of the
two equations
{ºt 2 y” = 2 z”,
a" – 2 y” = 2 w";
for, by doubling these, we have
2 a.” + (2 y)* = (2 2)*,
{. a” – (2 y)* = (2 wy”,
which we have seen are impossible.
(4.) By a very similar mode of reasoning it may be
proved, that the two equations
* + 2 gy” = wº,
** *E. 2 gº - 2*,
are both impossible with the same values of a and y,
as are also the two equations
2 tº + y” = wº,
2 * — y” = z*.
(5.) In this way the following table of impossible
forms in pairs have been deduced, viz.
r” + y^ = 2*, ſa" + y^ = 2 zº,
1. º 2 2 2. 2 2 — 2
a" – y = wº. Ur — y” = 2 w".
- (2 x* + y^ = 2*, a” + 2 y” = 22°,
3. 3. * — nº — alsº 4. 4..." 2 — 2
2 w" — y” = w”. a" – 2 y” = 2 w".
5. {. + 2 y” = 2*,
6 {:tº 2°,
a” – 2 y” = w”. *
2 a.” — y” = 2 w".
{i,ji. s. (..I...I. tº
ſº a” + 2 y” = w”. gº TUzº sºms 2 y” = w”. minate
gº –– gy* = 2”, I0 a” — gy” -: 2*, Equations.
© \-y-
a" + 3 y” = w”. tº Isº
w” + 2 y” = 2*, a” – 2 y” = 2*,
ll. a * + 3 * = wº 12. * + 3 y” = w”
gy" - wº. a" + 3 y” = w”.
* — * - 2 2 2 – 2
13 {. 3/ º: 14. {. -- #. º:
a” + 2 y” = w”. a” – 2 y” = w”.
&c. &c.
And, generally, the pair of equations
a" + c y” = 2*,
a” + y^ = w”
are impossible, if the two equations
m* + c n” = (c – 1) p”,
m” + n’ = (c – 1) q”
be impossible; and, conversely, if these two be pos-
sible so also are the former.
46. The difference of two biquadrates cannot be
equal to a square, or the equation a “ — y” = 2* is im-
possible. For
£4 gºme y” t- (* + y”) (* mº 9°),
and since as and y are prime, or may be supposed prime
to each other, these factors are either prime to each
other, or have only the common measure 2; and, there
fore, if their product be a square we must have either
...tº {..t.
OF
a” — y” = s”, a” — y” = 2 s”,
for otherwise their product would not be a square, or
they would have a greater common measure than 2.
But these are both impossible forms, by the last
article, therefore the equation
a" – y' = 2*
is also impossible.
DEDUCTIONs.
(1.) In a similar way it may be shown, that
a" + 4 y' = 2*
is impossible.
(2.) And that wº—H y' = 22° is impossible.
47. The sum of two biquadrates cannot be equal to
a square ; or the equation
a' + y' = 2*
is impossible.
For (art. 54,--2) if a 4 + y” be a square, we must
have either
gº — tº — *} {. = tº — w”,
9. OT 2
Ay” = 2 t u, r” = 2 t it,
which are similar expressions; it will therefore be suf.
ficient for our purpose to prove that either pair of them
are impossible, and, as we may suppose aſ and y prime
to each other, (art. 41,) it follows, that t and u are also
prime to each other; and, consequently, since 2 tu = y”,
one of these quantities must be a square, and the other
double a square ; let then t = 2 w", and w = y”, whence
t” – wº H 4 wº. — y”; that is, 4 a” — y” = *. Or,
making t = r" and w = 2 y”, the equation becomes
a" – 4 y' = a ". We have, therefore, to examine the
tWO CaSeS
658
T H E O R Y O F N U M B E R. S.
a" – 4 y' = a,
4 w” — y' = aº,
one of which conditions must obtain, if the original
equation be possible.
Now these are resolvable into
I. r" – 4 y' = (r” + 2 y”) (w" – 2 y”).
2. 4 w” — y' = (2 w" + y”) (2 w" — y”).
And since a is prime to y, and t to u, it follows, w is
prime to y', and therefore these factors are prime to
each other, or can have only the common measure 2.
And, moreover, as their product is a square we must
have either
a" + 2 gy” = r", a'? -- y” = 2 r^*,
w” — 2 gy” - s”, Or a'? * y” - 2 s”,
in the first case, and
2 w" – y” = r", 2 *-ī-y” = 2 r",
2 a.2 — gy” = s”, OI’ 2 a.” – gy” = 2 s”,
in the second.
But each of these forms, taken in pairs, has been
demonstrated to be impossible, consequently the ori-
ginal equation, whence they have been derived, is
impossible also.
DEDUCTIONs.
(1.) Hence also it follows, that the two equations
..a a" – 4 y' = z*,
4 & — y' = 2*,
as is evident from the preceding investigation.
(2.) Since a' + y' = 2* is impossible, a fortiori,
* + y' = 2* is impossible.
48. The area of a rational right angled triangle can-
not be equal to a square.
For this would require the two equations
a”-- y” = z*
# a y = wº
to be both possible together.
Multiply the latter by 4, and add and subtract it
from the first, and we shall have
2* + 4 wºe (r-i-y)*,
: 2” – 4 w” = (a – y)”;
but these are impossible ; therefore the area of a
rational right angled triangle cannot be a square
number.
IXEDUCTION.
In a rational right angled triangle
a" + y^ = 2*,
we must therefore have
a = r" – s”,
y = 2 r s. }
2 sº
wº 7" —
And, consequently, if in the fraction T2 rs" OF
2 ºr S
++, the numerator and denominator be taken for
ºr" — S
the sides of a right angled triangle, it will be a rational
one ; and in these expressions we may give any values
at pleasure to r and s. If, in the second fraction Sect. VI.
2 r S º
sº-ºmºmºmºmº :- tº Igºne T
===, we make r = s—H 1, it becomes #.
2s + 1 TT 2 s -- 1 °
and in this expression, by making successively
s = 1, 2, 3, 4, &c.,
we have the following remarkable series,
S I 2 3 4 5 6
— tº 1 — C’s g- 2, 3 5 5. 6 — , &c.;
*-F 5-H = 15, 2; 37, 45 ° nº "is
each of which expressions, reduced to an improper
fraction, gives the sides of a rational right angled
* — s”
l
triangle. And if in the fraction we make s = I,
2 7° S.
and r = 2n + 2, cur expression becomes
4 n” + 8 n + 3 4 m + 3
4; H = ** Tº H
and here, making m = 1, 2, 3, 4, &c., we have this
other series,
4 m + 3 7 - 11 15 . 19
º 5- :, . T.T., 3 3-, 4-, 5
* + i = 1; *i; 37; 4;
which has the same property as the former.
23
24”
&e.,
*
VI. Of the possible and impossible forms of Cubes and
Higher Powers.
49. All cube numbers are of one of the forms 4 n or
4 m + 2.
Every number is of one of the forms
h 4 m, 4 n + 1, or 4 m + 2,
therefore all cubes fall in one of the forms
(4 m) tº 4 m
(4 m + 1)* = 4 m + 1
(4 m + 2)* = 4 m.
Therefore all cubes are of one of the forms 4 m or
4 m + 1.
DEDUCTIONs.
(1.) By subdividing these, we deduce the forms to
modulus 8, as follow. All cubes fall in one of the
forms
8 m, 8 m + 1, 8 m + 3.
(2.) Therefore, conversely, no numbers of the form
4 n + 2, 8 m + 2, 8 n + 4,
can be cubes.
(3.) In a similar way we may deduce the possible
forms of cubes to the moduli 7 and 9, viz.
7 m, 7 m + 1, 9 m, 9 n + 1.
50. All cube numbers are of the same form to any
modulus a as the cubes
0°, 1*, 2*, 38, &c. (a – 1)".
For every number may be reduced to the form a m + r,
such that r shall be less than a. Consequently,
(a m + r.)” divided by a will leave the same remainder
as a, but r is either zero, or some number less than a,
whence the truth of the proposition is manifest.
T H E O R Y O F N U M B E R S.
659
Theory of
Numbers.
DEDUCTIONs.
(1.) By means of this general proposition, the pos-
sible forms of cube numbers to any modulus are easily
deduced. If we essay modulus 10 we find all the follow-
ing possible forms, viz. *
10 m, 10 m + 1, 10 m + 2, 10 m + 3, &c. 10 m + 9,
no number is therefore excluded by this modulus, con-
sequently, a cube number may terminate with any digit.
(2.) To modulus 6 all cubes are of the same forms
as their roots, consequently, the difference between any
cube number and its root is divisible by 6.
51. The equation (4 p + 2) tº + 4 q u% = wº is
always impossible in integers, while q is prime to 4.
By art 41, the three cubes tº, wº, w8 may be consi-
dered prime to each other; and since all cubes are of
one of the forms 4 m, or 4 m + 1, and 4 q u” is always of
the form 4 n,
(4 p + 2) tº + 4 quº
must be (when tº is of the form 4 m) of the form
(4 p + 2) 4 m + 4 q vºt-F 4 m,
that is, tº and wº are both of the form 4 m, which is
absurd, because they are prime to each other.
And if tº be supposed of the form 4 m + 1, then the
equation is of the form
(4 p + 2) (4 n + 1) + 4 q u" ºr 4 n + 2,
which is an impossible form.
Therefore (4 p + 2) tº + 4 q u% = wº is impossible,
q being prime to 4.
DEDUCTIONs.
(1.) By giving different values to p and q, we obtain
the following impossible forms
2 pº =E 4 wº, 2 pº -- 12 w",
6 pº H. 4 wº, 6 pº -- 12 w",
10 pº =E 4 wº, 10 pº + 12 w".
&c. &c.
(2.) In a similar way we may show, that
(7 p + 2) tº + 7 g wº = wº,
(7 p + 3) tº + 7 quº = w”,
(9 p + 2) tº + 9 quº = wº,
(9 p + 3) tº + 9 quº = w8,
(9 p + 4) tº + 9 quº = wº,
&c. &c.
q being in the first two prime to 7, and in the latter
three prime to 9; and from these an indefinite number
of impossible forms may be deduced.
52. All 4th powers are of the same form with regard
to any number a as a modulus, as the 4th powers
0°, 1*, 2*, 3, &c., (4 a)',
when a is even ; and as
— lº
04, 14, 24, 3°, sc.( 2 )
when a is odd.
For every number whatever may be represented by
the formula a n + r, where r never exceeds; a, (art. 10.)
But -
(an -E r) = a nº =E 4 a.ºnºr + 6 a”nºr” + 4 anrs + r",
and all the terms, but the last, of this expression, being
divisible by a, the whole quantity is evidently of the Sect. VI.
same form, with regard to a as a modulus, as the last Cube; and
term r"; but r never exceeds 3 a, therefore every 4th
Higher
Powers.
power to modulus a is of the same form as the 4th ^-y-Z
powers.
, 0", 1", 2’, 3, &c. (, al", a being even,
1N4
) a being odd.
By means of which result, tables of possible and im-
possible forms of both powers may be obtained to any
indefinite extent, and amongst other curious results it
will be found by examining these series, that all 4th
powers are of one of the forms 16 m, or 16 m + 1.
53. The two indeterminate equations
a' + y' = z*,
4:4. + (8 yº - 2*,
are both impossible.
For we have seen, (arts. 46, 47,) that the equation
a" + y' = 2* is impossible in integers; and therefore,
a fortiori, the equation a' + y' = z* is also impos-
sible.
Again, we have, by transposition, in the second
equation,
f
Q, -
0, 1, 2’, 3, &c. (
w" – 2* = (a y”)*,
which is also impossible, (art. 46;) and, consequently,
the two given equations are impossible in integers.
DEDUCTIONs.
(1.) Hence it follows, that the equation
a 4 + 4 y' = 2*
is impossible; and, in like manner,
w” -- y4 = 2 z*,
{ acº — y = 24,
4 wº — y” = 2*,
are all impossible equations.
(2.) In a manner very similar to that employed in
the case of squares and cubes, it may be demon-
strated that
(5 p + 2) tº + 5 q wº = wº,
(5 p + 3) tº + 5 q u% = wº,
(5 p + 4) tº + 5 qu' = wº,
&c. &c.
are all impossible equations, as is also the general
form
(16 p + r.) tº + v v^ = w”,
r and v being so taken that r + v < 16.
54. Every 5th power is terminated with the same
digit as its root. Or all 5th powers are of the same
form, with regard to modulus 10, as the roots of those
powers.
For all numbers to modulus 10 are of one of the fol-
lowing forms:
(10 m )* = 105 m" ºr 10 m.",
(10 m + 1)* = 10 m' + 1 = 10 n" + 1,
(10 m + 2)" ºr 10 n' -– 2* = 10 m." + 2,
(10 n + 3)* = 10 nſ –H 3% = 10 n”-H 3,
(10 n + 4)* == 10 m/+ 4* = 10 m." + 4,
660
T H E O R Y OF N U M B E R S
Theory of
Numbers.
\—y-*
(10 m + 5); # 10 n' + 5° tº 10 m." + 5,
(10 m + 6)* = 10 n' + 6% == 10 m” + 6,
(10 m + 7)* = 10 n' + 7% ºf 10 n” -- 7,
(10 m + 8)* = 10 n' + 8* == 10 m." + 8,
(10 m + 9)* = 10 m/+9° tº 10 m." + 9.
Where the latter formulae are evidently the same as the
first; and, consequently, the powers have the same
forms to modulus 10 as the roots of those powers, or
they are terminated with the same digits.
DEDUCTION.
It has been demonstrated, (art. 50,—2,) that all
cubes have the same forms as their roots to modulus
6; and, in the above proposition, that all 5th powers
have the same forms as their roots to modulus 10; and
the same is universally true for prime powers, namely,
that they are of the same form as their roots to modul
lus double the exponent of the power, viz. all 7th
powers are of the same form as their roots to modulus
14, and 11th powers of the same form as their roots to
modulus 22: and so on for any other prime powers. .
VII. Of the divisors and forms of the Integral Powers
of Numbers.
55. The difference of two equal integral powers is
divisible by the difference of their roots.
Let a and y be two numbers, then will
a" – y”
a – y
a" – y" tº M (a — y).
Let x = y + d, or a – y = d, then we have to
prove that
(y -- d)" – y”
d
Make l, n, m, p, &c, n, l,
to represent the integral coefficients of y + d, raised
to the m” power, then the above numerator is ex-
pressed by
d" + n d"-" ºf + m d"-" y” + p d"-8 y”, &c.,
every term of which is obviously divisible by d, and,
consequently, the whole number is so, that is,
= M, an integer,
OT
= M, an integer.
a" – y” is always divisible by a - y,
a" – y” H M (a — y).
56. The difference of two equal integral powers is
always divisible by the sum of the roots, if the index of
the power be an even number ; that is
a" – y” ºr, M (a + y)
when n is an even number.
Make
then, as in the preceding proposition, writing
Or,
a + y = s, or a = s — y,
1, m, m, p, &c. m, 1,
for the coefficient of s — y ”, we have
(s — y)" – y” =
s" — n s"- y + m s”- y” – ps"-3 y” + &c.
-- m s” wº- — n sy” + y” — 3y".
In which, as the last two terms destroy each other, and
the others are each divisible, by s, the whole quantity is
divisible by s, that is
a" – y” tº M (a + y)
when n is an even number. -
57. The sum of two equal odd powers is always
divisible by the sum of their roots, or a " + y”
H M (a + y) when n is an odd number.
Make a + y = s, or a = s — y, then a " + y” be,
(s — y)" -- y” =
s" – m s”- y + m s”-*y? — &c. – m s” y” + m sy"-
— y” + y”.
In which, as before, the last two terms destroy each
other, and each of the remaining terms is divisible by s,
and therefore the whole remainder is divisible by it;
that is,
COIſles
a" + y” tº M (a + y),
when m is an odd number.
DEDUCTION.
By means of the above propositions, we are also
enabled to ascertain the divisors of the sum or differ-
ence of unequal powers of the same root, viz.
(r" — r") = M (a – 1), and M (a + 1),
when m – m is even, or of the form 2 m', for
a" – a "= r" × (w"-" — 1),
and since m — n = 2 nſ, therefore,
(r"-" — 1) = (x^* – 12") = M (a – 1), and M (a + 1);
and, consequently, - -
a" × (r"-"–1) = (r"—r")= M (a – 1), and M(x + 1).
Again, if n – m be odd, or of the form 2 n' + 1, then
(r" — r") = M (1 – 1), and
(a" + æ") = M (a + 1).
For
(a" — r") = a," × (w"-" — 1), and
(r" + æ") = a," × (r"-" + 1);
also, since m – n = 2 m' + 1, therefore,
(w"-" — 1) = (x^** – 1*) = M (a – 1), and
(r"-" + 1) = (v* + 1".H) = M (r-H. I.);
and, consequently,
a" x (r"-" — 1) = (r" — r") = M (a – 1),
a" x (a"-" + 1) = (r" + æ") = M (a + 1).
58. If m be a prime number, and a any number not
divisible by m, then will the remainder arising from
the division of a by m be the same as that from the
division of r" by m. -
It is necessary first to show, that if m be any prime
number, each of the coefficients of the expanded bino-
mial (a + 1)” is divisible by m, except the first and
last. For each of these coefficients is of the form
m . (m – 1) (m. – 2) (m. – 3) &c.
fººms
1 .. 2 . , 3 . 4 &c.
an integer, or
777, X (m – 1) (m. – 2) (m. – 3) &c.
2 . 3 . 4 &c.
Sect. Vlſ.
Integral
Powers.
~~~
T H E O R Y O F N U M B E R S. 661
". of Where the quantity in the parenthesis is obviously (2.) Since a "-" — 1 is always divisible by m under Sect. VII.
*" integral, because the whole quantity is so, and m is not the limitations of the proposition, therefore r"-" tº i.
y divisible by any of the factors of the denominator. We a m +1, and, consequently, every power whose expo- Jº ,
have, therefore, m N = p, consequently, p is always ment plus 1 is a prime as m, will be of the form a m or
divisible by m when m is a prime number.
This being premised, make a = a + 1, then we have
r"= (+/-ī- 1)" = r" + ma"-i-i- m a r"---|- &c. m. a 4-1.
And since each of the terms of this expanded binomial,
except the first and last, is divisible by m, it follows,
that the remainder from the division of (r^+ 1)" by m,
is the same as the remainder from the division of
a" + 1 by m, which, by rejecting the multiples of m,
may be expressed thus,
a" = (+/-ī- 1)" – r" + 1.
Making now a' = a " + 1, we shall have, on the
same principles,
w" = (r' + 1)" = w” + 1 = (a" + 1)" -- 1 = r" + 2.
Again, let a " = a " + 1, and we obtain
a” – alm + 1 = rſ" + 2 = a " + 3.
And thus, by continual substitutions, we have
r" = r" + 1 = r" + 2 = 4" -- 3 = &c.; or,
{r- (a. * 1)" + 1 = (a: ºme 2)" -- 2 = (a – 3)"—-3
&c. (w — ar)" + r,
the last of which terms is equal to a ; whence it fol-
lows, that the remainder arising from the division of 1.
by m is the same as that from the division of a "
by m.
59. If m be a prime number, and a any number not
divisible by m, then will the formula a "-" — I be divi-
sible by m, or, which is the same,
(r"-i – 1) = M (m).
For, by the foregoing proposition, the remainder of
7?!
£ . g JC
– is the same as the remainder of — ; and, conse-
77? 772
quently, the difference r" — a is divisible by m. But
a" – a = r (r"-" — 1), and since this product is divi-
sible by m, and the factor a is prima to m, it must be
the other factor, viz. (a "T" — 1), that is divisible by m.
DEDUCTIONs.
(1.) Since a "-" — 1 is always divisible by m, if a be
prime to m, and m itself a prime; there are necessarily
on – 1 values of a less than m that will satisfy the
equation
*~1 — I
= e, an integer,
7??,
that is, a may be any number in the series
l. 2. 3. 4, &c. m. — 1,
because all these numbers are necessarily prime to m ;
and since m – l is an even number, we shall have also
m — 1 values of a comprised between the limits – ; m
and + , m, that is, a may be any number in the
series,
7m — l
+ 1, + 2, + 3 + &c. E-g-,
so that in both cases we have m – l values of a < m,
which render the equation
dr"-1 — I
= e, an integer.
7??
VOL. I.
a m + 1, and we may thus ascertain the forms of many
of the higher powers ; thus
a" ºr 5 m, or 5 m + 1,
a" ºr 7 m, or 7 m + 1,
a'0 tº 11 m, or l l n + 1,
al” ºr, 13 m, or 13 m + 1,
&c. &c.
Again, since m is a prime number, if it be greater
than 2, it is an odd number; and, consequently, m – 1
an even number ; and, therefore, &
m - 1 *1 - 1
4"-" — 1 = #11)×(-)).
and, since this product,
(; , ) (F-1)
is divisible by m, and m is a prime number, one of
these factors must be divisible by m ; that is,
m — I
a" ºr m a + 1 ;
and, consequently, every power, the double of whose
exponent plus I is a prime number, as (m), is of one
of the forms -
a m, or a m + 1 ;
and hence, again, we derive the forms of many other
higher powers ; thus,
a 3 -t: 7 m,
a:3 tº l l n,
arº –– 13 m,
are tº 17 m, or
a 9 ºt; 19 m,
all ºr 23 m, 23 m + 1,
&c. &c.
(3.) And hence we have the following forms of all
powers from 2 to 12, the 7th power only excepted,
which cannot be introduced, because neither 7 -H 1, nor
2. 7 -i- I, is a prime number. gº
7 m + 1,
11 n + 1,
13 m + 1,
17 m =E 1,
19 m + 1,
Or
Or
OT
Or
OT
Table of the possible forms of Powers from 2 to 12.
a * = 3 m, or 3 m + 1 + 5 m, or 5 m + 1,
a’ tº . . . . . . . . . . . . . . == 7 m, or 7 m + 1,
a' = 5 m, or 5 m + 1 + . . . . . . . . . . . . . .
a" ºr . . . . . . . . . . . . . . tº 11 m, or l l n + 1,
a” ºr 7 m, or 7 m + 1 + 13 m, or 13 m + 1,
a" F . . . . . . . . . . . . . . == 17 ºn, or 17 m + 1,
a" ºr . . . . . . . . . . . . . . ºf 19 m, or 19 m + 1,
ac" -t, ll m, or l l n + 1 + . . . . . . . . . . . . . .
w" is . . . . . . . . . . . . . . a 23 m, or 23 m + 1,
ar” +, 13 m, or 13 n + 1 + . . . . . . . . . . . . . .
By means of the above table, we may frequently
prove the impossibility of equations of the form
a r" + b y” = d 2";
4 R
662 T H E O R Y O F N U M B E H. S.
Theory of but it does not follow of course, if they fall within the VIII. Of the products and transformations of Alge- Sect, viii.
Numbers, possible forms, that they are actually resolvable in in- braical Formulae referable to the forms of Numbers. Algetºical
\-N-'tegers; thus Formulae.
a' + y^ = 23, 61. The product of the sum and difference of two Y-V->
a 4 + y' = z*,
are impossible, and generally
a" + y" = z”
is impossible if n be greater than 2, although these
may not fall under the impossible forms of the table.
60. If m be a prime number, and P be made to re-.
present any polynomial of the nº degree, as
P = a + a w- + ba'---|- cº-s + Q,
then there cannot be more than n values of a, between
the limits + 3 m, and — ; m, which render this polyno-
mial divisible by m.
For let k bet he first value of a, which renders P divi-
sible by m, so that *-
A m = k" + a k"-" + b k"---|- c k" -- . . . .
then, by subtraction, we have
ſP – A m = ( – k") + a (r"-" — k" ) +
U 5 (r"-" — k"-s) + &c.
But the latter side of this equation, being divided by
a — k, (art. 55,) we shall have for a quotient a poly-
nomial of the degree n – l ; which, being represented
by P', gives
P – A m = (x-k) P', or P = (r. —k) P' + A m.
Let now k' be a second value of r, which renders P
divisible by m, then it follows, that (a — k) P' + A m
is also divisible by m ; and, eonsequently, (a — k) P'
divisible by m, but the factor a – k, which now be-
comes (k' – k), cannot be divisible by m, because both
k' and k are less than 4 m ; therefore P cannot be
divisible a second time by m, unless P’ be divisible
by m.
The polynomial P is therefore only once more divi-
sible by m than the polynomial P'; and, in the same
manner, it may be shown, that P", of the degree n – 1,
is only once more divisible by ºn, than P" of the m – 2
degree, &c.; and hence it follows, that P being a poly-
nome of the m degree, there can be only n different values
of a, comprised between the limits + 3 m, and – 4 m,
which renders it divisible by m.
q :
DEDUCTION.
We have seen, that if m be a prime number, the for-
mula a "T" – l has m – 1 values of a, between the
limits + 3 m and – 3 m, which renders it divisible by m.
Now this being put under the form
m-1 in-2
(. 2 + i)x (* *º- ..)
it follows, that each of the factors has m – 1 values of
it, between the limits + 4 m and – 4 m, which renders
them divisible by m. For neither of them can have
7?? –
l g
more than such values, by the foregoing propo-
sition, and since their product has m – 1, it is obvious
they have each the same number of values of a between
the above limits, and that this number is therefore
m — l
2
quantities, is equal to the difference of their squares.
For (a + y) x (a – y) = a – y”,
as is evident.
62. The product of the sum of two squares by dou-
ble a square, is also the sum of two squares, or
(a + y?) × 2 z2 ºr a '* + y”.
For (tº + y?) × 2 z* = (x + y)* 2° -- (a — y)* 2°,
which is evidently tº aº -- y”.
DEDUCTION.
Hence, if a number be the sum of two squares, its
double is the sum of two squares; and if N be the sum
of two squares, 2" N will be so likewise.
Thus 5 = 2* + 13, 5 × 2 = 10 = 38 + 1°,
10 × 2 = 20 = 4* + 2*, and 40 = 6*-i-2°.
63. The product arising from the sum of two
squares by the sum of two squares, is also the sum of
two squares.
Or (a" + y”) (r” + y”) = x". -- y”.
For
Q (a w' + y y')* + (a y'— a y)*,
2 tºº fº '?\ –
(e-woo'-y)={..., tºº.
as will appear from the developement of these expres-
sions, and, consequently,
(aº -- y”) a'? -- y”) tº ac'/2 + gy”.
DEDUCTION.
IHence the product may be divided into two squares
two different ways. And if this product be again mul-
tiplied by another, that is the sum of two squares, the
resulting product may be divided into two squares four
different ways ; and, generally, if a number N be the
product of n factors, each of which is the sum of two
squares, then will N be the sum of two squares, and
may be resolved into two squares 2" different ways.
5 = 2* + I*
13 = 33 -H 2*
For example,
then the product 65 = 8* + 1*, or 7* + 4*.
Again, 17 = 4*-ī- 1°
1105 = 32°–H 9°– 33°-H 4* = 31* +12”
= 24”-- 23°.
And this resolution of the given product into square
parts, is readily effected by the foregoing theorem ;
for
(8” + 1) (4* + 1*) = (4.8 + 1)* + (8 . 1–4. 1)* ==
(4.8 – 1)* + (8. 1 + 4. 1)*, and
ſ(7°4-4*) (4 + 1) = (4.7+ 1.4)*-ī- (4.4–7. 1) =
U (4.7 — I : 4)* + (7. 1 + 4.4)*.
And in the same manner may any other product,
arising from factors of this form, be resolved into its
Square parts.
the product |
T H E O R Y O F N U M B E R. S.
663
Theory of 64. The product of the sum of three squares by the
Numbers sum of two squares, is the sum of four squares ; or
J- (* + y” + zº) (w" + y”) as w” + æ" + y” + z”.
For -
(a” + y + zº) (a' + y”) =
(ra' + y y)* + (ry’ — ya')*-i- a” z*-i- y” zº,
as will appear from the developement of these formula,
and, consequently,
(aº + y” + 2*) (a.” –H y”) tº w” + a/? —-y” + 2/8,
Thus 14 = 3” -- 2* + 1"
5 = 2*-ī- 1°
70 = (3.2 + 2. 1)* + (2.2 – 3. 1)*
+ 2* + 1 = 8* + 1* + 2* + 1°,
and a like decomposition may be effected on any other
similar product.
65. The product of the sum of four squares by the
sum of two squares, is the sum of four squares ;
that is,
(w" + a *-H y” + zº) X (r” +y”) His op” + 1//* + w”--2”.
For
the product
(w” -j- a") (r” + y”) tº: 20//* + a",
(y” + 2*) (r"+ y”) = y” + z”;
consequently,
(w”--a” + y” +- g”) X (* +y”) tº My/* + a/8 + y” + 2”.
66. The product of the sum of four squares by the
sum of four squares, is also of the same form ; or
ſ(wº + 4*-ī-y” + zº) (w” -- w” + y^* + 2*) ==
l w!” + æ" + gy” + 2”).
For
(w' + 1 + y” + 2*) (w” + r" + y” + 2*) =
(ww' + ar' + yy' + z2')* + (wa'—arw' + yz' – zy')?--
(wy'— a 2" — y w” + za')* + (w z' + ay’ – yr" – 2 w")*,
as will appear immediately from the developement of
the above formulae ; and, consequently, the product in
question tº: (w!" + a's + y” + 2”).
DEDUCTIONs.
(1.) As in this product there are only complete
squares enter, we may change at pleasure the signs of
the simple quantities; and, consequently, there will
result several different formulae equal to the same pro-
duct, and each equal to the sum of four squares; and
in so many different ways may any number that arises
from the product of the factors of the above form, be
resolved into the sum of four squares.
(2.) This proposition may be rendered more general
by the following enunciation:
The product of the two formulae,
(wº-ba'-cy” + b c 2") (w”— b w”— cy”-- b c 2*) ==
(w!” – ba" – cy” + b c 2").
For
(w" — brº – cy” + b c 2") (w/* – br” – cy” + b c 2".) =
(ww' + b a' a' + c y y' =E b c z z')” —
b (w a' + w'a -E c y 2' + c y' 2)” –
c (w y’ — baz' + y w! HE b : a)” +
be (ry’ — wa' + 2 w" + y r*.
as will appear from the developement, and, conse- Sect. VIII.
quently, the product in question is of the same form as Algebraical
each of its factors. Formulae.
67. The product of the two formulae (a" – a y”) and N-V-2
(r” — a y”) is of the same form as each of them.
For -
w (ra'+a y y')*—a (ry'+y 2’)”
— or alº /* — *) –
(*-*W) ("-ºy") = i.e.,L....') ...,' ...
consequently,
(* tºº a y”) (r” — a y”) == r" — a y”.
Hence the product of any number of factors of this
form is of the same form as each of its factors.
68. The two formulae
a” + y” + 2*, and a " + y” + 2 zº,
are so related to each other, that the double of the one
produces the other; that is
2 (a + y” + 2*) = w” + y” + 2 2",
2 (a" + y^+ 2 z*) = z* + y” + 2*.
For
2 (a^+ y” + 2*) = (x + y)* + (p – y)* + 2 zº, and
2 (tº + y” + 2 z*) = (a + y)* + (a — y)* + (2 2)",
as is obvious.
For example, 14 = 3* + 2*-ī- 1”
multiplied by 2
= 28 = (3 + 2)” -- (3 – 2)" -- 2. 1"
the product = 5* + 1* + 2
And 15 = 32 + 2* + 2. 1"
multiplied by 2
º = 30 = (3 + 2)” –- (3 — 2)” –– 2*
the product = 5* + 1* + 2*.
And the same of all other numbers of these forms.
69. The formula a” – 2 y” may be always trans-
formed to another of the form 2 a.” — y”, and this last
may be converted into the former; that is,
a” – 2 y' = 2 a.” — y”,
2a" – y” -5 a.” – 2 y”.
For
a” – 2 y” = 2 (c-E y) — (p + 2y)* = 2a:” – y”, and
2 * – y” = (x + 2y)” – 2 (a + y)* = w” – 2 y”;
as is evident from the developement of these formulae;
and, consequently, a number that is of one of these
forms is also of the other.
14 = 2. 3” – 2* = 4* – 2. 1*;
28 = 6* – 2 . 2” – 2.4° – 2*.
And the same of any other numbers of either of these
forms.
70. The formula a” — 5 y” may be always trans-
formed to another of the form 5 a.” — y”, and this last
may be converted into the former; that is,
a” – 5 y” ºf 5 a.” — y”,
5 a” — y” tº r" – 5 y”.
For example,
also,
For
a” – 5 y” = 5 (r-E 2 y)*— (24 + 5 y)* =5 w”—y”, and
54"– y”= (5 a -E 2 y) — 5 (24 +y)* = r" — 5 y”;
aud, consequently, any number that is of one of these
forms is also of the other.
4 R 2
664
T H E o R Y of N UM BER S.
Theory of For example 29 = 7°–5.2°= 5.11% – 24” = 5.38–4*;
Numbers sºmé * — 29– 10°– 2 — I l? — 2
Jººl, and © 41 =3.3 *=10 5 . 82 - 11° – 5 - 42.
And a similar transformation may be made on any
other number falling under either of the above forms.
71. If a be any number of the form b% + 1, then
will the formula aº — a y” be resolvable into another of
the form a rº — y”; and, conversely, this last may be
transformed into the former ; that is,
r" – (b? -- 1) yº is (bº + 1) w” – y”, and
(bº + 1) as — y” ºr w/º – (b? -- 1) y”.
For
r? – (b2+-ly?=(bº-H1)(x-Eby)*— { ba-E(bº-H1)y #2,
and
(bº-H1)+2-y2={(bº-H1) witby }*— (bº-H1)(ba-i-y)*,
the first of which transformed formulae is evidently
# (b? -- 1) a” – y”; also the latter
++, a "2 – (b? -- 1) y”; and, consequently,
aft — a y2 = a wº — y”, and
a w? — y2 = w” — a y”, when
a tº b° + 1.
PEDUCTION.
These general formulae furnish us with many par
ticular cases, which have the singular property of being
convertible from one to the other; such are
a” – 2 y” <= 2 ac” – y”,
2 a.” — y” ºr a " — 2 y”.
{ a" – 5 y” ++, 5 a.” – gy”,
5 tº º y” & a/8 * * * * 5 y”.
a’ – 10 y” ºt; 10 a." – y”,
10 a.” — y” = w” – 10 y”.
a” – 17 y” == 17 a” — y”,
17 a.” — y” ºr a " — 17 y”.
&c. &c.
72. If m and n be the two roots of the quadratic
equation (p2 — a g + b = 0, then will the product of
the two formulae (a + 'm y), and (x + n y), be equal
to a 2 + a a y + b y”.
This is evident from the actual multiplication of the
factors (a + 'm y) and (a + n y).
OT -
(a + 'my) (a + 'm y) = a + (m. -- m) a y + m n y”;
and, since m and m are the roots of the equation
Ø” — a p + b = 0, we have, from the nature of equa-
tions, m + m = a, and m n = b ; and, consequently,
the above product becomes
a” + a a y + b y”.
DEDUCTION.
Hence, conversely, every quantity of the form
a" + a w y + b y? may be considered as the product
arising from the multiplication of two factors, (a + my)
and (a + n y), m and n being the roots of the qua-
dratic equation
Ø? — a j + b = 0;
or, which is the same, m and n being such as to answer
the conditions, m + m = a, and m n = b. ,
73. The product arising from the multiplication of Sect. VIII,
the two formulae - Algebraica,
C) ! .../ ! Formulae.
a" + a w y + b y”, and a " + a a 'y'-- b y”, \
is of the same form as each of them ; that is,
(* + a w y + b yº) (a" + a w'y' + b y”) ==
(a”-- a r" y" + b y”).
For
a” + a a y + b y” = (x + m y) (a + n y), and
w” + a r" y' + b y” = (a' + m y') (a' + n y');
and, therefore, the product in question is the same as
the continued product of the four latter factors.
Now,
(a + m y) (a' + m y') = a a' + m (~ y + a'y) + m2 yy',
but since m is one of the roots of the equation
Ø — a q + b = 0,
we have mº – a m + b = 0, whence m” = a m – b ;
and substituting this value of m”, in the above formula,
it becomes -
a w' – b y y’ + m (a y' + aſ y + a y y').
And if, in order to simplify, we make
X = a a ' – b y y',
Y = a y' + y a' + a y y',
the product of the two factors,
(a + 'm y) (r' + m y') = X + m Y;
and, in the same manner, we find
(a + m y) (a' + m y') = X + m Y:
and, consequently, the whole product will be
(X + m Y) (X -- n Y) = X* + a X Y + b Y’;
that is, the product
(a”-- a a y + b y”) (a" -- a w' y' + b y”) ==
(*!" + (Z ºrſ' gy" +- b gy”).
DEDUCTION.
Hence it follows, that the product of any number of
factors of this form ; as
a"-- a a y + b y”,
w” + a a y' + b y”,
aſ/? + a p" gy" + b y”,
&c. &c.
will always be of the same form as those factors.
Therefore if we make a = a ', and y = y', we shall
have X = x* – b y”, and Y = 2 a y + a y”; and, con-
sequently,
(a" +- a a y + b y”)* = X* + a X Y + b Yº.
And, therefore, if it were required to make a square of
the expression
X* + a X Y + b Yº,
we shall only have to give to X and Y the preceding
values, whence we readily obtain for the root of the
square required the formula
a” + a a y – b y”,
where c and y may be any numbers at pleasure.
Example 1. Find the values of a and y in the equa-
tion
a” + 3 a y + 5 y” = z*.
Here a = 3 and b = 5, therefore the general values of
a; and y are
T H E O R. Y O F N U M B E R S. 665
§ of {. = tº — 5 wº, therefore, by subtraction, sº º
Numbers. - / *=== CUral Cº.
- gy = 2 tº u + 3 w”, pr’ – q” = p r — q”, É.
where, for distinction sake, we write t and u, in the where these quantities will always have the same sign. S-N-"
above formulae, instead of a and y. Whence, by
assuming successively, *
t = 3, 4, 5, 6, &c.,
w = 1, 1, 1, 1, &c.,
we shall have the following corresponding values of
a; and y: -
a = 4, 11, 20, 31, &c.,
gy = 9, 11, 13, 15, &c.
Erample 2. Find the values of a and y in the
equation
a” – 7 a y + 3 y” = z*. -
Here, since a = — 7 and b = 3, the general values of
aſ and y are
{..I t? – 3 u%,
= 2 t w – 7 w”.
t = 4, 5, 6, 7, 8,
w = 1, 1, l, 1, 1
And making now
&c.,
&c.
3. 2 3. 3.
we obtain
a = 13, 22, 33, 46, 61, &c.,
y = 1, 3, 5, 7, 9, &c.
Each of which corresponding values of a and y answer
the required conditions of the equation ; and it is
manifest, that an infinite number of other values might
be obtained, by changing those of t and u.
IX. On the Quadratic Divisors of Algebraical
Formulae.
74. If in the indeterminate formula
p y” -- 2 p q y z + r 2* = @,
the coefficients p, q, and r have not all three the same
common divisor, and y and z be any numbers what-
ever prime to each other; and if 2 q > p; or > r, this
formula may always be transformed to a similar one,
p' y” + 2 p' q' z' y' + r. 2” = ©,
which shall be equal to the same quantity ºff, and in
which 2 q' shall not exceed either p' or r'.
Let us suppose, first, 2 q > p; and in the case in
which also 2 q > r, let p be the least of the two num-
bers p and r, abstracting from their signs.
Make y = y' – m z, m being an indeterminate
coefficient; and, substituting for this value of y in the
given equation, we have
p (y' – m z)* + 2 q 2 (y’ — m/z) + r z* = @, or
p y” – 2 (p m – q) y' z + (p m” – 2 q m + r.) z* = @,
where we may always take the indeterminate m, so
that + (p m -- q) < p. Calling therefore + (p m – q)
= q', and (p in” – 2 q m + r) = r", the transformed
formula will be
p y” + 2 q' y' z + r' 2" = @,
in which 2 q' < p (thºs sign not excluding equality)
and in which p r" – q” = p r — q", for
q” = p^m” – 2 p q m + q”,
pr' = p^ m” – 2 p q m + rp,
Since, then, we have 2 q > p, 2 q' < p, it follows
that q' < q. Hence we have now an equation
p y” + 2 q' y' 2 + r' 2" = @,
in which the mean coefficient 2 q' does not exceed the
extreme coefficient p; and if at the same time it does
not exceed the other extreme coefficient r", the formula
is transformed as required. But if 2 q, although 3
than p, be > r, we may proceed, in a similar manner,
to obtain a new transformation, in which the mean
coefficient (which we may denote by 2 q") shall be less
than q', and so on again for others, in which the mean
coefficient 2 q" shall be less than 2 q". But the series
of integers
q, q', q", q”, &c.
cannot go on continually decreasing, without becom-
ing finally less than the extreme coefficients; and,
therefore, by continuing these transformations, we must
necessarily arrive at that which admits not of any fur-
ther reduction ; and which will be consequently such,
that the mean coefficient is less than either of the ex-
tremes, or at least not greater than the least of them ;
for with any formula in which this is not the case further
reduction may be made. Therefore every formula
p y” + 3 q y z + r z*
in which the mean coefficient 2 q exceeds either, or
both, of the extreme coefficients, may be transformed
to another in which the mean coefficient 2 q' shall be
less than either of the extreme coefficients, or at least
not greater than the least of them.
DEDUCTIONs.
(1.) In the successive transformations of the for
mula -
p y” + 2 q y z + r 2°, to
p y” + 2 q' y' z + r z*, to
p' gy” + 2 q” $y' 2' + r z”, &c. 5
we have always
p r — q” = p r" — g” = p/r' — q”, &c.,
each of these quantities having the same sign, as is
obvious from the form of the preceding transforma-
tions.
(2.) As an example of the reduction stated in the
foregoing proposition, let it be proposed to transform
35 y? -- 172 y z + 2102* = @,
in which the mean coefficient 172 is greater than the
extreme coefficient 35 to another equal and similar
one, in which the mean coefficient shall be less than
either of the extremes.
First, put y = y’ — m z, which value of y, being
substituted in the given formula, gives
35 y” — (70 m – 172) y': + (35 m” – 172 m + 210) z”.
And now, in order that 70 m – 172 < 35, take m = 2,
which reduces the above to
35 y” + 32 y' 2 + 62° = @,
in which the mean coefficient 32, though 335, is still
> 6 ; and, therefore, we must proceed to another
similar reduction.
666 T H E O R Y O F N U M B E R. S.
Let, then, z = z' — m y', and the second transformed divisors of tº + a u", and, consequently, any factor or sect. Ix.
Theory of
Numbers, formula will become divisor of the formula tº + a u” is of the form *śl
*~ - rº n las.
YT 62” – (12 m – 32) y' z' -- (6 mº – 32 m + 35) y”. p y” + 2 q W u + r u”. sº
And here, taking m°= 3 in order that 12 m – 32 < 6,
we obtain
6 * – 4 z' y' – 7 y” = @,
and this last formula has the required conditions; be-
cause 4 × 6 and < 7. -
And moreover, in these transformations, we have
p r sms g” = p r" -> q* = p/r' - q”, Or
35. 210 – (66)* = — 46,
35. 6 – (16)* = — 46,
– 6. 7 – ( 2)” = — 46,
all equal, and with the same sign, as observed in the
foregoing deduction.
75. Every divisor of the formula tº -- a wº, in which
t and w are prime to each other, and a any integer
number whatever, positive or negative, is also a divisor
of the formula q” + a.
For let p represent any divisor of the formula
t” + a u", so that
tº + a u” = p p",
then it is evident, that p is prime to u, for otherwise t
and u must have the same common measure, which is
contrary to the hypothesis, because t is prime to u;
we may, therefore, find two other numbers, q and y,
such that t = p y + q u, q being + or – as the case
may require : and if now we substitute this value of t,
in the above expression, we obtain
p” y” + 2 p q y u + (q” + a) w” = p p’;
or, dividing by p, we have
q* + a I
py-ºnvu +(−)-e-p'.
and, consequently, since p' is an integer, (q + a) u” is
divisible by p, but we have seen that w is prime to p,
and, therefore, it must be the other factor, (q + a),
that is divisible by p, therefore, if p be a divisor of the
formula tº + a u”, t and at being prime to each other, it
is also a divisor of the more simple formula q” + a.
Hence, conversely, if p be not a divisor of the for-
mula q*-ī- a, in which there is only one indeterminate
quantity q, it cannot be a divisor of the more general
formula tº + a u", in which there are two indetermi-
mates prime to each other. -
76. Every divisor of the formula tº -- a w”, in whic
t and u are prime to each other, is of the form
p y” + 2 q y u + r u”; and in this formula p r — q* = u
2 q → p and < r, or not greater than p or r
By the foregoing proposition we have
py'--2 gyz--(* –– )*=p'
p
q” + a q*-ī- a
p p
= r, then
and since is an integer, make
the above becomes
p y” –– 2 q y u + r u = p/;
that is, the factor
p' = p y” + 2 q y u + r u";
but p' may equally represent any one of the factors or
q*-ī- u
p
seen how every indeterminate formula
py” + 2 q y u + r u"
may be transformed to a similar and equal formula,
so that 2 q >< p or < r, and in which p r — q* is always
equal to the same constant quantity. Consequently
every divisor of the formula tº + a u” has the property
stated in the head of the proposition.
= r, p r — q” - a, and we have
And, again, since
DEDUCTIONs.
(1.) Because 2 q → p, and 2 q > r, independently of
the signs of these quantities, we have 4 q” < p r ; and
since p r — q = a, it follows, that when a is negative,
p or r, that is p r is also negative, for otherwise
p r — q” would not have the same sign as a ; which
we have seen is always the case in every transforma-
tion. Hence
(2.) Every divisor of the formula tº -- a u”, when a
is positive, may be represented by the formula
p y” —- 2 q y z + r z”,
in which p r - q = a, 2 q → p, 2 q > r, and, conse-
quently, 4 g” < p r, and therefore p r – q" = a > 3 q",
a . .
or q × V a, as is evident.
(3.) And every divisor of the formula t” -- a u" may
be represented by the formula p y” + 2 q y z – r z”, in
which p r – q = — a, or p r + q = a ; and here, since
p r < 4 q", we must have q >< V;
(4.) We may have cases in which p = r = 2 q, as,
for example, when p = 2, q = 1, and r = 2; for then
2 ” does not exceed either p or r, neither are p, q, and r,
d. , 12. Lie by the same number, which condition is, there-
fore, strictly within the limits of the proposition; and
hence it follows, that we must not consider the sign 3
in the two expressions q 3 V; and q × V; to
exclude equality.
77. Every divisor of the formula tº + u", t and u
being prime to each other, is always of the same form
9°-H 2". Or the sum of two squares, which are prime
to each other, can only be divided by numbers that are
also the sum of two squares.
For by deduction 2 of the foregoing proposition, every
divisor of the formula tº + a u” is included in the for-
mula
p y” + 2 q y z + r z”,
and in which q > V; and p r – q* = a.
Now in the present case a = 1, therefore,
T
q ‘ V+, or q = 0, there being no integer
T
< V; and, since p r — q* = 1, we have pr = 1
T H E O R. Y O F N U M B E R S.
667
Theory of and therefore p = 1, and r = 1; and, consequently, the
Numbers. above formula, which includes all the divisors of tº + w”,
* becomes
gº + zº ;
that is, every divisor of the formula tº + wº is of the
form y” + 2*, or every divisor of the sum of two
squares, prime to each other, is also the sum of two
Squares.
DEDUCTIONs.
(1.) As an example, 65 = 64 + 1, or 8* + l’ is only
divisible by 13 = 3°-H 2°, and by 5 = 2*-H 1.
(2.) And 50 = 78 + 1* is only divisible by
5 = 2* + 1", by 10 = 3*-i- I*, by 2 = 1* + 1", and
by 25 = 4* + 3”; and the same obtains with the divi-
sors of every number that is the sum of two squares
prime to each other.
78. Every divisor of the formula tº + 2 w", t and u
being prime to each other, is of the same form
gy* + 2 z*; or the divisors of the sum of a square, and
double a square, are also each equal to the sum of a
square and double a square.
For every divisor of this formula tº + a w’ is con-
tained in the formula
p y” + 2 q y z + r *,
in which q >< V; and p r — q* = a, (art. 76,-2.)
Tº
But in this case a = 2, therefore q > Vºw- 0;
also, since p r — q* = 2, we have p r = 2, whence
p = 2, and r = 1, or p = 1, and r = 2; therefore, the
above formula becomes
{* y”–H zº, in the first case, and
y” —– 2 zº, in the second,
which are two identical forms, by changing y into z,
and z into y; consequently, every divisor of the formula
t” + 2 w" is also of the same form as itself.
With regard to the divisor 2, it can only be of the
form y” + 2.2°, when y = 0 and z = 1; so that, in this
case, we have 0° -- 2. 1°.
As an example to this proposition, we may take
99 = 1 + 2.7°, which can only be divided by
3 = 1° -- 2. 1°,
9 = 1° -- 2.2°,
1 l = 3* + 2 . 18,
33 = 5* + 2. 2";
and it is the same with every number that is contained
under the above form.
79. Every divisor of the formula tº — 2 w", t and w
being prime to each other, is of the same form
y” – 22*.
For since every divisor of the formula tº — a wº is
contained in the formula
p y” + 2 q y z – r zº,
in which p r + q = a, and also q >< V; Or
• 2.
< V. (art. 76,-3,) it follows, that in this case
q = 0, whence also p r = 2, and therefore p = 2,
r = 1, or p = 1, and r = 2; consequently, the above Sect. IX.
formula becomes either Algebraical
2 y” — 2*, or y? – 2 23, sº
which two forms are the same; because
2 y? – 22 = (2 y + 2)” – 2 (y-E 2)",
which is the same form Therefore every divisor of the
form tº — 2 w? is of the same form, or the divisors of
the difference between a square and double a square is
also the difference between a square and double a
square.
Thus, 98 = 10° – 2. 12, has for divisors
2 = 22 — 2. 12,
7 = 32 — 2. 12,
14 = 42 – 2 . 12,
49 = 92 – 2.42;
and the same obtains with
the form tº — 2 w?.
80. Every odd divisor of the formula p” + 3 uº is of
the same form, viz. y? -- 32°.
For since all its divisors are contained in the for-
mula
all numbers falling under
p y” + 2 q y z + r z”,
in which p r — q2 = a, or p r — q” = 3, and also q =
3
or < V; we must have q = 1, or q = 0; therefore,
in the first case, since 2 q is not greater than p, or r,
and p r — q2 = 3, we must have p = 2, and r = 2,
whence the formula becomes
2 y” + 2 q y z + 2 z*;
but as this is evidently an even divisor, it does not be-
long to the class at present under consideration, which
only relates to the odd divisors of the given formula.
In our case, therefore, q = 0, and, consequently,
p r — q* = 3, or p r = 3; therefore p = 3 and r = 1,
or p = 1 and r = 3, whence the above formula is
reduced to
3 y? + 2*, or y” + 3 z*;
which are identical as to their form, and therefore
every odd divisor of the formula tº + 3 w” is of the
same form y + 3 z”.
DEDUCTION.
When the divisor = 3, then y = 0, but in all other
cases y and z are real quantities. .
For example, 133 = 5% + 3.6°.
{ 19 = 42 + 3, 12,
7 = 2* + 3. 12.
its divisors
Again, 1209 = 33 – 3. 20°.
ſ 13 = I* + 3.2°,
31 = 2 + 3.3°,
its divisors, < 39 = 6* -- 3.1°,
lº = 92 + 3.2°.
&c. &c.
81. Every odd divisor of the formula t” – 5 w” is
also of the same form y” — 5 z*.
For all its divisors are contained in the formula
p y” + 2 q y z – r z”,
668 T H E O R Y OF N U M B E R S.
Theory of in which — p r – q = - a, or p r + q2 = 5, and and from these arise, by way of exclusion, the three Sect. X.
Numbers. 5 following: Prime
S-- q = or < V; ; and, consequently, q = I or 0; but (4.) No number of the form 4 n – I can be repre- ***.
5 sented by the formula y” + 2*.
the first case gives only even divisors, the same as in
the foregoing proposition ; and the latter case of q =
0 reduces the above formula to
5 gy’ — 2*, or y” — 5 z*,
which are identical forms; because
5 y” – 2* = (5 y + 2 2)” – 5 (2 y + 2)”;
and, consequently, every odd divisor of the formula
tº — 5 w” is itself of the same form.
As examples, we have
95 = 102 – 5 . 12.
e & 9 5 = 5° – 5 . 22,
its divisors { wº
19 = 72 – 5 . 22.
Again, 395 = 20° – 5. 22.
it & divrº core ſ 5 = 5? – 5. 28,
its divisors TU 79 — 182 – 5 . 72.
&c. &c.
DEDUCTIONs.
(1.) From the foregoing proposition it appears, that
all numbers which are comprised in the following
formulae,
2
tº + w }. – 2 ..}and t” – 5 wº,
t? -- 2 w") tº + 3 wº
t and u being prime to each other, are of the same form
as the numbers they divide, excepting only the two
latter, tº -- 3 w” and tº — 5 u”, when these are the
doubles of an odd number.
(2.) It frequently happens, that a number falls under
two or more of the above forms, in which case its
divisors are also of the same double or treble forms ;
and in some cases we have numbers that belong to
each of the forms above given. Thus,
24, 1 -: 15% 2 = 2i.” — I ()2
15* + 4 2}* — I } = 312 – 5. 122
241 = 132 + 2.6*) = 78 + 3.8%
X. Of the classification of Prime Numbers, according
to their quadratic forms.
82. We have already treated of the linear forms of
prime numbers, but there are several curious properties
of these numbers, depending on their quadratic forms,
which ought to find a place in an article of this kind;
several of these are the immediate consequence of
some of our preceding propositions, and others dedu-
cible from them. Of the former, the following theorems
may be enumerated, which, however, applies to all odd
numbers whatever.
(1.) Every odd number which is the sum of two
squares, is of the form 4 m + 1 ; that is, every odd
number represented by the formula
y” + 2* = 4 m + 1.
(2.) Every odd number represented by the formula
gy* -- 2 z* = 8 m + 1, or 8 m + 3.
(3.) Every odd number represented by the formula
gy” – 2 2* = 8 m + 1, or 8 m + 7;
(5.) No number of the form 8 n + 5, or 8 m + 7,
can be represented by the formula y” + 2 2*.
(6.) No number of the form 8 m + 3, or 8 m -- 5,
can be represented by the formula y” — 22°.
83. Every prime number of the form 4 m + 1 is the
sum of two squares, or is contained in the formula
y” + 2*. -
For let m represent a prime number of this form, or
m = 4 m + 1 ; then (art. 59)
(a"-" — 1) = M (m), or (r" – 1) = M (m).
But a “ — 1 = (x^* + 1) (a” – 1), and each of these
factors has 2 n values of a contained between the
limits + 3 m and — ; m, that render them divisible by
m, (art. 60,) whence the factor wº" + 1 is divisible by
on ; but a " + 1 is the sum of two squares, and there-
fore its divisor m is also the sum of two squares ; be-
cause every divisor of the formulatº—H w” is itself of the
same form.
DEDUCTIONs.
(1.) As the form 4 m + 1 includes the two, 8 m + 1
and 8 m + 5 ; therefore every prime number con-
tained in these two latter forms is also the sum of two
squares.
Thus, 5, 13, 17, 29, 37, and 41, are prime numbers
of the form 4 m + 1, and each of these is the sum of
two squares; for 5 = 2* + 1°, 13 = 3* + 2*, 17 =
43 –- 1, 29 = 5* + 2*, 37 = 6*-ī- 18, and 41 = 5* + 4*;
and so on for all other prime numbers of this form.
(2.) We have seen (art. 63) that every number,
which is produced from the multiplication of factors
that are the sums of two squares, is itself of the same
form, and may be resolved into two squares different
ways, according to the number of its factors ; and
hence we may find a number, that is resolvable into two
squares as many ways as we please, by multiplying
together different prime numbers of the form 4 m + 1.
84. Every prime number 8 n + 1 is of the three
y” + 2*, y” + 2 2*, y” — 2 z*.
Let m be any prime number of this form, or
m = 8 m + 1 ;
and as the first case has been demonstrated in the pre-
ceding proposition, we need here only attend to the two
latter.
Since (a"-" — 1) = M (m), or aº" — I = M (m),
(art. 59,) we may put this under the form
(a" – 1) (a" – 1),
and each of these factors will have 4 m values of a <
} m that render them divisible by m, (art. 60;) there
are, therefore, so many different values of a that render
the binomial arº" + 1 divisible by m ; but this may be
put under the form
(r" – 1) a + 2 rº";
and m being a divisor of this formula, it is itself of the
same form y” – 2 sº, (art. 78.) We may also put the
same quantity a " + 1 under the form (w" + 1)” –
2 a.”, and ºn being also a divisor of this formula is
itself of the same form y” – 2 2*, (art. 79.) Hence
every prime number of the form 8 n + 1 is of the three
y” + z”, y” + 2 2*.
forms
forms
T H E O R Y O F N U M B E R S.
669
Theory of
Numbers.
v-y-'
ſ41 = 5* + 4* = 3 + 2.42 = 7s — 2.2%
Thus 2 W. -
U73 = 8* +3* = 1 + 2.6* = 9 – 2.2.
85. Every prime number 8 m + 3 is of the form
gy* + 2 z*.
Let m be a prime number of this form, or m = 8m ––3,
then we have (art. 59)
(r"-i – 1) = M (m), or a "+" — 1 = M (m).
And there are 8 m + 2 values of a less than 8 m + 2,
which render this formula divisible by m.
Now 28:42 – 1 = (2* + 1) (2** – 1) = M (m),
therefore one of these factors is divisible by m, and it
cannot be the latter, because
24*1 – l = 2. 24" — 1 => 2 #2 — w?, or tº — 2 w”,
and if m were a divisor of this it would be of the same
form, or m = y” – 2 zº, but this formula cannot repre-
sent any number of the form 8 n + 3, (art. 82.) Con-
sequently, m must be a divisor of the other factor
2"+ + 1. But
2* + 1 = 2.2" + 1 + 2 tº + wº.
Consequently its divisor m is of the same form; that is,
m = 2 y” + 1 z* = y? -- 22°.
As examples, we have
11 = 3% - 2. 13, 19 = 1 + 2. 33,43 = 5* + 2. 3., &c.
86. Every prime number 8 m + 7 is of the form
y” – 2 z*.
Let m = 8 m + 7, then we have
a"-" — 1 = rº"+" – l = M (m).
Hence, therefore, as above
2** – 1 = (2* + 1) (2*– 1) = M (m),
one of these factors is divisible by m ; and, conse-
quently, m will also be a divisor of one of them when
doubled; that is, it is a divisor of one of the two quan-
tities
2 (2* + 1), or 2 (2** – 1),
which two expressions thus become
2* + 2. 18, and 2" — 2. 13,
and m is necessarily a divisor of one of them. But it
cannot be a divisor of the first, because this being of
the form tº —– 2 w”, if m were a divisor of it, we should
have m = y2 +22°, (art. 78;) but m = 8 m + 7, and no
odd number of the form y” – 2 z* is of the form 8 m + 7,
(art. 82 :) since, therefore, m is not a divisor of this
factor, it must necessarily be a divisor of the other fac-
tor 2" — 2 . I?, which is of the form t” – 2 w”; and,
consequently, its divisor m is also of the same form,
(art. 79 ;) that is, m = y” — 22°.
For example, 31 = 7” – 2. 34, and 47 = 7” — 2. 1*;
and the same of all other prime numbers in this form.
DEDUCTIONs.
From the last four propositions we may draw the
following theorems:
(1.) All prime numbers of the form 8 n + 1, and
8 n + 5, are, exclusively of all others, contained in the
formula y” + 2*. -
(2.) All prime numbers of the form 8 m + 1, and
8 n + 3, are, exclusively of all others, contained in the
formula y” + 2 zº,
WOL. I.
(3.) All prime numbers of the form 8 n + 1, and
8 m + 7, are, exclusively of all others, contained in the
formula y” – 22°.
Sect. X.
Prine
Numbers.
(4.) All prime numbers of the form 8 m + 1, are at ‘-w
the same time of the three forms
gy* + 2*, y” + 22°, y” — 22*.
87. If a be any prime number, and the series of
Squares
a – 1 Nº
2
be divided by a, they will each leave a different positive
remainder.
This is, in fact, only a particular case of the general
proposition demonstrated (art. 38;) for, by making
% = 1, the series of squares,
q}*, 2? q}”, 3” Ø”, 4? qº, &c., (. º ') p°,
1*, 2*, 3°, 4”, &c.,
becomes -
a – IN.”
2 2
each of which, when divided by a, will leave a different
remainder, as is demonstrated in that article.
1*, 2*, 32, 42, &c.,
DEDUCTIONS.
(1.) The same is evidently true of the negative
remainders, which arise from taking the quotients in
eXCeSS.
(2.) Hence, also, we may see in what cases the posi-
tive and negative remainders are equal to each other,
for then it is evident, that a will be a divisor of the sum
of two squares, and we shall have
s” tº
= e, an integer.
Therefore when a is not a divisor of the sum of two
squares, the positive and negative remainders are all
different from each other, and include every number
from 1 to a - 1.
(3.) When a is not the divisor of the sum of two
squares, that is, when all the positive and negative
remainders are different from each other, then some
of each of these remainders are greater and some
less than , a. For all the consecutive squares under a
will be found amongst the positive remainders, and
some of these squares must necessarily be greater and
some less than # a ; and since the positive and negative
remainders together include all numbers from 1 to
a — 1, the same is manifestly true of the negative
remainders.
88. If a be a prime number, it is always possible to
find four squares, wº, aº, y”, z*, the roots of each of
which shall be less than , a, such that their sum may
be divisible by a, or the equation
& w” + a "-j-y” – 28 = a aſ
is always possible, a being any prime number what
ever. -
First, when the prime number a is a divisor of the
sum of two squares, the proposition is evident; and it
will, therefore, only be necessary to consider the case
in which a is not a divisor of the sum of two squares,
and, consequently, when all the remainders of the
consecutive squares are different from each other
(art. 87, −2.) -
Now, in this case, we shall find some of the positive
4 S
670
T H E O R Y O F N U M B E R S.
Theory of
Numbers.
*-----, º 2
remainders greater, and some less, than , a , and the
same of the negative remainders, (art. 87,-3.) It is,
therefore, always possible to find two squares, such
that each being divided by a, the positive remainder of
the one shall exceed the negative remainder of the
other, by unity: and also two other squares in the
same series, such that each being divided as before, the
negative remainder of the one shall exceed the positive
remainder of the other, by unity; that is, the equa-
tions w” + æ" — 1 = m a, and y” + 2* + 1 = m a, are
always possible, which may be demonstrated as fol-
lows:
Let p be the least negative remainder, then p + 1
must be found amongst either the positive or negative
remainders ; if it be found amongst the positive re-
mainders, we have at once a positive remainder, that
exceeds a negative remainder, by unity; and if it be
not found amongst the positive, them p + 1 is still
negative : and p + 2 must be either a positive or
negative remainder; if it be positive, we have a posi-
tive remainder exceeding a negative one, by unity, but
if not, p + 2 is still negative, and p + 3 must be either
positive or negative; and proceeding thus, we must
necessarily (as some of each of these remainders are
greater and some less than ; a) arrive at that negative
remainder p’, such that p' -- I shall be a positive one ;
and, consequently, the equation wº + ar" – l = 'm a is
always possible ; and, in the same manner, the possi-
bility of the equation y” —- z*-ī- 1 = n a may be de-
monstrated. Having thus proved the possibility of the
equation w” + æ — 1 = m a, and y” + 2* + 1 = n a,
we have
w” + as + y” + 2*
(!,
or the equation
w” + a " + y” + 2* = a aſ
is always possible.
= m + n, an integer,
DEDUCTION.
(1.) It is obvious from the foregoing demonstration,
that the roots w, a, y, z are each less than 4 a, because
we have only considered the squares contained in the
series
-º-º: 2
1*, 2*, 3°, 4°, &c., (: 2 y
But independently of this limitation it may be readily
shown, that if a be the divisor of the sum of any four
squares w” + a 4 + y” + 2*, each of which is prime to
a, that it is also a divisor of the sum of the four
squares -
(w — a a)* + (a — 8 a)*-- (y – ’) a)* + (2 – 8 a)*,
in which it is obvious, that a, B, Y, 8, may always be so
taken as to make the roots less than 4 a.
89. Every prime number a is the sum of two, three,
or four squares.
For, by the foregoing proposition, the equation
w” - a 2 + y^+ 2* = a aſ
is always possible, each of the roots of these squares
being less than , a ; and, consequently, each of the
squares less than + a”, whence we have a aſ < a”, or
a' < a. Now, if a' = 1, it is evident that
w? -- alº –– y” + 2* = a,
and the proposition will be demonstrated.
But if a > 1, then, because a' is a divisor of the
formula
w” + a” -- y” + 2*,
it is also a divisor of the formula
(w — a a')* + (a – 6 a.)” -- (y – Ya')? -- (2 – 8 a')*,
where each of the roots is less than ; aſ, (art. 88,-—1;)
assuming, therefore,
(w—a a')*-ī- (a — 8a')*-H (y—qa')*-ī- (2–8 a')* = a'a',
we shall have, for the same reason as above,
a" aſ < a.”, or a!" < a.'.
Now, by means of the formula (art. 66,) if we mul-
tiply together the values of a a', and a' a', we shall
find a product that is the sum of four squares, and of
which each is divisible by a”; and having performed
this division, we obtain
a"a = (a — a w–6a – Yy–82)*-ī-(aw–6 w =|- Y 2–6 y)*
-- (a y — ºf w = 3a – 3 z)*-H (a 2–8 w -- 3 y – Ya)*;
or, for the sake of abridging this expression,
w” -- a " + y^* + 2* = a” a ;
and here we have a” < a.'. If now a' = 1, the above
becomes
w” + a " + y^* + 2* = a,
and the proposition will be demonstrated ; but if a”,
though 3 a', be > 1, we may proceed, in the same
manner, to find a new product,
w” -- a” + y” -- z", reaſ" a,
and in which a' 3 a”; and by continuing thus the
decreasing series of integers a, a', a!", a'", a”, &c.,
we must necessarily, finally, arrive at a term a "' equal
to unity, and then we shall have a equal to the sum of
four squares.
90. Every integral number whatever is either a
square, or the sum of two, three, or four squares.
This follows immediately from the foregoing propo-
sition, and the formula, (art. 65;) for every number is
either a prime, or produced by the multiplication of
prime factors; and since every prime number is of the
form
(w” + as + y” + 2*),
and the product of two or more such formulae being
still of the same form, (art. 65,) it necessarily follows,
that every integral number whatever is of the form
(w”-- a”-- y” + 2*).
But it is to be observed, that no limitation in the
course of the demonstration of the foregoing proposi-
tion was made, that could prevent any one or more of
these squares from becoming zero; therefore, every inte-
gral number whatever is either a square, or the sum of
two, three, or four squares.
DEDUCTIONs.
(1.) All that has been proved in the foregoing pro-
position for integral numbers, is equally true of frac-
tions; for every fraction may be expressed by an equi-
valent one having a square denominator; therefore,
every fraction is of the form .
w” + º + y” + 2*_ Q02
777° 77??
ſp? * 2?
+ ··· +
772.
gy” e
*tº
Sect. X,
Prime
Numbers.
N-V-'
T H E O H. Y O F N U M B E R S. 671
Theory of
Numbers.
this curious property, therefore, extends to every
rational number whatever.
(2.) The theorem that we have demonstrated, in the
two foregoing propositions, forms a part of a general
property of polygonal numbers, discovered by Fermat;
which is this, “Every number is either a triangular
number, or the sum of two or three triangular numbers.
A square, or the sum of two, three, or four squares.
A pentagonal, or the sum of two, three, four, or five
pentagonals. And so on for hexagonals,” &c. Or the
same may be more generally expressed thus: If m
represent the denomination of any order of polygonals,
then is every number N the sum of m polygonals, of
that order; it being understood that any of these poly-
gonals may become zero.
Let, therefore, N be any given number, and ar, y, z
indeterminate quantities; then the different parts of the
general theorem may be detailed in the following
order:
a" + r , y?--y 2* + 2* .
2 ––
–H 2 2 ”
2d, N = w? -- a " + y? -- 2°;
1st, N =
3 uº — u 3 wº — w 3 tº — a 3 y” — y
3d, N =
N rººf -- 2 + 2 + 2
3 z” — z
+. 2 y
4th, &c. &c.
The second form which relates to the squares has
been demonstrated in the foregoing proposition, and
Legendre has also demonstrated the first case, for
triangular numbers; but all the other cases, past the
second, still remain without demonstration, notwith-
standing the researches and investigations of many
of the ablest mathematicians of the present time,
and of others now no more : amongst the former we
may mention Lagrange, Legendre, and Gauss; and of
the latter, Euler, Waring, and Fermat himself; the
latter of whom, however, as appears from one of his
notes on Diophantus, was in possession of the demon-
stration, although it was never published, which cir-
cumstance renders the theorem still more interesting to
mathematicians, and the demonstration of it the more
desirable.
We have demonstrated the second case, but this car-
ries us no farther, whereas, if we had demonstrated the
first, the second would flow from it as a corollary; and
it may not be uninteresting to show in what manner
these different parts of the same theorem are connected
with each other. -
First, let us suppose the possibility of the equation
a” —- a * -- y” z* + 2
N = #1, #2+ 2
to have been demonstrated, from which may be drawn
this,
8 N + 3 = (24 + 1)2 + (2 y + 1)2 + (2 z + 1)”, or Sect. X.
8 N + 3 = w” + y” –– 29, or Nº.
8 N + 4 = w” + y^2 + 2* + l; S-V-
and since these four squares are all odd, the numbers
a' + y', aſ — y', aſ + 1, and 2" — 1, are all even ; and
hence we have, in integers,
4 N + 2 =
r' + y^* , /a/ — y^* (2' + l\* (z' – 1\?
(*#): (*)-(+)-(+).
or, for the sake of abridging,
4 N + 3 - w”2 + q/2 +y” + 2's ;
of which squares two are even and two odd, for other-
wise their sum could not have the form 4 N + 2 ; we
may therefore write
4 N + 2 = 4 r8 + 4 sº -i- (2 t + 1)2 + 2 v -- 1)*;
from which we deduce -
2N + 1 = (r-Hs)*-ī- (r—s)* + (t + v.-- 1)2 + (t–v)?;
that is, every odd number is the sum of four squares,
and the double of a number, that is, the sum of four
squares, is itself the sum of four squares, for
ſ2 (m2 + nº + p + q’) =
l(m+n) + (m – n) + (p + q) + (p → q)";
and, therefore, every number is the sum of four
Squares.
If, therefore, the case which relates to triangular
numbers was demonstrated, that which relates to
squares would be readily deduced from it; but the
converse has not place ; that is, we cannot deduce the
first case from the second.
The third case gives
3u?—w 3v’—w 3*-* +3y'-y 32°–2
N ===-|--|--|--> 2 g-or
24 N + 5 =
(6u – 1)*-H(6 wit 1)*-i- (63 – 1)*—H·(6y – 1)*-ī-(62 – 1)*.
So that the enunciation of this particular part returns
to this,
Every number of the form 24 N + 5 is the sum of
five squares, of which each of the roots is of the form
6 m — 1.
The fourth case returns to this,
Every number of the form 8 N + 6 may be decom-
posed into six squares, of which the roots are of the form
4 m — 1.
And, in general, the proposition is always reducible
to the decomposition of a number into squares, and all
the partial propositions that we have considered are
included in the general form,
8 a N + (a + 2) (a – 2)” =
(2.aa – a +2)*-ī- (2 a y – a ––2)*-ī-(2 a z—a + 2}*-ī- &c.
the number of squares on the latter side of the equa-
tion being (a + 2).
Trigono-
metry.
S-V-->
T R I G O N O M E T R Y.
TRIgoNoMETRY, (Tptywwouetpča, from Tptywyðs, a triangle, and uerpée, I measure,) the Science of Trian-
gles, the branch of Mathematics which treats of the application of Arithmetic to Geometry. The term was
originally restricted to signify the science which gives the relation of the parts of triangles described on a
plane or spherical surface; but it is now understood to comprehend all theorems respecting the properties
of angles and circular arcs, and the lines belonging to them. This latter department is frequently called the
Arithmetic of Sines.
In the application of Mathematics to Physics, no branch is more important than Trigonometry. It is
the connecting link by which we are enabled to combine, in their fullest extent, the practical exactness of
Arithmetical calculations with the hypothetical accuracy of Geometrical constructions. Without it, the former
could never have been applied to Physics, and the limit of the errors of the latter would have depended on the
skill of the practical Geometer. By the substitution of numerical calculations for graphical constructions, we
are enabled to obtain results to any desired degree of accuracy. With Trigonometry, in fact, Astronomy first
received such a degree of exactness as justly to merit the name of Science ; and every improvement that has
been made in Trigonometry to the present time, has been attended with corresponding improvements in all parts
of Physical Science.
The following will be the arrangement of the present Treatise: The first section will contain the definitions
of the terms most frequently in use; in the second will be given the principal theorems relating to Trigonome-
trical lines; the third will explain the use of subsidiary angles; the fourth will contain all the most important
propositions of Plane Trigonometry ; the fifth, those of Spherical Geometry; and the sixth, those of Sphe-
rical Trigonometry. In the seventh will be given formulae for small corresponding variations of the parts
of triangles; and the eighth will contain some theorems which require for their investigation a more refined
analysis. The ninth will treat of some expressions peculiar to Geodetic operations; and the tenth will
xplain the construction of Trigonometrical tables.
SECTION I.
Definitions.
(1.) LET A B (fig. 1) be a circular arc, of which C is the centre, and let C A, C B be joined. The arc
A B is proportional to the angle A C B, and either of these can therefore be used as the measure of the other,
provided the arc A B is less than half the circumference, or the angle A C B less than two right angles.
Since this holds with regard to all the angles of triangles, we shall, in treating of them, use indifferently the
terms arc and angle to express the inclination of two lines.
(2.) But in the higher parts of the science it is by no means a matter of indifference which term we
employ. It is evident, that an arc can be conceived to exceed, not only half a circumference, but even a
whole circumference, or any number of circumferences; while an angle cannot be greater than two right angles.
Much obscurity has frequently arisen from neglecting to observe, that when we speak of an angle greater
than two right angles, we mean merely an arc greater than half a circumference ; and that, when we consider
trigonometrical lines as functions of such an angle, we intend nothing more than that they are functions of the
corresponding arc of a circle. The reader, therefore, will be careful to recollect, that all trigonometrical lines are
considered to be functions of the circular arc to which they correspond, the radius being given; and that
there is no limit whatever to the extension of this arc.
(3.) The circumference of the circle has usually been divided into 360 equal parts, called degrees; each of
these subdivided into 60, calledminutes ; each of these into 60, called seconds ; the seconds are sometimes
divided each into 60 thirds, the thirds into 60 fourths, &c., but they are more usually divided decimally. But
in most of the French treatises lately published the circumference is divided into 400 equal parts, or grades,
each grade into 100 minutes, and each minute into 100 seconds. Degrees, minutes, and seconds are com-
monly marked *, ', '', grades and their subdivisions sometimes thus, 8, , ". Thus, 38° 17' 22" is read
thirty-eight degrees, seventeen minutes, twenty-two seconds; 448 V6 27", or 445,7627, is forty-four grades,
seventy-six minutes, twenty-seven seconds.
(4.) In most of the following investigations we shall consider the radius of the circle as the unit of
linear measure. The semi-circumference is then = 3,141592653590 ; its logarithm = 0,49714987.26;
one degree = 0,017453292520 ; one minute = 0,000290888209; one second = 0,000004848.137;
their logarithms increased by 10 are 8,2418773675; 6,4637261171, and 4,6855748667. One grade =
0,0157079632679; its logarithm increased by 10 = 8,196.1198769; from which the values for a minute
and second are immediately found. The number of degrees contained in the radius is 57,29577; the
number of grades is 63,66197. The value of the semi-circumference to radius I is generally denoted by
77 ; + is therefore the value of the quadrant, and 2 7 that of the circumference.
(5.) The defect of an arc from 180° is called its supplement; its defect from 90° is called its complement.
(6.) Join A B, (fig. 1;) draw B D and C F perpendicular to A C ; at A and F draw lines touching
the circle, which will therefore be parallel to CF, CA; produce C B to cut these lines in E and G. Then AB
672 - -
Sect. [.
Definitions.
`-y-
TRI Go No M ET R Y. 673
Trigong-
metry.
\-V-
is the chord of the arc AB, BD is the sine, C D is the cosine, AE is the tangent, C E is the secant, FG
is the cotangent, C G the cosecant, A D the versed sine. D H has been called by some the suversed sine.
(7.) These definitions suppose the arc to be less than a quadrant. If it be greater than a quadrant and less
than a semicircle, as A B', the same construction gives for the sine, versed sine, cosecant, cosine, tangent,
secant, and cotangent, the lines B'D', A D', C G', CD', AE', C.E.", FG'. The four last of these, it will be
observed, are measured in directions opposite to those in which the corresponding lines for arcs less than a
quadrant were measured, and are therefore considered negative.* We shall show that, by this convention,
formulae which have been found to be true for arcs less than a quadrant may be made to apply to arcs
greater than a quadrant.
(8.) If the arc be greater than two quadrants, and less than three, as A FH B", (fig. 2,) making the same
construction, we find that the sine, cosine, secant, and cosecant, are negative. And if the arc be greater than
three quadrants, and less than four, as A F H B", it appears that the sine, tangent, cotangent, and cosecant
are negative. The remark at the end of (7) applies to these. The versed sine and suversed sine are posi-
tive for all values of the arc.
(9.) Thus it appears, that, while the arc increases from 0 to a quadrant, the sine increases from 0 to
radius, (its greatest value,) and the cosine diminishes from radius (its greatest value) to 0. While the
arc increases to a semicircle, the sine diminishes to 0 ; and the cosine, whose sign is now negative, in-
creases in magnitude till it = — radius. As the arc increases to three quadrants, the sine is negative, and
its magnitude increases from 0 till it = — radius, while the negative value of the cosine diminishes till it
= 0. From three quadrants to four the sine, still negative, diminishes its negative value till it = 0,
while the cosine, become positive, increases till it is = radius, as at first. -
T
(10.) The tangent, while the arc increases from 0 till it is 2 increases so as to become greater than any
g e 7- 3 ºr e ſº º g
assigned quantity; when the arc = a , or -a-, there is really no tangent, as the lines, by whose intersection
the tangent is defined, do not meet; then, until the are = T the tangent is negative, and diminishes from a
value indefinitely great to 0; then, for the third and fourth quadrants the values are the same as for the first
e g 7- g ſº
and second. And the secant, while the arc increases from 0 to º, increases from radius to a value greater
than any assignable; it then becomes negative, and diminishes from a value indefinitely great to radius,
which it reaches when the arc = ºr; for the third and fourth quadrants its values are the same as for the first
and second, with the sign changed. *
(11.) If the arc, instead of being = A B, were = A B increased by any number of whole circumferences,
the values of the several trigonometrical lines would be the same as those for the arc A. B.
(12.) The definitions of the complement and supplement, without some extension, will not apply to arcs
greater than 90° or 180° respectively. It is only necessary to consider the defect of the arc from 90° or 180°
as being negative when the arc is greater than either of those values; and all the theorems relating to these
defects will be comprehended under the same formula. *
(13.) Since we have considered positive arcs as measured from A towards F, we may consider negative
arcs as measured in the opposite direction. Let A B, A B' (fig. 3) be equal arcs positive and negative; their
sines B D, B'D will evidently be in the same straight line; AE'- A E, FG'= FG, C E'= CE, CG'= C G.
Hence for a negative arc, the cosine, versed sine, and secant, are the same as those for an equal positive arc;
the sine, tangent, cotangent, and cosecant, are equal in respect of magnitude, but are affected with different
signs. Our figure supposes A B less than a quadrant, but it will be seen that the same is true if A B be
greater than a quadrant.
(14.) The whole of what we have assumed with regard to the signs to be affixed to the expressions for
lines according to their directions, is purely arbitrary. Its utility is this: a single formula, as we shall show
by induction, will now comprehend several cases for which as many separate formulae would otherwise have
been necessary. This, we conceive, is in all cases the true foundation for the use of the negative sign.
* SECTION II.
Relations of Trigonometrical Lincs.
(15.) IN the succeeding articles we shall use the abbreviations Sin, Cos, Tan, Sec, Cot, Cosec, Vers, to denote
the sine, cosine, &c. to the radius r ; and sin, cos, &c. to denote them supposing the radius = 1.
(16.) If C K L be drawn perpendicular to A B, (fig. 1,) A K = KB, the angle A C K = B C K, and the
arc A. L = B L, therefore A B = 2. A K. But A K is evidently the sine of A L, or ; A B. And the straight
line A B is the chord of the arc A. B. Hence Chord A B = 2. Sin A B , and chord A B = 2 sin Aſ *
sº
(17.) A D = AC – CD, or Vers A B = r – Cos AB, and therefore vers A B = 1 — cos A.B. By the con-
vention established with regard to signs it will be found, that this equation applies to arcs terminated in all
quadrants of the circle.
* The secant is negative, because it is not measured from the centre in the direction of the radius through the extremity of the are, but
in the opposite direction
Sect. II.
Relations of
Trigono-
Inetrical
Lines.
S-N-7
Fig. 2.
674 T R. I G O N O M ET. R. Y.
Sºº-yº”
}
Trigono-
metry.
g
4
(18) By similar triangles, (GEOMETRY, book iv. prop. 20,) A E = pºº, or Tan AB = º, rº
and tan AB = #. tº º
(19.) By similar triangles, FG = cº, or Cot A B = **** and cot A B = #. ſº S-N-2
(20.) Multiplying together these expressions, Tan A B × Cot A B = rº, and tan A B cot AB = 1.
(21.) By similar triangles, C E = cº , or Sec A B = cºw and sec A B = sº o
(22.) By similar triangles, C G = º, or Cosec A B = sº and cosec A B = mºn g
(23.) Suppose H B = A B ; then A B', or 180° – H B', is the supplement of A B. And B'D' = BD,
C D' = C D, A E = A E, C E = CE, F G' = F G, C G' = C G, A D = H D’. Hence the sine and cosecant
of any arc are the same as those of its supplement; the cosine, tangent, cotangent, and secant, are equal in
magnitude, with different signs; and the versed sine of one is the suversed sine of the other.
(24.) If A b = F B, and b d, C eg, be drawn as before, it is plain that b d = CD, C d = B D, A e = FG,
F g = A E, C e = C G, C g = C. E. But b d, C d, Ae, Fg, Ce, C g, are the sine, cosine, tangent, cotangent,
secant, and cosecant of A b or B F; and B F is the complement of A. B. Hence the sine, cosine, tangent,
cotangent, secant, and cosecant, of the complement of an arc, are respectively equal to the cosine, sine, cotan-
gent, tangent, cosecant, and secant of the arc.
(25.) All these theorems have been proved for arcs less than a quadrant. If, however, we make use of the
convention established with regard to signs, it will be found that they apply to every case. For example, when
the arc, as A F H B", fig. 2, is greater than three quadrants and less than four, the sine is negative, the cosine is
sine
positive; therefore the tangent = (18) ought by the formula to be negative; which from the figure it
cosine
appears to be. The magnitude is determined by the same proportion as before, and cannot be erroneous. The
- I *
Secant E (21) ought to be positive; and the cosecant = + (19) ought to be negative; as they are found
SIlle
cosine
to be. The same, it will be found, is true for every other case.
(26.) By similar triangles, the following proportions will easily be verified. Radius: Sin A B :: Sec A B :
T. Tan A B tan A B
Tan A B ; therefore Sin A B = , and sin A B = Radius : Cos AB . . Cosec A B . . Cot A B;
Sec A B sec A B'
r. CotAB __ cot A B
therefore Cos A B = Cosec A B' and cos A B = cosec A B'
(27.) Since (Sec A B)* = r^+ (Tan A B)*, (GEOMETRY, book iv. prop. 13,) or secº AB = 1 + tan” A B, and
tan A B w/ secº AB-1
v I-E tang AB T sec A B
cosec” A B = 1 + cot” A B, we may thus express these values; sin A B =
cot A B *ºm v'cosecº A B — I
cos A B = — = . And the equations of (21) and (22) may be thus expressed;
v I-E cotº A B cosec A B
l
cos A B = — . sin A B = ==
- v'1+ tan” AB ' W 1 + cot” A B
(28.) In the same way, observing that sin” A B -- cosº A B = 1, we find from (18) and (19), tan A B
in A B w/T- cosº AB w/ Tsing AB
== sin A tº :- COS ; cot A B = __cos AB - ... :-- 1 — sin AB. These are the
W I - sing AB cos A B v. 1 — cos’. A B sin A B
principal formulae of the relations of trigonometrical lines belonging to one arc.
(29.) We proceed to one of the most important propositions of Trigonometry. To find the sine and cosine of
the sum and difference of two arcs in terms of the sine and cosine of the simple arcs. Let A B, fig. 4, be the
longer arc = A; B E = B F = B; then A E = A + B, A F = A — B. Draw E G, F G, perpendicular to
C B, which will meet at G and be in the same straight line, and will be equal ; also draw B D, E H, FK, GL,
Terpendicular to AC, and GM, FN, perpendicular to E H, G L. Then E H or G L -- E M = sin A + B;
FK or G II – GN = sin A – B ; CH or C L – G M = cos A + B ; C K or CL + F N= cos A – B. Now
the angle E G M = 90°– M G C = C G L = C B D ; also E M G and C D B are right angles, therefore the
triangles E G M, B C D, are similar, and C B : C D : : E G : E M, or Radius : Cos A :: Sin B : E M
. Si SI . Si
- Cos A.Sin B = GN. And C.B.: B D. : E G : G M, orradius sin A::sin B. GM =**** = FN. Also
*º-º-º-º- ºr--------
r
T R I G O N O M E T R Y.
**
675
Trigono-
B D. C G Sim A. Cos B g
CD. C G Sect. II.
... C B : B D :: C G : G L = C B 7. ; and C B : C D : : C G : C L = GB ºf
t } rigono.
Cos A. Cos B &=mºm-º-º-º gº - . Si trical
vos A. vos . Substituting these values Sin A —- B = Sin A. Cos B -- Cos A. Sin B. Sin A — B º
7" 7" \-y-Z
Sin A. Cos B — . Si ſºmºsºm-s-s & B — Sin A . Si *a---
:- sm os B – Cos A sin B, Cos A -- B = Cos A . Cos b. Sin A . Sin B. Cos A – B
r
C © Nº 2 J \, in A. Si
:- os A cºnfisms Sin B. Or, if the radius be the unit of measure,
sin A+B = sin A. cos B + cos A. sin B;
cos A+B = cos A. cos B – sin A. sin B; cos A – B
(30.) It is here supposed that A is greater than B, and that A is less than 90°.
sin A — B = sin A. cos B — cos A. sin B ;
= cos A. cos B + sin A. sin B.
If these conditions should
not hold, it would still be found that, by virtue of our conventions with regard to the signs of arcs and straight
lines, the same formulae would apply. We shall leave it to the reader
case, and shall, merely as an example, suppose A greater than 180°, B greater than 90°.
Make the same construction in every respect as before ; then E' H’= E/M'— G'L'
= A; B/E' = B'F' = B.
ºn CD’. E'G' B' D’. C. G
- TGBT --GB7- -
CD'e - Cos A; E' G' = Sin B; B' D'a- Sin A; C G = – Cos B;
_ – CosA.Sin B-Sina.Cos B Sin A. Cos B -- Cos A.
/
But, by (7) and (8), since A F B'E'
to examine in this manner every distinct
Let A F B'
A+ B, E H is - – sin ATB,
thus the equation becomes — Sin A + B
Sin B
tº- , or Sin A + B =
7° 7"
And the same will be found to be true for every different case.
; the same as for arcs less than 90°
(31.) From these expressions, sin A + B – sin A — B = 2 sin A. cos B,
sin A – B – sin A — B = 2 cos A. sin B,
cos A + B + cos A – B = 2 cos A. cos B,
cos A – B – cos A + B = 2 sin A. sin B.
(32.) Let A + B = C ; A – B = D ; then in c-sin D = 2 sinº coºr".
sin C — sin D =2 coº sinº,
cosc-cos D=2 cos ºf P coºr”.
cos D – cos C-2 sinº sinº.
(33.) Let B = A; them sin 2 A = 2 sin A cos A; and cos 2 A = cosº A — sin.” A =
cos 2 A
; cos A=v/ºf
versin 2 A
º 1 — 2 A
From these, sin A = Vi-gº = \,
*=ºmºsºsº-sº-sºuse- 1 — VI – sin”:
"T-ºxº-as-a-v. *; sin” 2 A
= 4 (w/TF sin 2 A -- VT-sin 2 A).
cos A sin A cosº. A + sin” A
tº in, cot A + tan A = −. — tºº — tº:
(34.) Again, cot A+ tan sin A cos A sin A cos A
cos” A -- sinº A 2 cos
sin A cos A
= 2 cosec 2 A. Similarly, cot A — tan A =
V. + cos 2A"
1 - tan” A
1 — cos 2 A
1 + taig A
Hence cos 2 A =
T sin 2 A
(35.) Since sin A = A/ 1=ºsa, and cos A = VIEE
cos 2 A
-—-----, we have tan A =
1–2 sin” A = 2 cos” A — i.
2 If in these values we put
= 4 (VT-F sin 2 A – MTsin 2 A); cos A
l 2 2
sin A cos A T 2 sin A cos A sin 2 A
O
-s)
A
= 2 cot 2 A.
sin A
cos A T
2
676 T R I G O N O M ET R Y.
Trigono-
metry.
\-
g gº tan A I -
(36.) Since sin A = – “t--, and cos A = –––. (27) sin 2 A = 2 sin A cos A º
w/ I + tan’A v/ 1 + tan? A Trigono-
2 tan A metrical
:= —. Lines
1 + tanº A w º
(37.) l — cos 2 A 2 sin” A ** = an A. similarly-tº-- tan A
, j –— — — :- -: g TIV. --—- = e
sin 2 A 2 sin A cos A cos A al J. T.T.cos 2 A
(38.) From (31,) sin A+ B = 2 sin A : cos B — sin A — B. Let A = n B ; then sin n-Ei B =
2 sin n B. cos B – sin n – l B. Making n successively = 2, 3, &c., we form the following table:
sin B = sin B,
sin 2 B = 2 sin B cos B,
sin 3 B = 3 sin B — 4 sin” B,
sin 4 B = (4 sin B – 8 sinº B) cos B,
sin 5 B = 5 sin B – 20 sinº B + 16 sin” B,
&c. &c.
Again, from (31,) cos A + B = 2 cos A. cos B – cos A – B. Let A = n B, therefore cos n + 1 B =
2 cos n B. cos B — cos n – l B. Making n successively = 2, 3, &c.
cos B = cos B,
cos 2 B = 2 cosº B – I,
cos 3 B = 4 cosº B – 3 cos B,
cos 4 B = 8 cosº B — 8 cosº B -- 1,
cos 5 B = 16 cosº B — 20 cosº B –– 5 cos B,
&c. &c. -
(39.) sin A + B. sin A — B, by (31,) (putting A + B for A, and A – B for B) = } (cos 2 B – cos 2 A)
= 4 (1 – 2 sinº B – 1 + 2 sin” A); by (33,) = sin” A-sin"B, or = cosº B – cos” A. And cos A+B. cos A–B
= } (cos 2 B + cos 2 A) = 4 (1 – 2 sinº B + 2 coss A – 1) = cos"A — sin"B, or = cos’ B – sin” A.
(40.) sin A+ B sin A cos B + cos A. sin B — tan A + tan B Or tº: cot B -- cot A
sin A – B sin A cos B – cos A. sin B tan A – tan B' cot B — cot A
COS A-EB cot B – tan A O cot A – tan B
* = r = –
cos A – B wº- cot B -- tan A Bººm! cot A + tan B
; and similarly,
2 sin A+ B cos^i º tan A+ B 2 cos A+ B cos “tº
sim A + sin B sº 2 *—— 2 s cos B + cos A 2 2
(41.) sin A — sin B T A -- B ... A – B T A — B' cos B — cos A T . A + B . A — B
2 cos Sl]] tan 2 sin SIIl
2 2 2 2 2
A + B A — B
:= cot . cot &
2
sin A sin B sin A cos B + cos A sin B sin A + B
cos A cos B T cos A. cos B T cos A. cos B'
in A- B . — e ſº-º-º:
SIIl #3 cot A + cot B = ****., cot B – cot A = sin A — B
(42.) tan A + tan B = Similarly, tan A — tan B
T cos A. cos sin A. sin B sin A. sin B'
(43.) To find an expression for the tangent of the sum or difference of two arcs: tan A + B = sin A + B
cos A + B
_ sin A. cos B + cos A. sin B
Tº cos Á cos BTsin. A sin B? which, dividing the numerator and denominator by cos A. cos B, and
sin tan A + tan B
1 — tan A. tan B'
tan A — tan B
1 + tan A. tan B'
-:
A *º-ºº-ººs - memºry mºº
observing that * = tan A, gives tan A + B Similarly, tan A–B =
COS
2 tan A
If B = A, * ------.
A, tan 2 A l — tan? A
&ºm====ºm-m-m-m-ºsmºs- B –– tan C t sº- º ... tan C
(44.) Hence, tan A + B + C = tan A+ i. * * = tan A -- tan B + tan C — tan A. tan B. tan &e
I – tan A+ B. tan C 1 — tan A. tan B – tan A. tan C — tan B. tan C
— fans *==ºmsºmºmºsºm
C = B = A, tan & A–º º If A + B + C = T, tan A + B – C = 0, (10;) hence in that case
" * - º La Il
we have this remarkable equation, tan A + tan B -- tan C = tan A. tan B tan C.
If
T R. I G O N O M ET. R. Y. 677
Trigono-
Inetry.
N-a-w
(45.) These are the most important relations that subsist generally between different arcs. As there are some Sect. II.
which depend upon the numerical expression for the lines belonging to particular arcs, we shall proceed to *::::: of
investigate their values. - rigono •
metrical
(46.) Let B C D, fig. 5, be half a right angle, or AB = 45° = +: therefore the angle C B D = half a right ſº Lines.
- . A 7T 7- ... o. 7ſ . Tr l Fig. 5.
angle = B C D, therefore B D = CD, therefore l = sin” T + cos” T = 2 sin? T” therefore sin T = wº
> COS
T 7 7- 7. --- 7.
— : tan — = 1 = cot -- : Sec - E v. 2 = cosec — .
4 * 4 4 * 4 4
(47.) Let A E = 60° = ; ; then, since the sum of the three angles of the triangle A C E = two right angles
= 180°, the sum of those at A and E = 120°; and as they are equal each = 60° = # ; therefore the triangle
o g " _ " . ... " — l w/3 7- tºº-º-e
is equilateral, and C F = A F. Hence cos 3 T 2 ° sin a = V I — T =-3-3 tall 3 = w/3;
t Tr l ar 2 7- 2
CO't — E — ; Sec — t ; COSec - = —.
3 w/3 º 3 C 3 w/3 -
(48.) Let A G = 36° = 2 x 18°; then the complement of A G = 54° = 3 x 18°; therefore, (24.)
sin 2 × 18 = cos 3 × 18°, or 2. sin 18° cos 18° = 4 cos' 18 – 3 cos 18°, by (38;) or, dividing by cos 18°,
2 sin 18° = 4 cosº 18° – 3. Let sin 18° = a ; therefore 2 r = 1 – 4 a.”, from the solution of which
e — 1 + V 5 * * 1 + v 5 .
equation a or sin 18° = *** = cos 72°; cos 36° = 1 — 2 sin” 18° (33) = tº = sin 54°.
cos A. -- sin A
(49.) From these values, sin 45° + A = sin 45°. cos A –H cos 45°. sin A = ; cos 45°-H. A
w/ 2
cos A – sin A tan 45° + tan A 1 + tan A
; tam T456 TA - ; tan 459 — A,
= cos 45°. cos A – sin 45°. sin A =
M2 1–tan 45°. tan AT I – tan A
1 — tan A e ----- *-*-*ms --m -------------------- 4 tan A
similarly -: Hº: from which tan 45° + A — tan 45° — A = T– tan? A = 2 tan 2 A, (43.) Also
sin 60°-- A – sin 60° — A = 2 cos 60°. sin A = sin A. And
: ºo Tº : -, -ºo Tº <x C ~ : v 3 – 1 .
sin 72°–– A – sin 72° — A = 2 cos 72°. sin A = g—sin A,
: ... Sºo TTA ...;_ EZ-STR O ~ * v 5-H 1 .
sin 36°-H A — sin 36° — A = 2 cos 36 sin A =–3– sin A.
Subtracting the upper from the lower, and transposing
s sºm-mm-a-wºrmvºus
sin 36°-H A -- sin 72°– A = sin A -- sin 36° – A -- sin 72°-i-A.
If we had taken cos 72°–H A + cos 72° – A, &c., we should have found
cos 36°-H A + cos 36°– A = cos A -- cos 72°– A –i-cos 723 – A.
(50). These are the principal formulae of the Arithmetic of Sines. Many of them may be proved geome-
trically, but we have preferred the algebraical investigations, as less cumbrous, and not less satisfactory.
(51.) The values of the trigonometrical lines which have occurred in these theorems, (the numerical calcula-
tion of which we shall treat of hereafter,) for different arcs, have, with their logarithms, been collected in tables.
The sines, tangents, &c. themselves are very seldom used, almost all calculations being now conducted by means
of their logarithms. With regard to these it is necessary to observe, that the sines and cosines of all arcs, and
the tangents of arcs less than 45°, being less than 1, their logarithms are negative ; the use of which would be
extremely inconvenient. To avoid this, the logarithms of the tables are made greater by 10 than the real
logarithms of the numbers; which it is always necessary to keep in mind in using the tables. For instance,
A
(using 1 for the true logarithm, and L for the logarithm of the tables,) since tan A = .
os A’ therefore 1 . tan A
= 1. sin A–l. cos A, therefore L. tan A – 10 = L sin A — 10 — L cos A + 10, or Ltan A = L sin A–L cos A + 10.
The natural sines, &c. are usually given to radius 10000, but upon removing the decimal point four places to
the left they are adapted to radius 1.
(52.) In all expressions involving the length of an arc, deduced from operations by the differential calculus,
or from series in terms of the sines, &c., radius is supposed to be the unit of measure. To obtain the number
of seconds, we must divide the length by 0,000004848137; or add to its logarithm 5,3144251 to find the
sogarithm of the number of seconds.
WOL. I. 4 T
678 T R I G O N O M ET. R. Y.
Trigono-
metry.
Sect. III.
- Subsidiary
SECTION III. Angles,
S->
On the use of Subsidiary Angles.
(53.) THE possession of trigonometrical tables, ready calculated, frequently enables us to shorten very much
numerical calculations which have no relation whatever to Trigonometry. The angles which are used in this
process, being employed simply to expedite a calculation, are called Subsidiary Angles. Their use will be best
elucidated by examples.
(54.) Suppose it is wished to calculate a = v a” — bº, and suppose that the logarithms of a and b have
TāT
º b g
already occurred in our operations. Here a = a v/ 1 — Tº If + were the sine of an angle 6, a would
b -
be a x cos 0. Determine 0 therefore by the condition + = sin 6, or L, sin 6 = log b + 10 — log a (51,)
and having found 6 in the tables, a will be found from the expression log a = log a + L cos 6 - 10.
(Z g
(55.) It is required to calculate the expression a = a cos ?) -- b sin ºff. If we make -- = tan 6, this can
b. sin 6 —-
cos 6 o
b
be put under the form b (tan 0 . cos @ + sin (p) = cos 0 (sin 6 cos @ -- cos 6 sin (b) =
Determine 6 by the equation L tan 0 = log a + 10 — log b, and then log a = log b + L sin 0 + $5–I, cos 6,
or = log b + L sin 9-H p + L sec 0 – 20. - -
(56.) It is required to find the logarithm of a + b, the logarithms of a and b being known. If a and b are
ſº b b
of such a nature that both are in all cases positive, a + b = a ( ++). make 7 = tan” 6, then a
b -
( + #) = a secº 9. In logarithms, 2 L tan 6 = log b + 20–log a , log required = log a + 2 L sec 6–20
If, however, a and b may be sometimes positive and sometimes negative, the following method must be used:
a + b = x^2 + + . = M2 . (a cos 45°-i- b sin 45°). Let * = tan 6, or L tan 6 = log a + 10 -- log b ;
- w/ 2 b- w/>
**-*, sºmºsºmº ~ b g .* e b v. 2
then a + b = v 2 . b. (tan 9. cos 45° + sin 45°) = V 2. cos 6 (sin 6. cos 45°-i- cos 0 sin 45°) = cos 6
sin 0-F 45°, and log a + b = ,1505150-- log b + L sin 6 + 45° – L cos 0.
(57.) In Physical Astronomy the following expression occurs : P = (1 + e). (1 + e"). (1 + e'") . &c., where
_ 1 – VI – e' , ... I — VI – e’.
e' = −, c// = = , &c.
1 – Vºl -- e. 1 + WTJ2
º * 1 – A/1 — e” I — cos 6 6
Let e = sin 6, V 1 — e” = cos 0, w/ dºº-e -- tan’; ;
1 + VIII, T 1 + cos 0
6 - G e 6) & ! I 6/ wº
I + e! = secº Tº Similarly, making e! or tan” # = sin 6', 1 + e' = secº 2 &c. Hence, iog P =
6 6/ e º o
2 (L see; + L sec Tº + &c. — 10 – 10 — &c.) This computation would be almost impracticable in any
other way.
(58.) The roots of the quadratic aº – p r – q = 0, being
2 *mmºn 2
# == Vºtº, o ºſ-e- * +1}, let *= = cot 6:
2 4 lz Vº 4 q " ' ' ' 2./J
**** -s-sm. **-*=== gº 6)
the roots are v q. (cot 0 + cosec 0) = x/ q . sº I = by (37) — vº , tan #and v q, cot Tº "
SIIl n
. t - 4 4 o
The roots of rº . p r + q = 0, being # ( + v/ * # , let # = sin” 6, and the roots are
p 6 6)
+ (l + 6) - * — ge tº-
2 ( tºº COS ) p cos 2 and p Sin? tº
T R. I G O N O M E T R Y. 670
Trigono- (59.) The possible root of the cubic z9 – q a - r = 0 is
Sect. IV.
metry. 3 ſ ==mºmºmºmºmºs H tº:
J r r? q° 3 r v/ r? Q rigono-
\-N- V!; † – #} + VH-V4–; metry.
* 3 sºmeºm-s- *-*-*** 3 Z *-m-m- • 2----------
q V/ 27 r? 27 rº - V 27 rº 27 rº
= V . . \ V V ºr + V TH − 1 + -Tº- – M -Tº-- 1 J’
Q Q Q Q
27 jº
Let # = cosec” 6, the root

– V3 V/Hºrv/Hº = y/4 |W/ t * Vº #}
tº-º-º- 3 sin 6 sin 6 § { cot a + an a j .
§ *-*-*g *==
6 4 S
Let Va. # = tan p, the root = V; (cot p + tan ºp) = v/#! . cosec 2 @. If # be greater than
9.
r &} 4 as 3 a. 3 aº
T' let a be assumed = a cos 6, or z = cos 0; then cos 3 0 = is a by (38,) or tº — Jº —
(I,
a 3 s Gº? a? - ©
4 cos 3 0 = 0; making this coincide with the given equation, -- tº Q, 4 cos 3 6 = r, which determine
a and 6 ; and a cos 0, or a, is then immediately found. The equation will also be satisfied by making a = a
2 4 7t e 3 dº aş *E*ºmºsºmºmºmºmºs a 3
cos 0 + 3 T' or a = a . cos 0 + -ā-, for these give wº – + r equal to + cos 3 0 + 2 T and T cos
—— 3 -
3 0 + 4 T, which by (11) are each equal toº. cos 3 0.
Cºmmºmºmºmºl
SECTION IV.
Plane Trigonometry.
(60.) A TRIANGLE consists of six parts, viz. three sides and three angles; and if any three of these be
given, the triangle is completely defined. The case must be excepted in which the three angles are given;
as then the proportion only of the sides can be found, the absolute magnitudes remaining unknown. To
determine in number the values of three parts from those of three given parts, is the special object of
Plane Trigonometry. * ,
(61.) Suppose the triangle right-angled, let a and b be the sides containing the right angle, c the third side,
A, B, C the angles opposite, (fig. 6.) If the hypothenuse and the angle B be given, describe a circle D E Fig. 6.
to radius 1 ; draw D F and E G perpendicular to B C ; then D F is sin B, B F is cos B, E G is tan B. And
A B : B C : : D B : B F, or c : a : 1 : cos B, therefore a = c cos B. Also A B : A C : : D B : D F, or c : b : :
1 : sin B, therefore b = c sin B. And the angle A = : — B.
(62.) If a and B be given, B C : C A : : B E : E G, or a b :: 1: tan B, therefore b = a tan B. And
b
B C : B A : : B E : B G, or a c :: l ; sec B, therefore c = a sec B. If b and B be given, a = miſs = b
al
b
cot B ; c = −r = b cosec B.
sin B
(63.) If a and c be given, b = vºc aº, cos B = * = sin A. If a and b be given, c = w/aº -- 52,
C
tan B = * = cot A.
Q,
(64.) Now, suppose the triangle to be any whatever, we shall first prove this general proposition: The sides
of a triangle are in the same proportion as the sines of the angles opposite. In fig. 7 and 8 draw B D a Fig. 7, S.
perpendicular from B on A C, or A C produced; then B D = A B sin A, (61,) and B D also = C B : sin
BC A, whether B C A be greater or less than 90°, (23;) therefore A B. sin A = C B. sin B CA, or A B : C B
: : sin B C A : sin B A C.
(65.) Suppose the three sides of a triangle given, to find the angles. In fig. 7, B A* = B Cº -- C A* –
2 A. C. CD, (GeoMETRY, book iv. prop. 16;) in fig. 8, B A* = B Cº -- C A* + 2 A. C. C. D., (GFom ETRY,
book iv. prop. 15.) Now in the former case, by (61,) C D = B C : cos C; in the latter, C D = B C : cos
T – C = – B C cos C, (23;) therefore, generally, A B*= B Cº -- C A2 – 2 A. C. B C , cos C, or cº = a” +
4 T 2
680 T R. I G O N O M ET R Y.
Trigono-
metry.
2 2 — 22 - •
b” – 2 a b : cos C. Hence cos C = *...* This formula is very inconvenient for logarithmic com- sºy
2 (2. - e
putation. - º
(66) 1 + cos C, or 2 cosº # (33) \-y---
a + b + c a + b + c
= *** = c – 4 + 2 + 6-º-Hº- a 2 ſº 2 – C
2 a b &== 2 a b * a b
a + b + c C S ST 6 C c” – a – bº
Let — = S, theref s” — = ~~ *º in? — –
2 relore cos 3 a b Also 1 — cos C, or 2 sin 2 (33) 2 a b
c + a - b. c – a + b S — b . S – & C S – b . S — a tº tº
-: tº 2 . º * — = g g
2 a b 2 a b ; therefore sin 2 a b . Dividing this by
C S — b . S — C C
the last, tan” — = − Multiplying the product of sin” — and cos” -- by 4, since sin C =
2 S . S – c 2 2
S.S — a , S – b. S — c
All these expressions, but more particularly the second,
C C º
2 sin a cos g, (33,) sin” C = 4 a” b”
are very convenient for the application of logarithms. If two or three angles were required, the formula for
C gº g
tan? 2. would probably be most convenient, as the same numbers would be used for the three calculations;
or when one angle is found, the theorem of (64) may be applied. -
(67.) From the last expression we derive the formula for the area of a triangle in terms of the sides. For
C A. B. D b sin C
the area = —g— (GEOMETRY, book iv. prop. 8) = a o sin v v/s . S – a . S – b . S – c.
(68.) Suppose now two sides and the angle they contain (a, b, C) to be given, to find the other angles.
tan A + B
sin A (Z sin A + sin B a + b 2 a + b
By (64, E -i-, •e º----→ = ––F, or (41,) —— = −.
y (64) is = -, therefore ºx-ºn = a +5 or ( '...AFB a — b
2
B C
Now A + B + C = T, therefore A + - # – 3. therefore, by (24,) tan A + B = COt #. and therefore
A — B a — b C º tº ºn tº sº º
tan — - a Tº COt When the logarithms of a and b are known, the operation is facilitated thus.
(3 a — b tan 6 – 1 tan 9 — tan 45° *---------------º-º-ma- — B
- - 6 — = —— “º: -- — AiR,9 2. YT --
Let b tan 6, therefore a + b tan 6 + 1 - 1 + tan 6 . tan 45° tan 6 45°, and tan 2
*-m-m-m-mº-º-º-º: - b tº-º sº
= tan 6 – 45°. cot C. Or thus, if b be less than a, let – = cos @, then a – b - _1 - cos P. = tan” ºb.
2 (Z a + 1 + cos @ 2
C A + B A —
A —
(35,) and tan = tan” # . cot Tº 2 and 2
and their difference that of B. The third side may be found by the proportion of (64.)
(69.) Sometimes, however, it is desirable to find the third side without finding the two remaining angles.
In this case, by (65,) cº = a” + bº — 2 a b cos C = a + 2 a b + b” – 2 a b (1 + cos C) = a + 5°
4 a b C 4 a b , C g *sºmºmºmºmºs
{ cos . . Let I5), cos” + = sin” 0; then c = a + b ... cos 0. Or cº = a” – 2 a b
being known, their sum gives the value of A,
T (a + b)* (a + 2
- 4 a b C b
-- bº + 2 a b (1 – cos C) = a – lº. {1 + *; in #}. letº; in . = tan” 0, then c =
* Q, “-
ammºmeºmºmºsº C , , C C C
a – b sec 0. Or, since cos C = cos” - – sin” 27, by (33,) and 1 = cos” 2 + sinº, cº - a - b]* . cos #
2
—-N C b \, 2 C b C
+ a + ... in =zF coe (1-(#; , tan? %) let:+, an i = an 0, then c = a-
tº Kºmº- J Q. --
... COS + sec 9. All these are easily calculated by logarithms.
(70.) If two sides and an angle opposite one of them (a, b, A) be given, the angle B is found by the pro-
portion a b . . sin A : sin B, (64;) then C = r – A – B, and a c : , sin A : sin C. * >
(71.) If c, A, B be given, C = r – A – B ; and the three angles and one side being known, the other sides
are easily found by (64.)
T R. I G O N O M ETR Y. 681.
f - º - - ſº (, ſe sin B & e si C S t. V.
º: (72.) If A, B, a be given, C = ºr – A – B, and b = −X-, c = - * These forms comprehend sºil
- º SIIl SII] . - Geometry.
all the cases of Plane Trigonometry.
(73.) In using these formulae we must, however, observe, that we shall in certain cases arrive at results, the \
meaning of which is apparently doubtful. These are called the ambiguous cases. We proceed to distinguish
those in which the ambiguity is apparent, from those in which it is real.
(74.) First, then, we may observe, that the lengths of lines determined by the formulae above, since they are
the results of simple multiplication and division, and are not given by the solution of quadratic equations, are
perfectly free from ambiguity.
(75.) In the next place, an angle when determined by the value of its cosine, versed sine, tangent, cotangent,
or secant, is not ambiguous. For the values of the tangent and cotangent, which correspond to the arc A,
correspond also to the arc ºr + A (10) and to no smaller arc ; the values of the cosine, versed sine, and secant,
belong to the arc 2 T – A, and to no smaller arc, by (9) and (10 ;) and these being greater than T, or 180°,
cannot be used in calculations of triangles. -
(76.) But if an angle be determined by the value of its sine or cosecant, since these by (23) belong equally
to the are A and T – A, both of which, when the sine is positive, are less than 7, the value of the arc is appa-
rently doubtful. We will examine every case in which these expressions are found.
(77.) In right-angled triangles the angles must be less than +, and there is therefore no ambiguity. When
*
- C C
the angle C in (66) is found by the expression for sin T2 . since C must be less than 7, Tº must be less than
+. and there is no ambiguity. If found by the expression for sin C, it must be observed that C is greater or
less than # , according as cº is greater or less than a” + bº. In the case of two sides and an angle opposite
one being given (70,) if a be greater than b, there is no ambiguity; for in the triangle A C B (fig. 9) the angle Fig. 9.
B must be less than A, and must therefore be less than +, (as if A be greater than #, sin B being less than
sin A, of the arcs corresponding to it one is less than # , the other greater than A.) But if a be less than
b, the angle A being less than +. (fig. 10,) there is nothing to determine whether B is greater or less than Fig. 49
+ ; that is, whether the triangle A C B or A C B' is to be taken. In this case, then, and in this alone, there is
&l
real ambiguity.
SECTION V.
Spherical Geometry.
IN our Paper on GEOMETRY, book ix. a comprehensive Treatise of Spherical Geometry has been given. As it is
necessary, however, for our present purpose, to state some of the propositions with slight alterations and
additions, and as a small number only are wanted here, we have thought it best, at the risk of some repetition,
to premise all the Geometrical propositions that may be necessary.
(78.) A sphere is a solid bounded by a surface of which every point is equally distant from a point within
it, called the centre. A straight line drawn from the centre to the surface, is called a radius; if produced both
ways to meet the surface, it is a diameter.
(79.) Every section of a sphere by a plane is a circle. Let AB (fig. 11) be any section of a sphere made by Fig. 11
a plane; from the centre O draw OC perpendicular to this plane; take D, K, any points in the section, and join
CD, O D, C K, O K. Since O C is perpendicular to the plane, it is perpendicular to every line which meets it in
the plane; therefore O C D, O C K are right angles, and C D = WOD" - O Cº., C K = MO K*-O Cº. But
O K = O D, therefore C K = C D, or the section is a circle of which C is the centre.
(80.) A great circle is one whose plane passes through the centre of the sphere; a small circle is one whose
plane does not pass through the centre. Hence a radius of a great circle is a radius of the sphere. Two circles
are said to be parallel when their planes are parallel.
(81.) A great circle may be drawn through any two points on the surface of a sphere, but not generally
through more than two. For the plane of a great circle must also pass through the centre of the sphere ; and a
plane may be made to pass through any three given points, but not generally more than three, (GEOMETRY,
book vi. prop. 2, cor. 1.) A small circle may be drawn through any three given points.
(82.) Two great circles bisect each other. For the intersection of their planes, being a straight line passing
through the centre, is a diameter of the sphere, and is therefore a diameter of both circles; and the circles are
the refore bisected.
682 a gº
T R. I G O N O M ET. R. Y.
Trigono-
Fig.
(83.) The inclination of two great circles, is the angle made by their tangents at the point of intersection.
Sect. W.
metry. . Since each of these tangents is perpendicular to the radius in which the planes of the circles intersect, the same jº.
^* angle measures the inclination of the planes of the circles, (GEOMETRY, book vi. def. 4.)
Wig. 11.
(84.) If through the centre of a circle, whether great or small, a straight line be drawn perpendicular to its
plane, the point in which, if produced, it meets the surface of the sphere, is called the pole of that circle. Thus,
in fig. 11, F C E being perpendicular to the plane of A B D, and passing through its centre C, E and F are the
poles of A DB. From the demonstration of (79) it is evident, that this line will always pass through the centre
of the sphere. In a small circle the term pole is more usually applied to that point only, as E, which is nearest
to the circle.
(85.) If a great circle be made to pass through D and E, and another through K and E, and if the chords
D E, K E be drawn; then, since C D is equal to C K, and C E is common, and the angle E C D = E C K, both
being right angles, the chord E D is equal to the chord E K, and the arc E D = are E K. Hence the pole of
a circle is equally distant from every point of that circle; the distances being measured by arcs of great circles.
(86.) If E be the pole of the great circle G H, since the centre of this circle is the same with the centre of
the sphere, E O G is a right angle, and E G is a quadrant. The distance, therefore, of every point of a great
circle from its pole is a quadrant of a great circle. Since E O is perpendicular to G O H, the plane E O G is
perpendicular to the plane G O H, and the angle E G H is therefore, by (83,) a right angle. And the tangent
of G M at G is perpendicular to the tangent of G E ; and it is also perpendicular to GO, therefore it is
perpendicular to the plane E O G, (GEoMETRY, book vi. prop. 4;) so also is the tangent of D B at D, which is
parallel to it, (GEOMETRY, book vi. prop. 7;) therefore the tangent of D B at D is perpendicular to the
tangent of D E.
(87.) The inclination of E G, E H, which is measured by the inclination of the tangents at G and H,
since these tangents are parallel to O G and O H respectively, is also measured by the angle G O H, or the
arc G. H.
(88.) Since a line which is perpendicular to two lines meeting it in a plane is perpendicular to that
plane, if a point E can be found such that its distance, measured by a great circle, from each of two
points G and H not in the same diameter, is a quadrant, that point is the pole of the great circle passing
through G and H.
(89.) If in a plane perpendicular to another plane a line be drawn at right angles to their common inter-
section, it will be perpendicular to the second plane, (GEoMETRY, book vi. prop. 17.) Hence, if G E be
drawn, so that E G H is a right angle, and G E be made = a quadrant, E will be the pole of the circle
L G M.
(90.) If D K be a small circle parallel to G H, the line O C is perpendicular to both their planes, and
therefore, by (84,) E is the pole of both. And the angle D C K is equal to the angle G O H. Hence D K, the
part of the small circle AB intercepted between the two great circles E D G, EK H, passing through their common
pole : H G, the part of the great circle LM intercepted in the same manner :: O G : C D :: radius : sin E. D.
If the radius of the sphere = 1, this ratio becomes 1 : sin E D.
(91.) A spherical triangle is a portion of the surface of a sphere contained by three arcs of great circles.
(92.) Any two sides of a spherical triangle taken together are greater than the third. For the arcs A B, BC,
C A, fig. 12, being arcs of circles whose radii are equal, are measures of the angles A. OB, B O C, COA, at the
centre; and when a solid angle is formed by three plane angles, any two of these taken together are greater than
the third, (GEOMETRY, book vi. prop. 19;) hence, any two of the sides A B, BC, CA, taken together, are
greater than the third. .
(93.) The sum of the three angles A O B, B O C, CO A, is less than four right angles, (GeoMETRY, book vi.
rop. 20;) and, consequently, the sum of the sides A B, B C, CA, is less than a whole circumference, or 2 m.
(94.) The surface of the sphere included between E G F, E H F, fig. 11, is proportional to the angle H E G.
For if the angle H E G be repeated any number of times, it is quite evident that the area will be repeated as
often, and therefore the whole area will be proportional to the number of the repetitions, or to the whole angle.
Hence the area E H F G E is to the whole surface as H E G is to four right angles, or 2 r. Now the surface of
a sphere whose radius is r is 4 T. rº; hence the surface E H F G E = 2 rº x H E G.
(95.) Produce all the sides of the spherical triangle A B C, fig. 12, so as to form complete circles; let D, E, F,
be the points of their intersections. Now, (82) the arc Ald = semicircle = CF, therefore A C = D F.
Similarly, AB = D E, B C = E F. And the angle at A = the angle at D, since (83) each of these is the same
as the inclination of the planes A B D, A C D ; similarly, the angles at B and C are equal to those at E and F
respectively. Hence the triangle A B C is in every respect similar and equal to DEF, and therefore encloses
an equal surface. Similarly, A FE = B D C, B F D = A E C. Let the area of A B C or D E F = x : that of
B D C or A FE + P; that of A E C or B F D = Q ; that of A FB = R. Then, by (94,) since aſ and P
together make up the space included by A B D, A CD, we have
a + P = 2 r° × A.
Similarly, a + Q = 2 r" x B,
- a + R = 2 rº X C.
(A being taken to represent the arc corresponding to the angle at A, to radius 1.)
Adding them, 2 r + r + P + Q -- R = 2 rº x (A + B + C). But a + P + Q -- R = E D F + A FE +
B D F + B F A = area defined by B D E A = surface of hemisphere = 2 ºr rº, therefore 2 z + 2 r rº = 2 rº
(A + B + C), therefore a = r^ (A + B + C – ºr). If r = 1, a = A + B + C – T. The area of a spherical
eometry.
T R I G O N O M ET R Y. 683
Trigono- triangle, therefore, is proportional to the excess of the sum of its angles above two right angles. This is usually Sect. VI.
metry. , called the spherical excess. Spherical
-N-' (96.) Suppose great circles E.F, FD, DE, fig. 13, to be described, of which A, B, C are respectively the poles; Tº:
they will intersect in points D, E, F, and form a spherical triangle, called the polar or supplemental triangle. J–
Now, since A is the pole of E F, the are joining A and Fis a quadrant, by (863) since B is the pole of D F, the Fig. 3.
arc joining B and F is also a quadrant; hence Fis the pole of A B, (88.) Similarly, D and E are the poles of
BC, AC, and therefore the triangle A B C is the polar triangle to D E F.
(97.) Produce the sides of A B C, if necessary, to meet the sides of the polar triangle. Now, D being the pole
of K B C, D K = quadrant; similarly, E H = quadrant, therefore DE = DK + E H – H K = semicircle —
H. K. But as C H and C K are each = a quadrant, H K is the measure of the angle at C, by (87;) hence the
sides of the polar triangle are supplements of the angles of the original triangle. Similarly, since the relation
between the triangles is reciprocal, the angles of the polar triangle are supplements of the sides of the original
triangle. -
§ The sum of the sides of the polar triangle and the angles of the original triangle = 3 T. Now, the
sides of the polar triangle must have some magnitude, and their sum (93) is less than 2 ºr ; hence the sum of the
angles of the original triangle must be less than 3 T, and greater than T.
(99.) A right-angled spherical triangle is a spherical triangle having at least one of its angles a right angle.
(100.) If we describe the polar triangle corresponding to a right-angled triangle, one at least of its sides will
- #. (97.) This is called a quadrantal triangle.
(101.) Let A B C be a triangle right-angled at C, fig. 14; produce the sides A B, CB, to D and E, making Fig. 14.
A D = C E = #: join FD, and produce it to meet A C produced in F; E B D is called the complemental
triangle. Since E C =+. and A C E is a right angle, E is the pole of A C, and F A = E F =#, by (89)
and (86.) And because A E = A D = #. A is the pole of ED, and A F = --. Since AIF and A D each
- #. DF measures the angle A, (87.) But E D is the complement of D F, therefore ED is the complement
of A. Similarly, the angle E is the complement of A. C. And the side B D is evidently the complement of
the hypothenuse A B. The angle A D E being a right angle, the complemental triangle is also a right-angled
triangle.
SECTION VI.
Spherical Trigonometry.
(102.) The sines of the sides of a spherical triangle are proportional to the sines of the opposite angles. Let
A B C, fig. 15, be any spherical triangle : from C draw C D perpendicular on the plane A O B, meeting it in D : Fig. 15
and from D draw in that plane D E, D F perpendicular to A O, B O, and join CE, C F. D. O. Now, CE* =
C D* + D E2 = CO2 – O Dº + D E° (since C D being perpendicular to the plane A O B is perpendicular to
DE, D O) = CO2 – O E°; therefore the angle C E O is a right angle, and the angle C E D (83) = A, and
C E is the sine of A. C. Hence C D = CE . sin C E D = sin A C. sin A. Similarly, C D = sin C B. sin B.
Hence sin C A . sin A = sin C B . sin B, or sin C A : sin C B : : sin B : sin A. -
(103.) To find the cosine of one angle of a spherical triangle when the three sides are given. Let A B C,
fig. 16, be the triangle; draw CD, CE, tangents to CA, C B, and O D, O E secants; join D E. Then (83) Fig. 16.
the angle made by DC, EC, is the angle C ; also, the angle D O E is measured by A. B. Now, D E2 = D C2
+ C E” – 2 D.C. C E cos D C E, (65,) and D EP = D O2 + O.E. – 2 D O. O.E. cos D. O.E. Comparing
these values, and substituting for DC, &c., tan” A C + tanº B C – 2 tan A C . tan B C . cos C – secº A C +
secº B C – 2 sec A C , sec B C. cos A. B. But secº A C = 1 + tanº A C, secº B C = 1 + tanº B C ; subtracting
from both sides tan” A C -- tan” B C, - 2 tan A C tan B C cos C = 2 – 2 sec A C . sec B C cos A C ; or
2 sin A C . sin B C . cos C 2 cos AB cos A B — cos A. C. cos BC
tºº cos A C. cos BC - - ... KG. Ho ; from which cos C = sin A C . sin B C . It
is convenient to denote the sides opposite to the angles A, B, C, by the letters a, b, c ; then cos C =
cos c – cos a . cos b
sin a . sin b
(104.) This is the fundamental formula of Spherical Trigonometry: the theorem of (102) may be deduced
from it, but as the process is rather long, and as the geometrical proof is very simple, we have preferred esta-
blishing it on an independent demonstration. We shall now proceed to investigate the formulae best adapted for
the logarithmic computation of spherical triangles; the general problem being, as in Plane Trigonometry, from
any three given parts (sides or angles) to find the other three. And we shall begin with right-angled triangles.
684
T R I G O N O M E T R Y.
Trigono-
metry.
S-v- C =
(105.) Let A B C, fig. 14, be the triangle, having the angle at C a right angle. By the formulae of (103,) cos
cos c – cos a . cos b
sin a . sin b
; but C = 90°, cos C = 0, therefore cos c = cos a . cos b.
(106.) Hence in the complemental triangle E B D, which is right-angled, cos d = cos b”. cos e : by the relation
given in (101) this is immediately transformed into sin a = sin c. sin A; similarly, sin b = sin c. sin B. This
might have been proved by (102.)
(107.) Since sin b/- sin d . sin B, we have cos A = cos a . sin B. And cos B = cos b. sin A. Multiplying
these equations, cos B. cos A = cos b. cos a . sin B. sin A, or cot A. cot B = cos c.
(108.) Hence cot E. cot B = cos d, or tan b. cot B = sin a.
Similarly, tan a . cot A = sin b.
(109.) From this, tan e. cot E = sin b’, or cot c. tan b = cos A; and cot c. tan a = cos B.
(I 10.) These equations comprehend every case of right-angled 'spherical triangles; that is, if any two parts
besides the right angle be given, any one of the remaining parts can be found by a short logarithmic calculation
In the opinion of Delambre (and no one was better qualified by experience to give an opinion) these theorems
are best recollected by the practical calculator in their unconnected form. For common purposes, however, a
technical memory has been invented, under the title of Naper's rules for Circular Parts, which we shall now
describe.
(Ill.) The five circular parts are the two sides, the complement of the hypothenuse, and the complements of
the angles.
the two remaining ones the opposite parts.
Any one of these is called a middle part ; the two next it are then called the adjacent parts, and
The two rules are then as follows: the sine of the middle part =
product of tangents of adjacent parts; and the sine of the middle part = product of cosines of opposite parts.
(112.) These rules are proved to be true only by showing that they comprehend all the equations which we
have just found. We shall leave to the reader the labour of examining every case.
(113.) It was observed in (100) that the polar triangle, corresponding to a right-angled triangle, is a quadrantal
Naper's rules then may be applied to quadrantal triangles, if we take for the circular parts the com-
triangle.
plements of the sides, the complement of the angle opposite the quadrant, and the two angles.
But as there is
some difficulty in the determination of the signs, it will, perhaps, be found more convenient to make use of the
general formulae of (102) and (103,) which for this case are always much simplified.
(114.) We shall now examine whether any of these solutions are ambiguous.
before, we shall attend only to those whose values are given by the values of their sines.
And for this purpose, as
Now it is easily seen,
that if A and a be given, B, b, and C are all given by their sines; and this case therefore is ambiguous, there
being nothing which will enable us to determine whether the smallest corresponding arcs, or their supplements,
In fact, the triangles A B C and A/B C, fig. 17, will equally satisfy the given conditions, since
the angle at A' = that at A. -
Fig. 17.
(115.) If A and c be given, a is given by its sine.
are to be taken.
Since, however, tan a = sin b . tan A, and the tangent
becomes negative when the arc is greater than 90°, and since sin b is always positive, (as b must be less than
180°,) a must be greater or less than 90°, as A is greater or less than 90°, which removes the apparent ambiguity.
If a and c be given to find A, the same remark applies.
(116.) We proceed to find formulae of solution for all spherical triangles.
angles. We have seen (103) that cos C =
calculation. But 1 + cos C, or 2 cos” --- =
2 sin
a + b + c
2 g
a + b – c
2
sin
or 2 sin”
º
2
to . C
them together, since sin C = 2 sin —a cos
sin a . sin b
cos c – cos a . cos b
sin a . sin b
cos c – (cos a cos b — sin a . sin b)
Given the three sides to find the
This formula is not adapted to logarithmic
cos c — cos a -i- b
(*s
a + b + c
—g
, or putting S =
sin a . sin b
C
, cos" —- =
2
C (cos a , cos b -- sin a . sin b) — cos c cos a — b – cos c
sin a
º C sin S — a . sin S — b
SIII" — =
sin a . sin b
. sin b
Dividing this by the former, tan”
sin a . sin b
2
C
sin a . sin b
sin S . sin S – C
. Again, 1 — cos C,
a + c – b
n —;
sin a . sin b
b -- C
•- ſº,
2 sin
**
sin a . sin b
sin S — a , sin S -- b
sin S , sin S — c
. Multiplying
4 . sin S. sin S — a . sin S – b . sin S – c
C .
---, sin” C =
2
these forms logarithms can conveniently be used.
(117.) Given two sides (a, b) and the included angle (C) to find the other parts.
V"... ." . "
sin S. sin S – a
. With all
sin” a . sin” b
$ºmºsºs
From the expressions just
B
* = V/
2 T
A
found, tan Tø
and
lººm:
º
sin S – a . sin S — c
; tan
sin S. sin S – b
*
A
, therefore tan —-
#
Sect. VI. ,
Spherical
Trigono-
metry.
S-N-2
T R L G O N O M E T R Y. 685
Trigono-
metry.
A B - - sin S — b sin S — a
tan — tan – ... --— *ºmºsºmmemº —-- e . --—
an-, + tan H. = V/*#: MH+ sin S – 6 V+=
1 – an *. an” sin S 1 sin S - ? ©mº sin S – a . sin S – 5
- - 2 2 - — ——º - - -
- sin S
2 sin C cost a — b
sin S – b –- sin S – a C . 2 2 cos —; C ... . A — B
= cot T2 . -- b a = — T. cot T2 . Similarly, tan g— =
sin S — sin S – c - 2 cos ? in — co, “it
2 2
t A t B in tº
an — — ta.In —- ... --—r ... --—- S
2 C sin S – b – sin S — a 2 C *
2 A. B = cot ºf x — ––– = T.I., cot;. The sum and difference of A and
I -- tan Tº tan 2 - sin S + sin S – c sin ă
B being thus found, A and B will be determined. The third side will be found by the proportion of (102.)
(118.) It is, however, very frequently desirable to find the third side without finding the angles. Now, cos c
(103) = cos a . cos b + sin a . sin b. cos C, or versin c = 1 — cos c = 1 — cos a cos b — sin a . sin b . cos C
= 1 – (cos a . cos b + sin a . sin b) -- sin a . sin b ... versin C = versin a – b –- sin a . sin b. versin C
= versin a – b . (1+
versin a – b ... secº 6. Or thus, cos C = 2 cos? 2 - 1; therefore cos c = cos a . cos b — sin a . sin b +
sin a . sin b ... versin C
sin a . sin b ... versin C *
) Make = tan? 6 ; then versin c =
C
versin a – b versin a – b
º e C q=> Eº-ºp g C . . C
2 sin a sin b. cost-a-- cosa + b + 2 sin a sin b. cos' g : therefore 1 — cos c, or 2 sin” + = 1 –
—- o e C . .. 6 a —— b e C © G
cosa + b – 2 sin a . sin (b. cos” 2 ” and sin” + = sin? ; — sin a . sin b . cos” a Let sin a . sin b
C o g C . . Cl b (Z b . a b
. cos^ - E sin” 6 ; then sin” —- = sin” + — sin” 6 =, by (39,) sin + + 6. sin + 2 – 6.
2 2 2 Q, 2
º e cos a - cos b . cos c
(119.) The following theorem is frequently useful. We have found cos A = in b ... si ; also cosc
SID 0 , SIII C
= cos a . cos b + sin a . sin b. cos C; substituting this in the numerator, cos A =
cos a — cos” b. cos a — sin b . cos b. sin a cos C cos a . sin b – cos b. sin a cos C
sin b . sin c sin c 2
ſe sin c. sin a L cos a . sin b – cos b , sin a . cos C gº
and sin c = ſº , therefore cos A = sin § e , or cot A. sin C =
sin A sin C : sin A
cot a. sin b – cos C. cos b. This formula is chiefly useful for finding the corresponding small variations of the parts
tº º º cot a © cos C
of a spherical triangle. It may also be used to determine A: thus, cot A = # X ( sin b – COS º).
in C COta.
cos C COſ, a g g cot a . sin b – 6
let “tº - tan 6, then cot A = −r (sim b cos 6 — cos b sin 6) = −. —-e
COt a sin C : cos 6 sin C. cos 6
(120.) Suppose two angles and the included side (A, B, c) given, To find the remaining parts. Take the
polar triangle; let a', b', c', be the sides of which the points A, B, C, are the poles; A', B, C, the opposite angles.
a' — bº
A! -H B' c. *-5 a + b
Then, (97,) c' = r — C, C' = r – c. Then tan = COt T2 T.T. (117,) or — tan 2
COS sº
cos B-A - A — B a'— b'
=tanº 2 tanti-º- tan : cos—g Similarl Aſ — B' tº: sin —;
= an;. AIE , or an–a–s an g : A-H B imilarly, tan 2 E CO 2 . ... a' + b'’ Or
- COS COS SIII
2 2 2
. A – B
— b C sin —;
tan : g—- tan a TATE - The sides being thus found, the third angle may be found by the pro-
sin *
VOL. I. 4 U
Sect. VI.
Spherical
Trigono-
metry.
S-N-
2 .4
686 T R. I G O N O M ETR Y.
Trigono- portion of (102.) If it be wished to have the third angle independently, the formulae of (118) may be adapted in sect. VII.
metry
the same way. * - Y;
- of Triangles,
- tant tº tan A+ B S-N-2
(121.) If we divide one of the equations in (117,) or (120,) by the other, we find − = -RTB.
tan tan
(122.) If two sides be given and an angle adjacent to one, then another angle is found by (102,) and the
third side by (120,) or the third angle by (117.) In this case the solution is ambiguous under the same cir-
cumstances as in the corresponding case of plane triangles. If two angles and an adjacent side, B, C, b, fig. 18, Fig. 18.
be given, the process is the same. In this case, when C is greater than B, either of the triangles C A B, C A B'
(in which B'A produced makes A D = A B) satisfies the given conditions. These are the only ambiguous cases
of oblique-angled spherical triangles. . . -
(123.) If the three angles be given, the formulae of (116) may be applied to the polar triangle, and the sides
of the given triangle may be found. This, however, is a case which never occurs in any applications of
Trigonometry.
SECTION VII.
On small corresponding Variations of the Parts of Triangles.
(124.) It is frequently desirable to ascertain the effect which will be produced on one part of a triangle by
the variation of another, all the rest remaining unvaried. To estimate the probable effect of error in observa-
tion; to reduce observations made in one situation to what they would be in a situation little distant; to take
account of refraction, parallax, &c., this theory is absolutely necessary. We shall, therefore, give the general
method of finding these corresponding variations. •
(125.) In almost all cases expressions may be conveniently found by writing down two equations, one of
which results from giving to the quantities contained in the other the variations which they are supposed to
undergo, and then taking their difference. And this method has the advantage of showing precisely the
magnitude of the error made by any farther simplification. It will be best illustrated by examples.
(126.) The height of a building is found by measuring a horizontal line from its base, and at the extremity
observing the apparent altitude ; and the angle is liable to a small error of observation. In this case, if a be
the measured distance, 6 the angle, a the height, we have a = a . tan 6. And if giving to 6 the variation 8 6
would produce in a the variation 8 r, we have a + 8 w = a, tan 9 + 66. Subtracting the former equation,
sin 8 6
cos 0. cos 0 + 36
ð a = a (tan 6 + 6 6 — tan 6) =, by (42,) a . Now, if we suppose 60 to be very small, we
a 5 6
may put 60 instead of sin 80, and cos 0 instead of cos 0-F 30, without sensible error; then 3 r = cosº. 9
Here 6 6 is supposed to be expressed by the length of the corresponding arc to radius 1. If it = n seconds,
a . m . 0,000004848
cos” 9 very nearly.
then for 60 we must put n x 0,000004848, (4,) and 3 a =
(127.) If it were wished to determine a, so that the error should be a minimum, it must be observed that a
( . g 3 9 2 a 3 9
though determinate is not constant, but = a, , cot 9, whence 3 w = + 3.2 = —
sin 9. cos 9 sin 29
, which is least when
sin 20 is greatest, or 2 0 = #, or 9 =+.
(128.) Suppose in a right-angled spherical triangle, C being the right angle, A is given, To find the variation
of a when c receives a small variation. Here (106) sin a = sin A. sin c; hence sin a + 3 a = sin A.
º = , e - e - - - - © • *mme Q * 2: Te — . - - 3.
sin c + 3 c : taking the difference, sin a + 3 a - sin a = sin A (sin c + 3 c – sin c), or 2 cosa--": ... sin *#
w g C
sºam-sºº's ºw-ºgºm sin A. cos c + = -
= 2 sin A. cos c + C in” and s dºm 2 . 3 c g
$º-sº & ; : sin -à, and sin g = - TY., sin g, or if 3 a and 3 c be very small,
cos a + =
sº
TRI G O N O M ETR Y. * 687
Trigono- sin A. cos c s • - tan & s ºn { } Sect. VII.
: & a = − 8 c = sin A. cos b . g c, or = 3 c. If m be the number of seconds in 8 a., n that in 8 c, vii.
COS (Z tan C ... •
\-/-/ • - of Triangles.
tan & ~ N-y-Z
T tan c
(129.) The consideration of particular cases of this last problem shows that we must be cautious in applying
to any extent the simplifications which were there introduced from considering 3 c as small. Suppose c = 90°:
&
g © C
sin A. cose + ,
it would seem that 3 a = 0. Taking, however, the original expression sin *: === 3. ... sin *
(Z - 2
cos a + Tº
we may observe that, when c= #. cos c + º: * — i., by (23) and (34;) therefore sin * -- – sin A Sin? ; g
coS a + 3 a
2
g - |2 si A.
Making 3 c very small, 34 º- sin A 3 c , or 3 a = . SlT1 3 c) *. Here then m = —
2 2 COS (Z 4 2 cos a -
gin. A x 0,000004848
2 cos a
(130.) Given two sides (a, b) of a spherical triangle, and the included angle (C) to find the variation pro-
m* = — tan A x 0,000002424 × n°, since a now = A.
duced in c by the variation of C. Here cos c = cos a. cos b + sin a . sin b. cos C, (103,) and cos c + 3 c
= cos a . cos b + sin a . sin b. cos C + 3 C. Subtracting the latter, 2 sin c + º: . sin ; = 2 sin a . sin b .
sin C —- #. in *g If 3 C be small, and if C or c be not small, then sin c. 3 c = sin a . sin b. sin C. & C
sin a . sin b. sin C
nearly, or 3 c = sin c × 3 C = sin B. sin a . 8 C. If m and n be the number of seconds in 3 c and
3 C, m = sin B. sin a . m . If C = 0, then 2 sin c + # . Sin º = 2 sin a . in sinº, and supposing SC
ſº ſº *
sin a sin b -- 2
, or 3 c = ++ 3C)', or m =
sin a . sin b x 0,000004848 X
2 sin c
4 2 sin c
(131.) With the same data, to find the variation in A. Here (119) cot A. sin C = cot a . sin b – cos C.
cos b, and cot A+ 3 A. sin C-FSC = cot a. sin b – cos. C-F 3 C. cos b : subtracting the former, cot A-E3A.
sin C+ 3 C – cot A. sin C = cos b. (cos C – cos C-FSC). Now cot A+ 3A. (sin C+ 3 c – sin C)
n?.
l ſe 3 c © g
very small, sin c. T2' = sin a . sin b .
====ms TSG § C mºmºmºmºmºu “ sº-summº g sin . 3 A
= cot A + 3 A . 2 cos C + + . sin -; also sin C (cot A + 3 A — cot A) = - sin C F ;
2 2 ( + sin A : sin A + 3 A
g-º-º-º-º-º-º-º-me wºmºsºmsºmºsºms mºre–sº * *mmamººsm-tº-mºre 3 ... 3
adding these together, cot A + 3 A. sin C + 3 C — cot A. sin C = 2 cot A + 8 A . cos C + *; © sin .
sin 3 A
ºmº- ºf-> —Fa C tº º
** And cos C – cos C + 3 C = 2 sin C + º . Sin *g, substituting in the equa-
sin A. sin A + 3 A */ 2
tion, and supposing 3 C and 3 A very small, cot A. cos C. 3 C – # 3 A = cos b. sin C. 3 C, and
sinº A sin” A
* A = ** (cot A cos C – cos b. sin C). 3 C ; or if p be the number of seconds in 3 A, p = −F
sin C - - sin C
º o gº gº tº g ſº - sinº A
(cot A cos C – cos b. sin C) × m. Putting for cot A its value, this is easily changed into p = - sin C
cot B sin b n = — tº A. cos B
SIIl (), sin C
(132.) The principle and the mode of its application is now sufficiently evident. We must, however, remark
that in many cases the corresponding variations may be easily found by geometrical considerations. Thus, for .
the problem of (130,) let A B C, fig. 19, be the triangle, and by the variation of C let it be changed to A B'C': Fig. 19
- 4 U 2
688 T R. I G O N O M ETR Y.
Trigono- if B a be supposed to be drawn perpendicular to A B', then Ar will ultimately = A B, and therefore a B' = 3 c. Sect. VIII,
metry. . Now 3 c = B B'sin B' B a = B B'sin C B A (since C B B' is a right angle, as C is the pole of B B', (86,) and A.
Y-v" therefore C B B" = a B.A.); but B B" = sin a . B C B' by (90) = sin a .5 C, therefore 3 c = sin a . sin C B A. 3 C Sº"Yº:
= sin B. sin a . 3 C, as in (130.) And if the variation of A were required, we should have 6 A = #.
- Sl
B B' ſº B' B g & º 8 C ſº A e B ©
- cos * — sin a cos B , or = sin a cost & C for the quantity by which A is diminished, as
SIII C Slil C sin C - .
in (131.)
(133.) The geometrical method then can be applied with great ease to those examples in which the variation
of one element is expressed in terms of the first power of the variation of another element, but it can very sel-
dom be applied to those cases in which as in (129) the variation of one depends on the square of the variation
of the other. Another method will hereafter be described, not however preferable in general to the first given
here. -
SECTION VIII.
Investigations requiring a higher Analysis than the preceding.
(134.) THE preceding sections have referred to nothing more difficult than the most common propositions of
Plane Geometry and Algebra, and one or two theorems of Solid Geometry. In this section it is proposed to
comprehend some of those expressions which require for their demonstration some of the higher parts of
analysis, particularly the Differential Calculus, and the Calculus of Finite Differences. -
(135.) To express generally cos n aſ in a series proceeding by powers of cos w. If we observe the manner
in which the expressions of (38) are successively formed, we shall easily see that cos n ar, m being a positive
integer, will always be expressed by this form, 2"-" cos" a + a cos"-" a -i- b cos"-‘a’ + &c.; a, b, &c. being
functions of n. Also there will be no second term till m = 2; no third term till n = 4, &c. Let 2 cos n a = u, ;
2 cos w = p ; then u, H = p. u, - ul-I. Assume then u, - p" + A, . p"** + B, . p" *-i- C, p"T"-- &c., An, B, &c.
being functions of n to be determined; then u, º, -= p^* + Ant, . p"T" + B, E, . p"-" -- C, H. p"** + &c.; u,-,
= p^* + An-1. p"T"-- B, , . p"-3 + C, i. p"~7 + &c. Substituting these expressions in the equation above,
and equating the coefficients of similar powers, we have these equations; Ant, E= A, - 1 ; B, E = B. - An-, ;
C.E. = C, - B, 1, &c.; or since Anil - An = A. A., &c.; A. An = – l ; A. B., E – An- ; A C, - - B, 1, &c.
Integrating the first, we have A, z= — m + C ; and since 2 cos 2 a = 2 cos al” — 2, we must have As = – 2,
therefore C = 0, and An = — n. (It will be remarked, that we have not found the correction by making n = 1,
because the equation. As = A, - 1 is not true, the value of u, or 2 cos 0 being not l but 2. After this, however, the
equation A, H = An-1 always holds; A, therefore is the first quantity to which the general value of A, can be applied.
The other equations B, E = B, - An-1, &c. are true without any exception.) Hence An-1 = — n – 1, therefore
sms- wº-vº-mº — 2 . m — 3 e
A B, E m – l = n – 2 + 1 ; integrating, B. = * ;" + n + C'; making this = 0 when n = 3,
n – 2 . n – 3 , — m . m — 3 — m — 1 . m – 4 — m — 3 . m — 4 * — 4
B, - –––– — 3 = gºmºn -- -: tº ºr *=ºmmºgº
2 + m 2 Hence — B,-, 2 2 - 2 2
- – 3. n – 4 . m — -s it , ſ , - ‘Ā’A. --→ mº
= A. C., ; therefore C, E — 7. * - 4 - ? – 5 m - 4. n *= _ n : n - 4 - ? 3. which needs no
2. 3 2 2. 3 g
correction, as it vanishes when n = 5. Continuing the process, we find D, = * : * † . † # ... ?? -- 7. &c.;
- 1. † = 7. . m — 3 , ºt, -
hence u, - p" — m. p"~" + 2 p” – &c.; or 2 cos n x = (2 cost)" – m (2 cos ryº-" + " * 3
g * = m . m — 4. m. – 5 77 . n – 5 . m – 6. m. – 7
(2 cos ar) "-" — 2 . 3 (2 cos a)"-6 + 2. 3.4 (2 cos a)"-" — &c. Of this important
theorem we believe this is the simplest demonstration that has yet been given.
. m. – 1 . m — 2 . . . .
(136.) If n be even and = 2 m, the last term will be + 2 m . ºn 2. 3 777, 770, * = 2; the last but one
2 m . m . m – l . . . . 4.3 sº-º-º: wº
= + *-*- (2 cosz) = + me. (2 cost); the last but two = +***t it." 1. . 6.5
2. 3. . . . m – 1 * 2. 3. . . . . m – 2
2 mº . m? — I - 2 º * —
(2 cos a)* = + T2 ... a T (2 cos r)", &c. Hence cos n w = + { * H. cos” a + #.
n”. nº -- 4. nº – 16 ..., º
cost ac – I. a. 3. T. s. 6 °w + &c.) the upper sign to be taken when m is even or n divisible by 4
T R I G O N O M ET R Y. 689
ionna- - , 27. F.T. m.m.- i.... —— Sact. X.
Tº: If n be odd or of the form 2 m + 1, the last term will be +** + m . m – 1 #2 cost = + 2 m + 1. Higher
2. 3. . . . m. Analysis.
* : 1. I . m . m – l . . . . 4 '-º'-
2 cos a = + n . 2 cos a ; the last but one = + 2 m + 1 . 7m + * : * ~ * (2 cos ry”
2. 3. . . . . m — I
* —
=# ***** ... ???, (2 cos w}* = + *# 2 cosº r, &c.; hence when m is odd cos n a
. 718 – 1 - . m” — 1 . ºn” – 9 - g g
=~ +} n cos º – *H: cos r +* #. cos" aſ — &c. }. the upper sign being taken when n
*
is of the form 4 s -- 1, and the lower when of the form 4 s + 3.
7- © 7, 7t 7, 7- ... m. Tr
(137.) Let r = g-y; then cos a = sin y, and cos n a = cos -ā-- ny)= cos + cos n y + sin-g
sin my. If n be divisible by 4, this = cos n y; if only divisible by 2, it = — cos n y. Hence in all cases,
m” ..., , , n° (nº – 4)
Tº sin", + T-5-1
= sin n y; if of the form 4 s—H 3, cos n a = — sin n y. Hence in all cases, n being odd, sin n y = n sin y
2 -
— l
-****, + se
n being even, cos n y = 1 –
sin” y — &c. If n be of the form 4 s + 1, cos n r
(138.) Differentiating the first equation of the last article we find, n being even, sin n y = cos y {n S]n gy
* (* - 4).sos n (nº – 4) (n° – 16)
Hººsin'y---Hº-5
other formulae.
sin” gy — se} By similar operations we may from these deduce
(139.) Let n y = z ; then (n even) cos 2 r 1 –
1 — 4. . ſ. 1 — 16 sin" y
26 77° 77°
gy" + &
4
sº-c =mme ir 4
2” sin” y + 24 ( #) sin” g
l g 2 g y” l & 2 o 3 tº 4 G yº
- T5. TET6 : c. Suppose now n to be increased without limit; the
4 16 sin y
expressions 1 — Tº 1 — T. &c. approach to 1 as their limit; the fraction also has 1 for its limit.
22 24 26
Hence cos z = 1 - H3 + Iga I – Iga I; a + &c. -
- - sin y 25 ( º +) sin” y 25
140.) Agai dd te = e – -— . sºme-a-m-m---------e-----e.
( ) gain, (n odd) sin 2 2 gy 1. 2 .. 3 gy” * Ig. 3 T 3
&c.
( - )(-)ºw . . . . 28 25
ę gy” 72, * — & 2
c. Increasing m without imit sin z = z — — &m-ºmºnº-ººmsºmºsºme ºmº
ng n without imit sin z = z – Ha + H+.
(141.) Now we may remark, that if we expand e” v- and e- vº- (e being the base of Naperian logarithms
= 2,7182818) in the same way in which we expand e”, we have
- a W – 1 3.2 a" V – 1 3.4
e V* = 1 + +*— — — — — — 4 — tº
1 + = T5 - Tº a # Tai i + &c
6. l Ta-H Tºº + Tää-1 - &c
v-T L 2 -r V-1
* * * V-T -* V-i — tºº q2 £4 ** dº º – “ + e -
Adding them, e V-1 + e =2( I5 it I: ETA &c. ) = 2 cosa, or cos x = —g—. Sub
* «-» * *º-º a;3 a 5 Gººg
tracti , et V-W – e-" V-i = 2 v - sº mºsºm ºmºmº- *-*---------------—- sºme } = v — 1 . si e
racting, e 62 i{. Ia 3 + i. 2.3.I.F &c 2 1 . Sin ar, or sin r
er v-, -e-z V-T
== ==-. These expressions are to be regarded as having no other meaning than this; if expanded
2 vTI
according to the rules by which we expand possible algebraic quantities, they would produce the series for cos a
and sin w.
690 T R I G O N O M E T R Y.
Trigono- (142.) From these equations we have e” V-i = cos r + W — 1. sin a 5 e" V* = cos w — W – 1 . sin r. º
- - er
try. * gººmsºmºrºs sº tº-º-º-º: wº-
metry Similarly, e” V-1 = cos v – T. sin w; multiplying this by e V*, e” V-1 = (cos a + V-1 sin a Analysis.
º 3/ 3/ > plying y
(cos y + V - 1. sin y). But dºw) V-F = cos r + y + W — 1. sin a + y : hence we have this very remark- M
able formula, (cos r + W — 1. sin r). (cos y + v — 1. sin y) = cos a + y + w/Ti. sin a + y. The same
result will be obtained by actually multiplying together the two factors. If we suppose y successively
= r, 2 x, &c., we shall have (cos r + V-1. sin ry" = cos n + + v – 1. sin n r, which implies that n is an
integer. Or thus, e” v- = (e" vº)", that is, cos n a + V - I. sin n w = (cos r + v — 1. sing)", whether n
be whole or fractional. Similarly, cos n a – V – i. sin n w = (cos r — v — 1. sin a ". This theorem is
due to Demoivre. - - -
(143.) Expanding the two last expressions, adding them together, and dividing by 2, we have
7) . 77 – 1 g & &m. © — 2 c 77 – 3 se
cos n ºr E COS" a - — cos""" a . sin” a + m . m – l . m . . cos"4 a. sin” a - &c.
2 2. 3. 4
n . n – 1 77 . m — I ... m. – 2 e ºn — 3 l
= cos" ac & 1 — — tan? a tan” a - &c. º.
| 2 nº r + 2. 3. 4 | ſ
Subtracting and dividing by 2 v — 1,
p-mºmºmº-º
... m. — 1 . m – 2 º
sin n w = n, cost- r. sin a – “tº #;" cos"-8 a. sin” a + &c.
... ?" – l . ºn —
- == corºn tan a – 7? - ??, 2 㺠*tan's + &c.).
Dividing the latter by the former, t
o dº I e e- 2
n tan a – “tº 72. tan” a + &c.
2 . 3
tan n a = - c
n . n – 1 77 . Tº — 1 . n – 2 . m — 3
1. – – tan” as tan‘a’ — &c.
+ 2. 3. 4
(144.) In (142) suppose such a value to be given to n a that sin m a = 0, cos n w = 1 ; then n w = 0, or 2 m,
2 4 2
or 4 T, &c., and a = 0, or *z, Or *z, &c. Hence the following equations are true, Il" = 1; (co- * T +
71, 70, 7?,
r— . 2 7TY” 4 7. — . 4 TN." G = G tº s ºf Q
M – 1 sin **) = 1 ; (or T. + V — 1. sin **) = 1, &c. The quantities within the brackets are there-
fore roots of the equation 2" = 1, or 2" — 1 = 0. Hence we have for simple divisors of that equation, z – 1,
2 ºr 2 ºr 2 7- — . 27 tº e g ge
z – cos − — V – T. sin “, z – cost-" + v — 1. sin º-, &c.; or grouping in pairs the corresponding
72. 72. 72, 71.
2 ºr - 4 ºr g º
factors, the factors of 2" – l are 2 – 1, 22 — 2 z. cos + + 1, 2 – 2 s. cos º--1, &c., to be continued till
the number of dimensions = n. If n be even, the last factor will be z + 1. In a similar way, we have for
- 3
the factors of 2" + 1, 28 – 22 cos ºr + 1, 22 — 22 COS + -j- 1, &c. to n dimensions. If n be odd, the
- ** - 7?
last factor will be z + 1. -
2
(145.) If we put . for 2, we have w" — a” = (w – a . (*-2 macos; -- a) e (* — 2 w a
4 g G ſº
cost" -- a) &c. to m dimensions, the last factor being w – a if n be even. And w” + a” =
70,
(e. – 2 wa cos : + •) o (*- 2 w a cos 37 + •) &c. to n dimensions, the last factor being w -- a if
77. 71,
n be odd. This is called, from the inventor, Cotes's theorem.
(.46.) It is required to express (cos a)" by the cosines of multiples of w. Here (cos a)" = + (ex- + e-" W-Tºn
l &= s = ſe
** + &c. 4 ++!
=== |-ºra.cº.<r
* * -º 2–
e-º-º: V-i + n . e-n-gº V-7 4- ~~} 71.
— l
77 . m – 1
e o l . ..., T . --> - * * - mº ſº wamº flººms mgºs *E=
being àIl integer, := 27- {-> +-e-" A/-1 + n(e"-z" w/-i + e-n-gº VF) + 70, * - 1 (e-e V-T-i- e-n-4” V-1) + se)
T R I G O N O M ETR Y. 69]
Trigono- I - f 77 . m — I Sect. VIII.
metry. = #3 or m r + n cos n — 2. a + cos n – 4. a + &c. &. The coefficients are the same as Higher
\-y-' 2n-1 ( 2 Analysis.
those in the first half of the expansion of (a + b)"; but if n be even, the last term, which does not multiply a S-
cosine, is half of the middle term in the expansion of the binomial.
(147.) As this formula is demonstrated entirely by means of imaginary symbols, we shall endeavour to explain
how it happens that operations conducted by imaginary expressions can give correctly a real result. We know
: $y •y\n *s * 2 4. º,
that (*#) =#. {e" + e-” + n (e"-" + e-"-*) + &c. }, or ( –– #, + =*= + se) ==
1 .. 2 . 3 - 4
H n° yº nº gº m – 2)” . y? | n – 24 . yº | } ſº
tºº * s=sºmº tºmsºmºsºme º ſº o ſº -
2n-1 {1 + 4*. H Tâi-H &c. 4-m (1 - H+++++++&c.) + &c.;. Now this
s true for all values of y; and, consequently, if both sides were expanded, the expanded expressions would be
identically equal; and therefore there would still be an equality if instead of y” we put – r" and operated upon
ºn 2 a.2
l .. 2
tº - a 2 a;4 º, l {
bra. g e — — — — — — &c. = — - 1 –
it by the rules of common algebra. This would give us ( I .. 2 + I . 2. 3.4 se) 2n-1
714 tº & m 212 a.2 n T214 r. } s" — 4 {
+++ -at ºn ( -º-Hº-se)-se , or (cosa) = g- cos n a + n
* . m – 1
cos n – 2 z + —;
COS 71, –4 a + &c.).
(148.) In this formula for a put # – gy; then (sin y)* =
1
2n-1
7, Tr 7?
cos-a- - ny + m. cos
Tr n . n – 1 n — 4 tr- }
tºmº tº- - —- . sº-sº- = * tº- &c. g
2 m – 2 y + 2 COS 2 m — 4 y + &c
Let n = 4p; then cos". – my = cos 2 p r – n y = cos 2p ºr . cos n y + sin 2 p it. sin n y = cos n y;
G
COS *** – n = 3 y = cos 2 p – 1 . . . cosm – 2 y + sín 27–1 r. sin n-2 y = — cosm – 2 y, &c.;
w l *=º 7. . m — 1 ººmymºmºmºmº
therefore in this case (sin y)" = 2n-1 {so n y – m . cos n – 2 y -- —g- cos n – 4 y – &c.).
I,et n = 4 p + 2; then in the same manner it is found, that
(sin y)" =
l mºm n . n = i l
#}-cony 4 m. coin-29– 2 cos n – 4 y + &c.;
Let n = 4 p + I; cos". – my = cos 37-F# * : cosny--sin 373 s r. sin my = sin my;
— 2 *g ºmºgmºmºmºmºsºm *mºmºsºmºmºmºs *===º ſºmeºmºmºmºmºmº
cost 2 t –7 - 2 y = cos 27-3 r. cos n = 2, y + sín 27 - # 1 . sin n – 2 y = — sin m – 2 y, &c.;
and therefore in this case
{inny-nºmºv tº sin m — 4 y – se).
(sin gy)" -: 2n-1
Let n = 4 p + 3; then in the same way
l - cº-ººm' a ſº, - *== -
(sin y)" = #{-inny--n, ini — 2. 3) – ***. sin m — Hy-se).
n . n - i. . . . . . #-Fi l
(149.) When n is even, the last term in the expression for (cos a)" and for (sin y)" is Tº 27
- 1. 2 . . . . . . . . . . tº-mºm,
2
_ 1 1. 2. 3. . . . n – 1 . m . . l. 2. 3. . . . n - J - n_1 - 3. 5 . . . . n - 1 __--~
= 37 (i. #) - (3.4 ........ m)” T 2 . 4.6 . . . . . . 70,
(150.) One of the principal uses of these expressions is the simplification of integrals taken between two
values of a or y that differ by a circumference. Since f. cos p r or ſ cos p a . da, as well as ſ... sin p r,
692 - g T R I G O N O M ET. R. Y.
Trigono- (p being an integer,) always vanishes between two such values, it appears that through a whole circumference Sect. VIII.
º J. (cos ar)” or ſ. (sin & )" is = 0 when n is odd, and = 2 tr.
Higher
1 - 3. 5 . . . . m – 1 Analysis.
2 . 4. 6 . . . . . 71,
(151.) Since Ø (cos a) can generally be expanded in integral powers of cos a, it can generally be expanded
in cosines of multiples of w. This in most cases can be effected with the greatest ease by particular
artifices, and especially by the use of the imaginary expression for cos a, &c., as we proceed to show by
examples. - -
(152.) Suppose tan p = n tan 0; it is required to find a series for ºf in terms of 6. If in tan ºff and tan 6 we
e? V-1 — e-ºv-T e'vº — e-e V-1 e” V-T— I
mºm-mºme
when n is even.
put the values for sin Ø, cosº, &c., found in (141) we have ºf IE -ºvº. = 77. , ºv=ill ºf Or e” v-i – 1
ºm tº- 7? -- I e-8 V-1
esº V-1 — I sº m + 1 77 – 1 -
= n. --—, whence e^* v- = es' V*. H . Let; T = k, and take the logarithms of
e” v-1 + 1 - l _ !! ~ e2° V- 71 -H
n + 1
both sides; then 2 ºf V-1 =
20 MTI + log (1 – k. e-” vº) – log (1 – k. e” vº) =
ſºmºmºmºmº 2 3 sºme
20 & I+ (ex---~~~) + (ex-i-cº) + (ex- - -ºvº. 1 &
* • k2 . k3 . tº
Dividing by 2 v – i, q = 6 + k sin 26 + 2 sin 46 -- 3. sin 6 6 + &c.; a theorem of great utility. The
truth of the process is to be proved as in (147.) In the same manner we might find a series if
n sin 6
—, m being less than 1.
1 — m cos 6 eing
(153.) To expand (a” – 2 a b cos 6 + bº)" in a series proceeding by cosines of multiples of 6, b being less
than a. Since 2 cos 0 = e V* + e-'vº, this expression = {(a – b . e V-5). (a – b . e-" wº) } •
= @9" . 1 – " evº ". 1 - " - FY
(l, (l,
tan j =
rººmsº
b? m . m — 1 . m – 2 bº
Now || || – * ex- "= 1 - nº ev-, -- * : * ~ * º e” V-i – . . . e” V-1 + &c.
(M, (Z | 2 a? 2 . 3 a?
b 9 J-i º b 8 V-T n . n – 1 b? 39 V-1 m . m – 1 . m – 2 b3 -89 V-1
— — GT" VT * : * 1 – ?) -- 6°" VT —- . — eTºº Y " " — , - . &c.
( 2 * ) 1 – nº e-º- ++, 2. “ 2 .. 3 is . * + &c
The product of these (observing that e” V-i + e-s’ V-7 – 2 cos 26, &c.) =
b% m . m – 1 N2 bº m . m — 1 . m – 2\? b%
2 º ſº-º-E-
1 + m, * +( 2 ). +( 2 - 3 ). ++ &c.
a;
b m . m — I be . m . m — 1 m . m – 1 . m — 2 b"
& ºs fººms ... --— . – — . ... -- g 6)
n = +n. =: 0.8 2 2. 3 a 5 +&c.)2 co,
© — l b2 g ºmmºng o – 2 b4
(+'. ; +n.” 72. sº 71, e ++se)*
n . m — 1 . n – 2 b8
*º- - -— tº - 6
2. 3 aß +&c.)2 cois
+ &c.
Multiplying this by a "we have the series required.
(154.) To find log (1 – n cos 0), m being less than 1, in a similar series. Let 1 – n cos 6 =
(a – b. e. V-1) (a – b . e-" F) = a + 52– 2abcos 0, therefore a-i-b= VT-Fºn, a-b = v 1 - m. The log
b b gº *= 2 -
= 2 log a -i- log (-; eº)+log(1-#~~)= alosa – . b e” V-i – &c.
6 V - 1
62
* =====
2 a.2
2
– "...--F – ". e-º" V-i – &c.
(l, 2 as
T R. I G O N O M ET. R. Y. 693
Trigono-
metry.
Q
2 b ... 2 b *m- *===mmº- *=
2 los a -º coso- + cos 20 — so. And a = 3 (vi-F-H (IFI), a = { q + TFR), =
M1-ET, - v1. Tº 72. 1 + VI – nº
–7–---— = — ; whence log (1 – m cos 0) = log. ——— - 4.—“—
VT + n + VI – n 1 + v I-72 g ( ) g 2 * IEC/H.
2 3
cº-i(--72, ) cº-i(H+= ) co-so-se
*1 + v 1 — nº 1 + VI – nº
(155.) To express sin a by a continued product. We have seen in (145) that a* — a” = + - a .
2 - 2 s = + 2 . a 2 – 2 2 T 2 — 1 : dividi b *º- 2n -
Jº a w cos . Qº . Ç ar. cos-i: -j- a” . . . . (m terms) . . . . a + a ; dividing by a - a, acº"-1
2n-2 7t 2 7r 2 2 27- - -
+ æ a + &c. -- a”-* = a – 2 a w cos -- + a”. a” – 2 a w cos - + a”. . . . (n − 1 terms).... rºa;
72 7.
this is true if a differ from a, however small the difference may be. By making that difference very small, and
making a = 1, we have this equation for the limit of that above; 2 n = 2. 1 — cos ". . 2. 1 — cos 27. G
n 70,
a mammºmºsºm-ºsmºs . .. 2 . .. 3 *-
(n − 1 terms) . . . . 2 = 2*-* . sin? # e sin #. . Sin? : . . . . (n − 1 terms). Again, let w = 1++,
= 1 — —: t 2 2 ~~ — " ". * * = ** || – U.
(Z 2 m." hen tº + a - 2 + 2 ( #) 2 a. a. 2 – 2 (#). and the first equation becomes
2 N2n 2 \?” 2 2. 7- 2: N2 7" 2 ºr
1 + -— ) – ſ 1 — — ) = − . 2 | 1 — cos — — ) . I - E. s. Eºmº emºnes
( +- ;) ( 2 m 2 m. ( cos r + (; + cos #) 2 (l cos :
wamºmºmºmº-º-ºmrºw 1 + cos 2.
2 N2 2 *º-
+(...) 4-wºº).... GE termo.... x 2. Or, since 1 – cos− = 2 sins ºf , and ** =
7? 71. 77. 2 m. 7
1 — cos —
º 2 m.
7r 2 \2n 2 N2n 7- 2 ºr 3 ºr &ºmº- 2
tan * –, l -º-º-º- - tº ºmºmºm :- an . ... n.2 - . 5.2 — . in 2 - . . . . - -
COU8, # ( ++) ( 2 m 2”. Sin 3. sin a sin g : (n − 1 terms) . . . . 2 m
1+(ºf 'coº- l 2 N2 tº ." ºn ~ I t hich the f tº
2 n. 2 m / + 2n . cou" ; . . . . (m. – erms), which the former equation reduces to
7b,
2 N2n z \2n 2 N2 Tr 2 Nº 27-Y mºm-
1 + - ) — ( 1 – 3- ) = 22 ( 1 + i + ) . cotº - ). ( 1 + ( :- * — *- *
( 2 ;) ( 2 ;) 2-( ( ;) Cotº 2 ;) (l (. ..) cot 2 ;) ... (n − 1 terms). Now suppose
1
1 — —
21, ºmºe 2
m indefinitely great; since ( + #) = 1 +2 n.ſ. + *** ‘. . + &c., or = 1 + 2 + H* 2*
-*-
+ &c., the limit of the first side is 2(4. — +== + se) since + , cot -º- = + . 2 m.
1. 2. 3 l . 2. 3. 4. 5 2 m 2 m tr Tr
tari 2n
- 2 2 -
= ultimately #. the limit of the second side is 22 ( + :) s ( ++.) . &c. indefinitely continued.
2 4. º 2
Dividing both by 2 z, 1 + Tºš -- Tai-Tº-F &c. =: ( +.) ( + #) . &c.; therefore, as in
a” º gº º e - tº e
(147,) 1 – T. 2 3 + T2. 3. 4.5 T &c. =( tº- #) o ( - #) . &c.; and multiplying both sides by ar,
º q2 gº gº tº gº tº
SIIl ſº tº: -( *- #) t (1 - #) ſº ( - #) . &c. ad infinitum,
N
(156.) To express cos a by a continued product. By (145,) w” + a” = { a } – 2 a w cos # + •) º
3
(* - a . to #: + 2). &c. to n terms. Let a = 1, a = l ; then 2 = 2 ( º cos 5: ... 2
VOL. I. 4 x
(f
Sect. VIII.
Higher
Analysis.
N-V-’
694 T R I G O N O M ETR Y.
Trigono- - 3 ºr fi ...? tr . . 3 tr g 2 2 . Sect. VIII
metry. ( – COS #) . . . . (n terms) = 2*. sinº 4 m sin' L. . (n terms). Again, let r = 1 + 2.' " = 1 – 27, Higher
\-v-' - Analysis.
then as before ( 1 + 2. ** l 2 N2n 2*. sin” ºf sin.” Tr (n terms) × (1 + (ºf 2 cot-f
— gººms mºsºms -: º — . * — . . . . [. e e s & *=== • COL" —
2 m 2 m 4 m. 4 m. 2 m 4 m
2 N2 3 tr * ºn g 2 N2n z \ºn
. ( 1 + ( - ) . cot” 7–1 | . . . . (n terms), and the equation just found reduces this to ( 1 + - ) + ( 1 — ºr
2 m 4 m 2 m. 2 m
2 2 3
= 2 ( + (#) Cot? #) ( + (...) COt? #) . . . . (n terms); and taking the limit of each side when n is
2 24
indefinitely increased, 1 + = + -º- + &c. = (l + 4 2* I 4 2% &c., therefore 1 as
y in 3. 1 .. 2 1 . 2. 3.4 s = Trº t 5. . &c., therefore – H +
tº 4 ºr” 4 ºr?
— — & C. tº I — — — — . . &c. = s
1 . 2. 3. 4 &c ( #)(l 9 trº C. E. COS tº
(157.) Taking the differential coefficient with respect to r of the logarithm of the expression for cos w we
8 ºr 8 ºr 1 2 ºr
find tan a =
+ &c. Similarly, from the expression for sin w, cot a = − – —; 2
- tº
Tr” – 4 a.” 97 T42 Tr” — ſº
2 a. -
- Lºz-H &c.
(158.) The following theorems we shall find useful hereafter, a " – 2 w"cosa-i- 1 = (w" — cos a + 'V-I. sin a]
. (r" – cos a — ^^ - T. sin a). If we solve the equation a " — cos a + v — 1. sin a = 0, we have
- - l
a = (cos a + V - I. sin a)"; the different values of which, as will be seen upon applying the theorems of
(142,) and (11,) are cos . -H v - i. sin; COS 2 + a + V-I. sin *::: 2. &c.; and the factors of
-º-º-º-m-m- 2
+ V — 1. sin +",
— . C. : — . a. 2 m + a
ar" — cos a + ^^ - 1. sin a are therefore r - cost + A/- 1. sin —, a - cos
- - 72, 71, 72.
º º gººmsºmºmºsºme º (1. mºmºmºsºm-us ſº £1.
&c. Similarly, the factors of a " — cos a — ^^ - T. sin a are r - cos + – w" — 1 . sin ; : –
*mº 2 Cl, u º tº tº
– V – 1 . sin 2 * + 2. &c. Combining the similar factors, a " — 2 w". cos a + 1 = ( a." — 2 a.
70,
2 tr-H a
7,
- 4
COS : +1) ſº (e-2-colº + a + 1) e (e – 2 a. . cos ++ i) & to n terms.
COS
72. 77.
(ſº,
(159.) Now let a = 1; a`" – 2 a.". cos a + 1 becomes 2 – 2 cos a = 4 sin g; a" — 2 * cos; +1 becomes
, 2 m + a
2–2 cos
* cos 2 m.
Cº. g º º ... o d tº C. &
~ 4 sim” 27 &c., and the equation is changed to this; 4 sin' a rt 4". Sin? a sin
| 71, 72.
;
sin' tº #
2 m.
s . Cl . & . 2 7ſ — a . 4 Q.
. . . . (n terms), or sin -a = 2n-1. sing; sin *; , SIIl # g (n terms). Let -*-
2 7.
mºmºmºsºmºm
e . © 7T . 2
= 6; then sin n 8 = 2*-*. sin B. sin B + +, sin 3 + . . . . . (n terms).
(160.) In the equation wº" – 2 w" . cos a + 1 = (e —- 2 a. . cos + + 1) (e – 2 a... cos *:::: * + .) &c.,
4
** tº + cos tr + a
7t, 72.
the coefficient of a 2"-1 must = – 2 (or 7, + cos + &c. (n terms) ). But this coeffi-
2 4
cient = 0; therefore, putting Y for #, cos y + cos ( + **) + cos (, + #) + &c. (n terms) = 0. If
n be even, this is an identical equation. If n be odd, the terms are all different, and observing that the cosine
of an arc is the same as that of its defect from 2 tr, the equation, supposing ºf less than #, may be put under this
form
T R I G O N O M ETR Y. 695
º --------" amºsºmm mºmsºmº-ºms Sect, VIII,
Trigono- - - 2 4 tr tº
* ~" - - - is ,
2 4
+cos: - ++ cos’. — y + &c.
where each line is to be continued to that value of the are which is next less than r. By transferring to the
second side those terms that are negative, this is easily changed into the following,
2 4 3
cos y + cost; + i + cosº- +&c. cos; +, + cosº. --, + &c.
a T H- -: fººmsmemºmmº are-smºs-
+ cost – y + costſ — 4 &c. + cost – º – cos” – a 4 &c.
70, 72, 70, 72,
in which, Y being supposed less than * each series is to be continued till the angle reaches its greatest value
72.
next below 90°. If n be made = 5, it will easily be seen that the last theorem of (49) is but a particular case
of this.
(161.) In (124) and the following articles we explained a method of finding the corresponding small varia-
tions of parts of triangles. This may sometimes be abridged by the Differential Calculus. For if a. a function
d a dº a (3 c)*
+ 3 c + i ; ; ; ;
of c receive the variation 3 a in consequence of c receiving the variation 3 c, then 3 a =
d a d a d? & 3 c)3
+ &c. If 3 c be very small, then 3 a = do 6 c nearly. If, however, H. = 0, then 3 a = dºg ep.
g tº * te g d a tº d a sin A. cos c
nearly. Thus, in the case of (128,) sin a = sin A. sin c ; cos a . -- = sin A . cos c, or − = −.
d c d c COS (I,
in A. cos c e
therefore 3 & = *...*** nearly. This is 0 when c = # ; taking the second differential coefficient,
d? d a N2 gº dº º ... sin” A . cos” e g
cos a – sin a. **Y = — sin A : sin c, or cosa. ** = ** SII] cost c* — sin A. sin c. Make
d C2 d c d cº cos” a
dº e dź } c)? g
C = +, a = A; cos A. #: = — sin A; #= — tan A; and 3 a = — tan A . ( 2 nearly, as in (129.)
(162.) This example sufficiently illustrates the use of this principle. For the cases in which the first diſ-
ferential coefficient does not vanish, and in which the neglect of the other terms will certainly introduce no error,
it is convenient; but when a particular value makes the first differential coefficient vanish, or when it is neces-
sary to examine the terms after the first, the method of (125) is generally preferable.
(163.) In our solutions of triangles it will be remarked, that we have frequently given several formulae for the
same case. The reason is, that in particular cases the value of an angle cannot at all by the tables be found
exactly from its logarithmic sine or cosine; and in other cases it cannot be found exactly without much trouble.
To provide, then, for all cases several formulae are sometimes necessary. We shall now show in what cases
these difficulties occur. -
(164.) The ratio of the small variation of any function of an arc to the variation of the arc being ultimately the
d : log sin 6
d 6
of common logarithms = 0.43429448. Now when 6 is near 90°, cot 6 is very small, and a large variation of the
are is attended by a small variation of its log sine. A small error then in the log sine will produce a great
error in the arc; or if the tables be not carried to many decimals, the same log sine will correspond to several
successive values of the arc. Consequently an arc cannot be found accurately from its log sine when it is
near 90°.
(165.) If now the arc be very small, M. cot 6 becomes large; the second differential coefficient also
(= — M cosec” () is very great. It may happen then that the second differences of the log sines (of which the
differential coefficient, we shall have 3. log sin 6 = 3 0 nearly = M cot 9. § 6, M being the modulus
dº . log sin 6
d 6”
differences. This, however, is commonly avoided by constructing tables for a few of the first degrees of the
quadrant to every second, or to smaller.intervals than the rest of the tables; 3 6 is thus made so small that the
second differences are seldom sensible. But it is still better avoided by the use of a small table giving the
sin 6 sin 6 62 04
for a few degrees. For 6 * by (140,) = 1 — T2 3 + I. : : TE - &c.; its loga-
expression is (3 6)" -- &c.) become large ; and we must have the labour of interpolating by second
logarithm of
g 6? 64 tº 3
rithm = — M(; + 180 + se), the differential coefficient of which, or — M. (; -- f + &c.) is very
4 x 2
696 T R. I G O N O M ET. R. Y.
Trigono- small when 6 is small. And if 0 = n”, log 0 = log m + log 1" = log n + 4.6855749, of which the first part Sect. VIII
metry, can be found to any accuracy by common tables, and the second is constant; thus, when 6 is small, log sin 6 , Higher
\-y- sin 6 X 1// Analysis.
can be found accurately. The most convenient tables contain a table of log —g— ; let the number in this Yº
table corresponding to n' be a, then log sin m” = a + log n. -
(166.) Conversely from a given value of log sin 6, 6 when small is found with great ease. For subtracting
from log sin 6 the logarithm of 1", or 4.6855749, we have, nearly, the log of the number of seconds, by which
in 6 in 6 × 1// º º
we find in the table the log sy or the log ** * ; and though the number of seconds is not theoretically
{} i º sin 6 tº g º
exact, yet from the very slow variation of log . -º-, the error in the result will not be sensible. Then log true
sin 6 × 1"
number of seconds = log sin 9 – log 6
in 6 62 62 N. i
(167.) In the want of such tables, this method is convenient, º: = 1 — T6 nearly = ( T 1.2 )= (cos 6)*
sin 6 gº
nearly ; therefore log –H– = # log cos 6 nearly. Hence, log sin 6 = log 0 + $ log cos 0 nearly = log 0 –
# arithmetical complement of log cos 0.
sin 9
(168.) The same remarks in all respects apply to the tangent of a small arc. The series for the tan 9 = cos 9
OS
9
( &=º sºm-º. + se) *
6 2 tan 6 Q 62 N –
F. = 9 1+++&c. , therefore àIl = 1 + 4 = 1 — — º, and log tan 9 =
92 3 6) 3 1.2
1 — — —H. &c.
2 -
log 0 + 3 ar. comp. log cos 9 nearly. These expressions can be used without sensible error till 6 = 8°. Since
M &
is never small, we can never meet with difficulties in
the differential coefficient of log tan 9 ( = −;
sin 9. cos 0
the use of it like that mentioned in (164.(
(169.) In this way, then, we find that an arc cannot be determined accurately from its sine or cosecant when
it is near 90°, from its cosine or secant when very small, or from its versed sine when near 180°; but from its
tangent it can always be found with accuracy. Of the expressions, therefore, in (66) and (116) the first must
C
not be used when T2 is small or C is small ; the second must not be used when C is near 180°, nor the fourth
when C is near 90°. The third may always be used. In (63) cos B = a which is inaccurate if B is small ;
C
e I — cos B B -
but this expression may then safely be used; H.H. or tan? † = . + . In (70) if B is near 90°, let B =
b
b Jº 1 — a sin A
900 + a ; then cos a = Ta sin A, and tanººg = —–. Now — in all cases of difficulty will be greater
1 + — sin A Q,
(Z
- l (), wº Jº sin 6 – sin A 6 — A 6 —– A e
than 1, and less than sin A ; let T = sin 9 ; then tan? Tº -: sin 9-E sin. A = tan 2 ” COt *A. which
can be calculated with accuracy. In (105) if a. and b be very small, (a case which often occurs,) c cannot be
a ſl (M, tan b
. t g *
accurately found from that formula; we must therefore take tan A = in b’ and tan c = cos A’ by which c is
Sl O
tan b 1 — cos A tan c – tan b
f . In (109, A = ; i , --— — 3.
ound to the greatest accuracy. In (109,) cos tan C if A be small 1 +-cos A tan c + tan b
A -
or tan”---
2
in c – b ſº g
= * * – , which is not liable to inaccuracy. In (118,) if c should be near 180°, use this expression, 1 +
m=ºmmº
sin c + b
g e tº C tºº e
cos c = 1 + cos a cos b + sin a . sin b — sin a . sin b (1 – cos C,) or cos' g = cos' 2 — SII] (t ,
g C - . C C — b • r a — b
sin b . sinº-5- make sin a . sin b . sin” g = sin” 6, then cosº, = cos? – sin° 0 = cos . + 6
T. R. I G O N O M ET. R. Y. 697
Trigono- a — b • e . w ſº Sect. IX.
metry. COS – 6. We have given, we believe, the most important cases; but in any others the same principle Geodetic
\-y-' 2 Operations,
may easily be applied. '-º'-
(170.) We shall conclude our remarks on this subject with the solution of the following problem: To find how
far the tables are sufficiently exact. This will be done by giving to the arc the variation 1", or any other, accord-
ing to the degree of accuracy required, and finding at what limit the corresponding variation of the tabular
numbers is equal to one unit in the last place of decimals. Thus, for log sines : by (164,) the variation of log
sin 6 for 1" = 0.4343 x cot 6 × 0.000004848. If the tables be carried to 7 decimals, cot 6 at the limit =
0.0000001 : if to 10, cot 6 = 0.0000000001
0.4343 × 0.000004848 ' ' 2 T 0.4343 × 0.000004848’
gives 6 = 89° 59' 50’’; and beyond these the tables of log sines cannot be trusted to seednds. The same
principle may be applied to any other tables.
The former gives 6 = 87° 17'; the latter
SECTION IX.
Formulae peculiar to Geodetic Operations.
(171.) THE Trigonometrical surveys, which have been carried on for the two objects of mapping an extensive
country, and determining the figure and dimensions of the earth, afford the best exemplifications of most of the
theorems both in plane and in Spherical Trigonometry. For some of the reductions, however, they require
peculiar formulae; these we shall give, after describing generally the course of operations.
(172.) The first part is the measurement of a base, for which a plain of four or five miles in extent is generally
chosen; the line is measured with the most scrupulous exactness. In England, rods of deal, tubes of glass,
and steel chains, have been used ; the temperature being always noticed, and the proper correction applied for
expansion. In the late surveys in France, the measuring rods consisted each of a rod of platina and a rod of
brass, lying one upon the other, and connected at one extremity; the expansion of these metals being different,
the difference of the expansions was observed, and the whole expansion of one bar found by a simple proportion.
Other bases are measured in different situations, called bases of verification, and their measure, compared
with their length, as found by calculation, serves for a criterion of the correctness of the observations. Thus,
for the French surveys of 1740, 17 bases were measured; but in the late surveys there, two only were used;
and in the operations in Hindoostan, carried over a greater extent of country, five only were employed.
(173.) Proper situations for signals being selected, the country is divided into triangles by lines joining the
stations; and the angles of the triangles, that is, the angles which two signals subtend, as seen from a third,
are measured, (the first observation being made from the extremities of the base ;) and here the nature of the
instruments used, modifies the calculation in a considerable degree. For the late French survey, repeating
circles were employed, by means of which the angle between the two signals was observed; but since the signals
are seldom seen exactly in the horizon, a calculation is necessary to find from this the horizontal angle. In
England and India the horizontal angle was observed immediately by a theodolite.
(174.) In all the principal triangles each of the three angles is observed; and the error, if it is found from
their sum that any exists, is divided among them in the most probable proportion. The sum of the errors in
the nicer observations has seldom amounted to 2". For the smaller triangles it is sufficient to measure two
angles. -
(175.) Beginning now with the measured base, we have the length of the base and the observed angles at its
extremities, to determine the distance of a signal from its extremities, and the angle at the signal ; that is, we
have one side and the two adjacent angles to determine the other parts. Thus we determine A C, B C, fig. 20. Fig. 20.
Similarly, A D, BD, are found; then C D is determined. Then in the triangle C E D we have similar data.
And this process we extend to any number of triangles, till we arrive at a base of verification.
(176.) It is generally thought proper to choose the stations such that the sides of the triangles are greater
than 10 and less than 20 miles. In the English survey, however, the distance from Beachy-Head to Dunnose,
which formed one side of a triangle, is more than 644 miles. And in the extension of the French survey to
Spain, to connect Iviza with the continent, a triangle was formed, of which one side was nearly 100 miles. The
calculations are verified either by comparing the calculated length of a base of verification with the measured
length, or by comparing the distance between two signals as calculated from two chains of triangles, beginning
either from the same base or from different bases. Thus, in England by a series of triangles, extending more
than 200 miles, from Dunnose in the Isle of Wight to Clifton in Yorkshire, it was found that the error in a line
of 22 miles does not exceed six feet. And in some of the English bases of verification of four or five miles in
length, the difference between the computed and measured lengths has not exceeded one or two inches.
(177.) The latitudes and longitudes of the principal stations (those of one being known) are then determined
accurately, and those of the minor objects which have been observed by a more expeditious method. This is for
the purpose of mapping ; if it is intended to ascertain the length of a degree of latitude, the distance of two
places in the direction of the meridian must be ascertained, and the latitude of each must be observed. This was
the object of the late French survey; their purpose being to determine the length of the terrestrial quadrant, of
698 T R. I G O N O M ET. R. Y.
Trigono- which the 10,000,000th part, or metre, was made the standard of linear measure. For the determination of a
metry.
Pig. 21.
Fig. 22.
degree of longitude (a calculation which implies the spheroidal form of the earth) methods are used of which it
would be foreign to our purpose to treat. -
(178.) This is a general explanation of the usual process : we shall now give the mathematical theorems
connected with it. In the French bases, the line measured was not straight, but consisted of two parts, as a and
b, fig. 21, forming a small angle 6, (when largest it was 49'.) To find the correction, cº = a” + b – 2 a b .
_*===== & tº 62 e-magam-msº
cos T – 6 = a2 + b% + 2 a b cos 0; but since 6 is very small, cos 0 = 1 – T2T nearly; therefore cº = a + bº
a b
2 (a + b)
ſº a b ... n.”
0,000004848, the correction = a + b × 0,000,000,000,01175.
(179.) Supposing the three angles of a triangle observed, and one side, as a, known, To find its figure, that
the lengths of the other sides may be least affected by the errors of observation. Let A be the observed angle
opposite to a, B and C the angles adjacent, and b and c the sides opposite to them. Suppose the errors of A, B,
and C, to be & A, 6B, and 8 C.; then, as the sum of the angles (if erroneous) is supposed to be corrected by
altering each of the angles by the same quantity, 6 A = — (6 B + 8 C). Then the true value of c is a .
sin C + 8 C
sin A – 8 B – 8 C
a similar expression for the denominator, and observing that sin B = sin 7 – B = sin A + C = sin A cos C +
sin C : sin B sin C. cos A sin B
in A Tiº +* B) or the enor oria (; , C --
— a b . 6°, and c = a + b - 6° nearly, or the correction is 6°. If 0 = n seconds = n x
a b
2 (a + b)
; but sin C + 8 C = sin C : cos & C + cos C , sin & C = sin C-H & C. cos C nearly ; putting
S
cos A. sin C, we find c = a (
sin C. cos A g J s sin C sin B . cos A
sin” A 8 B) Similarly, the error of b is a(; A & B -- sin.” A
assign exactly the chances of the errors & B, 6 C, and 8 A, or — (8 B + 3 C), and our reasoning must therefore
be vague. It is evident, however, that sin A must not be small ; it is largest when A = 90°. But it is equally
evident, that there is a greater chance that the signs of 8 B and & C are different, than that they are the same ;
since in the three pairs that we can form of 3 A, 6B, 8 C, two will have errors of different signs, and one will have
errors of the same sign. And if & B and 6 C have different signs, the errors of b and c will be diminished by
giving cos A a positive value. A therefore ought to be less than 90°; and if & B and 6 C are probably not very
different, B and C should be nearly equal. These conditions will be satisfied by a triangle differing not much
from an equilateral triangle.
(180.) If two angles only, A and B, be observed, the expression for the errors will be as above; but we have
now no reason to think them of different signs rather than of the same sign. In this case, then, we shall
probably have our errors smallest, if cos A = 0, or A = 90°; the remaining angle of the triangle ought there-
fore to be as nearly as possible a right angle.
(181.) The elevations or depressions of signals being small, the correction to be applied to their measured
angular distance in order to obtain the horizontal angle is thus found. Suppose O A, O B, (fig. 22,) to be the
directions in which two signals are seen from O ; the angle A O B is measured. If a sphere be supposed
described about O as centre, and if through Z the point vertical to O great circles O A C, O B D be drawn, and
C O D be the horizontal plane, then C O D or Z, since Z is the pole of C D, is the horizontal angle required.
cos A B — cos Z.A. cos Z B
r tºº * tº-wº tº &ºme — si
|Now cos Z = sin Z A . sin Z B . Let A B = D, Z = D + r ; cos Z = cos D. cos a — sin D
8 c) . Now it is impossible to
Q /2
... sin a = cos D — sin D . a nearly , let A C = h, B D = h", then cos h = 1 – #. cos h’ = 1 -*- sin h
— h h"
= h, sin Ji' = h’, nearly ; and the equation becomes cos D – a sin D = -***'. nearly = cos D-hh'
1 – 3--->
cos ID h h! cos ID -
h” -- h’?): theref :- * 2 fº 'E p : h — h’ = q :
+ 2 (h” + h^*); therefore a sin D 2 sin D (h” + h^). Let h + h p; h — h q; therefore
2 — n2 2 Q } 2 — nº (p” + q”) cos D
h h' - p q g 2 h/2 F p –– Q tº - p q symms
- ; h° + 2 ” and a 4 \ sin D sin D
_ 1 / ... 1 — cos D ... 1 + cos DY - 1 * tan *. ‘cot ...)
T 4 p sin D — q sin D = }(p 2 q" co 2 / .
For observations with the theodolite, this is not necessary.
(182.) The horizontal angles being thus found, all our triangles are converted into spherical triangles, the
sides of which are Sinall compared with the radius of the sphere. For the solution of these triangles, three
different methods are used. The first is to solve them as spherical triangles, taking for the sines of the sides
Sect. IX.
Geodetic
Operations.
T R. I G O N O M ET. R. Y. 699
Trigono
metry.
the expressions in (165) and (167.) Knowing nearly the radius of the earth, the angle subtended at the centre à.
sin a
Operations.
by an arc of given length is known, and hence log can be taken from a table where a is expressed in
feet or toises; adding log a, log sin a is found. This method is, by Delambre, preferred to the others. The
second is to find from the angles of the spherical triangles the angles formed by their chords, and to solve this
as a plane triangle. Let C be one spherical angle, C — a the angle contained by the chords, then cos C — a
... a 0. . . b te
sin” - + sin” - — sin”
(chord of a)2 + (chord of b)* – (chord of c)* 2 2 Tº
t C
2 chord of a . chord of b . 0, . But cos
2 sin – . sin —
2 2
1 – 2 sin – (1–2 in #) (1–2 in
_ cos c – eos a , cos b 2 2 2
*mºs sin a . sin b Tºmºs 4 sin (Z COS Cº. sin b COS b
2 2 Tº 2
sins tº + sin” * – sin' . sin “... sin
- 2 2 2 2 2 ſºmeºmºs 67, b •
= ſº ; : therefore cos C — a = cos -a cos g cos C + sin
(Z
COS - . COS —
2 2
;
... a . b (Z
2 sin - sin – cos – cos
2 2 2
a . b -
a sin 3- = cos C + a sin C ; therefore a sin C = sin #. in # *ge ( dº tº cos # ... COS #) cos C =
b 2 -- b% 2 — fº
** – & ; cos C. Let a + b = e, a - b = f; therefore a + i = +/-, an = ** and
* e' – f' e” + fº 1 1 — cos C 1 + cos C - -
- *— - C, = − e” –——º- — f". T--~ l = — e” &ºm=&
a sin C 16 ja-cos v. or *=TE (. sin C J*.*.*. ) Ta (etan 3
— fº cot #). All these expressions suppose the angles to be expressed in numbers considering the radius
72,
number of feet in radius’ if a = m seconds, for a we must put
as l; if e = n feet, then for e we must put
m x 0,000004848. This method was used in the English surveys.
(183.) This principle of the third method is, by applying a correction to the angles of the spherical triangle
to treat it as a plane triangle. Let a, b, c be the sides to radius r ; then
C (Z s" l cº + 04 l a? + a 4 l b? + b%
— —- (*OS – COS - &=º sºme ** = &= sº-º-mº-e sºmº-º-º-º- assum, sº- *-*
cost-cos = cos; 2 rº 24 7-4 ( 2 * 24 7-4 2 * ' 24 rº
cos C = (, b -- a b I (Zº l b?
r * rº 6 rº 6 rº
sin – . sin –
nearly = 12 rº | . But if C — a be the angle
2 a b
a” + bº — cº
2 a b
* + º--e-r- {2 * * * * *e-2 we- *-*-*}
; therefore a sin C =
in the triangle considered as plane, then cos C — a. or cos c + a sin C =
24 Fab {2 a bº-H 2 a” cº 4-2 b" c' – a – b – c'}. The part within the brackets = 4 a” b” – (a” + 5° – c.)”
= {2 a b + a” + b – c’ #. {2 a b – a + b – c. } = {(a + b)” – c. 3. [ cº — (a – b)* } = (a + b + c)
(a + b – c) (a + c – b) (b + c – a) = 16 (area of triangle)”. But a b sin C = 2. area; therefore a =
area of triangle . area of triangle
3 +2 9 3 rº x 0,000004848 °
area of the triangle be found in feet, the logarithm of the divisor is 9,8038940, a degree on the earth's surface
being considered = 365.155 feet. This is due to General Roy. The dimensions of the triangle are always
known accurately enough to find the area with sufficient exactness. The correction is the same for each of the
angles; it is therefore one-third of the excess of the sum of the three angles above 180°, commonly called the
spherical excess. The spherical excess seldom amounts to 5"; in the largest triangle joining Iviza with the
coast of Spain it amounted however to 39".
or if a = n seconds, m = This is Legendre's theorem. If the
700 T R. I G O N O M ETR Y.
Trigono- .
metry.
Fig. 23.
Fig. 24.
(184.) The sides and angles of the triangles being found by some of these methods, and the relative situation
of the signals being found, it is necessary to determine the angle which some one of the lines makes with the
meridian. In the English surveys this was done by observing with the theodolite the horizontal angle between
a signal and the pole-star, at the time when it was known to be at its greatest azimuth. Let Z, fig. 23, be the
zenith, P the pole, S the pole-star, ZS a great circle. Then cot Z. sin ZPS = cot S. P. sin Z P – cos Z PS
. cos Z P. Suppose a small error in the time, this would create a small error in the angle Z PS. Now, as in
(131,) we find that the simplified expression for the error of Z vanishes when cos S is 0, or S is a right angle.
Returning then to the original expression, and observing that cos ZP = cos Z. cos P; and putting for cot
sin Z. cos Z (3 P)*
Z + 6 Z, &c. their approximate values, we find at length & Z = — ºf F- - -a-. Now with the
Il
pole-star sin Z is small, and 6 P very small; hence a small error in time will not produce a sensible error in the
azimuth. r
(185.) In the French surveys the azimuth was found by observing the angle between the signal and the sun
when near the horizon; also by taking the angular distance of the signal from the pole-star when nearest to the
signal, or farthest from it. To allow the observations to be repeated, a correction was investigated not very
dissimilar to that of the last article, to be applied to the observations made when the pole-star was a little
removed from the point nearest to, or farthest from, the signal. From this distance the azimuth is found by a
right-angled spherical triangle. But in Spain, a transit instrument being adjusted to a mark nearly in the
meridian, the intervals of the transits of different stars were observed: comparing these intervals with those that
ought to have been observed in the meridian, the azimuth of the mark was determined by a formula common in
practical Astronomy. From this the azimuth of any signal was easily found.
(186.) The direction of one side being known, we have now to solve this problem. Given PA, fig. 24, the
colatitude of A, and the angle PA B, and the length of A B ; to find P B the colatitude of B, and the angle B,
and the difference of longitude P; A B being small (seldom = 1°.) Here cos B P = cos A. P. cos AB + sin
gºssmº-se
A P. sin A B . cos A; let B P = A P – a ; cos A P – a – cos A P, or 2 sin A P — ; : sin # = sin A P.
sin A.B. cos A – cos AP (1 — cos AB) = A B . sin A P. cos A – cos A P. AB nearly ; therefore sin #
A B2
A B . sin A P. cos A — cos A P
2 o a . A B . cos A tº s a
- *sºmºsºmy —. An approximate value of + is ++--- ; substituting
ſº tº 2 2
2 sin A P — -
2
& , sin”
this in the denominator, a = 2 sin T2 nearly = AB cos A – COt ar, sinº A A B*. If greater accuracy is
desired, this may be again substituted in the denominator; then A B* must be taken in the numerator; and
S in?
observing that # = sin #. –– l_ sin + nearly, a = A B cos A — cot A P. sin? A A B2
2 2 6 2 2
- 2 g 4. © º g tº
(1 + 3 cot”A P) sinº A. cos A A B3. Then sin P = SIIl A B ſº in A. and sin B = * AP - SlT1 A. Or, if a
6 Sin P B sin P B
* A B . sin A A B*. sin A . cos A. . cot P A O
series be preferred, P = Tsin PAT ~ sin PA — &c.; B = 180° – A –-
A B : sin A. in **tº
2
sin PB
(I87.) For the points of less consequence, which have been observed from two stations, the distances being
found considering the triangles as plane, the value a = A B cos 0 is sufficiently accurate; and then P =
A B . sin A
sin PA
These are the principal formulae of Trigonometry that are used for surveys on a large scale. We have
treated of them at some length, as we know not any book in the English language in which any account of them
is to be found. We have confined ourselves to what appeared to be strictly connected with the subject of this
Treatise; for the explanation of the methods used in different hypotheses of the figure of the earth, and for
the results deduced from them, we refer to our article on the Figure of the Earth.
nearly.
Sect. IX.
Geodetic
Operations.
T R I G O N O M ET. R. Y. 70] .
Trigono-
metry.
\-y /
*
SECTION X.
On the Construction of Trigonometrical Tables,
(188.) THE construction of tables naturally divides itself into two parts: the first is, the determination of
values of the function to be tabulated for certain values of the arc, at large intervals; the second is, the filling
up of the tables by inserting the values included between these. In this order we propose to consider the
formation of tables of the values of Trigonometrical lines and their logarithms.
(189.) The method which first suggests itself for the determination of natural sines, is to take some arc whose
sine and cosine are known, (as 30°, 45°, 18°, 54°, &c.) and determine the cosine of half the arc by the formula
COS (M = V l + º: 2 2. and after repeated applications of it to determine the sine by the form sin a =
v/Fº— cos 2 a. . Or the sine and cosine may be determined by the formula sin a = } { V 1 + sin 2 a -
2
w/T- sin 2 a 3, cos a = } { V 1.-- sin 2 a + 'V 1 – sin 2 a #. This method, when 2 a is small, is more
1 — cos 2 a.
accurate than the former. For when —g— is very small = v, suppose a to be the error to which it is
liable, or the value of the figures rejected ; then its square root will be liable to the error . w/o.
2 A/ v
when v is small, is very considerable. On the contrary, in the other method, 1 + sin 2 a, and 1 — sin 2 a, being
nearly = 1, upon extracting their roots we are not liable to the same error. In this manner find the sine of
30°
–37 = 52" 44" 3" 45". Now by observation of the sines of this arc, and of the double of this arc, it will
be seen that the sines of small arcs are nearly as the arcs ; and therefore 52'44" 3” 45" : 1’: : sine found :
sine of 1'. From this the cosine of 1" is found; and the sines and cosines of 2',3', 4', &c. are found by the
formulae of (38.) .
(190.) But the same thing may be done in this manner, with fewer (though more laborious) operations, and
without the proportion used in the last article. It was found that sin 5 a = 5 sin a – 20 sin” a -i- 16 sins a ;
conversely, the solution of the equation 5 & – 20 as + 16 as -- sin 5 a will give the value of sin a. Thus, from
sin 15° (found by bisection) we may by approximation find sin 3°. Again, sin 3 b = 3 sin b – 4 sin” by solving
this equation we have the value of sin b from sin 3 b, and therefore from sin 3° we find sin 19. By a repetition
of the same operations we descend to sin 30', sin 15', sin 3', sin l’; and then ascend as before. In the same
way we might have begun from 18°, or any arc whose sine is known.
(191.) But in a process of this kind, where an error in the calculation of one number would affect all the
following ones, it is clearly desirable to compute independently some numbers in the series at convenient
intervals to serve as verifications for the rest. Thus, from sin 30° we may by trisection find sin 10°; from this
nearly, which,
we get cos 10° or sin 80°; then sin 20° = 2 sin 10°. cos 10° is found; then since sin 60°-- A — sin 60° — A =
sin A, we have sin 80° — sin 40° = sin 20°, whence sin 40° is found ; thence sin 50° or cos 40° is found ; and
sin 70° = sin 50°-- sin 10°. The sines for every 10° of the quadrant being found, those of every degree should
then be calculated as verifications for those of every minute, &c. The following is the best method of
performing these calculations: sin m + 1 b = 2 cos b. sin n b — sin n – 1 b, therefore sin n + 1 b — sin m b = sin
m b — sin m — 1 b – (2 — 2 cos b) sin m b. But sin m + 1 b — sin m b = difference of sin n b ; sin m b — sin
m — 1 b = the preceding difference; hence the difference is less than the preceding difference by (2 – 2 cos b)
* b b e g te
sin m b, or 4 sin” 2 sin m b ; that is, the second difference is — 4 sin” 2 sin n b. Now, since sin n b is
- b
already found, this can be calculated ; and the operation will not be long, for the multiplier 4 sin” Tº
being the same every time, a table of its products by the 9 digits may be prepared. Thus then we have sin 12°
— sin 11° = sin 11° — sin 10° – 4 sin° 30'. sin 11°, &c. In this way the sines for every degree may be found;
if the values for sin 10°, sin 20°, &c. are not the same as those found before, it shows that there is some
error in the computation.
(192.) But the natural sines for these arcs, at least for 10°, 20°, &c. or more conveniently for 9°, 18°, &c.
3
may be calculated independently thus. We found for sin w the series a — –tº– 3:5
. . . I. 3.3 * Ig. 3. IB
WO L. I. 4 Y
Sect. X.
Construc-
tion of
Trigono-
metrical
Tables.
702
T R I, G O N O M E T R Y.
q2 º
~ 1.3 ºf I. a 3.4
7-
7??,
— &c. we have cos. – . — =
70, 2
1,000000000000000 wº # × 1,23370.0550136170
4. 6
++ x 0.25866950790.1048 – “... x 0,020863460763353
774 706
8 - - 10
++ x 0.000919260274889 – “... x 0.000025202042373
778 7,10
M2 14
+* x 0.000000471087478 – “... x 0.000000006386603
7112 7.14
16 18
#. × 0,000000000065660 sº #. × 0,000000000000529
20
+ #: × 0,000000000000003
g g te 7??, g
The cosine of an arc being the sine of its complement, + will never exceed #; and a few terms of these
70,
series will give the natural sines with great ease to 15 decimals.
(193.) When the sines for every degree are calculated, they should be verified; and for this purpose the last
equation of (160) will be found extremely useful. By giving to Y and n different values, we may with great
ease examine the accuracy of as many calculated sines as we wish. d
(194.) The sines for degrees being found, those for smaller divisions, as minutes, are generally found by
differences. And a remarkable relation between the differences of successive orders enables us to determine the
differences with which we must begin our table, from knowing the two first of them. Let it be supposed
that the arc w is formed by successive additions of h; then A sin a = sin a + h – sin a = 2 sin → . cos
h te . h. 3 h. h . . h . —
* ++ ; A” sin a = 2 sin Tº (co. JC ++ — cos w -- #)= — 4 sin"; . sin r + h; and, consequently, A*
º º h º e gººmsºmºmºmº & iſ o *ºmmºne
sin a -- h = — 4 sin” -- . sin a. Hence A4 . sin a – h = — 4 sin': A* sin a = 16 sin” -: , sin a + h, and,
therefore, A4 sin . a - 2 h = 16 sin" 2. ' sin a. Similarly, A*. 2 sin a – 3 h = — 64 sin: . sin a, &c. Also
cos a + *
g . . h. *sºmºmºsºm-m h h
A” sin a = — 8 sinº To , therefore A*. sin a – h = — 8 . sinº # , cos a + 2 : similarly, A5
, –- . . h. h - g 6
sin a – 2 h = 32 sinº # , cost + 3’ &c. Now if we arrange these in tables in the usual order, as below,
sin a – 3 h
Sºº-
A sin a – 3 h
sin a – 2 h.
sin ºr — h
A sin a – 2 h
A” sin ºr T37.
A sin a – h
A? sin a -3%
A* sin a – 2 h
sin a
sin a + h
A sin a
A* sin a -- h
A* sin a – 3 h
A3 sin a – 2 h
A4 sin a – 2 h.
As sing – 3 h
A” sin a – h
A* sin a – 2 h
* * Sect. X
º: let r = + e #: then + being found by the differential calculus to a 1,570796326794897, we have sin dº.
º tion f
S-N-2 97, ºr tº:
2 m º
onn 8 ables.
+ x 1,570796326704897 – 4 x 0.645964097506246 S-N-2
1775 ºm?
++ x 0,079692626246167 - -, * 0.004681764135319
7779 ºn 1:
++ x 0,000160441184787 + x 0.000003398843235
777.18 ºn!"
ºn 8 0,000000056921729 – ºr x 0.000000000668804
m17 777-19 -
ºr x 0,000000000006067 – ºr x 0.000000000000044

T R I G O N to M E T R Y. 703
Trigºnº- we shall remark that sin w, A*. sin a – h, A“sin a – 2 h, &c. are in one horizontal line, and that A sin w, A* sin Šº.
metry. —- | - g g ſe g g g Construc-
y , a - h, A* sin a – 2 h, &c. are also in a horizontal line. Hence the numbers in each horizontal line form a tion of
tº e tº e g h g e . —- - Trigono-
geometrical progression, whose ratio is – 4 sin” Tº Knowing then sin a and sin a + h, we can calculate all º
- 3.D. 162S.
the differences as far as are necessary, and all our sines are then formed by addition and subtraction. If a = 0, S--~~~
we have but one series of differences to calculate.
(195.) By a slight alteration in the enunciation of this relation of the differences, we may avoid using any
g e & h s tº-º-º-º-º- tº tºº-rººmsºmº & e
more numbers than are absolutely necessary. Since A* sin w = — 4 sin” -: , sin a + h, and sin a + h = sin a
º g h e tºg gº ** te - º
+ A sin ar, therefore A* sin a = – 4 sin” 2 (sin a + A sin a ; taking the n – 2" difference of each side
A" sin a = — 4 sin” Tº (A*-* sin a + A*-* sin ry, a formula which gives any difference in terms of the two
differences immediately preceding.
(196.) One important point we must not omit to notice, namely, the number of decimals to which these
differences ought to be calculated. For this investigation we shall consider each of them as liable to the
same error in the last figure used, (it will never exceed half an unit, if we increase the last figure by 1 when
the first rejected is equal to or greater than 5.) Now it is useless to take one difference to so many decimals,
that the error from it will be much less than that from any other; we shall then make them as nearly as
possible equal. Suppose, now, there are m sines to be calculated by the differences, before our operations are
verified by one of the previously calculated sines. The theory of finite differences gives us for the n-FIth sine,
e w . 77 – 1 . —--> . m. 1. nº lº . —- . . m – l . 1. m. – 2
sin r + n A sin a + “ º: A. sin ºr i +++,+,++ A. in F7 4-tº-: ****
*º-mm-mm, ... m. – 1 . 1 . m — 2 . 2 . —- tº g
A*. sin 3–2 h-- * 70, #. #. m –– A°. Sin a – 2 h -- &c. The error of each difference will
be multiplied, in the nº sine, by the multiplier of that difference. If then n = 59, the first difference should be
carried to 2 figures more than the sines, the second to 4, the third to 5, the fourth to 6, the fifth to 7, the sixth
to 8, the seventh to 9, the eighth to 10, the ninth to 11, the tenth to 12. Or, if we make use of the differences
tº-º-º-º-º-º- . m. — 1 te ... m. — 1 . n – 2
calculated in (195,) as the n + 1" sine = sin a + n A sin a + ++ A* sin a + 70, ... ??, 72. A° sin a
2 . 3
+ &c., we may in a similar manner find the number of decimal places to which each of these must be calculated.
In adding any number to, or subtracting it from, any other number which has not so many decimals, we must not
use the superabundant figures, but increase by 1 the first figure used, if the first of the superabundant figures
be not less than 5. The sines with which we begin should be taken to 2 or 3 figures more than it is intended
to preserve in the tables. In this way we can calculate with great accuracy and without any unnecessary labour.
(197.) To interpolate for smaller divisions, as seconds, it is convenient to have a formula for finding the
differences for the smaller divisions, by means of the differences for the larger ones. Suppose, now, the smaller
tº º º 1 th tº - wº
divisions to be each p of the large ones. Let A', A", &c. be the 1st, 2d, &c. differences for minutes, and 3'
8", &c. those for seconds. Then, by the common formula, we have
sin a = sin a
—, . I I I N A// I I 1 All!
in a + 1" = — A' — — ſ 1 — — 1 – + – l l — — ſ 2 — — . . — —
sin a + ins-- }( p #4 ( })(? }) T; ; – &c
º 1. l l A" 2 3 1 All/ 6 IJ , 6 I
--- l, — A' — ſ — — — . . . --— * Exº ºm-se - . - – I — — — as—- smºs -
sin a + p (. ...) l #4 (; p? +;) 1 .. 2 .. 3 (; # TF p"
X Alth 24 50 –H 35 10 + l A///m
I, a 3.4 °F p p” pº p' ' p") 1.2. 3. 4. 5
- *mºs º-mºmem-sº-sº-s-s 2 3. ſº
and the sines of a + 2", a + 3", &c. will be found by putting P’ p , &c. for #. Upon taking the differences
l
of these successive values, it is clear that the numerator of in the mºh difference will be A". 0” + multiplied
Tn
by its factor in sin a + 1". Thus we find, (going as far as the 5th differences,)
ººms
* By A". 0n is meant the first term of the nº order of differences of the series 0", 1", 2", 3", &c.
4 Y 2
704 T R T G O N O M E T R Y.
*mme
*:::: 3 = A' p *An LP - 1 - 2p - "Am P-1.2 p-l • *P-' An EP-1 *P-1 : *P-1 : *P-'Amr, ë.
© p 2 pº 6 pº 24 p. 120 pº tion of
\f -* Trigono-
37 - 1 A// º p - 1 All! + £ tº º l g Il p sº 7 Alliſ tºº p immy l c 2 p sºmº l * op — 3 Alllll. º
p° 12 p. 12 pº S--
3//’ = * A” – 3 . # l A/" + p — º p — 5 Alth.
p p
3” – 1 A" — 2. p – 1 Aſſiſ.
p" p"
S//l/ - 1. A/////
5
These expressions * are quite general; from the relation among the differences of natural sines mentioned in
(194,) it is not absolutely necessary to calculate more than the first of them; but even there it will be more
convenient to use the formulae.
(198.) The sines up to 60° being calculated, those above 60° will be found by simple addition, from the
formula sin 60° -- A = sin 60° — A + sin A. Thus the sines are found for the quadrant; and, consequently,
all the cosines are known.
(199.) The tangents will be found by dividing the sines by the cosines. After 45° they may be found by the
formula tan 45°-i- A = tan 45° – A – 2 tan 2 A.
(200.) The tangents may also be found independently in the following manner. If we expand every fractional
term, except the first, of the first series in (157,) and add together the coefficients of similar powers of a, and for
a put * & #. we have the following expression, tan + o + ==
*** x 0,6866197723575318
7?” – 770
777, m3
++ x 0.297556782039784 ++ x 0.018588650277880
7775 7m?
++ x 0.001842475203510 ++ x 0.000197580071520
7779 m*
rºy rºyº te
+ + x 0,000021697737325 + iſ x 0,000002401136991
777.18 777.15
++ × 0,0000002664.13303 + iſ x 0,000000029586468
77,17 77,19
++ x 0.00000000328,788 + ºr x 0.000000000305175
7nº1 7m 23
r x 0.000000000040754 ++ x 0.000000000004508
ºn 25 tº 7mº w
++ x 0.000000000000501 ++ x 0000000000000056
777.29
+, × 0.000000000000006
.* The demonstration in the text is the most simple, but the law may be found more generally in this manner. The problem is, from the
given differences of the series, u, , 2,4-p, u, Hap, &c. to find the differences of ur, urºl, ur-Hº, &c. Let p (t) be the Generating Function
º & 1 H 2
of we ; the generating functions of A ur, A* ur, &c. are ( 7- - 1) © (t), (+ sº 1) . p (t) &c.; and those of 3 u, , 3* u, , &c. are
H I 2 I n . l e
(+ *º 1) •(, , (+ dºms 1) . © (t), &c. For 3" . wr, then, we must express (+ Eºs 1) in powers of (; ſº- 1). Let + - 1 = z:
1 l * l 7. I 1 \ n 1 ſº
F = (1 + 2); ; (+–1)=(IFF} - 1), le. this = A an -- B z* + &c.; therefore (+ - ) =A. (+ – ) +
1 n+1 I wº 1 1, I n+1 " – . e
B. (# tº 1) + &c., and (+ tºº 1) . p (t) = A. (+ sº 1) © (t) + B. (+ — 1 . p (t) + &c.; and taking the quantities
of which these are the generating functions, 3" . us = A. A." . we -- B. A." H. u, + &c., where A, B, &c, are the coefficients of 2", 2*h,
I re
&c. in the expansion of ( Tz' F — 1) e
T R. I. G O N O M E T R Y. 705
g Sect. X.
Tº: Similarly, from the second series in (157,) cot + & + --- dº
* tion of
\-y- 7, - - 4 m. m. - - - - Trigono-
— x 0,6366197723675813 × 0,318309886.1837907 metrical
777, 4 nº — m” Tables.
S ,-smººr’
- + x 0.2052888394.4508 – 4 x 0.0065510747ssals V
5 - 7
tºyº #. × 0,000345029255397 wº- + × 0,000020279106052
9 ll
gº +. × 0.000001236652718 imº +. × 0,000000076495882
13 15
gºmºs # × 0,000000004759738 tº º #. × 0,000000000296.905
17 19
ſº # × 0,000000000018541 EEsº # × 0,000000000001158
7m 21 7m 28 -
– ºr x 0.000000000000072 — = x 0.000000000000005
The first fractional term in each expression is not expanded, as the series by that means are made to converge
772
much more rapidly than if it were. It will never be necessary to take Tn greater than }.
(201.) It is plain, however, that this process is too laborious to be applied to every one of the small divisions,
and that it cannot with ease be extended farther than to every degree. But the calculation of differences of
tangents admits of none of those simplifications which assisted us so much in forming tables of sines; we proceed,
therefore, to give a method which applies to all cases whatever.
(202.) Let u be a function of a = 7 (a); suppose a to receive the increments h, 2 h, &c.; then
@ (r) = u. -
* 7t h” dë u h9
d u h d
* (* + h) = u + H+. ++. H4+. . Ha + &c.
G - d u 2. h. d” at 22. h4 dº w 2°. h."
* (*--2h) = u + H+. Hº-H. H.-H.. Hºa
Upon taking the differences it is evident, that (observing that A". 0" is 0 when n is greater than m) A". w =
A” e 0" d" A” © 0n+1 dº-Fl e A” . 0" A” . 0.4-1
te g 1, •= * #. h"+" + &c. Now the numbers — h", *-*-*-a---
1 .. 2 .... n d a 1.2 . . . .7, TT da". 1 .. 2 . . . . n l. 2 . . . . m + 1
h"+", &c., can be conveniently calculated first (as they will be the same for every difference calculated thus)
+ &c.
d”
h being expressed supposing the radius = 1; and the formula for d º/,
can also be found, and it will only be
fº
necessary at each calculation of differences to substitute numerical values in the expression for dº' For the
9 a.
tangent, u = tan ar, # = Secº ar, #: = 2 secº a . tan ar, &c.; and for intervals of a minute each,
Jº º
& " . 0m -
h = 0,000290888208665721596. The following table contains the values of Hº: from n = 1 to n = 12,
© tº e º e 77.
and from m == i to m = 12.
706 T R I G O N O M E T R Y.

Irigono-
metry.
S-V-7
01 || 02 || 03 04 05 06 07 || 08 09 010 Oil 012
TIEI5.3|I.5.3.III BIGIF|IBI. 911. . . . . . 10|] . . . . . . I lil. . . . . . . . 12
A || 1 | * l l l l I l l 1 } l
2 || 6 || 24 120 | 720 5040 40320 || 362880 || 3628800 39916800 || 47900 1600
...Tº a TI, II; 17 73 31 2047
T2. 4 360 40 20160 | 12096 || 259200 604800 239500800
...T 1 | * 5 || 3 || 43 23 605 3.11 2591 437
2 4 4 120 | 160 12096 20 ! 60 604800 403200
... T I a | 18 5 8] 37 6821 265 5559]
6 2. 80 72 30240 3024 1814400
A5 1 5 10 25 331 45 2243 1045
2 3 *m. 144 32 3024 3024
*mmºs. 30083
A6 1 a 19 21 1087 259 || 30088
4 4. 240 80 15120
* I sºmeºmºsºme 939 TT 4753
A7 I 7. 77 49. 1939 ºm-ºs-
2 12 6 240 720
-—
- 25 4819
AB l 4 zo | 2 sºmsºmºsºm
3 360
9 21 135
A9 I tº-º-º-º: ºs-ºs. “ i ºmºmº
2 2 8
A10 5 155
12
| ll.
All | I tº-ºr
| 2
A12 . l
(203.) The same cautions as in (196) must be observed with regard to the number of decimals. And for
the calculation for smaller divisions, as seconds, the formulae of (197) must be used. Thus the table of tangents
is completed. -
(204.) The secants are calculated from the formula tan A + cot A = 2 cosec 2 A. This gives the
cosecants or secants only for every second division; but the interpolation for every division will be sufficiently easy.
(205.) Thus then our tables of natural sines, tangents, and secants, is completed. The tables of their loga-
rithms might be formed by taking from logarithmic tables the logarithms of these numbers; and many writers
have considered this as being upon the whole the easiest way. As they may, however, be found independently,
and therefore free from all errors of previous computations, and as the method appears to us the easiest, we shall
give it here. - - :
q22
2 2
(206.) It has been seen (155) that sin a = a ( *_*: #)( sºms #) ( enºm, #) . &c., and therefore log
ºr 4 m.9 9 m2 /
e a;2 grº
sin a = log a +log(–3)+ log(1– #)+ log(l wº-
the first, and putting M for the modulus of common logarithms, log sin a = log a + log ( sº #)
2
# J + &c. Expanding all the fractions but
T e
ſ * , 1 * , 1 ºf
| Tº t 3 16. " 3 ſº H; +&c.
- wº do? 1 º 1 3:6
&= -ºmº - e - -f- — . — &c.
ſ + 5. T a . sº ºf a 7a: ... + ° f
U + &c. J
Adding the coefficients of similar powers of a, and putting + tº ; for a, we find the following series,
Tr
log sin #. # = log m +log (2n = m) + log (2n + m)–3 log n + 9,594059885702190
71.
Sect. X.
Construc-
tion of
Trigono-
metrical
Tables.
N-V-2
T R. I G O N O M E T R Y. 707
Trigono- - - 27.2 - 70% - - - Sect. X
metry. - Tº X 0,070022826605902 — — ºr x 0.001117266441662 Construc.
J 71. -
--~~ 7ng 77.8 tion of
- Tº x 0.000039229146464 --> x 0.000001729370798. .
77,10 ºn 12 Tables.
- + x 0.000000084362986 --in X 0.000000004348716 . ." \-N--
77,14 16
--ºr x 0.000000000231931 tº- #. × 0,000000000012659
7%.18 - 7.20 •
-- ºr x 0.000000000000703 – + x 0.000000000000040
7m 22
- + x 0.000000000000002.
And similarly log cos + .# = log n – m + log n + m – 2 log n.
7772 7704
- + x 0.101494859341898 – ºr x 0.00318729406545i
––. × 0,0002094.85800017 &- + × 0.00001684884850s
* : * 4. × 0,000001480193987 ºsmº 4. × 0,000000136502272 º,
gmºgº 4. × 0,00000001298.1715 — 4. × 0,000000001261471
— 4. × 0,000000000124567 sº-> 4. × 0,000000000012456
º-º º: × 0.000000000001858 * -º-; +: × 0,000000000000 128
–4. × 0.000000000000018 tº- + × 0,000000000000001.
2
(207.) If # be small, the first terms in the last expression, which together = log 1 – #, may be expanded
~ 70,
. e e ºn? l 7??? 8
into the series – 2 M × 272 – 72 —— Ta #) + &c )where M = modulus = 0,43429.4481903252.
To make the logarithm positive, 10 must be added. This makes our operations entirely independent of
logarithmic tables. - -
(208.) It is sufficient to find the log sines of the arcs between 45° and 90°, or the log cosines of arcs less
than 45°. The remainder may be found thus, log sin A = 10 + log sin 2 A — log cos A – log 2 =
log sin 2 A — log cos A+ 9,698970.004336019. By properly applying this theorem we may descend succes-
sively from log sin 45° to the log sines of all arcs less than 45°. By this method then the log sines and log
cosines, and consequently the log tangents (since log tan A = 10+ log sin A — log cos A) may be calculated
for every degree.
(209.) If, however, the log sines be calculated independently for harger intervals, as for every 10°, the
differences for every degree may be thus found, log sin a + h — log sin a = log sin (, ; b) ---
SHII (?”
º rºſh 4-ºxº g l © gºmºsºme tº- º 3
2 M #. aſ + SIII ºf + + . (i. * + k, SIIl ;) + se} a series which converges rapidly. Or when one
sin a + h + sin a 3 sin a + h + sin a
first difference is thus found, the second differences may be calculated by this series, A*logsina = log Sin & Sin T +
sin” a + h
sin” a + h — sin a . sin a + 2 h l (######
= - 2 M { sºmº — —H + ſº-º-º-º-
sins a -- h-- sin a . sin a -F2 h ’ 9 \sinº r +h -- sin a . sin a +2 h
3 l
) + &c.; ; which, since
sin” a + h — sin a . sin a + 2 h sin” h • g
e —= = - , converges much more rapidly.
sin” a + h -- sin a . sin a + 2 h cos” h -- cos 2 a + h -
708 T R. I G O N O M ETR Y
Trigono- (210.) Before proceeding farther, it will be proper to verify the numbers already calculated; and here the Sect. x
metry. formula of (159) will be found very useful. For taking the logarithms of both sides of that equation, log sin n 3 Construc-
S-N-7 - †. of
ammºmº" 3 r - rigono-
= n – 1 x 0,3010299956639812 + log sin 6 + log sin 3 + + + logsin 8 + º: + &c. to n terms, where n and º
e - • ables.
B may be taken at pleasure. S-N-2
(211.) It is then best to fill them up by differences; and the differences may be calculated in the same
in (202.) H ** = M cot dº u_ M (1 + cot? dº u 2M cot, (1 + cotº wo, &c
manner as in (202.) ere g = coºrs g = (1 + co w) ; ; ; = 2 tº
The calculation of the differences is rather tedious, but the tables are formed then with great ease, and the
certainty that any error will be discovered at the next place of verification makes this method superior to any
other. -
(212.) For the smaller divisions, the differences will be found from these differences by the formulae in (197.)
Thus our tables of logarithmic sines, cosines, and tangents will be completed. : -
(213.) It is unnecessary to examine by any formula of verification the accuracy of the numbers for the small
divisions of the arc. It is scarcely possible to have a better verification, than the agreement of the last of a
series of numbers computed by differences with one which has previously been calculated by an independent
process. -
(214.) In (165) we have alluded to tables of the logarithms of º: for a few degrees. These are calculated
6 39.4. 5 34. 5
has no significant figure in the ten first decimal places. For tables to 10 decimals the first term is sufficient up
to 1*, and the two first terms to 5°. For tables to 7 decimals, the first term is sufficient, as the second term
produces I in the last place when a = 5°. This therefore is easily calculated by second differences. If the
tan ºr o ... tan aſ sin a l tan ac sin a
log be required, since == e , we have log = log
& *} tº COS ſº * *
can be calculated in the same way.
ſº 2. 6
very easily from the series log º # = — M {; + Q:4 + tº 7 + se! When r = 5° the third term
+ ar. comp. log cos a 5 or it
(215.) Since sec a' = cos a ' its logarithm will be immediately found. And since versin a = 1 — cos w =
2 sin? 2. ' the natural and logarithmic versed sines are found. They are seldom inserted in tables, except in
those employed in Nautical Astronomy.
(216.) The principal tables commonly in use are the following: Sherwin's, containing, besides the logarithms
of numbers, sines, cosines, tangents, &c., natural and logarithmic, for every minute, to 7 decimals; Hutton's, •
containing the same, with an interesting and valuable Introduction; Gardiner's, with log. sines, &c. for every
10 seconds to 7 decimals; Taylor's, with log. sines, &c. to 7 decimals for every second ; of these, the most
common is Hutton's. Many smaller collections of tables are in use. Of the foreign tables, the best are Vega's,
containing the logarithms of numbers and log. sines, &c. for every 10" to 10 decimals; Callet's logarithms of
numbers, log. sines, &c. for every 10" to 7 decimals, with some tables for the decimal division of the circle.
This is a very convenient and useful collection. An abridged form of the Tables du Cadastre, revised by
Delambre, has (we believe) been edited by Borda ; and must form a useful collection for the decimal division.
(217.) Trigonometrical tables have generally sines, cosines, tangents, cotangents, &c. up to 45°; the cotangent
of an arc being the tangent of its complement, &c. What is gained by this arrangement, except perhaps in
the use of subsidiary angles, it is not easy to say; and in taking out the sine, &c. of an arc greater than 45°, or
greater than 90°, there is frequently some confusion. We should prefer the more natural arrangement of sines,
tangents, &c. up to 90°; these read in the reverse order (as shown by the figures and titles at the bottom of the
page,) would give the cosines, cotangents, &c. . --
TRIGONOMETRY. - Plate I.
E.
2 5
IF. .3
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(> F G" N. F G’
N R
N E
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Analytical
Geometry.
A N A LY TIC A. L G E O METR Y.
THE Application of Algebra to Geometry forms two
distinct branches of Science. The object of the first
is to investigate the Theorems, and resolve the deter-
minate Problems, of Elementary Geometry; that of the
second, to assign the Figure, and determine the Pro-
perties of Curves and of Surfaces. The first of these
is of very limited extent, and of comparatively trifling
importance; we shall, therefore, confine our attention
to the second, which is of great use, as an instrument
of investigation, in various departments of Pure and
Mixed Mathematics. This branch of the subject is
usually distinguished by the name of Algebraic, or
Analytical, Geometry. It may, with propriety, be
divided into two parts, of which the one will embrace
the Theory of Curves, and the other, the Theory of
Surfaces.
tºne sº-º-º-mºm.
PART I.
ON THE APPLICATION OF ALGEBRA TO THE THEORY
OF CURWES.
(I.) Geometrical magnitude may be represented by
the characters of Algebra. -
For let A and B be any two straight lines which are
to each other as a . 1, a being an abstract number.
Then A = a B, or if B be taken = 1, A = a, that is,
the straight line A is represented by the algebraic cha-
racter a. The line B thus assumed equal to unity is
called the linear unit.
Similarly, if the square and cube described upon B
be taken as the respective units of surface and solidity,
any abstract number which expresses how often either
of them is contained in any proposed surface, or solid,
may be conceived to represent the surface or solid
itself.
Hence, if a, b, c represent any three straight lines,
a × b will represent a rectangle whose area is a b
times Bº, and a × b x c will represent a rectangular
parallelepiped whose solid content is a b c times B",
(2.) A variable quantity in Algebra may be repre-
sented in Geometry by an indefinite straight line.
Let a be any variable quantity, and X X" an indefi-
nite straight line, (fig. 1.)
In XX' assume A as the point from which the lines
are to be measured. Then any finite portion A P may
be taken to represent a given value of w. Thus, if the
point P fall upon A, the distance A P will correspond
to a = 0; and by increasing A P we may evidently
represent all the determinate values of r.
It is immaterial whether the values of a be measured
to the right, or to the left of the point A, since the line ex-
tends indefinitely in both directions. But if we begin
to measure the positive values of a to the right, then
the negative values must be measured to the left, of A.
To illustrate this, let A' be the point from which the
VOL. I.
values of a second variable a' are to be measured:
take
A/P = a, A'P' = a-', and A P as before = r.
Then a' = a + ar,
Or a = a – a.
Now if a 'be positive, and less than a, the values of r
will plainly be negative ; but in this case the point P
falls to the left of A, as at P'; hence the negative
values of a ought to be measured to the left of A.
We may therefore lay down this general rule, “When
distance is to be estimated from a fixed point, along a
straight line given in position, if the positive values
of any quantity be measured in either direction from the
fixed point, the negative values must be measured in
the opposite direction from the same point.”
(3.) The application of Algebra to the theory of
curves is founded on this principle, that an indeter-
minate equation between two variables is capable of
being represented by a geometrical locus, and con-
versely.
Let f(x, y) be any indeterminate equation between
v and y; a any arbitrary value of a, and b the cor-
responding value of y.
Draw two straight lines AX, AY of indefinite length,
at right angles to each other, and meeting in A, (fig. 2.)
In A X take A M = a, and in AY, A N = b ; through
M and N draw M P and N P parallel respectively to
A Y, AX, meeting in P; then the point P corresponds
to the solution of the proposed equation.
Since the equation admits of an unlimited number
of solutions, the points P furnished by each solution
will also be infinite in number ; and their assemblage
will therefore form a certain line, straight or curved,
which is called the locus of the equation f(x, y) = 0.
When the equation admits of only one solution, it
represents a point; and when it has no real solution, it
indicates an imaginary curve.
(4.) Of the two quantities a and b which represent
A M and M. P, the former is called the abscissa, the
latter the ordinate of the point P; they are both in-
cluded under the general appellation of the coordinates
of that point.
The lines A X, A Y are called the aves, and the point
A the origin, of the coordinates. We have supposed,
for simplicity, that the axes are at right angles to each
other, or rectangular ; but they may have any inclina-
tion whatever.
When the point P is not given, its coordinates are
represented by the letters v and y, of which the former
denotes an abscissa measured along A X, the latter an
ordinate measured along AY. Hence, A X is usually
called the axis of ar, and A Y the axis of y.
If a point be situated on the axis of z, then y = 0;
if on the axis of y, then a = 0; and if it coincide
with the origin, then a and y each = 0.
By applying the conventional rule with respect to
the signs laid down in Art. 2, it is evident, that if the
4 z 709
Part l,
2.
710
A N A L Y TH C A L G E O M ET. R. Y.
Analytical
Geometry. values of a to the right of A be supposed positive,
those to the left of A must be considered negative.
TT. In like manner, if the values of y measured along
A Y be positive, those in the direction A Y’ must be
reckoned negative.
(5.) Let a curve be now supposed to be traced
upon a plane, then, reversing the process by which the
locus of a given equation is determined, we may deduce
from some known property of the curve, the relation
subsisting between the coordinates of any one of its
points. The equation which expresses this relation,
supposing it to be the same for every point, is called
the equation to the curve.
It is convenient to distinguish curves by the generic
appellation of lines. They are divided into orders,
according to the dimension of the equation by which
they are represented.
Thus, a line of the first order is the locus of the
equation
a y + b a + c = 0.
A line of the second order is the locus of the equation
a y” + b a y + c a' + d y + e c -- f = 0;
and so on.
(6.) The position of a point upon a plane may be
determined in a manner somewhat different from that
which has just been explained; namely, by means of
its distance from a given point, and the angle which
that distance makes with a line given in position. The
given point is called the pole, and the variable distance,
the radius vector.
Thus, referring to fig. 2, A is the pole, and A P the
radius vector r. The angle which A P makes with the
line A X given in position, is usually denoted by w.
The quantities r and w are called polar coordinates,
and the equation which expresses the relation subsist-
ing between them at any point in a curve, is called the
polar equation to the curve.
ON THE STRAIGHT LINE.
(7.) To find the equation to a straight line.
Let B Z be a straight line of indefinite length,
and suppose it referred to the rectangular axes A X,
A Y, (fig. 3.)
Assume any point in it P, draw PM parallel to A Y,
meeting A X in M ; through B draw B Q parallel to
A X, meeting PM in Q.
Let A M = a, M P = y, A B = b.
Now in the right angled triangle B QP,
PQ sin P B Q
QB T cos PB Q
... PQ = Q B tan P B Q.
= tan P B Q ;
But y = MP = M Q -- Q P,
- = A B + QP,
and •. -- b + Q B tan P B Q.
Now as the position of BZ, with respect to AX, is
supposed to be given, the angle P B Q or PC X will
be known. Hence, denoting tan P B Q by a, we have
Ay = a + -- b.
The same relation may be shown to subsist between Part I.
the coordinates of any other point in the line. Hence S-V-
the equation required is
3) = d a -j- b.
(8.) Cor. 1. When the straight line passes through
the origin, b = 0; therefore the equation becomes
- 3y = a T.
(9.) Cor. 2. The general equation of the first degree
between two variables is
| A y + B a + C = 0;
or dividing by A, and transposing,
B C
y = --X w. --R.
B C
Let - —- - ~ -— aſ
€ R = a --R = 0,
then g = a a + b ,
which coincides with the equation deduced in the last
article. Whence it appears, conversely, that the locus
of the general equation of the first degree between two
variables is a straight line.
(10.) To draw the straight line which is the locus of
any given equation of the first degree.
Since two points serve to fix the position of a
straight line, it will only be requisite for the solution
of the proposed problem to find in each case the points
in which the line meets the axes. This will be done
by making a and y successively e 0 in the general
equation.
The equation in its most general form is
y = + (a a + b).
1. Let gy = a a -- b.
Then if a = 0, y = b,
b
if = 0, a = — — .
and i 3/ , Jº Q.
Hence in A Y (fig. 4) take A B = b, and in X A pro-
Fig. 4.
duced, A C = #. join C, B, then C B Z is the line
required.
2. Let y = a a - b.
Then if a = 0, y = — b,
and if
y = 0, a = Ta
Hence in YA produced take A B' = b, and in AX,
b e
A C' = a join B', C', and C' B' Z' is the line re-
quired.
3. Let
y = — a a + b.
Then, as before, take A B = b, A C' = # , and the
line required is C/B Z.
4. Let
If A B' be taken = b, and A C =
quired will be CB' Z.
(11.) The general equation to a straight line is y =
a r + b, which involves two constants, a and b. The
equation will therefore occur under various forms, cor-
gy = — a a - b.
b
—, the line re-
(7.
A NAL YTI C A L G E O M ETR Y.
7II
Analytical responding to the conditions which serve to determine
Geometry. , these constants. Now a straight line is given in posi-
--~~' tion when it passes through two given points, or when
it passes through one given point, and makes a known
angle with another straight line. We shall investigate
the form of the equation in each of these cases.
(12.) To find the equation to a line which passes
through two given points.
Let the coordinates of the given points be a ', y',
and a ", y”; then the general equation is
gy = a a + b . . . . (1,)
and since the given coordinates must satisfy this, we
also have
Ay' = a a' + b . . . . (2,)
and y" = a r" + b . . . . (3.)
Subtracting the second from the first, and the third, suc-
cessively, we have
Sy – y' = a (r – a ') . . . . (4,)
and gy" – y' = a (w" — a ') . . . . (5,)
y” — y'
but ſrom (5,) a = +7—#7-; substituting this in (4)
Jº — ſº
we have for the equation sought
y” —
y-y' = + , (, – 1).
which may easily be reduced to the same form as the
last.
Cor. Equation (4,) y – y' = a (r. — r")
is the equation to a straight line passing through one
given point (r', y'); in which the coefficient a is inde-
terminate, since an indefinite number of lines can be
drawn passing through the same point.
(13.) To find the equation to a straight line which
passes through a given point, and makes a given angle
with a given straight line.
Let the equation to the given line be y = a x + b,
then the form of the equation required will be
(Art. 12)
Ay — y' = a' (a – a ') . . . . (1,)
in which a' is to be determined.
If A M, A N be drawn through the origin parallel to
the two lines, (fig. 5,) the angle contained between them
is M A X – N A X ;
. . . tan M A N =
Fig. 5
tan M A X – tan N A X
1 + tan MAX tan N A X
or assuming tan M A N, which is supposed to be
given, F m,
a — a
1 + a aft'
ºn + m a a ,
(1 + a m) a',
G - 771
1 + a m '
hence, by substitution in (1,)
, a - m
y-y = THz
which is the equation required.
(14.) Cor. 1. If the two lines are perpendicular to
770, Ec
. . . a - a' =
... " . CL - 777, E
‘. a' =
(a — ar'),
tº € g l
each other, m is infinite, therefore a' = — Ta’ and the
Part 1.
--~~
equation becomes
I
— ar' = — — emº
gy — y + ( – ’).
(15.) Cor. 2. If the two lines are parallel, then m = 0,
therefore a' = a, and the equation becomes
y – y' = a (a – a ').
Observation. If p, p' denote the angle of intersection
of two lines p and p", whose equations are
y = a a + b,
and gy = aſ a + b/;
we then have
a — aſ -
t , p = ——- . . . . ( 1.
an p, p" 1 + a a/ (1.)
In like manner,
tº a — aſ
, p' = . . . . (2,
in P. p = 7&i Hºyºſ Hºo (2)
and
I I a'
cos p, p' = + + a (3,)
A/ (1 + a”) (1 + a”)
in which the positive sign is to be used when the angle
is acute, and the negative when it is obtuse.
(16.) The equation to a straight line may be expressed
in terms of the perpendicular let fall upon it from the
origin.
Let p represent this perpendicular, then if the sym-
bols p, r and p, y be taken to denote the angles which
the straight line forms with the axes of w and y, we
have p = b sin p, y = b cos p, ar,
. . . b = p 3.
COS p, a
therefore, by substitution in the general equation,
Q) = a a + b,
we have y = a + + p 3.
COS p, q}
sin p,
but a = tan p, a = ºn p. r.
COS p, a
__ a sin p, a -- p
3/ cos p, a '
. . . . p = y CoS p, a - a sin p, a . . . .
which is the equation sought.
Again, if the angles which p makes with the axes be
denoted by p, a and p, y, we have
(1,)
cos p, a sº sin p, ar,
sin p, r = — cos p, ar,
. . p = y sin p, a + a cos p, v.
(17.) To find the length of the perpendicular let fall
jrom a given point upon a given straight line.
Let P be the given point, (fig. 6,) B Z the given line,
and PQ the perpendicular whose value (p) is to be
expressed. •
Let the coordinates of P be a ', y', and the equation
to the given line
Fig. 6.
y = a a + b.
Through P draw PR parallel to BZ, and from A let
fall the perpendicular Ap upon R, meeting Q B pro-
duced in q.
Then, since P R is parallel to B Q, the equation to
4 z 2
712
A N A LY T I C A L G E O M ET R Y.
Analytical
Geometry.
P R is y = a a + b', (15,)
, and since (r', y') is a point in it,
Fig. 7
y' = a x' -- b',
whence, by the last article,
A p = y' cos p, a - a 'sin p, a ;
A q = b cos p, a 5
. . . A p — A q, or p = (y' - b) cos p, a - a 'sin p, t,
but
sin p, a
= COS p, *{y-º-º: y)
- f), @
= cos p, a 4 y’ – a c' – b )
|
y’ – a c' – b
= . . . H −.
vºl -i- dº
b gº) Cor. 1. If the line pass through the origin,
gy' m== a ar'
v 1 + a”
(19.) Cor. 2. If the given point be the origin, then
a' and y' = 0', and
‘. p = +
p = + —— ,
v 1 + a”
according as the line is situated below or above, the
axis AX.
(20.) To find the analytical value of the line which
joins two given points.
Let P and Q be the given points, (fig. 7,) join
them. Draw PM, Q N parallel to A Y, and P R
parallel to AX, meeting Q N in R.
Let the coordinates of P be ac', 'y', and those of Q
, y", and assume PQ = r.
Then in the right angled triangle PR Q,
PQ? = P R2 –– R. Q”,
M N* -- R Q2,
(AN — AM)* + (NQ — PM)”,
. . . (c" sm a/), + (y" * - y')*,
... " .. 7° rºt + W { (p"— a')* + (y"— 9')*},
which is the expression required.
(21.) Cor. If either point coincide with the origin,
the coordinates of that point will = 0, and the above
expression will be simplified.
Then let P coincide with A, then a ' and y' = 0, and
r or A Q = yzſº + y”.
(22.) To find the coordinates of the point of inter-
section of two lines.
Let the equation to the first be y = a a -i- b. . . . ( ..)
that to the second y = a' a + b'. . . . (2.)
Two lines which cut each other have evidently the
same coordinates at their point of intersection. The
coordinates sought will therefore be found by supposing
r and y to have the same value in both equations, and
a"
*ms
*=e
*=as
~
then eliminating them.
Hence subtracting (2) from (1) we have
b -- h"
a — a ''
(a – a') a + b – b = 0,
. . . T = —
*
'b — b'
. . . y which = a x + b, a 0 – 0 a.
y
ſ! - 0,
hence the coordinates required are found.
When the lines are parallel, the coordinates are infi-
nitely great, therefore the denominators of the above
fractions = 0, or a = a', which is the condition of
parallelism already established in (15.)
(23.) For the sake of simplicity we have hitherto
employed rectangular axes, but it is frequently con-
venient to suppose them inclined at any angle what-
ever. We shall therefore present in a tabular form the
foregoing results adapted to this hypothesis, leaving the
investigation to be supplied by the reader.
1. The equation to any straight line is
gy e. a a + b,
sin p, a
sin p, y
in which a =
, and b = the ordinate drawn
at the origin.
2. The equation to a line passing through one given
point (a!", y') is
y – y' = a (a – a ').
3. The equation to a lime passing through two given
points (r', y') and (a", y") is
*/
3/
gy — y' = a" —
y'
p' (a — a ').
4. If p, p' denote the angle of intersection of two
lines whose equations are
\ 3/
3/ -:
Then tan p, p' = sin r,
a r + b,
a'a, + b'
a — a'
9 I + (a + a') cos r, y + a d"
5. The equation to a straight line drawn through a
given point (r', y') at right angles to a given line
y = a a + b is
y – y' = –
tºms
1 –– a cos a, iſ
a-F cost, y
When the lines are parallel,
y – y' = a (r – a '),
as in the case of rectangular coordinates.
(a — ar').
6. The equations to a straight line in terms of the
perpendicular dropped upon it from the origin, are
(1.) p = y sin p, y - a sin p, ºr,
(2.) p = y cos p, y + a cos p, a.
7. The value of the perpendicular (p) let fall from
a given point (r', y') on the line y = a a + b, is
p = (y'— a w' – b) sin p, y . . . . (1,)
= (y – b) sin p, y – a 'sin p, a . . . . (2.)
8. The analytical value of the distance (r) between
two given points (a', y') and (a", y") is
r = W { (t"– a)* + 2 (r" — wº) (y" – y') cos r, y
+ (y" – y'); };
and when the point (r', y') coincides with the origin
r = y +” + 2 w"J" cos r, y +y”.
(24.) We shall now exemplify the principles laid
down in this chapter, by applying them to the follow-
ing propositions:
OT
Part I.
A N A L YT I C A L G E O M ET. R. Y.
713
Analytical
Geometry.
\-N-"
1. Required the equation to a line which bisects the
angle contained by two given lines.
Let A X, A Y be rectangular axes, (fig. 5.)
Draw through the origin two lines A M, A N pa-
rallel respectively to the given lines, then M A N is
the angle to be bisected. Let A P be the line whose
equation is required.
Suppose the equations to A M, A N, A P respectively
to be
in which m is to be determined.
The tangent of the angle PA N =
y = a ar, y = aſ a, and y = m a.,
m — aſ
1 + m a'
(, – ...???,
The t t of the angle PAM = —
e tangent of the angle 1 + m a '
but these being equal, by hypothesis,
ºn – a' a — m
i-Im a T T-E ºn a
m + m2 a - a'— m a. a' = a + m a. a' — mºa',
... m” (a + a') + m (2 — 2 a. a') — (a + a') = 0,
1 — a a!
TTE a aſ
Whence two real values of m may be found. Two
lines therefore may be drawn, one of which bisects the
angle itself, the other its supplement.
The equations of these lines are
Or
".. m” – 2. 7m — 1 = 0.
3/ = 7m ſc,
and y = - 1 r.
therefore they are at right angles to each other.
2. Through a given point P in a given angle Y A X to
draw a line MN, such that the triangle so cut off may
be of given area, (a”,) (fig. 8.)
Assuming A X, AY as oblique axes, draw PQ
parallel to A Y, and let A Q = a-', Q P = y'; and
A M, which is unknown, - c.
Then the equation to M N is
Fig. 8.
_ _Sin P, & (r. — ºr,)
Sin p, 3/ * / 12
J
3/
- ... —— (JG – J.
a' — a ( D
y’
Let now a = 0, ... y or AN = -2 ºn
– w/
... area of the triangle A M N = }; A M . A N sin A
a. * */ tº
= } – 4 *— sin A = a”, by the question,
a — a
... a y' sin A = 2 a” r" – 2 a” r,
... a,” y' sin A + 2 a” a, = 2 a” r",
2 a.2 _ 2 a w'
y sin A. *1 =
whence a value of a) may be obtained.
If it were required to draw M N such that A M may
= A N,
2
". Tº —H·-– ,
f y' sin A
/
tº
We should then have — g *— = r),
a — v.
..". - y’ = r" mºm d'ſ,
Or Q P = Q M.
3. If the sides of a plane triangle be bisected, and
lines be drawn from the points of bisection at right
angles to the sides, they will meet in the same point.
Let the lines M m, N m, Pp, be drawn at right
angles to the sides of the triangle A B C from the
points of bisection M, N, P, (fig. 9,) these lines will
intersect in the same point.
The triangle being referred to rectangular axes A X,
A Y, originating at A,
Let the coordinates of A be a ', y', and those of
f f
B, a ", 0; then the coordinates of N will be #. #.
f ſy f
and those of P Jº + 2 %
In order to prove the proposition we shall find the
ordinates of the points in which the lines N m, Pp
meet M m ; and shall then show that these ordinates
are identical.
Now N n being drawn through the point N (#. #)
its equation is of the form
f 2/
J.
rº
3/
9 - 5
but N m is supposed to be perpendicular to A C, whose
! #
inclination to A X = tan" ". . . . & F
g g $y' — aſ a' \
the equation to N n is y — g = Ty * - a J. . . (l.)
Similarly, the equation to P. p is
Sy’ * a"— ac' a"+ a' º
Let N m, P. p be now supposed to meet M. m. ; in
which case, æ in both equations will become A M or
q,"
g-; making this substitution therefore, we have for
a'
~7 . . . .
3/
the ordinates at the point of intersection,
y' A.
a' a' — aſ
y - a = y’ 2
in the first case, and
$y' a" — aſ
y – 2. sº. — /
3/
in the second ; but these values are evidently the same,
therefore N n, Pp, and M m, intersect in the same point.
On precisely similar principles the two following
theorems may be proved :
(i.) The perpendiculars let fall from the angular points
of a triangle on the opposite sides intersect one another
in the same point.
(ii.) The lines drawn from the angles of a triangle to
the middle points of the opposite sides intersect in the
same point.
4. To prove, by reference to the figure of Euclid, I. 47,
that the lines A M, N B, and C P, meet in the same
point, (fig. 10.)
From M and N let fall the perpendiculars M m, N m,
on A B produced; let the figure be referred to rectan-
gular axes originating at A, the axis of a being supposed
to coincide with A. B.
Let the coordinates of C be a ', y'; then if A B = r",
P B will = a,” – a '.
... aſ
f
|
* By the expression tan- º, is meant the angle whose tangent is
J.'
Part I.
Fig. 9.
Fig. 10.
7 14
AN A LYT I C A L G E O M ETR Y.
Analytical
Geometry.
<-y-
Fig I l, 12.
Fig. 13.
Now the triangles A n N and A P C being evidently
equal, A n = C P = y', and N n = A P = a'.
Similarly, B m = y' and M m = a," — a '.
M. m.
... A * =
N
B N isy = - . ( – ’) =
a" — ar'
7-Ey” te (1)
Again the equation to AM is y =
and that to
a'
~ –Ziy ( – ’)...(?)
Now let A M and B N meet C P; in which case, w = wº
in each equation.
" — aſ
f
7T, ".
a' ** t"
and (2) 9 = — ºr—H a',
a" + y
which are identical, therefore the three lines meet in the
same point.
Then (1) becomes y =
ºmsºmºsºmsºmº
ON THE TRANSFORMATION OF COORDINATES,
(25.) The position of a point with respect to a given
system of ares being known, to find its position when
referred to a new system of awes parallel to the former.
Let P be the point, A X, A Y the old, A/X', A'Y'
the new, axes, (fig. 11 and 12,) draw PM parallel to
A Y meeting A/X' in M', and produce X'A', Y'A'
to meet A. Y., A X in the points C, B.
Let the coordinates of P when referred to A X, A Y
be a, y, and when referred to A/X', A’Y', a ', y!; also
assume AB = a, B.A' = b.
Then M A = M B + B A
= M'A' + B A,
a = a + a
y = y' + b . . . . (1.)
Hence if f (a, y) = 0 be the equation to the point
P when referred to the old axes, we have only to sub-
stitute in it for a and y the values just obtained, and
we shall have the equation to P when referred to the
IleW axe S.
The signs of a and b will depend on the position of
the new origin A'; this circumstance being attended
to, the formulas (1) are quite general.
Or
Similarly,
(26.) The position of a point with respect to any
system of aires being known, to find its position when
referred to any other system originating at the same
point with the former. -
Let A X, A Y be the primitive,
A X', A Y', the new axes, (fig. 13;)
r, y the coordinates of the given point when referred
to the former, a ', y' its coordinates when referred to the
latter. -
It may be remarked in general, that whatever be the
axes to which a curve is referred, the nature of that
curve must remain unchanged; since the object for
which axes are employed is merely to determine the
relative position of the points of any line. Hence it is
evident, that in passing from one system of coordinates
to another, the new ones must be linear functions on
the old ones; for, otherwise, the degree of the equation
by which the curve is represented, and therefore the
nature of the curve itself, would be altered.
We shall assume, therefore, that the relation between
the old and new coordinates may be thus expressed,
a = m a' + n y! (l,)
$/ I. Tº • * * V + 2
m, n and mº, n' being independent of either system of
coordinates.
In order, therefore, to determine these quantities:
Let y' = 0, in which case the point will be situated on
A X', as at P; draw PM parallel to A. Y.
Part I.
and
a A M sin a'
Then a = m a.', or m = H, = +H = H • 3/
a' A P sin w, y
PM sin a', a.
and m' = 9. = −3 = +.
Jº A P sin w, y
In like manner, by supposing a = 0, we obtain
o : tº f
SIIl SII] TV". Jº
= ** ***, and n = ** * * *;
Sln r, ly Sin ar, 3/
hence, substituting in (1) for m, n and m', n' these
Talues, we have
l
sin r, y,
I
sina, y,
{ aſ sin a', y + y' sin y', y |
{ a 'sina', a + y' sin y', a 3.
3) =
If, therefore, these values of a and y be substituted
in f(x, y) = 0, the equation to the point P when re-
ferred to the new system of axes will be found.
The general problem being thus resolved, we shall
consider the following particular cases: -
I. Let the primitive axes be rectangular, and the
new ones oblique.
Then sin r, y = 1
e f ſº 7-
sin a', y = Sin # - ", , = cos ar', r
sin y' y = sin (; + y', a j = cosy', a ;
therefore the formulas to be used in this case, are
w = a 'cos a', a + y' cos y', a
W = a 'sin a', a + y' sin y', a.
2. Let both systems be rectangular.
Then these formulas become
a = a 'cos w!, a - y' sin a', a
Ay = a sin a , a + y' cos a ', a.
3. Let the primitive axes be oblique, and the new
ones rectangular.
Then sin y', y = cos ar', y
sin y', a = cos aſ, a ,
therefore the general formulas become
I
sin a y
{a/sin a', y + y'cos wº, y :
3) = { a 'sin a', a + y' cos a', ºr ; .
sin r, y
If the origin and direction of the axes be both
changed at the same time, we have only in the preced-
ing formulas to add the new abscissa to the value of
z, and the new ordinate to the value of y.
A N A L YT I C A L G E O M ET RY
715
Analytical
Geometry.
-N/~!
Fig. 14.
ON THE CIRCLE.
(27.) To find the equation to the circle
Let D P be a circle, (fig. 14,) P any point in its cir-
cumference ; and let it be referred to the rectangular
axes A X and A. Y.
Let A B, B C be the coordinates of the centre
C, and A N, N P those of the point P.
Let A B = a ', B C = y'
A N = p, N P = y, and C P = r ;
then C P2 = C M2 + M P2
= (A N – A B)* + (PN – BC)*,
therefore, by substitution,
r” = (a — a')*-ī- (y – y')*,
which is the equation required.
(28.) This equation may be simplified as follows:
1. When the axis of a passes through the centre,
then y' = 0, and the equation is
g” + (v – a ')* = rº.
Similarly, when the axis of y passes through the centre,
* + (y – y) = r".
2. When the origin is on the circumference, then
a" + y^* = r", and the equation therefore becomes
* + y” – 2 a. a' — 2 y y' = 0.
3. When the origin is on the circumference, and
either axis passes through the centre,
a? -- y” — 2 r a = 0,
OI’ a’ + y” — 2 r y = 0.
* When the origin is at the centle, a ' and y' both
9 ... a 2 + y^ = re.
(29.) The general form of the equation to the circle
when referred to rectangular coordinates is
a" + y? --A a + B y + C = 0.
Let it now be required to assign the position and
magnitude of the circle to which this belongs.
Comparing it with the general equation
(a — a ')2 + (y – y')* = rº,
that is, with
a * + y” — 2 a. a' – 2 y y' + æ" + y” — r" = 0
we have
A
A = — 2 a.', or a ' = — –
2
B 5
B = – 2 ', ' = — —
3/ 3/ 2
therefore the coordinates of the centre, in other words,
the position of the circle, is known.
Again, C = a " + y” — r",
... r" = a " + y” – C = } (A* + Bº – 4 C),
... r = 4 MA. H. Bº – 4 C,
therefore the value of the radius, or the magnitude of
the circle C is found.
ExAMPLE.
Find the position and magnitude of the circle whose
equation is
3
gy* -- tº + 2 y – g º – 5 =0.
Comparing this with
9° -- alº – 2 y y' – 2 a. a' + w(2 + y” — ri = 0
#
We have y' = – 1
a' 3
vºnºmy 4 *
* + y”—r = -4.
2 ”
9 l
**s __ == 2
+ 15T a 3.
v 33
, " .. 7" E —
4
Part I.
Assume A X, A Y as axes, (fig. 15,) A B = # and Fig. 15
B C = — 1, then from C as centre with rad = the
3 © g g
nearest whole number to describe a circle, and it
will be the circle required.
(30.) The general equation of the second degree
between two variables is
A y” + B w”-- C w y + D y + Ea' + F = 0,
which differs from the general equation to the circle. It
will afterwards become an important inquiry, to ascer-
tain what class of curves is represented by that equa
tion.
(31.) To find the polar equation to the circle.
Let any point S within the circle be assumed as the
pole, and draw through it S Z parallel to C X, meeting
M P in N, and let the ordinate of S meet C X in Q.
Let S P = p, angle P S Z = w, and the coordinates
of S, a and b.
Then a = a + p cos w,
and gy = b + p sin w.
Therefore by substitution of these values in the equa
tion
as + y^ = rº,
there results
(a + p cos w)” + (5 + p sin w)* = rº,
. . . p” -- 2 (a cos w -– b sin w) p + a” + bº — r" <= 0,
which is the equation required.
If the point S be without the circle, then the polar
equation is
o” – 2 (a cos w -H b sin wy p + a” -- bº — r" = 0.
(32.) To find the equation to a tangent applied at a
given point (a', y') of a circle.
Let the equation to the circle be tº + y' = r".
Now the equation to any line is
p = a cosp, a + y cos p, y.
Let the line touch the circle at the point (r', y) then
p = r, and
y
g 3/
= — and , a = –,
COS p, a ; an m p r
aſ a: ſ
• ?" → ** + 99, or
7" r
ra' + y y' = rº,
which is the equation required.
716
A N A L. YT I C A L G E O M ET. R. Y.
Analytical
Geometry.
'-SA-Z
(33.) If the given point be without the circle, let a, y,
be the unknown coordinates of the point of contact.
Then since the point (ro, yo) is on the circumference
a,” + y,” = r" . . . . (1,)
and since the point (a', y') is on the tangent
a'a, + y'y, = r" . . . (2,)
thereforea, y, may be found by elimination between these
two equations. This process, however, which is tedious,
may be superseded by an operation founded on the
principle, that elimination between any two equations
corresponds to the intersection of the geometrical loci
which they represent.
The points of contact are therefore determined by the
intersection of the loci whose equations are (1) and (2).
But the locus of (1) is the given circle, and the locus
of (2) is a straight line; and since the points in which
it meets the circle are the points of contact, equation (2)
7must be the equation to the line joining those points.
Its position is thus found:
7.2
Let *E*
f
&mmer
*-ºs
a’, E 0, . . . yo = A C, (fig. 17,)
7.2
gy, = 0, . . a, = − = A B.
J.
Join B, C meeting the circle in Q, P; these are the
points of contact required.
(34.) The points of contact may be found in a dif-
ferent manner, as follows:
Subtracting (2) from (1) we have
gy,” — y, y' + æ,” — a, a = 0 . . . . (3,)
which (art. 28) is the equation to a circle, the co-
g a' vº gº
ordinates of whose centre are 75' #, and whose radius
= , Virº-Fºy”. Hence the locus of (3) is the equa-
tion described on C T as a diameter; and its intersec-
tion with the given circle determines the points of
COntact.
This is the construction of Euclid, iii. 17.
(35.) To find the equation of a common tangent to
two circles. -
Let 3 be the distance between the centres of the two
circles, r and r’ their radii, and suppose the axis of a to
pass through the centres of both circles.
Then the equations to the circles are,
a” + y^ = r". . . . (I,)
r” – 3)2 + y” = r^2 . . . . (2.)
The equation of the tangent to the first is
a c' –H y y' = r" . . . . (3,)
and in order that this line may touch the second circle
also, the perpendicular dropped upon it from the cen-
tre must = r".
a 3 — b
Now the length of this perpendicular = — — —
- v'1+ a”
A 2
—- ſº r
but a = −7- and b = -7,
3/
* * =s* 3 w” — r" 3 vſ – r2
ºp=====<---
8 r" – ?"
º fººmsº * * — *N &s
r = – –F–. r r = — 3 a' -- rº,
_ 3 r – (r – r") a
T v 4 » – (r-7),
/
(r — r)
. . . dº :-
* . $/
;
hence by substitution in (3) º (r. — r") + y y' = rº,
which is the equation required.
(36.) If the axes to which the circle is re'erred be
inclined at any angle whatever to each other, then
l. The general equation is
rº – (c. — r")* + 2 (a – a ') (y – y) cosa, y + (y – y')*,
and when the centre is the origin,
r° = a + 2 a y cos ar, y + y”.
2. The equation to the tangent drawn at a given
point (w', y') of the circumference is
{ y' -- a 'cos r, y | y + 4 a.' + y' cos a, y | r = rº,
the origin being at the centre.
ON LINES OF THE SECOND ORDER.
(37.) The general equation of the second degree
between two variables is,
a y” + b a y + ca" + d y + e a +f- 0,
in which a, b, c . . . . are independent of a and y.
The locus of this equation is called a line of the
second order. In the following investigations we shall
use oblique axes, unless the contrary be specified.
The characteristic property of a line of the second
order is, that a straight time cannot intersect it in more
than two points. To prove this,
Let the curve be supposed to be cut by a straight
line whose equation is
gy = m r + m . . . . (1,)
then the points of intersection will be determined by
eliminating y between this equation and the general
equation
a y” + b a y + c a' + d y + e a + fe 0. . . . (2.)
Hence, substituting in (2) the value of y derived from
(1,) we have
a (m a + n)* + ba (m a + n) -- ca" -- d (m a + n)
+ ea + f = 0,
or developing the terms, and arranging the result
according to the powers of a,
(a m” + b m + c) tº + 3 (2 a m + b) n + d m + e r
+ a n” + d n + fe- 0.
This equation being of the second degree can have only
two roots, which, when real, represent the abscissas of
the points of intersection. Whence it follows, that a
straight line cannot cut the curve in more than two
points.
If the roots be imaginary, the straight line does not
meet the curve ; if they be equal, the two points of
section coincide, and the line touches the curve.
Definition. A straight line being supposed to cut a
line of the second order, the portion of it contained
within the curve is called a chord.
(38.) To find the locus of the middle points of any
number of parallel chords.
Let Q P q (fig. 18) be conceived to represent a
portion of a line of the second order; and let it be
referred to any oblique system of axes A X, A.Y.
Part [.
Fig. 18.
AN A LYT 1 C A L G E O M ETR Y.
717
Analytical
Geometry.
Through the origin draw any line A Pp, cutting the
curve in P, p ; then its equation will be of the form
$y = ma . . . . (1)
Let Q q be any chord parallel to A Pp, bisect it in O,
and draw O M parallel to A. Y.
Then the object of the problem is to determine the
relation between A M and MO, the coordinates of the
point O.
Assume A M = w, M O = y'.
Let the origin be now transferred to O, in which case
we shall have to substitute a + æ' and y + y' for a and
gy in the general equation ; we have therefore
a (y-º-y)" -- b (p + x') (y-- y’) + c (x + 1)*
+ d (y – y') + e (a + æ") + f2– 0 . . . . (2,)
which is the equation of the curve. -
Now when the origin is at A the equation of Q q
which is drawn parallel to A Pp is y = 'm a + m', but
when the origin is removed to O, the equation of Q q
will (Art. 8) be
3y = m a.
Hence, the points in which Q q intersects the curve
will be found by eliminating y between this and equa-
tion (2,) whence we have
a (m a -i- y’)* + b (a + æ") (m a + y') + c (x + æ')
+ d (m a + y') + e (a + r") + f2- 0,
which becomes on reduction (a m” -- b m + c) tº
Fig 19.
+ { (2 a m + b) y' + (b m + 2 c) a' + d m + e a
-j- a y” + b a' y' + c c”—- d y' -- ea' +f- 0.
But since Q q is bisected in O, it is plain that the roots
of this equation are equal, with contrary signs; there-
fore the coefficient of the second term must = 0.
Hence, suppressing the accents which were only
employed to distinguish the coordinates of O from
those of any point whatever, we have -
(2 a m + b) y + (bºrn + 2 c) a + d m + e = 0 . . . (3.)
The relation between a and y being thus expressed
by an equation of the first degree, it follows, that the
locus of the point O is a straight line.
The straight line which bisects any number of
parallel chords is called a diameter, and each of the
points in which it meets the curve is called a verter.
(39.) Cor. If the equation to any other chord be
y = m/a,
then the equation to the corresponding diameter
will be
(2 a m' + b) y + (5 m' + 2 c) r + d m' + e = 0.
Draw any two chords m n, p q, (fig. 19,) and their
corresponding diameters M. N., P Q ; then if either
chord be parallel to the diameter of the other, reci-
procally the diameter of the first will be parallel to the
chord of the second.
For if y = m r + n be the equation of m n,
and gy = m/a-H n' that of p q,
- b m + 2 c d m + e
then * T 2 a.m. -- b T 2 a.m. --5
will be the equation of MN, and
b m' + 2 c d m' + e
gy = - 2amſ-ES*T 2 a.m/+ b
that of PQ.
V O L. l.
Let m n be now supposed parallel to PQ,
b m' + 2 c -
2 a m' + b'
... 2 a m m' + b m = — b mſ — 2 c,
b m + 2 c
2 a m + b”
whence p q is parallel to M N, (Art. 15.)
In like manner, if p q be supposed parallel to MN,
it may be shown that PQ will be parallel to m n.
Whence it appears that each diameter bisects the
chords drawn parallel to the other. Diameters thus
related to each other are called conjugate diameters.
If y = m, a + n be any diameter, the equation of the
diameter conjugate to it is
bn + 26, d m + e
2 a m + b 2 a m + b”
whence it is evident that an infinite number of pairs of
conjugate diameters can be drawn.
We shall now investigate whether any of these systems
can be at right angles to each other.
Suppose, for the sake of simplicity, that the axes are
rectangular, and let *
gy = ma + n,
then 777, it
Or mſ =
gy =
gy = m'a + 'n'
be any system of conjugate diameters.
m' = — b m + 2 c
2 a m + b”
and since the conjugate diameters are by hypothesis at
right angles to each other,
l -
m' = — -, (Art. 14,)
772.
Then
b m + 2 c I
2 a.m. Th T nº
‘. b m” + 2 c m = 2 a m + b,
C - Cº,
b
C – C. v/ c - a N2
I 3.
fºr vi +(**)
a quantity which is manifestly always real.
Let m and A be the two roots of this equation; then,
since its last term = — 1,
... m” + 2
7m = 1,
". 772. E –
7m X A = – 1,
I to /
... a = –1 = ... m.,
hence it appears that m and m' are the roots of the
same quadratic : wherefore there can be only one system
of rectangular conjugate diameters.
These are called the principal diameters.
(40.) To find the form which the equation to lines
of the second order assumes, when the awes of coordi-
mates are parallel to a system of conjugate diameters.
Let y = 'm a + n be the equation of any chord,
then -
b m + 2 c d m + e
— Jº — —
2 a m + b 2 a m + b
will be the equation to its corresponding diameter.
3y = —
Part I.
5 A
718
A N A L YT I C A L G E O M ET R Y.
Analytical
Geometry.
Suppose now that the chord is parallel to the axis of
a ; then m = 0, and the equation of the diameter be-
COIſles
2 C 6
y = --- * – H-.... (1)
Again, let the chord be parallel to the axis of y, then
m is oo, and the equation of the corresponding diame-
ter is
. . . . (2.)
Hence, when these diameters are conjugate to each
other, and the axes are parallel to them, the first
will bisect the chords parallel to AX, and ought there-
fore to involve a alone; and the second ought, for a
like reason, to involve y alone ; therefore in each
case b must equal 0. -
Hence, when the axes of coordinates are parallel to
a system of conjugate diameters, the coefficient of the
Second term vanishes, and the general equation assumes
the formi -
a y? -- ca." + d y + ea + f = 0.
(41.) To find the coordinates of the centre.
The centre being the point in which any two dia-
meters cuſt each other, we have, eliminating y between
(1) and (2,) in Art. 40.
2 C * — b d
––– a – † = – g º – 5: .
Or *– 4 & 6 – 24 e- 'd
– 2 a b 2 a b '
o 2 a e – b d
... + = -R-III-,
tº e 2 c d – be
and similarly, 9 = a – T.
Cor. If the axes be parallel to a system of conjugate
diameters, then b = 0, and
ſº-Eºn 2 a e tºº 62
w = −1 == 2 c *
2 c d ſº d
-. Tº 2 a. o
(42.) Let the origin be now transferred to a point
(a, 8), which is done by substituting a + a and y + 3
for a and y in the equation,
Th a y + c aº + d y + c q + f = 0.
€Il
a (y -- 3)?-- c (x + a)2 + d (y-º-B) + e (a +2) + f_0,
therefore, developing, and arranging the result,
a y” + c a” + (2 a 3 + d) y + (2 c a + e) a + a 32 +
c a” + d g + e a + f2- 0.
Now since a, B are arbitrary quantities, we may fix
their value by making the coefficients of y and a = 0,
we thus have
2 a 3 + d = 0, 2 c a + e = 0,
3/ — 4 a c
62
a = — –,
2 c
d
3. It g 2 a.
which (41, Cor.) are the coordinates of the centre,
Then
Hence the general equation is reducible to the form Part I.
a y” + c tº + fº = 0. . . . (1.)
This reduction is only practicable on the supposi-
tion that the equation contains both the terms involv-
ing a y” and ca’; for if either of them, as
62
caº be = 0. then a = — TO E CO,
and the term e a cannot be taken away, and the equa-
tion therefore assumes the form
- a y” + e a + f = 0.
Now, by taking away the term involving y, we have
determined only one of the quantities a, B; we may
fix the value of the second, a, by supposing the last
term to = 0. This supposition is always possible,
because c a” vanishing, the last term is only of one
dimension in a.
The equation thus reduced will be of the form
a y” + e a = 0. . . . (2.)
Hence, Lines of the second order are divisible into
two classes, according as they have or have mot, a centre,
the corresponding equations being
a y” + c a' = F,
and - a y? -- e a = 0.
(43.) In the first of these equations the coefficients
of y” and a 2 may have either the same or different
signs, the constant quantity F being supposed indeter-
minate.
I. Let them have the same signs, and
- 1. Let both be positive.
Then, according as F is negative or positive,
A y” + Caº = F. . . . (1,)
Or A y” + C cº = — F. . . . (2,)
but since the sum of two quantities essentially positive
cannot equal a negative quantity, the line represented
by this equation must be imaginary.
2. Let both be negative, then
— A y? — Caº = F,
— A y? — Caº = — F;
therefore, changing the signs of the terms in each
equation,
fººms
tºº
and
A y” + Caº = — F,
A y” + C w? = F,
which are identical with (1) and (2.)
º II. Let them have different signs, and
1. Let A be positive and C negative.
A y? — Caº = F. . . . (3,)
and A y” – C a” = – F. ... (4.)
2. Let A be negative and C positive.
Then C a” – A y” = F,
and C 19 – A y” = — F;
or, changing the signs in both equations,
A y” — Caº = — F,
A y” — Caº = F,
which coincide respectively with (4) and (3.)
Lines of the first class, therefore, may be subdivided
into two species. -
sº
tºmsºmº
A N A LY TIC A. L. G E O M ETR Y.
719
Analytical
Geometry.
\-y-Z
The first, represented by the equation
A y” + C as = F,
is called the Ellipse.
The second, represented by the equation
A y” — Ca' = F,
Caº — A y” = F,
is called the Hyperbola.
It hence appears, that the equation to the hyperbola
is deduced from that to the ellipse by changing the sign
of a * or of y".
OY
ON THE ELLIPSE.
(44.) To find the equation to the ellipse, in terms of
a given system of conjugate diameters.
Let CP, CD be the given semi-conjugate diameters,
(fig. 20,) and the ellipse be referred to these as axes.
Let C P = a, C D = b'.
Then the general equation being
A y” + C a " + F.
F F
Let y == 0, . . . a" == G= C P2 – -". a'?, ..". C = d.º.
•=0,...y=# = CD* = ... b”, ... A = ;:
therefore, substituting these values of A and C in the
above equation, and dividing by F, we have
2 2
# + · = 1.... (1)
# * iF =
OT a'? y” + b” tº - a” b/*.... (2,)
either of which is the equation required.
(45.) Cor. 1. If the origin be transferred to P, we
must substitute in (2) a' — a for a ; the equation
therefore becomes -
a's yº -i- bºwº – 2 aſ b's a = 0,
12
* —
3/ * a's
Or (2a"w — wº). . . . (3.)
(46.) Cor. 2. Let 2 a, 2 b represent the principal
diameters, then the equation of the ellipse becomes
1. When the centre is the origin,
gy” dº? gº
ºr + · = 1.... (1)
Or a y” + be a = a bº. . . . (2.)
2. When the extremity of 2 a is the origin,
bº
y? + (2 as – º ... (3)
(47.) To find the equation to the tangent drawn at
a given point (w', y') in the ellipse,
If a straight line be drawn cutting the ellipse, and
the two points of section be then supposed to coincide,
the secant will become a tangent.
Now the equation to a secant drawn through the
given point is
y— y' = m (r. — w). . . . (1.)
But a', y' being the coordinates of a point in the
Curve,
a y” + ba aſs = a b-;
and, in general, -
a yº + bºas – as bº,
... as (yº — y”) + 5° (tº — wº) = 0,
... a” (y -- y) (y – y') = b (a - a) (a + æ),
or, substituting for y—y' its value in (1,) and dividing
each side by a - aſ,
a” m (y -- y’) = b (a + x').
If the points of section be now supposed to coincide,
a = a-' and y = y', and the secant becomes a tangent;
- b2 g/
— — . -Z,
3/
0.2
therefore, by substitution, the equation to the tan-
gent becomes
"... ??? :=
b?
a' G.
gy — y' = — a? " # (y-y)....(?)
Or a y y' + bºw a' = a” b°. . . . (3.)
(48.) To determine the figure of the ellipse.
Resuming the equation a y” + b a' = a, bº, we
have •
b ,—
-: vº-º.
3/ +. 0. & Jº
Now let y = 0, ... (fig. 21) a = -E a = C A or CW,
a = 0, ... y = + b = C B or C b.
So long as a remains positive, and increases from 0 to
a, y is real, and decreases from b to 0. -
When a > a., the values of y are imaginary, and the
curve therefore extends to the right no farther than A.
Let a be negative, then since aº is positive, it may in
like manner be proved that the curve does not extend
beyond W to the left. 2 : . .
Again, * = ++ v ( – yº),
and by a process similar to that which has just been
followed, it may be shown, that the curve does not
extend beyond B or b.
Hence the ellipse has the form assigned to it in the
figure, and is wholly contained within the parallels
M N, PQ and M P, N Q.
Of the two principal diameters AV, B b, the former
is commonly called the major, the latter the minor,
a X1S.
(49.) Definition. The focus is a point in the major
axis, such that its distance from any point in the ellipse
is a rational function of the abscissa.
To determine the focus.
The curve being represented by the equation
a yº-H bºaº – as bº.
Let the abscissa of the focus be a ', its ordinate being
necessarily = 0.
Then if r denote the distance of the focus from any
point (a, y) of the curve, we shall have
73 - (a — a ')* + y”,
2
= <-2 = x + 2 +. (a” — "),
Part I,
\-v- '
Fig. 21.
5 A 2
720
A N A L YT I C A L G E O M ETR Y.
Anal vtical - - b2
&º. = , = 2 = 2 + 4 + b – ºr r,
- a” — bº
= 0.8 a" – 2 a. a' + 1/* + bº.
Now in order that r may be rational, the quantity on
the right must be a perfect square; therefore we have
a” — b%
4 a" (bs + x') = 4 wºr",
Or - (as – bº) (bº + æ") — a r",
•. as bº + a” ºr” – bº — b% whº = a” aſ?,
•. (a” †-º-º-º: bº) b? — b% a's,
... a' = + V aº – bº.
Whence there are two foci, on opposite sides of the
centre, and equidistant from it by the quantity
Aſ a” — bº, which is called the excentricity
Assume V a” — bº = a e = C S or C H, then S and
H are the foci, (fig. 22.)
Also S P := a – e ar.
Similarly, if H be the other focus,
H P = a + e ºr.
S P + PH = 2 a ;
or, the distances of any point in the curve from the foci
are together equal to the major aris.
Cor. a e = va” – 5°,
a” e” = ah — b%,
bº
F
ig.
2
2
Hence
Since
therefore the equation to the ellipse becomes, by sub-
stitution, gy* = (1 — eº) (a” — a ").
(50.) To find the value of the ordinate passing
through the focus.
- b?
In general gº = Taº (a” — wº) . . . . (1,)
but a' = a” — bº,
• – ? 2 º 2
... y = ± a — (a" – bº) }
bé
Cº-sº y
0.8
- b”
... y = + T.
. 2 b2
twice this quantity, or --- is called the principal para-
meter; let it be denoted by 2 p, then the equation to
the ellipse in terms of its principal parameter becomes
by substitution in equation (1)
P .s
2 =
g” = 2 p a ... r.
(51.) To find the polar equation to the ellipse.
The pole may either be the centre or one of the
oci.
1. Let it be the centre.
Assume C P = p, angle P C A = w, (fig. 22.)
Then p” = z* -- y”, - Part I.
= z* + (1 e”)(a” — wº) (Art. 49, Cor.) S-
= e” as + a” (1 – e’), -
but a = p cos u",
... p” = e” p” cos” w -- a” (1 – e’),
* Art, dºs Vº
- p = 0. 1 – e' cos” w
2. Let the pole be the focus S.
Assume S P = r, angle PSA = v.
Then r = a – e ar, (Art. 49,)
but a = C M = C S – S M,
= a e -- r cos v,
‘. r = a – e (a e -- r cos y),
‘. r (1 + ecos v) = a (1 — e”),
1 — e”
iT. . . . . (1.)
+ ecos v.
Similarly, if H be the pole, and H P = r", angle
PH A = v', then
°, 7" E. C.
1 — e”
f
ºr = 0. * tº
1 — e cos v'
ON THE HYPERBOLA.
(52.) The same notation being retained, the equa-
tions to the hyperbola, deduced from the correspond-
ing equations to the ellipse, are
I. When the axes are a given system of conjugate
diameters,
# - # = –1
* – 4 = –1 . . . . (1,)
a/* b's Cºmº
/* ai" — h’º r" — — aſ: h"
Or º, "... ºt ...} . (2,)
b/2
y = ±, ("-2 a o ... (3.)
gy” gº E-º
# – H = - 1
2 Q . . . . (1',)
* – 4 = - 1
F – F –
OF a” y” — bºr" = — a bº A
e is tº a 27,
º, I., II; ( )
2
and y = }; G – 2 a 5.... (3)
III. The equation to the tangent, applied at a given
point (a', y') of the hyperbola, is
a’yy’ — bºw w = — a bº.
(53.) To determine the figure of the hyperbola.
Taking the first of equations (2)
a” y” — bºa" = — a” b”,
A NALY T I C A L G E O METR Y. 72I
Analytical
Geometry.
Fig. 23.
we have 3) = + + M (a" — a”),
(fig. 23.)
Let gy = 0, ... a = + a = CA or CV,
a = 0, ... y = + b v –I.
Hence it is evident that the axis of y can never meet
the curve.
Let r < a., then the values of y being still imaginary,
no part of the curve can lie between C. and A.
Let a > a 5 the values of y are now real, and to each
assumed value of a there correspond two equal values
of y with opposite signs.
As a increases, y also increases; when a is supposed
infinite, the values of y are also infinite. Hence, to
the right of C the curve extends indefinitely, and con-
sists of two branches AZ, A z symmetrically placed
with respect to the axis.
In the same manner it may be shown, by supposing
a to be negative, that to the left of C the curve has two
infinite branches V Z’, V c'.
Again, taking the second equation,
b” wº tºº as y” = mºme a*b*,
we have
* = + - My W.
Let a = 0, ... y = + b = C B, or C b, (fig. 24,)
y = 0, ... a = + a W — 1.
Hence it is evident that the axis of a cannot meet
the curve.
Nor can any part of the curve be situated between B
and b; for so long as y is less than b, the values of
a are imaginary.
The investigation being conducted as in the last
case, it will be found that there are two infinite branches
B U, B u ; b U", bu', on each side of the centre, sym-
metrically situated with respect to the axis of y.
The two hyperbolas represented by the figures 23
and 24, are said to be conjugate to each other.
Since the line B b, in the first case, and A V in the
second, never intersect the curve, they cannot, correctly
speaking, be called diameters. They are so named in
order that the analogy between the hyperbola and
ellipse may be preserved. •
2
(54.) To find the coordinates of the points in which
any diameter meets the curve.
Let the equation to any diameter be
y = m a.,
and that to the curve
a's y” — b” w” = — alº b”,
then, by elimination,
a” m” a “ — b'2 tº - -- a” b”,
g” b/ *
•". a" Sº T-7-5,
b's — a' mº
aſ bº
. . . a = ~—— ?
w/ba – a'* m”
/ ; /
m aſ b
and ... y = -E —.
w/bº *E= a/* ºn.”
So long, therefore, as b's aſ m” is positive, the diameter
gº b' tº Part I.
meets the curve; if b/* < a” mº, or m > aſ ' the dia- g º
º
meter does not meet it ; if b^* = a” m”, or m
- b! & g g - g
= -r, the diameter intersects the curve at an infinite
Q,
distance from the centre. t
Let P p, D d (fig. 25) be any two conjugate dia- * *
meters to which the curve is referred as axes; through
P draw Q q parallel to and equal to D d ; join C, Q :
C q ; then the lines C Q, C q being produced to Z,
2 will meet the hyperbola at an infinite distance.
The lines C Z, C 2 are called asymptotes; and their
b'
equation is y = + 7 *.
The asymptotes may be considered as separating
those diameters which meet the curve from those which
never meet it. * *
(55.) It may be proved, as in the ellipse, that there
are two foci S, H situated on the transverse axis at a
distance = va”-H bº from the centre. And in like
manner it may be shown, -
1. That S P = e r — a,
- H P = e a + a,
and therefore that the difference of the focal distances
equal the transverse aris.
2. That the polar equations of the hyperbola are
(1.) When the centre is the pole,
e? — I
p = a — .
e” cos” w – 1
(2) When the focus s is the pole,
e” — 1
7° tº O. — .
1 + ecosy
(3.) When the focus H is the pole,
e” 1
1 — ecos v
The equation to the hyperbola in terms of its prin-
cipal parameter is -
* = —
y = 2 p ≤ ++, ».
ON THE PARABOLA,
(56.) The equation to Lines of the second order,
when the centre is infinitely distant, is
a yº-H ea = 0,
Or gy* = m a ; if m be taken = - #.
The curve which is the locus of this equation is called
the parabola. -
(57.) To find the equation to the tangent drawn at
a given point (a', y') of the parabola.
If a straight line be drawn cutting the parabola,
and the two points of section be then supposed to
coincide, the secant will become a tangent
722 A N A L YTI C A L G E O M ET. R. Y.
Analytical Now the equation to a secant drawn through the tº e 77.
Čº. given point º § Cor. 1. The distance of any point P from S = a + T'
^*~~
3y — y' = a (a – a ') . . . . (1.) Cor. 2. Let S L be perpendicular to A X, then
But a ', y' being the coordinates of a point in the since y' = m w, we have
Part I.
curve, y” = m a.'; - SL* = * ,
and, in general, y” = m a., - 4
... yº — y” = m (r – a '), ... s L = +,
... (y – y') (y –- y’) = m (a – a '); 2 S L 2
& tº Or t= ???, ?
or, substituting f —w' its value 1, d dividing: 2 te
... º º gy' its value in (1,) and dividing the quantity 2 S L is called the lutus rectum, or prin-
- 5 ( ^ — cipal parameter.
a (y -- y’) = m, (60.) To find the polar equation to the parabola.
tºº 7??, G Let - A S P = w, S P = r,
y -H y' 777,
If the points of section be now supposed to coincide, then r = a + T.
r = a and y = y', and the secant becomes a tangent;
therefore, by substitution, the equation to the tangent = * – r cos tº),
becomes 4
777, T
v — y' = + (a — a ') . . . . (2,) r = — ”
2 y' r 1 + cos w '
— —” ſ which is the equation required.
Or 3) = 2 y (a + ar') . . . . (3.) (61.) The parabola may be considered as a species
of the ellipse, or hyperbola, and its equation deduced
(58.) To determine the figure of the parabola. from that of either of these curves, by supposing the
Since gy* = m a. centre removed to an infinite distance.
9 Thus the equation of the ellipse and hyperbola, in
... y = + W m r, terms of their principal parameters, is
therefore for each assumed value of a, there are two equal º P …,
values of y with opposite signs, therefore the curve is gy* = 2 p r FF ... ".
Fig 26 divided into two equal parts by the axis AX, (fig. 26.) 2 2
— nº
Let a = 0, then y = 0, Now p = * = *-** =
therefore the curve passes through the origin A. (Z (Z
Let a be supposed to increase, then y also increases; (a -- a e) (a — a e)
let a become infinitely great, then y is infinitely great (Z * .
also.
Let a be negative, then y being imaginary, no part But a -a e = As, and a +- a e = 2 a, when the centre
of the curve is situated to the left of A. is at an infinite distance,
Hence the parabola consists of two infinite branches "... p = A S 2 a 2
A Z, A 2, symmetrically placed with respect to AX. ..". p = a - A S,
(59.) The focus being defined as in the ellipse and • a a º
hyperbola, let it be required to find its position. therefore, by substitution,
2 A S
Let S be the focus, A S = x', (fig. 26,) and let the 9° = 4 A. S. a FE → *,
(Z
coordinates of any point in the curve be a, y; then
7-2 = g” + (a. * a')*,
= m, a -- a " — 2 a. a' + æ”, -
º + fº and may therefore be neglected,
= z*-ī- (m. – 2 a.') a + a”.
tº c g tº a ºn ... y” = 4 A S. a.
Now as this is to be a rational quantity, it must be a
complete Square ; , - By comparing this with the equation y” = m r, it
... 4 rºa' = r" (m. – 2 a.')" appears that the constant quantity m is equal to four
* * *- ...” times the distance of the vertex from the focus.
... 4 a.” = (m – 2 w")*, For the analytical investigation of the properties of
2
but a being infinitely great,
is infinitely small,
... 2 r = m – 2 a.', Lines of the second order, the reader is referred to the
772. - works on Analytical Geometry enumerated at the end
... a' = T - of this Article, and particularly to Dr. Lardner's Alge-
braic Geometry, vol. i., which contains a variety of
Hence there is only one focus in the parabola. Problems resolved with great elegance and simplicity.
A N A L YTIC A. L. G E O M ETR Y.
723
PART II.
APPLICATION OF ALGEBRA TO THE THEORY OF SURFACEs.
Analytical (62.) THE application of Algebra to the Theory of
Geometry. Surfaces is founded on this principle, that an indeter-
\-v- minate equation between three variables may be repre-
sented by a geometrical locus, and conversely.
Let f(x, y, z) = 0 be any indeterminate equation be-
tween a, y, and 2 ; let X A Y, XAZ, YA Z (fig. 27)
be three planes, each of which is at right angles to the
other two, and AX, A Y, A Z the lines in which they
intersect. In AX take A M equal to any arbitrary
value of z, draw M N parallel to AY, and equal to any
arbitrary value of y ; then if N P be drawn parallel to
A Z, and equal to the resulting value of z, each point
P so determined will correspond to a solution of the
equation f(x, y, z) = 0. The assemblage, therefore,
of all the points P will form a surface, plane or
curved, which is called the locus of the equation
f(x, y, z) = 0.
The lines AM, M N, N P are called the coordinates
of the point P, and A is said to be the origin, A X,
A Y, A Z the aires of the three coordinate planes
X AY, X A Z, Y A. Z.
The coordinates are usually denoted by ar, y, z re-
spectively; whence A X is called the axis of a, A Y
that of y, and A Z that of z, also X.A. Y is called the
plane of a y, X A Z the plane of a z, and Y A Z the
plane of y z.
The equation which expresses the relation between
the coordinates of any point of a surface is called the
equation to the surface.
(63.) Complete the rectangular parallelepiped A P,
then it is evident that A M = P m = the distance of P
from YAZ, estimated in the direction A X, and also
that MN, PN are respectively equal to P's distance from
the planes X A Z, X A Y measured in the directions
A Y, A Z. Hence it appears, that the position of a
point in space depends on its distances from three rec-
tangular coordinate planes estimated in the direction of
the lines in which they intersect.
(64.) The points N, n, m in which the lines PN,
Pn, Pm meet the planes of a y, a z, and y z are called
the projections of the point P upon these planes re-
spectively.
It is manifest, that if any two of these projections be
given the third will be known. Hence the position of
a point in space is determined when its projections on
any two of the coordinate planes are given.
In like manner, if the several points of a straight
line be projected upon any plane, the line so formed
is called the projection of the given line, and the
plane in which the perpendiculars are situated is called
the projecting plane.
Surfaces, in the same manner as lines, are divided
into orders according to the dimension of the equations
by which they are represented.
Thus, a surface of the Érst order is the locus of the
equation
a a -- o y + cz + d = 0.
Fig. 27.
equation
Part II.
\-N-
A surface of the second order is the locus of the
a aº + b y” -- c 22 + 2 a' y z + 2 b' r z + 2 c a y
+ 2 a." ~ + 2 b" y + 2 c” z + d = 0,
and so on.
ON THE STRAIGHT LINE IN SPACE.
(65.) If the projections of a straight line upon any
two of the coordinate planes be given, the position of
the line itself will be determined ; because it will evi-
dently be the intersection of the two projecting planes.
}%e may hence find the equations of a straight line
2n space.
Let PQ be the given line, p q, p q' its projections on
the planes a 2, y & respectively, (fig. 28.) Also let Fig. 28.
a = a 2 + a be the equation to p q,
and y = b z + 3 be the equation to p' q'.
Now, since the first of these is independent of y, it is
the equation not only to p q but also to every line in
the projecting plane p PQ q. In like manner, the
second equation is the equation to every line in the
plane p' PQ q'. Therefore the system of equations
a = a 2 + a,
y = b z + 3,
being common to the two projecting planes, must also
be the equation to PQ, which is the line of their inter-
section.
The quantities a, b denote the tangents of the angles
at which p q, p' d' are inclined to AZ; and a, B repre-
sent the portions of A Z intercepted between A and the
points in which the same lines intersect AZ.
(66.) To find the equations to a straight line passing
through a given point.
Let the coordinates of the given point be a ', y', 2'.
Then, since they must satisfy the general equations
a = a z -- a
y = b z + 3 . . . . (1,)
z' = a z'+ a, and y' = b z' -- B,
... a = a – a z', and B = y' – b z'.
Substituting these values of a, B in (1)
a = a 2 + x' – a z',
gy = b z + y' – b z',
Or a — a ' - a (2 – 2'
and ;Iyº-3 . . . . (2,)
which are the equations required.
we have
724
G E O M E T R Y.
A N A L YTIC A. L.
Analytical
Geometry.
(67.) To find the equations to a straight line passing
through two given points.
Let the coordinates of the second point be a', y”, z".
Substituting these for a, y, z in equation (2) in the
last article, we have
a" – ac'
a" – a = a (2” – 2'), ... a = 7-7,
gy’ — y'
y" ** gy' = b (z" sºme 2'), ... b =
!! — ºf *
gº
therefore replacing a and b by these values in the same
equation, we have
a" — aſ
a — aſ :- z" — z' (2 – 2'),
!! — a 1/
y – y = #F# (2 - 2),
which are the equations required.
(68.) To find the coordinates of the point of inter-
section of two straight lines.
Let the equation of the lines be -
a = a 2 +- a, y = b z + 6,
a = a' z + a', y = b' z + 3'.
When the lines intersect, the coordinates at the point
of their intersection will be identical ; therefore sub-
tracting the latter equations from the former,
and
(a — a') z +- a - a' = 0,
(b — bº) z + 3 – 3' = 0;
whence, eliminating z,
a — aſ _ 8 —£3'
a — aſ T b - b. '
which equation expresses the condition under which
the two lines intersect.
ſ
Q," - O. . ſº e
we immediately obtain
b £3' – b//3
a — aſ
Cor. It thence follows, that when the lines are
parallel
Now z being =
a a' — a a
ºr =:
and y =
a — a'
a = a' and b = b'.
(69.) To express analytically the distance of a given
point (a', y, z) from the origin.
Let P be the given point, (fig. 29,) A M, MN, N P
its coordinates; join A, N.; and let A P = p.
Then the triangles A N P, A M N being evidently
right angled in N and M, we have from the first
AP* = A N* + N Ps,
= ... A M* + M. N*-ī- N P and from
... p" = a " + y” + 2*.
Cor. In a rectangular parallelepiped, the square of
the diagonal is equivalent to the sum of the squares of
the three edges.
(70.) To express analytically the distance between two
given points.
the second,
Let a ", y", 2" be the coordinates of the second point
Q, and take PQ = 8.
Then PQ is evidently the diagonal of a rectangular
parallelepiped, whose three contiguous edges are
a' — a', y' — y", and 2' — z";
we have, therefore, by the last article,
8* = (a/ — a ")* -- (y' — y")* + (2 – 2")”.
Cor. If A Q = p", we have, by expanding the value
of 8*, •,
S” – ar.” –– y” + 2*-i- r” + w” -- 2//* — 2 {a' &// +
gy' y' -- 2'2''}
= '.' p” -- p” – 2 (w' a' + y' y' + 2'2'').
(71.) Given the equations to a straight line, to find
its inclination to each of the aves.
Draw through the origin a straight line
& parallel to
the given line, and let its equations be t
a = a 2, y = b 2.
In the triangle A PM,
A M = AP cos PA X,
Or & E p COS p, a ,
g – “ — Jº ſºmeº CZ 2
... cos ?, r = | – vº-Ey-Ez” ſºmºmº Vºivºrº)
hºmºmº (?,
w/I-E as-E 5* .
In like manner,
cos p, y = + b 3.
T M (TH aſ E 55
and cos p, 2 = l
+ — .
T M (1 + a”-- bº)
The line p forms with each of the axes two angles which
are supplements of each other; hence in the above
formulas the positive sign indicates the acute, and the
negative sign the obtuse angle.
Cor. 1. Squaring these values, and adding the results,
we have
cosº p, a + cos" p, y + cos” p, z = 1.
Cor. 2. If the angles which p makes with the
planes a y, a z, y z be respectively denoted by the
symbols p, a y : p, a 2, p, y 2, we shall have by the
last article
sin” p, y z + sin” p, a 2 + sin” p, a y = 1.
(72.) To find the inclination of two lines in terms
of their separate inclinations to the aires.
Through the origin draw two lines respectively
parallel to the given lines. In these take any two
points P and Q, join P, Q, and let A P = e, A Q = f';
then by Art.
PQ* = f° + p" — 2 (r'a"—H y y” + 2'2"),
but PQ’ = P + tº 2 p' cost, p.
Woodhouse's Trig. ch. ii. Prob. 1; Lardner's Trig.
Art, 75,
... pp'cos ??' = f'." + y^y" + 2'2''....(1)
But r" = p cos p, a 3 y' = p cosp, y, z = p cos p, 2.
Part II.
A N A L YT I C A L G E O M ET R Y.
725
Analytical Similarly,
Geometry, a " + p cos p, a , y" = p' cos p, y; 2"
= p COS p, & 5
therefore, substituting in (1) and dividing by p p", we
have
cos p, p' = cosp, a cos p', a + cos p, y cos p', y +
cos p, 2 cos p', z.
****
b=s
Cor. When the lines are at right angles to each
other,
cos p, a cos p', a + cos p, y, cosp', y + cosp, 2 cos p', z
s= 0.
(73.) The equations to two lines being given, to find
their mutual inclination. -
Draw two lines through the origin parallel to the
given lines, then their equations will be
a = a 2, y = b 2. . . . (l,)
w = a'z, w = W z. ... (2)
Now, by last Art.
cos p, p’ = cos p, a cos p', a + cos p, y cos p', y +
cos p, 2 cos p', 2 :
but
O. a/
v I-E as E 5. w/I-E aſ E 5*
and similarly with respect to cos p, y, cos p, 2, &c.;
therefore, by substitution,
cos p, a E ; cos p', a =
1 + a a' + b bl
y (1 + a” + bº) (1 + a” + b”) '
which is the expression required.
COS p, p' --
ON THE PLANE.
(74.) A plane is generated by a straight line which
moves parallel to itself, along a straight line given in
position. -
Of these straight lines the former is called the gene-
rating line, the latter the directriar.
The equation to a plane may be obtained by express-
ing analytically the mode in which it is generated.
Let the equations to the generating line be
a = a z + a, y = b z + 3. . . . (1,)
and the equation to the directrix, which we shall sup-
pose to be in the plane of a y, -
Y = m X + m, ... (2.)
Now, since the generating line is always parallel to
itself, its equations in any position will be
a = a z + a', y = b z + 3'.
But because it passes, by hypothesis, through a point in
the directrix whose coordinates are X, Y, 0, we shall
X = a, Y = 3’;
a' = a – a 2, and B' = y – b z,
have
but
... X = a – a z, Y = y – b z.
Substituting these values of X, Y in equation (2)
we have,
gy – b z = m (a – a z) + n,
.*, *) – m a + (m a - b) z – n = 0
which is the equation required.
WOL. I.
A more symmetrical form may be given to the equa-
g - & *=== A —
tion by assuming T = - m, G = b – m a, and
A amº. Yº
D T 72.
Then we have
A r + B y + C 2 + D = 0,
for the general equation to a plane.
(75.) Cor. 1. When the plane passes through the
origin, D = 0, and the equation becomes
A a + B y + C z = 0.
(76.) Cor. 2. If the plane meet any one of the
axes, for example AZ, then aſ and y = 0, therefore
D
- G.
If the plane be perpendicular to A Z, then each of its
points is equidistant from the plane a y, and therefore
z is constant.
2 :=
If the plane be parallel to A. Z, then
C being
infinitely great, C = 0.
(77.) Cor. 3. If the plane meet any one of the
coordinate planes, a y, for example, then z = 0, and
the equation to their intersection is
A a + B y + D = 0.
(78.) The intersection of a plane with any one of
the coordinate planes is called the trace of the given
plane.
If the plane be perpendicular to a y, then since it
must be parallel to A Z, C = 0, therefore the equation
to the trace is A a + B y +D = 0. -
As the same reasoning is applicable to the remain-
ing two coordinate planes, we conclude that when a
plane is perpendicular to any one of the coordinate
planes, its equation is that of its trace upon the same
plane.
(79.) If the plane be parallel to that of a y, then the
coordinates of its intersection with A X and A Y,
A
fore A and B each = 0, hence, the equation becomes
C z + D = 0.
namely, and
will be infinitely great, there-
(80.) To find the equation to a plane in terms of the
perpendicular (p) dropped upon it from the origin, and
the angles which that perpendicular forms with the
QºI'éS.
Let the plane meet the axes in the points B, C, D,
and take A B = a, A C = b, A D = c : then, by the
last article,
hºmſº P. b = D C = D
a = --x, 0 = - F. c = - a-
But the general equation is A a + B y + C z
lºmºnºgms y O A B — z = }
r – Bº – B y - B - = 1,
therefore, by substitution, # + # + # = 1 . . . . (1,)
- 5 B
Part II.
N-N-
726
A N A L Y T I C A L G E O M E T R Y.
Analytical
Geometry.
~~~~
but it is evident that p = a cosp, ar, or a = —t- in
COS p, a
like manner, b = p , C = p ;
COS p, y COS p, 2
therefore, replacing a, b, c, in (1) by these values, we
have a cos p, a + y cosp, y + 2 cosp, z = p,
which is the equation required.
(81.) Cor. Cosºp, a + cos” p, y + cos” p, z
p? p? p?
= x + i + . .
p?
=# (A + B-C) = 1,
- D
. . . p = + º 5
w/AT Bº-E C3
A A
amº-º-º
D T w/A2TBT Öº
. . . COS p, & E – p .
In like manner,
B
COS p, y = w/A, IE BETC,
C
cos p, 2 =
w/A* + Bº + C3'
(82.) To find the equations to a perpendicular let fall
from a given point (a', y', 2') upon a given plane
A a + B y + C 2 + D = 0.
By Art. 63 the equations sought will be of the form
a – a ' - a (2 – 2')
gy — y' = b (2 – 2') . . . . (1,)
in which a and b are to be determined.
Suppose the perpendicular and the plane to be pro-
jected upon any one of the coordinate planes, then these
projections will evidently be at right angles to each
other; because the projecting plane of the perpendicu-
lar being at right angles to the given plane, their inter-
sections with any of the coordinate planes, in other
words, the projections in question, will also be at right
angles to each other.
The given plane, then, being projected on the planes
of a z and y 2, the equations to its traces are
Aa -H C z + D = 0, or s = -k2 - . (2.)
- C D ſº ºf 2
B y + C z + D = 0, Or v= - F * – F
But since the projections of the perpendicular are at
right angles to these traces, we have (Art. 14)
A
a = + and b = 3;
therefore the equations required are
-2 = *(s-).
y-y=#G-2).
(83.) To find the length (p) of the perpendicular
dropped from a given point on a given plane.
Conceive a plane drawn through the given point
parallel to the given plane, and let fall upon it from
the origin A a perpendicular A Q meeting the given
plane in P; then PQ will = p.
Now A Q = a 'cos p, a + y' cos p, y + z'cos p, z,
= + Aa' + B y' +C2'.
T + v MIBTC.'
. . p = A Q – A P = A Q – p,
+ A* +By + °4'- P
WTA II BTC;
(84.) To find the inclination of a given straight line
to a given plane.
Let the equations to the line be
a = a z + a, y = b z + 3,
and the equation to the plane Aa. --By-H C2 + D = 0.
Now the inclination of a line to a plane is the angle
contained by the line and its projection upon the plane,
and is therefore equal to the complement of the angle
formed by the line and a perpendicular let fall from any
point of it upon the plane.
Let the equations to the perpendicular be
a = a' z + a', y = b/2 + 3',
B
A C’ by (Art. 82.)
a' = TG’ and b' =
Let p be the given line, p' the perpendicular dropped
from any point of it on the given plane, and let the sym-
bol p, II denote the angle at which the line is inelined
to the plane.
then
*sºred 1 + a a' + b b'
T.V (1+a+bº) (1+a”--b”)
g f
Then in general cos p, p
(Art. 73.)
Therefore, substituting for a, b, their values obtained
above, we have
A a + B b + C
W (1 + a”-- bº) (A* + B"-- C')"
(85.) Cor. When the line is parallel to the plane,
then A a + B b + C = 0.
sin p, TI =
(86.) To find the inclination of two planes, in terms
of their separate inclination to the aves.
Draw through the origin two planes parallel respec-
tively to the given planes; then if two lines p, p' be
drawn from the origin at right angles to the planes,
their inclination p, p' will equal the angle II, II", and
their equations will be
a cos p, a + y cos p, y + 2 cos p, z = 0,
a cos p', a + y cos p', y +z cos p', z = 0.
Now in general,
cos p, p' = cosp, a cosp,' a + cos p, y cos p', y +
cos p, 2 cos p',.2 . . . . (1,)
&º
ſºmeº
but cos p, a cos II, y z 3 cos p, y = cos II, a z ;
cos p, z = cos II, a y, and so on ; therefore, substi-
tuting these values in (1,) we have
cos II, II'- cos II, y z . cos II', y2 + cos II, a 2 cos II', w 2
+ cos II, a y cos II', a y,
which is the inclination required.
(87.) To find the inclination of two planes whose
equations are -
Part II.
\-N-
A N A L YTIC A. L. G E O M ETR Y.
727
Analytical
Geometry.
A z + B y + C z + D = 0,
A'a -- B'y + C'2 + DV = 0.
The inclination required will equal that of two planes
drawn through the origin parallel to the given planes.
In general,
cos p, p' = cos p, a cos p', a + cos p, y cos p', y +
cos p, 2 cos p', z. . . . (1 ;)
t A. B
but cosp, r ====, cos P,3) = +===,
Aſ A* + Bº + Cº. A/A*-i-B” + Cº.
and so on; therefore, by substitution in (1,) we have
d / A A' + B B'-- C C/
cos II, II'- M (A* + Bº + C*) (A” + B's -- Cº)"
(88.) Cor. When the planes are perpendicular to
each other,
A A' + B B' -- CC' -- 0.
ON THE TRANSFORMATION OF COORDINATES IN SPACE.
(89.) The position of a point with respect to a given
system of planes being given, to find its position when
referred to a new system of planes parallel to the
former.
Let a, b, c be the coordinates of the new origin: and
w, y, z, a y', 2' those of any point P when referred to the
old and new system respectively.
Then it is evident, that in order to obtain the equation
of P in relation to the new system, we have only to
substitute for a, y, z in the given equation
j (r, 3/, z) = 0,
the quantities a' + a, y' + b, 2 + c.
The position of the new origin relatively to that of
the old one will be indicated by the signs of a, b, and c.
(90.) The position of a point with respect to any
system of planes being known, to find its position when
referred to any other system whatever, originating at the
same point with the former.
Since the new coordinates must evidently be linear
junctions of the old ones, let us assume
a = m a' + n y' + p 2',
y = m/a' + n' y' + p'2',
2 = m'a' + n" y' + p" 2';
the quantities m, m/, m" . . . . being independent of
ū, 3/ . . . .
In order to determine their value, Let y' and 2' each
equal 0, in which case the point is situated on A X'.
º ſ
º Sln º', aſ 2
Then m = |* = *** * *
aſ T sin w, y z'
sy sin ºr', a z
a Sin 3/, a 2
m" = # = ** a', a y
* 7 – º º
a sin z, a y
In like manner, supposing the point to be successively
on the axes A Y’, A Z', we have
sin y', y z
n = ** * * *
_ sin 2', y 2
Tsina, y z'
sin r, y z'
e f w ~! s
| Sin V, a 2. , ... sIn 2', a 2
* º y - — .
Sln 3y, a 2 Sin 3/, a 2
º y -> /
nºr = ***, *2 p = *****
sin 2, a y' sin z, a y
Hence we have, by substitution,
1
! --- ~/ * . . / !---, -/ º
& E — 3 º' Slnæ 2 Słrl ºy', iſ 2 2' SIIl 2 2
iſſºt , y z + y'sin y', y2 + , y2 #;
gy tº { a 'sina', a 2 + y'sing', a 2–H 2'sin 2', a z };
sing, a z
l ! --- - --/ ! ---> 2./ / -- ~'
= — 3 & Slna', a. Slnºy', ſº 2' SIP) 2 , J J } .
*==== { y–H y'sin y', a y + v}
(91.) Such are the general formulas to be used in
passing from one oblique system to another. We shall
now deduce from them the following particular cases:
1. Let the old axes be rectangular, and the new
ones oblique.
Then the denominators become each = 1 ; also in
the first line,
sin a', y z = cosa', a ; sin y', y z = cos y', a ;
sin z', y z = cos 2', a 5
the remaining two lines being in like manner modi-
fied, we have
a = a 'cos r', a + y'cos y', a + z' cos 2', a
9 = a 'cos ar', y + y' cosy', y + z' cos 2', % ... (I ;)
2 = a 'cos a’, 2 + y'cos y', z + z'cos 2', 2
but because the primitive axes are rectangular, the fol-
lowing equations also hold true,
cos’ r", a + cos” w!, y + cos” ar', z = 1
cos” y', a + cosº y', y + cos” y', z = 1 -. . . . (2.)
cos” 2', a + cos” 2', y + cos”z', z = 1
It appears, therefore, that of the nine angles involved
in the formulas (1) six alone are independent, since
three of them are evidently determined by equa-
tions (2.)
2. Let both systems be rectangular.
Then, since each two of the coordinates a ', y', 2' are
at right angles to each other, we have, by (Art. 72;)
cosa', a cosy', a +cosa', y cosy', y + cosa', 2 cosy', z = 0
cosa', a cosz', a + cosa', y cosz', y + cosa', 2 cosz', z = 0
cosy', a cosz', a +cosy', y cosz', y + cosy', 2 cos 2', z = 0
. . . . (3,)
by means of which equations the six angles that enter
into the formulas (1) are now reduced to three.
Whence it follows, that in order to pass from one
rectangular system to another, three independent angles
alone are required.
ON THE SPHERE,
(92.) To find the equation to a spherical surface.
Let a sphere whose radius is r be referred to any
system of oblique axes; suppose a', y', 2' to be the
coordinates of the centre, and ar, y, z those of any
point on the surface.
Now, since all the points on the surface are equi-
distant from the centre, we have
Part II.
\sºvº’
5 B 2
728
A N A L YT I C A L G E O M ETR Y.
Analytical (a — a ')* + (y – y')* + (2 – z')* + 2 (a – a ') (y – y')
**, cost, y +2 (c-a') (2 – 2') cos r, 2 + 2 (y-y') (2–2)
cos y, z = r".
(93.) Cor. Let the origin be at the centre; then
a', y', 2' being = 0, the equation becomes
a" + y” + 2* + 2 + y cos r, y + 2 a z cos a, 2
+ 2 y2 cos y, z = r^.
The general equation to the sphere may, as in the
case of the circle, be simplified by changing the origin
and direction of the axes.
Let the axes be now supposed rectangular, then the
general equation becomes
(, – r)--- (y – y')*-ī- (2 – 2) = rs. ... (1)
and when the origin is at the centre
a" + y” + 2* = rs. . . . (2.)
Let the origin be on the surface of the sphere, then
since
gº'? + gy” +- 2/2 - r?,
equation (1) becomes
w” + y” + 2*— 2 a. a' – 2 y y' – 22 2' = 0 . . . . (3.)
Let the origin be on one of the coordinate planes.
If it be upon the plane of a y, then 2' = 0, and the
equation becomes
(a – a ')*--(y – y')* + 2* = r^ . . . .
Let the origin be upon one of the axes.
If it be upon the axis of a, then y' and 2' = 0, and
the equation becomes
(a – a ')2 + y” + 2* = 0 . . . . (5.)
(94.) The general form to the equation to a sphere
when referred to rectangular coordinates is,
a” + y” + 2* + A a + B y + C z + D = 0 . . . . (1.)
Let it now be required to assign the position and mag-
nitude of the sphere which it represents.
(4.)
Comparing equation (1) with the general equation
(a – a ')2 + (y – y')* + (2 – 2')* = rº,
that is, with
a” + y + 2* – 2 a. a' – 2 y y’ – 2 z z' + x' -- y”
+ 2* — rº = 0 . . . . (2,)
we have
A = — 2 a.', ors = – #,
B = — 2 y', or y' = — B.
2
C
= — 2 z', 2' = — -;
C 2". Or 2
also D = a " + y^* + 2* — rº,
= 4 (A*-- Bº + Cº.) — rº,
... r = + , v(A*-H B2 + C° – 4 D).
Hence it follows, that equation (1) belongs to a sphere
whose radius is - # W {A* + B2 + C* – 4 D #, and
the coordinates of whose centre are — #, tº ,- #
(95.) To find the equation to a tangent plane, drawn
through a given point (a!, y', 2') of the sphere.
The origin being at the centre, the equation to the Part II.
sphere will be S-N-
a" + y” + 2* + 2 a y cos ar, y + 2 r z cos w, 2
+ 2 y z cos y, z = r". . . . (1.)
Now if a secant be drawn through the given point,
its equations will be
/ — — »’
a — a ' - m (z – 2') . . . . (2.)
gy — y' = n (2 – 2')
But since the given coordinates a', y', 2
satisfy equation (1) we have -
a" + y^* + 2* + 2 a.' y' cosa, y + 2 +/2] cos w, 2
+ 2 y' z' cos y, z = r" . . . . (3,)
and
' must
whence subtracting this from (1)
a” – a " + y” — y”-- ** – 2° + 2 (a y – aſ y') cos r, y
+ 2 (a z – a '2') cos ar, z + 2 (y 2 – y'2')cos y, z = 0
* . . . . (4.)
But a' — a 4 = (a, -i- a ') (a — a ') = (x + æ') m (2 — 2'),
gº — y” = = (y-H y') m (2 – z'),
22 — 2% = = (2 + 2') (2 — 2'),
also,
a y – aſ y' = a (y – y') + y^ (a — a')
= a n (? – 2') -- y' m . (: — z'),
= (2 – 2') (m y' + m ar),
a z – a '2' = (2 – 2') (a + m z'),
y z – y' z' = (2 – 2') (y -- n 2').
therefore substituting in (4) and dividing each term of
the result by 2 — 2', we have
m (a +a')+n (y-H y')+(z+2') +2 (my'+ na) cosa, y
+ 2 (a + m z') º
+ 2 (y –- m 2') cos y, z
= 0.
Suppose now that a = a-', y = 'y', and z = z', then the
points of section coincide, and the secant becomes a
tangent; we have therefore in this case, after dividing
each term by 2,
m a.' + m y + 2'-H (m y' + m a.') º
= 0,
+ (a' + m z') cos w, 2
+ (y' + n z') cos y, z
or collecting the terms involving m and n,
(a' + y' cos a, y + z' cos ar, z) m
+ (y' + æ' cos r, y + z' cos y, z) n
+ 2 + x' cos a, z + y' cos y, z = 0;
eliminating m and n by means of equation (2)
a — ac'
{*-i-y cos x, y +2 cost, 2 z'
gy — y!
2 — 2'
+ (y' -- a 'cos a, y + z' cos y, z)
+ 2 + x' cos a, 2 + y' cos y, z = 0,
which on being reduced by means of equation (3) be-
COIſles
{ a' + y^ cos w, y + z' cos ar, z } a
+ { y' + a 'cos y, a + z' cos y, z } y
+ { z' + a 'cos 2, a + y' cos 2, y : z = rº,
which is the equation required.
AN A LY TI C A L G E O M ET R Y.
729
Analytical
Geometry.
\-y-
(96.) Cor. When the axes are rectangular, this
equation becomes
a w' + y y’ + 2 2' = r".
As this is the equation commonly used, we shall inves-
tigate it by a method analogous to that employed in
the case of the circle.
The equation to the sphere is
a 2 + y? —– 2* = rº . . . . (1,)
and the equation to any plane is
* cos p, r + y cos p, y + 2 cos p, z = p . . . . (2.)
Now when this plane touches the sphere, we have
a' / z'
p = r, also cos p, a = +. COS p, y = º, cosp, a = +,
a', y' and 2' being the coordinates of the point of con-
tact; hence, by substitution,
+ (x,y--yº-H = 2 ) = r,
Or a w' + y y' + z z' = rº,
as before.
In like manner, if the equation to the sphere be
(r. — a + (y – 8)* + (2 – )* = rº,
the equation to a tangent plane applied at a point
a', y', 2' will be
(a – a) (a’—a) + (y-B) (y'–6) + (2 – ) (z’— y) = rº.
(97.) To find the equation of a plane that shall be a
common tangent to two given spheres.
Let the axes be rectangular, and let us suppose for
simplicity that the plane of a y passes through the cen-
tres of the two spheres, and that the axis of a coincides
with the line joining their centres.
Hence if r, r" be the radii and of the spheres, and 6
the distance between their centres, the equations to the
spheres will be
a's + y”-- z's = r" . . . . (1,)
(c" – 3)”-- y” + 2" = r^*. . . . (2.)
The equation of the tangent plane to the first sphere
will be
a w' + y y' + z z' = r" . . . . (3.)
And in order that this plane may also touch the second
sphere, the perpendicular let fall upon it from the cen-
tre of the latter must = r".
Now the coordinates of the second sphere being
a = 6, y = 0, 2 = 0,
and r’ being = p, we have
3 ar' — nº ëº
w/r's + y” -- z” r
but as the spheres are situated between the tangent
plane and the plane of a y, the lower sign must be
taken,
r' = + 8 ºr' — rº
2
A 8 aſ — r?
*... ºr = — º
r
... y = + ( – ’). ... (4)
therefore, substituting this value of r" in (3) and trans-
posing, we have
zz'- r" – y y” — # (r — r") ar,
but z' = V r" — a " — y”,
2 = 3 (r” – y y') — r (r – r") r
M (r" — a * – y”) '
or substituting for a ſº its value in (4,)
_. 8 (r" – y y') — r (r – r") a
TV sº (rº - yº) = r(r – r').
which is the equation required.
If z be made = 0, we obtain
_ 3 r – (r – r") r
T v 5 - (r. 7)2
which was proved in Art. 35 to be the equation of the
common tangent to two circles.
2
3/
ON THE CYLINDER.
(98.) To find the equation to a cylindrical surface.
A cylindrical surface is generated by a straight line
which moves parallel to itself, and with its extremity
describes the perimeter of a given curve.
The straight line is called the generating line, and the
given curve, the directrin, or base.
Let the equations to the generating line, when in
any position, be
a = a 2 + a
. . . . (1,
y = b 2 + 3 (1,)
in which a, B are variable, and a, b constant, since
the line is supposed to move parallel to itself.
Let the equation to the directrix, which we shall
assume in the plane of a y, be
f (X, Y) = 0 . . . . (2.)
Then, since the generating line always moves through
a point of the directrix (a = X, y = Y, z = 0), we
have
X = a, and therefore from (1) X = a – a z,
Y = {3, and therefore Y = y – b z,
whence, by substitution in (2),
f { a - a 2, y – b z } = 0,
which is the equation required.
The surface generated is said to be that of a right,
or of an oblique cylinder, according as the generating
line is perpendicular, or inclined, to the plane of the
directrix.
Example. The equation to an oblique cylinder whose
base is a circle X2 + Y2 = 2 r X,
(a – a z)*-i- (y – b 2)” = 2 r (a – a z),
the origin being at the extremity of a diameter.
is
ON THE CONE.
(99.) To find the equation to a conical surface.
A conical surface is generated in the same manner
as a cylindrical surface, except that the generating line
instead of moving parallel to itself always passes
through a given point. The given point is called the
vertea of the cone.
Let the coordinates of the vertex be a ', y', 2'; then
the equations to the generating line will be
a — a ' - a (2 — 2') -
gy — y' = b (2 – 2') . . . . (1.)
Let the equation to the directrix, which we shall sup-
pose, as before, to lie in the plane of a y, be
f (X, Y) = 0 . . . . (2.)
Part II
S-N-
730
A N A L YT I C A L G E O M ET. R. Y.
Analytical
Geometry.
*-v-
Then, since the generating line must always pass
through a point of the directrix (p = X, y = Y, z = 0)
we shall have
X — a ' - – a 2’, or X = a," – a z',
Y — y' = — b 2', or Y = y' – b z',
therefore by substitution in (2)
f{a' – a z', y' – b z' } = 0,
Or f (a, b) = 0;
f
but = tº be *=%
2 - 2, 2 – 2
mº / emgºng f
therefore f ſt – a 9 - 9 | = 0,
U2 — 2" z-z'
which is the equation required.
Since the generating line may be extended inde-
finitely upwards, the conical surface will be composed
of two similar portions, one above, and the other
below the vertex; each portion is called a sheet, this
term being understood to bear the same relation to
surface, that branch does to curve.
The surface generated is said to be that of a right or
of an oblique cone, according as the generating line is
perpendicular, or inclined, to the plane of the directrix.
Ea'ample 1. The equation to a right cone whose
base is a circle
(X – c')* + (Y — y')* = rs,
is (; – ) + (v-Vy-º. ( – ’).
Example 2.
base is a circle
(X - a)2 + (Y – 3)” = rº,
The equation to an oblique cone whose
is
{ z r"—a z'—a (2–2)*} + 3 2 y'— y 2’-3 (2–2)*}”
= rº (2 – 2')”.
ON SURFACES OF REVOLUTION.
(100.) To find the equation to a surface of revolu-
tion.
A surface of revolution is generated by a curve
which revolves about a fixed line or aris, in such a man-
ner that each point of the curve may describe a circle
whose centre is on that line, and whose plane is per-
pendicular to it.
Hence if the surface be cut by a plane perpendicular
to the axis, the intersection will be a circle. The sur-
face may, therefore, be considered as formed by a circle
of variable magnitude, which moves parallel to itself
and meets the generating curve.
Let the equations to the generating curve be
f(x, y, z) = 0 - . (1.)
f' (a', y', 2') = 0} ‘’’
Then if r', y', 2' be the coordinates of any point in the
axis of revolution, the equations to the axis will be
a — r" - a (2 – 2')Y
s g º º 2.
9 – y' = b (2–2) (2.)
Hence the equation to a plane perpendicular to
axis is (Art. 82) -
a r + b y + z = c. ... (3)
and that to a sphere whose centre is (r', y', 2')
is (a — aſ)* + (y – y')*-ī- (2 – 2)” = r”.
the
Now we may conceive the circle which results from the
Part II.
intersection of this sphere by the plane (3) to be one S-v-
of those which compose the surface. Therefore c and rº,
or their equals, must be constant or variable together,
for the same points. In other words, one of them
must be a function of the other. Hence
(a — a ')* + (y – y')*-ī- (2 – 2')* = F(a a + b y + 2)
will be the equation required.
(101.) Cor. Let the axis of revolution coincide
with the axis of 2, then the equation to the variable
circle will be
2 = c, a " + y^ = r".
Whence the equation to the surface of revolution be-
COIlleS
a" + y^ = F(z).
In like manner, the equation to the surface will be
a" + 2* = F (y),
gy* + 2* = F(r),
according as the axis of revolution coincides with the
axis of y or of a.
(102.) Let the generating curve be
Example 1. A parabola, a = 2 p 2.
Then the equation to a paraboloid of revolution is
a” + y^ = 2 p z.
Example 2. An ellipse, a2 2* + bi wº = a” bº.
Then according as the revolution is performed about the
major or the minor axis, the equation to the ellipsoid of
revolution, or of the spheroid, will be
b° tº + a” (y” + 2*) = a” b” . . . . (1,)
Or, a”z" -- bº (P + y”) = a bº. . . . (2.)
The spheroid is of two kinds, the prolate and the
oblate; the former is represented by equation (1,) and
the latter by equation (2.) -
Example 3. In like manner, the equation to the
hyperboloid of revolution is
bº (as - yº) – a”2* = a bº.
Or
ON SURFACES OF THE SECOND ORDER IN GENERAL.
(103.) The general equation of the second degree
between three variables is, -
a rº-H b y” + c z*-H 2 aſ y z + 2 b'a z + 2 c' a y
+ 2 a” a + 2 b'ſ y + 2 cº 2 + d = 0.
The surfaces which are the loci of this equation are
called surfaces of the second order.
In the following investigations the coordinate planes
are supposed to have any inclination whatever, except
in those cases which are expressly mentioned.
The characteristic property of surfaces of the second
order is, that they cannot be intersected by a straight
line in more than two points. -
For let the surface be cut by the straight line
a = m 2 -j- a,
y = n z + 3.
Then at the points of intersection the coordinates of
the line and surface are identical ; therefore by substi-
tuting the values of a and y in the general equation
a aº + b y” + c 2* + 2 a' y z + 2 b'a 2 + 2 c. ry
+ 2 a." ~ + 2 b" y + 2 c” z + d = 0.
A N A LY TIC A. L. G E O M ET R Y. 73]
Then, since (h) and (k) are each of them parallel Part II
to (l) we have S-y-
m' (a m + c n + b') + n’ (c' m + b n + a')
Analytical we shall obtain a quadratic equation, which can only
Geometry, have two roots; hence the surface cannot be inter-
-v- sected by a straight line in more than two points.
Def. The portion of the line intercepted between
the two points of section is called a chord.
(104.) To find the locus of the middle points of any
number of parallel chords.
Let a = m 2, y = n. 2 . . . . (g)
be the equations to any straight line drawn through the
origin, and cutting the surface in the points P, p; take
O (a', y', 2') the middle point of any chord Q q parallel
to P p ; then the object of the proposition is to find the
relation between a', y', and 2'.
Let the origin be transferred to the point O, then the
equation to the surface becomes
a (a + r.)* + b (y –– y')* + c (2 + 2)” –– 2 a' (y -- y')
(2 + 2) + 2 b' (a + x') (2 + x') + 2 c'(r. -- w') (y -- y')
+ 2 a." (a + æ") + 2 b" (y –– y") + 2 c” (2 + 2")
+ d = 0,
and the equations to Q q will then become
a = 7m 2, 3) = n. 2.
Now the points in which Q q intersects the surface
will be found by supposing the variables ar, y, z iden-
tical in the two equations; we thus have
a (m2 + x')*-i- b (n 2 + y')* + c (2 + 2)” --2a' (n2+y')
(z + 2) + 2 b' (m. 2 + æ') (2 + 2') + 2 c (m z + æ')
(n 2 + y') + 2 a!' (m z + æ") + 2 b" (n 2 + y')
+ 2 c” (2 + z') + d = 0.
But since the chord Q q is bisected in O, the two
values of z in this quadratic equation will be equal,
with contrary signs; therefore the coefficient of the
second term will vanish ; whence collecting the terms
involving z we have
2 a ma' + 2 b m y' + 2 a'n 2' -- 2 a' y' + 2 b'm 2'--
2 b'a' + 2 c na' + 2 c'm y'+ 2 a."m + 2b"n–H 2 c"= 0,
therefore, dividing by 2, suppressing the accents of the
variables, and arranging the result with reference to
a', y, and 2, we have
(a m + c n + bº) r + (c’m + b n + a') y +
(b'm –– a' n + c) z + a!' m + b" n + c" = 0. . . . (1)
the equation to a plane, which is therefore the locus
required.
That plane which bisects a system of parallel chords
is called a diametral plane.
(105.) In like manner, if there be two other chords
a = m/z + a', y = m/z + 3'.... (h)
a = m." z + a!", y = n." 2 + 3". . . . (k)
the corresponding diametral planes will be
(a m' + c n' + b) a + (c/m' + b n' + a') y + (b’ m'
+ a'n' + c) z + a” m' + b" m' + c” = 0. . . . (2,)
(a 'm' + c'n' + b) a + (c' m" + b m” + a') y + (b/m"
-- aſ m” + c) z + a'm' -- b” n" + c = 0. ... (3.)
The direction of the plane (1) depends on the direc-
tion of the chord (g) which was drawn at pleasure.
We shall now fix the relative position of the other two
chords (h) and (k,) by supposing, first, that each of
them is parallel to the plane (1,) and next that either
of them (k) is parallel to the diametral plane (2) of
the other.
+ b'm + a' n + c = 0,
m" (a m + c'm + b/) + n' (c' m + b m + aſ)
+ b'm + a' n + c = 0;
Or,
b’ (m –– m') + a' (n + n’) + c (m n' + m mſ)
+ a m m' + b n n' + c = 0. ... (4,)
b' (m -- m/) + a' (n + n") + c' (m n" + n m”)
–– a m m' + b m, n' + c = 0. . . . (5 ;)
and since k also is parallel to (2,)
b' (m' + m") + a' (n' + n") + c'(m’ m" + n’ m")
-j- a m' m” + b nſ' n'+ c = 0. . . . (6.)
But since (k) is parallel to (1) and (2) it must be parallel
to the line of their intersection, therefore the diametral
plane (3) of the chord (k) bisects all chords which are
parallel to the intersection of the other two diametral
planes. -
And since (4,) (5,) and (6) are equations of symme-
trical forms, the diametral planes of (g) and (h) will
bisect the chords which are parallel to the respective
intersections of the planes (2) and (3,) and (I)
and (3.)
It appears, therefore, that each of these three dia-
metral planes bisects the chords which are parallel to
the intersections of the other two. 4.
Diametral planes thus related, are said to be conju-
gate to one another, and the intersections of each two
of them are called conjugate diameters.
The point in which any three diametral planes inter-
'sect one another is called the centre.
(106.) To find whether any system of conjugate
diametral planes can be rectangular.
In this problem we shall suppose, for the sake of
brevity, that the coordinate planes are rectangular.
When three diametral planes are conjugate to one
another, their equations are -
b' (m. -- m^) + a' (n + n) + c' (m n' + n m')
+- a m m' + b n n’ + c = 0
b' (m + m") + a' (n + n”) –– c' (m n" + n m")
+ a m m' + b m, n' + c = 0 -
b' (m/+ m'') +a' (n'+n") + c (m/n"+n’m")
-j- a m/m" + b n' m” + c = 0
Now, if these be supposed rectangular, the following
equations must hold true, Art. 88, :
1 + m m' + n n' = 0 -
1 + m/m"+ n/n" = º ... (2.)
1 + m m' + m n" = 0
The object therefore is to derive from equations (1) and
(2) the values of m and n, of m' and n', and of m”
and m'.
Multiplying the first of equations (1) by n", and the
last by n', and taking the difference of the products, we
have
(c -- b'm + a n) (n" — m/) + (a + c m + b m)
(m/n"— n' m”) = 0.
... (1)
732
A N A L YTIC A. L. G E O M ET R Y.
Analytical
Geometry.
The same operation being performed on the first and
last of equations (2,) there results
n" — n' + m (m/m"— m/m") = 0;
therefore eliminating n”— m/ from the last two equa-
tions, we have
f
b' + a m + *....(3)
c + b'm –– aſ a
In like manner, if the first and last of equations (1)
and (2) be successively multiplied by m", m', and the
difference of the respective products taken, we shall
have
m =
_ a' + c'm + b n
T e + b m + aſ n
From (3) is derived
, m (a + 2)-H b’ (I — mº)
a' m — c
70
... (4)
. (5,)
and from (4)
a' m” + (b' m + c – b) n – c'm — a' = 0.
If the value of m in (5) be substituted in this equa-
tion, the result is a cubic equation which must contain
at least one real value of m, to which there corresponds
a real value of m deducible from equation (3.)
It may be proved, in like manner, that there must
exist at least one real value of m' and m', and of m'
and m”. -
Now the cubic equations involving m, m' and m”
will be found identical, as may at once be inferred from
the symmetrical form of equations (1) and (2;) there-
fore m, m', 'm' are the three roots of the same cubic
equation. Hence it follows, that there can be only
one system of conjugate diametral planes that are rec-
tangular.
The intersections of each two of these planes are
called the principal diameters, and the points in which
they cut the surface are called the vertices.
(107.) To find the form of the equation to sur-
faces of the second order, when the coordinate planes are
parallel to a system of conjugate diametral planes.
The equation to any diametral plane is
(a m + c n + b') a + (c' m + b n + a) y +
(b'm + aſ n + c) 2 + a!' m + b% m + c' = 0.
If m and n, successively, be now supposed first to be
infinitely great, and next to be equal to 0, the result-
ing equations will be the equations to the diametral
planes which bisect the chords parallel to the axes of a,
of y, and of 2.
Hence when the coordinate planes are parallel to a
system of conjugate diametral planes, of the three
equations
a a + c y + b” 2 + a' = 0,
c' a + b y + a' z + b" = 0,
b'a 4- aſ y + c 2 + c' = 0;
the first ought only to involve ar, the second y, and the
third z ;
... a' = 0, b' = 0, c' = 0;
wherefore the general equation becomes
a r" + b y” + cz + 2 a!" a + 2 b" y + 2 c'' z + d = 0,
which is the equation required. -
Let the origin be now transferred to a point (a, B, 7)
which is effected by substituting in the last equation
a + a, y + £3, 2 + y,
for a’, Q/, 2 3 hence
a (p + a)* + b (y -- B)* + c (2 + y)* + 2 a." (a + a)
+ 2 b" (y –- B) + 2 cº (2 + q) → d = 0;
or, developing, and arranging the result,
a aº + b y + c z* + 2 (a a + a”) a + 2 (b 3 + b") y
+ 2 (c y +c/) 2 + a d”-- b 6* + c 0°-H 2 a” a + 2b B"
+ 2 c” y + d = 0;
hence if the last term be represented by f.
a w’ + b y” + c 2* + 2 (a a + a!") a + 2 (b 3 + b") y
+ 2 (c y + cºl) z + f = 0.
Now a, B, Y being arbitrary quantities, we may fix
their value by supposing that the coefficients of a, of
gy, of z = 0; we thus have
a a + a' = 0, b (3 + b" = 0, c y + c” = 0,
a/ b” c!!
a = — — ; 3 = ---, * = – H-.
which are evidently the coordinates of the centre, Art. 105.
Hence the general equation is reducible to the form
a z* + b y” + c 2* + f_ 0.... (1.)
This reduction has been effected on the supposition
that the general equation contains the terms a wº, b y”,
c 2*; if any one of these, a r" for instance, be wanting,
then since a = 0, we have
therefore, since the term 2 a” (a + a) cannot be taken
away, the equation assumes the form
b y” + c 2*-i- 2 a!' a + f = 0.
Now, by taking away the terms involving y and z, we
have determined only two of the three quantities a, B,
and Y ; we may fix the value of the third, a, by sup-
posing the last term to equal 0; a supposition which
is always possible, since a w” vanishing, that term is
only of one dimension in a. The equation thus
reduced will be of the form
b y” + c 22 + 2 a” a = 0. . . . (2.)
Hence surfaces of the second order are divisible into
two classes, characterised by equations of the form
A 29 + B y”-H Caº + D = 0,
A z* + B y? -- E a = 0.
Or
Cº, º –
and
ON SURFACES OF THE SECOND ORDER WHICH HAVE
A CENTRE.
(108.) In order to ascertain the different species of
surfaces represented by the equation
A 2* + B y” + Caº + D = 0,
we shall make successively r, y, and z equal to some
constant quantity, which amounts to the same thing as
cutting the surface by planes respectively parallel to
the coordinate planes. Now the nature of the inter-
section will depend, as was shown in Part I., on
the signs of the coefficients A, B, C. Hence, by as-
signing to these quantities all the varieties of sign
which they admit of, the above equation will assume
the following different forms:
(1.) A z* + B y” + Ca" + D = 0,
when A, B, C are all positive.
(2.) A 2* + B y” – C 2" —- D = 0,
A N A L YT I C A L G E O M ET R Y. 733
Analytical when two of the coefficients are positive, and one
Geometry, negative.
•-N2-’ (3.) A z*— B y” — Ca"-- D = 0,
when one of the coefficients is positive, and the other
two negative.
(4.) — A z* – B y? — Caº +, D = 0,
when the coefficients are all negative.
We shall now discuss these equations in succession.
(109.) I. When A, B, C are all positive, the equa-
tion is -
A 22 + B y? -- Caº –– D = 0,
in which the last term D may be negative, positive, or zero.
First. Let D be negative.
Then the equation is
A 2* + B y? -- Caº = D.
Let the surface be cut by planes parallel respectively
to the coordinate planes; then if a = a, the equation
becomes
A z* + B y” = D – C a”. . . . (1,)
which is the equation to the section made by a plane
parallel to the plane of y 2, In like manner, the
equations
A z* + Caº = D – B 8*. ... (2,)
By” + Ca" = D – A yº. . . . (3)
belong to the sections made by planes parallel respec-
tively to the planes of a z and a y.
These sections therefore are ellipses; which become
imaginary when
>+ V% = -1 v/? * Vº
* C * *Eºs H-, * > * A ’
and are reduced to a point when
T) DT mº
•= + V3, as a V#4–1 Vº
This surface is limited in all directions, and is called,
from the nature of its sections, the ellipsoid.
(110.) To find the traces, or principal sections of the
ellipsoid. -
These are determined by making r, y, and z succes-
sively equal 0 in the general equation, whence we
have
A z*-ī- B y = D,
A z* + C arº = D,
B y” + Caº = D.
It appears, therefore, that the principal sections are
ellipses.
(111.) To find the points in which the surface inter-
sects the three aces.
Referring to the last article,
In the first equation, let
gy = 0, . . . 2 = + Vº- O C, (fig. 30;)
in the second, smººt-mººn -
a = 0, ...y= + V; = on.
in the third,
T)
- 0, • . :- VR -- o
2 a = + A. O A
The lines A a, B b, C c are called the principal
diameters, or aires of the ellipsoid.
V () I., I
(112.) To express the equation to the ellipsoid in
terms of its principal diameters.
Let O A = a, O B = b, O C – c.
D D D
Then A = +, B = ºr, c = . ;
hence we have, by substitution,
22 y” drº
+ + + + z = 1. ... (1)
Or -
a” b” z” + a” cº y” + be cºa' = a” be cº .... (2.)
If any two of the coefficients be equal, for example,
those of aº and y”, then the equation becomes
b° 22 + c (tº + yº) = a cº,
which is the equation to an ellipsoid of revolution about
the axis of c. In like manner, if a = c, or b = c, the
resulting equation will be that to an ellipsoid of
revolution about the axis of b, or of a.
If a = b = c, the equation will be
2* + y” + æ" = a”,
which is the equation to a spherical surface.
(113.) Secondly. We have hitherto supposed D to
be negative, let it now be considered positive, then
A z* + B y” + C aº = — D,
which is impossible ; therefore the surface is in this
case imaginary.
(114.) Thirdly. Let D = 0,
..'. A z*-i- B y” —- Caº = 0,
which is the equation to a point.
Hence, the first species of surfaces that have a centre
is an ellipsoid, which in particular cases becomes an
ellipsoid of revolution, a sphere, a point, and an imagi-
nary surface.
(115.) II. When A and B are positive and C nega-
tive, the equation becomes
A 2* + B y” — Caº –– D = 0.
First. Let D be negative.
Then A z* + B y” – C tº = D.
Let - \
a = a, . . A 2* + B y” = D + C a” . . . . (1,)
y = 3, ... A z* – C w” = D – B 3*. . . . (2,)
2 = y, ... B y” – C at = D — A y” . . . . (3.)
It appears, therefore, that the sections of the surface
made by planes parallel to y z are in all cases
ellipses; the two remaining sections are hyperbolas.
Hence the surface is continuous, and is called the
hyperboloid of one sheet.*
The principal sections of the surface are
1. An ellipse, whose equation is
A 2* + B y” = D.
2. An hyperbola, whose equation is
A 2* – C are = D.
3. An hyperbola, whose equation is
B y? — Caº = D.
(116.) To express the equation to the hyperboloid of
one sheet, in terms of its principal diameters.
D Vº D
:- – b = *=== - fºssº
* For the explanation of the term sheet, see Art. 99.
6 c
Part II.
S-N-7
734
G E O METRY
A N A L YT I C A L
Analytical
Geometry.
S-N-º'
then the equation becomes, by substitution,
a” b” z* -- a” cºyº – be cº r" = as b% cº.
If in this equation b = c, we shall have
a? (2* + y”) — be as = as bº,
which is the equation to a hyperboloid of revolution.
about the axis of a.
(117.) Secondly. Let D be positive.
The equation then becomes
A 2* -- B y” – C as = – D.
Let
a = 2, . . . A z* + B y” = Ca2 – D . . . . (1,)
g = 3, ... A z* – C tº - – B 8°–D. ... (2)
2 = 7, ... B y” – C tº - – A * – D. ... (3.)
It appears from equations (2) and (3) that the sections
them.
of the surface by planes parallel to the planes of a z
and a y are hyperbolas. The section parallel to the
plane of y z is an ellipse so long as a” X- #. Or a >
V/P.
+ C
Hence, if two planes be drawn parallel to the plane
of y z at the distance + Vº. and — Vº
the surface will have no point situated between those
planes, but will extend indefinitely above and below
This surface is composed, ºtherefore, of two
distinct parts, and is for that reason called the hyper-
boloid of two sheets.
The principal sections are
1. An imaginary line, A z* + B y” = – D,
2. An hyperbola, A z* – C a” = — D,
3. An hyperbola, B y” – Caº = — D
These hyperbolas have evidently a common transverse
axis which coincides with the axis of a.
(118.) Proceeding as in the former case, we obtain
for the equation of the hyperboloid of two sheets re-
ferred to its principal diameters,
a? be z* + as cº y? – bº cºa” = — a” b” cº.
Of the three diameters 2 a, 2 b, 2 c, the first alone
meets the surface.
Uf in this equation b = c, we shall have
a? (2* -- y”) — b% a” = — a” b”,
which is the equation of a hyperboloid of revolution
about the axis of ar. -
(119.) Thirdly. If D = 0. * -
Then the equations of the two species of hyper-
boloid become
A 2* + B yº – C as ºr 0,
y” – C A.
2
Q?
2”
T. B
... }=f(#)
whence the surface becomes that of a come.
The equation to the two species of hyperboloid be-
comes, therefore, in particular cases, the equation to an
hyperbola of revolution, and to a conical surface.
Or
2
ON SURFACEs which HAVE NOT A CENTRE,
(120.) The general equation is
A 2* + B y” + E a = 0,
in which A and B may have the same, or different
signs.
1. Let A and B be both positive ; and since E
may be either negative or positive : first, let it be
negative. -
Then the equation is
A z*-i- B y” = E a.
Let r = a, ... A 2* + B y' = E a . . . . (1)
gy = 3, ... A 2* = E a - B 8° . . . . (2,)
2 = Y, ... B y” = E a - A Y" . . . . (3.)
The section made by the first plane is evidently an
ellipse, which is real so long as a remains positive.
When a = 0, the ellipse is reduced to a point, and it
becomes imaginary when a is negative. The surface,
therefore, extends indefinitely to the right of the origin,
in the direction of the axis of a, and is limited towards
the left by the plane of y 2, which it touches. The
remaining sections are evidently parabolas,
Let
E
y = 0, then 2* = Tº º
E
*
B
Hence the principal sections by the planes of a z and
a y are parabolas.
This surface is called the elliptic paraboloid.
Next, let E be positive, then the equation is
A z*-i- B y” = — Ea',
which becomes, if a be supposed negative,
A z*-i- B y” = E a ;
hence the surface is the same as before, only it now
extends indefinitely to the left of the origin.
When A = B, the equation becomes
* +y=# ,
which belongs to a paraboloid of revolution about the
axis of ar.
(121.) 2. Let A and B have different signs, the
equation will then assume the form
A 2* – B y” = Ea.
2 = 0, 3)" = J).
Let a = a, then A z* – B y” = E a . . . . (1,)
9 = 3, A 2* = B (3* + Ea. . . . . (2,)
2 = Y, B y? = A y” – E a . . . . (3.)
The first of these equations is that to an hyperbola,
whose transverse axis is parallel to the axis of a or of
y, according as a is positive or negative. The remain-
ing two equations are those to parabolas whose axes
are parallel to the axis of ar.
The principal sections have for their equations,
(1) A z* – B y” = 0,
(2.) A z* = Ea,
(3.) B y” = — Ea.
The first equation is that to two straight lines which
intersect at the origin, and the remaining two are the
Part II.
X-V-
GEOMETRY,
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g A N A LY TIC A. L G E O METR Y. 735
Now the principal sections made by the planes of a y, Part II.
Analytical equations to parabolas which have a common origin,
a z are in the ellipsoid, ellipses; and, in the hyper-S-N-
Geometry, and whose axes are A X, A X'.
S- This surface is called the hyperbolic paraboloud.
f (122.) No section of the hyperbolic paraboloid made
by a plane can be an ellipse.
For if a be eliminated between the equation to any
plane, and the equation
A z* – B y” = E w,
no term of the resulting equation can involve y z.
Hence, since the terms containing 2* and y” are of dif-
ferent signs, the equation cannot represent an ellipse.
It thence follows, that this species of paraboloid can
never become a surface of revolution.
(123.) We shall conclude this discussion of surfaces
of the second order with proving, that
The equation to the paraboloid may be deduced from
the equation to surfaces of the first class.
The equation to the ellipsoid, or hyperboloid of one
sheet, is ſº
a” y” zº *
* + š, H: ; = 1.
Let the origin be transferred to the extremity of the
diameter 2 a, which is done by substituting a - a for a ;
then the above equation becomes
3.2 y? 2° 2 a.
a? b? C? a '
therefore, multiplying each term by a, we have
2:3 Q, Cº.
... + iyº #: "= 2*.... (1.)
boloid of one sheet, an ellipse and hyperbola. Let m
and m! denote the distances from the foci of these sec-
tions to the vertex of the diameter 2 a.
Then if the centre be supposed infinitely distant,
2
Jº b2 G2
— = 0, also – = 2 m, and – = 2 m/.
Q. (7, (º,
Therefore, by substitution, equation (1) becomes
m! Jº HE m 2* = 4 m m'a,
which is the equation to the elliptic or hyperbolic para-
boloid, according as the upper or lower sign is used.
For farther information on the subject of Curves and
of Surfaces, the reader is referred to the following
works : - - - -
Annales de Mathématiques; Biot's Essai de Géomé-
trie Analytique, sixth edition; Bourdon's Application
de l'Algébre à la Géométrie ; Boucharlat’s Théorie des
Courbes et des Surfaces du Second Ordre ; Correspon-
dance sur l’Ecole Polytechnique ; Cramer's Introduc-
tion à l’Analyse des Lignes Courbes Algebriques; Euler's
Introductio in Analysin Infinitorum, tom. ii.; Gar-
nier's Géométrie Analytique ; Hamilton's Principles of
Analytical Geometry ; Journal de l’Ecole Polytechni-
que; Lacroix's Traité Elémentaire de Trigonométrie
Rectiligme et Spherique, et d’Application de l'Algèbre à
la Géométrie ; Lardner's Algebraic Geometry; Maclau-
rin's Algebra ; Monge's Application de l'Algébre à la
Géométrie; Peacock's Examples on the Differential and
Integral Calculus ; Poullet-Delisle's Application de
l'Algèbre à la Géométrie; Reynaud's Traité d’Appli-
cation d'Algèbre à la Géométrie.
5 (; 2
Comic
LINES OF THE SECOND ORDER,
C O NIC
on
SECTIONS.
THE principal properties of Lines of the Second
Sections. Order, or, as they are more generally termed, of Conic
S-N-' Sections, may be derived with great facility from their
Fig. 1.
equations, which have been already obtained in the
Article entitled Analytical Geometry. But with a view
of rendering the following Treatise independent of any
previous investigation, we propose to deduce those
equations from a general definition first assumed by
Boscovich,” and subsequently adopted by other writers
of celebrity, as the basis of geometrical systems of
Conic Sections.
(1.) Definition. A Conic Section is the locus of a
point, whose distances from a fixed point and a straight
line given in position, are to each other in a constant
ratio.
Thus, let S be a fixed point, Kk a straight line given
in position, P any point; join P, S, and let fall the per-
pendicular PQ upon Kk; then, if P be always taken
to PQ in the same constant ratio, the locus of P will be
a conic section.
The fixed point S is called the focus, and the straight
line Kik, given in position, the directria.
(2.) The particular species of the conic section will
depend upon the constant ratio of P S : P Q, which
may be either a ratio of equality, or of lesser or greater
inequality.
1. Let P S = P Q.
Then the locus of P is called the Parabola,
2. Let P S < PQ.
Then the locus of P is called the Ellipse.
3. Let PS > PQ.
Then the locus of P is called the Hyperbola.
ON THE PARABOLA.
CHAPTER I.
ON THE PARABOL. A. REFERRED TO ITS AXIS .
(3.) To find the equation to the parabola.
The parabola is the locus of a point whose distance
from the focus is always equal to its perpendicular
distance from the directrix.
Let S be the focus, K k the directrix, P any point in
the parabola ; through S draw the indefinite line ESX
* Element, Univers. Mathes, tom, iii. p. 1.
736
perpendicular to the directrix; from P let fall the per-
pendiculars PM and PQ on E X, Kk respectively,
and join P, S.
Then, if E S be bisected in A, the point A is, agree-
ably to the definition, a point in the parabola; from A
draw A Y at right angles to A X, and assume A X and
A Y as the rectangular axes, to which the parabola is
to be referred.
Let A M = a, M P = y, and A S =.m.
Then S P = PM*-ī- M S = y^+ (r. — m)*. . . . (1,)
but SP2 = P Q* = E M2 = (EA + AM)?
= (m. -- r)*. . . . (2.)
Whence, equating these two values of S Pº,
y” + (a — m)* = (a, -i- m)*,
. . . y” = 4 m a.,
which is the equation required.
(4.) To determine the figure of the parabola from
its equation.
The same axes being employed, the equation to the
parabola is
gy? = 4 m a.,
OI’ y = + 2 vºma.
Let a = 0, ... y = 0;
therefore the curve passes through the origin A.
Let a be supposed to have any positive value.
Then, for each assumed value of a, there are two
equal values of y, with contrary signs; as a increases,
the values of y increase; and when a is taken indefi-
nitely great, the values of y will also become indefi-
nitely great.
Let a be now supposed to have any negative value.
Then the values of y being in this case imaginary, it
is plain that no part of the curve can lie to the left
of A.
The parabola consists, therefore, of two infinite
branches A Z, A z situated to the right of the point A,
and symmetrically placed with respect to the straight
line A X.
The point A is called the verter, and the line A X the
aris, of the parabola.
Cor. 1. The parabola can have but one focus, and
one directrix.
Cor. 2. To find the value of SL, the ordinate pass-
ing through the focus.
At the point S, a = A S = m,
. . . y? = 4 m”,
... y or S L = + 2 m.
Parabola.
~~
C O N I C S E C T I O N S.
737
Conic
The double ordinate Ll passing through the focus,
Sections, is called the principal parameter; or latus rectum, of
u-y- the parabola.
Cor. 3. Hence, if P be any point in the parabola,
P M* - L. l. A M ;
that is, the square of the ordinate is equal to the latus
rectum multiplied into the corresponding abscissa.
(5.) To find the intersection of a straight line with
a parabola.
Let the equation to the proposed straight line be
y = a a + £3. . . . (1.)
Then the coordinates of the point or points of inter-
section with the parabola will be determined by com-
bining this equation with the equation
y” = 4 m a... . . (2.)
Substituting, then, in (2) the value of a derived from
(1,) we have
gy” – 4 m y - 8.
Cº,
4 4
Or y – tº + ** -- 0
This quadratic gives two values of y, which, substituted
in (1), furnish two corresponding values of a , there-
fore the coordinates required may be determined.
When the two roots of the quadratic are equal, the
points of section coincide, and the straight line Pp will
then touch the parabola; and when the two roots are
imaginary, the straight line P p falls entirely without
the parabola.
Hence it appears, that a straight line cannot cut a
parabola in more than two points.
That part of the straight line contained within the
parabola, is called a chord; when it passes through the
focus, it is then called the focal chord.
(6.) To find the equation to a straight line that
touches a parabola in a given point.
Let a', y' be the coordinates of the given point,
and a ", y” those of any other point in the parabola near
the first.
Then the equation to the straight line drawn through
these two points, and cutting the parabola, is
y” — y'
a' — aſ
gy – y' = (a – a ') . . . . (1.)
But these points being in the parabola, we have
y^2 = 4 m a.',
and y"2 = 4 m a.",
... y” — y” = 4 m (r" — w),
gy" *º $y' tº 4 7??,
FIT - WI7.
therefore equation (1) becomes, by substitution,
4 m
gy — y' = 7Tyſ (a – a ').
Let the points (r', y') and (r", y") be now sup-
posed to coincide, then r" = wº, y' = y', and the
secant will become a tangent.
Hence the equation to the tangent is
2 m.
gy – y' = # ( – ’).
in which a', y' are the coordinates of the point of con-
Parabola.
tact, and r, y the variable coordinates of any point S-
whatever, in the tangent.
Cor. The equation to the tangent may be presented
under a more commodious form; for multiplying each
side by y' we have
y y' — y” = 2 m a - 2 m a',
Ay” = 4 m a',
... y y' = 2 m a + 4 m a' – 2 m a',
= 2 m (a + æ"),
which is the equation most commonly used.
(7.) To find the intersection of the tangent with the
(L.J. Z.S.
In the equation y y' = 2 m (a + æ).
Let y = 0, as at T, then a + æ' = 0,
Or a =r – a ',
that is, A T = A M ;
the negative sign merely implying that A T must be
measured in the contrary direction to A. M.
Cor. 1. Hence M T = 2 M A.
Def. The line M T intercepted between the foot of
the ordinate, and the point where the tangent meets the
axis, is called the subtangent.
It appears, therefore, that the subtangent is equal to
twice the abscissa.
Cor. 2. Hence is derived a simple method of drawing
a tangent to a parabola at a given point.
Let P be the given point, and A M, M P its coor-
dinates; in M. A produced take A T = AM, join TP,
then T P touches the parabola in P.
Def. The straight line which is drawn from the point
of contact at right angles to the tangent, is called the
normal.
but
(8.) To find the equation to the normal.
Let T P touch the parabola in P, and from this point
draw Pg at right angles to PT.
Then, since P g is at right angles to PT, whose
equation is
ſy 2 m,
y – y = ± G-:)
the equation to Pg will be
|
y – y' = — # (, – 1).
(9.) To find the intersection of the normal with the
(1,172S.
When the normal cuts the axis, as at G, then y = 0,
f
ſ $/
* — = — — — m/
... — y 3. Gº a',)
‘.. a - r" = 2 m ;
that is, A G – A M, or M G = 2 m.
Def. The line M G, intercepted between the foot of
the ordinate and the point where the normal cuts the
axis, is called the subnormal. *
Hence it appears, that the subnormal is equal to half
the latus rectum.
We have considered the normal P g as an indefinite
line, but it is customary to give that name to the straight
line P G intercepted between the point of contact and
the point in which Pg cuts the axis.
Fig. 4.
738
C O N I C S E C T I O N S.
Sections.
Conic
5
(10.) To draw a tangent to a parabola from a given
point (r", y”) without it.
Let a', y' be the unknown coordinates of the point of
contact.
The equation to the tangent being in general
2 m,
v= ºr (a + a'),
and the point (a!", y") being, by hypothesis, a point in
the tangent, we have
- gy” = +G'+*) . . . . (1.)
Also, the point of contact (r', y') being in the para-
bola, y” = 4 m a' . . . . (2 ;)
hence, by means of these two equations, the coor-
dinates a', y' of the point of contact may be deter-
mined.
Since the equation which results from the elimina-
tion of a between (i) and (2) is of the second degree,
it follows, that there are two points of contact, or that
two tangents may be drawn to a parabola from a given
point without it.
In general, the values of a, and y, obtained by elimi-
nation between any two equations, are the coordinates
of the point or points in which the loci of such equa-
tions intersect. We may at once, therefore, in the
question under consideration, determine the position of
the points of contact by constructing the loci of (1)
and (2) in which a’ and y' are the variable quantities.
Now, the locus of (2) is the given parabola, and that
of (1) is evidently a straight line, whose position may be
assigned by making a' and y' successively - 0. ANA-
LYTICAL GEOMETRY, Art. 10.
//
If tº
a' = 0, then y'- 2 m 7°
gy' = 0, then a' = — a ".
Hence, take A T in the opposite direction to A M
a."
= a”, and in AY take A B = 2 m +7-; join T, B, and
f/ >
let T B cut the parabola in the points P, p ; these
will be the points of contact required.
Cor. 1. Since the straight line which has just been
constructed determines, by its intersection with the para-
bola, the points of contact, it follows that the equation
Ay" y' = 2 m (r' + æ"),
in which a' and y' are variable, is the equation to
the indefinite straight line joining the points of
COntact.
Cor. 2. Since AT is independent of y”, it will remain
the same for all points whose abscissas are = w!", that
is, for all points in the indefinite line Q q drawn through
Q parallel to A Y. Hence the following theorem:
If from the several points of a line, perpendicular to
the aaris, pairs of tangents be drawn to a parabola, the
chords joining the points of contact in each case will all
pass through the same point.
Cor. 3. If the given line be the directrix, then
a" = — m ; therefore A T = m = A S, and all the
chords will in this case pass through the focus. Hence
the equation to the focal chord of contact is
y” y = 2 m (r' — m).
Parabola.
CHAPTER II. N-V-2
ON THE PARABOLA REFERRED TO THE FOCUS.
(ll.) To find the polar equation to the parabola, the
focus being the pole.
Let P be any point in the parabola, PQ and PM Fig. 6.
perpendiculars on the directrix and axis respectively,
and join P, S.
Let PS = r, angle A S P = w.
Then S P = PQ = E M = E S + S M,
... r = 2 m + r cos PS M,
2 m — r cos w,
2 m.
1 + cos w
770,
Tºmm
. (1,)
•". r =
Or
. (2.)
tº )
cos” —
2
Cor. 1. If P S be produced to meet the parabola in p,
and Sp be denoted by r", then since A S p = r — w,
r' = 2 m,
T I – cosº. '
777,
Or -:
in? {R}
l -
S1 (l 2
Cor. 2. Hence
l I 1 + cos w 1 — cos w 2
++ · = −5; + -g- = gº
} I 2
* SP + sp=si.
That is, the principal semi-parameter is an harmonic
mean between the segments of any chord drawn through
the focus.
r + r"
1 | 2
Cor. 3. Since — — — — = y wº-º-º-º-º-º:
r */ 7° 7'
d also =
and also = g;
2
... r r" = m (r. 4- r"),
that is, S P. Sp = m. Pp.
(12.) If from the point of contact two straight lines
be drawn, one parallel to the aris, and the other to
the focus, they will make equal angles with the tangent.
Let T P be a tangent, from P draw PX' parallel to
A X, and join PS; the angles t PX' = S PT.
For since A T = AM,
S T = S A + AT = m + r = ... SP, (11,)
therefore angle S P T = angle STP,
= t PX’,
because PX' is parallel to AX.
Fig. 7.
(13.) The tangent at any point, and the perpendicu-
lar let fall upon it from the focus, intersect AY in the
same point.
Let TP be a tangent at P, from S let fall the per-
pendicular S Q upon it, to prove that Q is a point in
A. Y.
The equation to PT is
2
v=#&# *) . . . . (1.)
Fig. 8.
C O N I C S E C T I O N S.
739
Comic
and the equation to S Q drawn from S (w = m, y = 0)
Sections. perpendicular to T P is
Fig 9.
Fig. 9.
f
y = - #– G – m) .... (2)
2 m.
Now when PT, S Q meet A Y, a must = 0 in both
equations, we have therefore from (1)
a’ 3/* 3/
= 2 m – E 2 . -—, E - ,
3/ 777. y ”. In Ay’ 2
ſ
and from (2) y = % ;
and as these values of y are identical, PT and S Q
meet A Y in the same point.
Cor. Hence S T . SA = S Q?, or since S T = S P,
S P. SA = S Q2.
(14.) If two lines be drawn from the focus, one to the
point of contact, and the other to the point in which the
tangent meets the directria, they will be at right angles
to each other.
Let the tangent at P meet the directrix in Q, then
drawing SP, SQ, it is required to prove that S P is
perpendicular to S Q.
The equation to the tangent being
2 m.
9 = −7– (w + a').
3/
When it meets the directrix, a = — m,
2 m,
/ (2' ſº 7m).
... y or E Q =
Now the equation to S Q is
gy = — tan Q S E (a — m),
Q E
* -e- --—
2 ºn (a — m),
a' — m
H (a — m) . . . . (1.)
3/ -
Also, the equation to S P is
y = tan PS X (a - m),
º
*===w s e
/
$/
a' — m
*-* * > (a — m) . . . . (2 ;)
therefore comparing the coefficients in (1) and (2) it
follows (ANALYºric AL GEOMETRy, (Art. 14,)) that SP
is perpendicular to S Q.
The proposition may be at once proved by taking
the equation to the focal chord of contact. For that
equation being
2
! --> # (r' — m), Art. 10, Cor. 3,
and the equation to S Q being
Q E
y = - is (, – m),
gy"
= - a . (a — m), since Q E = y”,
it follows, that S Q is perpendicular to S.P.
CHAPTER III.
ON THE PARABOLA REFERRED TO ANY DIAMETER.
(15.) To find the locus of the middle points of any
mumber of parallel chords.
Let P p be any chord, O its middle point; from the
points P, O, p let fall the perpendiculars PM, ON,
p m on the axis A X ; then if the equation to P p be
3y = a a + £3,
the equation containing the values of y at the points
P, p will be
4 m.
* — —
3/ (I,
4 m (3
y + → = 0. Art, 5.
Now, since in any quadratic equation the coefficient
of the second term with its proper sign is equal to the
sum of the roots with their signs changed,
4. -
** = P M + p.m.
0.
But O being the middle point of Pp
PM + p m
2
. 2 m
Gºº
*-s º e
O N =
O.
Now m is constant, and 2 remains the same for all
chords parallel to Pp, therefore this value of O N is in-
variable ; in other words, the equation to the middle
points of any number of parallel chords is
3y = constant,
therefore (ANALYTICAL GEoMETRY, Art. 4) the locus
required is a straight line parallel to the axis A X.
Def. 1. The straight line which has just been shown
to bisect any number of parallel chords is called a
diameter.
Def. 2. Each half of the chord, so bisected, is called
an ordinate to the diameter bisecting it.
Cor. 1. The diameters of the parabola, are parallel
to the axis, and intersect the curve only in one point.
The truth of the first part of the corollary is evi-
dent from the proposition ; that of the second may be
thus proved.
The equation to any diameter is
*/ = c, c being a constant quantity;
therefore the intersection of the diameter with the
parabola will be determined by combining this equation
with the equation
we therefore have
y” = 4 m a ,
cº = 4 m r,
2
C
".. ºr = "e
4 m.
Hence there is only one point of intersection,
Cor. 2. If the equation to any chord be
y = a a + 3. . . . (1,)
the equation to a diameter passing through any point
(w', y'), and bisecting that chord, will be
2 m.
w’ = — .... (2)
Conversely, since & º ——
Parabola.
Fig. 10.
740
C O N I C S E C T I O N S.
Comic
Sections.
S--~~~~
Fig. l I
the ordinate to a diameter passing through the point
(*', y') will have for its equation
a + 3 . . . . (3.)
Cor. 3. Comparing equation (3) with the equation
to a tangent at the point (r', y'), (6) it appears that the
tangent applied at the vertea of any diameter is parallel
to the ordinates of that diameter.
(16.) To find the equation to the parabola, when
it is referred to any diameter and the tangent at its
verter, as a res.
Let PX' be any diameter, and PY’ the tangent at
its vertex; draw any chord Q q parallel to PY' meet-
ing PX' in V; and let P W = a, W Q = y
Then since the chord Q q, being parallel to the tan-
gent, is bisected in V, V Q = V q, that is, for any as-
sumed value of a there are two equal values of y with
opposite signs. As the same thing holds true for all
other chords drawn parallel to PY', the equation re-
quired must necessarily be of the form
gy* = M. r. . . . (1,)
in which M is some constant quantity.
In order to determine M, let Q'S q' be the position
of the chord when it passes through the focus, join
PS, and produce W P to meet the directrix in O.
Then PS and PV making equal angles with PY',
and therefore with Q' q', P W = PS = PO, therefore
OV is bisected in P and r = PS'.
Also, Q q = sum of the distances of Q and q from
the directrix = 2 O W = 4 O P = 4 PS,
... y = 2 P S.
Substituting these values of r and y in (I) we have
4 PS2 – M. PS,
gy = 2 m.
9'
... M = 4 P S ;
therefore the equation required is
3/2 = 4 P.S. r.
Cor. 1. Hence if m denote the distance of the origin
from the focus, the equation to the parabola is always
gy* = 4 m a.
Cor. 2. It appears that the ordinate passing through
the focus is equal to four times the distance of the
vertex from the focus, this quantity is called the para-
(meter of the diameter.
Cor. 3. Hence, at any point of the parabola, the
square of the ordinate is equal to the parameter mul-
tiplied into the corresponding abscissa. -
(17.) The equation to the tangent, when the para-
bola is referred to any diameter, is of the same form as
before, namely,
2 ºn
y ==#-G + 2)
denoting in this case the ratio of
the coefficient ſ
the sines of the angles which the tangent makes with
the axis of a and y.
Cor. 1. When the tangent meets the axis of a, then
y = 0, . . . a = — a '; -
that is, the subtangent is bisected by the curve, whether
the coordinates are rectangular or oblique.
Cor. 2. Hence also, whatever be the inclination of the Parabola.
axes, the equation of an ordinate to a diameter passing Q-y-
through any point (r', y') is
2 m.
y = --> + 3.
See Art. 15, Cor. 2.
(18.) If from the several points of a line given in
position, pairs of tangents be drawn to a parabola, the
lines joining the corresponding points of contact will all
pass through the same point.
Let M N be the given line, and AX the axis of the
parabola ; through any point in A X draw a chord
'm n parallel to MN, let PX' be the diameter which
bisects this chord, and at the vertex P apply a tangent
PY’, which will be parallel to M. N.
Then the equation to the parabola when referred to
the oblique axes PX' and PY' will be
y” = 4 m a. . . . (1,)
and if from any point (a!", y") in M N a pair of tangents
be drawn to the parabola, it may be shown precisely
as in Art. 10, which is a particular case of the question
under consideration, that the equation to the line join-
ing the points of contact is
Ay” y' = 2 m (a + æ"). . . . (2,)
in which a' and y' are the variable coordinates of the
point of contact.
Let the chord (2) cut the axis of a, then y' = 0, and
therefore a' = — ar";
hence the point of intersection will be the same for all
points whose abscissa = a,", that is, for all points in
the given line M. N.
(19.) If from the point of intersection of two tangents
a diameter be drawn, it will bisect the line joining the
points of contact.
From the equation to an ordinate to the diameter
passing through (a", y") is (15, Cor. 2,)
2 m.
y = +, +3,...(1)
And the equation to the line joining the points of
contact is
2
y' = # (a' + æ") . . . . (2,)
therefore the latter being parallel to the former is also
an ordinate, and consequently bisected. "
(20.) If through any point within or without a para-
bola two straight lines, given in position, be drawn to meet
the curve, the rectangle contained by the segments of the
one will be to that contained by the segments of the other
in a constant ratio. -
Let O be any point within or without a parabola,
and let any two straight lines drawn through that
point meet the curve in the points R, r and Q, q; to
prove that the ratio of
O R. Or; O Q. O q is given.
Through O draw the diameter P X', then the equation
to the parabola when referred to that diameter, and the
tangent at its vertex, is y” = 4 m a..... (1.)
sin r, a sin r,
Let O R = r, PO = S, i. --- , * * * = q.
Sln ar, y Sln ar, º/
Fig.
º
12.
C O N I C S E C T I O N S.
74.1
Comic
Sections.
\-N-"
Fig. 13.
Then y = p r,
a = 8 – q r,
therefore by substitution in (1)
p” r" = 4 m 3 – 4 m q r,
4 m 3
pº
in which the two values of r are O R and O r ; therefore
by the theory of equation
4 m q
2
... ré +
7" —
=- 0,
tºyºs 3
O R., O r = 4 m. In like manner, if O Q = r^, and
sin r", a sin r", – 4 m 8
e :- **, * = q', O Q. O q = F-;
Sln a, gy Sln aſ, ºy p
... O R. Or : O Q. O q :: p”: p”;
but the directions of r, r' being by hypothesis given, the
quantities p” and p” are known, therefore these rect-
angles are to each other in a given ratio.
CHAPTER IV.
MISCELLANEOUS PROPOSITIONS.
(21.) A parabola being traced upon a plane, to find
the position of its aris.
Draw any two parallel chords Pp, Q q, and bisect
them by the line MN; that line will be a diameter.
Art. 15.
In this diameter take any point, and through it
draw R r perpendicular to MN, meeting the para-
bola in R, r ; then if R r be bisected in O, and A O X
be drawn parallel to MN, it will be the axis required,
as is evident.
(22.) Let Pºp be any chord cutting the aris in O, and
let A M, A m be the respective abscissas of P and p, to
prove that
A M . A m = A O°.
Let the equation to P p be
y = a a + b . . . . (1,)
then the abscissas AM, A m will be found by elimi-
nating y between this equation and
y? = 4 m a . . . . (2;)
we therefore have
(a a -i- b)* = 4 m a.
But at the point O where Pp cuts the axis, y = 0,
. . . & E — b. = A O,
(,
2
...Ao’ - = ... AM. Am.
as was to be proved.
(23.) In the aris A X of a given parabola to find a
point O such that if any chord whatever P O p be
drawn through it, the angle PA p may be a right angle.
vol. 1.
Since the proposed property is, by hypothesis, true of Parabola,
all chords whatever drawn through P, we shall take
that which is at right angles to the axis.
Let POp, therefore, be perpendicular to AX, and
join A P, A p.
Then at the point P, if a., y be its coordinates,
gy* = 4 m a. . . . . (1.)
But since A X evidently bisects PA p, the angle PAO,
and therefore also A PO, is half a right angle,
... A O - O P,
therefore substituting a for y in equation (1)
OT
a” = 4 m w,
. . a = 0, and = 4 m.
The first value of a corresponds to the origin, the
second to the point O; whence it follows, that a point
whose distance from the vertex is equal to the latus
rectum, has the property above mentioned.
(24.) If pairs of tangents to a parabola be always
supposed to intersect at right lines, to find the locus of
their intersection.
Let = a a + £3. . . . (1)
be any line cutting a parabola
gy* = 4 m a. . . . . (2,)
then the equation which contains the values of (y) at
the points of intersection is (5)
4 m. 4 m /3
* — — — `" *
Q/ y + H+ = o' ... (3)
but when these roots are equal, the intersecting line be-
comes a tangent; hence equation (3) is in this case a
perfect square, the criterion of which is, that four times
the product of the extreme terms is equal to the square of
the mean ; we have therefor
m 3 7m2
16 —- = 16 —,
Cl, a?
7??,
a = 3 = y – a v,
. m = a y – a” ar,
as — 4 a + -- = 0,
tº £
in which equation the values of a are the trigonome-
trical tangents of the angles which the two tangents to
the parabola make with the axis; therefore the pro-
*ś
*
777, tº
duct of these values = - , and also — I, since by
º
hypothesis the tangents are at right angles to each
other,
OF … a = - 7m,
hence the locus of their intersection is the directrix.
5 D
742
C O N I C S E C T I O N S.
Conic
Sections.
\-y-Z
Fig. 2.
O N T H E E L L IPS E.
CHAPTER I.
ON THE ELLIPSE REFERRED TO ITS AXIS.
THE ellipse is the locus of a point whose distance
from the focus is always less, in a given ratio, than its
distance from the directrix.
(25.) To find the equation to the ellipse.
Let S be the focus, K k the directrix, P any point in
the ellipse; through S draw the indefinite line E S X
perpendicular to the directrix; from P let fall the per-
pendiculars PM, P Q on EX, K k respectively, and
join P, S.
Let the given ratio of P S : P Q be as e : 1, e being
less than I ; then if S E be divided in A, so that
S A : A E :: e : 1, A is a point in the ellipse.
From A draw A Y at right angles to A X, and
assume A X, A Y as the rectangular axes to which the
ellipse is to be referred.
Let A M = a, MP = y, and AS = m. ty
Then S P = PM2 + M. S2 = y2 + (r. — m)?.... (l,)
but S P = e2 . P Q2 = e” (A E + A M)*
*(* + 2) ...e.)
gmºm
wimsºn
therefore equating (1) and (2)
9° + (a — m)* = m2 + 2 m e a + e” wº,
... y” = 2 m (1 + e) a - (1 — e”) a”,
2 m, )
a — atº ,
º gºgº 2
= (1 – e’) ( – 6
be assumed = a,
y? = (1 — e”) (2 a a - wº),
which is the equation required.
Cor. I. In A X take A a = 2 a, and bisect A a in C,
then at this point, a = a,
... y” = (1 – c’) a”,
... y = + a v1 – e’,
which is always real, since e < 1.
Hence if B C b be drawn through C at right angles
to A a, and C B, C b be each taken E a w/T e°, B, b
will be points in the ellipse.
Cor. 2. Let B b be denoted by 2 b,
or if
".
then b = + a vºl - e”,
-º-º-º-º-º-º-º-º: b
... VI – e' = + +,
~ 0.
therefore by substitution the above equation becomes
y = + % w/2 air – º .... (1.)
Def. The straight lines A a and B b, represented by
2 a. and 2 b, are called respectively the major and
ninor aires; the points A, a, B, b in which they meet Q-J-
the ellipse are called the vertices; and the point C in
which they intersect each other, the centre.
(26.) To find the equation to the ellipse, when the
coordinates are measured from the centre.
Let P be any point in the ellipse, let fall the perpen-
dicular PM on A a, and assume C M = a '.
Then the equation to the ellipse when the coordi-
nates originate at A is
2
g” #(ear – a "). . . . (1,)
a = A M = A C -- C M,
= a + æſ;
therefore substituting this value for a, we have
b%
- i. {2 a (a + æ') — (a + x')* },
b?
— (a) – a "). . . . (2,)
(2
tºº
ſººms
but
gy”
which is the equation required.
Cor. Suppressing the accent, which was only
used to distinguish the new from the old abscissa, we
have by multiplication and transposition
a” y” -- bºa” = a” b”. . . . (3.)
If each term be divided by a” b%, we have
2 *
# + · = 1.... (4)
Of the three last forms of the equation to the ellipse,
the equation marked (3) is the most frequently used.
When a = b, these equations represent the circle,
which is therefore a species of the ellipse.
(27.) Equations (1) and (2) when translated into
geometrical language, express a property of the ellipse.
For if P be any point, we have |
2 aa – a – (2 a - a) a = A M. M. a, and
a” – a " – (a + æ) (a — w!) = A.M. Ma,
B C2
ra, AM. M. a.
Or A M. M. a M P :: A C*: B Cº.;
that is, the rectangle contained by the segments of the
major aris is to the square of the ordinate, as the square
of the semiaris major is to the square of the semiaris
minor.
(28.) To determine the figure of the ellipse, from its
equation.
•. M P2 -
Resuming the equation a” y” + b x* = a” bº,
we have either
Ellipse.
C O N I C S E C T I O N S.
743
Conic
Sections. y = ++ waſ Tº.... (1)
\-ev-Z * — 0. *
Or * = + , 'W·V....(2) •
1. In equation (l,)
Let a = 0,
then g = + b = C B or C b.
Let y = 0,
then a = + a = CA or C a.
Let t a 3 + a,
then for each value of a there are two equal values
of y with contrary signs.
Let a = + a,
then y = + 0, that is, the ellipse cuts the axis of a at
the points A and a,
Let tº > + Q,
then the quantity under the vinculum being nega-
tive, the values of y are imaginary, and no point of the
ellipse can lie beyond A to the right, or a to the left.
It appears, therefore, that the ellipse is a continuous
curve, returning into itself, and divided by the major
axis A a into two equal parts.
In the same way it might be shown by discussing
equation (2,) that the ellipse has the form just assigned
to it, and that it is divided by the minor axis B b into
two equal parts.
(29.) Cor. To find the value of the ordinate passing
through the focus.
When the ordinate passes through the focus
a = m = a (1 – e),
b?
... y = x : 20 ( – ) – a ( – ) ,
= bº (1 — e) {2 – (1 — e) },
= b” (1 – e’),
b%
= z:
2 bº
b?
... y = + a ’ therefore the latus rectum =
The double ordinate passing through the focus is
called the principal parameter, or latus rectum,
2 2
therefore the latus rectum. a
Def. The line S C = a e, is called the eccentricity of
the ellipse.
(30.) To find the intersection of a straight line with
the ellipse.
Let the equation to the proposed line be
= a a + 8 . . . . (1.)
Then the coordinates of the point or points of inter-
section with the ellipse will be obtained by combining
this equation with that to the ellipse
a” y + b a' = a” b”. ... (2.)
Substituting, then, in (2) the value of a derived from
(l) we have
- fºx2
gº y” -j- b” (**) -: a” bº,
Cº.,
... (a” as + bº) y” – 2 bºg y + bº 6* = a bºa”,
• nº 2 bºg b° (32 – an as) - 0
• 3/ as a " + b” a” a” + bº -º-ºw Wºº &
From this quadratic are obtained two values of y,
which substituted in (1) furnish two corresponding
values of a ; therefore the coordinates required may be
determined.
When the two roots of the quadratic are equal, the
points of section coincide, and the straight line touches
the ellipse; and when they are imaginary, the straight
line falls entirely without the ellipse.
Hence it appears, that a straight line cannot cut an
ellipse in more than two points.
Def. The portion of the straight line contained with-
in the ellipse is called a chord ; when the chord passes
through the focus it is called the focal chord.
Elſpse.
(31.) To find the equation to a straight line which
touches the ellipse in a given point.
Let a', y' be the coordinates of the given point, and a ",
y" those of any other point in the ellipse near the first.
Then the equation to the line drawn through these
points is (ANALYTICAI, GEOMETRY, Art. 12)
" — ar'
y – y' =# (n − 2) ....(1)
But these two points being in the ellipse, we have
a” y” + bºa” = a” bº, and
a”y" + 5°r' = a, bº,
... a” (y” º 9°) + b” (r's tº- r”) : G,
te g" tºº gy” tº-ºn b?
e ap” — ºp'? tº-- * 2 ”
... (y"+y') (y" – y') ºf
(a' + æ") (a" — aſ) a?’
y" – y' .
. . --—r , ems tº
therefore equation (1) becomes by substitution
b° a '' –– a '
a; #. (a — a').
Let the point (a!", y") be now supposed to coincide
with (a', y'), then a' = a ', y' = y', and the secant will
become a tangent at the point (a', y'); hence the equa-
tion to the tangent is
3y — y' = —
2 |
f tº
y – y' = = , G-20,
Q,
in which r, y are the variable coordinates of any point
whatever is the tangent.
Cor. This equation may be presented under a more
commodious form, for multiplying each side by a y',
we have
a” y y' — a” y” = — bºr w’ + bºa”,
therefore transposing
a y y’ + bºw a' = a”y” + bºa”,
= ... a2 b",
which is the equation most frequently employed.
Cor. Let a = b, then the ellipse becomes a circle,
and the equation to the tangent at a point (r', y') in
the circumference is
3/ y! + aſ aſ E a”. %
(32.) To find the intersection of the tangent with the
ares of a and y.
5 D 2
744
C O N I C S E C T I O N S.
Conic
Sections.
S-N-sº
Flg. 15.
The equation to the tangent being
a” y y' + bºa w' - a” bº,
let it cut (1) the axis of ar, as at T ;
a”
a/ 3.
then
gy = 0, ... b” a w' = a” b%, ... a =
C A2
C M
(2) Let the tangent cut the axis of y, as at t;
Or C T ~
2
then a = 0, ... a y y' = a, bº, ... y = +,
C B2
C m
Whence it follows, that each of the semi-ares is a mean
proportional between the abscissa of any point, and the
part of the aris intercepted between its intersection with
the tangent and the centre.
OF C t =
a”
C T = −,
Jº
... M T – C T – C M,
2
= + – w
m 3.
a'
a? * - a'a
Cor. I. Since
a’
Def. The line MT intercepted between the foot of
the ordinate, and the point where the tangent meets
the axis, is called the subtangemt.
Cor. 2. The value of the subtangent being indepen-
dent of the ordinate y', it will remain the same for all
ellipses described upon the same major axis A a 5
now the circle is a species of ellipse, (26, Cor. 1;) hence
if on the major aris a circle be described, and the
ordinate MP be produced upwards to meet the circum-
ference in Q, the tangents applied at P and Q will inter-
sect the aris A X in the same point T.
This may be directly proved ; for the equation to a
line touching the circle at Q, is
3y y' + a w' = a”.
Let this line cut the axis of a, then y = 0,
9. -
* = CT,
£
* , º, tº
as in the ellipse.
Def. The straight line which is drawn from the point
of contact at right angles to the tangent is called the
normal.
(33.) To find the equation to the normal.
Let T P touch the ellipse in P, and from this point
draw P G at right angles to P.T.
Then, because P G is drawn through the point
(a', y') at right angles to PT, whose equation is
b2 º'
v- w = - # 7 (, – 20
the equation will be
(tº /
y – y' = + g + ( – ’),
in which aſ, y are the variable coordinates of any point , Ellipse.
whatever in the line PG, considered as indefinite.
(34.) To find the intersection of the normal with the Fig. 15.
ares of a and y. -
The equation to the normal being
a" y'
y – y' = +, g + ( – ’),
let it first cut the axis of a, as at G -
a? f
then y = 0, and – y' = r. # ( – ’).
b%
tº — a' = - Tºº a',
bº
a' — a = → a',
OT C M – C G,
b2
that is, M G = − a '.
(l,
Next, conceive the normal to cut the axis of y, as
at g; then a = 0,
a” f
... y – y' = — - . *r r',
3/ 3/ b? a’
a” ...,
- - Tº 3/ 2
a8 – bº
•". 3/ tº — a” gy'
The negative sign implying that the point g lies below
the axis A X.
Def. The line M G intercepted between the foot of
the ordinate, and the point where the normal cuts the
axis of ºn, is called the subnormal.
(35.) To draw a tangent to an ellipse from a given
point (a!", y") without it.
Let a', y' be the unknown coordinates of the point
of contact.
Then the equation to the tangent being, in general,
a” y y' + b c a' = ah b',
and the point (a!", y”) being by hypothesis a point in
the tangent, we have
a” y” y’ + bºr" a” – as bº. ... (1 ;)
also the point of contact (a', y') being in the ellipse,
a” y” + b x's = a” be . . . . (2;)
hence, by means of these two equations, the coordinates
a', y' of the point of contact may be determined.
Since the equation which results from the elimination
of a between (1) and (2) is of the second degree, it
follows that there are two points of contact; in other
words, that two tangents may be drawn to an ellipse
from a given point without it.
Instead of going through the operation of eliminating
we may, as in the case of the parabola, (Art. 10,) find
the position of the points of contact by constructing
the loci of (1) and (2,) in which aſ, y' are the variable
coordinates.
C O N I C
745
S E C T I O N S.
Conic Now the locus of (2) is the given ellipse, and the
Sections. locus of (1,) which is an equation of the first degree,
is a straight line, whose position is determined by
making a' and y' successively := 0.
Fig. 16. If, then, in the equation a” y' y' + bºa' a' = a” b”,
b%
7.
a' = 0, then y' =
2
gy' = 0, then a' = #.
$ b”
Hence, take C R = # and C r = T. join R, r, and
let R r meet the ellipse in P and p, these will be the
points of contact required.
Cor. 1. Since the straight line R. r, which has just
been drawn, determines by its intersection with the
ellipse the points of contact, it follows that the equa-
tion a y” y' + bºa," a = a” b”,
in which a' and y' are the variables, is the equation
to the indefinite line joining the points of contact.
Cor. 2. Because C R is independent of y" it will re-
main the same for all points whose abscissas are = w”,
that is, for all the points in the indefinite line Q q
drawn through Q parallel to C. Y. Hence, the follow-
ing theorem:
If from the several points of a straight line perpen-
dicular to the aris C X, pairs of tangents be drawn to
the ellipse, the chords joining the points of contact in
each case will all pass through the same point.
CHAPTER II.
ON THE ELLIPSE REFERRED TO THE FOCUS.
(36.) To find the distance of any point in the ellipse
jrom the focus. .
Let S, H be the foci, P any point in the ellipse, to
find the distance of P from S.
Let fall the perpendicular PM on C.A.
Then SP2 = S M* + M. Pº,
= (C S –CM)” –– M Pº,
= (a e – a Y” + y”,
= a” e” – 2 a e r + wº—H (1 – e’) (a’—a"),
= a” – 2 a e a + e” atº,
..". S P = a – e ar.
In like manner it may be proved, that
H P = a + ea.
Cor. Hence, by addition,
S P + H P = 2 a t- A a.
In other words, the sum of the focal distances is equal
to the major aris.
From this property the equation to the ellipse may
be deduced, as in the following article:
(37.) To find the locus of a point whose distances from
two fired points are together always equal to a given
guantity 2 a. *
Let S, H be the two fixed points, P the point whose
locus is required.
Join S, H, bisect SH in C, let fall the perpendicular
PM on SH, which produce indefinitely towards X ;
from C draw CY at right angles to C X, and assume
C X and CY as the axes of coordinates.
Ellipse.
Let C M = a, M P = y, and S C = c. Fig. 18.
Then S Pa = y2 + (c – a jº (l,)
and H P = y + (c + æ)” • * * V + 9
... H P – S P = (c -j- a)” – (c — ty”,
Or (HP+ SP) (H P – S P) = 4 ca,
- ... HP – S P = ***,
2 a.
2 c + .
- a
but H P + S P = 2 a.
... H P = a + -tº,
(º,
and SP = a – ºt.
Q,
Squaring these values, and adding the result,
- C2 a.2
SP” -- H P = 2 (e –– a? )
= 2 (y” + c + aº) from (1,)
2 gº
... y” + c + æs = a + º: 3.
and also
C2 a.2
& * - zºº — — rº
... y = a” — cº -- a? 30°,
a” — cº
— ſº,
0.2
= a” – c’ –
a? — C3
=–F– (a” — a "),
which is the equation to an ellipse, whose major axis
= 2 a, and minor axis = 2 Maº - c.
If a = 0, then y” = a” — cº = bº if b = the ordi-
nate drawn from C.
(38.) To find the polar equation to the ellipse, the
focus being the pole.
(1.) Let S be the pole.
Let S P = r, angle PS X = w ;
then (Art. 36,)
but
T = a – e ar,
a = C S – S M,
= a e – r cos (ºr — to),
= a e -- r cos w,
.*. r = a – a €” – e r cos w,
... (1 + e cos w) r = a (I — e”),
ºr --> a (1 — e”)
I + ecos w”
which is the equation required.
(2) Let H be the pole.
Let H P = r", and P H X = w ,
then r' = a + er,
746
C O N I C S E C T I O N S.
but
Conic
Sections.
S-V-'
a = C M = H M – H C,
= r cos w! — a e,
... r' = a + er'cos w! – a 6°,
... r" (1 – e cos w') = a (1 – e’),
... we & G = 2,
1 + ecos w'
which is the equation required.
Cor. 1. If P S be produced to meet the ellipse in p,
then since the angle A S p = 7 — w, we have
a (l – e’)
S p = —— .
p 1 — e cos w
Cor. 2. Hence
l l
*-is
Sp
_ 1 + ecos w
T "a (T- ºy
gººgº 2 2 e
T a (1 — e”) TST. "
tnat is, the principal semi-parameter is an harmonic
7mean between the segments of any focal chord.
l 1 S P + Sp
S P +-s; = S P. Sp
_ 2
- a (ITE)
I + ecos to
a (1 – e’) '
amº-ºrº
S P
ſº
*===º
Cor. 3. Since
and also
... sp. sp = + ( – e (SP + sp).
(39.) To find the polar equation to the ellipse, the
centre being the pole. -
Let C P = p and the angle P C A = v.
Then p” = a + y”,
a” + (1 – e”) (a” – a”),
e” a " + a” (1 – e”),
= e” p” cosºv -- a” (1 – e’),
... p" (1 — e” cosº v) = a” (1 — e”),
- - - I — e”
. . p = d. •ºmº-º-º-º-º-º-º-º-º-
1 — e” cos” w”
which is the equation required.
Pig, 17.
=
(40.) To prove that the focal distances of any point
make equal angles with the tangent at that point.
Let T P t be a tangent at the point P (w', y), draw
the normal PG, and join S, P and H, P
a? — b”
2
Fig. 19.
Then C G = a' = e2 aſ, (34,)
. S G S C – C G a e—e r" a – e r'
H G T S C-H C G T ae-Ees ºf T a Hez'
SP
- - HF:
‘. angle S P G = angle H P G. (Euc. vi. 3.)
But G P T = G Pt, ... S P T = H Pt,
as was to be proved.
(41.) To find the locus of the points in which a per-
pendicular from the focus upon the tangent at any point
*ntersects the tangent.
Let PT be a tangent at any point P (r', y'), and Ellipse.
SY a perpendicular let fall from S on PT, meeting it S-V-2
in Y, to find the locus of Y. Fig. 20.
From C let fall the perpendicular C Q on TP pro-
duced, and draw S q parallel to PT meeting C Q in q.
Then CY = CQ"-- QYs = C Q”--S g”,
= C Tº sinº T-H CS2 cos"T;
2
but C T ~~ + (32) and C S = a, e,
JC
aft
tº ; : º
. C. Y2 = Tºys sin” T + a” escosº T,
4.
- + (1 — cosº T) + a” e” cos"T,
4. 2
* . * #. (a” – e’aº) cos"T. ... (1.)
2 A
Nowtant--". # (31) = — b_ ſº === ,
a" ºy O, vº – aſ
b2 g/?
... 1 -- tan” T = I -- as (as -ºj
_ a” — (as – bº) wº
0.3 (a” tº- a's) 2
a' — a” eº ar's
T a” (a" — r")
a? — e” ar's
= −R-,
a? — a "
".. cosº T - 2, 3
a? — e? a "2
therefore by substitution in (1)
aft a” a” — ar's
tºº. *=º sº tº-ºº: 2 sº **-- -
C Yº -- a's a's (a” — tº) a” – ex r" '
05° g? d f
tººl *º sºmºsºmº * === 9.
T aſs r^2 (a” — a”),
*
a?
== { a, -o- + 4*) = a,
... CY = + a,
therefore the locus of Y is a circle whose radius is a,
and which is therefore described on the major axis A a
as a diameter.
(42.) The rectangle contained by the perpendiculars let
fall from the foci upon any tangent, is equal to the square
of the semi-aris minor. .
For if the perpendiculars SY and HZ be let fall Fig. 20.
from S and H on the tangent PT, then
SY = S T sin T,
but
2
ST = CT – c s = +- a e = + (a – ex).
... SY = + (a – er) sin T
tº — = . ea') sin T.
Similarly, H Z = # (a + er') sin T,
•. S Y
Hz = + (a” – e' r") sin” T. ... (1.)
$;
C O N I C S E C T I O N S.
747
Comic &” — wº
Sections. Now cos? T = - gº — e” a'2 º
... sin” T = 1 — cosº T,
a? — a "
= 1 - ------. F.,
dº — e” (C
* – e r".
= ~...T.
therefore by substitution in (1)
0% alº (l — e”)
sy. H z = + (2 – “r”). ==
= a” (1 – e’) = bº.
CHAPTER III.
GN THE ELLIPSE REFERRED TO ANY SYSTEM OF
CONJUGATE DIAMETERS.
SECTION I.
ON Conjugate DIAMETERS IN GENERAL.
(43.) To find the locus of the middle points of any
number of parallel chords.
Fig. 21. Let Pp be any chord, O its middle point, and X, Y
its coordinates.
From the points O, P, p let fall the perpendiculars
ON, PM, p m on the axis A X, then if the equation
to P p be
3y = a + +- £3,
the equation containing the values of y at the points
P, p will be
2 b” & , , bº (8° – a gº)
* — ---
3/ a? a” + b? 3/ a2, a3 + b2 0, (Art. 30.)
Now, since in any quadratic equation the coefficient of
the second term, with its proper sign, is equal to the sum
of the roots with their signs changed,
2 b2 &
==H = PM + p.m.;
but O being the middle point of P p,
on = **t P”,
2
ſe *_º b° 3
... Y - zºº. ... (1)
Now x = -(y –8),
C!,
a” a 3 te
- = H ... (*).
To obtain the relation between X and Y we must eli-
minate 8 between (1) and (2,)
... — — Y = –––X,
$2
... Y = — — — X
Q (t.
Now, a remains the same for chords parallel to Pp,
therefore the equation just found expresses the rela-
tion between the coordinates of their middle points,
and being of the first degree, the locus required is a
straight line.
Def. The straight line which has been proved to be
the locus of the middle points of any number of pa-
rallel chords is called a diameter, and the points in
which it intersects the curve are called the vertices.
2
(44.) Cor. The equation Y = — X is the equa-
a” a
tion to a line passing through the origin, which is in
this case the centre ; hence every diameter must pass
through the centre.
(45.) A diameter being drawn through a given point,
to find the equation to any one of its ordinates.
If a ', y' be the coordinates of the given point, the
equation to the diameter drawn through it will be
f
y = + ..... (1)
Let gy = a a -i- B . . . . (2,)
be the required equation to any ordinate,
* gy’ b”
then T.T. * - a” a (44.)
b2 º'
... a E — — — . —-,
- C." Wy'
therefore any ordinate to a diameter passing through
(r', y') has for its equation
b2 ºf
— . 7 –– 6.
y = - i.
Cor. Comparing this equation with the equation to
the tangent, it appears that the tangent applied at the
vertex of any diameter is parallel to the ordinates of
that diameter.
(46.) Any two diameters being given, if the ordinates
of one be parallel to the other, the ordinates of the latter
will be parallel to the former.
Let 3) = a + . . . . (1,)
3y = a'a. . . . . (2,)
be any two diameters CP, CD, then by the last article
the equations of any ordinates MN, Q R to the first
and second, respectively, will be
- * + 3.... (1)
y = —
a” a
2
b
y = - ºr r + 6.... (2)
Let the ordinate MN be now supposed parallel to the
diameter C D,
b%
Then
OT — —-
a” aſ '
therefore the equation to Q R becomes by substitution
in (2} 3y = x + -ī- (3',
that is, Q R the second ordinate is parallel to the first
diameter C P, which was to be proved.
Whence each of these diameters is parallel to the
ordinates of the other.
Ellipse.
\-N-7
ig
2
2
748
C O N I C S E C T I O N S.
Conic
Sections.
Diameters, thus related, are said to be conjugate to
each other.
\-y-. Cor. 1. Hence, when the two diameters
Fig. 23.
3y = a ar,
g = a'a,
are conjugate to each other,
Cor. 2. Therefore if
gy = a a
be any diameter,
5%
a” a
dºml
3/ * *-*.
will be the diameter conjugate to it. The number of
pairs of conjugate diameters is therefore unlimited.
If a = 0, or the first diameter be A a, then
b%
a” 0
therefore the diameter conjugate to A a, being at right
angles to it, is B b ; or, the aves of the ellipse are con-
jugate diameters. ©
Cor. 3. If (x', y') be any point in the ellipse, the
diameter passing through it is
£ t OO . Jº,
*-ºl
3/
v=# ,
* * * * * b2 g/
y = - 7.7
is the corresponding conjugate diameter.
But the equation to a tangent drawn through
(w', y') is
a’
gy — y' = — - . Sy’ (a — w'), (31,)
whence it follows, that the tangent applied at the verter
of any diameter is parallel to the corresponding conju-
gate diameter. -
(47.) It has just been shown in Cor. 2, that the axes of
the ellipse are conjugate diameters, we shall now prove
that the aires are the only pair of conjugate diameters
which can be at right angles to each other. -
For, if possible, let C P, CD be a pair of rectan-
gular conjugate diameters different from the axes,
and let -
angle P C A = 0, angle D C A = 0'.
Now 9 = D C A = D C P + P C A,
S- + + 6 by hypothesis;
b”
but — —z- = a a' (46, Cor. 1) = ... tan 6 tan 0",
(7,
_. sin 6 sin 6!
T cos 6 cos 9'?
... a” sin 6 sin 6' + ba cos 6 cos 0 = 0 . . . . (1,)
but
sin 6' = sin (; —— 6) = sin (; *º- •) = cos 6,
2 2
os 9= cosſ- 6) * - 0
C -: *s-s-s-s jº sº- — — 0 \ = —-si
2 + cos \-g sin 6,
therefore by substitution in (1) -
(a” — bº) sin 6 cos 6 = 0,
Or # (a” — bº) sin 20 = 0.
Now, since a is > b, this equation can be satisfied
only by supposing -
sin 26 = 0,
... 2 0 = 0, or = ir,
Ellipse.
ºr
6 = 0, or = −,
r=-3
and 6' = +, or = 0;
C P and C D must coincide with C A and C B respec-
tively.
(48.) To find the equation to the ellipse when it is
referred to any two conjugate diameters as awes.
Let C be the centre, CP, CD, a given system of
conjugate diameters, of which the former is supposed
to be the axis of a, the latter the axis of y. -
Take any point Q in the ellipse, and draw Q q
parallel to C Y meeting CX in V.
Let C W = a, V Q = y; also C P -- a', C D = bl. Fig. 24.
Then, since the chord Q q is bisected by C P in V,
V Q = V q.; and since every other chord parallel to
CY is big cted by CX, it follows that for each assumed
value of ºt" there are two equal values of y, with con-
trary signs. In like manner it may be shown, that for
each assumed value of y, there are two equal values
of a with contrary signs; therefore the equation re-
quired will be of the form
M y” -- Naº is P.
It now remains to determine the values of M, N, and
P. When the axis CX cuts the ellipse,
y = 0, and a = C P = aſ,
... Na’ = P = N a”,
... N = º
(Z
When the axis CY cuts the ellipse,
a = 0, and y = C D = b',
... M y” = P = M b”,
P
... M = Tºſſ-
Substituting these values of M, N, P in the above
equation, and dividing each term of the result by P,
we have
y” tº
+ + i = 1 .... (1)
Or a” y” + b” as = a” b/2 ....
either of which is the equation required.
(2,)
/ summemºmº
Cor. 1. Hence y = +4 A/a/* — a 2.
* Q.
Cor. 2. To find the form of the equation when the
coordinates originate at P, the vertex of the diameter
C P.
Let PM = a', then v = C P – PM = a' – aſ
Substituting this value of a in Cor. 1, we have
b/ .
lº ſ
Taſ vº aſ 7-aſs,
Q) = +
C O N I C S E C T I O N S.
749
Conic
Sections.
N-V-2
Fig 25.
or suppressing the accent of r
f -º-º: -
y = # 7 M2 aſ a Tºº,
which is the equation required.
Cor. 3. The equations (1,) (2,) and (3) are of the
same form as the equations in terms of the axes,
and express, when translatºg into geometrical lan-
guage, a property of which that in Art. 29 is only a
particular case.
For a” – a = (a! -H r) (a' — a = P V. V G,
and 2 a'r – º – (2 a' – 4) v = P V. V G,
vo = ** Pv. v G
tº G tº-s C D” º
PV. V. G. : Q Vs :: P C*: C D* :
that is, the rectangle contained by the segments of any
diameter is to the square of the ordinate as the square
of the semi-diameter is to the square of its semi-con-
jugate.
Or
(49.) It appears from the preceding proposition, that
the equation to the ellipse, whether the axes be rectan-
gular or oblique, is always of the same form ; whence
it follows:
(1.) That if the equation to the major axis A a be
3) = a £,
bys
y = — —7– a
then
a” a
will be the equation to the minor axis B. b. And
(2.) That the equation to the tangent will be
a's y y' + b's ra' = a” b”,
\ }
(50.) To find the intersection of the tangent with any
two conjugate diameters considered as awes.
Let a tangent applied at any point Q meet C P
in T and C D in t, and draw the ordinates Q V, Q v.
The equation to the tangent being
- a's 3/ y' + b” Jº a' == aſ? b's,
let the tangent meet C X as at T, then y = 0,
ſº 2
Cº. C P
... a = -r-, or C T =
Jº
C V
Let the tangent meet CY as at t, then r = 0,
- - - wº or C t = C D*
..". Jy = gy'' T C 0 °
whence the points of intersection required are found.
(51.) If from the several points of a line given in
position, pairs of tangents be drawn to an ellipse, the
lines which join the corresponding points of contact will
all pass through the same point.
Let C be the centre of the ellipse, MN the given
line.
Draw any chord m n parallel to MN, and bisect it
by the diameter C X; from C draw C Y parallel to
m n or MN, then C X, CY are conjugate diameters;
and if the ellipse be referred to these as axes, its
equation will be
a" y” + b's a' = aſs bº. . . . (1.)
From any point (a!", y”) in M N let a pair of tan-
gents be drawn to the ellipse, then it may be shown
as in Art. 35, which is only a particular case of this
WOL. 1.
and ..". T =
proposition, that the equation to the line joining the Ellipse.
points of contact is q -
a/2 y" y' + b/2 a' 4." F. a/? b/2. * G - (2,)
in which a', y' are the variable coordinates of
point of contact.
Let the line (2) cut the axis of a, then y' = 0,
aſ?
T;
hence the point of intersection will be the same for all
points whose abscissa = a,", that is, for all points in the
line M N, as was to be proved.
Cor. The point of intersection is situated on the
diameter conjugate to that which is parallel to the given
line.
(52.) If from the point of intersection of two tangents
a diameter be drawn, it will bisect the line joining the
points of contact.
the
f
For the equation ºto an ordinate to the diameter
passing through (w", y") is (45)
b° aſ:
~7, . # *-*... (1)
(2.
and the equation to the line which joins the points of
contact is
a’’ b'2
gy" *-i-....(?)
hence the latter, being parallel to the former, is also an
ordinate, and is therefore bisected.
y = —
b'2
f sºmº * > &=º
3/ g”
(53.) If through any point within or without an
ellipse, two straight lines, given in position, be drawn to
meet the curve, the rectangle contained by the segments
of the one will bear a constant ratio to the rectangle
contained by the segments of the other.
Let O be any point within the ellipse, through which Fig. 26.
draw the two lines P. p and Q q, whose position is sup-
posed known, meeting the ellipse in the points P, p and
Q, q 5 to prove that
O P. O p : O Q . O q in a constant ratio.
Through O draw the diameter CX, and let C Y be the
diameter conjugate to it; then if the ellipse be referred
to these diameters as axes, its equation will be
a”y” + b” as = a” b”. ... (1.)
Through P draw PM parallel to CY, and let
O P = r, CO = 3;
P M _ sin POM sin r, r
then = p (suppose)
P O T sin PM OT sina, y
tº dºmº • * ~ 1: º sin r, y_
... y = p r s in like manner if sina, ai = 1.
a = CO + O N,
= 3 + q r ;
therefore substituting these values of a and y in (1,)
a"pºr" + 5° 43' + 2 3 q r + q2 r" } = a” b”,
... (a's p" + 5° q”) r* + 23 q bºr + b% (3* — a”) = 0,
28 abº r + 5° (? – ?) =
aſ? p° + b” q° a/* p? + b'2 g” p
in which the values of r º P, Op,
..". " +


750
C O N I C S E C T I O N S.
Coni - —b"(3* —a”)
sºil. ... O P. O p = aſ pºR. Sºº
"T" In like manner,
_ – bº(3°– a”)
99.9q= z:x:y.
Fig. 27.
therefore
O P. Op: O Q. Q q :: a” p” -- bºg”. a's p” -- b” g”,
which is a constant ratio, as was to be proved.
SECTION II.
ON THE PROPERTIES of Conjugate DIAMETERs.
(54.) A diameter being drawn through a given point
(r', y') to find the coordinates of the point in which the
diameter conjugate to it meets the ellipse.
Let CP, CD be any two semi-conjugate diameters,
then the equation to C P being ,
gy = ºr a .... (1)
the equation to CD will (46, Cor. 2) be
b? a'
therefore the coordinates of the point D in which CD
cuts the ellipse will be determined by combining (2)
with the equation
a y” + 5°wº = a bº. ... (3.)
Hence, substituting in (3) the value of y in (2) we
have
ap'º
–* : * + w! 2- ºr
a' ' Jº + = a- 0°,
or, dividing by b",
2 /2
(; , ; + )*= a,
(I,
... (b" w” + as y”) as = a y”,
... as b2 tº — a y”,
2
(Z"
". a’ = + y”,
OT a: = + *— aſ
- L b 3/",
b2 ac'
therefore also gy = — a2 y dº,
~ * b a'
= + + r.
The signs of a and y being different, as they ought
to be.
(55.) The sum of the squares of any two semi-
conjugate diameters is equal to the sum of the squares
of the semi-aves.
Let CP, CD be any two semi-conjugate diameters ;
then CP2 = CM*-H M P = x/2 + y”,
CD2 = C mº –– ºn D = * * + . r”
b? 3/ (tº y
• , , ...? 2
. . CP” + C D* = (*#. r)+(~4. #. *).
mº b° tº + a gy” a. * g” + be a " Ellipse
=–#------→-, -
on bº gº bº
— — —- -a-,
= a 2 + bº.
(56.) If at the verties of any two conjugate dia-
meters tangents be applied so as to form a parallelogram,
the area of all such parallelograms is constant.
Let P p, D d be any two conjugate diameters, and Fig. 28.
let the tangents applied at P and p, D and d be pro-
duced to meet, then it is plain (45, Cor.) that they will
form a parallelogram.
From P and T let fall the perpendiculars PF, T Q,
on D C produced. -
Then the area of the whole parallelogram is equal to
four times the area of the parallelogram P D
= 4 PC . CD sin PC D,
= 4 C D . P. F. ... (1 ;)
but 2
w & ... 0. m D -
P F = TQ - CT sin T C Q = +. Hºa, (82)
Y a” p
... P.F. C D = + . m D.
2
= * : *-*. (34)
Jº (Z
= a b . . . . (2,)
therefore by substitution in (1,)
The area of the whole parallelogram = 4 a b,
and is therefore constant.
Cor. I. By equation (2) P F. C D = a b ;
but C D = b', and PF = PC sin P C D = aſ sin y if
q = PC D, -
".. a b = a' b'sin Y.
Cor. 2. Hence the value of PF may be found.
* a b
For PF=-air, -
but C D* = a” + b% — a”, (55,)
.*. PF -: a b
aſ a T5 - aſ g
(57.) To find the magnitude and position of two
equal conjugate diameters.
a' + b = a” + bº
a' = b',
•". 2 a." = a” + b ,
". * = + Vº # . . . . (1,)
therefore the magnitude of the equal conjugate, diame-
ters is found.
Again, their position may be determined.
For a b = aſ bº sin Y,
= ... a's sin Y, when a' = b',
In general,
Let
a b
... sin Y. – –F– , .
C O N I C S E C T I O N S.
751
Conic
Sections.
Fig. 23.
Pig. 49.
2 a b
5- ŽT; ... (2,)
which is their mutual inclination.
Also, their inclination to the major axis may be found,
beeause, being equal, they are symmetrically placed with
respect to the major axis, and are therefore equally
inclined to it; but in general -
2
tan PC X. tan D C X = — º,
5, 2
2
Or
—z- = tan” PC X,
a”
tan PC X tan D C a =
- b
... tan PC X = + +.... (3)
whence it follows, that the equal conjugate diameters are
parallel to the lines B.A, B a.
(58.) Of all systems of conjugate diameters, those that
are equal contain the greatest angle.
a b
aſ b’’
therefore the angle PC d is a minimum, or P C D a
maximum when the product a' b' is a marimum ;
that is, when a' = b', as was to be proved.
Cor. Hence it may be proved, that of all systems of
conjugate diameters the sum of those that are rectan-
gular is the least, and of those that are equal, the
greatest.
For
For, in general, sin Y =
a' + b = y (a” + b” + 2 aſ b/),
= V/(2 + º- #).
Therefore (1) a' + b is a marimum, when sin Y is
a minimum, that is, when a' = b'.
(2) a' + b' is a minimum when sin Y is a maximum,
T e e
that is, when ( := -g- or tne conjugate diameters are
rectangular.
(59.) The rectangle contained by the distances of any
point from the two foci is equal to the square of the
corresponding semi-conjugate diameter.”
Let P be any point, C D the semi-diameter conju-
gate to CP, join P, S and P, H.; to prove that
SP - H P – C D2.
C D* = ax + b” – C Pº,
a" + b – (rº + y”),
a? -- bº — a " — (1 — e”) (a” — rº),
a8+ b° — a " — a”--a”-- e. a-- er",
b'+ d'é – a wº,
a”-- e a ". . . . (1.)
a" – e' tº - (a — ea) (a + er)
= S P. H. P., (36,)
... S P : H P = C D*.
For
But
(60.) Let CP, CD be any two semi-conjugate dia-
meters, and let a tangent at P meet the aires of the
ellipse in T and t, to prove that PT. P. t = C D*.
If CP, CD be assumed as the axes of coordinates, Ellipse.
then the equations to CA, C B are respectively º
gy = a +, ig. 39.
/2
y = — T.J. J.T 37. (49.)
Let a = aſ or C P, then y or P T = a aſ in the first,
/?
and gy or P t = — in the second,
a' a
... PT . P # = — bº = C D*.
The product PT. Pt is negative because PT, Pt
being situated on opposite sides of the axis A X have
different signs.
SECTION III.
ON SUPPLEMENTAL CHoRDs.
Def. If from the vertices of any diameter two
straight lines be drawn to any point in the ellipse, they
are called Supplemental Chords.
(61.) Any two supplemental chords being drawn, and
the equation to either of them being given, to find the
equation to the other.
The ellipse being referred to any two conjugate dia- Fig. 31.
meters, its equation will be
a's y? -- b% aº = a” b” . . . . (1)
Through any point P(+', y') draw the diameter P p, and
let PQ , p q be any two supplemental chords; then the
equation to PQ being
y – y' = a (a — r').... (2)
it is required to find the equation to p Q.
The coordinates of P being r', y', those of p will be
— a ', - y', therefore the equation to p Q will be of the
y + y' = a ' (a + ar') . . . . (3,)
in which a’ is to be found.
Since the limes whose equations are (2) and (3) inter-
sect at Q, the coordinates of Q will be identical; there-
fore considering r and y as the same in these equations
we have by multiplying them together
gº — y” = a a' (rº — wº),
form
Q **** gy? tº º gy” g
tº ... a a” = †-..... (4)
but aſ and y being the coordinates of Q, a point in the
ellipse, they will satisfy equation (1,)
... a” y” + b” r2 = a's bº.
Subtracting (1) from this, we have
a” (y” — y”) + b” (rº — wº) = 0,
... yº–y” b°.
. . .T. = - in ;
therefore by substitution in (4)
b/2 b's
* * — " " - — —
• * = - ºr, . • &L = a/*a*
and the equation to p Q becomes by substitution
b/2
3y. - 3/ := - a'a (, – 40.
752
C O Nº I C S E C T I O N S.
Conic
Sections.
S-N-2
Fig. 33.
Fig. 34.
Cor. Let Pºp coincide with the major axis Aa, then
the equation to a Q drawn through the point (– a, 0)
will be
gy = a (a + a),
therefore the equation to AQ drawn through the point
A (+ a, 0) will be • ‘7.
(62.) If two diameters be drawn parallel to any sup-
plemental chords, they will be conjugate to each other.
The equations to any two supplemental chords being
v — w” = a (a – a ') . . . . (1,)
+G+2)....(2)
T J.T.,
let any diameter be drawn parallel to (1,) then its equa-
tion will be
3) + y' =
and
= a ac;
therefore the equation to its conjugate being
b/2
3/ := - a’” 0. • 30,
it follows that the latter is parallel to (2), as was to be
proved.
Cor. 1. Hence may be drawn a diameter which shall
be conjugate to a given diameter. -
Let P p be the given diameter, and
1. Let the major axis of the ellipse be given.
From a draw a R parallel to Pp, and join R, A; then
if D d be drawn through C parallel to RA, it will be
conjugate to Pp.
2. If the major axis be not given.
Draw any diameter whatever, R r ; through r draw
r Q parallel to Pp, join Q, R.; then if D d be drawn
through C parallel to R. Q, it will be conjugate to P p.
These conclusions are evident.
Cor. 2. Hence also is derived a very simple method
of applying a tangent at a given point of the ellipse.
Let P be the given point, and
1. Let the major axis be given. *
Draw PC, and the chord a Q parallel to it, join
Q A ; then if PT be drawn parallel to Q A, it will
touch the ellipse at P.
2. Let the major axis be unknown.
Draw any diameter whatever R C r, join P, C; draw
r Q parallel to PC, join Q, R.; then if PT be drawn
parallel to Q R it will be a tangent at P.
(63.) To find the angle contained by the supplemen-
tal chords, drawn from the extremities of the major
(ZºS.
Let the point Q (a', y') be the intersection of the
two chords A Q, a Q, and suppose the ellipse referred
to its axes. -
Then if the equations to Q a, Q A be
y = a (a + a),
a' (a — a),
smº
*
a'- 0.
tan A Q a = I-Faº (ANAL. GEOM., Art. 13)
_ a - a © f - - b”.
— — . . . . (l,) since aſ := as a
1 – F
Now a' = tan Q A X = — tan Q A a = - a — r"
gy'
and a = tan Q a X - a -- "
G f *º- / l + I
...*-* = − y \, = + Tay,
y' .
a- * zºº, 2 a.
_ 2 abº -
a” y' 2
therefore by substitution in (1)
2 a bº
tan AQ a = y’ (a” – bº) '
therefore, since the sign of this quantity is negative, the
angle is always obtuse.
Cor. 1. The angle A Q a will be the greatest pos-
sible when y' is so, that is, when y' = b, or the point Q
coincides with B, the vertex of the minor axis. At this
point the supplemental chords are equal, and their incli-
nation to the major axis is
= tan" —.
(l,
Cor. 2. Hence, the conjugate diameters which are
parallel to these chords are also equal, and contain the
greatest possible angle. See Art. 57.
(64.) To draw two conjugate diameters making a
given angle with each other.
The ellipse being referred to its axes, let 2 a', 2 b'
denote the conjugate diameters required, and ºf the
angle at which they are inclined to each other.
a" + bº = a + b . . . . (1,)
a b
a b' = −.
Sin Y
Then, since
and
we have, by adding twice the second equation to the
first,
2 a b
f ſ? / A/ — »” 2
a's -- b” + 2 aſ b' = a + b Tinº
therefore extracting the square root,
2 a b
/ ! — * hº •
2-v-1 Veº Tinº,
In like manner,
2 a b
a' — b' = + V- b” — * a 2.
Sin Y
therefore by addition and subtraction successively,
- 2 a b Vº 2 a b
2–4; Verº; + # *** - i.
v= + , Vº
a b
sin ºf
2 a b
* -i- h" — -
+ V. b sin ºr "

C O N I C S E C T I O N S.
753
Conic
Sections.
N-y-Z
Fig. 35
Fig. 36.
therefore the magnitude of the required diameters is
determined.
Again, since P C A = D C A — D C P,
tan D C A – tan D C P
1 + tan D C A tan D CP’
or retaining the notation already used
tan P C A =
a' + tan Y
a = 7–H– ;
I — aſ tan Y
b2 b?
but a al = — — . . . a' = — — — .
a? a? a
therefore by substitution
b?
Eº as a –– tan Y
a :- º
1 + hº t
all
(tº a n \
Q b? b?
* . u% — as tan Y. a = – # - a tan 7,
b? b”
9.
Or a” — | 1 - - ) tan Y. a = — —,
( #) ºy 0.8
a? – bº
l -
tan Y =E 2 as v (as -bº), tangº – 4 a.º. be,
(T. sº 2 Q?
therefore the position, also, of the diameters is deter-
mined.
The problem would be impossible, if
4 gº b%
(a *- bº)"
2 a b
tan Y 3 2 - 52"
But Y being an acute angle it will be a minimum
when the diameters are equal, and in that case
2 a b
º
bº
tan” ºf 3
O I'
tan Y =
(Z
therefore tan ºf can never be less than : (Z º and there-
fore the problem is always possible. tº
The same problem admits of the following geome-
trical solution. -
Draw any diameter whatever, R r, and upon it de-
scribe a segment of a circle containing an angle equal to
the given angle, and cutting the ellipse in Q; join QR,
Q r, and parallel to these draw the diameters Pp, D d,
these will be the diameters required. .
For being parallel to the supplemental chords Q R,
Q r, they are conjugate to each other, and the angle
PC D = R. Q r, and therefore equal to the given angle.
The problem admits of a second solution : for the
circle will cut the ellipse again in some point Q';
draw therefore the supplemental chords Q'R, Q'r; then
if P' p and D'd be drawn through the centre parallel to
Q' R, Q' r, they will be the diameters required.
For they are evidently conjugate to each other, and
P" CD' = r — R. Q' r, and is therefore equal to the
given angle.
CHAPTER IV.
MISCELLANEOUS PROPOSITIONS.
(65.) An ellipse being traced upon a plane, to find its
centre and awes.
1. To find its centre.
Draw any two parallel chords M N, PQ, and bisect
them in the points m, p respectively, join m p and pro-
duce it to meet the ellipse in R, r ; then, because
m p passes through the centre it is a diameter, and
therefore C, the middle point of R r, is the centre re-
quired.
2. To find the ares.
Assume any point P in the ellipse, and
From the point C, just found, as centre, with distance
CP, describe a circle cutting the ellipse in p, draw
P p and bisect it at right angles by a straight line
A C a meeting the ellipse in A and a ; then A C a is the
major axis: and the minor axis is obtained by drawing
B C b at right angles to A a.
(66.) To find the locus of the ertremity of a straight
Wine which moves on two lines at right angles to each
other, so that the parts intercepted by these lines may
always be of the same given length.
Let A X, A Y be the given lines, Q R P any position
of the line, the locus of whose extremity is sought.
Assuming A X, A Y as the axes of coordinates, let
fall the perpendicular PM on A X, and produce it to
meet in N a parallel to AX drawn through the
point Q.
Let A M = a, M P = y, Q P = a, P R = b ;
then Q P2 = Q N* + N Pe. . . . (1 ;)
but Q N = A M = a,
Q P _ &
and NP = #P. M P = + y,
therefore by substitution
2 0.2
a} = r + ºr y',
OF a” y + bºaº = a” b”,
which is the equation to an ellipse.
Therefore the locus of P is an ellipse of which A is
the centre, and 2 a. and 2 b the axes.
Cor. 1. Hence may be derived an easy practical
method of describing an ellipse by means of an instru-
ment called the Elliptic Compasses.
Let Q P be assumed equal the semi-major, and NP
equal the semi-minor, axis; and let the line Q N P be
turned round so that the points Q, N may always re-
main upon the axes of coordinates ; then the point
P will describe an ellipse, as is evident from the fore-
going investigation.
Cor. 2. By a method precisely similar to the above,
it may be proved, that if the axes are inclined to each
other at an angle 6, the equation to the locus of P
will be
a” y” + bºr” + 2 a b cos 0. A y – a bº = 0.
(67.) In the major aris A a of an ellipse to find a
point O, such that if any chord whatever P O p be
Ellipse.
S-N-"
Fig. 37.
Fig. 38.
Fig. 39.
Fig. 40.
754
C O N I C S E C T I O N S.
Conic
Sections.
drawn through it, the angle P A p may be a right
angle.
S-M-' Let the equation to A P be y = a w, then that to Ap
Fig. 41.
g I º
will be y = — — a ; therefore the coordinates of
0.
P (r', y') and p (a", y") will be determined by elimi-
nating (y) between the above equations, and the equa-
b?
tion to the ellipse y” = 2. (2 aw — wº); we thus have
_ 2 bº a , 2 bºa a
Taº a 2 + 52' agmº *a* + 5°
'— 2 b° a 22 " - – 2 b" a a
- as I ºf 9 - T aſ Tº
therefore, denoting 2 b% a by c, and the denominator in
the first and second lines respectively by m and m, we
have for the equation to Pp
777, 71, C
y = -a++( – ’).
(tº 777 - 7. 7??,
Let P p now cut the axis as at O, then y = 0, and
C C Cº 777 – ??
Jº — — — —
77?, n m + m '
a? -- 1
n —H m
therefore, substituting for m and m, and reducing,
c . 2 b" a
(68.) Pairs of tangents to an ellipse being always
supposed to intersect at right angles, to find the locus of
the points of intersection.
If the straight line
y = a a + £3. . . . (1)
be drawn to cut the ellipse
a? y” + bºa' = a” b%. . . . (2,)
the ordinates of the two points of section will be ob-
tained from the equation
(aº a” + bº) y? – 2 b" B y + b (3*— a” a”). Art. 30
Let the secant be now supposed to become a tangent,
then the two roots of this equation are equal, and the
equation being therefore a perfect square,
4 (a? a” + b°) bº (3. – a” a”) = 4 b% 8°, Ellipse.
Ol' (a? a” + b”) (3* — a” a”) = b” B",
... a 2 a.” Bº – a” a” + b% a” a” = b%,
... a2 a” (32 = a aº + b% a” a”,
... a” as -- b% = 8* = (y – a r)” from (1,)
= y” — 2 a y a + a” a”,
... (a” -- wº) a” + 2 + y . a + b% — y” = 0,
2 - ar?
2 a y b * = 0.
Or a? *-------
-- a? — a 2 a” – arº
Suppose a', a” to be the roots of this equation, then
they denote the trigonometrical tangents of the angle
which the tangents to the ellipse form with the axis of
a, and by the theory of equations
b° — y?
a? — a 2"
but, by hypothesis, the tangents intersect at right
angles,
a a' =
‘. a. a' = — 1 ;
bº — y?
hence ty 3/ = — 1,
a” — ºr?
... b” — y” = — a” + wº,
... y” + a” = a” -- b%,
which is the equation to a circle.
Hence the locus required is a circle whose radius
= Maº -F 53.
Cor. In the same manner we may find the locus of
the intersection of pairs of tangents which are always
parallel to conjugate diameters.
b?
For in this case a a' = — –,
(12
bº — y” b?
' ' …T. - T is .
... a” b% — a” y” = — bº as + b r",
... a” y” + b” as = 2 a” b”.
which is the equation to an ellipse.
Hence the locus required is an ellipse whose centre
is the same as that of the original.
To find its axes.
Let r = 0, ... as y” = 2 a” b”,
. . . y = b W2 = the semi-minor axis;
and, in like manner,
r = a M2 = the semi-major axis.
C 6) N I C S E C T I O N S
755
Hyperbola.
ON THE HYPERBOLA. *
Conic
Sections.
\-y-
CHAPTER I.
ON THE HYPER BOLA REFERRED TO ITS AXIS.
THE hyperbola is the locus of a point, whose distance
from the focus is always greater, in a given ratio, than
its distance from the directrix. -
(69.) To find the equation to the hyperbola.
Fig. 42 Let S be the focus, K k the directrix, P any point in
the hyperbola, through S draw the indefinite line E S X
perpendicular to Kºk, and from P let fall the perpen-
diculars PM, P Q on A X, K k respectively, and join
P,S.
Let the given ratio of P S : P Q be as e : 1, e being
> 1 ; then if S E be divided in A, so that S A : A E . .
e : ], A will be a point in the hyperbola.
From A draw A Y at right angles to A X, and
assume A X and A Y as the axes of coordinates.
Let A M = r, M P = y, A S = m ;
then SP2 = PM2 + M. Sº = y2 + (r. — m)”. . . . (1,)
'but SP2 = e2 . P Q” = e” (A E -- A M)*
2
= e(? +.)...go
therefore equating (1) and (2,)
y” -- (a — m)* = m” + 2 m e a + e” a “,
... y = 2 m (1 +e) r + (6 – 1) wº,
2
= (e? – 1) (. * a + r)
be assumed = a,
"y” = (e” – 1) (2 a a + aº),
which is the equation required.
777,
if
or r ==
Cor. 1. In X. A take A a = ##, bisect A a in C
then at this point * = — 0,
4 ... y? = (e.” – 1) a - a”,
... y = + a W-i . We – I,
which is always imaginary, since e > 1.
Hence, if B C b be drawn through C at right angles
to Aa, and C B, Cb each taken = a v e-l, the
points B and b are not points in the hyperbola.
Cor. 2. Let B b be denoted by 2 b, then
b = + a We” — 1,
b
tº * — = + — ; ©
... We ! - - (Z *
therefore, by substitution, the above equation becomes
y = + + Vă a Fr.
&
Def. The straight lines A a, B b represented by 2 a.
and 2 b are called, respectively, the transverse and the
conjugate, axis; the points A, a in which the former
meets the hyperbola, are called the vertices ; and the
point C, in which the axes intersect each other, the
centre.
(70.) To find the equation to the hyperbola when the
coordinates are measured from the centre.
Let P be any point in the hyperbola, let fall the
perpendicular PM on A a, and assume C M = a '.
Then the equation to the hyperbola, when the coor-
dinates originate at A, is -
b2
y = ±(2a, +, ) . . . . (1,)
a = A M = C M – C A,
= a – a.
Substituting this value for a, we have
v=#2 a G-04 (2–0)}.
but
b”
=# ("-a")....(2)
which is the equation required.
Cor. 1. Suppressing the accent, which was used only
to distinguish the new from the old abscissa, we have
by multiplying and transposing,
A a”y” — bºa” = — a” bº.... (3.)
If each term be divided by a bº, we have
gy” q2 -
º – A = — 1. ... (4.)
Of the three last forms of the equation to the hyper -.
bola, that marked (3) is the most frequently used.
Cor. 2. These equations when translated into geome
trical language express a property of the hyperbola.
For if P be any point, we have
2 a r + a = a (2 a + a) = AM. M. a,
and a” — a” = (w' — a) (a' + a) = A M. M. a,
... M P = ** A.M.M a
C A2 ---,
Or A. M. M. a. : M P2 :: A C* : B Cº,
that is, the rectangle contained by the segments of the
transverse aris is to the square of the ordinate, as the
square of the semi-transverse aris is to the square of the
semi-conjugate. .
Cor. 3. Let a = b, then equations (1) and (2) be-
y” = 2 a. a + a”, -
gy* = z* — a”.
COIIlê
º -

756 C - O N I C
S E C T I O N S
Conic
Sections.
\-V-'
The hyperbola represented by these equations is called
equilateral, or rectangular, and is to the common
hyperbola what the circle is to the ellipse.
By comparing the equation to the hyperbola
as y” — bººs = – a bº.
with the equation to the ellipse
a2 y? -- b” as a aº bº,
it is manifest, that to pass from the one curve to the
other we have only to change + b into - bº, or b
into b M-1.
(71.) To determine the figure of the hyperbola, from
its equation.
Resuming the equation
gº y” ** b2 42
we have either
b *-*mmam mºss
3/ = + -ā- w/ cº- a”. . . . (1,)
wº-mº
*===e
$
— a” b”,
OT
- * = + + Vºf 5....(?)
I. In equation (l,)
let a = 0,
then v = + b v — 1 = C B or C b.
Let y = 0,
then a = + a = C A or C a.
Let a < -E a,
then the values of y are imaginary ; therefore no
point of the hyperbola is situated between A and a.
Let a = + a,
then Ay = + 0 ;
that is, the hyperbola cuts the axis A X at the points
A, a.
Let a > -- a,
them for each value of a there are two equal values
of y with contrary signs.
As a increases, the values of y increase ; and when a
becomes indefinitely great, the values of y become so
likewise.
The hyperbola, therefore, consists of two equal and
opposite branches extending indefinitely to the right of
A and to the left of a, and symmetrically placed with
respect to the axis XAX'.
II. The discussion of equation (2) would lead to the
same result. -
Observation. In the equation a” y” — bºa” = — a” b”,
let a be changed into y, and y into a ; in other words,
let the abscissas be now reckoned along CY and the
ordinates along C X; we then have
gº tº smºgº bº 9° = gººms a” bº,
which represents the same hyperbola as before, but
differently placed.
Let a = 0, ... y = + a,
y == 0, ... a = + b W – 1,
therefore the transverse axis is now B b, and the con-
jugate axis A a. -
This hyperbola is called, relatively to the former, the
conjugate hyperbola.
Cor.
through the focus.
When the ordinate passes through the focus,
a = m = a (e – 1),
therefore by substitution in (1), Art. 70,
2
b
y = {2 a ( – 1) + æſe – 1) #,
= b” (e – 1) {2 + e – 1 },
= b” (e” – I),
b4
= z: (Art. 69, Cor. 2,)
b”
... " .. 7 = + –.
3/ (Z
The double ordinate passing through the focus is
called the principal parameter, or latus rectum,
º
therefore the latus rectum =
&
Def. The line S C, represented by a e, is called the
eccentricity of the hyperbola.
(72.) To find the intersection of a straight line with
the hyperbola.
Let the equation to the proposed line be
3y = a r + 8 . . . . (1)
Then the coordinates of the point or points of inter-
section with the hyperbola will be obtained by com-
bining this equation with that to the hyperbola
a? y” — bºw” = — a” b” . . . . (2.)
Substituting, then, in (2) the value of a derived from
(l) we have
* 2
a? y? — b% (*#)- — a” b°,
ſº,
... (a” a” — bº) y + 2 bº 6 y – bº (32 = — a” be a”,
2 bº 6 bº (8° – as a”)
| 9 || T. LT, T-
from this quadratic are obtained two values of y, which
substituted;
‘.. y” + 0;
a” a” —
fº
y
㺠furnish two corresponding values of
a, therefore "the coordinates required may be deter-
Imined.
When the two roots of the quadratic are equal, the
points of intersection coincide, and the straightline then
touches the hyperbola; when they are imaginary, the
straight line falls entirely without the hyperbola.
Hence it appears, that a straight line cannot cut an
hyperbola in more than two points.
Def. The portion of the straight line contained with-
in the hyperbola is called a chord; when the chord
passes through the focus it is called the focal chord.
(73.) To find the equation to a straight line which
touches an hyperbola in a given point.
st tºy.
Let a', y' be thé coordinates of the given point, and
a", y” those of any other point in the hyperbola near
the first.
Then the equation to the line drawn through these
points is - - -
— , – 9"-9" | Y
3/ y'- ºr-., (, — »)....
To find the value of the ordinate passing Hyperbola.
C O N I C S E C T I O N S.
757
Conic
Sections.
N-y-'
Fig. 13.
But these two points being in the hyperbola, we
have * -
a” y” — b% z* = — a” bº,
a? gy” – bºr's = — a” b”;
therefore by subtraction
a? (y” — y”) = b (r"? — wº),
(y" + y') (y" – y') bº
' ' (r" + x') (r" – a ') T a .
y” — y' b° 4" + r.
ºſ- º' a 7- y’
and equation (1) becomes by substitution
be a " + x'
3) — y' = . . .7-L,
a” y”-H 3/
Let the point (a!", y') be now supposed to coincide
with (r', y'); then a' = a ', y' = y', and the secant be-
comes a tangent at the point (w', y'); hence the equa-
tion to the tangent is
(a — r").
Y.
y-y=*.
a?
o # (a – a '),
in which a' and y' are the coordinates of the point of
contact, and r, y the variable coordinates of any point
whatever in the tangent.
Cor. This equation may be simplified, for multiply-
ing each side by a” y',
a” y y' — a” y' = b% a w' — bºw”,
... a y y' — bºa w' = a” y” — bºa”,
tº- aft b°,
:-
which is the equation most commonly used.
When a = b, the hyperbola becomes equilateral, and
the equation to the tangent is
gy y' — a w' - — a”.
(74.) To find the intersection of the tangent with the
aves of w and y.
The equation to the tangent being
.#
a” y y’ — bºa w' = — a” bº;
let it cut
1. The axis of a, as at T.
2
Then 3) = 0, ... a = #.
º C A2
Or C T – C M"
2. The axis of y, as at t.
Th 0 b2
€Il * = U, . . ºf E —7,
y = 7
C B” .
# = o
Or C Cºn
Whence it follows, that each semi-aris is a mean pro-
portional between the abscissa of any point, and the
part of it intercepted between its intersection with the
tangent, and the centre.
** @ ſº
Cor. Since C T = Tº
wº
WOL. I.
... M T -- C M - c T, Hyperbola
dº
= r^ —
a' 2
* :4 a's sº a?
*—F-
Def. The line MT intercepted between the foot of
the ordinate, and the point where the tangent meets
the axis, is called the subtangent.
Def. The straight line drawn from the point of con-
tact at right angles to the tangent, is called the
normal. -
(75) To find the equation to the normal.
Let PT touch the hyperbola in P, from which point
draw the line Pg at right angles to PT.
Then, because P g is drawn through the point
(r', y') at right angles to the line,
b8 a'
y-y= x . ; G – )
its equation will be
- 0.2 y'
tº- ' = tºgmm mºme-ºs - U.T., -
gy — y ; # (, — »).
in which w', y' are the coordinates of the point of con-
tact, and r, y those of any point whatever in the inde.
finite line Pp.
The term normal is usually confined to the line PG
See Art. 9.
(76.) To find the intersection of the normal with the
aves of a and y.
The equation to the normal being Fig. 43.
a” y'
y – y' = – i. ºr ( – ’).
let it intersect
i. The axis of a as at G.
2
Then y = 0, and — y' = — #.4% (a — r'),
b”
... " - w = + r = M G.
2. The axis of y, as at g.
Then r = 0, ... y – y' = 73 ºf a ,
a” ...,
7. 9.
a" -- bº
... y = -; 9'.
Def. The line M G intercepted between the foot of
the ordinate, and the point where the normal cuts the
axis of ac, is called the subnormal.
(77.) To draw a tangent to an hyperbola from a
given point without it.
The equation to the tangent being in general
a y y' – bºrr' = — as b”,
and the point (r", y") being by hypothesis a point in
the tangent, we have
5 F
…tº
758
C O N I C S E C T I O N S.
Conic a”y” y' – bºr"a' = — a bº. . . . (I ;)
sections, also, the point of contact (w', y') being in the hyperbola
a” y” — bºa's = — a bº. ... (2,)
hence, by means of these two equations, the coordinates
r', y' of the point of contact may be determined.
Since the equation resulting from the elimination of
a' between (1) and (2) is of the second degree, it fol-
lows, that there are two points of contact; in other
words, that two taugents may be drawn to an hyper-
bola from a given point without it.
But the position of the points of contact may be
directly found by constructing, as in Arts. 10 and 35,
the loci of equations of (1) and (2,) in which a' and y'
are variable.
Now the locus of (2) is the given hyperbola, and the
locus of (1) is a straight line whose position is deter-
mined by making a' and y' successively – 0.
If, therefore, in the equation
a? y” V' – bºa" a' = — a” b%,
b°
a' = 0, then y' = — ºr,
3/
= 0, then º'- – “.
3/' tº: 5 16. Il Q, — — .7.
2 2
- - b -
Hence if C R be taken = -, and C r = #, the
3/ • Jº
line joining R, r will cut the hyperbola in the points of
contact required.
Cor. 1. The equation to the chord joining the points
of contact is
a y” y/ — b” ar" ar' = — a” b”.
Cor. 2. Since C R is independent of y", it follows
that if from the several points of a line perpendicular
to C X pairs of tangents be drawn to the hyperbola,
the chords joining the points of contact, in each case,
will all pass through the same given point.
CHAPTER II,
ON THE HYPERBOLA REFERRED TO THE FOCUS.
(78.) To find the distance of any point in the hyper-
bola from either focus.
Let S, H be the foci, P any point (a, y) in the hyper-
bola, to find the value of SP, or H. P.
I. Of S. P.
In general, the distance between two points (t, y)
and (a', y') is
= MG — p")*-E (y – y')”;
but the coordinates of S, since it is a point on the axis
of a, are a' = a e, y' = 0,
... S P = (a — a e)” + y”,
(a — a e)* + (e” – 1) (a" – a”), -
a” – 2 a e a + a” e” – e’aº – e’aº – a " + a”,
a” – 2 a e a + e” aº, -
..”. S P = e a - a.
2. In kºke manner,
H P = e a + a.
Fig. 44.
* sº
fººms
-
assº
*=s
Cor. Hence, subtracting S P from HP,
H P – S P = 2 a.
In other words, the difference of the focal distances we
equal to the transverse aris.
The distance of any point from the focus is called the
focal distance.
(79.) From this property the equation to the hyperbola
may be deduced, as in the case of the ellipse.
Let S, H be the two fixed points, P the point whose
locus is required.
Join S, H, S, P, and H, P; bisect S H in C; let fall
the perpendicular PM on S H, which produce inde-
finitely towards X; from C draw CY at right angles to
C X, and assume CX and CY as the axes of the coor-
dinates.
Let C M = a, M P = y, and S C = c.
Then ...I.I.I.I.} (1,)
H P = H M*-ī- M P = y” + (c. -- a)*J “”
... H P – S P = (c + æ)” – (c — ar)”,
Ol' (H P + SP) (H P – S P) = 4 cr;
but H P – S P = 2a,
Hyperbola.
4 c ºr 2 g ºr
. . . P = = — ,
H P + S 2 a. (Z and
H P – S P = 2 a,
C tº
... HP =++ a
C tº
and S P = — — a,
(Ž
squaring these values, and adding the results,
cº gº
H P -- S P = 2 Tº +2)
and also = 2 (y” + c + a ")
by adding equations (1,)
cº tº
... y + º-º- ++a,
2
C“ （:
2
+ a” – c’ – aº,
• 2 ..?
• 3/ a”
gº
; (e– a + (a' – e’),
c” – a”
º
(* — a”),
which is the equation to an hyperbola whose transverse
axis = 2 a, and conjugate axis = 2 W cº — a”.
If a = 0, then y” = — (cº — a”) = — bº, if b be the
imaginary ordinate drawn from C. -
(80.) To find the polar equation to the hyperbola, the
focus being the pole,
1. Let S be the pole.
Let S P = r, angléA S P = w.
Then
but
Fig. 44.
T = e a - a,
a = C S + S M,
= a e -- r cos (ºr — w),
= a e - r cos w,
r = a e” – e r cos w — a
C O N I C S E C T I O N S.
759
Conic * . P = 0, e” – 1
Sections. . . . . -- ~ 1 + ecos w’
which is the equation required.
2. Let H be the pole.
Let H P = r", angle PH A = w';
then t’ = e a + a,
but a – C M = H M – H C
= r" cos w' — a e,
... r" = e r" cos w/ - a e” + a,
... ?’ - gºmºs e? – 1 f
1 — ecos w'’
which is the equation required.
Cor. 1. Produce P S to meet the hyperbola in p,
then because A Sp = 7 — w,
e” – 1
l – e cos w
1 1+ ecos w
Sp" aſe-T)
* 2
T a (es – 1)'
– 2
~ SL’
therefore the principal semi-parameter is an harmonic
mean between the segments of any chord drawn through
the focus.
S p = a
1 — ecos w
H
Cor. 2. Hence SP + a GTI)
... I 1 S P + Sp
Cor. 3. Since — — — — = ——f
Or * sp-Fs, * sps;
& ºms 2
Ta (e? – 1)”
... S. P. S. p = 4 a (e” – 1) (S P + S p).
and also
(81.) The focal distances of any point make equal
angles with the tangent at that point.
Fig. 45. Let T Pt be a tangent at any point P (r', y') draw
the normal PG, and join S, P and H, P.
2 . 2 º:
Then CG = } # a’ = e, aſ, (76.)
Q,
S G C G – C S e” aſ — a e
H G T C G + C S Teº aſ + a e
ea' – a S P
= — = Hº, (78,
ea' + a HP (78,)
therefore P G bisects the angle S Ph. Euc. vi. Prop. A.
Now the right angle G PT = G Pt,
G P S = GP t,
therefore the remaining angle S P T = h P t = H PT,
that is, S P and H P make equal angles with the tan-
gent at P, as was to be proved. .
and angle
(82.) To find the locus of the points in which a per-
pendicular from the focus upon the tangent at any
point intersects the tangent.
Let PT be a tangent at any point (w', y'), and S Y Hyperbola-
a perpendicular let fall from S on PT, to find the locus
Of Y.
From C let fall the perpendicular C Q on PT, and Fig. 46.
draw S q parallel to PT meeting C Q in q.
Then CY = C Qº + Q Y*,
= C Q2 + S g”,
= CT2 sinº T -- C Sº cosº T;
but C T = + (74) and C S = a e,
... C Yº = -r, sinº T -- a” escosº T,
= z: (1 — cosº T) + a” e” cos” T,
a 4 a" f 2 -
* * Q
= −F + º (es r" ) cosº.T. '#' (1.)
— bº
But tan T = • T, ,
Il & 9'
a'? — a”
therefore, as in Art. 41, cos? T = Hi-, ;
and substituting in (1)
, a.4 a” . . . a"— a”
CY" = + + →r (e” ar" – a”). --→
a'2 a/* ( ) e” aſſº * g”
0.4 a”
= is +: (* - a)
0.4 aft
= −7; + a” — —zº,
ap/2 aſſº
= a”,
J. C. Y = + a,
therefore the locus of Y is a circle described on the
transverse axis.
(83.) The rectangle contained by the perpendiculars
let fall from the foci upon the tangent at any point, is
equal to the square of the semi-conjugate aris
For S Y = S T sin T,
-: 0.2 Oſ,
but ST= Cs-CT= ae-j =# (ex- a),
Jº
. . . SY = # (ea' – a) sin T. In like manner,
(Z
Hz == (ea' + a) sin T,
2
...sy.Hz=#. (eº rº — aº) sin” T;
e” aſ 2 – arº
& ſº In 2 Hºmº
but, as in the ellipse, sin” T = egºs – º'
(e” – 1)
a? A
... SY. H Z = gº (** – d’). ". . ;--
= a” (e? – 1) = bº.
5 y 2
C O N I C S E C T I O N S.
Conic
Sections.
N-y-
Fig. 47
CHAPTER III.
ON THE HYPERBOLA REFERRED TO ANY SYSTEM OF
CONJUGATE DIAMETERS.
SECTION I.
ON CONJ UGATE DIAMETERS IN GENERAL.
(84.) To find the locus of the middle points of any
two parallel chords.
Let Pp be any chord, O its middle point; from the
points O, P, p let fall the perpendiculars O N, PM,
p?n on the axis A X.
Let A N = X, N O = Y ;
then if the equation to P p be
* y = a + + 6.... (1)
the equation containing the values of y at the points
P, p will be
- 2 2
** – fºr y – º – “...
a? at – b” a? a” — b%
Now since in any quadratic equation the coefficient
of the second term with its proper sign is equal to the
sum of the roots with their signs changed,
= 0.
2 b% 8
But O being the middle point of P p,
O N = PM-H pm.
2
– bº 6
.*. Y = a” a” — b” . . . . (2.)
Now X and Y satisfy equation (1,) since they are the
coordinates of a point in P p, therefore
x = + (Y-5),
— a” a 3
gººms '' zºº T; ' ' ' ' (3.).
To obtain the relation between X and Y we must eli-
minate fl between (2) and (3,)
a? a? * - b”
... → - Y = −r.
b”
a” a
a” a” — bº
*-*m
X,
”. Y =
Now a remains the same for all chords parallel to
P p, therefore the equation just found expresses the
relation between the coordinates of their middle points,
and being of the first degree, the locus required is a
straight line.
Def. The straight line which has been proved to be
the locus of the middle points of any number of parallel
chords is called a diameter, and the points in which it
intersects the curve are called the vertices.
The letters X and Y are introduced to distinguish
the two sets of coordinates, and the equation to the
diameter bisecting any chord,
3y = a a + £3,
may always be written Hyperbola.
- b?
a” a
3) = £.
From the form of this, it is plain that every diameter
passes through the centre.
(85.) To find the intersection of any diameter with
the hyperbola.
The equation to any diameter being Fig. 48,
= d. Jº,
and that to the hyperbola
a’yº — bºws = – a bº,
the coordinates of the points of intersection will be
obtained by combining these two equations; we thus
have (a” a” — b”) ar" = — a” b”,
b
. Tº = Cº., = C M or C m,
T vſ 52 – 22 a.
and . . y = + a b a = PM or p m,
* A/ 5* * a” a”
the coordinates required.
Cor. 1. Since AM = A m, and PM = p m, it follows
that every diameter is bisected at the centre.
Cor. 2. In order that the diameter may meet the
hyperbola,
b° must be > a” a”,
Or + b must be > a. a,
b
therefore a must be < =E a
From the vertex A draw A E and A e perpendicular
to A C, and each equal b ; join C E, Ce, and produce
them indefinitely towards Z and 2.
Then since tan Z C X = FA = * , Fig. 49.
- A C ſº
e A b
and tan 2 C X = A GT – . .
it follows that the diameters C Z, Cz will never meet
the curve at any finite distance.
The lines C Z, Cz are from this circumstance called
asymptotes.
(86.) A diameter being drawn through a given point
to find the equation to any one of its ordinates.
If a ', y' be the coordinates of the given point, the
equation to the diameter drawn through it will be
... (1.)
J
*-*..
y = -7
Let gy = a + + 3. . . . (2,)
be the required equation to any ordinate, then
y’ b”
a/º a” a '
b” ºf
• Q =
therefore the equation to any ordinate to a diameter
passing through (r', y') is
2 a’
y = + , a +- 8.
C O N I C S E C T I O N S.
761
Conic Cor. Comparing this equation with the equation to
Sections the tangent, it appears that the tangent applied at the
\-v- verter of any diameter is parallel to the ordinates of that
diameter.
(87.) Two diameters being drawn such that the ordi-
mates of one may be parallel to the other, to prove that
the ordinates of the latter will be parallel to the former.
Let C P, CD be two diameters, and MN, Q R.
chords bisected by each respectively; then if M N be
supposed parallel to CD, we are to prove that Q R.
will be parallel to C P.
If the equations to CP, CD be
= a +. . . . (1,)
gy = aſ a . . . . (2,)
then the equations to MN, Q R, respectively, will, by
Art. 84, be
Fig. 50.
a + 3. . . . (1',)
3/ = a” a
y =# * +3,...(?)
But if M N be parallel to CD, then
b?
a8 a.
b?
therefore by substitution in (2) the equation to Q R.
becomes
a' =
º
gº (i. mº
3) = a + + 3',
that is, Q R is parallel to CP, as was to be proved.
Whence each of the diameters C P, CD is parallel to
the ordinates of the other. -
Diameters thus related to each other are called con-
jugate diameters.
Cor. 1. Therefore when the two diameters
3y = a ar,
3/ = a/ Jº,
are conjugate to each other,
bº
a al = — .
a?
Cor. 2. Hence if 3y = a +
be any diameter,
b2
3/ tº mº a” (i.
will be the diameter conjugate to it.
Cor. 3. Since (a) may have any value between 0
and r, the number of pairs of conjugate diameters is
infinite. -
If a = 0, or the first diameter be the transverse axis
A a, then
b2
a2 . 0
therefore the diameter conjugate to A a being at right
angles to it, is the conjugate axis B b ; whence the axes
a! :-
E CO ,
are conjugate diameters ; and it may be shown, pre- Hyperbola.
cisely; as in Art. 47, that they are the only conjugate S-'
diameters which are at right angles to each other. --
Cor. 4. If (r', y') be any point in the hyperbola, the
diameter passing through it is
3/
y = −r r,
b2 º'
- ?/ = &2 g/
is the corresponding conjugate diameter.
But the equation to a tangent applied at the point
(r', y') is
2 ac'
a y'
y – y' = (a — a '),
whence it follows, that the tangent at the verter of
any diameter is parallel to the corresponding conjugate
diameter. -
(88.) Of any two conjugate diameters, only one can
meet t he curve.
For let 3y = a ar,
3y = aſ a
be any two conjugate diameters.
It was shown that no diameter can meet the curve
unless
b
a 3 — .
(3
Suppose, then, in the given system, that the first
diameter meets the curve, then
b
a 3 —,
Cl, .
b”
2
a”
a aſ :-
but
f b
".. a' > —,
Q,
and consequently the second diameter cannot meet the
hyperbola.
(89.) To find the equation to the hyperbola when
it is referred to any two conjugate diameters as a res.
Let C be the centre, CP, CD a given system of Fig. 61
conjugate diameters, of which the former is supposed
to be the axis of r, the latter the axis of y.
Take any point Q in the hyperbola, and draw Q q
parallel to CY, meeting CX in V.
Let CV = a, W Q = y, C P = a'; and since CY does
not meet the hyperbola, let C D = b v-1. Because
the chord Q q is bisected by C P in V, V Q = V q,
and since every other chord parallel to C Y is bisected
by C X, it follows that for each assumed value of r
there are two equal values of y with contrary signs.
In like manner it may be shown, that for each assumed
value of y there are two equal values of a with con-
trary signs; also, when a = 0 the values of y ought
to be imaginary, and when y = 0 the values of a are
real ; therefore the equation required must be of the
form My' – Næ" = – P.
We are now to determine the values of M, N,
and P. . . . . . .
762
C O N I C S E C T I O N S.
Conic
Sections.
N-J.--
and
When the axis CX meets the hyperbola, y = 0, and
r = C P = aſ, "$5.
... Naº – P = N a”,
P
aſ?
tº , ,
. . .'; º' -
. . . N =
When a = 0, the axis C Y does not meet the curve,
but gy or C D = b v — 1,
M y” = – P = — N b”,
- P
... M = b/2
Substituting these values of M and N in the above
equation, and dividing each term of the result by P,
we have
g" a" .
Or a' y? — b” a” = — a!” b”. . . . (2,)
either of which is the equation required.
b' *ºmºmºm,
Cor. 1. Hence 3y = + Tºſ w/º: Taº.
Cor. 2. To find the form of the equation, when the
coordinates originate at P, the verter of the diameter
C. P.
Let P M = a-', then a = C P + PM = a + ar'.
Substituting this value of a in Cor. 1, we have
b/
y = + +w (*-i- a)' – d'.
or suppressing the accent of ar,
| -
y = + , /* F27s,
which is the equation required.
Cor. 3. The equations (1,) (2,) and (3,) are of the
same form as the equation in terms of the axes,
(Art. 70,) and express a property of the hyperbola.
For a * — a' = (a + a') (a — a') = P V. V G,
2 aſ a + ak = (2 a' + æ) a = P V. V G,
P C2
.*. W Q? = . V. G.
Q C D2 P V
or PV. V G : Q Ve . : P C2 : C D*
that is, the rectangle contained by the segments of any
diameter is to the square of the ordinate as the square
of the semidiameter is to the square of its semicon-
jugate.
(90.) It appears from the preceding article, that the
equation to the hyperbola retains the same form, whether
the axes of coordinates be rectangular or oblique.
Whence it follows, when the axes are oblique,
(1.) That if the equation to the transverse axis A a
be y = a ar,
the equation to the conjugate axis B b will be
b's
3) F- a/2 £1,
( % That the equation to the tangent at any point
1", ºy .
a” y y' — b” a r" = — a” b"
(91.) To find the intersection of the tangent with any Hyperbola.
two conjugate diameters, considered as a res. S-N-
Let a tangent applied at any point Q (r', y') meet Fig. 52.
C P in T, and C D in t, and draw the ordinates Q V,
Q v. Then the equation to the tangent being
ºf
/ * — #2
a's y y! — b” a w’ = — a” b”,
Let the tangent meet C X as at T, then y = 0,
a'2 C Ps
and * = +, or C T =-av-
Let the tangent meet CY as at t, then a = 0,
— bº C D*
... y = -º- or c = −av.
Whence the points of intersection are known. See
(74,) which is only a particular case of this Article.
(92.) If from the several points of a straight line given
in position, pairs of tangents be drawn to an hyperbola,
the lines which join the corresponding points of contact
will all pass through the same point.
Let C be the centre of the hyperbola, M N the
given line, draw any chord m n parallel to M N, and
bisect it by the diameter C X ; from C draw C Y
parallel to m 'm, or M. N, then C X, C Y are conjugate
diameters, and if the hyperbola be referred to these as
axes, its equation will be
— a” b”. . . . (1.)
From any point (a!", y") in M N let a pair of tangents
be drawn to the hyperbola, then it may be shown,
that the equation to the line joining the points of
contact is
a'? y” * b's 2 -:
a” y” y' — b/*a'a' = — a” b”.... (2,
in which a', y' are the variable coordinates of the point
of contact.
Let the straight line (2) cut the axis of x, then
Sy = 0,
and
hence the point of intersection will be the same for all
points whose abscissas equal ar", that is, for all points
in the line M N, as was to be proved.
Cor. The point of intersection is situated on the
diameter conjugate to that which is parallel to the
given line.
(93.) If from the point of intersection of two tangents
a diameter be drawn, it will bisect the line joining the
points of contact.
For the equation to an ordinate to the diameter
passing through (a", y”) is (86)
bº a"
ºf Z * + £3. . . . (l,)
and the equation to the line joining the points of con-
tact is
&=º
*=s*
ſº
b/2 aſſ/ b/2
y = z: ; * + 7 © tº ſº gº (2,)
therefore the latter being parallel to the former is also
an ordinate and consequently is bisected.
C O N I C S E C T I O N S.
763
Con10.
(94.) If through any point within or without an
Sections. , hyperbola, two straight lines, given in position, be drawn
S-' to meet the curve, the rectangle contained by the segments
Fig. 53
of the one will bear a constant ratio to the rectangle
contained by the segments of the other.
Let O be any point within the hyperbola, through
which draw the two lines P p, Q q, whose position is
supposed known, to meet the hyperbola in the points
P, p and Q, q, to prove that
O P. O p : O Q . O q in a constant ratio.
Through O draw the diameter CX, and let C Y be the
diameter conjugate to it; then, if the hyperbola be
referred to these diameters as axes, its equation will be
a” y” — bºwº – – a” b”. ... (1.)
Through P draw PM parallel to C Y, and let O P = r,
P M sin P O M
sin r, a .
P OT sin PM OT sin r, y' ' ' " T
C O = 3; then
sin r, a
gº r
Sin T, y
º ºg sin r, y
Similarly, a = 8 –H +++ r,
SIn £, Q/
or, denoting the coefficient of r by p in the first case,
and q in the second, and substituting these values of a
and y in (1,)
a” pºrº — b” { 8°-H. 23 q r + q r ) = — a” b?,
23 q b” bº (3* – a”)
a” pº – bl2 *" a/* p” — b/* q” - v 3
in which the values of r are O P, Op,
ºmsº
b” (Sº — a ")
... O P . Op == Zºº Yº
In like manner, if O Q = 7", and p' and q’ denote
ge f w f b? 8? — reſº
*** and ***, oo, o q = º
Sin a y Sln 3, 3/ a” p" — b” q
therefore
O P. O p : O Q. O q :: a” p” — b/2 q”: aſ pº — b” q*,
which is a constant ratio, as was to be proved.
SECTION II.
UN THE PROPERTIES OF conjug ATE DIAMETERs.
(95.) A diameter being drawn through a given point
(w', y') to find the imaginary coordinates of the extre-
mity of the diameter conjugate to it.
Let C P, CD be any two conjugate diameters, of
which the former is drawn through the given point P
(a', y'); then the latter C D will not meet the hyper-
bola.
ſ
If y = # a. . . . (1) be the equation to CP, then
b2 º'
y = ºr 7 * . . . . (2) will be the equation to C D ;
therefore the imaginary coordinates of the point D,
will be found by combining (2) with the equation
a” y” — bºrº = — a” b% . . . . (3.)
Hence, substituting in (3) the value of y in (2,) and
dividing the result by b", we have
b?
(–; ;
Q.
*-mºmº, 3 — »2
* /* + |). = @”,
... (a” y” — bºw") tº — a y”,
OT — a” bºrº - a y”,
(96.) The difference of the squares of any two semi-
conjugate diameters is equal to the difference of the
squares of the semiaxes.
Let C P, CD be any two semiconjugate diameters,
then denoting them by a' and b v - I respectively,
a/* = a,”-- y”,
2 2
— b” = — # y” +. 2
b? a”
Q 2
f2 /2 — waſ ſ
a/* – bº = ºr'? — 3/2 + y” — -2°,
b° a 's – a y” + a” y” — bºr.”
b? a”
(t? b2 a? b”
b2 a2 °
= a – bº.
(97.) If at the extremities of any two conjugate
diameters, tangents be applied so as to form a parallel-
ogram, the area of all such parallelograms is constant.
Hyperbola.
**s:
Let P p, D d be any two conjugate diameters, and Fig. S4. ,
let the tangents at P and p, D and d, be produced to
meet, then it is plain, (Art. 87. Cor. 3) that they will
form a parallelogram.
From P and T let fall the perpendiculars PF, TQ
on DC. Then the area of the whole parallelogram is
equal to four times the area of the parallelogram P D
– 4 P C : C D sin PC D,
= 4 C D . P. F. . . . (1.)
aft, m, D
But PF = TQ = CT sin T C Q = T BE (95.)
... P. F. C. D
Jºãº
º-s
a b v- I. ... (2;)
therefore by substitution in (1) the area of the whole
parallelogram = 4 a b v — 1, and is therefore con-
stant. The imaginary quantity involved in this expres-
sion indicates that the parallelogram does not, as in the
case of the ellipse, circumscribe the curve.
Cor. 1. From equation (2) PF. C.D = a b w/– 1,
but C D = b v — 1, and PF = P C sin PC D,
= aſsin Y, if P C D = y,
‘.. a b = a' b'sin Y.
764
C O N I C S E C T I O N S.
Conic
Sections.
g - - * W -
- s' ‘. . :- • 2
N. , , , -
4.
Fig. 55.
Cor. 2. Hence the value of PF may be found; for
a b a b
C D T w/aſs – (a” – bº) e
Cor. 3. Since a” — bº = a” — b%, the conjugate
diameters cannot be equal to each other in the hyperbola.
(98.) The rectangle contained by the focal distances
of any point is equal to the square of the semidiameter
conjugate to that which passes through the proposed
point.
Let P be any point, C D the semidiameter conjugate
to CP, join P, S and P, H.; to prove that
S P . P H = C D2.
C P* – C D* = a 2 – bº,
... C D* = C P – a” —– b”,
a" + y” — a” –H bº,
a" + (e” – 1) (r.” — a”) — a”-- bº,
e” as — es a” + bº,
e” a “ — a”,
(ea – a) (e a + a),
... S P . H. P.
(99.) Let C P, CD be any two semiconjugate diame-
ters, and let a tangent at P meet the axes of the hyperbola
in T, t , to prove that PT. P. t = C D*.
If CP, C D be assumed as the axes of coordinates,
then the equations to C A, C B are respectively
For
:
3) = a T,
bº
y = ± 4. -
Let r = a' or C P, then y or PT = a a' from (1,)
/?
and gy or P t = (2,)
a’” Q,
... P.T. P. t = 5'2 =: C D*.
SECTION III.
ON SUPPLE MENTAL CHORDS
Def. If from the vertices of any diameter two
straight lines be drawn to any point in the hyperbola,
they are called supplemental chords.
(100.) Any two supplemental chords being drawn,
and the equation to either of them being given, to find
the equation to the other.
The hyperbola being referred to any two conjugate
diameters, its equation will be
a's yº – b” a” = — a” b”. . . . (1.)
Through any point P (w', y') draw the diameter P p, and
let PQ, p Q be any two supplemental chords, then if
the equation to PQ be -
v – y' = a (r – r').... (2)
it is required to find the equation to p Q. . .
The coordinates of P being a ', y' those of p will be
— a ', - y', therefore the equation to p Q will be of the
form
gy + y' = a' (a + x').... (3)
in which a' is to be found. -
Since the lines whose equations are (2) and (3) Hyperbola.
intersect at Q, the coordinates of that point will be S-N-
identical in both; therefore considering a and y as the
same in these equations, we have by multiplying them
together
gy” — y” = a a' (r" — wº),
/ 3y" – y
”. (L. (L. E — .
a 2 — a *
º H2
... (4,)
but because a', y are the coordinates of P, a point in
the hyperbola, they will satisfy equation (1,)
... a” y” — bºw” = — a” b”.
Subtracting this from (1) we have
a” (y".- y”) — bº (wº — w”) = 0,
3/* *sº gy” b/2
". a’” — t’s - a/* >
therefore by substitution in (4)
b/2 bº
(I, a' - —F-, and a' = | }
a'? a” a
and the equation to p Q becomes by substitution in (3)
#2
y -- y' = (a + æ").
Cor. 1. Let P p coincide with the transverse axis A a,
then the equation to a Q drawn through the point
a (— a, 0) will be
a's a
= a (a + a),
therefore the equation to A Q drawn through the point
A (a, 0) will be
b/2
3/ :- ...(* wº- a).
Cor. 2. If the hyperbola be referred to its axes, we
have only to substitute a and b for a' and b' in the
above equation.
(101.) If two diameters be drawn parallel to any two
supplemental chords, they will be conjugate to each
other.
The equations to any two supplemental chords being
gy — y' = 2 (a – a '). . . . (I,)
*2
b
y-H y = ± (, + x).... (2)
let a diameter be drawn parallel to the chord whose
equation is (1,) then its equation will be
3) = a aſ,
and
therefore the equation to its conjugate being
b/2
3) = ... *
it follows that the latter is parallel to (2,) as was to be
proved. -
Cor. 1. Hence may be drawn a diameter which shall
be conjugate to a given diameter.
Let P p be the given diameter, and
First, Let the transverse axis be given.
From a draw a R parallel to P p, and join R. A.;
then if D d be drawn through C parallel to R A, it will
be conjugate to Pp.
Secondly, 1ſ the transverse axis be not given
Fig. 55.
C O N I C S E C T I O N S.
765
Conic Draw any diameter whatever R r, through r draw
Sections, r Q parallel to Pp, join Q, R.; then if D d be drawn
S-y-' through C parallel to R. Q, it will be conjugate to P p.
Fig. 56. These conclusions are evident.
Cor. 2. Hence also is derived a very simple method
of applying a tangent at a given point of the hyper-
bola.
Let P be the given point, and
First, Let the transverse axis be given.
IDraw PC and the chord a Q parallel to it, join
Q, A.; then if PT be drawn parallel to Q A, it will
touch the hyperbola at P.
Secondly, If the transverse axis be not given.
Draw any diameter R C r, meeting the hyperbola in
R, r, join P, C, draw r Q parallel to PC, join Q, R.;
then if PT be drawn parallel to Q R, it will be a
tangent at P.
(102.) To find the angle contained by the principal
supplemental chords.”
Fig. 57. Let the point Q (r', y') be the intersection of the
chords A Q, a Q, and suppose the hyperbola referred to
its axes ; then if the equations to Q a, Q A be
3y = a (a + a),
= a (ar + a),
a' — a
tan A ill – —- "
an A Q a wi 1 + a a!'
b2
or since a d' = −r,
(l
tan A Q a = + =; . . . (I.)
1 + →
Now a' = tan Q A X = 3/
a' — a
y'
d - t = −7–,
8. Il Ol. an Q a X a' -- a
g f l l
• Cº - O - . I — — —T-- . 1,
* = y : U → - TT,
2 a.
=: 9'. a’2 tº- a?”
therefore by substitution in (1)
2 a bº
t tº —--—- .
an A Q a gy' (a” + bº) '
and since the sign of this quantity is positive, the
angle is always acute.
Cor. When y' = 0, tan A Q a = ob, therefore the
angle is a right angle.
When y' = a, , tan A Q a = 0, therefore the angle
is = 0; hence the acute angle contained by any two
supplemental chords in the hyperbola may pass through
gº Tr
all states of magnitude from 0 to T2 .
(103.) To draw two conjugate diameters making a
given angle.
Fig. 58. The analytical solution of the problem is similar to
that for the ellipse, except that the reducing equation
will be a quadratic of the fourth degree. We shall
therefore proceed to give the geometrical construction.
* The chords drawn from the vertices of the transverse axis to any
point in the hyperbola, are called the principal supplemental chords
WOL. I.
meter meets the curve; but when a = +
Draw any diameter R. r, meeting the hyperbola in Hyperbola.
R, r, and upon it describe a segment of a circle con- -
taining an angle equal to the given angle and cutting
the hyperbola in Q, join Q R, Q r, and parallel to these
draw the diameters Pp, D'd ; these will be the dia-
meters required
For being parallel to the supplemental chords Q R,
Q r, they are conjugate to each other, and the angle
PC D = R. Q r, and therefore equal the given angle.
The problem admits also, as in the ellipse, of a
second solution. See Art, 64.
In the case of the ellipse, the given angle formed by
two conjugate diameters must be confined within cer-
tain limits, but in the hyperbola no such restriction is
necessary.
From the principles already laid down, the reader
will have no difficulty in adapting the miscellaneous
propositions on the ellipse, ch. iv. p. 753, to the case of
the hyperbola.
CHAPTER IV.
ON THE ASYMPTOTES OF THE HYPER BOLA.
It was shown in Art. 85, Cor. 2, that certain dia-
Emeters of the hyperbola meet the curve only at an
infinite distance, and are for that reason termed Asymp-
totes. Since the asymptotes, therefore, pass through the
centre, and are inclined to the transverse axis at an
b
angle whose tan = + -, their equation will be
(Ž
b
it: -i – T.
y = + ,
(104.) Let it now be required to find the position
of the asymptotes when the hyperbola is referred to any
two conjugate diameters.
For this purpose it is only necessary to find the in-
tersection of any diameter
= a + . . . . (l,)
with the hyperbola
a”y” — b% w” = – a” b% . . . . (2.)
Eliminating y between (1) and (2)
(a” a” — bº) tº - aſ 5'2,
6 L/
. . . a = + -— ** . º
*==g b/2 *= a'3 a? - Fig. 59.
f I./
and ‘.. y = + aſ b' a
vºbº - a's Jº
|
Now so long as b% > a” a”, or a 3 + %. the dia.
Q,
b'
a’’ the dia-
meter becomes an asymptote.
Hence, if through P the line E e be drawn equal
and parallel to D d, and C E, Ce be joined; the lines
C E X', C c Y’ will be asymptotes.
- f
The equation to C X' is y = … ".
— b
CY' is y = iſ *.
5 G
and that to
766
C O N I C S E C T I O N S.
Conic Cor. 1, Since E e touches the hyperbola at P, it fol-
* lows that the part of the tangent intercepted by the
STYT asymptotes is bisected at the point.
Cor. 2. If P N, Pn be drawn parallel to C X', C. Y',
then since e P equal PE, e N will equal NC, and
C m = m E.
(105.) The equation to the asymptotes may be
deduced from that to the curve; for we have
b' - a -—-
y = ++ wr’ – d'.
in which y is the ordinate to the hyperbola, and a the
corresponding abscissa. Now in tracing the figure of
the hyperbola from it sequation, it was shown that for
each value of a, however great, there are two equal
values of y with contrary signs. If a therefore be as-
sumed infinitely great, the ordinate to the curve ought
to coincide with that to the asymptote.
Hence in the above equation, expanding the value of
3y, we have S,
v=# ( -4. *
- # = ( – ) -a.e.)
= + , , * . .
Let x = G0, therefore all the terms containing aſ in
the denominator vanish, and we have
b'
!y = ++ £,
which is the equation required,
(106.) The asymptote may be considered as a tangent
to the hyperbola at a point infinitely distant.
For the equation to a tangent at any point (r', y') is
a/º y y' – b” a r" = — a” b”,
* _ b% w! 5%
Oi y = z: ; , --, … (1)
b'
Now y’ = + +, Va.” — a”
Suppose a' to be infinitely great, then aſ” vanishes when
compared with a 'º,
b'
•". y’ ~ + aſ a',
therefore by substitution in (1) the equation to the
tangent, when the point (r', y') is infinitely distant,
becomes
b’ — aſ bº
y = + ··· + F = ,
f
or since —F = 0,
º
f
J = + aſ £,
which is the equation to the asymptotes whence the
truth of the proposition. .
(107.) If any chord of the hyperbola be produced to Hyperbola.
meet the asymptotes, the parts of it intercepted between S-N-2
the curve and the asymptotes will be equal.
Let the chord Q q be produced to meet the asymp- Fig. 50.
totes in R, r, to prove that
Q R = q r.
Bisect Q q by the diameter C X, and draw CD con-
jugate to it; then the equation to the hyperbola being
b’ ,—
3y = # iſ va" – a” . . . . (1,)
that to the asymptotes will be
b/
y = + , , .... (2)
Now to the same abscissa C M, we have
M Q = M q
from the first of these equations, and
M R = M r
from the second ;
therefore by subtraction
R. Q = r q,
as was to be proved.
Cor. Hence PR. Prº- (M R — MP) (MR + MP)
= M R2 – M P2,
b/g
but M R2 - —z. a’,
aſ 2
b/2
and M P2 = aſ: (a” – a”),
|2
... MR – M P = . { w — a + a” #,
- b/* = C D*,
... P. R. . P r → C D*.
(108.) To find the equation to the hyperbola when
referred to its asymptotes.
Let P be any point whatever in the hyperbola, join Fig. 59.
C P, and draw the conjugate diameter D d 3 through
P draw R P r equal and parallel to D d, and join C R,
C r, which produce indefinitely towards Y and X, then
CY, C X are asymptotes. Assuming these as the
axes of coordinates, it is required to find the equation
to the hyperbola.
From P draw PM, P m parallel to CY, C X, respec-
tively, and let C M = a, M P = y, angle R C P = 0.
Then C 7 – 2 C M = 2 ar,
and C R = 2 C m = 2 y,
... Cr. C R = 4 a y . . . . (1;)
but C. r. C R sin 26 = twice the triangle R C r
twice the parallelogram DP
= 2 aſ b'sin y = . . 2 a b,
2 a b
... C R. Cr-imag:
2 a b a b
. . . . (1.)
Jº Q/ º – sº —-
hence 3/ 4 sin 26 4 sin 6 cos 6
Now tan 6 =
3.
(2
C O N I C
767
S E. C T I O N S.
Sin? 0 - sinº. 6
cos 6 1 — sin” 6
".. b3 — b% sin” 0 = a 2 sins 9
Conic
Sections.
S-V-
= tan” 6 -
*
(Z
. . sin 0 = — .
a/as + b3
Q,
cos 0 = —
Similarly y
va; + b%
therefore substituting in (1)
a b 2 2
** = Ti, (a”-- bº),
a” + bº
4 °
^ =s g
which is the equation required.
(109.) Having given the equation to the hyperbola in
terms of its ares, to find the equation when it is referred
to the asymptotes.
Let C X’ be the transverse axis, C X, C Y the
asymptotes, P any point in the hyperbola ; let fall the
perpendicular P on C X', and draw P N parallel
to C. Y.
Let C M = a, M P = y; C N = a ', N P = y', and
X C Y = 2 6, it is required from the equation between
aſ and y,
Fig. 60.
a” y” — bºa” = — a” b° . . . . (1,
o deduce that between a' and y'.
From N and P, draw N m and P n parallel to PM,
and A M, respectively. *
Then y = P M = N m — N m,
= N C sin N C m — N P sin N P m,
= (N C – NP) sin N C m,
= (a' — y') sin 6.
In like manner,
a = (a' + y^) sin 6,
... a” y = (a' — g')* a? sin” 6 = (r' — gy')* a? b% -
© g Cºsmº a*-i- bº'
b*a* = (x + ')*b* sin” 0 = (x + '). a” b%
ſº-º $/ * 3/ a” ºil b2’
... ey-ºº-ºº:: G-y) - (< 4 y))
a” — bº
a” be
- iº 4 º'y';
but a” y” — bºa” = — as bº,
... 4 a.' y' = a + bº,
Or ** = ***
4 '
which is the equation required.
(108.) The asymptotes being assumed as ares, to find
the equation to the tangent at a given point (r', y').
Any other point (a!", y") being taken in the hyper-
bola near the first, the equation to a line drawn through
(w', y') and (a", y") is
! – ar'
v-y=%–% (r – a '). . . . (1 ;)
f
*
and
but because these points are in the hyperbola, we have Hyperbola.
S-N-
a' Sy' - m°,
ac" y" tº: m”,
... w” y” — aſ J' = 0,
f
. ... ºv
Q 3/ wºme a'ſ 3.
f ... ."
! .. 4 3/
==# – y,
... y” — y
/
3/
, y” – y' y’
aſ a T T 27'
therefore by substitution in (1) the equation to the
secant becomes
f
y—y = – (, — »).
Let the point (a!", y") be now supposed to coincide
with (a', y'), then a' = a ', y' = y', and the secant be-
comes a tangent, therefore the equation to the tangent is
y
y – y' = — % (a — a').
Cor. Multiplying each side by a ',
y w' — a y' = tº-º-º: a y' + y'a'
... y + + a y' = 2 aſ y',
= 2 m.º.
(111.) To find the intersection of the tangent with the
asymptotes.
The equation to the tangent being
3/ a' + a y' = 2 m”.
Let the tangent cut the axis of r, as at T,
2 mº
7 5
then y = o and a =
and when it cuts the axis of y, as at t,
2 m2
a’ ”
4 4 m.4
Cor. Hence C T. C t = 4 m. = **
then
a = 0 and y =
Wyº 7.
... }, CT. C t x sin T C t = 2 m” sin T C t,
that is, area of the triangle T C t = 2 m” sin T C t.
In other words, if the tangent at any point be produced
to meet the asymptotes, the area of the triangle so cut
off will be constant.
= 4 m”;
(112.) Having given one point in the hyperbola, and
the position of the asymptotes, to find the direction and
magnitude of the transverse and conjugate diameters.
Let C be the centre, CX', CY' the asymptotes, and P Fig. 60.
the given point in the hyperbola.
1. To find the direction of the axes.
Bisect the angle X' C Y' by the line C X, and the
angle X'C v', the supplement of the former, by the line
C Y; then C X, C Y will evidently be the direction of
the transverse and conjugate axes, respectively.
2. To find their magnitude. -
If the coordinates of the given point P be (w', y) we
5 G 2
768
C O N I C S E C T I O N S.
Conic h
Sections. Plave
\-y-
but
a? -- bº
3/ y
a' y' =
+ = an 3, X'CY' = tan 9,
ſº b? _ sin” 6
a” cos” 0
or bº cosº. 6 = a sin” 6, therefore substituting for bº its Hyperbola.
value in (1) -
(4 x' y' — a”) cos’ 6 = a sin” 6;
... 4 aſ y' cos” 0 = a” (sin” 6 + cosº. 6),
= a-,
... a = + 2 cos 6 v aſ y,
and b = + 2 sin 6 Ma'y,
therefore their magnitude is found.
C O N I C S E C T I O N S.
769
Comic
O N T H E S E CT I O N S OF T H E CON E.
Def. Let C be a fixt point above the plane of a given
***, circle B E D, and B & Z an indefinite straight line
Fig. 61.
Fig. 62.
which always passes through C, whilst its extremity A
moves over the circumference B E D ; then B C Z will
describe by its revolution a solid figure called a cone.
The point C is called the verter, the circle B E D the
base, and the line CO, which joins the vertex with the
centre of the base, the aris of the cone.
The cone is denominated a right, or an oblique cone,
according as the axis is at right angles or inclined to
the plane of the base.
The surface of a cone is composed of two similar
pprtions, one above, and the other below the vertex;
each of these portions is called a sheet.*
It is evident from the manner in which a come is
generated, that every section made by a plane parallel
to the base is a circle.
(113.) To find the nature of the curve which results
from the intersection of a right come by a plane.
Let A Pp be the curve formed by the intersection of
a right cone by a plane; through the axis CO draw a
plane B C D perpendicular to the given plane, then
their intersection will be the straight line A. a. In A a
take any point M, through which draw a plane parallel
to the base, then its intersections with the cone and the
given plane will be, respectively, the circle N P Q and
the straight line M P, which being perpendicular to
A a and N Q, will be a common ordinate to both
CUlrWeS,
Assume Ala as the axis of a, and A Y, at right angles
to A a, as the axis of y, and let A M = a, M P = y;
also take A C = 8, angle B C D = a, and angle
C A a = 6.
Th A a sin A C a sin a
eIl ACT sin A a C sin (a +6)'
(7, tº sin a 8
T sin (a + 6).”
..'. M. a = A a - a 8 sin a
* TCTD) - Jº . . . . (1.)
Now, by the property of the circle,
MP or y” = N M. M. Q;
sin N A M sin C A a
but NM = M Air Fºi = "jºi
exº~ * a sin 6
*= e e a ’
cos;
º sin A a C . sin (a + 6)
and Mo = Maiº. = “sin Nºd
* Sheet is to a surface, what branch is to a curve
a-ºº º 3 sin a sin (a +6).
T \sin(a-E 3) T } cos & a
therefore by substitution
, a sin 6 sin (a + 6) 8 sin a *}
* > cosº, a cos ?, a sin (a + 6) y
sin 6 . e
=;{*ina. – sin (a + 6) tº },
which is the equation required.
1. Let the plane be parallel to CD,
then a + 6 = T, therefore sin (a + 0) = sin r = 0;
also sin 6 = sin (7 — a) = sin a,
therefore the equation becomes
, 6 sin” a 48 sin”; a cos” # a
(E.
*== o a' = 4 3 sin” — a
cos”; a cos”; a 2 ”
which is the equation to a parabola whose latus rectum
=43 sin’ tº -
º 2 ”
If the plane pass through the vertex of the cone, then
3 = 0, and the equation to the section becomes y” = 0,
which is the equation to the line C D.
2. Let the plane meet C B and C D,
then a + 6 × 7", and therefore sin (a + 6) is positive;
therefore the equation to the section is
sin 6
g” – {3 sin a a — sin (a + 6) r" },
cos” # a
which is the equation to the ellipse.
Comparing this with the equation
2
y = z: {2 a w — aº ; , or with
2 b” bº
* = – a – -- arº
Q/ (Z ... *,
2 b e in 6
we have — or latus rectum = *in ºn
& COS # a
= 28 tan : sin 9
2
3 sin a
t = —-,
Le (Z 2 sin (a + 6)
•. * = . . 28 tan; sin 0,
3 sin a 8 t “sin 9
- –— . In - SIIl ty,
3 in Kº Taj : * * *
- 88 sin”: º sin 6
2 sin (a + 6)"
Cl, sin 6
... b = 8 sin – —.
* 3 V sin (a TE 55
If the plane pass through the vertex, then 8 = 0, and
the equation becomes
sin (a + 6) sin 6
cos' a
y” = —
2
Cone.
770
C o N I C S E C T I O N s.
Conic
Sections.
-N-
Fig. 63.
which is the equation to the point C, since the equation
can be satisfied only by a = 0, y = 0.
3. Let the plane meet both sheets of the surface.
Then a + 6 × 7", and because sin (a + 6) is negative,
therefore the equation to the section is -
sin 6 & e s
* ~~ gºat 3 sin a a + sin (a + 6) a "$,
which is the equation to the hyperbola.
The latus rectum and axes of the hyperbola may be
determined in the same manner as in the ellipse.
If the plane pass through the vertex, then 3 = 0, and
the equation becomes
, sin 9
gy' = cos”; a sin (a + 9) aº,
v' sin 9 sin (a + 6)
*= cos ?, a
which are the equations to C B, C D ; hence the section
becomes in this case the two generating lines of the
COne. . . - -
It appears, therefore, that if a right cone be cut by a
plane, the section will be -
1. A parabola, when the plane is parallel to the
generating line. -
2. An ellipse, when the plane meets only one sheet
of the cone.
3. An hyperbola, when the plane meets both sheets
of the cone.
(114.) To find the nature of the curve which results
from the intersection of an oblique cone by a plane.
Let A Pp be the curve formed by the section of an
oblique cone by a plane. -
The construction is the same as before, excepting
that the line M P is no longer perpendicular both to
A a and N Q, but only to the latter; we shall assume
therefore, as oblique axes, A a and A Y parallel to M P.
8 sin a
sin (a + 6)"
8 sin a
M. a = #F#, -
3/2 = N M . M. Q;
..". 3) = +
5
Hence, as before, A a =
a . . . . (1 ;)
also
sin 6
but NM = *... a.
tºmº sin (a + 6)
and M Q = M a sin (a + B)'
=#| || 3 sin a – , ,
sin (a + B) \sin (a + 6) ſ'
sin 6
•". :-- in a . a -si |-0 9
y? in Biº-TE) {*** a – sin (a +6) tº 3
which, according as the given plane is parallel to CD,
or meets one or both sheets of the cone, is the equation
to a parabola, ellipse, or hyperbola, referred to oblique
axes. * -
Cor. To find in what cases the section is a circle.
Having put the equation under the form .
, sin 6 sin (a + 6) ſ 3 sin a 8 U.
9 ºf sin B sin (a + B) \sin (a + 35* - £"?,
it is evident that the section will be a circle when the
coefficient • *
sin 6 sin (a + 6) l
sin B sin (a + B) T ‘’
Or sin 6 sin (a + 6) = sin B sin (a + B),
or cosa – cos (a + 2 0) = cos a — cos (a + 2 B),
... cos (a + 2 6) must = cos (a + 2 B),
‘.. a + 26 = a + 2 B . . . . . (1),
Or = 2 r — (a + 2 B). ... (2.)
First, if ...-a -- 26 = a +2 B, . . . . . .
6 = B, . -
that is, when the plane is parallel to the base the
section is a circle.
Secondly, if a + 26 = 2 r — (a + 2 B),
2 a + 2 6 -- 2 B = 2 r,
Or a + 0 + B = r = ... a + D + B,
... 6 = D ;
hence, when the angle C A X is = C D B, the section
is also a circle. This is called the subcontrary section
of the cone. - -
Cone.
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IDIFFERENTIAL AND INTEGRAL CALCULUS.
PART I.
DIFFERENTIAL CALCULUS.
Differential (1) In the investigations of the relations which exist between several quantities, those which are supposed to , " "
Calculus, retain the same value are said to be constant, and those to which several values may be assigned are said to be
-vº-' variable. The first are usually represented by the first letters of the alphabet, and the others by the last. The
words constant and variable are also frequently used substantively, to express constant and variable quantities.
(2.) When variable quantities are so connected that the value of one of them is determined by the values
ascribed to the others, that variable quantity is said to be a function of the others. Thus, for instance, the sum
of the terms of a geometrical progression is a function of the first term, of the ratio, and of the number of the
terms. In a like manner, when an equation subsists between several variable quantities, any one of them is a
function of all the others. - ~
To express in a general manner a function of one or more variables, one of the letters F, f, ºff, pr, &c. is usually
prefixed to the letters by which the variables are represented, enclosing them at the same time between paren-
theses, and separating them, when there are several, by commas. Thus, F (x), @ (r, y, z), signify, the first a
function of the variable a, and the second a function of the three variables a, y, z. Another notation, also
employed, consists in placing simply the variable on the right side of, and a little below, the letter U, or any
other. Thus, U, Z, y denote, the first a function of a, and the other a function of a and y.
(3.) A function of one or more variables is said to be explicit, when the operations, to be performed on the
variables, to obtain the value of the function, are immediately expressed by means of algebraical signs, or by
means of notations previously defined. But when the relation between a function and the variables is only
expressed by means of an equation, it is said to be an implicit function, as long as the equation is not
resolved.
(4.) Functions receive different denominations, according to the nature of the operations which produce them.
Those which are formed by means of the operations of Algebra, viz. addition, subtraction, multiplication,
division, involution, and evolution, are called algebraical functions: those which contain variable exponents are
called exponential functions; they receive the name of logarithmic functions, when they contain variable loga-
rithms; and they are designated by the name of circular or trigonometrical functions, when some of the ope-
rations of Trigonometry are required to form them. All those which cannot be reduced to some of the preceding
are called transcendental functions. - -
(5.) Algebraical functions are again divided into rational and irrational functions; the first being those which
contain only integral powers of the variables, and the last containing fractional powers of the variables, or
radical quantities, under which the variables enter. An integral function is a polynom which contains only
integral powers of the variable; and the quotient of two such functions is a fractional function.
(6.) Different values of a function often correspond to a set of values of the variables. In the function
A (a — a)} + B, for instance, two values correspond to every value of a, except to the value a = a ; in the
function log. a, an infinite number of values correspond to every value of a, one of which is real, and all the
others imaginary, when a is a positive quantity; and all of which are imaginary, when a is negative. The arc
being considered as a function of its sine, is another instance in which, to every value of the variable, corresponds
an infinite number of values of the function, but in this case all the values are real.
In establishing any conclusion with respect to any particular function, it is always necessary to examine whether
it is true for every value of the function, and if not to state for which of the values it obtains.
(7.) When a function of one variable f(r) takes a single and finite value for every value of a equal to or greater
than a, but less than, or, at most, equal to b; and that, at the same time, for every value of a between these limits,
the difference f(x + h) – f'(a) may be made less than any assignable quantity, by taking h sufficiently small,
f(a) is said to be a continuous function, between the limits a and b.
A function is also said to be continuous for values differing but little from a particular value a, when it is
continuous between two limits, nearly equal, the one greater and the other less than a.
(8.) A quantity A is said to be the limit of a function of one variable a, when the values of that function
corresponding to a series of increasing or decreasing values of the variable, continually approach to A, and
that a value of a may always be assigned such as to make the difference between the limit and the function less
than any given quantity. O
The function A + Ba, for instance, has obviously for its limit A, for decreasing values of a ; and A + +
has the same limit, for increasing values of the variable,
771
772 D I FF E R E N T I A L C A LC U L U.S.
Differential It is not always easy to find the limits of a given function of a, but various simple remarks frequently Part I.
Calculus. facilitate their determination. If for every value assigned to a, for instance, the value of f(a) is always S-2
Y-v- included between the corresponding values of two other functions of the same variable, which have for their
common limit A, it is evident that A will equally be the limit of f(r.) -
It follows also from the above definition, that if A and B represent the limits of two functions of a, them
A + B, A – B, A B, #. will respectively be the limits of the sum, the difference, the product, or the quotient
of the two functions.
(9.) All functions can undergo, without changing their values, an infinite number of transformations; from the
comparison of some of which their properties arise. When they are transformed in a finite or an infinite series of
terms connected together by a certain law, they are said to be developed, and the series is called the developement
of the function. Among the various developements of a function, that which proceeds according to the powers of
the variable has been most considered, and appears to be of a greater importanee than any other.
The binomial theorem, demonstrated in Algebra, furnishes examples of finite and infinite developements of
functions according to the powers of a variable.
In order to render the nature and object of the Differential and Integral Calculus better understood, we
shall begin by demonstrating the following theorem, relative to the transformation or developement of a
function of the sum of two variables into a series of terms containing the successive powers of one of them.
(10.) Let u represent any function of x, and uſ what that function becomes when in it x is changed into
x + h. Then, provided x remains an indeterminate quantity, u' may always be developed in a series of the
following forms :
w – Ph. -- Q hº + R h’ + S h" + &c.
where P, Q, R, S, &c. do not contain h.
Let us first suppose w! = N + Ph" + Q h" + Rh' + Sh’-- &c. . . . . . (a,)
N, P, Q, R, S, &c. being unknown functions of a, and a, 6, 7, 8, &c. indeterminate exponents, arranged in
ascending order.
It is first obvious that all these exponents must be positive; for if any of them were negative, the supposition
h = 0 would render u infinite, while by that hypothesis it becomes equal to u. The supposition of h = 0, in
both sides of equation (a,) proves now that N = u, since it makes the left side equal to u, and the other equal
to N. The equation (1) will therefore have the form
w! = u + Ph" + Q hº + Rh' + Sh’ + &c. ... (b.)
Let us now change h into h -- k, and let u" represent what w/ becomes by that substitution, we shall have
w" = u + P (h+ k)* + Q (h -H k)” + R (h -- k)" -- S (h -- k)' + &c. ... (c.)
But if in equation (2) we change r into a + k, ul will also become equal to w!"; for the result of this substitution
will be again the same function of a + h + k, as u is of w. The quantities u, P, Q, &c. which are functions
of a, will become functions of a + k ; and we may represent their developements according to the powers of this
l st quantity respectively by
w -- P k" -- &c.
P + P' k" + &c.
Q -- Q' k” + &c.
R + Rſ k” + &c.
&c.
The first differing only in the developement of uſ in equation (b) by the change of k for h. Thus by the sub-
stitution of a + k for h in equation (b) we shall have
w" := u + Ph" + Q h" + R. h7 -- S h’ + &c.
+ PK-4- P’ k” h’ + Q' k” h’ + R/k” h’ + S'k” h’ + &c. § . . . (d.)
+ &c. -- &c. + &c. -- &c. + &c.
The two values we have obtained for u", must be equal; let us first compare them in the supposition of k = h,
where they become respectively
w" = u + P 2" h" + Q 2°h" + &c.
w” = u + P h" -- Q h" + Rh' -– S h’ + &c.
+ P hº + P'h' +* + Q' hºt" + &c.
+ &c. -- &c. + &c.
And these cannot be equal unless the terms which multiply the same power of h be separately equal to each
other, since the equality must subsist, h remaining an indeterminate quantity. We shall have, consequently,
O P. 2* = 2 P, hence 2* = 2, 2* ~ * = 1, and a = 1.
The equation (b) will thus become,
- u' = u + Ph + Q h’ + R h’ + Sh’ + &c
D IF F E R E N T I A L C A L C U L U. S. 773
Differential and therefore the second term of the developement of any function u' of the sum of two variables, according to Part I.
Calculus. , the powers of one of them, contains the Jirst power of that variable.
STNT (11.) The immediate consequence of this proposition is, that the exponents a!, a!", a”, &c. are each equal
to unity.
This understood, the equations (c) and (d) will become respectively,
u" = u + Ph -- Q h" + Rh' -– S h’ + &c.
+ P k + 3 Q hº-i k + q Rhº-1 k + 8 Shº-1 k + &c.
+ &c.
u" = u + Ph -- Q hº + Rhy -- sh’-- &c. -
+ P k + P'h k + Q’ hº k + Rſ hº k + S'h' k + &c.
+ &c.
The first lines of these two values of u" are the same; the second lines are composed of all the terms which
contain the first power of k, they must consequently be equal. Dividing each of these lines by k, and suppress-
ing P, which is common to both, we shall have the following equation,
£3 Q hº- + y R hy-1 + 3 S h’-i -- &c. = P' h + Q' h’ + R'h' + Sº h’ + &c.
in both sides of which the exponents of h being in ascending order, the terms of the same rank must be equal
to one another, and therefore we shall have -
g Q hº- = P h, , Rhº-i = Q'h', 3 sh’-- = S h", &c.
e *N P/ Q’ s S!
From which we get {3 = 2, Y = 3, 3 = 4, &c. Q = -a-, R = +, s = -i-, &c.
Substituting the values of the exponents, the equation (2) becomes
w’ = u + Ph. -- Q h” -- R h’ + Sh* -- &c. . . . . . (7,)
which proves the theorem stated (10,) and shows, moreover, from the above values of Q, R, S, that Q is equal
to half the coefficient of the second term of the developement according to the powers of h, of what the
function represented by P becomes when a is changed into a + h; that R is equal to the third of the coefficient
of the second term of the developement according to the power of h, of what the function represented by Q
becomes when a is changed into a + h, &c.
We are indebted for this very important theorem to Dr. Brook Taylor. We shall soon see with what elegance
it may be analytically expressed by means of some notations we shall now proceed to explain.
(12.) The difference u' – u between any function of one variable x represented by u, and the value u' assumed
by that function when in it x is changed into x + h, is called the DIFFERENCE of the function u, and is represented
by A u. So that, according to what precedes,
A u = Ph. -- Q h” + R h9 + S h -- &c.
The first term Ph of this difference is the DIFFERENTIAL of the function u, and is designated by d u. Thus
- d u = Ph.
According to these notations we shall have A a = h and da = h, since h is at the same time the whole difference
between the function a and a + h, and the first term of that difference.
The coefficient of h in the differential of a function, or the coefficient of h in the first term of the develope-
ment of the difference, is called the differential coefficient of that function. It is therefore equal to ;: Or
d º ę º | 8 º
#, since h and d a represent the same quantity. This understood, equation (7) may already be written in the
Jº
following manner,
d u d P h? d Q h3 d R. h4
| = mºssºmsºmºsº *-***** *==mºs *-sº gº-º-º-ess-mº as & & a s g a tº 9 g
7/, ºu + d a h –– d a 1.2 + d a 1.2.3 H d a 1.2.3.4 + &C (8.)
d u
(13.) The differential coefficient of a function of a is generally another function of a, which has also a
differential and a differential coefficient. By means of the agreed notations they will respectively be represented
d u
d w d a ſº d? u
by d. da: and —HI-. It has been agreed upon to write the first a ’ and consequently the second
dº at
d a " ' It should be remembered, that in these expressions the figure 2 placed a little above d is not an exponent,
but only indicates the differential of a differential; and that d tº does not signify the differential of wº, but the
º
d2 u.
g e he f º
square of d ar. The function d r*
, or the differential coefficient of the differential coefficient of the function u,
is called the second differential coefficient of that function. It has also a differential, and differential coefficient,
VOL. I. b H
774 D I FF E R E N T I A L C A L C U L U. S.
Differential d d” at Part I.
Calculus. d2 u. º T.I.F 3 ºf dº ºu.
S—— which will be expressed by d . d arº and — Pi— , or in using the preceding notation by d a " ' and d anº
This last quantity is the third differential coefficient.
It is now easy to understand what is meant by the fourth, &c. or generally by the nº differential coefficient of
the function u, and that they may be represented by
dº nº d4 u d"-" tº d" at
d r3 ° d tº ' ' ' ' ' ' ' ' ' ' d r"-" d ar” "
(14.) We may now make use of these notations, to express more simply the developement of the dif-
ference of the function u. It is plain from the relations we have found between the successive coefficients
P, Q, R, &c. of that developement, that
d u d” at d? M.
d a ’ Q = Ol ºr? . R = d a 3 *
d4
s = **, &c.
º
P =
d
and consequently that
d at h d” u h” dº at h” d" tº h4
d a l + ºr 1 .. 2 d as 1. 2 .. 3 d a 4 l. 2. 3. 4
This may still receive another form, by substituting d r for h, and writing w in the left side of the equation. It
becomes then
w' = u +
+ &c. . . . . . (9.)
w" — u = A w = ** + d” at + = +
- tº- l I . 2 ! .. 2 .. 3
which expresses the difference of a function by means of its successive differentials.
(15.) The solutions of a great many important and interesting questions have been found to depend upon the
differential coefficients of functions. This has given rise to a separate branch of Analysis, the object of which
is, first, to show how the differential coefficients of functions may be obtained ; and, secondly, how, from the
knowledge of the differential coefficients, or, ſrom known relations between the functions and their differential
coefficients, the values of these functions may be determined. The methods hitherto discovered to resolve the
different cases of this double problem, constitute the Differential and Integral Calculus.
To obtain the value of the differential coefficients, various considerations have been used ; sometimes that of
the rate of increase of functions for increasing values of the variables; sometimes that of limits, &c. Each of
these views may be employed exclusively, to establish the principles of the differential calculus; and hence have
arisen the divers methods which have been proposed, each possessing some advantage in particular cases, but all
arriving at the same end, though by different means.
(16.) From what has already been stated, we may deduce a general method to find the differential coefficient
of any explicit function. It will be sufficient to substitute a + h for a in the function, then to develope
according to the powers of h, and the coefficient of the first power of that letter will be the quantity required.
If, therefore, we knew how to find the developement of every such function, the problem of the differentiation of
explicit functions would present no difficulties. When this cannot be done easily, the value of the differential
coefficient may be determined by means of the following proposition.
(17.) We have found
+ &c.
d u d” it h” dº it h9
- – — — — — — — — — &c.
d a it is is d : I.3.3 ºf C
Aſ
u' – u
w' – u du du A di u ę.
h" T dºr T dº? I, 2. is Tālā -H &c.
Hence
r & g g g te . . . d u d u . © gº g
The right side of this equation has obviously for limit dº (8), therefore H. is also the limit of the left side.
T -
g e . . d u Q is e -
Thus, the differential coefficient da, of any function u is equal to the limit of the ratio between uſ — u or the
difference of the functions, and h or the difference of the variable.
We shall now proceed to the investigation of the value of the differential coefficient of the various explicit
functions of one variable.
(18.) The differential coefficient of u + A, A being any constant quantity, and w any function of r is the
same as the differential coefficient of u ; and the differential coefficient of A u is equal to the differential
coefficient of u multiplied by A. -
If in w we change a into a + h, we shall have
d u. d2 u h
H h -- H. riº
2 d a 2 1 .. 2 -- &c.
u' == u +
. JC 1 .. 2
D I F. F. E. R. E N T I A L C A L C U L U S.
75
Pºw
l
Differential Consequently the developements of what the functions u + A become, and A u, when in them a + h is
Calculus. substituted for a will be
\-N-
d u d2 u h;2
e-Eº *— — —H &c.
u + A + ih + i = H, + C
d at A dº uſ h?
*E*E****º- — — —H &c.
A u +AH;h + H+ H+&c
And since the coefficient of the first power of h in the first is da,
have
d (u + A) d u nd dA * = . A dº
Pi—= H, and -H = Hi-
(19.) When two functions of the same variable are equal, their differential coefficients are also equal.
Let u and v be two equal functions of ar.
results of this substitution, we shall have u' = v'.
w-u-4. A +
Hence d w
but u = v, and as h remains indeterminate, the coefficients of the terms which contain the same power of that
quantity in both sides of the equation must consequently be equal.
du d v
d a T d p’
(20.) The reciprocal of this proposition is not true, that is to say, that from the equality between the
differential coefficients of the same rank, of two functions of the same variable, we cannot infer the equality of
the functions. If, for instance,
dº u_ dº v
d a 4T d arº’
are equal to each other ; but we cannot say any thing about the equality of the preceding terms, and, con-
We shall be able, hereafter, to give the form of their difference.
sequently, about that of u and v.
(21.) The differential coefficients of a function, composed of the sum or difference of several functions of the
d” at
hº
dº T. 2
I . 2
dº wº h2
- . — , — &c. =
u + i. h 4 ± is + &e
d’u dº v
da:3 T d tº
If in each we change a into a + h, and represent by u' and v' the
But, by Taylor's Theorem,
d v
-- &c., and v' = 0 + H+. h. -- H.
2
d” v #5-se
da, d r* I .
d v d? v
– . h. — .
* + 7. Târ,
Therefore
d? at d? v
d a 2 T d r2"
&c.
, it will result, it is true, from the preceding proposition that
h2
I 2 + &c.
same variable, is equal to the sum or difference of the differential coefficients of these functions.
Let u = y, + y, - 2, - 2, u, yi, y, z, zz, being functions of a.
nate by u', yi', &c. what these different functions become, we shall have
ſ — A / f ſ
w' = y,’ + y, - 2,'— z's,
and by Taylor's theorem
Therefore
d tº
u + —
And, consequently,
du d° ºf h?
u' = u + º-º: . — . º
+ . h + i ; H, 4 &e
d d? h2
y = y + #. A + i. i*i; +&c.
&c. &c. &c.
d° at /.2 d y, dº y, h”
i; "+ . . Tºg-H &c. = y, + ... h--- . 1–5 -- &c
d dº h”
+y, + ... h-i-H. Tºg--&c
d zi d”z, hº
- 2 - # h – : . H. - &c.
d 22 d”z, h2
- -, -ā- i. i*i; - &c.
du d y, d ya d z, d 2,
... = ... -- * – ; ; – , ,
tº da, d an dr d a
**, and in the second A4. we shall
and, consequently, that in the developements of u' and v' all the terms, beginning with the fourth,
If for a we substitute a + h, and desig-
Part I.
Sºº-y-Z
776
D I FF E R E N, T I A L C A L C U L U.S.
Differential and also - Part 1.
Calculus. dºw - dº y, dº y, * d’2, * = **. S-N-2
d a 2 d x * ' d arº da;" d anº
&c. &c.
(22.) The differential coefficient of the product of two functions of the same variable, is equal to the sum of the
products of each of them by the differential coefficient of the other.
Let u = y, y, we shall have u' = y,' ye!, but
d u
d” at 2
! — * tº-sºmºsº
w = u + +...h-Fi: , H +&c.
dy dº y, hº
y' = y, # h -- i. º H; +&c.
dy dº y H.”
* — 2 2
yº-y, + i. h + i. H +&c. -
Multiplying the two last equations, it is easy to see that in the product of the two right sides, the coefficient of
the first power of h will be
gº-º-º-mº
d y d y
3/1 ; –– 3/2 d º ;
and since this product is equal to the developement of u', h renaining an indeterminate quantity, we shall have
du e- d y, dy,
d : T 9, d. * d a
If we suppose w to be the product of three functions yi, y, y, the preceding proposition will give
d u ..., d. 939, d y,
i...— V, -ī; + 9, V, i.
d 3/33/s gºsº d 3/a dys
but # = y, i. # 9, i.
Hence by substitution d d
- d u dy, 3/2 $/l
da: T 3/1 9, i. + 3/13/s da, + y, y, da, ’
and, generally, if u = y, y, y, . . . . y,
d u d y d y
; := y, y, . . . . y i. --V, V, . . . 9. i.
d y,
Jº # -- . . . . V, V, . . . V.- ...
This equation and the preceding ones may receive another form, by dividing both sides by u. The last becomes
then
+ £- #-F#-...... + 1 39.
2/, a y, d a 9, d a
J., d r"
(23.) The differential coefficient of a fraction whose numerator and denominator are functions of the same
variable, is equal to the denominator multiplied by the differential coefficient of the numerator, less the product
of the numerator by that differential coefficient of the denominator, the whole divided by the square of the
denominator.
Let w = 3/1
, where y, and z, are functions of r.
2, .
R
Multiplying both sides by z, we shall have w x = y,
d it z, d y, du z, ... d z, d u u d 2, d tº d y,
consequently Jº dº,” but dº, T “ dº + z, d w' therefore +
*
* ... F., and hence
d 9. d y, d 2
- -** — u → d y, d z
du 2, d a da "d a 9, Jr.
da T da T g
*=
l
(*
l
2,"
This last value may be written under another form by multiplying and dividing it by 3/
&=º
---. It them becomes
- 3,
d. 9
# =}}} d y, 1 #}
da z, ly, da z, d a j
D I FF E R E N T I A L C A L C U L US 777
Differential - . (l, dz, Part I.
Calculus. - d 2. (Z da, \-y-’
\-y-. If the numerator is constant, equal to a for instance, then + = — 9 *
- º &
º -
(24.) If u is a function of y, and y a function of r, then the differential coefficient of u considered as a
function of a, is equal to the differential coefficient of u considered as a function of y, multiplied by the
differential coefficient of y considered as a function of a.
Let u = F(y), and y = f(r). To prove the truth of this proposition, we must show that when a is changed
into a + h, the developement of the corresponding value of u according to the powers of h, has for the coeffi-
g ... d d : G - e. -
cient of the first power of that quantity # e # . In that supposition let y' be what y becomes,
D
dy dº y h
! --- gºmº º 7-
3/ = f(x + i) =y++. h + i. © H; +&c.
d d? . h2
Letº. , h +. T. 2 + &c., the increase of y corresponding to the substitution of r + h for r, be
represented by k. Then if we change in the function u, y into y + k, we shall have the value of that function
corresponding to r + h. Let u' be this value
d u d2 u k?
f — º tº-º-º-º: &m-s ºms
'll, = FG + i) = u + iy k++; H +&c.
tº & o © a * : * tº d y dº y hº
It is easy to see, now, that if we substitute in this developement for k its value da, h -H dº? I, 2 + &c., the
e g g ſe du dy
only term which will contain the first power of h will be dy dº Therefore
du du dy
da T dy dºn’
When u is a function of y, reciprocally y may be considered as a function of u, and an immediate consequence
of the proposition just demonstrated is, that the product of the differential coefficient of u considered as a func-
tion of y, by the differential coefficient of y considered as a function of u, is equal to unity.
(25.) The differential coefficient of the function a r" + b is equal to m a w”-", for every value of m, positive or
negative, integral or fractional. º
This results evidently from the binomial theorem demonstrated in Algebra. For if we change a into (a + h)
in the function we shall have by that theorem
2
h
a (a + h)" + b = a a " + b + m a a "-" h + m (m — 1) --" 2
a developement in which the coefficient of the first power of h is equal to m a w”. Therefore if we suppose
w = a a " + b, we shall have
+ &c.,
d u. d’t m (m = 1),...,
------ ** m = 1 arms T –2, .
H. = ma "Tº T. -T3- aw",
d"w m (m – 1) (m. – 2). . . . . . (*~ * + 1) a cº-
I. - Ta-à T. 7?,
If m be an integer, it is obvious, from these formulae, that the m” differential coefficient will be equal to a,
and, consequently, all those of a higher order equal to nothing.
It will be easy, by means of the preceding rule, to find the differential coefficients of every algebraical func-
tion of one variable. We shall apply it to a few examples.
Example 1. Let u = V (a + b x + ca” + &c.)”. Assume a + b x + c a' + &c. = 2, then u = V2" = z*.
d
2. du m *-i d u du d z
- b o d — ºr — z º. o — = — . –
—H 2 ca -H, &c., an d 2 ... * But by (24) d a dz d r" therefore
-
d
Jº
d u m lº- e & a a º º
— = 7. ,7- (b + 2 c a + &c.); or in substituting for z its value
d a
#=# (b -- 2 cr + &c.) V (a + b c + cz + &c.)"-".
If m = 1, n = 2,
du (b+ 2 cr-H &c.)
dº. T 2 ya-i-ba -- ca.”--&c.)
778 D IF F E R E N T I A L C A L C U L U.S.
*
º Ea'ample 2. Let u = (a + b x + c wº)" (a' + b a + c' wº)". Part I.
J. We shall have, by (22.) S-N-2
d u , .., d. (a' + b'a -- c'aº)" / f , d. (a + b a + c aº)"
# = (a+b++ or ) º d a + (a' + b' a' + c aº)". d as gº
But, -
d (a' -- bº of £2)" tº b QY?t
(a' -- # a") =n (b'+2 cl a) (a’--b' w—Fc'a")"-" and d. (a. #: + ca”) =m (b+2 ca) (a+b w—- ca.”)"
substituting
#= (a + b x + c wº)"-" (a' + b'a + c'a")"-" { (b' + 2 c'a) (a + b x + c wº) + (5 + 2 ca) (a' + b/w -- c'aº) }.
_ (a + b x + c wº)"
T (a' + b x + claſſ).”
we shall find, in applying the rule given (22,)
du (a + b a + c wº)"-" { (b' + 2 cl r) (a + b x +ca") + (b + 2 ca) (a' + b/w -– c' wº) }
Erample 3. Let º,
d a (a' + b'a + c'a")"+"
4 8 b
Eacample 4. Let u = V/a-º-we-ro). Assume a - ºr + V (c’ – a “) = y; then
8. - d u 3 -
*— 4 iſºsºm – = - 4
u = 4/y” = yº, and herefore #: 4 $/ '
a( – 4 +ve-ro) – a ſº
but d y_ v w – V* + d V6 – ?)
Ul dºr T da, ºr sº d a d a 2
a + - *
d w/a d b a * |- bri = — b
8|| d ºr T d a T 2 T 2 a y
I
*V(º-º) = *(*-*); - ?-6 – , )-; — 2 a. E — 2 a. {º
d ſt d a 3 3 V(cº – wº)*
ſº d d d
And since tº - tº y we shall have, by substitution,
a r T ây dº
3 b 2 ºr
32 vº. T 337(cº-rºy,
d
* Wºw…]
We shall give two more examples, in which we propose to find the second, third, and differential coefficients
as well as the first. ~
Eacample 5. Let w = a + b a + c a' + da:*-H. . . . . . ta", then
** = b +2 or +3 art a 6 g º s is s is m a "-1,
da,
# = 2 +2.3 as + e e e s G G = & m (m – 1) a”-º,
** = 2 3 d -H, m (m – 1) (m. – 2)a"**
d as T tº Kºº Wºº e º is e º O p → 9
# = m (m – 1) (m. – 2). . . . . . . . l.
Example 6. Let u = (a + b a + c aº)", and let it be required to find the n” differential coefficient of u.
Instead of calculating successively the first, second, third, and differential coefficients, it is obvious from
Taylor's theorem, that we shall obtain the n” differential coefficient at once ; if, after having substituted a + h
for a in the function u, we can find the coefficient of h" in the developement. For it will be sufficient to multiply
it by 1.2. 3.... n to have the value of the nº differential coefficient. The result of the substitution of (a + h)
for a in u, gives w' - (a + b x + b h + ca" + 2 ca, h -- ch”)".
Assume a + b a + c a' = p, and b + 2 c r = q,
D IF F E R E N I T A L C A L C U L U. S. 779
Differential then u' e (p + q h + ch”), Part I,
Calculus, and by the binomial theorem
r (r – 1 * (r – 1) (r. – 2
w'- (p + q h)" + + (p + q h)” chº + “H” (p + q h)"-*.cs h" + . ( l }. * *(n+qh)" cºh9 + &c.
If we develope now the powers of (p + q h) which are indicated in this series, and collect afterwards under
the same coefficient, the terms which contain the same power of h, we shall have the developement of uſ accord-
ing to the powers of that letter. But since we only want the coefficient of h", it will be sufficient to calculate
the coefficient of h" in the developement of (p + q h)"; that of h"-" in the developement of (p + q h)r-, since
that binomial in the above series is multiplied by h”; that of h"-4 in the developement of (p + q h)^*, &c.
These coefficients are respectively,
r. (r — 1) (r. — 2). . . . . . (r – m + 1) pr-" q"
1. 2. 3. . . . . . . . 72, Q",
(r —- 1) (r – 2) . . . . . . . . (r — n + 2) pr-rº gº.--
l. 2. 3. . . . . . . . 77 – 2 Q" ",
(r — 2) (r. — 3) . . . . . . . . (r – m + 3) pr--- gº-
1. 2. 3. . . . . . . . 77 — 4 2
&c.
Hence the value of the coefficient of h" in the developement of u', will be the sum of these quantities, which
being multiplied by 1. 2. 3. . . . m will give
d" it
1. * r (r-1) (r-2)... (r—n + D p'-- r|. +
n (n=1). p + (º- D (n-3) viº ºr
r — n + 1 gº I.2 (r-n + 1) (r-ºn-F2) ºr
n (n − 1) (n − 2) (n – 3) (n — 4) (m. – 5) cºp”
+ I.3.3 (in Eijº-jia) in I; ſº * *
d" u,
d ar"
of analytical transformation.
The value of may be put under a simpler form, which it will not be useless to give here, as an example
We have, first, w = (p + qi + cºy-r( ; ; * +4; w).
But p = a + b a + ca", hence 4 p c = 4 a c + 4 b c r + 4 cºa”,
q = b + 2 ca, hence q* = b + 4 b c ºr + 4 cºa", *
therefore 4 p c – q^ = 4 a c -- b”. Assume 4 a c – b% = e, then 4 p c = q.” + e”,
substituting in the value of u', we shall have
2 q q* -- e” r {( Q Q e? h"
! — ont t ? ) - 707 *mº 2 y .
w = r(1 + # * + 4 pº .*) p * 1+; a —H· *}
4 pº
Hence
' = , ſ 1+-4– "... ." ‘l *-** He “H”( _9_ )"; r (r — l) (r. 2)
QM, r|( Tân *)-f( : , ) Tº " I . 2 1+3, '' Tº ºf 1 .. 2 .. 3
-q 3r-6 eg hº |
I —— — — h — — — &c. º.
( + ºr ) #F# &
Developing each of the binomials, collecting the terms which multiply h", and multiplying their aggregate by
1. 2. 3. . . . m, we shall have
*" = 1 2. 3 ..ſ 2 r (2 r-1)... (2 r-n + 1) q" r (2 r – 2) (2 r – 3).. (2 r – n + 1) q"-* e”
# = | *.*, *ſ-La-. 70, 2"p" 1 .. 2. . . . . . . . m -- 2 27-7. Tº
• U - - 2 r — 4) (2 r — 5). . . . (2 r — l * - 4
-- " : " .. 2 (2 r ) (2 r – 5) (2 r – n + 1) q' > -º- +&c.
1 .. 2 l. 2. 3. . . . . . . . m — 4 2” “p"-- 4 p. J
This value may be written in the following manner,
# = 2 (ºr - D (2r-n + 1) + p"
d r" t tº e º E 2”
r m (n − 1) 6. r (r – 1) m (n − 1) (n − 2) (n − 3) e? }
{1+ I 2 r (2 r - 1) ++ 1. 2 2 r (2 r – 1) (2 r – 2) (2 r – 3) q" —|- &c
m + 1
g tº sº 77 s e
When n is an even number this series has 2. + 1 terms, and when n is odd. This formulae and the other
found before, were first given by Lagrange. They led to important and curious results, when various values
are assumed for u and n. (See a collection of examples on the application of the Differential and Integral
Calculus, p. 12.)
780 D I F F E R E N T I A L C A L C U L U. S.
Differential We shall now proceed to investigate the rules to find the values of the differential coefficients of the expo- Part I.
Calculus. , mential, logarithmic and trigonometrical functions. S-N-
(26.) The differential coefficient of the function a” is equal to a” la, l a being the hyperbolic logarithm of the
base. - &
I,et a be changed into a + h, the difference of the function will be a” — a” = a” (a" – 1), and it is the
coefficient of the first power of h in the developement of that difference that we are to determine. Assume
a = 1 + b, then a' = (1 + b)', and therefore the difference of a” takes the form a” { (1 + b)" — 1 }
Expanding (1 + b)" by the binomial theorem, we shall have
Jº — ... ( * h (h – 1) , , , h (h – 1) (h — 2)
et a 4-y-1} = -(+. +**** ** I . 2 .. 3
If we arrange now the terms between the parenthesis according to the powers of h, we shall have for the coeffi-
cient of the first power of that quantity the following series,
b° -- se).
d a”
d aſ k a
dº a” is ... dº a” § 2.8
Hence d a " - k” aº, d a 9 k” a”, &c
Therefore by Taylor's theorem
- h 3. 8
* = or * . — Ş ſº a . e
Cº. a” + k a 1 + k” aw H + k a H + &c
Dividing both sides by a′,
k h k”. hº kg h9
h — Eºmºsºme gºsºsº ºn ===s**º- *
a = 1 + + + H++ -īā-ā- + &c
\
This equation being true for every value of h. Assume h = Tº it will become
+ l I I
k - -->4 º ºmºmºmºmº &mºmºmºmºmºmºmº *=º &c.
As a' = 1 + 1 + H++ H+, + H+ i + &c
The ratio of two successive terms of this series decreases rapidly. Therefore we can approximate indefinitely to
its value. The ten first terms equal 2.71828.18. Let the whole be represented by e, then
l I R
e = 1 + 1 + H+ + Ha + -Hai- +&c.
1
and a k
tº 62.
The number e is of frequent use in analysis; and it will not be useless to prove, before we proceed, that it is
incommensurable. First, e cannot be a whole number; for, evidently,
I I } I l I
T= + T = a + -īga T--F &c. . . . . < — — — — — — —H. &c.
but the last series is equal to one. Hence
I I l
&=º m. me am-m-m-tºss- mºſºme ºsmºss-s-- ... < 1.
T3 + T = a +Ta a 4 + &c. “
Therefore e is not an integer, and its value lies between 2 and 3. Secondly, no fractional number can be equal
to e ; for, if possible, let + = e, n being an integer less than m, but greater than one. Then
777, l I l I
+ = 1 + 1 + H + Ha + ...... + +
— — — &c.
1. 2. 3. . . . m. TālāTHE TI, + °
Multiplying both sides by 1. 2. 3. ... n, it becomes
l I
. 2. 3. . m – 1 - 7m = 1 . 2. 3. . l. 2. 3. . 4. 5.6. . . . -- ' — o
1. 2. 3. . m – 1 . m = 1. 2. 3. . n + 1 .. 2 * + 4.5 . 6. . m + +n+1+; I F & II) (a F 2) + °
The left side being an integer, the other side should also be one. This cannot take place unless
l 1
77 –H l * G.II) (n + 2)
I + 1
m + 1 (n + 1)*
mensurable number.
+ &c. be a whole number. But it is impossible, for the series is obviously less
l tº sº ºn 1
than + (n + IY + &c., which is equal to †. Therefore e cannot be equal to a com-
D IF F E R E N T I A L C A L C U L U S. 781
Differential
Calculus.
I
We resume now the investigation of the value of k. We have found a k = e. Taking the logarithms of both Part I.
L
-N sides, we find L. a = k L e, or k = 14. and since a and e are known, k is known. If a be the base of the
Le
system of logarithms, L. a = 1, and therefore k = # . If e be the base then L e = 1, and k = L a.
€
The logarithms corresponding to the base e, are called Naperian or hyperbolic logarithms. They are of great
use ; and it will be found convenient to denote them in a particular manner. We shall therefore prefix the
letter l to a quantity to express its hyperbolic logarithm, and the letter L to represent the logarithm related to
any other base. Thus the value of k will be represented by l a , therefore
d a”
da:
By substituting a for h, and for k its value, in the series we have obtained for a” we shall have
(! a)' , , L (! a)”
*H. . " + Tºa
=a’ l a.
* = 1 +*.* + . as + &c
If a = e, this series becomes
tº a 2 gº
€” - — — — — — — — — — &c.
1 + + + H+ H+, + &e
(27.) The differential coefficient of L x is equal to º , m being the modulus corresponding to the base of the
system of logarithms; that is to say, equal to one divided by the hyperbolic logarithm of that base.
w d
Let u = L w, and a be the base. Then a = a”, and therefore, by (26), #: = a "... l a = a l a. But, by (24),
d it ** = . connºlently “E”- l
d a du 2 d a T d a T a la
dº La – m d". L & 2 m dº La – 2.3 m
d.T - T.T. T.I.T. - . . . . . ~~ 4:4.
_ m m bei ual to l
T a . eing eq l a
Hence, , &c.
If the logarithms were hyperbolics, m = # would be equal to one, and therefore
Q,
d l a I dº la — I dº la 2 dº l r – 2, 3
dº, T , " Tº . ~ T. Jºs - T.s Tº 2, T Tz
Having thus found the values of the successive differential coefficients of the functions L a. and l ar, we may
apply Taylor's theorem to the developements of L (~ + h) and l (a + h). We shall find
tº ºe h, 1 hº 1 h9 1 hº )
LG-1) = 1.4 m (4-4.4% + -ī = + &c.)
h. 1 hº I h9 1 hº
l --- mm mºus smºs =st sº - gººm a sm mº, º mº &
(*-ī-h) = 1 = + , -ă = + , ; – H = +&c
Or, assumin = 2
9
*
LG-1) - L. = L (+* = L (1 + 2) = n (# – , 4-, -: +&c.)
l (a + h) ! ºr - l *)= 1 a 2) = * + 2. * + &e
tº-sumº a JT + 2) = z 2 3 4 e
The two preceding rules, combined with those previously explained, will enable us to find the differential
coefficients of any function in which logarithms or exponentials, depending on the variable, enter.
Erample 1. Let it be proposed to find the differential coefficient of u = l (a + v (1 + æ")).
Jº
Assume a + v (1 + æ) = 2, then u = i < * = } -Fºy and # = | + Wi-Fi-
w/(1 + a ") + æ
y(1 + 4*) , hence
d w d u d 2 l
dr T a 2 dr Tv (1-Eas)
d u n-l -- d ? – 1
Erample 2. u = (1 z)". Let la = 2, then w = 2", dz = n :- = n (la)", dr T a " and therefore
du n (l r)".
dºr T a
WOL. I. 5 1
782 D IF F E R E N T I A L C A L C U L U. S.
Differential
Calculus.
S-N-r’
d u l 1 d 2 l d u l
e ==- e -: {º :- ! 5 — = - :- sº- 3 --- :- time 3. *ºmº -: – a
Example 3. u ! (lay. Let la: = 2; then u 2 d 2 2 l a da Jº and consequently d aſ a lar
d d
Erample 4. u = abº. Let bº = z, then u = a”, ** = a" l a = abº la, ** = b” l b, hence du = abº bº l a lb.
d 2 da, d a
Erample 5. Let u = 29, 2 and y being any functions of a. Taking the hyperbolic logarithms, we have
l u = y lz, and therefore
1 d u y dz 1. *
7, da T 2 dº i., or
d u y d 2 d y
# = u ž ## 1 º
If y = a and z = x, this formula becomes
d tº
da,
(28.) The differential coefficient of the sine of an arc, considered as a function of the arc itself, is equal to the
cosine of the same arc ; and the differential coefficient of the cosine of an arc, is equal to minus the sine of the
SQ7776 (17°C.
= a + (1 + la).
To prove this proposition we shall make use of the property of the differential coefficient demonstrated (17),
viz. that it is the limit of the ratio between the difference of the function and the difference of the variable. It
is necessary to show previously, that the limit of the ratio of the sine of an arc to the arc itself, the arc being sup-
posed to decrease indefinitely, is equal to unity.
First, it is obvious, that the arc is always greater than its sine; for it is greater than the chord, and the chord
is an oblique with respect to the sine. Secondly, the arc is always less than its tangent; for the product of the
arc by half the radius is the surface of a sector contained in the triangle measured by the product of the tangent
S11] ſº
by half the radius. Hence, if we designate by a any arc, the ratio will always be included between the two
si tº g * e
. * = l, and # = cos a, since they have all the same numerator, and since the denominator of the first
IIR ſº a.
is greater than that of the second, and less than that of the third. But the second is equal to unity, and the third
SI IT
= cos ar, has clearly for limit one. Therefore *. included between the two has also the same limit.
This understood, let w = sin ar, the difference of the function is sim (a + h) — sin a and we must determine
sin (a + h) — sin a
h g
sin (a + h) — sin a = 2 sin # h cos (a + 4 h).
n 4 h - º , , , , , sin 4 h
# = cos (t + 4 h). But we have just proved that the limit of ;- was equal to
the limit of the ratio We shall observe to that effect, that
ſº si
Hence the ratio becomes
# h ne-o-º:
g tº a e . . . , SIIl (Jº — ſº, ) — J’.
unity. The limit of cos (a + 4 h) is clearly cos w; consequently the limit of + p sin "is equal to cos ar.
d si
Therefore - *** = cos w.
d aſ
. . . . g ... / Tr . Tr
To find the differential coefficient of cos ar, we observe that cos a = sinſ a T ..) and assuming 2 - * = 2,
tº ºr g d : sin 2 ºr d 2
Slºl || -- - ? ) - SIIl 2, — = cos 2 = cos ( → — a J, and − = — 1.
£ d 2 2
Tr
& d x
a sin(; – )
sin(; tº _ d cos a
* * Tr g
Therefore *=ms - - COS (; — ſº tºº – SIIl ſº.
da, d a N. 2
The second, third, and differential coefficients of cos r and sin r, may now be easily calculated. We shall find
d” sin a d cos a dº sin a d sin a d" sin a d cos a
= - = — sin ar, E — — = — COS ar, – E – = sin ar, &c.;
da’ d a d a " da, da:4 da, 9
d d2 cos ºr d sin a dº cos as d cosa, si d” cos a d sin a cos ar, &c
all Cl —------- - - = — coS ºr, == — — — = Sin ºt, —- = = 5 * ~ * > *
d a 2 da d a 3 d aſ da: da,
These values, combined with Taylor's theorem, give
h h2 hº e hº
o = sin a ... + — si * * – COS tº , –– sin a . s—. — &c.
sin (a + h) = sin a + cosa. 1 SIIl tº I . 2 1. 2. 3 l. 2. 3.4
2 º h9 h4 &
a -F sin” . i. 3.3 + cos a . T.I - &c.
g }, h
cos (a + h) = cos a – sin w. T – cos a i
Part I.
S-N-2
D I FF E R E N T I A L C A L C U L U. S. 783
Differential If we change h into – h, in the first, it becomes Part I.
Calculus. *-*. h h” hº hi S-V-2
-- sin (a' – h) = sin a – cos a . T – sin a . H, -- cost. T2 .. 3 -j- sin a . T2 ... a 4 T &c.
Subtracting this last equation from the first, and dividing by 2 cosa, we obtain
h. h9 h5 h7 &
sin h = i - T53 + Tää-TE – Tai-TEF -- &c.
By the addition of the same equations, and in dividing by 2 sin a,
h? + h4 hº
1 .. 2 1. 2. 3. 4 1. 2. 3. 4. 5.6
(29.) The differential coefficients of the other trigonometrical lines, considered as functions of the arc, may now
easily be found.
cos h = 1 —
+ &c.
sin a
Ist. Let u = tan ar. Since tan r = , we shall have, by (23),
Jº
sin a C d sin a © , dcost
© OS ſº - SIIl *mºnº g
du cos a da, d tº (cos a)* + (sin a Y” . l
d a T (cos a)* T (cos ar)* fººms (cos r)* T (cos rj"
& COS ſº & tº e
2d. Let u = cot al. Since cot a = −, we shall find, in a similar manner,
SIIl º
du 1
d a T (sin a yº'
I
3d. Let w = sec w. Since sec a' = cosa." by (23) we shall get
d u sin a = tan & Sec ºr
d x (cosº). T tº
- g l
4th. Let u = cosec v. Since cosec w = —, we find
S11
d u — cos a Cot a COSeC &
d a T (sin a y T tº
(30.) The differential coefficient of an arc considered as a function of its sine, is equal to one divided by the
square root of the difference between one and the square of the arc.
We shall represent the arc whose sine is equal to a by sin” ar. This manner of denoting such a function
results from a notation lately introduced in the higher branches of analysis, to express the repetition of the
operation indicated by the nature of a function upon the function itself. It has been proposed to represent
such functions as l l ar, sin sin ar, tan tan tan tan ar, by l’a, sin” ar, tan” ar, the index denoting the number of times
the operation must be repeated. In general, f* (v), fº (a), . . . . f* (v) will be equal respectively to f(f(a)),
f(f(f(a))), &c. An immediate consequence of this notation is, that f" (f" (r)) = f" (r). To find the
meaning of such expressions as fº (a), frº (a), it will be sufficient in the last formula, first, to make n = 0 and
n = 1, and, secondly, m = 1 and m = — 1. The first supposition gives f(fº (a)) = f(r), and, consequently,
fº (a) = r. The second supposition gives f (f-* (v)) = f°(r) = r. Let x = f(y), and let the value of y,
derived from this equation, be F (a), then F will be the inverse function of f; but in substituting for y its value
F (a) we have a = f(F(r)), which equation compared to f(f- (r)) = a gives fri (a) = F(a), therefore f
denotes the inverse function of f, and, consequently, sin-1 a, cos" ar, tan" w, will stand respectively for arc
whose sine = a, arc whose cosine = a, arc whose tangent = w. The symmetry of this notation, and above all
the new views it opens of the nature of analytical operations, seem to authorize its universal adoption.
º gº da:
Let therefore u = sin” ar, then a = sin w and Hi + cosu ; therefore, by (24),
du 1 l
d a T cosu T V (1 – rº)"
tº e • - da, s
In a similar manner, if we suppose w = cos" a. we shall have a = cos u, consequently I = - sin it, and
70,
smººn * ºnymgmyºnº
ſº
(31.) The differential cofficient of an arc, considered as a function of its tangent, is equal to a fraction whose
numerator is one, and whose denominator is one plus the square of the tangent.
5 I 2
784 DIFF E R ENT I A L C A LC U L Us.
Differential da: I - Part I.
Calculus. Let u = tan” ar, we shall have a = tan u, hence + = −., and therefore N-V-2
du (cos u)*
S-N-2
du (cos u)* = l
J. = (6 T 1 + 4*
In the same manner, if u = cot"; a , we shall find
d it — I
da, T T-H a
(32.) The differential coefficient of an arc, considered as a function of its secant, is equal to a fraction whose
numerator is one, and whose denominator is the product of the arc by the square root of the difference between
one and the square of the arc. - -
da:
Let u = sect” r, we shall have a = sec u, therefore 3. * tan u sec u, and
7ſ,
d u — 1
d a T a V (rº-I)”
In a similar manner, if u = cosecT'a,
d u l
d a T a V (rº – 1).
With the help of the preceding formulae, we are enabled to find the differential coefficient of any function of r,
containing sines, cosines, tangents, &c. We shall only give a few examples, which will be sufficient to show
how to proceed in other cases.
ſº ſº d u g
Example 1. Let u = (sin c) . Assume sin a = 2, then u = 2", hence T. = n 2"*" = n (sin z)"-", there
2
d u
fore H = n (sin a Y”-1 cos a.
da,
du
Example 2. u = l cos w. Let cos a = 2, then u = l z, and H = − = −, hence
d z 2 COS ſº
dw sin a
– = - t - tan T.
da: COS ſº
1 + sin & 1 + sin a 1 + sin r du 1 1 — sin a
F. le 3. - – l = } liſ → l. –— t :, t — = 5 - E → ,
rampte 7A, V(, — sin a *(; — sin j Assume 1 — sin a hen F. #; 1 + sin a
d z 2 cos aſ d u I
d – = --— — `. ge
and H. = T-in a)* therefore d a cos r
l ſb + a cos a b -H a cosa,
E le 4. tº — T' & — ». t -——- = 2, th
a'ample 4. u M (a” — b%) COS ###} et; Fºr = , then
d u I | fº (a + b cos w) _ , , (a + b cosa)
d : T V (as - 5) T V (I-25) T T 7(º- 5).73 ag- 5-(a = 5) (cosa); ; T T (aſ-5) sin º'
d z (a” — bº) sin a d it I
we shall have also da, * – (a + b cosa)?' therefore; ºr T a + b cosa'
(33.) Hitherto we have only considered explicit functions of one variable, we shall now examine those which
contain two or more independent variables.
Let u be a function of any number of variables a, y, z, &c. The successive differential coefficients of u with
ſº º d u dº u
respect to the variable a alone, the others being considered as constant, are still represented by da' d'aº' &c.,
- d u d?
and are called the partial differential coefficients of u with respect to r. In like manner, #; # &c. are the
e sº sº g s g wº d u dº u & * * g. te G
partial differential coefficients of w with respect to y; and d 2' dºzº &c. are the partial differential coefficients
of u with respect to z.
Each of these partial differential coefficients will, in general, be a function of the same variables, and therefore
it becomes necessary to be able to represent their own partial differential coefficients with respect to either of
d w
them. According to the notations already explained, the partial differential coefficient of d a
with respect to y,
D IF FE R E N T I A L C A LC U L U.S. 786
Differential d #) G3 (#: * Part I.
Calculus. G2 * 1 S-L-->
\-v- would be expressed by -** the third partial differential coefficient of #. with respect to y, by -**
tº
, (d" u
dm ºf d (#)
dy”
and the n” partial differential coefficient of dºn with respect to y, by To simplify these results, it
has been agreed to represent them respectively by the following symbols;
dº? u dº at d”u
dy da' dy" drº' dy” da"
which, by means of the numerical index in the numerator, and the exponents of d y, d w, in the denominator,
dº nº
d y” da'
ought to take the first partial differential coefficient of u, with respect to a, and then the second partial
can leave no doubt with respect to their real meaning. Thus, Ior instance, to form the value of We
G3
differential coefficient of this result with respect to y. To find the value of in the order of the operations
should be inverted.
dºwn diºgy” #. we should first find the m” partial differential coefficient of u, with
respect to a, and afterwards the nº partial differential coefficient of this result with respect to y. And to form
In general, to form the value of
}
the value of the order of the operations should be the reverse.
70,
drº dyn
No farther explanation will be necessary, to understand the meaning of expressions such as
dºn-Hº-Hº w
drºdy" dzº
where u is supposed to contain the three variables a, y, z, and to extend the same notation to any number of
variables. -
The determination of the values of these various partial differential coefficients, can present no difficulty, each
operation being performed in the supposition that all the variables but one are constant, and being consequently
assimilated to the case of functions of one variable.
(34.) Let f(x, y, z, &c.) be a function of any number of variables, if we change a into a + h, y into y + k, &c.,
h, k, &c. being indeterminate quantities, it becomes f (a + h, y + k, &c.) and f(x + h, y + k, &c.) – f'(x, y) is
called the difference of the function. In making use of the preceding notations, and of Taylor's theorem, this
difference may be developed, under a symmetrical form, in a series of terms containing the successive powers
of h, k, &c.
We shall first consider the case of a function of two variables, and then it will be easy to extend the results
we shall obtain to a function of any number of variables.
Let u = f(r, y), and substitute a + h for a, we shall have, by Taylor's theorem,
d u d? u h? dº nº h.”
f(x + h, y) = u + i. h -- i. i*i; + j : , T: ;
Change now y into y + k in both sides. Each of the coefficients of the powers of h in the right side of the
equation will become a function of y + k, and may, consequently, be developed according to the powers of k.
Thus u, # &c. will give rise to the following series respectively:
u tº k +}; tº ###. ſº H+&c.
#: #. +++. H4+ H+&c.
#: # * ++ . . . we
#: # * + + . . 4 sº
+ &c.
786 HD IF F E R E N T I A L C A L. C. U L U. S.
Differential
Calculus.
S-V-7
and, in general,
Hence we shall have
fe + ºv-º) = u +} +#. ####: tº, H+ &c.
++. A + ##########4 se
2 Q . S - 2
++. #, +H:#; #4 &c.
++++a+se
- + &c.
We have obtained this developement in changing first & into a + h, and afterwards y into y + k ; but we
might have inverted the order of these substitutions, and begun with that relative to y. Then u would have
become successively -
d u dº ng k” dB iſ k?
f(x, y + 9 = u ++, K+ iy, H+5; H = +&c.
and -
du d;2 at h9 dº nº h9 -
f(z+h, y + k) = u ++, ++, Fºg -- H. Hi +&c.
d u d” w dº u, hº
* j,” + i.i. - + = is k--ee.
d” w łº d8 w k?
+ i. i. 3 + H+... " Its # *.
ds / kº -
+; Tāſā 4 &e.
+ &c.
This second series representing the value of f (a + h, y + k) must obviously be equal to the first,
independently of any particular value of h and k, therefore the coefficients of the terms which contain the same
F.
powers of h and k must separately be equal. Hence
dºw dºw dºw _ _ dºw d'u dºu
dyār T âq dy’ dyār T dºdy’ a y dº; T ârſăy’
dºn-Hº tº _ dºt-w
- Tjºa; - Târâyn
Therefore, if we take m + n successive times the differential coefficient of a function of two variables, n times
with respect to a, and m times with respect to y, the result will be the same in whatever order these m + m,
operations are performed.
(35.) The two first terms # h and #: * of the difference of a function u of two variables are called, the
º -
one the partial differential with respect to x, and the other the partial ºnal with respect to y; their sum is
called the total differential, or simply the differential of the function. Thus
d w d u
Or, since h is equal to the differential of a, and k to the differential of y,
d u d u
In this expression, it is necessary to recollect that
is not the differential of u, divided by the dif-
ferential of a, but is intended to represent the partial differential coefficient of w with respect to r ; and, conse-
d u g º o
quently, that drº da is not equal to d u. When it becomes necessary to express the quotient of the
differential of u by the differential of a., it may then be written
# d u, as it has been proposed by Fontaine.
(36.) We may now, without difficulty, find the values of the successive total differentials of u, or d'u,
dº w, &c. To form the second differential, for instance, we shall take the differential of d u, or of # h ––
: Part I,
\er-S-
D IF FER E N T I A L C A L C U L U.S. 787
Differential d tº tº w & ſe tº tº . e o º g - o Part I.
Calculus, dy k, and this will be obtained by determining the first partial differential coefficient of dw with respect to r, Q-N->
and that with respect to y, multiplying the first by h, and the second by k, and adding the two results.
We shall have successively
*
d (d u) d” u d2 u. d (d u) d2 u. d” at
:= * — = — h + → k,
da, # h 4- da d y "' d y d y d a + d y?
e d (d u) d (d u) dº u dº u, dº u
d : : — k = --—- h” ——º- h k + → k”;
8|| - - d” w d a h + d y k d a " h”-- 2 d a d y H. K. -H d y”
or, substituting d a for h, and dy for k, we find -
• * dº nº d” at dº at
ſ - . §2 - d *.
dº nº Tdº d at +2-#- as dy-Hir 3/ *
The value of dº w will be obtained in a similar manner: first,
d? d (d2
a u = *** A + 4** *
da: d y
d (dºw) dº w is 2 de M. dº ºu
but da, = I. h -- d a "d y ºf driy k
d (dºw) * * * 2 da u d" u ,s
'ſ sº * * * == - — — — h k -j- – -- k",
and d y H; i.”-F d a dy” h k -j- d y”
dº w 3 dB at 3 d5 u dº w
8 7/ = 3 -i- — hº —— h k” kº,
therefore dº nº d a 9 h° + da" d y h” k + d a dy” + d y”
dº u 3 da M. , 3 d5 Qd dº u,
* QL = — * —l— — d rº —- d a d v" -- - d v".
OF dº nº H. d ≤ + Hiſ dº y + Hy J. y-Fi 3/
The analogy of the numerical coefficients and exponents of h and k, in the expression of the successive
differentials of u, to the coefficients and exponents of the same letters in the developements of the first,
second, and third power of the binomial h + k, is obvious. We may prove that the same analogy subsists for
any order. For let us suppose
d" u A d" u B d" wº C d” aſ
Wł tº $º - n = 1 . A, ri -2 * -8 . - |C.
d” M = d a " * + -ār-rig- h ** =# * *+7= * k” -- &c
We shall have an u → ***) h : * (**)-k.
da: d y
d (d" u) d"+, u , A d" u wa- B d” u , , , C d” u , a_a
\ | -: _ _ ]ºº $8 - I % = †- d . &c.
But da, da"t -- da"d y h * + = * k*-i- da"-*d yº h"-" kº. -- &c
d (dºw) dº u , , Ad"w B d"+i w , C dº nº
and == !! — h” — h"** — h"T8 c
alſ] d y dy da." + da"—" dy? k –H da"—" dys hº-2 Kº -- da"—a dy. h"-s k" + &c
Multiplying the first series by h, the second by k, and adding them, we find
d"+, at dº w d"+1 u, d"+, it -
An-Fl = n+1 • • ſº *=- fl-l - *1 = 2 º
** =# *-(A+1) #;"|-G-A)=# *-(c4 B)== ** + sº
From the manner in which this last series has been obtained, it is evident that the numerical coefficients and
exponents are precisely the same as if we had multiplied the value of d" u by h -- k. But we have already
proved that for the first, second, and third differentials, these coefficients and exponents are the same as in the
developements of the three first powers of k + h, therefore for the nº differential they will be equal to those
of the developement of (h+ k)". Thus
d" u, 77 d" u n (n − 1) d" u n – 1) (n − 2) d" u
*-** +”
in t= — h" in-l . n-3
d" u #;"+H=High k -- 1.2 da"-"d y” 1. 2 .. 3 JF-17" k" + &c.
d” at 'm d" u n (n − 1) d” at
" u = — º mºmºmºmºs n = 1 * - 2 cl a 12 e
Or dºw # ** + airi, da"- d y + 1 .. 2 da"-*d y” da"-* dy +&c
(37.) The foregoing expressions on the successive differentials of a function of two variables, will enable us
to give a very symmetrical form to the developement of the difference of that function. For that purpose let
us bring to the same denominator, in the developement of f(x + h, y + k), the terms in which the sum of the
exponents of these two letters is the same. We shall have
788 D IF F E R E N T I A.L C A L C U L U. S.
Differential
Calculus.
\-, *
1 /d w du
fe + , , ; ) = u ++(#4 +} )
l d” u , , , 2 dºw dº nº 2
+H: (#4 +######e)
I d" u is 3 dº u , 3 dº u 2 dº no
- H Hº (; * +izi; ºr Hº, he ++ e
+ &c.
To form the nº horizontal line of this series, we must collect all the terms of the developement of f(a + h,
y + k), in which the sum of the exponents of h and k is equal to n : or, which is the same thing, those which
contain the partial differential coefficients of the nº order. These will clearly be the first term of the deve-
J’art I.
d” h"
lopement of # 1 .. 2 m ' when in it y is changed into y + k ; the second term of the developement
cin-1 76. h" I dº-2 QM, hº-2
f ſe tº a tº w º .
* Lºſ T. 2 ºn II : * the same supposition ; the third of the developement of d a "~" l. 2 . . m – 2 0,
- º 1 e
&c.... down to the (n + 0)” of the developement of f(n, y + k). Therefore if I . 2 .. 3 70, is considered
as a common factor to all these terms, the n” horizontal line will be
I d” w m d” w m (n − 1) d" u d" at
ſº ——— h"T" mº- . . . . — k"
Hi-F (Fºr H=Hy-º" + -i-, - Fig. 4. Tº )
If we compare the developement of f (a + h, y + k) under this form, to the values we have given for the
successive differentials of u, we shall immediately observe that the last are equal to the quantities enclosed
between parentheses in the first. Hence
d” at dº w dº w
A u = d u + T-5 + T = + H + &c.
which is the same formula we have obtained in (14), only applied to functions of two variables.
(38.) All that has been said with respect to functions of two variables may easily be extended to functions of
any number of variables. -
Let u be a function of n variables a, y, z, &c. There will be n first partial differential coefficients repre-
sented by
d w d it d u
d a ’ d y ' d'z
and the partial differential coefficient of the mº order will be expressed generally by
d p + q + r +&c. º
da” d y' d 2’ &c.’
, &c.
where p + q + r + &c. = m.
If a, y, z, &c. are changed into a + h, y + k, z + 1, &c.; and if u' represent the value assumed by u in
that supposition, uſ – u will be the difference of u, and we shall be able to develope it in a series containing
the successive powers of h, k, l, &c. by substituting first a + h for a, in u, then developing by Taylor's theorem,
and changing in the developement successively y into y + k, 2 into 2 + 1, &c.; and after each substitution
developing each term by means of the same theorem.
It is obvious that the terms which will multiply the first powers of h, k, l, &c. will be
d w d w d u
*= — k -- - l –H &c.
# * ++; H = t4 &
They are respectively the partial differentials with respect to w, y, z, &c., and their sum constitutes the total
differential, or simply the differential of u. Thus
du d u d u
.7/. C — s==amº — l g
du d a h + d y k + d z + &c
du d at du
Or du = + d r + # dy-H; d = +&c.
The developement of u' must remain the same, whatever be the order of the substitutions of a + h to r,
y -- k to k, 2 + l to 2, &c. Hence we shall infer, that if we take p times the partial differential coefficient of w
with respect to a, q times with respect to y, r times with respect to z, &c., the result will be the same whatever
be the order of these successive operations.
The formation of the successive differentials of u will present no difficulty. It will be sufficient to operate
upon du, dº u, dºu, &c., precisely in the same manner as we have operated upon u to form du. Thus, we
da: d y dz ! + &c.
D IF F E R E N T I A L C A LC U L U.S. 780
Differential
Calculus.
f
.*
and if we observe that we shall have to multiply the partial differential coefficients of
- dºw du - dw
tº — Cºmºmº – l —— &c.
du da, h -- d y k + d 2 + &c
successively by h, k, l, &c. it will appear evident, with a little attention, that the numerical coefficients and
exponents of h, k, l, &c. in the value of d” u, will be the same as in the product of (h -- k + l + &c.), by
(h + k + l + &c.), or in (h – k + l + &c.)”. Hence, in using a similar reasoning, we shall conclude that
in the value of dº u, these coefficients and exponents will be the same as in (h -- k + l + &c.)”, and generally
in the expression of d" w the same as in the developement of (h -- k + l + &c.)".
The comparison of the values of the successive differentials of u, with the developement of u', after having
written in a line the terms in which the sum of the exponents of h, k, l, &c. is the same, will lead, as before,
to the formulae d
w d” at dº w
l + 1.5 + l. 2 .. 3
which therefore is general, whatever be the number of variables of the function w.
(39.) We have considered hitherto all the variables ar, y, z, &c. which enter the function u, as independent
of each other. Let us suppose now that some of them are functions of some of the others; and, first, let
gy, z, &c. be all functions of w. Then u = f(a), y, z, &c.) will be a function composed of functions of a ; and
when a is changed into a + h, &c. y, z, &c. will become -
A u E + &c.
d y dº y hº dº y hº *
w t + 4 + as tº ++, +++ as
d z dź 2 h2 dº z A3
* + -ī; h + = Tºs---ji= Ti 4 &c.
But by (38) we know that generally if we change in u, a into a + h, y into y + k, z into z + l, &c. and
develope, the terms of the series containing the first powers of h, k, l, &c. are
d at
da:
In the present case, the substitution of a into a + h in u will make these terms assume the following form,
d u d u / d y dº y hº du / d z d2 2 hº )
# * ++, (# h + i. Tºg + sc)+ dz (#: + hiº Hi + &c.) + &c.
Hence the coefficient of the first power of h, or the differential coefficient of u, is
d d
* +++++ 1 + &
l d w d u dy d u d 2
jr. du = d r dy dº +-I. # + &c.
d d d d
But tº $/ * : **-, &c. are by (24) the partial differential coefficients of u with respect to
d y d a ' d z d a ’
y, z, &c., these variables being considered as functions of v, therefore the differential coefficient of any
junction of x, y, z, &c. in which y, z, &c. are the representatives of functions of x, is equal to the sum of
the partial differential coefficients of that function with respect to x, y, z, &c. separately.
d d
This rule applied to the first differential coefficient, in which + 2 #, &c. are to be considered as new
variables, functions of a, will give the second, and then third, fourth, &c. differential coefficients.
The manner in which the partial differential coefficients of it may be obtained, in any other supposition,
relative to the dependency of the variables a, y, z, &c. is now sufficiently indicated by the preceding inves-
tigation.
(40.) A few examples will be sufficient to show the application of the rules, to find the values of the differ-
entials and differential coefficients of a function of several variables
Erample 1. Let u = (a" + y” + 2*)", then
d u d w d u
# = Gº-y-ºy" mr-, #;=r (, ; , ; 2)-ny-, + = e^+y+2) p."
and d u = r (r" + y" + 2*)" (m. 1"-" d a + n y”- d y + p 28- d 2).
Example 2. Let u = (a + b r")” (c -i- d y”)", then
d u f, * Tº Y o = _, d u m ** {
# = 64 avy p (a+b+)" ºn “” i = (a+bºy a gravy” any-,
d d u = *n \ p - 1 is Y G = } º frt n-1 if
aſl vol. " (a + b r")*-* (c + d y”)" (** arºu. ºne tºo, 'd y H.
Part I.
790 D IF F E R E N T I A L C A L C U L U. S.
Differential Evample 3. Let u = a ". Then
- Part I,
Calculus .
& d u d u / 3/ *V-
- = w a "T", = ºrº - *
d2 at d? 7.
# = y Q - D - # = *-ya- tº # = sº a sy.
d’u = y (y – 1) w” da' + 2 w"-" (1 + y la) da d y + wº (la)” dy",
= r^* { y (y – 1) da" + 2 a. (1 + y la) da d y + æ (la)” dy”; .
Example 4. Let u = a sin y + y sin a then
** = sin + y COS a. d u , e
d : T gy + y cos #;= * cos y +sin.
dº aſ
e Gº ?!, º 2
— ºf - Slil 30, -—e tº - ºr SIIl
His = - vsnº, Hy 3/,
#; = cos y + cos w.
d” u = 2 (cos y + cosa) da d y – y sin a da” — a sin y d y”.
Example 5. Let u = (x + 1 + + e” + sin a]”. *
Assume la = y, e = 2, sin a = v, then u = (x + y + 2 + v)", and
I d u , d u d y d u d 2 du d v
by (39) # ** = +, + iy Hi + j = i+ i ==
m (x + y + 2 + v)"-" {1 ++ + c + coºr}.
The two last examples we propose to give, will afford a verification of a theorem of considerable importance,
relative to homogeneous functions of several variables, and which for that reason we shall first demonstrate.
(4.1.) If n be the sum of the exponents in each term of an homogeneous function u of the variables x, y, z,
&c., then
d w d u d u
n u = f; w + #; y + H+&c.
Let us change the variables a, y, z, &c. into a + g”, y + g”, &c., or a (1 + g), y (1 + g), &c. The function w
will become (1 + g)" u. Hence
d u dº nº gº anº
(1+d) u = u + His y + # * +&c.
d a 2 1 .. 2
d w d? at
*= 2
d u dº u gº y?
+ is t iſ ºr
— &c. + &c.
— l – 1 — 2
But a +9) we u(1 +ns 4 +” ºr * }. e-se)
The terms which multiply the same powers of g in these two developements of (1 + g)" u, must be equal;
d u ... d u , du
therefore n u =# * + #; y + H+&c.
2 2 2
and also n (n − 1) u = ** 2. 2 dº at d2 u.
*º- *mºmsºmº — arº
... * –– ãºd, "9t ãº" + &c.
The relations between a function and its partial differential coefficients, is sometimes called the theorem of
homogeneous functions, it was discovered by Fontaine ; the preceding demonstration was given by Lagrange.
Example 6. Let u = —###- where m = 2. We shall find
a + y + 2
d tº (* + y + 3) y z – a y z du - Cr H y + 2) a 2 - w y z du – Gº H y + 2) a y - r y <
d an (x + y + 2)* 'd y T (p + y + 2)? 'd 2 T (a + y + 2)*
d u d u d u 2 a y”
Hence tºms
*
3
Example 7. Let u = (x + y) v (, — y); where n = 3 We shall have
du — — , , -l ºf 9) #4 – – A – “tº
F = x/(r y)+; º; #;= ve 9) 2 y ſz – y)
D I F F E R E N T I A L C A L C U L U. S. 79]
<- d u d u ...— (a + y) (r. — y) 3 , _ 3
P. Hence # * + d y y = (a + y) w(a – y) + 2 v(a — y) T 2 (t + y) wºº — y) = a u.
u-y- We shall find also, -
d” wº l a + y d’u a + y d’u – 1 a + y
7.35 V(x, -y) T4 (p = y) V(x, -y) dºdyT 4Q – y) MG — y) dy. Twº-y) T 4QTJ) 7(? – y)
Hence
d" u , dº nº d” w a" – y” (a + y) (a — y)* 3 3 / 3
ºustºmsº 2 —- u° E — ſº- * — I, - tº -- — —- ſº
(42.) When two variables a and y are connected by an equation, such as f(x, y) = 0, either of them may be
considered as a function of the other ; y, for instance, as a function of a ; and if the equation cannot be resolved
with respect to y, then, as we have before stated, y is said to be an implicit function of a.
We shall now examine how, in that supposition, we may determine the values of the successive differential co-
efficients of the function of a represented by y, or rather how we may discover the relations which subsist between
2
ar, y, and the differential coefficients #. 5 #.
Let f(x, y) = u = 0 be the proposed equation, and let ºf (a) be the function of a which y represents; that is
to say, let us suppose that @ (r) is the value we would obtain for y, if we were able to resolve the equation
f(x, y) = 0. If we substitute ºff (a) for y in f(x, y), we shall have therefore f (ºr, Ø (a)) equal nothing inde-
pendently of any particular value of a. Consequently, if we change a into a + h, we shall also have
j (a + h, q} (a + h)) equal to nothing. Now, since ºff (a) is the value of y, by Taylor's theorem
Yºº d y dº y hº dº y hº
* G + h) = y ++ h-H; H + + H+, + &c. or = y + k,
* * e d y dº y hº dº y hº
In assuming à." Tājā ( , t . I. 3
So that f (a + h, y + k) = f { a + h, j (a + h) } is equal to nothing, whatever be the values of a and h,
when for y and k the above values are substituted. But by (34),
d u h d” tº hº dº nº ha
+ &c. = k.
f(t+h, y + k) = u + H++++ H++. H + &c.
d u k d? at dº at h”
+ jji Tä, ä, " " + dºi. Tº &
d" tº * + d’”. A *
d y 1. 2 d y” d a 1. 2'
dº nº lºs
Jºy T. 2 .3'
+ &c.
Or, in substituting for k its value,
. d u , d u d y
f(x + y + k) = u + (; ; ++; #:
dº u , 2 dº w dy
+(#####, #4
+ &c.
Therefore this series must be equal to nothing, whatever be the value of h; hence the coefficients of the different
powers of h must separately equal nothing. Consequently the following equations will obtain
d’u d y” du dº y h”
d y”d a” d y daº) 1. 2'
w = 0,
d u , d w d y .
# , t , , ; 3. * *
d” at 2 dºw d y d’u d y” +** * 4 =
dº ºf dy dº dº. " dº drº T a y dº T
0.
The first is only the proposed equation f(x, y) = u = 0. The second is the expression of the relation which
exists between a, y, and the first differential coefficient #. ; and from it we may determine the value of that dif.
º © a g gº * o d
ferential coefficient in function of w and y. The third expresses the relation between a, y, # , and the second
Jº
Part I.
5 k 2
792 D I FF E R E N T I A L C A L C U L U. S.
Differential .. © e G2
Calculus. differential coefficient d
--~~~
- - Part I.
#. and would give the value of the last quantity in function of the three others, or S-2->
d
simply in function of w and y, if we had previously determined the value of #, by means of the second equation.
e G 9 & 9 e dº y dº y
The following equations would, in a similar manner, give the values of &c.
dra’ diri’
gº º d u , du d y e e. &
A very simple rule may be given to form the equation dºn + dy I. = 0; for, by multiplying both sides
d -
by d ar, it becomes # d a + #. d y = 0, and then the left side is obviously the total differential of u.
It follows from this remark, that to find the value of the first differential coefficient of an implicit function
represented by y, and connected with the variable x by an equation u = f(x, y) = 0, we must find the total
differential of u, as if the two variables x and y were independent of each other, then write this differential
equal to zero, and deduce from the equation so formed, the value of #.
Jº
(43.) The equations, by means of which the values of the second and successive differential coefficients of the
implicit function y may be determined, can easily be derived from the preceding.
g }* h9
We may observe, that the left sides of these equations are the coefficients of h, T2' T 2, 3. &c., in the
developement off (a + h, y + k), - f'(a + h, j) (a + h)), that is, in the developement of a function of a +h.
- 2
Hence it will result, from Taylor's theorem, that the second, which is the coefficient of
h
9 * must be the dif-
g tº Yº ſº se h3 e ſº
ferential coefficient of the first ; that the third, which is the coefficient of I. 2 .3’ must be the differential
• 2 .
coefficient of the second ; and, in the same manner, each succeeding one the differential coefficient of that which
precedes it. But in taking these successive differential coefficients, it must not be forgotten that a is the only
& tº | d y d” g E * * * * ** a tº
independent variable, and that y, 7 in#, #. &c., are only the representatives of implicit functions of a. There-
fore we should operate as in the case of functions of functions (39), that is to say, after having taken the partial
differential coefficient of the quantity under consideration, with respect to r, we ought to take the partial differ-
d y dº y
r' d wº
each of them respectively by the differential coefficient of each of these variables with respect to r, that is, by
d u d” w dà tº * * * e º
º, ºl. a y, &c., and then add all these partial differential coefficients together.
d a da” d as
Let us form, according to this rule, the equation upon which depends the determination of the value of
ential coefficients of the same quantity, with respect to each of the other variables y, &c., multiplying
2
#. We know that the equation from which the value of the preceding differential coefficient may be derived, is
tº
d u , d w d
&_º-ºm- * * * 4 = 0. . . . . . . * g g g s = º (a).
d a d y d in
o º © & , dy dº y d p
Hence, if we suppose the left side of this equation equal to u', 3 ºr = p, and consequently dº. T dº we shall
have to determine this last quantity from the equation
d w! d u' d y d w! = 0 :
i. i* Tay - d ... " do d 5
d w' dºw d” w d y dºw' d” w
but ă. Tº aſ " dyā; ài T ~ āyā; * dº à: IP = I,
d? d
Substituting these values, and #4 instead of **, we find
d tº da:
d’u a dºw dy dº u d y” d u dº y
... it dy dº da, 㺠dº * dy da,” T is sº g tº e s s a tº e g º e ∈ (b)
an equation identical with that obtained in (41).
3
In following the same process, we shall find that the equation expressing the relation between #. the pre
- dy
dź e • * ~ *
ceding differential coefficients, #. daº the function y, and the variable a is
D IF F E R E N T J A L C A L C U L U. S. 793
º dºw , 3 da u d y , 3 dº u d y” dº u d y” 3 dº u dº y 3 dº u d y dº y dº d'y o (c) Part I.
Calculus. His āyā; aſ " a y Ạdà H. J. T. J. J. G. "Tº H. as Tājār, - " " --
Mºrº",
The formation of the equations relative to the differential coefficients of higher orders can present no difficulty.
(44.) All these equations, and the equation u = 0, would be verified; that is, that in each, the left side
would become identical with the right side, if we were to substitute for y the function of r, it represents, and for
the differential coefficients of y, the differential coefficients of that function. This is expressed by saying, that
these various equations subsist or obtain together. Hence, by combining them in any way whatever, other
equations will be formed, which will subsist or obtain with them.
(45.) An equation which contains one or several differential coefficients is called a differential equation ; and
a primitive equation is that which does not contain any.
A differential equation of the first order is that which contains no other differential coefficient than the first,
and generally it is said to be of the n” order, when the n” differential coefficient is the highest it contains.
The degree of a differential equation is the highest power of the differential coefficient, which marks its order,
it contains. Thus a differential equation of the n” order in which the highest power of the n” differential
coefficient would be the m", would be of the m' degree.
(46.) From the remark we have made in (44), we already perceive that several differential equations of the
same order may correspond to the same primitive equation. Thus, it is obvious that from each combination of
a differential equation of the mº order, with the differential equations of the preceding orders, will result another
differential equation of the m” order. But among the various differential equations of the same order which may
be so obtained, some require a peculiar attention, because they express more general relations between a, 3), and
the differential coefficients of y, than the others.
We must first observe, that by differentiating a primitive equation between a and y, that is, by applying the
& d
rule given (42) to form the equation which gives the value of #, it may happen that one of the constants con-
ū;
tained in the equation should disappear. It would obviously be the case, for instance, with respect to the con-
stant a, if the primitive equation had the form f (a, y) = a ; and if a were not contained in f(r, y). But in
all cases, by combining the primitive equation with the differential equation of the first order, so as to eliminate
one of the constants, it will always be easy to obtain a differential equation of the first order, containing one
constant less than the primitive equation.
Such a differential equation does not only correspond to the proposed primitive equation, but to all those
& d
which differ from it by the value of the constant. Hence it expresses a relation between r, y, and # In 10Fe
general, than a differential equation of the first order containing that constant.
If the constant eliminated enter the primitive equation in a degree higher than the first, the result to which we
shall arrive will contain the differential coefficient of the first order in a degree higher than the first.
(47.) These considerations may easily be extended to differential equations of higher orders. We shall be
able, for instance, to eliminate two constants between the primitive equation, the differential equation of the first
order, and that upon which depends the value of the differential coefficient of the second order; and the result
will he a differential equation of the second order containing two constants less than the primitive equation.
Generally, we see that we may obtain a differential equation of the m” order, containing m constants less than
the primitive equation. -
(48.) Instead of eliminating constants between the primitive equation, and its differential equations, they
might be combined so as to make other quantities disappear in the result. The variables as or y, for instance,
or any function of them entering the primitive and differential equations might be eliminated.
We shall now apply the foregoing rules and observations relative to implicit functions of a, or to equation
between the two variables a and y, to a few examples. :
Earample 1. Let it be proposed to determine the values of the first and second differential coefficients of the
implicit function of a which y represents in the equation
a y” + b x^ = c a y + d.
We shall have, by (41), to determine the first differential coefficient
d d
3 a y || 4 3 bºa e = } + cy e g is 6 a º e º e º a tº (a),
d y 3 ba" – c y
ſlence --- sº -— .
d r 3 a y” — ca.
To find the second differential coefficient, we shall take the differential coefficient of both sides of (a), consi-
d
dering y and # as implicit functions of w. We find
... dº y d y? dy dº a d y
3 a y #4 + 6 a y} + 6 b + = e º 4 c = } + c +4.
794 D IF FERENT 1 A L C A LC U L U.S.
Differential
Calculus,
S-V-'
d
and, supposing # = p,
d -
dº y – (6 b a + 6 a y p" + 2 op).
d tº T 3 a y” — ca. 2
or substituting for p its value ‘. - -
dº y — {6 b x (3 a y? — ca)* + 6 a y (3 b a' – c y)* – 2 c (3 b wº – c y) (3 a y” — ca.) }
T. T (3 a y” — ca.)" º
Erample 2. Let the proposed equation be y” – 2 m a y + 4* — a” = 0.
d
We shall have to determine a y
da,
d y dy $2 rv —
2 y H. – 2 m r +: 2 m y + 2 x = 0;
d 772, QM – º –
and, consequently, # - * 9 - *. &
J) y – m z
In this case the primitive equation, containing no higher power of y than the second, may be resolved with
respect to that variable. It gives
y = m, a -i- w (a” — a -i- m^a).
d
Substituting these values for y, in the expression of #. we shall find
d y — a + m” a
da = m + w/ (a” — a " + m” wº) '
It is easy to verify that the two values we have thus obtained for the differential coefficient of y are identical with
those we mightsderive from the value of y.
tº g g d g
There is still another manner in which we might arrive at the value of * 9. expressed in terms of a alone.
d a
We might eliminate y between the primitive equation, and the differential equation of the first order, by
taking the value of y in the last, where it enters only in the first degree, and substituting it in the other, we
shall have, by this process, the following equation,
d y? d y a" – m? as — a” m”
—º- – 2 m –– -
d a 2 m #; + a 2 — a” — m” wº 0,
d
which, being resolved, will lead to the same values of # as before.
Example 3. Let the primitive equation be
a sin y + y cos a = a,
ſº dy d y g
then in y + , convair coºr – vºn = 0. *
and dy a y sin a – sin 9
da, cos a + a cos y
Example 4. Let the primitive equation be
y” = a a + b,
dy
th | 71 — tº ſº. 5
eIl 2 y da, (2 3
and by eliminating a we find for differential equation of the first order independent of a
d y
* – 2 -º-º- — b = 0.
3/ * 9 d. 0
Differentiating this equation, b will disappear, and we shall obtain a differential equation of the second order
independent of the two constants a and b,
d y” dº y
daº + y d x ~ 0.
It is easy to verify that this equation is satisfied by the value of y given by the primitive equation. For this
l d d? - -3.
value is y = (a a + b)*, hence # = a (a a + yº and #. = — ; a” (a a + b) *, which being substi-
tuted in the differential equation of the seeond order, makes one side identically equal to the other.
Part H.
D IF FE R E N T I A L C A LC U L U.S. 795.
Differential
Calculus.
Example 5. Let the primitive equation be
- Ay” — 2 a y + a” = a”,
where the constant a enters in the second power. We find
d y
(y – a) # + r = 0,
taking the value of a in that equation, and substituting it in the primitive equation, we shall have
d
d y”
(* – 2 y) # – 4 y} - r = 0,
for the differential equation of the first order, independent of the constant a.
Example 6. Let the primitive equation be
y” + y = (a” + æ)",
we shall have for the differential equation of the first order
d y m* * -1 m (a”-- *):
2 -º- = − (a” —– a ") ". . . 2 a = - Y ---. . 2 a.
(3 y” + 1) d a m (a” + æ") Jº m (a”-- a ") 30
We may now substitute in this equation, instead of (a” + aº)", its value taken in the primitive equation,
and then we shall obtain a differential equation independent of that irrational function of r,
d y m (y” -- y)
* —- 1) →- = —-tº-:4– . 2 a.
(3 y” + ). n (a” + aº) Jº
Example 7. We shall take for the last example an equation containing logarithmic, exponential, and trigono
metrical functions; and we shall propose to eliminate them by means of the differential equations.
Let the primitive equation be
gy -- l y + et" + sin a E c, $3.
dy 1 dy
d ſºms *=e mºm-a-mº smº - ſº -: *
we shall fin da: 3y da. e-* + cos w = 0.
and by differentiating again -
dº y l 1 d y” = º to
#(i++)- ##4 + sin a = 0,
subtracting this from the primitive equation, the functions e-" and sin a will be destroyed. We shall have
d? l 1 d vº
v-ty-H (1 ++)-; #=0;
d q." gº daº T
and, it is obvious, that by a new differentiation l y will disappear.
(49.) When m variables are connected together by m — 1 equations, any one of them may be considered as
the independent variable, and all the others as implicit functions of it. Hence it may be required to find the
values of the differential coefficients of these implicit functions.
Let u = 0, v = 0, w = 0, &c. be the proposed equations between the variables t, r, y, z, &c., in which t is
supposed to be the independent variable, and ac, y, z, &c. implicit functions of t. Then w, v, w, &c. may be
considered as functions of t, and therefore their first differential coefficients, with respect to that variable, will
be respectively (39.)
d u d u d a d w d y d u d 2
— — — — — — — — — — —; — + &c.
# 4 + H+++++++++++ &
d v d v d ºr d v dy d v d 2
tºº mºm-sºº &c.
# +++ -ā; ++, ++++ -ā; +*
d w d w dºw d w d y d w d 2
*—mm–sum ºn-ºn, autº mºsºme mºme sº-c = -s º " -mºus mea--- &c.
# +++ -ā- +++ -ā- +++++ + &
But the functions of functions of t, denoted by u, v, w, are equal to zero; since by the hypothesis, if we
substitute for a, y, z in them, the functions of t they represent, the equations u = 0, v = 0, w = 0 must be
verified. Therefore the differential coefficients of these functions must also be equal to zero. Hence
d d -
###### 4 ++ #-se–0.
# ++++} +++++&c=0,
+ # ##### # #4 se – 0.
&c.
796 D I F F E R E N T I A L C A L C U L U. S.
Differential
Calculus.
\-y-Z
Equations by means of which the values of #, #, #.
The formation of the equations upon which depends the determination of the differential coefficients
#. #. #. &c. will present no difficulty. It is clear that the differential coefficients of the second
order of the functions u, v, w, &c., considered as functions of functions of t, will be obtained by taking the
differential coefficients of their first differential coefficients, in which, it must be remembered, #, #. #.
&c. may be determined.
&c. are new variables representing functions of t. These differential coefficients of the second order must, as
well as those of the first, be equal to zero. Hence will result a sufficient number of equations to calculate the
d” ºr dº y dº 2
d tº ' d' tº * d ?? " &c., which quantities they will involve in the first degree. That which
values of
te dº a dº y de 2
should be done to determine the values of d tº ' d tº ' d tº * *
higher orders, are sufficiently indicated by what precedes, and require no further explanation.
- (50.) The observations which have been made before in the case of a single equation between two variables,
with respect to the combinations of the primitive equation, and its differential equations, apply clearly here.
Between the m – 1 primitive equations, and the m – 1 differential equations of the first order 2 m – 3, constant
or variable quantities may be eliminated ; and generally between the m – 1 primitive equations and n m — n
differential equations of the n first orders, (n + 1) m — n – 2 quantities may be eliminated.
Let us take, for example, the two equations
gy” + 3 a t w = b cº,
&c., and the differential coefficients of still
Part I.
a 3 + 3 c t y = ah b. - - - . . . "
We shall have to determine dº, and dy, the two following
d t d t
d y da:
8 - 2 - mºmºmº-º-º: :-
d a d y
º := U.
* † C t d tº + c y = 0
And by taking the differential coefficients of the left sides of these equations, we shall have
a dº y d y” d” a. d a
* —º- —– 2 lºssºme + = 0,
## 4-2 viº; + at ºf 4-2 a ji
d2 a. d wº dº y d y
2 em-sº —- tºt U.
* + + 2 + i + c t +, + 2 c + = 0
g º dº y dºr
Equations by means of which we shall be able to find the values of Tº dº
(51.) We shall now proceed to examine implicit functions of two or more variables.
Let u = 0 be an equation containing three variables a, y, z. Either of them may be considered as a function
of the other two ; 2, for instance, as a function of a and y. Let us propose to find the values of the partial
differential coefficients of z.
This will be very easy; for if z were expressed by an explicit function of a and y, to determine # we should
consider y as a constant in that function, and then differentiate it as a function of a alone. Hence, in the
present case, we shall first suppose that z and a are the only variables in u, and we shall have by (42)
d u d u d 2
d as d z da T
d
Equation which will give the value of +: ſº
d
Secondly. We shall consider w as a constant, and we shall have to determine the value of # the equation
d u d u d 2
dy * g.
3/
d? 2
#. #;" # will present no difficulty. To find the first we shall
= 0.... (b).
The research of the values of
D I FF E R E N T I A L C A LC U L U.S. - 797
Differential take, as in (43), the differential coefficient of the left side of (a), y being still supposed to be a constant, but Part I.
Calculus, d d” V :
# being considered as a variable, we shall have to determine #: the equation, - -
- d? ut 2 ** d 2 d" u d 2* d u ** = 0
d a 3 d 2 da; d a d 2* d acº d a daº T *
d -
Operating upon (b) in a similar manner, a being then the constant, and y, z, #; he variables, we shall find
d y” ”
for the equation which gives the value of
d” u dº w d 2 d” tº d 2* d w dºz
zºº tº dº, j-Fi: … + j j =
3/ 2 a y a y 2” dy” d y dy
, we may take either the differential coefficient
to - d”
To obtain the equation upon which depends the value of da: i 3/
of the left side of (a) with respect to y, or the differential coefficient of the left side of (b) with respect to r.
The two results will be found to be
d” w dº w d 2 d” w * * * dºw d2 d = 4 du d” 2
d a d y d 2 d y d a d 2 d a d y d z* da d y d z d a dy
No farther explanation is required to understand how the partial differential coefficients of a superior order may
be determined. -
(52.) If instead of one equation between three variables ar, y, z, we had m equations between m + n variables,
it is obvious that any m of them could be considered as implicit functions of the m remaining. -
Let w; y, z, &c. represent the n independent variábles, and w', y', 2', t', &c. the m variables which are con
sidered as functions of them. Each of the m equations may be differentiated in the supposition of a being the
- - - d ar' day' d 2'
only independent variable, and lead to m equations involving the differential coefficients * , 43/ 2
d a
d tº & ſº
dr " and sufficient to determine their values. The same operation may be repeated on the given equations,
= 0.
d a ’ d a ’
y being then considered as the independent variable, and lead to m new equations, by means of which the
~! A f
values of the m differential coefficients d an 9 d y 3. d 2 y
d y d y d y'
process we shall find the m n partial differential coefficients of the first order. It will be sufficient to differen-
tiate the m n preceding equations in a similar manner; and in considering the partial differential coefficients
already involved in them, as new variable functions of a, y, z, to arrive at new equations which will give the
partial differential coefficients of superior order.
These various equations would be verified, as well as the proposed equation, if for a ', y', 2', &c. the functions
of a, y, z, which they represent, were substituted. Hence they may be combined in any way, and lead to
new equations which will also be satisfied by the same values of ac, y', 2', &c. Consequently, constant or
variable quantities may be eliminated between them. - g
The denominations of partial differential equations of the first, second order, &c., and the degree of a partial
differential equation of a given order, can be easily understood from what has been said (45), and do not
require any further explanation.
The elimination between partial differential equations, presents important results, which we shall now
€Xa.II)] Ile. *...* *
Let u = 0 be an equation between three variables r, y, z, and lett denote a certain function of r and y, a
function of which f(t) = s is involved in u. So that if t = @ (r, y), w may be represented by F (s, a, y, z),
or by F (f(q) (r, y)), a., y, z). Hence, if we apply to the equation u = 0 the rules for differentiating functions
of functions, we shall have .
&c. may be obtained. In following the same
du d u d 2 d w d's d t
iſ tax d a d's dº dz
d w d u d 2 d u d's d t
d *=mºse — — —- — ; – ... --
AIl ây Tà: ăy " as d t d y
. . . . d's tº º • , a . . - ſº
These two equations contain s and Tº therefore by combining them with the proposed equation u = 0, the
d - -
two quantities s and +; may be eliminated. The result will be a partial differential equation of the first
order, not containing s = f(t), and which therefore will be verified by the primitive equation u = 0, whatever
be the form of the function of t designated by f. Thus it appears, that by means of the two partial differential
VOL. I. 5 L
798 D I FF E R E N T I A L C A L C U L U. S.
Differential equations derived from u = 0, it is always possible to eliminate a function of a certain function of z and y in-
Calculus. d z
\-v-2 volved in u, and to obtain a relation between ar, y, dy
to that function. -
A similar reasoning would prove, that, generally, m arbitrary functions may be eliminated, with the assistance
of the m n partial differential equations of the first order derived from m equations between m + n variables.
(53.) We cannot however infer, by analogy from what precedes, that a partial differential equation of the
second order may, in all cases, be obtained, containing two arbitrary functions less than the primitive equation.
Let us suppose that the equation u = 0 between the three variables ar, y, z, involves two functions s and s',
the first of t and the other of t'. The two partial differential equations of the first order will contain the quantities
, and H, true for every form which may be assigned
2
* j.
d d ſ o g º g e g tº & g gº
#: and # . Differentiating again, we shall obtain three partial differential equations of the second order, in
& tº g & e d’s dº s' {} a & , d's d s'
which will be found, in general, besides the two coefficients dº ’ d ?'s" the quantities s, s', d : and d?' ' Thus,
to make s and sº disappear, we should eliminate these six quantities, between the primitive and the five dif-
ſerential equations, but this will be generally impossible. We should have recourse therefore to the partial dif-
ferential equations of the fourth order. These will be four in number, and will only contain the two new arbitrary
d d5 s d3 s'
functions dºs’ and dºſs'
quently we shall be able to arrive at two partial differential equations of the fourth order, entirely independent of
the arbitrary functions s and s'.
The same considerations will easily show what order of partial differential equations it is necessary to use, to
eliminate any given number of arbitrary functions, and how many differential equations of that order may be
obtained independent of those functions. In the case of m arbitrary functions to be eliminated from a given
equation between three variables, it will be easy to see that the partial differential equations of the (2 m – 1)"
order must be used, and that m differential equations of that order may be obtained independent of those
functions.
The order of differentiation indicated by the preceding observations, is the highest which can be required, to
perform the elimination, but it may happen that such relations should exist between the terms of the proposed
equations, that the arbitrary functions might be made to disappear without having recourse to it.
Let us take for first example the equation
2 = (x + y)" q (wº — y”),
and let us represent the differential coefficient of Ø (r" – y”) taken with respect to the function between the
parenthesis, considered as a variable, be denoted by p' (p" — y”). We shall have for the two partial differential
equations of the first order,
We shall have, then, ten equations between eight arbitrary quantities; and conse-
# = n (n + y)"-" p (i." – y”) + 2 + (2 + y)"#" (" – yº),
#; = n (a + y)"- p (tº — yº) – 2 y(t+ y)" (p' (a" – y”).
Eliminating º' (r” — y”) between these two equations, we find
3/ #4 Jº # = m \r + y)" () (a" – y”).
Substituting now for q (wº y”) its value taken in the primitive equation, we shall have
d z d 2
$/ dº -- at dy i •
for the partial differential equation of the first order, independent of the function º, and expressing therefore a
d g
#. #. verified by the equation z = (x + y)" | (a" – y”), and by all those which
- $/
differ only from it by the form of the function p.
5 m 2,
relation between ar, y, z,
Example 2. Let the equation be
– 9: l
* = 3 +2(; + logy)
d : l I d 2 l l
Ther #= -2 (Hº). #-, -º (; +*);
€11. da, © (. +logy) a’’ dy 9 + º' (T. 3/
1
Eliminating º' (. + log w) between these two equations, we get
d 2 d z
* H+y #-y=0.
Part l.
D IF F-E R E N T I A L C A L C U L U. S. 799
Differential
Calculus.
N-y-Z
Example 3. Let the equation be
* z = b (y-º-r) + r y \, (y – wy,
containing the two functions ºff (y –- a) and ºff (y – 3), which are to be eliminated by means of the differential
equations. To simplify, we shall write @, Y, º' and Yº', &c., instead of p(y -- a), ºr (y – a j, q' (y -- a),
Yº' (y – r), &c. We shall find -
d? — ..., * , d 2 f /
# = 2 +vy-rvy, j = 0 + º-'vy.
Between these two equations, and the primitive equation, we cannot eliminate the four quantities ºff, \, gy,
and ſº, and therefore we proceed to the partial differential coefficients of the second order. We shall have
e d’ 2 - ..., f m dº º – an f //
# = ? 2 y \, '+ a y \!", # = % + 2 + \,' + a y \, ',
d° 2
# = *-* +o, -ow-ºvº.
These three new equations contain two new indeterminate functions @" and Jº', so that we have six equations,
and six quantities to eliminate, which is impossible. We shall therefore determine the partial differential co-
efficients of the third order. We get -
# = y + ayy-ºvº" dº z = @" — 2 \,' — (2 y – a J \" + a y \,"
d wº ' da” dy
de z --- ~k/// | // /// dº 2. ---, -}. If? f/ I//
ării = ? + 2 \r' + (y – 2 a.) ſº — a y \,”, i; = % + 3 + \r" -- a y \r".
We have now ten equations and only eight arbitrary functions @, ºp', ºft", p", \r, \", \", \", therefore the eli-
mination is possible, and will lead to two partial differential equations of the third order. The values of the
differential coefficients of the third order give, by adding the two last and subtracting the two first,
de z dº z dº z dº z d? 2 d? 2
º &= * f •– —- sº ',
Tº t d y” d a drºdy a rº 4 y', but dy” d tº 2 (a + y) \,
dº z dº 2 d” 2 d8 2 dº 2 dº 2
h ** * g-mº. — — — — . = 0.
€Il Cé 2 (#. #) (a + y) iſ ºf d y” d a da” dy #)
The other equation, independent of the arbitrary functions, would be obtained in making use of the differential
coefficients of the first order, but it is much more complicated than that just obtained, and is useless to our
present purpose.
Erample 4. We shall take for the last example an equation containing two arbitrary functions, which will
disappear in making use only of the partial differential coefficients of the second order
Let * = • , (4) + V (%)
we shall have in writing again q and y instead of Ø (# ) and \e (*).
d 2 & d – & M." d 2 | I f .
# = % * * is Y, # = 0 + = W
3/
ºp' and P' may be eliminated at the same time from these two equations, by multiplying the second by a
and then adding. We find thus
d 2 y dz ©
d a ' a d y T "
Taking the partial differential coefficients of this equation, first with respect to r, and then with respect to y, we
shall have
* : * * * – – 4 # = – 4 º',
d anº a d y d a wº dy a"
d"2 y dº? 1 d = l gy
d a d y a dy” a d y T a " '
multiplying the last equation by # , and adding it with the first, we shall eliminate &, and find
dº & d? 2 d? 2
* ...; 2 a y d a dy $/ d y” 0
A partial differential equation of the second order, which is satisfied by the equation z = a *(* ) + \r (#)
whatever be the forms of the functions @ and \r.
Part I.
\-N-Z
5 L 2
800 D IF F E R E N T I A L C A L C U L U. S.
Differential
Calculus.
Plate I.
Fig. 1.
Fig. 2.
(54.) We have investigated the rules to determine the values of the differential coefficients of every given
explicit or implicit function of one or more variables. To complete the subject, it remains only to show how, in
some cases, the value of the differential coefficients may be determined, although the relation of the function to
the variables is not known either explicitly, or by unresolved equations which connect them together.
This can be done in various cases by means of some circumstances which cannot be expressed analytically,
and which, without any knowledge of the nature of the function, allow us to determine the limit of the ratio be-
tween its difference and the difference of the variable. We have had already an example of this method, in the
manner which we have used to find the differential coefficient of sin ar.
Let us propose, for another example, to investigate the value of the differential coefficient of the area A B M P,
included between the axis of the abscissae, two ordinates A B, MP, and a curve AM. This area is clearly a
function of the abscissa O P = a, since the point A being supposed a given point, the value of A B M P will be
determined for every value assigned to w. Let the ordinate be called y, and let y = f(r) be the equation of
the curve. If we suppose PP' = h, and if we change w into a + h, the unknown function of a that is repre-
sented by A B M P will become A B M'P', and the difference of the function will be M M P P'. Let us draw
the two lines M. N., M/N' parallel to O-P', it is obvious, that by taking h sufficiently small, the area PM P. M'
may always be considered as greater than the rectangular parallelogram P M N P', and less than PP'N' M'.
Therefore the ratio of the difference of the unknown function to the difference of the variable, that is,
PPſ M M! f P P. M. Nº P P’ M f Mr N ?
º is greater than rºs and less than TFET . But -FFTN = M P = y, and rºs* N
= M'P', which is the value of the ordinate corresponding to the abscissa w -- h, and consequently equal to
2 2
gy –F # h + % # 2 + &c., the limit of which, with respect to decreasing values of h is y. Hence the
ratio of the difference of the function A B M P, to the difference of the variable, is included between two quan-
tities, one of which is y, and the other has for its limit y 3 consequently the limit of that ratio, or the differential
coefficient of the area is also equal to y.
We may apply the same method to a function of two variables. Let D A B, D A C, C A B be three coordi-
nate planes, cut by a curve surface D C B H G M E F, whose equation is z = f(a, y). If by any point M of
that surface, whose coordinates M. M', M'P, A P, are respectively 2, y, z, two planes are drawn, F M H Q and
E M G P, parallel to the coordinate planes D A B, D A C, they will form a solid D H M G A PM Q, whose
volume is clearly a function of a and y. Let u be that unknown function, and let it be required to find the value
d's
of #, . Let P p = h and Q q = k, and by the points p and q let planes be drawn parallel to D A C and
D A B, and meeting in N N'. If in the function w we change a into a + h, it becomes D H mg A Q m/p
9 Yº
= u + #: h —- º: #; + &c., and the partial difference of u with respect to a is M. m. M/m' G g P p
_ du + dº w hº
T d a d a 2 1 .. 2
and will have for its own difference M N m n M'N'm' n', the first term of the developement of which will clearly
.72
be dº u h k. But we may easily see that the first term of the expression of M N m n M'N'm'n' is also
d a d y
z h k. For M M' = 2,
+ &c. Let us change in this difference y into y + k, it will become N n N'n' Gg Pp,
/ d 2 - d.2 2 hº , - - , d ? d? 2 k.”
mn-4 in 4 g. H. n n = ′ + #4 + H+ H++&c.,
d 2 d 2 d” 2 h”
^ = ºmasº — k -** *g e
N N = 2 + = A + i + + H+, +, + &
Consequently, if by the points M, N, m, n, we draw four planes parallel to the plane A B C, we shall form four
rectangular parallelopipedons having the same base M'N' m/ n' = h k, and the first terms of the expressions
of the volumes of which will all be 2 h k. Now, it is obvious, that the parallelopipedons are always some
d”
greater and some less than the solid N N m n M'N' m' n', therefore the first term #; k of the expression
of the volume of that last solid, must be equal to the first term 2 h k, common to the expressions of the volumes
e d” w
of the four parallelopipedons. Hence d x d y T 2 f (a, y).
(55.) It does not unfrequently happen that it becomes necessary to substitute for the differential coefficients of
one or several functions, with respect to one or more variables, involved in a formula, the differential coefficients
of the same function, with respect to other variables connected with the first by given relations. * ,
Let us first suppose that the formula contains only the differential coefficients of y with respect to the variable
a, and that it is required to substitute for them the differential coefficients of y with respect to another variable t,
tº wº d v d” º d iſ d? d a d” ºr
a being a function of t. The values of #. #, &c., in terms of #. #, C., d : " dºg' may readily be
formed, by means of what precedes.
Part I.
D IF F E R E N T I A L C A L C U L U.S. 801
Differential
Calculus.
*—Sy--"
We shall have first, by (24), Part I.
d y d y d a d y d t º
- Sº —- , — — = − , h
d t d a d tº and hence da, d a from which
- • *-*. d t
d”y d a dº a d y
dº y d t d t” d t
we get d wº = d acº 3.
dº
and, by taking again the differential coefficients of both sides of this equation with respect to t, we find
d y daº 3 *g dº a da; d y dº tº dº a d y d a
dº y ## TF ~ *āā āā āº t *ā; a FT is iſ ſº
d are T d as
d tº
In a similar manner, the values of the differential coefficients of higher orders may be found.
If, in these formulae, we suppose t = y, then -
d :) -- I dº y – a dº y – a .
# = 1, 3-, = 0 +} = 0;
and they become respectively,
dº r
d y_ I dº y dy”
Jr. T J . . . .” T dº' and
d y dy”
d” tº dº a dr
dº y – dy' dº dy
d as T d acº
d y”
With these values we shall be able to transform any analytical expression involving the differential coefficients of
gy with respect to w, into another, in which they will be replaced by the differential coefficients of a with respect
to y. -
(56.) No greater difficulty will be found to change the differential coefficients of any number of functions
gy, y, y, &c., with respect to any number of variables wi, as, als, &c., into the differential coefficients of the
same functions with respect to an equal number of variables z1, z, za, &c., connected with the first by as many
equations as there are variables. The first partial differential coefficients will be as before, -
dy, dy,
d y, d z, d| y, d 2,
[.. ººmsº
da, *s da, 2 d as da, y
d 2, d 2,
And, taking the differential coefficients of each of those with respect to each of the new variables, the values of
dº, == da, ’ da, F da, , &c.
d 2, d z,
dy, dy,
dy, - d. 2, dy, --- d 2, &c
the partial differential coefficients of higher orders will be obtained.
We have now explained all the general rules of the Differential Calculus, and sufficiently illustrated the meaning
of the notations which are used in it. We shall therefore proceed to the Integral Calculus, intending to show
afterwards the application of both to analytical and geometrical investigations.
802 I N T E G R A L. C. A L C U L U.S.
PART II
INTEGRAL CALCULUS.
Integral (57.) We have before stated, that the Integral Calculus was the inverse of the Differential Calculus, and had for Part II.
Calculus, its object to determine the value of a function, the differential coefficient of which is known, or, more generally, S-
S-V-” to discover the relations which exist between the variables and the functions, from given equations between them
and their differential coefficients. -
(58.) We shall first consider the simplest case; which is, to find the value of a function of one variable, when
the first differential coefficient is given explicitly in terms of that variable. -
* , g d ,
Let X be the given differential coefficient, and let y designate the unknown function, then #. = X, or
tº
d y = X d ar. The required function is generally represented by f X dr, the characteristic ſ denoting an
operation precisely the inverse of that indicated by d in the differential calculus. Hence, if the two characteristics
and d were prefixed to the same function u, they would neutralize each other, and we would have ſ d w = w.
It follows also from (30) that dº X d a would signify the same thing as ſ X d a ; and consequently that we might
dispense with the use of a new sign. But as it is universally employed, we shall retain it here. In the sequel
we shall have occasion to mention the origin of this notation, and also of the name integral, applied to f X d r,
or to the function whose differential coefficient is X. The operation, by means of which the integral of a given
differential is determined, is called integration. To integrate a differential, is to find the value of its integral.
These definitions and notations understood, we shall deduce without any difficulty from the observations and
rules stated in the differential calculus, the following results.
(59.) If y represent a function of a, and if d y = X d ar, then, from (18), / X d a = y + A, where A is an
arbitrary constant. Hence we may always add to the integral of a given differential a constant quantity, whose
value remains in general indeterminate. If however the value of the integral corresponding to a particular value
of ar, happen to be known, then the constant may be determined. Let us suppose, for instance, that we know
that the integral becomes equal to B, when a is assumed equal to b. Then, if we designate by C the value of y
corresponding to the same supposition, we must have C + A = B, and consequently A = B – C.
(60.) We shall also have, by (18), M being a constant,
ſ M X d r = M_ſ X d w = M y + A.
Hence, when a constant factor multiplies a given differential function, it may be written out of the sign of
integration.
(61.) Let yi, y, y, &c. be functions of r, and d y = X. daº, dy, = X, d r, d y, = X, d x, &c ;
then, by (21),
f (X, d a + X, d a — X, da) = ſ X, d r + ſ X, da – ſ X, dr = y, + y, - y, + A.
Hence the integral of the sum, or difference, of the several differential functions of the same variable is equai to
the sum or difference of the integrals of these differentials. s
(62.) The rule given (22) to find the differential coefficient of the product of two functions of the same variable,
will give -
J y, X, d x = y, y. - ſy, X, d r, or ſy, dy, = y, y. - ſy, dy.
This result shows, that when the differential function may be decomposed into two factors y, and X, da, and
that the integral of one of these may be obtained, the integration of the proposed formula will depend upon that
of another function equal to the product of the integral already found by the differential of the factor not yet
integrated. This method is called integration by parts. We shall frequently have occasion to make use of it.
(63.) Each of the rules given in the differential calculus, to obtain the differential coefficients of the functions
of one variable, we have examined, being inverted, will clearly lead to a corresponding rule of integration. In
consequence, to avoid repetitions, we shall write down the values we have determined for the differential coeffi-
cients of the various species of functions, and opposite to each, the integral formula which is deduced from it.
Thus we shall form the following tableau. -
da a” - 1 , a wº
We have found . d a := m a.”- ". . . . . . hence......ſa. * > iTi + ° g g º O Lº gº tº & o o ºs e e ... (a).
d a” ſº
† = a 1 a e e º e s a ſº is a s 6 tº g c s & P ſa d x = f; + s a s s e s s > s tº e s is a s ... (b).
d e” -
di F * . . . . . . . . a s - a e & e º a º º ſea. = 2 + . e e º e º e º 'º e º 'º º 'º e º G is e º & (c),
INTE G R A L C A LC U L U.S. 803
Integral
Calculus.
S-v-f
- m d a -
We have found d I, a m. as e º 'º º & hence . . . . . . . iſ º = L a + c . . . . . . . . . . . . . . . . . . (d).
d a Jº
d l aſ } d ar
t: - . . . . . . . . . . . . . . . . . * * * — = l as C , . . . . . . . . . . . . * * * * * * * (e).
d a Jº J.; +
*** = cos ... • * * * * * e s s s e º a * ſco. a = i < * , ... • e s e e - e º " " * * * (f).
d a :
*** = – sin a ............. ſin, are — cos r + c . . . . . . . . . . . (g).
d x.
d tan a l - d a (h)
Tº a T (cosº)? …; air = tan r + 6.....…
d cot a I d a g
— = -- . . . . . . . . . . . . . . . . --— = + c. . . . . . . . . . . . . . . [* J.
d x T (sin r)* ' ' ' (sin a Y” Cot & C (i)
º: *** = an see ........... ſtan see, a = < * + . . . … (9.
Jº
* - *** = — cot, conce......ſ' coºr core ar--comes ºf (p.
JC
d sin"' ac 1 -- da, l
--~~~~ ºf -- . . . . . . . . . . E. SlT1 " .22 -H. C. . . . . . . . . . . (m)
d a v(1 — wº) A/(1 – a ") +
d cost' a — 1 — d. a. I
— = -- . . . . . . . . . . . →- = cost' a' + c . . . . . . . . . (n).
d a w/(1 — wº) A/(1 — arº) ––
d tant* I d a
* f = …/ = tant" a + c . . . . . . . . . . . . . (0).
d a 1 + æ" ... - e 1 + aº
*** = H. e G tº g e º ºs e º e º 'º iſ F#. = cot" a + c. . . . . . . . . . . . . . (p).
- 1
d; * = ==T ~ſº- p = see" -- “… (q).
** = Pº-y e g a C & ſºn- cosect' a' + c . . . . . . . (r).
(64.) Each of these formulae is the analytical expression of a rule of integration. The first, which is one of
the most important, shows, that the integral of the m” power of a variable multiplied by the differential of that
variable, is obtained by increasing the exponent by one, then dividing by the new exponent and by the differential
of the variable, and adding an arbitrary constant to the result. This rule does not apply when m = – l ; that
o e g da, g
is, to the integration of T. but the formula (e) gives the integral in that case.
(65.) Whenever we are able by some transformation to change the formula X da' into one of the pre-
ceding, the value of ſ X d a will become known. Hence our object now must be to examine successively the
various forms X may have, and to endeavour to reduce each of them to one of those we already know how to
integrate.
(66.) When X is a rational and integral algebraical function of a, its most general form is
A a " + B a”-- C a." -- . . . . . . -- T,
a, b, c, &c. being positive integers. By comparing, in that supposition, X d a with the formula (a) of the
preceding paragraph, we shall have, obviously, -
A a." —- B a " —- Caº T) as - *** -LP ºf 19 tº T'a -- V
ſ(Aa" + + w" + * @ e g to ſº + T) * = i++++++++H + s ºn e s tº g +- a + V,
V being the arbitrary constant. -
It is not necessary that a, b, c, &c. should be positive integers; they might be negative or fractional, and
the integral would still be obtained in the same manner, except in the case in which one of the terms of
X should be of the form . , and then the corresponding term of the integral would be s la by (e), (63).
The same mode of integration succeeds when X = (A + B a)", or equal the sum of terms similar to that.
First, if a be a positive integer, the binomial (A + B a)" may be developed into a finite series of terms of the
form Ma'", and then X d a may be integrated as above. But we may obtain the integral in a simpler manner,
which has, besides, the advantage of being applicable whatever be the value of a. Assume (A + B a) = y, then
9°4'
d a = #, and (A + B a)* = y”; therefore X d a = 3/ #% and ſ Xd r = I (a TT) + c, substituting now
Part II.
\-y-

804 I N T E G R A L C A L C U L U.S.
G - & (A + B ar)*** ; - Part II.
for y its value, we shall find ſ X da = ſ. (A + B r)" da =*HR- + c, \--/
B (a + 1)
Integral
Calculus.
S-N- In the case of a = — 1, the same transformation gives
da, _ ! (A + B z) ſº #
(RTE) = -RT + c = l (A + Bay” + c,
This transformation will succeed, again, when X = (A + B wº)" a” d ar, whatever be the exponents a and b.
We shall assume A + B a' = y, then (A + B a”)* = y”, a "Tº da = # : therefore x d r = } d y
y B , and
X d ——º- + bstituting f it l
g-g > 2 Ul ſy" we 2
f Jº B (a 1) c; or, substituting Ior y its value tº)**
(A + Baº)***
= / (A gº)” arº-1 = — —- s
J X d r = ſ ( + Bºy. d a B (a + 1) C
and when a = - 1,
a *-* d a ! (A + Ba') l -
(67.) Let us next consider the case in which the function X is a rational, but fractional function. Its most
general form will then be t
A r"-" —H B rººs -- C wº—a -- . . . . . . . . T
* + Afrº- + Bº-s-E............T.
The denominator of this expression may always be put under the following form :
(r—a) (a –b) &c. . . . . x (w-a')" (a —b')' &c. . . . . x (23–2 aw—I a” + 3°) &c. . . . . x (irº –2 a' a + a” + 3°)' &c.
If we suppose, that by resolving the equation
a" + Aſ a "-" + B'a'-* + &c..... + T = 0,
we have found it had the unequal roots a, b, &c., p roots equal to a', q roots equal to b', &c., a pair of
imaginary roots equal to a HE 8 v — 1, &c., and r pairs of imaginary roots equal to a' + 8' v — 1.
The denominator of the fraction being so decomposed into factors, we may transform the proposed function into
the sum of the following simple fractions:
N N,
*Ea" . . + &c. - -
P P P
+ (a — a')* + (a — #y- +. . . . . . . . . . + (a: #y
Q - - dº
+ gºt a ºn + . . . . . . . . . . -f- tº
-j- &c.
a" – 2 a. a + a” + 6*
+ &c.
+ R. a. +- S + Ri a + S, + + R.-, a + S,-i
(a 2- 2 a'a + a” + 3”)" (a" – 2 aſ a -- a” + gº)- ' ' ' ' ' ' a *–2 aſ a +a” +3”
+ &c. - -
N, N', . . . . . . R,-i, S,-i, being constant quantities. To determine their values we should bring all these frac-
tions to the same denominator, which will obviously be the denominator of the proposed fraction. Then the
sum of their numerators must be equal to the numerator of X; and as this equality must subsist independently
of any particular value assigned to r, the coefficients of the same powers of that variable in both quantities must
be equal. This will furnish precisely the same number of equations as there are unknown quantities. For it is
easy to see, that n being the degree of the denominator of X, n – I will be the degree of the numerator of the
N,
a — a a – b ' -
that these last quantities will enter the equations only in the first degree, and, consequently, that their values will
be real, and that they may always be assigned. Therefore the transformation of the proposed fraction, indicated
above, may always take place, and the difficulty of its integration is reduced to that of the four following formulae,
N d 4 Pd* (K a + L) d a (R. a + S) d a
a – a ' (a – a')?' a' + 2 a r + a” + 38' (19 – 2 aſ a -i- a” + 8”)”
which include all the forms of the fractions in which it is decomposed.
um of the fractions
&c., and n the number of the unknown quantities. It is clear, moreover,
I N T E G R A L CAL CU L U.S. 805 .
Integrai (68.) The determination of the values of the numerators N, Ni, &c. of the partial fractions, by the method we Part II.
Calcuius. have explained, will, in general, be very laborious. It may be simplified in making use of the differential S-7
S-,-7 calculus.
Let us propose to find the numerators of the fractions
P P, P
p – 1 :
(w — a')?’ (r. — aſ)*-* ' ' ' ' ' ' ' ' ' ' ' ' ' ' a - a''
corresponding to the factor (a — a')". All that we shall say will apply to the numerators of the simple factors
a — a, a - b, &c. in supposing p = 1. To simplify, let us represent the numerator of X by U, and its deno-
minator by V; then we shall have -
* = z*-* + → H + Pº-l
W T (a — a')? (; Jy- tº E e g tº gº º º
Uſ
, + wº
U -
d being the sum of all the other partial fractions. Multiplying both sides of this equation by Q, and observing
that V = Q (r. — aſ)", we find •
U
Q {j — P – P, (a — a') — P, (a — a!)". . . . . . . . . — Pe-1 (a — dy-º!
* (a — a!)" -
The part of the numerator of this expression contained within the parenthesis, since Q is not divisible by
(a – a'), must be divisible by (r — a!)". It may therefore be represented by y (a — a')”, y being an integral
function of w. Hence all the differential coefficients of that quantity, till that of the (p – 1)" order inclusively,
* – C.
U. =
d “ dº tº
must become = 0, when a = a'. We shall have therefore in denoting by #. ++ ++ , &c., the values
assumed by º and its differential coefficients when a' is substituted for a
70, d . H d2 . l d +
P = + F = ++, P = +, ++, P. = ++, ++, &c.
d Q dº Q
The values of Q, dº ’ dº?
cients of V of the p” and following orders, corresponding to the same hypothesis, in using the relation
V = Q (a — a')". So that the determination of P, P., &c. may be made to depend upon the differential co-
efficients of the numerator U, and denominator V of the proposed fraction.
(69.) To find the values of the numerators of the fractions corresponding to the imaginary roots, we shall
proceed nearly in the same manner. In that case, we have
and for a = a', may even be derived from the values of the differential coeffi-
U - R. a + S gº. -- R, a + Sl + R.-a ºr H-. S.-, | 9 |
V T (rº- 2 aſ r-- aſ H. Bºy T (rs – 2 aſ r + aſ -- gº)-; ' ' ' ' ' ' ' ' ' a" – 2 o'r + a” -i- 6° Q
Reducing to the same denominator, and observing that V = Q (v* – 2 aſ a + a” + 6”)", we get
o 4% - (R = +s) – (R, ++s) (rº – 2 aſ a + a” + 3°) — &c.. . . . . . . . }
U. = c
(r" — 2 a'a. + a^* + gº)
The part of the numerator of this expression between the parenthesis, must be divisible by (18 – 2 a'a + a” + 3°)";
if therefore we suppose it equal to W, we must have
d W d” W dr-1 W
d r * d tº ” ' ' ' ' ' ' ' ' ' ' ' ' ' d r"-" "
equal zero when a is one of the values which makes a 2 – 2 aſ a + a” + 3% = 0.
By substituting for them W, and its differential coefficients, each of these quantities will assume two forms,
such as G + H v — 1, and G – H v — 1, which cannot be both equal to zero, unless G = 0 and H = 0.
Hence we shall have just as many equations as there are quantities to determine.
(70.) Let us now examine the four formulae, to the integration of which may be reduced that of any rational
fractional function, as we have seen (67).
W,
, it is sufficient to observe that the numerator is equal to a constant N multiplied
tº -- 0,
by the differential of the denominator, therefore by (e) (63),
To integrate the first
N d * = N i (r – a) + c = l (r – a)' + c.
tº *
P d Jº º & e a • ſº | sº
(71.) The second formula, (rI a'); is also integrated immediately. Assuming a - a' = 2, then da = d z.
Jº — (1.
WOL. I. 5 M
806 I N T E G R A L C A L C U L U.S.
Integral
Calculus.
Neº-V-->
& a a º P d 2 P. G. — P Part II.
Substituting, we find sº- = P z**d z ; hence, by (a) (63), ſ r; --> (p-T)27-; + c ; and, putting sº
again for z its value, - 2. p— 1)2
P dº * * * * — P + c
(ºr — a')” T (p — 1) (r. — a')?-, ſº
72.) To find the i te (K r + L) dr
(72.) ntegral of the third, a" – 2 a a + a” + 35'
(K 2 + K a + L) dz
We assume a - a = 2, then d x = d z, and
substituting, the formula becomes
This may be resolved into the two following,
2* + 3”
K z d z (K a + L) dz
zºº gº and zº + 3° 9
which may be written
g- 2
K. . . * * * * and K & H L a (;)
3 : FIEE * 6
1 + .
The first is equal to a constant multiplied by a fraction, the numerator of which is the differential of the denomi-
nator. Hence we shall have by (e) (63),
K 2 z d z * , , , 2\
ſº; #F# = 5 i (, -j- 8*) -- c.
The second is equal to a constant multiplied by a formula; which, being compared to (0) (63), gives
z \
fºrt *(i)_ K a + L
... --— –––.
2.
tan" - + c.
2
8 1 + + 8
Bs
Hence, by substituting for z its value, and adding the two results, we get
TO *
(K 4 + L) d a K Q 2 2 K a + L _1 r - a
tº − * → — t &º-º-º-mºms g
ſ a” – 2 aa + a” – 8° ; : (, 2 a r + a” + 3°) + g alſº £3 + c
(R. a + S) d a
(wº — 2 a'a, +- a” + 8”,)"
for the preceding ; it will then assume the form t
(R2 + Riº + S) d .
(zº + 3's) '
which may also be resolved into the two |
R. z d 2 (R * –– S) dz
g g ſº . R. 2 z dz
The first being written in the following manner, "2 Gº-Egy
ferential of the quantity, within the parenthesis, in the denominator.
R. 2 z dz — R.
Hence * * *
3 GT57) - 2 (FET) (=Egy-
Since (R's + S) is a constant quantity, to find the integral of the second part, it will be sufficient to determine
d z
that of (; -- Bºy'
Assume
(73.) To integrate the fourth formula , we shall use the same transformation as
W
, it is obvious that the numerator is the dif-
d 2 G z H d z
tºy = a-ºya # ſº
G and H being two indeterminate quantities. To find their values, take the differentials of both sides of this
equation, then bringing all the terms to the same denominator, and dividing by d z, we shall find
1 = G (2* + 3”) — 2 (r – 1) G z* + H (z*-ī- 8”).
The comparison of the terms containing the same powers of z, will give the two equations
1 = G 3” + H 8's, (3 – 2 r) G + H = 0.
I 2 r – 3
and H =
Trom which G -: (27 - 2) gº (27 – 2)3F
I N T E G R A L C A LC U L Us. 807
Integral.
Calculus.
^-y-Z
d 2 - 2 2 r — 3 Aſ d z
H -— = --a *-*-*—sº-º-º-º-º-º- *-*-*-*-mºm.
eil Cé J’ Grigºy - (57–5); G.T.Bºys, T âi- ..ſ (2*-ī-gs)"-"
e ſº c d z d 2
With this formula we shall be able to obtain the value ofſ.7-555- , if w determi 7-7–F–F–B–,
(z” + 3”)" II. We Can Cieterſnline (z' -- 8") r-
this might be made to depend, in a similar manner, on the integration of º and the same process
being pursued until the exponent of (2' + 3") shall be reduced to unity, we shall have, finally, to find the
integral of #r. which by (72) is equal to # tan ** #.
The value ofJºry being thus calculated, we shall add to it 2 (r — 1) . + 6")"— ” and substitu-
ting then for z its value a - a, we shall have the integral of
(R a + S) da
(r.” * 2 a." 4; -- a^* + g”)'
The last transformation we have used, and by which the value of an integral is made to depend on another,
must be noticed as being frequently employed, and often with success.
It follows from the preceding investigations, that every differential whose coefficient is a rational function of a.,
may always be integrated, and that the integral will be composed of rational functions, logarithms, and arcs
of circle.
(74.) To illustrate the rules we have already given, we shall apply them to a few examples.
Erample 1. Let X =
a" + 17 — a “ — as "
The factors of the denominator of this fraction are easily found. We have clearly
- a" + æſ – a “ — as = a- (x + 1) (r" -- 1) = irº (a + 1)" (a – 1) (* + 1).
We shall therefore assume
1 N P P,
4. -: + 2 —- smºm-º.
a" + aſ — a “ — as * — I (a + 1) a + 1
To determine the values of the numerators N, P, &c. we shall also use the formulae given (68.)
Let us first consider the numerator N, corresponding to the factor a – 1.
R Ri R, k a -- L
++ + i ++++++.
In this case U = 1, Q = a- (a + 1)*(x++ 1), therefore N=# = +.
To find P and P., we have U = 1, Q = 2* (4 – 1) (a^+ 1), and w = — 1. Hence
- º
– “ — T”, t- a + _ 9
– a – 4 * - Tº a T 8
To obtain the values of R, R., R, we must suppose U = 1, Q = (x - 1) (a + 1)*(a" + k), and r = c, and
we get
d w. dº #
* = + = - = —4-- a = −4– —#– = -
R = — = - 1, R, e d a l, R, - 1 .. 2 d ar" - l
Finally, for the numerator k a + L, we assume the proposed fraction equal to
* + 2 + U.
a" + 1 a" (a + 1)*(x - 1)
1 - (R r + L) wº (2 + 1)" (r-1)
- a" + 1 -—s
Hence the value of U, -
the numerator of this expression must be divisible by a′ + 1, and consequently should become nothing when
s" + 1 = 0, or when r = + v — 1. Hence the two equations 2 R + 2 L = i, and R = L, from which
I 1.
we get R = T. L = T-
Thus the differential X d r is resolved into the following
* - d - 4 1 dº + 2 - ** d; + dº – d. , , (r: 1) dz ,
8 ºr — l 4 (a + 1)* 8 a + 1 gº * T Tº 7 Trº II
Part II.
z –
5 M 2
808 I N T E G R A L C A LC U L U.S.
Integral The integrals of which are obtained without difficulty by (70), (71), (72), (73); and are respectively Part II
Calculus. 1 l I 9 I 1 l l : º m
\-y-Z &=º º-e *=ſ_º tº-g sº-ºmsºmº – — l — — l 2 l —t tant" a.
a ! (t 1), 4 (a + 1) ' 5 i (r-H i), 2 a.” a ' àº, 8 (a”-- 1), I- lan Jº
After reduction we shall find
d aſ 2 – 2 a. – 5 a.” I a" — l a –– 1
- -——----— sºme — —— l — — tan" o
J. Hº-3 4 a.” (1 -- a) + (−)-- ( J} ) 7 tan" r + c
- I l
Example 2. Let X = s= ©
(. : b
a + b a + c as • (++++++)
G C
When b – 4 a c is positive, # + 7 * -- a may be decomposed into the two real factors
b -- v (bº — 4 a c) - b – M (b" – 4 a c)
a + 2 c 2 a + 2 c 3.
and then we shall easily find
I *sº 2 c — I - 1. - ſº
a + ba -- ca.” T v (bº – 4 a c) 2 ca. -- b + v (bº — 4 a c) + 㺠TV-ºw-Tº
d - * - * — 4
and hence Jº - I 2 ca, -i- b – A/ (b 4 a c)
a + b a + c & T V (b? – 4 a c) 2 c r + b + v (b” – 4 a c)
But if b" – 4 a c is negative, the factors of # -- c -- 3:2 are imaginary; and instead of resolving the fraction
4 a ** h?
4 C°
tº e - tº . b a ba:
into two others, it is preferable to assume a + -ār = 2, then da = d 2, and — — — — — — a " + 2* +
* C C
and we shall have
d a tºº d z
H- — — b% N’
a + b a + c ºr • (* + “; )
4 C°
4 gº
and by putting again for 2 its value
but - d 2 ſº 2 tan"l 2 c 2
— tº-ºº! *mº — bº Jºãº, a - iºn’
Jºl 4 a c *) M (4 a c ) M (4 a c – bº)
**— = —“– tan-i –“iº-
ſi-ºria = w/ (4 a c – bº) w/ (4 a c – bº)
wº
Q?
Erample 3. Let X = *II.
All the factors of the second degree of the denominator are included in the general form
a" – 2 a cos ei: ). + 1,
in which i is an integer. Let us propose to find the numerator of the partial fraction corresponding to that
denominator. If we represent it by k a + L, we shall have to determine k and L,
a" k a + L U
-: g + 3 ,
* -- 1 -
w” -; * – 2 cos “tº + 1 Q
7.
and hence U - tº” – º &
*–2, co, “tº 4
The numerator of the value of U1 must vanish when a is equal to one of the roots of the equation a’ – 2 a cos
(2 i + 1) ºr (2 i + 1) ºr (2 i + 1) m
— — —- 71. &
The result of the substitution of the first of these two values in a " will be
S m (2 i + 1) ſt + A/ — 1 sin m (2 i + 1) tr
% 71
+ 1 = 0, that is to say, one of the two quantities cos + V – 1 sin
CO
I N T E G B. A. L. C. A. L C U L U. S. 309
- 2 : + 1) tr & ºn s º g - g
Č. but a " + 1 = Q ( a." – 2 ºr co, º Pi. + ..) hence, taking the differential coefficients, * Part II y
m **** = Q || 2 a - 2 cos 70,
\-y- ..] ..." 2 7 ––
elior), ſº 2-2, co, * * * 1)
Therefore the same value of a being put in Q will give
nºos (n − 1) (2 i + 1) tr + V — 1 sin g-pºir).
7? 70,
... (2 i + 1) ºr
sin–F–
2 y - I
and the numerator of the value U, will become by this substitution,
COS m (2 i + 1) T + V – 1 sin m (2 i + 1) tr
70, - ??,
-*.*.*, v-lºne; tº-nº-0 gºo. 1, ...6-metro-
72. 7?, 7? 7?,
2 . wº - * *
2 v-1 in “H” + 2 w - 1 in *::: **
This quantity must be equal to zero, as well as the result we would have obtained if we had put for a the other
value cos (2 * + 1) a1) "r – A/ – 1 sin (2 i + 1) tºi –– 1) tr We shall in consequence obtain the two equations
70, 70,
2 º' sº ſº L º ty *
cos ? (??-f 1) tr – t k sin n (2 : + 1) + n L sin (n − 1) (2i + 1)"
70, 2 71, 2 72. = 0
sin (** + 1) T sin (** + 1)"
70, 70.
2 : —- 1 * º e
sin m (2 i + 1) tr +4. keos " (** + 1) * + " [. COS (n − 1) (2 i + 1) ºr
º Q ſº e
72, 70,
From which we derive
2 tºmm ºf ºf , mºmmy te • *- te
k = – cos (n – m 1) (2i + i)x. and L = – cas (n m) (2i + 1) #
71. 7. 70, 7?
The partial fraction corresponding to the factor º – 2 r cos ei: ): + 1, is therefore
#(º e-r-petro. --e-ºg-to-)
70, 71, 72.
a” – 2 a cos erfor + 1

By comparing it with the fraction integrated (72), we shall have
(, co- (m. — m – 1) (2 i + 1) tr COS e-ºg-to-).
71, 7,
ſº ––
7. * – 2, cos “Hºt + 1
+ \cos (*= * = 2 (2 i + 1) ºr v(e- 2: … “H”: + 1
72
(*= n = 0 °ir or un- + c,
+ Sin Wº. - in (2 : + 1) 7
70,
Sl
- S (2 i + 1) ºr
(n – m – 1) (2 i + 1) "r tan"l
Tº. it b J
va – 1U DeCOI geS
71, º 2 : l) ºr
sin (** T Đ ".
adding to this formula the constant sin
7%
810 I N T E G R A L C A LC U L U.S.
I l sº-> &Eºst 2 : g Part II.
§. +{co. (n - m - 1) (?? -i- 1) ". ! A/{ a,” – 2 a cos (2 : + 1) a + ..) -
* ~ * 7, 72
a sin (2 i + 1) r
— m – 1) (2 :
-- sin (n – m – 1) (2 : + 1) ºr tan T” &FDF + c
71. 1 — a cos ––––
º
Let us first suppose n to be an even number, then by taking in this formula i = 0, i = 1, &c. ... i = + , it
2” d ºr
a" + 1
will give the integrals of the + partial fractions into which , may be resolved; therefore, by adding
these values we shall have
w” d ºr 2 (m + 1) ºr 2 7- )
— `` - – > --— gººms — — l º
a” – 1 ; cos 71. * V(, 2 * cos;
º Tr
Jº SII]. --
70
2
+ – sin on-ºp- tan T1
70,
Tr
l — a cos —
72
*mºnºg 2 cos 3 (n + 1)" / V(*- 2 a cos 3 ºr +1).
7. 7, - - 70,
a sin “”.
2
+ - sin 3 (m+ 1) ºr tan"? %2. º
72. 70, 3 ºr
1 – a cos —
-- 7.
2 5 (ºn l
* . * – 2 cos ? (* + 1) = 1 V(e-2s co, * +1).
72. 72 7.
a sin ºt
2 5 (m –– 1 f
+ — sin 5 (m+ 1) a tan"1 72 3.
7, 7. 5 ºr
1 — a cos
&c.
This series being pursued to the terms corresponding to the value i = a .
When n is an odd number, the series must only be calculated for all the values of i, from i = 0 to
— I tº ſº -
i = * a ; and it is necessary to add to it the integral corresponding to the real factor a + 1 of the deno
minator, which it is easy to see is ( – 1) º + 1) tº
e tº e a"d a
Similar steps would lead to the integral of +. 1” and we would find
t" 2 I
ſº-- *s cos ? (*# 9". I V( * – 2, cos tº +1)
a" — l 7, º 7.
2 . 2 (m -- 1) ºr *g a sin–
- — Sł fa —- tars" --— :*, *gºrk:y-ºr-r
2 ºr
1 — a cos —
7.
+ 2 cos 4 (m+ 1) + y V(e–2 coºr ).
7) 7. 7.
4 ºr
a sin —
2 4
3 ºn 4 (m+ 1) +
7 7.
tall." 71.
1 - r cos —
'W',
#
I N T E G R A L C A LC U L U.S. 811
Integral
Calculus.
~~
*** = + 2 • * ºr V(*- : * ~ *, + )
zº – l T 72 n - 70.
. 6 Tr
2
– sin 6 (m + 1) + tang" —a r º
+ &c.
- * e . m. – 2 º e
This series being continued, if n is an even number, to i = -ā-, and then adding to it the two terms
( – 1)
*-i-l -
l (a + 1), 1 ! (a – 1), which are the integral of the partial fractions corresponding to the two real
7. 72,
wº
factors of the denominator of a." 1 When n is an odd number the series is carried no farther than the term
– 1
corresponding to i = 71. , and l l (n − 1), which is the integral of the real factor of the denominator
71.
added to it.
Analogous formule might be obtained, in a similar way, for the integral of —tº
IM3,100'OUS IOTII) bliże I(\l € OOLaIIleCls I I a. T LIle IIltegºra, O
g e § 2 Y, gr a" – 2 a” a” cos @ -- a”
we know the general form of the real factors of the second degree of the denominator. -
Having proved that the integral of X d a may always be obtained whenever X is a rational function of ar,
any differential must be considered as integrated, when, by some transformation, we shall have been able to
reduce it to a rational form.
No general rule can be given for these transformations, they depend on the form of X. We must therefore
examine successively the few classes of functions, for which some means have been discovered to make them
rational. -
, since
m P
_ A r" + B a” + &c.
— — … --,-----.
Aſ a "+ B'a' + &c.
If we reduce the fractional exponents of a to a common denominator N, it is obvious that in assuming a = y”,
X will become rational. But, then d w = N yº-" dy, therefore X da will also be reduced to a rational form.
First, let X
Secondly, let X equal a rational function of r, and of terms such as (a + b x)". If we suppose all the frac-
tional exponents of a + b + to be reduced to a common denominator N, and if we assume a + b x = y^, we
N — N ºf"-" g
shall have a = *-i-º-, and d a = **, * ; therefore, by substitution, X dºn will become a rational
differential.
If instead of (a + b x), in the preceding function, we had ###, the same transformation would suc-
e a + b x -- a sº a' y” — a
ceed. For if we suppose 7-HT7 = y, we find a = 5 – Wyº
ſ ‘b’ & ºv b/ º b’ ' ar" —
and d a = n y” { a' ( à ) # # w" — a) } d y, which values being substituted in X da will reduce it
to a rational form.
(75.) We shall next suppose X to be a rational function of a, and v (a + b x + c a”). In order to make it
rational, we must distinguish two cases, that in which the roots of the equation a + b x + c a' = 0 are real,
and that in which they are imaginary. In the first supposition a + b a + c 2% may be decomposed into two real
factors, which we shall represent by p – q w, and p’ — q'a ; then let us assume
a + b a + c tº ::= (p — q a)* 2°.
We shall find in putting instead of a + b x + c wº, the product (p — q a) (p" — q'a),
_ p * - q' _2 (p' q – p q') z dz -, (p'q – p q') z
===#-, as ==####, and war º- + c 0 = ***.
Hence X d a will be transformed into a rational differential function of 2.
In the second case we shall suppose
a + b x + c a' = (a V c + 2)",
Part II.
812 I N T E G R A L C A L C U L U.S.
Integral which gives Part II.
Calculus. 2
= 47. 2 (b. 2 – 22 V c – a V c) dz b 2 – 23 y c – a V c >-N--
\-y-Z --- 2 (Z :- - 2Y - 2
* = y = g; 7: dº (b – 2 z V c)* , and V (a + b x + c a”) b – 2 z Vc
which values will reduce X d a to a rational form. This transformation would also apply, although the factors
of a + b a + c a” should not be imaginary, provided C be positive.
The same transformations will succeed to make the formula
S
a"~" da (a + b a' + c wºr)*,
i - 1
ty ſº & © º tº a ſº º e ir- gy' d 3/
in which i is any integer, positive or negative ; for if we suppose ar' = y, we shall have a "d r = r *
I . S
it will become 7 Sy” d y (a + b y + c y”)*,
which may be made rational by the means used above.
(76.) The next class of irrational functions we shall examine is represented by the formula
p
a"-" da (a + 0 a.") d ar.
We shall observe, that without making it less general, we may always suppose m and n to be integers. For if
they were fractional, they might be brought to a common denominator N ; and by assuming a = 2*, the
formula would be transformed into a similar one, in which the exponents corresponding to m – l and m would
be integers. We may also consider n as positive; for in the contrary case, it would be sufficient to make
l º g © a tº
— = 2, and then the exponent of the variable between the parenthesis would become positive. The formula
J}
p.
a""" da (a a' + b a”) is consequently included in the preceding, since it may be written in the following
manner,
on + pr_
1 p.
‘I a-(a -- b r).
This understood, let a + b a' == y?, we shall have
97 — a
(a + b ar") = p h -- * -— yº – a \ . d "-id *-*- q Q — l q # -
= y”, tº" - * -, * = (-; ) , and a * = i. 9 (y" — a)" "d y.
p.
So that the formula r"- d 1 (a + b r"), will become
q gº-tº-1 d (*** | - y
m b
which is rational when — is equal to an integer. .
71,
Secondly, let us assume in the same formula a + b x" = a "z". Then we shall find
p
(!, a 27 P. aſ zº a"
* = g-g, a + b x = ++, (a + b x") = —;, ºp” — —-,
(z" — b)" (:" — b)"
ºt. _”
r"- d r = – 4– a 2- (27 – b) " T'.
7t
These values being substituted. the proposed formula becomes
m p - m p
+ + º- - - - - - - 1
3 aſ "º 2-4-1 d2 (zº – b) " "
72.
* tº g 7??, ſp . g
which is rational when — + +- is an integer
7b, Q
There are, therefore, two cases in which we shall be able to transform the binomial differential r"~" d r
P.
(a + b a ") into a rational formula. They are the only two which have hitherto been assigned.
(77.) The formulae
P
º
f{ a "", (a + b x")7, (a + bºy, (a + bºoi, &c. } r"- d r,
f{-, (4+; ), (#), (º)}*a*.
l a' + b'a" a' + b'a" a' -i- b' r"
in which f denotes a rational function of the quantities contained within the parenthesis, may also be made
I N T E G R A L C A L C U L U. S. 813
Integral rational by a simple transformation. For the first, it is sufficient to assume a + b a' = y”* . From which Part II
Calculus.
We get S-N-2
Sºy-/ g *** - (Z **** — a N* g" &sc.
a" -: — —, a" — ( b ) 2 and a "~" da = gº- d y.
b gº
For the second we shall make #. = z** , and we shall find
º a - a' gºu &c. yºut Q — (M. 24tu &c. 17, † —s *gº q's u &c. zºu &c.- (a b' — Q. b)
ºp" = b' zºº &c. — b =(#). and r" | d a = (b. 2”. ==\m b)* y
which values, being substituted in the proposed formulae, will make them rational.
(78.) There is no other irrational form of X, besides those already considered, for which general rules may
be given to transform X d w into a rational differential. However, for various particular cases not included in
these forms, some transformations have been, or may be, found to succeed. Much practice in analysis enables
us to foresee, without going through every intermediate step, the result of a substitution, and consequently
indicates that which is most likely to lead to the sought result. In the examples which will be given of the
integration of irrational functions, will be found two or three instances of such cases.
When X da cannot be made irrational, its integral may not unfrequently be made to depend upon that of a
simpler formula. This, in many cases, is effected in using the integration by parts (62), which gives
J. V. dy, = y, y. - ſy, dy,.... (1).
(79.) We shall apply this method to the binomial differential, which, in order to simplify the calculation, we
shall write
w"~" d a (a + b a ")",
p denoting then a fractional number.
Let y = (a + b x")”, and dy, = "Tº da, then dy, = p n b r"-" (a + b r")*~1 day, and y, = +. These
77',
values being substituted in the equation (1) will give
ſwº-i da (a + b r") = ºp” a fºr gmga
b
* fºr- da (a + ba")"-". . . . (a).
A formula by means of which the integration of the proposed differential is made to depend on another, in
which the exponent of the binomial is less by one, and the expouent of a out of the parenthesis increased
by n.
Let us suppose now
y = r"-", and dy, = a "" (a + b al")” dr,
n\p-Hl
then dy, = (ºn — n) a "-" - d r, and ye = º-
We shall find -
2"-" (a + b x")** – (m – m).ſ r"-"- d 1 (a + b r")* (b)
*1 - 1 *YP -->
J a d x (a + b r")” = n b (p + 1)
A formula presenting a reduction of the exponent of a without the parenthesis.
In order to obtain farther reductions, we shall observe that
(a + b x")* = (a + b r") (a + b r")*-* = a (a + b r")*-* + b a "(a + b x")**'.
Hence frº- d r (a + b x) = a ſw"- d 1 (a + b r")?-, + b ſa"+"-" (a + b r")*-*.... (c).
This value being substituted in equation (a), gives, after reduction,
+ (a + b x) – a ſº-'dº (a + bry-
fºr- d a (a + b ar")*T* = — n 6 ' ' ' ' ' ' (d)
5 + 2* * -
7??,
We may, in this, change p into p + 1, and p — I into p, m into m — n, and m + n into m, and we fin
a"-" (a + b r")* — a (m. – m) ſa"-"- d r (a + b r")p ,
b (m. -- p m) . . . . (e).
Hence the integral of r" da (a + b r")” depends on that of rº-"- d r (a + b r")”; and if we change in (e.
successively m into m 'm, m – 2 m, &c. we shall have
WOL. I. 5 N
f r"- d 1 (a + b r") =
814 I N T E G R A L C A L C U L U.S.
Integral - a"-- (a + b a”)** – a (m. – 2 m).fa"-" da (a + b x")P Part II.
g. ſ:---- d. (a + b x) = ** m — f • S-N-2
- º a"-" (a + b a")** – a (m — 3 m) ſ a"-a"—" dir (a + ba')?
frº-r-, dr (a + b x) = w-r- p
b (m. – 2 m + n p)

. . . . . . . . . . . . . . . . . . . . . . * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
and generally tº s te g -
& r"-" (a + b a”)* — a (m — i n) ſºn-in- da (a + b x")P
ſº-c-ºn- da (a + b ar")P = e • 9
b (m — (? – 1) m + m p)
i being an integer.
If we substitute the first of these values in (e), then the second in the result of this first substitution, and so
on, we shall find that ſa"- dr (a + b r")” depends on the value of ſºn-in- da (a + b a”)”. The coefficient of
this last integral is m — i m, therefore it will disappear in the result, when Tº is an integer, and i = −. So
77.
that in that case, in which we have already proved that the integration of the binomial differential may be
effected, the above process will give us the general expression of the integral.
Let us now substitute in (c) the value of ſº- da (a + b ar")*T* obtained in (d). We shall find
a" (a + b r")” m a.ſa"-i da (a + b a')?-1
ſt"- dr (a + b x) = ( ) ##,
Changing p into p – 1, in this equation, we shall obtain
fºr- dr (a + b a')?-' by means of ſa"-'d r (a + b a')*-*,
and by each step we shall decrease the exponent of (a + b x") by one, until it becomes less than one, if p be
ſractional, or equal zero, if p be an integer.
The formulae (e) and (f) will thus enable us to reduce in every case the integration of
r"- dir (a + b r")P to that of wº" da (a + b wº)”,
in being the highest multiple of n contained in m, and r the greatest integer contained in p.
If the exponents m and p were negative, these formulae would increase the exponents of the factors of the
binomial differential instead of diminishing them; but it will be sufficient to invert (e) and (f) to obtain the
formulae answering to this case.
We derive from equation (e)
a"-" (a + b a')** – b (m = n p) ſa" Tº da (a + b a”)”
m — n - 1 ”)P -
fa. da (a + b r") a (m. — m)
y
and if we change m into – m + m, we shall find
= ??? b n\p-Hl b me —l- —m-Hn – 1 in
fr—"-" d r (a + b x")? = w-" (a + b r") (n – m + n p).ſ a da (a + b r")” ... (g).
- Q, QPQ,
Writing sºccessively in this formula – m + m, - m + 2 m . . . . — m -- in, instead of m we shall find that
ſ.r-º- da (a + b x")” depends on ſa-ºn- da (a + b x")”.
When p is negative, we shall take the value of . ſw" da (a + b r")” in (f), and we shall get
*n *YP – m - 1 y;
frº- d 1 (a + b r")*-* = a" (a + b r")” – (m -i- m p) frº-" dr (a + b x)".
a m p
Changing p into – p + 1, we find
ſa" d r (a + b wº)-P =
a" (a + ba") " – (m. -- n – m p).ſ r"~" dr(a + b r”)-r-ţi
a n (p − 1)
This formula, combined with the preceding (g), will reduce the integration of
w-"- d x (a + b x")-F to that of r-"+"-" da (a + b r")-p+,
in being the highest multiple of n contained in m, and r the greatest integer contained in p.
(80.) There are some cases in which these various formulae cannot be used, because their denominators
become equal zero, but then the binomial differentials may be easily integrated.
When m = 0, the denominator of (a) equals zero. In that case the binomial differential is reduced to
... (h).
da, (a + b ar")P sº se e
fº ( Jº ) which becomes rational by assuming a + b ſt" equal z raised to a power equal to the deno-
minator of p.
The denominator of (b) may vanish by three different suppositions, when m = 0, b = 0, or p = — 1. In
the two first the binomial difference becomes a "T" d a multiplied by a constant, and is therefore immediately
*n-1
a Tbaſ' that is to a rational fraction.
integrated In the third it is reduced to
I N T E G R A L C A L C U L U.S. - 815
Integral Part II
a tº 777,
Calculus. The supposition of b = 0, or p = — – makes the denominator of (d) equal zero. We have already See Il
- 72, ... .
\-N- . . . . . . 17,
what becomes the differential in the first ; in the second it is reduced to r"~" da (a + b x) 7; and by assuming
a – b ar" = a "z", it is transformed into a rational function. . 3. af b
The hypotheses which make the denominators of the other formulae (e), (f), (g), (h) vanish, are the same
as those we have examined. - - 2
(81.) Similar reductions to those which have been effected in (79) upon the binomial differentials, may be
also applied to some other functions. The integration of the general formula , may
a"Tº da (a + b a " + c wº" + ea.” + &c.)”,
may be made to depend on those
wn-º- d a XP, tº-ºn-, d a XP, a"-sº-1d + XP, &c.
where X is equal to a + ba" + c a”-- eas" + &c. -
The steps of the calculation are entirely analogous to those used in (79.)
(82.) The expression a "Tº da (a + b tº + c aº)" may sometimes be reduced to binomial differentials. Let
* -- b . l b \* -1 /4 a c – bº p -
us assume a " = y - 2 o' it becomes 7. 9 - a . n 4 c + c y”), and, consequently, will be reduced
e ... ???, e º-
to a limited number of binomial differentials, if 7, be an integer. This will also be the case, if -* — 2 p be
- 72.
a positive integer. For the proposed formula may be written a"*" da (a wº" + b a "" -- c)", and if we assume
b
a-" = y — 2 a.' it will be changed into
— I b \t"–2p-1 4 a c – bºx”
* -- -— ſº 2 gººm º
; (9 #) av(av -- 4 a ).
(83.) The integration of X dr, where X is a rational function of r and aſ a + b x + c re-E drº + e r", may
be proved to depend on that of the three following formulae,
d ºr a 2 d ºr d a
—H-, --Pi—, and — ;
R R. (a" -- a) R.
R being equal to w/a+ 3 a.º. Hºrs.
We cannot give here the details of this investigation, which has been the object of the labours of Euler,
Lagrange, and Legendre. We only mention it, to take the opportunity to observe, that with respect to the
integration of functions of one variable, in the present state of analytical science, it is principally the reduction
of differentials to a few really distinct formulae, which must be aimed at. The integrals of these must be
considered as new transcendants differing from logarithmic and trigonometrical functions; but which may be
equally important in analytical researches. -
We shall now give a few examples of the integration of irrational functions.
x= (lit: — wº) d ?
Example 1. Let
1 + zł
The common denominator of the fractional exponents of r is 6. Hence we assume w = y”, then dra- 6 y” dy,
max becomedy avº." --day(F-y-y-v-v – H. Integrati y
- 1 + y? I + ;) egrating each
term, and putting for y its value, we shall find
TO 1. 3.
1 –– a “ — arº — 3 # 6 º' 6 !. † -z-
ſº-Hº * + 2* + 2 - #2 + 2 +* – 6 r" + tan-" r" + C.
1 + r." 5
§
}
E le 2. Let X = 1 c bei d itiv w
arample 2. Le gºngmºm M (a + b c + ca”)' ing supposed a positive quantity.
Assume as in (75) a + b a + c a' = (a v c + 2)", then we shall find
2 d z 2 d 2 — 1
= +------, but I -- = −7– — 2
X da; b – 22 M c utſ-ºſ- y; t ( a ve)
substituting for z its value, we get
da,
— l
ſ x d r = Jari-Ha =#10 + 2*-* we v (a + ºr º)+c.
This value may be put under another form, in multiplying and dividing the quantity under the sign I by
b + 2 cr–H 2 v c v (a + b x + ca"). We shall have - - -
5 N 2
816 I N T E G R A L C A L U U L U. S.
Integral da: zº( b* — 4 a c )+c or equal t
Calculus. TZ/2 - # II ~ * --- TZT g 3. Ulal U.0
g | zerº-rº ze b + 2 ca. -- 2 A/c v (a + b a + c a”)
7:10 + 2 ca--2 v c. v (a + b x + c wº)) + C,
... a — 1
including in the arbitrary constant C the quantity Wo l (b" – 4 a c).
When b = 0, and a = c = 1, this formula becomes, in adding l 2 to the arbitrary constant,
da: *-*.
viº = l (r-- VI + wº) + C.
l
Example 3. Let X = A/ (a + ba – ca.”)
equation a + ba – ca.” = 0 are real and of contrary signs. The two factors of a + ba – ca" may then be
represented by p – qa, and p' + q'a, where p, q, pſ, q', are all positive.
Assuming as in (75), a + b a - ca.” = (p — ga)* 2°. We shall find
X d 2 dz and sinceſ. 2 d 2 2 tant" 2 V;
a: = —/, --> --> —, ,
q z* + q' q z*-H q' V q q.' q'
gº da, _ 3 - V(p' + q'a) Vg
then /xas-ſº-º-; ºn jºr Hijj + °.
, this value may be presented under the following form
2 tan a
Since tan 2 a =
1 — tan” a
d a _ ! - 2 Mgq'M (p' + q r.) (p → q +) ū
gººms tan" ----------, Æ + C ;
M(a + b a' — ca") w/ q q.' p q' – p’ q – 2 q q’ a
and in observing that q q’= c, p q' – p' q = b, and V(p'+ q'a) (p – q a) = V (a + b x – c.a.") it becomes
d as =-|-tan- *Mex/a+** = **) - c.
M(a + b x – c aº) T V c b – 2 c as
or, again ſ. d it =-|-cos- –*****= + c.
y 1) W(a + ba – ca”) V c y (bº + 4 a c)
When b = 0, and a = c = 1, then p = q = p’ = q' = 1, and the formula becomes
*H = 2 tan"l &#}+c = sin", r + C.
da,
This result agrees with the value of
in Erample 2, may be applied to Jºy by supposing in it b = 0, a = 1, c = — 1. It gives then,
in including º in the constant,
d c l
—- sº —- — 1 — nº o
Hence
1
7+16 J-1+wo-so)+c.
sin" ºr -
and if we suppose a = sin y
v=#| Govt V-iny)+c.
It is easy to see that this constant equal zero; for if y = 0, we have sin y = 0, cos y = 1, and l l = 0.
fore we shall have simply
l
y = A/– it (cos y +w - 1 sin y),
and by changing the sign of y
l
-v=#1 (co-v-v-1 in y).
If we suppose in the first of these two equations y = #, we get
, and suppose a and c to be both positive, so that the roots of the
, given (m) (63). We shall observe, that the formula given
Part II.
INTEGRAL CAL CU L U.S. 817
Integral
Calculus.
S-N-
m ly– 1 ºr tº-º-º: l (– 1) -
— = ++, or – – = A/– l l A/ – l = — 1) V-1, = \T *2 = – l (– 1) v-
2 A/– 1 or – a M-1 l V – l = 1 (V – 1) Or ºr M– 1 (– 1)
formulae, which may also be expressed in the following manner:
e i = (V – 1)*, or e− = (–1)--, or (–1) = 1 /= i.
The numerical values of e and r are known, it is easy to find that e F = .207879, hence
(V – 1)*-* = .207879.
These various singular results must be considered as the symbolic expressions of relations between infinite
series.
Example 4. We shall propose for the next example to find the value of
a.m. daº
V (1 — wº)
If we compare it with the binomial differentials, we find that here we have m instead of m – 1, n = 2, and
1. ſº 777, . e tº 770, tº
. := — Tº Hence if m be an even number – will be an integer, and if m be odd, 7, -- P. will be equal to a
7?, 2. Q
whole number. Therefore, provided m be an integer positive or negative, it will always be possible to transfer
the above formula into a rational one, and, consequently, to obtain its integral under a finite form. For that
purpose we shall make use of the formula of reduction given (79).
- I
By substituting m — 1 for m, and assuming n = 2, p = — ;-, a = 1, and b = - 1, the formulae (e) (76) will
2
give
a ºn d ºr =-º-º: “iº ſº; (
W (1 — wº).T 772. 772. M (1 — wº) ' ' ' ' ' a).
Making successively m = 1, m = 3, we shall obtain in
a d a
— = — A/ (1 — ac" C,
are d ºr ! ... 2 2. a da:
#5= - eva-º,++ſº
j a" d a * ... - * 4 anº d an
* @7 d ºr l 2 6 tº dº
†† = - eva-º,+; ſº,
&c. &c. &e.
Hence, by substituting in each the value of the preceding integral, we find
a da
Jæry--wa-º,+c.
a:3 d a I 1 .. 2 &
— - - ? – ?? -- – - ?”“
M (1 — wº) T (; e-H), a a") -- C,
a 6 d ºr I 1 .. 4 l. 2 . 4
— — — I — r* — r° – rº
M (1 – tº) "T (; ; ++++}#})/d a") + C,
ac'ſ d ºr 1 1 .. 6 1. 4.6 1 .. 2. 4 .. 6
-— ” --- — r" *mºnººmºsº — rº cºma-º-º-º-º-w-mºmºmºmºn - - wº
M (1 — wº) T (; "+} +}#}* +}#})/d r") + C,
&c. &c. &c.
The law of these values is very obvious, and we may easily form the general formulae.
*Fi dº r l I . 2 ºr 1 .. 2 r . 2 r - 2
—– º – Eºs — rº" Qre? - •4
M(1 — r") W (l *{#i. * @F-I) (ºr II) * + Grºsjöºf D (ºr IT,” +-
l. 2. 4. . . . . . . 2 r
e e g º O & Q & Q a +}####5}+c.
Part II.
*N*
818 I N T E G R A L C A LC U L.U. S.
Integral Let us assume 1:ow successively m = 0, m = 2, m = 4, &c. We shall have Part II.
Calculus. - '.' - N-N-2
Sºc-y--" Ho- sin- a + c by (m) (63), -
a” d ºr a V (1 – a “) l da,
* = -º,++, ſº
tº da as V (1 – 29) 3 a" d a
Jū-j = ~ TT T. A. JJT-sy
a" da *VG-r) + arº d aſ
M (1 — wº)"
and by substitution,
d a :- - 1
ſº - in a + C,
a da 1 1 — arº ' co- C
Jºãº)=-5 ºva-º,+ 3 sin". c.
a" d a I I ... 3 | . 3
— = — I — wº **-*. – rº sin"
M(1 – alº) (; , ; })/d r) + 3 + sin-ºr--C,
arº d'a, l 1 .. 5 1 .. 3. 5 1. 3 .. 5
—----, -, -- — rº — a 3 tº-m-- * — r? : rn -1
M(1 – 4%) (; "-H. +})/a *) + 3 +... sin", + c,
&c. &c. &c.
and, generally,
tº d ? ! ... 1. (2 r – 1) 1. (2 r – 3) (2 r – 1)
= — I — ºr” "l —Y------T’ r2"? *** -- . . . . . .
M(1 — wº) 2.” TGF-2).3% ºf ſº, (57–5)37 ° + -
1. 3.5 . . . . (2 r – 1) 1. 3. 5. . . . (2 r – 1) ...,
• e s " ——º º 2 —-º - 1 o
. . . . . . * g. TET. )wa *) + 3 + 3 + sin" ºr + C
When m is negative, the differential assumes the form ap" º a) ' and by substituting m for m – 1,
* - ſº
and making a = 1, b = – 1, n = 2, and p = — ;, in the formula (g) (76), we shall find
** – VQ - *) + = ſ. d r (b
rºy(I – wº) T (m. – 1) a” on — I J a”-* w/(1 — rº) ' ' ' ' ' ' ' ' ' ' ' ' ).
If we suppose m = 1, the left side of the equation becomes infinite. To obtain the integral in this case, let
I — a 2 = z*, then a = y( 1 – 2%) and d a = – 2 d = . Therefore we shall have
A/(1 – 2°)
dº = ſº = - # 1 H = + x'\! = 3-c
ſº-a, - 1 — 22 T 2 1 – 2 T 2 1 — A7(1 — wº) y
which, by multiplying both terms of the fraction under the sign l, by the numerator, becomes, after reduction,
| + V ( – ’) + c.
tº
— l
We shall therefore obtain from equation (b), by substituting successively for m the terms of the series of odd
numbers, the following results :
da, -* - 1 + V (1 . . a')
ſº-º- J. tº -H C,
d a
=-“...” + , ſ , ºr
ſº-,-- 3. * , a V(1 – 18)"
da, _ V(1 — wº) .. 3 d as
Jºãº--,-- 4 a.4 + T a"M(1 — wº)'
da, W(l ==2+ 5. b d a
ſº-º,-- 6 tº 6 a' M (1 — wº)"
&c.
and consequently by substitution,
I N T E G R A L C A L C U L U. S. 819
Integral ſ da: , 1 + V(1 — a ") . . . Part II.
Calculus. —77–R = — l + c, . . . . S-N-
k-v- a wºl — wº) a '. . . - • e
da, _ _ MCI - tº l 1 * if VQ - *2 + c -
&B W(i — wº) T 2 gº 2 Jº y
da, 3
*** - l 1 .. 3 l . 1 + V(1 — a ")
zº-, -- wa – º (; ; +...+)- 3.4 ° tº + c,
Tº da, l 1 .. 5 1 .. 3 .. 5 1 .. 3. 5 1 + V (1 — wº)
— tº: --- — r" -ammº-º-º-º-º-º: * * -- - --— —-—--- - - - - tº & -
J a 7 M (1 — wº) W(l a") (+++++ ...) 2 . 4. 6 + c.
J’
and generally -
d a - l 1. (2 r – 1) I. (2 r – 3) (2 r – 1)
* - - — rº —- -f- –— -T- --—— —T- . . . . . .
ſ as” v(1 — wº) w(l * (# + (gria) 3 ºr " (ºr: DGrfºr T
1. 3. 5. . . . (2 r — 1) 1.3.3. ... (2 r = 1), 1 + V(l − rº
tº e º e g º 2.4.6. . . . . 2 r zº") T 2.4.6. IT2 . - Q? -j- c.
We shall obtain in the same manner, by supposing in equation (b) m successively equal to 0, 2, 4, 6, &c.
d a — r? ~ *l
Q_e W(IT) = sin” a + c,
ſ 2 d a - - MQ – º – c.
a"A/(1 — aº) Jº
da, _ _ VQ - *2 + 3
-*.
2 f d a
a" M (1 — wº) T 3 * 3. a” v(1 - wº)
4
T
d ºr — — VQ - tº ſ d a
*=mmemº-s-summº-ºº: a JCT F)
a" w(1 — wº) T 5 wº +
&c.
dr times W(1 — tº)
Hence frº-ji=-“H” +,
^ da,
I 2
#5- - wa-º. (sº + +.)+ ,
da, ... ( . 4 2 .. 4
ſº-º;= – wa-e (+++++ H+)+.
and generally
d a tº 2 I - I . (2 r — 2) 1. (2 r — 4) (2 r – 2)
ſ r” MCI – a;=-wa-e) (a 7" - Dººl ºf ūrīājør-ijº ºf & Fjørtājār-I) ºf
1. 2. 4. ... (2 r – 2)
ſº tº e º e º 'º & 1. 3. 5. . . . (2 r =#) —H c.
£2" 1 d agº" d
If in the formula :* we suppose as e- #. we shall have da = 2 V2 c #. and **.
= + +4++, and in making the same substitution in the value we have found above, we shall
2 (2 c) A/(2 c y – y”) -
obtain that of y” d y , at which we might arrive in a direct manner by analogous reductions.
A/(2 c y – y”)
d * * g
Erample 5. Let the proposed differential be Jº ; : To make it rational, we assume a + b x = a y”.
a (a + b x)?
º 3 — 2 2,
We find d a = **** tº ºt •or p (a + b a)* = as y”, these values being substituted, we get for the
3 3 l + 2 * @
transformed differential 3 d y Then -. tº -- - ++++, but ſ a 9 = 1 (y – 1) + c,
2. y” — 1 y – 1 y” + y + 1 y — I
as (y” – 1)
- — (y -- 2) d y I _a 2 y + 1 .
—S*——-2 = – – l (u% l) — A/3 ta *-ºs-rººmsºms c. Theref
and ſ y? -- y + 1 2 (y” + y + 1) W3 tan M3 —- Tefore
- 2 ºu -- 1
ſº- =+ |o- J) – ; ! (y” + y + 1) — V3 tant* *}+.
as (y? – 1) aft
820 I N T E G R A L C A L C U L U S.
Part II.
Integral -
Calculus. g I tº- — 1 \g
St. US Observing that ! (y & Fºgº 1) dº +1 (w -- 3/ + 1) - l w( # # l - W (y D. - 3. ! & – 1 T, S-N-"
º 3/ 3/ ) (y” — 1); 2 (y” — 1)s
© 3 d I 3 — l
it becomes → ***— = -s. ſ: _3/ 1- — W3 tan" *. + c
as (yº – 1) as (y – 1)* W
Substituting now for y its value, we find
tº !, l, l |
d a | | 3 (a + b x)3 — a” 2 (a + b rhs as
./ f = |} i. —w8 tan-1” (*# ** + °) +.
a (a + b ar)* Q8 tº 3 as W3
We shall now give examples of some of the irrational formulae which may be integrated, although th di
rules cannot be applied to make them rational. y b 3. ougn the preceding
l 2m / ſs * --> tº
Example 6. Let X = and let ºt-2=y, we minºey--ºr
- Jº
ag"
(1 — a ") */(2 w" — 1)
* – 2 r" + 1 (r” – 1)*
a 2" * a;2" "
d r (1 – r")
and 1 — y” = Hence y”- dy =
Dividing both sides of this last equation, respectively by the sides of the preceding, we find
gy”- d y d x:
I-yn T (Trºy
and dividing again the left side of this by y, and the right side by the assumed value of y, we get
da: _ y”-- d y
(1 — w”) */(2 w" – 1) T 1 — y”
2m-
The integration of the proposed formula is thus reduced to that of º, which is rational.
tº-e 3/ ??!
arm-l
Harample 7. Let X = (1 — w”) */(2 w" – 1). we shall make *V(2 w" – 1) = y, from which we deduce
g” = 2 w" – 1, or w” = } (y” – 1).
Hence a "-" d r = yº"-" dy, and therefore
2 y”- d y r"-" dr
II ºr - TL F.
and dividing by y = *V(x" - 1), we find
a"-1 d ºr _2 y”- d y
(1 – rºy ºva" – l T 1 – yº" '
which last expression is rational. - -
(84.) The number of cases in which X d a may be integrated when X involves logarithmic functions of the
variable is very limited.
We shall first consider the function X d a (la)", in which X is supposed to represent a rational function of z.
iſ x d r may therefore be determined, and we shall represent it by y. We shall have, in integrating the pro-
posed formula by parts,
ſ x d r ( 1)" = y (la)" – a ſºvo."
If 3/ be again a rational function, we may, in the same manner, make the integration of d a y (la)", to
Jº
gºmºmºe
º
depend on that of another in which the exponent of l r will be still less by one. By continuing the same process,
if n be an integer, and if at each operation we may integrate the function which multiplies the power of l r, we
shall be able to effect completely the integration.
Let us suppose X = a ", we shall obtain
71. da: n- 1 rºm-H1 – a.m.-- ſº 71.
ſº (! ?)"-" r"+ = (la) -; H ſ. d x (la)*-*.
7m -H 1
arm-H
ſ r" d r ( 1)" = m-Hi (la)" T m
In the same manner,
wn-Fl 72
yº ſºl = trººm ſº -- — 1 17, ſº se
ſ r" dr (la) =; H ( ) * — #Tiſ- dr (la)"-",
arm-Fl 70,
wn fi - 2 -- n = 2 — 2 •º 10 me
ſ a d a (la) =; H Q ) - #Hiſ: da (la)"-",
&c.
I N T E G R AL C A L C U L U. S. 82ſ
Integral
Calculus.
\-y-'
And substituting each value; in the preceding equation, we shall get the general formula
ſ.r.º. da, (l a)" - arm-Fl |Q a)" gº- 70, (l r)- + n (n − 1) (l w)". *g
- m—HT U m + 1 (m + 1)*
n (n − 1) (n − 2) n-3 }
- (m. -- 1)* (l a.)"-" —H &c.; + c. . . . . . . . (a).
This series is limited when n is an integer. When m = — 1 the denominators become equal to zero, and con-
d a (la.)"
sequently the series cannot be used. But in that case the proposed formula is , and by supposing
n+1
d -
la: = z, we have *: = d. 2 ; consequently it is changed into 2" d z, the integral of which is Substi-
dr 2, ... ... (la)"
ſº a 9 *g 7, -ī- i + c
It is obvious that the transformation we have used here, would be equally successful for any differential function
X d a
tuting again for z its value, we find
, in which X would only involve la ; the supposition a = l r = 2, would make it algebraical. When n is
either fractional or negative, the series a is unlimited. If n = — , for instance, it becomes
*†" (, ºr I l 1 .. 3 1. 3 .. 5
M la -- F-- ; + ; + ; +&c e g º ºs & }+.
(m + 1) (la). 2 (m+1)*(la); 4 (m+1)*(la)” 8 (m + 1)*(la)*
We may obtain formulae of reduction, corresponding to the case in which the exponent of l r is negative, analogous
to those obtained above, and by means of which the integration is made to depend on that of differentials, in which
that exponent is less.
The expression X da (la)" may then be written X w. º: (la)", and by integration by parts,
da, - ?? ---, Xa, l l
ſº a 9 =-a-jº-ra. H5 ſºadcº
And by applying to this last integral the same decomposition, we shall reduce it to the integration of a formula
in which the exponent of (la) will be — n + 2 ; the same process being continued will lead by successive sub-
stitutions to the following expression,
ſ}; tº — *— – X, a tº-g X, ºr — &c.
(la.)" (n − 1) (la)"-" (n − 1) (n − 2) (la)"-" (n − 1) (n − 2) (m. – 3) (la)"""
in which d (X,E) = X, d w, d (X, ar) = X, d ar, &c. . . . . The last term of the series will be
l X. , da; 1 X. - d.
n – 1 , if gº O” 2 trº ºt 2
+a+Taºiſ lar if n be an integer and + o-º-º-; ſº; 3rº
if n be a fractional number, and m the greatest integer it contains.
Let X = a ", the above formula will give
a"d a rin-Fl (m. -- 1) r"+ (m. -- 1)2 w"Fl
ſº 2)7 T (n =T) (ºr T (n − 1)(n − 2) (; )* T (n − 1)(n − 2) (n = 3) (jºyº
-- (n + 1)"-" ſº d x
(n − 1) (n − 2). . . . ] , / la:
3.
m being supposed to be an integer.
This last integral may be reduced to a simpler form by assuming a "*" = y, for then a "d r = #:l”
la: ! y
which thus appears to be a new transcendant.
ºr, d * d
la: = º# I’ and hence, f r” dº -: ſ a y. No further reduction can be effected upon the expression ſº#.
d a
a ’ but then the integral is
The preceding method of integration would not apply to the differential
obviously l (la) + c. -
(85.) When X involves exponential functions of r, the differential expression X d a may also be completely
integrated in a few cases.
We shall first observe, that if X be an algebraical function of aº, X d a may be reduced to an algebraical
a dº du, da = la . d w
l a
W. C. L. I 5 O
differential expression of one variable. For by assuming a' = u, we get , and by
Part II.
S-N/~
822 I N T E G R A L c A Lc U Lus.
Integral substituting for a” and d as their values in X d ar, it will become an algebraical function of u, which may be Part II.
**, integrated by methods previously explained. S-N-"
When X contains at the same time the variable w, and a”, the expression Xd a may easily be transformed into
another involving only the variable, and the logarithm of that variable, by supposing a' = u. Then the rules
given in (84) may be applied to the new differential. In most cases, however, it will be simpler to integrate
without making use of this transformation.
The expression X e” da will be integrated immediately, whenever we shall be able to decompose X into two
parts, one of which shall be the differential coefficient of the other. Let Y designate one of the parts, and,
d
consequently, # the other, it is obvious that we shall have
d Y
ſ X et d w = Y++. e" da = Ye’ + c.
No general rule can be given now for this decomposition, the discovery of the transformations and artifices
calculated to facilitate it depends entirely upon the habit of analytical investigations. We shall find in the sequel
that such a decomposition may be effected by means of the integration of an equation ; but this means brings
back the difficulty precisely to the same point.
When X is an integral and rational function of a, the expression X a” da may be completely integrated. We
shall have first, by integrating by parts,
sº R l Jº
ſ X a d x = H. Xa – Tº ſa d X.
Let d X = X. daº, d X = X, d ar, d X, = X, d w, &c. We shall have successively,
I 1 • º l I 3 *
#, X, a - #2 ſa d X, ſº, a dº =#x, a -āſa d Xs, &c.
fx, a da = };
and by substitution,
l I
smºmmºn-gmºmº * + → X. a ". . . . — &c. -- c.
ū;*, * + (la)* a 0. &c. -- c
A series which will obviously be limited, since X being by supposition an integral and rational function of r,
one of the successive differential coefficients X1, Xs, &c. will necessarily be equal to nothing.
Let X = a ", m being an integer, we shall have
, ... , . . . . ſ a " ma""" m (m – 1) a "Tº + m (m – 1). . . . Il
fr Q, are a {i. Ta); (la)* * c v e s e e (la)"+" } + .
the sign of the last term being — when m is an odd number, and + when it is even.
Another transformation may sometimes be used to obtain the integral of X a dr. Let f X d c = X,
fix, da = X, ſix, da = X, &c.; and let us begin the integration by parts of the differential Xaº dr, by the
factor X dar, instead of the factor a "da, we shall have successively,
ſix a da = X, a” — laſ X, a da', ſ &, a da = X, a - la ſix, a dr, &c.
and, consequently, -
ſX and r = X, a” – l a X, a” + (la)? X, a . . . . . . . . + (la)",ſ X, a dr,
the sign of the last term being + when n is an even number, and – when it is odd.
Let in the last equation X = a- ", and it will become
ſº: &aid=t Gºr a.” l a a” (la)?
Tº T T (m-T) ºf T (m = O(n − 2) ºr T (m-T) (n − 2) (n-3) ºn-s’’ ‘’’
(la)”- a” d ºr
* (m = 1) (m. – 2) . . . . lſº
The last term of this series cannot be reduced any further, but it may easily be shown that it does not differ
d
from the new transcendant f # to which we have been led in (84), for if we suppose a' = y, we shall have
fxa as - #xa. smºgº
J a
a” d a dy
Jº ly
a” d r = #. * = #. and, consequently, ſ
We shall apply the rules for integrating logarithmic and exponential functions to two examples.
Example 1. Let X da = d; (H) This differential expression is included in the general formula
2. ‘’’ wº
w
Y dril Z, in which Y and Z are algebraical functions of a, and which, by integration by parts, is reduced to
d Z
l Zſ-Y dwiſ(? ſ Y d ..) When the quantity between the parenthesis happens to be an algebraical function,
I N T E G R A L C A L C U L U.S. 823
Integral the whole integration may be performed by the rules given for this kind of functions. In the example chosen Part II.
Calculus. this will take place. We shall have - S-N-2
\ºv-' y da: I 2 - / 1 2 d a
* (H = — -i (H)+/−. but
tºg gº w” (1 — a J
2 da; 4 d (*) 1 + *
J.T :- f IL. = 2 l ( i) + c, and therefore,
a" (1 - a) 1 — a - .
l
2
ſº; (H)=- (H)+2(H)+.
a. * • a 9 1 — a
e” a da * tº - e
Example 2. Let X da = (1 + x)' We shall try to decompose (ITT), in two parts, one of which shall
be the differential coefficient of the other. With a little attention we see that
‘C _ 1 + a 1 1 l
(TTE). (II), T (III), (II*) T (TTºjº
e” a da; et
(1 + æ), T 1 + z-Fe.
(86.) We proceed now to the integration of differential expressions containing circular functions of the
variable.
The formulae f g, h, &c. (63) will enable us to integrate any differential of the following form,
d x 4. A + B sin a + C sin 2 a + &c. . . . . . . . + A -- B, cos a + C, cos 2 a 4- &c. §
And therefore, since any rational and integral function of the sine and cosine may be transformed into series
similar to that between the parenthesis, we shall be able to integrate the differential X da whenever X will be
such a function. In many cases, however, it will not be necessary to make use of these developements. The
formula (sin a " (cos r)" da, for instance, may easily be integrated in several cases by the method of integration
by parts. We have first
f d a (sin a Y” (cos a)" = ſ dw sin a (sin a]"-" (cos ar)", but
and that under this last form the proposed decomposition has been effected. Consequently
n-l-l
J'd a sin a (cosa)" = — (cos oºt. since d cos a = — sin a day, therefore
m –– 1
n+1 ſci m = 1 — 1 -
Jº da (sin ay" (cos r)" = — (cos * º a) -- *Hºſa r (cosa)* (sin ar)”,
and because (cosa)"+* = (cos a)' (cos w}" = {1 - (sin a Jº (cosa)" = (cosa)" – (sin a Jº (cos r)", we shall
have by substituting, and then taking the value of ſ da (sin a j" (cos a)", a quantity which will be, in both sides
of the equation,
(sin ay” (cos ar)", , m — I
e ºl * = * *===mº º - m-2 " . . . . . . . . o
f dw Gins (cosa)" = 7m + n + 7m –H ..ſd a (sin a Y” (cos r) (a)
Operating upon cos w as we have upon sin w, we shall arrive by similar steps to the following expression,
: ºn anY ºn ſ , , , (sin a "+" (cosa)". n – 1 n-2 f..... <\ºm
J'd w (sina)" (cosa)" = m + m, + m –– n Jº da (cost)"-" (sin a "........ (b).
By means of the formula (a) the integration of d a (sin ac)" (cos w)" will be reduced to that of d a
sin a (cos ar)" if ºn be a positive odd number, and to that of d a (cos ar)" if it be a positive even number.
In the first case, the expression will be completely integrated, whatever be the value of n, since ſ d w
(cosa)"+,
T T. TI
positive integers, the complete integral of d a (sin a J." (cos a)" may be obtained by the use of the formulae
(a) and (b), for they will reduce the integration to that of one of the following differentials d ar, d a sin w, dr
sin a (cos a)" = + c. Similar remarks apply to the formula (b). If both m and n are
s g e g o sin ac)?
cos ar, d a sin a cos w, the integrals of which are respectively r, cos a, - sin ar, ( 2 ) If m + n = 0, these
formulae will be of no use, even in the supposition that m should be an odd number, because the coefficient
* – 1
7m + n, -
If in the formulae (a) and (b) we take the values of ſil a (sina)"-" (cosa)", and ſ da (sina)" (cos ry”, and
then substitute in the first m for m – 2, and in the second n for n – 2, we shall find
becomes then infinite.
5 O 2
824 I N T E G R A L C A L C U L U. S.
Integral
Calculus.
\-y-
tº (sina)” (cosa.)” m + n + 2 g
m f * - m-H2 ". . . . (c).
- ſ da (sin r)" (cosa) m + 1 + 7m + 1 ſda (sina)” (cosa) (c)
(sin r)" (cost)* m + n + 2
ſ d a (sin ry" (cosa!)" = — n -H 1 –– m + 1 Jº da (sin ry" (cosa)*.... (d).
The formula (c) will reduce the integration of da (sina," (cos wy" to that of d a (sin a j-" (cosa)", if m be a
negative odd number, and to that of d a (cos r)", if it be even. The first of these may easily be transformed into
te ſº . . . * — y” d
an algebraical and rational formula, for if we assume cos x = y, it becomes ###! , and therefore can always
be integrated whatever be the value of n. Similar remarks apply to the formula (d). If m and n be both nega-
tive integers, the formulae (c) and (d) will reduce the integration of da (sin a Y” (cos ar)" to one of the four
d d
following differentials, d ar, a *. Jº d aſ Of these we have only to consider the three last, and they
sin a 'cos a 'sin a cos a
may all be easily reduced to the same form. By Trigonometry we have sin a = 2 sin # a cosº, a, and
da,
Tr 7- tº ºr ºr da, 2
= sin ſ - – = sin ( --|- a la- 2 sin}ſ - – - a l cosº ( - + a therefore–H = −;-, and
COS ſº in (; ) in (; + 1(; –– ) #(; -- ) sin a sin # a cos 4 a.
da,
da, tº ºn &
. To integrate −, we divide both numerator and denominator
Siſl Jº COS ſº
da, tºº Tº
COS ſº in (; + a J cos; 4-H -
2 2
d a da:
9. 2
(cos r). == (cost)". Under this form, it is obvious that the numerator is the dif-
by (cos ry”, it becomes then sin a tan 3,
cos a
ſº g da,
ferential of the denominator, hence J.— = l tan a + c, and consequently
- S11] T COS (C
da:
da: 2
# = ſºrº: =tº-Fa and
d a
d a - 2 = 1. tan + (4- ..) C.
ſº. ſing +)-cºs(; +.) 2 g-H +
2 2
We have already stated that the formulae (a) and (b) were of no use to integrate d'a (sin ar)" (cos r)", when
m + n = 0, or n = - m. In that case, the formulae (c) and (d) may be employed with success if m be an
integer. But in that same supposition the differential may always be completely integrated, whatever be the
g º wº
value of m. It assumes then the form da (sin ay" = d a (tan ar)”, or di (cosi)" --- _d , . These two cannot
(cos r)" (sin a Y” (tan ar)”
be considered as distinct from one another, since m is supposed to be any quantity whatever. Therefore, we
te d
shall only consider the first. If we make tan a – */, we shall have da = Hº and substituting, the pro-
3/
posed differential will become y” d 3.
1 + y?
a formula which is rational, if m be an integer, or which may always,
without difficulty, be transformed into a rational one, if m be fractional. Therefore when m + n = 0 the dif-
ferential d a (sin º'" (cosa)" can always be integrated. We may even generalize this result; for since by means
of the formulæ (a), (b); (c), (d), the exponents m and n may be increased or decreased by any multiple of two,
the differential da (sin wo" (cos a)" can also be integrated if m + n equal, plus, or minus any multiple of two,
be an integer. If we recapitulate now the various cases in which we have proved
or in other words if
that the differential expression da (sin a Y” (cosa)” can be integrated, we shall find that they are all included in
the two following conditions: First, when one of the two exponents m and n is a positive or negative odd
an + 1 71 -- 1 7m -H m
number, or, which is the same thing, when g—, or 2 ” is an integer. Secondly, When 2 is an
integer.
Part II.
\-y-Z
I N T E G R A L C A L C U L U S. 825
Integral We would have arrived precisely to the same result if we had first transformed the differential Part II.
Calculus. d a (sin a]" (cos r)" into an algebraical expression. For that it would have been sufficient to make either S-N-2
d y
M1 – yº
n 1
these values the differential becomes y” d y (i — yº) 2. Under this form it is easy to compare it with binomial
differentials, and applying to it what has been proved (76), we find that it may be made rational when
7m -H, l, or * + n, are integers.
2 2
(87.) By substitutions similar to the last it will always be possible to transform a differential function of
trigonometrical lines into an algebraical one. They may also be transformed into exponential functions by
substituting for the trigonometrical lines their values in terms of the exponential of the arc. It requires much
practice in analysis to determine in each particular case upon the means which are most likely to lead to the
required result in the simplest manner. To complete what can be said here upon the integration of circular func-
tions, we shall give a few examples, chosen with the intention to show the various artifices which have been used
hitherto, and which will include nearly all the cases which have been integrated, besides the general differential
expression we have already examined in (86).
Erample 1. Let X d a = d a sin (a + b x) sin (a' + b x). The difficulty of integrating here, arises
from the circumstance that the sines of two different angles are multiplied; but we have generally,
Substituting
sin a = y, or cos w = y. In the first supposition we have cosa - v1 — y”, and d w =
sin y sin z = cos (y – z) — cos (y –F 2). Making use of this reduction the proposed differential will become
2
da cos { (a - a)+(5– b) w ł. d a cos (a + a') + (5 + 0') w ł
2 2 2
and hence we get -
º e i — aſ b — bº i + a') + (b -- b') a
fdº sin (a + b x) sin (a' + b x) = + { (a #; )* } – in {(g # ) ºr ; + c
Ewample 2. Let X da = a "d a sin a. Integrating by parts, we shall have successively,
ſa" da sin r = — a "cos as + n ſa"- d a cosa,
J a ** d a cos w = a "r" sin a – m — 1 ſa"-- dr sin ar,
&c. &c. &c.
and by substitution,
ſa" da sin a = – a "cos a + m a.""" sin a + n (n − 1) a” cos a — &c.. . . . . . . . -- c.
A series which will be limited when n is a positive integer.
d a si • * a
Eacample 3. Let X da = asºn: We shall again integrate by parts, but we shall begin with the factor
#. We shall find
d a sin r sin a 1 d a cosa.
− = — ;-R-E-H → M →H,
a" (n − 1) a n — 1 apº-1
ſº- COS tº l ſº
wº-1 T (n- 2) r"-* n – 2 an–2 2
&c. &c. &c.
and by substitution,
d a si e sin a -
ſ *** = — SI IT ſº 1 - COS ſº + ar - - &c.
cr" (m – 1) a "- (n − 1) (n − 2) a "-" ' (n − 1) (n − 2) (n − 3) a "Tº
If n be an integer, this series will have its nº term infinite, and, consequently, can be of no use to represent
d a sin ar,
the integral. The integration by parts shows, however, that the integral of —- may be made to depend
upon that of SRI) , n being an integer. If we substitute in this last expression for sin a its value
e” V-1– e-z V-1 d a sin a e” d a
2 vL i ' it will appear that the transcendant ſ Jº does not essentially differ from ſ F-, or
— l
d a
la
Eacample 4. Let X da = e “d a (sin a Y”. The integration by parts will succeed here, if m be an integer.
We shall find
826 I N T E G R A L C A LC U L U.S.
Integral - .* 97.8 1 ... c.: †, 777, gº e tº as Part ll.
Calculus. ſe” dr (sin r) =+ (sin a y – ſe d a cosa (sin wy"-", Q g
• z e l l l غ tº in-2 tº º
ſe” dr cos w (sin wyr-1 = Taº e” cos w (sin r)*-* — Ta Jºe d a { (m – 1) (cosa)* (sin w\"-" — (sin r)" ; ,
writing in this last integral for (cos a)* its value 1 — (sin a Y”, and substituting the value of
ſ e” d a cos a (sin a Y”-1, which will arise, in the equation above, ſ e” d a (sina)" will then be in both sides,
taking its value we shall find -
... sº, e” (sin a "-" (a sin a – n cosa) m (m – 1) ſ.a. !-- ~\mi-
ſe” d a (sin wy" = a? -- m3 a? -- nº e” dir (sima)"-". • .
The integral contained in the right side of this equation disappears when m = 1, and when m = 0, con-
sequently the integral of e^* d a (sin a y” is known in those two cases; and since the above equation shows that
this integral may always be reduced to one of these two cases when m is an integer, we may conclude that the
proposed differential can always be integrated in that supposition.
Example 5. Let X da = e^* d a (sin ar)" (cosa)". Here it is necessary to recollect that when m and
7, are integers, a series of terms, such as sin b ar, or cos ca, may be substituted for such an expression as
(sin a Y” (cos wy". The required integration will therefore be reduced to integrate differential expressions of the
form e^* da sin b w, or eº dº cos ca, which will be effected in the manner indicated in the last example.
d & tº te wº
Erample 6. Let Xd r = ziº ac' This example is very remarkable by the reductions it presents, and
the various manners to express the integral. An algebraical form may be given to the differential by assuming
I — w?
cos a = y, but to avoid radicals it will be simpler to make cos w = 3/ . Then we shall have da = 2dy.
d 1 + y? 1 + y”
da, 2 d y T -
d 3. -: g \º e º & e tº g º . tº 2
and consequently a T5 cost ºf a T5 + (a — b) y” Comparing this last differential with that integrated
Erample 2. (74), we find, immediately, the two following expressions for the integral,
2 d I — b) v — b? — a”
3/ --- (a ) y – wº ( a”) -- c, and
a + b + (a — b) y? W (b% — a”) (a — b) y + v (b? — a”)
2 d y 2 (a — b) y
... = tan" — — g
Jiříž-5, A/(a” — bº) alſ] jºij tº
e I — y” M (1 — cos ry *
But since cos a = -—ºr, h * — sº — e.
UIU S1 IT S º 1 + y” we nave y M (1 + cosa.) tan 2
Substituting these values, we find -
da, ſºmeºmº l , Mº i o) G + cos º-F ſº-ºº-ºoºº to
a + b cosa T v(b" – a”) V (5 + a) (1 + cosa) – V (b - a) (T-cos r)
Multiplying both terms of the fraction under the sign l, by the numerator, we shall have
da, &=ºm 1 . , *-i- a cost-Hsina V (º' - a) + .
a + b cos a T (V b”— a”) a + b cosa. * >
We shall have also by the substitution of tan . to y in the second of the two first values obtained for the
integral,
da, 2 a y (a — b)
-: ** *-*-4 tank tº
Jºãº. A/(a” — bº) tan w/ (a + b) an ºr + c
If for tan , w its value be substituted, the value of the integral becomes
C - 2 v/ (a — b) (1 — cos ar)
— t - 1 c
A/ (a” — b%) a. Il y (a + b) (1 + cos rj -- c
2 k
But twice the arc whose tangent = k, equal the arc whose tangent = 1 — k” therefore
da: l sin a y (a” — b%) I b -- a cos a
— — = — t = 1 :- —- * — ©
ſ:#; M (a” — bº) a]] b -- a cosa, + c V&E” Hi-Ho
0
These various values of the integral become O' when a s—: b.
In that case, we can integrate without any difficulty. We find
ſ da: _ ! ſ d r =}| ſ da: = 1 . *
a + a cosa T a J 1 + cost T a ( i) : an g + c.
2 cos :
I N T E G R A L C A L C U L U. S. 827
Integral dr (a' + b” cosa.)
Calculus. Erample 7. Let X da = (a + b cos r).” " We shall make use here of an artifice we have already had ++
occasion to employ. We shall assume * -
da (a' + b'cos a) *mº A sin a da, (B + C cos a)
(a + b cost)" T (a + b cos r)"-" -- (a + b cosa.)"
A, B, C being three unknown constant quantities which are to be determined so as to satisfy this equation.
If we differentiate both sides of this equation, and then divide by day, we shall find
a' + b'cos w = A cos w (a + b cosa) + (m – 1) A b (sin ry”-- (B + C cos aj (a + b cos r). *
Developing, and putting (sin a yº instead of 1 – (cos a)”, we shall have
(m – 1) A b + A a cos a -- A b (cos ar)* = 0.
+ B a + B b — (m – 1) A b
+ C a + C b
tºp a' – bº
Making equal to nothing the coefficients of similar terms, we shall get three equations, in which the coefficients
A, B, C enter only in the first degree, and from which we shall obtain the following values,
A = a b' – b aſ B = **-** _ (n − 2) (a b' – b a')
(n − 1) (a” — bº) ' T a” — bº " - TOTT) (ºr 5) '
and consequently
da (a' -- b'cos *) — (a b' – b a') sin a I
(a + b cos wy" T (n-1) (a, -bº) (a + b cosa)". + (n − 1) (a” — b”)
ſt da (n − 1) (a a' – b b) + (n − 2) (a b' – b a') cos r_
(a + b cosa)”-
By means of this formula, if n be an integer, the required integration will be reduced to that of a differential of
da (p + q cos r)
a + b cosa.
da (p + q cosa) q b p – a q – ? **ſ da:
ſ a + b cos a cºmme as {:}+;###5}=# = + b a + b cosa. "
(88.) Differential expressions containing the circular functions sin a, cos" a., &c. can also be integrated in
a few cases. The means by which the integrals are obtained are nearly the same as those which have been used
with functions of sines and cosines, &c. We shall, therefore, show simply upon some general examples, including
most of the formulae for which the integration may be completed, which are the substitutions and transformations
most likely to succeed.
the form - ; and this presents no difficulty, for we easily get
Example 1. Let the differential be X d a sin", r, and letſ X da = X, ; then, integrating by parts, we shall
{} X º
findſxas in-a-x, sin". -ſ i da
If, therefore, X, be an algebraical function, the integration
(1 — wº)
of X d a sin" aſ is reduced to that of an algebraical function. Let X = a ", for instance, we shall have
º *#1 . I a "+1 d ºr
ſz" dr sin-1 r = m + 1 sin" aſ – m + 1.J V (1 — wº) ' and when m is an integer this last integral is
obtained, as in Example 4. (83).
In a similar manner we shall have
wº-Fi l a"-Fi d ºr
a" d ºr tan", a s— tan" ºr ~ te
ſ º n + 1 a]]T * * Hiſ H.
g tº arº d aſ e
Erample 2. Let the differential be w(I-29 sin" ar. We have found before
* = tº
a:3 d ºr l 1.2
V (ITF) = ~~~ (; as + #) W (1 - wº); hence, integrating by parts, we shall have
3.5 d ? in "1 ºn - ! ... , 1.2 l 1.2
Ji-a; sin” . = - # * +}), a -º), ºr's iſ: 24 H. da,
and reducing o
a 9 d ºr sin-" l I. 2 ** 2 a.
*---------> I :- &ºº, *m. Q | o e ę
828 I N T E G R A L C A L C U L U.S.
Integral te a 4 0 ºr tº Part II.
Calculus. Example 3. Let the differential be V (I-72) sin-1 r. \-V-2
~~~ 4 d M (1 – a 1 - 1.3 1.3
— — — asses r" - 1 — r" -T sin" +,
We have foulid M (1 — wº) T (; *+ ai J.W. ) + 3.
hence we shall find by integration by parts
a 4 da: * ſ l 1.3 1.3 ... } : ... - 1
- 1 * — - i. 3 *- º 2) - — Si Il I J. S|[l tº -
Ji-Fi-sin'" |(} 2. +}} ), a *) - a .4 +
l I .3 1.3 d a . l
-- *3 tº- da — — — *1 a ,
ſ(; * + 2.4 •) * - 5.1 ſq ==) ***)
and consequently by reduction
a:4 da: tº { l l. 3 3 . }: l 3
===s**s sus-sº-sa • I -> = tºms S sm ºmºm, — a 2 e-me - 1 - 1 4. tºm-º. gº e
M (1 — wº) SIIl T * @ (; , ; : ), a a;2) is sin a sin * + -ī; a + TE + c
Erample 4. We shall take for the last example the differential da (sin w)". Integrating by parts, we shall
have successively
d
ſd w (sin-'a)" = a (sin- a)" — m +++ (sin- +)--,
M (1 — a ")
º (sin- a)"-" = — y (1 — wº) (sin", w)*-* + m — I ſaw (sin-'a)”,
&c. &c.
and by substitution
iſ d x (sin-1 r)" = a (sin- a)" + m / (I — wº) (sin- a)"-" — m (m – 1) a (sin- a)"-" — m (m – 1) (m. – 2)
w/ (1 – a ") (sin-'a)” + &c.
a series which will be limited when m is a positive integer.
(89.) By means of series it is always possible to represent the value of the integral of a differential
expression; and these, especially when none of the preceding rules can be applied, may sometimes be used with
advantage. -
From the theorem of Taylor, it obviously results that if we designate by y the integral of X dr, and by
y, the value of y when in it a is changed into ar, + h, we shall have
d X h” dº X }.3
y = y + X h –– d a 1.2 -- d a 3 l. 2.3
If in this series we change h into — ar, y, will become an arbitrary constant c equal to the value of the integral
corresponding to a = 0, and by writing y in the left side of the equation we shall have
d X are d” X as
d a 1.2 d a 1.2.3
This series has been given for the first time by Jean Bernouilli, and it is known under the name of the series
of Bernouilli. It may be obtained by applying to the differential X d v, the process of integration by parts.
We shall have successively
+ &c. . . . (a).
w = ſ. X da = c + X w. — &c. . . . (b).
d X d X d X aº d” X w8 d 4.
/xas -xx-ſ}. ar. = . * dº = i. T.; – ſº- TT. 2
d” X *d a d'Y a 3 ſº a 3 day &c
J. : T. 2 " ... • T. 2.3 T.J. J. T. 3.3 “
and by substitution
d X a." dº X a 3 d" X &n d ºr
* = X w. — - - - - - --— . . . . . . . . . . e
f X da * - i. i. 3 tº T.; +..] "J.T I.3.In
The arbitrary constant being included in the last integral.
Another developement of the integral may be derived from the series of Taylor. If in the above equation (a)
we suppose a = 0, and afterwards change a' into h, we shall have in representing by Zo, Z, , Zs, &c. the values
of y, X, #. #} , &c. corresponding to a = 0,
v=/Xar-z,--z, ž-+ z ř; + z ; := +&c,...
l "I . 2 * I. 2.3
* gº º g ge ... d X dº X
This series has the disadvantage of being only applicable when none of the quantities da’ dºr?’ &c. becomes
infinite on the supposition of a = 0.
I N T E G R A L C A L C U L U. S. 829
Integral (90.) It will always be easy to find a developement of the value of the integral of any differential X d ar, Part II.
Calculus. whenever we are able to transform the function X into a series of terms, each of which can be integrated. S-M-7
S-N-" This will be better elucidated by some examples. We shall begin with one or two differentials which we have
already integrated, in order to be able to compare together the results of various methods.
Example 1. Let #; be the proposed driential Here the function H is easily developed accord-
ing to the powers of w. We have
I l
––– = *-* +* – t + &c.
a + æ (Z a? a? aft
ànd eonsequently
da, l J. gº a:3 } Jº a 2 ars **
# = ſa, (;---, -}. &c. & = - + — — — — — &c. -- c.
a + 1, a" a T 2 a. 3 a 3 4 aft
Comparing this last series with one of those obtained in (27) we see that it is equal to l (r. -- a) - la. Hence
d
ſºQ, #: = l (a + a) – l a + c, or simply l (a + a) + c, including – l a in the constant. A result which if
identical with the known value of –tº– -
a + a
Eacample 2. Let d a be the proposed differential. We can develope I = (l *) ; b
the binomial theorem. We get
I l 1. 3 1.3.5 1. 3. 5. 7
— tº: — a " + – a 4 + – a " + -— º
zā-āj = 1 + 3 +++++ ###" + š, "--&e
da: * I as 1.3 ºb l. 3. 5 aſ
and Zi-ry = i + 3 g + 3 + 3 + 3 + 3 + + &c. + c.
d a
k that I — = sin"' tl
But we know tha A/ (1 — a ") sin"' a' + c, consesluently
e Jº I q.3 1. 3 a.5 1. 3. 5 a.7
- 1 -* — gms mºsºme mº-m ºmº- gºmºsºsºmsº mºsºme : \, g, G
sin" w = ++ = a + 3 + ··· + i + + &e + c
We arrive thus, in a very simple manner, to the developement of sin” a and by similar means we might obtain
those of cos" ar, tan" ar, &c., and generally of all those functions, the differential coefficient of which may be
developed according to the powers of the variable.
Erample 3. We have found that º = 1 (1 + V (1 + wº)) + c, We may easily get the deve-
lopement of this logarithm, according to the powers of a. For we have
l *** - I 1.3 l. 3. 5
— tº = 1 — — a " + – a 4 — — **=p tº
yū Hay = (1+*) 3 * + air – if r + &c
d an 1 at 1.3 as 1.3.5 aſ
Therefore yāi-j = 164 vote)) = *-ā ā-4 ; ; ; – ###-4 & 1 &
It is especially when the integral cannot be obtained under a finite form, that it may be useful to find its
developement. We shall take for the following examples differentials which cannot be integrated by the rules
previously given.
Example 4. In this example, the integral of each term of the developement of the differential will be composed
da V (1 – e’aº)
of several terms. The proposed differential is
M (1 — wº)
T 1. 1 1. 1.3 -
W — e” ºr”) = - -- e” r" -º-º-º: sºmº smºsºmsºmº e
e have first M (1 — e” wº) = 1 ge +gier a-Herº + &c
d a y (1 – e' a') iſ: da: { 1 , , 1.1 1. 1.3 }
— —r— * — $ 1 – =- as-s-sºº, — — e” a tº
and w/ (1 – a ") w/ (1 — alº) l 2 €” ºr + 3 + er' 2. 4.6 * + &c. &
each term of which may be integrated by means of the formulae given in Erample 4. (83.) Substituting these
integrals we shall find
WOL. I. 5 P
830 I N T E G R A L C A L C U L U. S.
Integral da V (1 – **) - ...-, Part II.
Calculus. W (ſ — as) = Sln ar, \-N-
I e” l a V (1 — wº) — 1 sin" *}
+ 3 2 2 2
. 1 { l l. 3 1.3 . }
* = sms - - - - 1 — * =e • 1
+ 3 −1 & (; * + 2.4 •), *) – #1 sin” r},
l, 1.3 { 1 1.5 1. 3.5 1. 3.5 }
mammºn-mº- º' Tº *-* . — arº –– —— ſº — — — sin" ,
+++ e (. *+ i + + 2. In •), a *) – gº sin" + 3,
+ &c. + c
1:
Example 5. We shall take for this example the new transcendant ſºa” d a We have found (26) that
JC
! a (la)? (la)*
* = 1 —- – -—-º- ºrº -- ——º- nº e
Q, ++++ -ij-, ++...+ x + &c
a” d a {} ! a (la)" (! a)* ..., }
Consequently ſ z- = ſ dw dº? +++++ - + -ā- + &c. -,
*_ la (la)* as (la)* as (! a)' at
= 1n + + r + +3= ++++ -ā--- T.; H + + &c. -- c.
If we substitute e for a, this formula becomes
e” d an I gº I q’s l **
f + = a + r +++ -ā- + T = + + H++ + &c. -- c.
e” d d
and making a = ly, in which cueſ tº tº =ſ+. we find
d y 1 (ly)* 1 (ly)* 1 (1 y)*
—?– = l l l *m ºmºmºnº-esº-m- &c. e
l y y -- ly-- H -- +H= -4–4. Tsai =#|- + &c. 4: c
a d
Example 6. Several series may be obtained for ſº tº By developing a” according to the powers of ar,
da: g e g a" da: tº p
the value of ſº would be expressed in a series of integrals of the form TI. By successive inte-
- . – 12
gration by parts and substitutions, the following developement may easily be found
*a* d a ..., I l 1 .. 2 1. 2. 3
J T-I = a {{= ºia T (Tºday tº Hº), T (IHºwa). T &c.} + c.
To find a series arranged according to the powers of a, we shall observe that
l = 1 + a + a” + a " + &c. and a' = 1 + a la + a" (la)* a" (la)* -- &c.
I — a l I .2 1. 2. 3
these being multiplied will give a product of the form
A + B a + Caº -i-Da"-- &c.
in which
ºs-_º sº ! a * -a- ! a (la)? &= -º (la)* (la)*
A = i, B = 1 ++, c = 1 +++++, D = 1 + la + -ā- + i + + &c.
Hence
a" d ºr l a \ a,” t a #) 4:3 la (la)* , (la)* \ r"
ſº = 1 + ( : *)++( : I--- i. ++( ++++; +%); +&c. He
Erample 7. We shall take for the last example the differential a "d v. Applying to a " the known develope-
ment of a”, we find
flº ºn a la: m*a*(la)? mº a.º. (la)*
* = 1 + →---→---→
We have integrated (84) differentials of the form a "d a (la)". If we make use here of these formulae, we
shall have
* | 1 1
+ &c.
1 1 a g 2 2 2. I
* gº *º-r ! m= mºm.
+ T : 5 m *(q.) ă (; ) + -ā- )
| 1 3 3, 2 3. 2. 1
—— — on 3 9 – — 2 *-º-º-º-º-º: *m.
+ 1.2.3 4 713 a.4 (a ſt) i (la) + 42 (la) 43 )
+. &c. -- c.
I N T E G R A L C A L C U L U. S. 831
Integral Or if the terms are arranged according to the powers of (la). ſ Part II.
++ ſ-a---( 1 — º: *::: -****) S-N-
+ &c. + c.
(91.) When a differential X da is decomposed into an infinite series of terms, which we shall represent
generally by
(A + Al Y + A, Yº + A, Yº -- &c.) Z da.
In which Y and Z are two functions of w. The integration of each term separately may be avoided when the
Z da;
di Us l *-** & & ©
ifferentia Tay can be integrated
Let U be this integral, and let its developement according to the powers of a be
a' V -- a V, + a” V, -j- aº V, -i- &c.
developed according to the powers of the same letter gives
Z da (1 + a Y -- a” Y” + aº Ya + &c.)
The differential
Z d a
1 — a Y
Therefore we must have
a0 V -- al V, -- as V, -i- &c. = ſ. Z da -- aſ ZY da + a ſ ZYº da + &c.,
and, consequently,
V = ſ Zda, º V, as ſz Y dr, V, e ſ ZYºda, &c.
Hence
ſix da = ſ Z da (A + A Y + A, Y′ + A, Yº + &c.) = A V-1-A, V, -í-A, V, + &c. -- c.
Thus after having integrated the differential #y
it will be sufficient to substitute, in this series, for the successive powers of a, the coefficients A, A, A, &c. of the
differential (A + Al Y -j- A, Y′ + A, Yº + &c.) Z da to obtain its integral.
, and developed the integral according to the powers of a,
* @ * d
Let us apply this to the differential #. a particular case of the differential # £
integrated in Erample 6. (90). We have
which we have already
Jº 22 a 3
3. -: l *==== & tº
€ +++ H+ Hi + &c -
e” d ºr JC arº Q8 a;+ da:
C H - 1 e-ºm *==s *ºmºmºmºsºm-º: *=-º-º-º-º-º-º-º-ºsmº g º &-
&l I) ( ſº: ſ(i++++++ IāT3 + 1.3.3.4 +&c)H
But instead of integrating each differential term of this series, we shall, according to the preceding remark,
l integrate first da:
I – a “*” (1 — a (l − a r)
d an i
ſº-H = H, (10 – a d-to-o: + c.
To develope this according to the powers of a, we have
observing that here Y = a, and Z =
We easily get
l 2 - 2 8
H = 1 +a+a+ c + &c., and 10 -a, - - *-*.* -** – &c.
Consequently the developement ofſ 1 — jä — a wy’ according to the powers of a, will be
– l (1 — w)
–– a {-1 ( – ) – #}
an?
+ º-o-o-; -;}
I 2
*E aſs a's
+ e{-10 – * = **m Hº-m ºr a s
(l ( – ) - H --, -āj
+ &c. . + c *
5 p 2
832 I'N T E G R A L C A L C U L U. S.
Integral And it is now sufficient to substitute, in this series, to the successive powers of a, the coefficients of the powers Part II.
Calculus of a in the developement
*} º gº act
1 + i + Tºa -- T., 3 + T = Ti-H &c.
and we shall find
* d
** = – 1 (1 – 3)
1 —
I a
++|- a -o-º:
1 22
+H:{-to-o-; – )
l tº gº q:3
+H={-to-o-º-, -}
+ &c. + C.
Or by putting for l (1 — a J its developement, arranging then the terms according to the powers of r,
e” d a tº 1 N ºr? I 1 N ºb } l I a 4
— tº — emºs amº-a I + — — — — — 1 + - - - – ------ || – te g
ſº ~ ſº #4 (4. #) 2 +( + T + 1.3)3 +( + T + H+++) 1 + &c. 4 e
A result agreeing with that obtained in Example 6. (90.)
(92.) In the applications of the Integral Calculus, it is not, in most cases, under the general and undeter-
minate form that we have obtained them, that the integrals of differential expressions are required. What is
wanted generally, is the difference of the values assumed by the integral, such as we have found it, when for
the variable two particular values are successively substituted. In taking this difference the arbitrary constant
disappears, and a result is obtained in which nothing remains undeterminate. This result is called a definite
integral, and the two quantities substituted for the variable, are the limits of the integral. Indefinite integrals,
on the contrary, are like those we have hitherto considered, in which the variable and the constant remain
m+1
Q}
undeterminate. Thus we have found that the general or indefinite integral of a "d a was m + 1 + c ; the defi-
nite integral of the same differential between the limits a and b will be the difference of the values
# + c, # + c, of the general integral corresponding to a 2- a and r = b, and is therefore equal to
am-Fi — bºn-Fl
7m + 1
To designate a definite integral the sign ſ is still used, and the two limits are placed by the side of it, the
one corresponding to the value of the integral which is substracted below, and the other above. Thus we
have
am+1 -º-º: bºn-Fl
fºr d r = 7m + 1
In the same manner we shall have
* G \, a ` m+1 l da: Tr
— ``. M, — = e” (e — 1 —H----, - — .
ſ. Q? (#).ſ. e” da = e” (e ſ. M (1 — wº) 4
(93.) When the indefinite integral is known, the determination of the value of the definite integral presents
no difficulty, since, to find it, it is sufficient to take the difference between the values of the indefinite integral
corresponding to the two limits. But, in many cases, the value of the definite integral may be obtained,
although that of the indefinite integral cannot. These determinations form one of the most important parts of
the Integral Calculus, and will be treated separately with all that relates to definite integrals. In this place we
shall limit ourselves to a few remarks which will be necessary to understand the analytical and geometrical
applications of the Differential and Integral Calculus.
(94.) Let X da be a differential of which it is required to find the definite integral between the limits a and b.
Let the indefinite integral be represented by f(r), and let a - b = h. We shall have by Taylor's theorem
h d X hº d? X h9
f(t+h) = f(r)+x++ i. i*i; + + H++&c.
if we make in both sides of this equation a = b, and if we designate by Y', Y", Y", &c. the values assumed
d X d2 X
y X, d a d rº’ &c. in that supposition, we shall get
h2 h9
f(5 + h)=f(0) = f(b) + y h 4 Y"; a + Y" Hara F &c.
and consequently
h2 hº
gº gº º º TV — V’ Žf f//
j(a) f(0) = ſ. X d w = Y' h + Y i.;4 Y Tālā + &c.
I N T E G R A L C A LC U L U.S. 833
Integral - ... d X d2 X g wº ſº tº ſº Part II.
Cºlºs. and if none of the quantities X, dº ’ diº has become infinite by making in them a' = b, we have a series \-->
* T representing the value of the definite integral, which, if converging, may be used to find its approximate
value.
(95.) It is therefore necessary to examine in what cases the series of Taylor is converging, or more generally
to determine the limits of the series beginning with any term. We shall first demonstrate the following
proposition.
Every function U of a which vanishes for a = 0, and the first differential coefficient of which, designated by U',
neither becomes infinite, nor changes its sign, for any value of the variable between the two limits a = 0, and
r = b, is of the same sign as the differential coefficient, if b be positive, and of a contrary sign if b be negative.
Let b be divided into any number of equal parts represented by i, and let
- 0, U1, Us, Us, &c. Uo', 'U', Us', Uſ, &c.
be the values of U and U' corresponding to
a = 0, - i, < 2 i, E 3 i, &c.
By Taylor's theorem, we have for the developement of what U becomes when a + i is substituted for a,
d U/ $2 d? U/ 73
U –– U' i -- - - -— —H − — — — &c.
+ + 7. I. : "F a. Tº ºf
and since the first differential coefficient does not become infinite for any of the above values of r, we shall have
in substituting them successively in this series, and representing by Vo it, Vi i", V, iº, V, iº, the corresponding
values of the part which follows the two first terms, in the following equations
U, a U, i + i Wo,
U, -U, a U," i + i Vl,
U, - Us = Us! i + tº V,
+ Ua tº e Un- -- U'a- + ** Vn-1.
We must first observe that the exponent & is necessarily greater than one, and secondly, that since when
a = 0, U vanishes, and consequently, that when i = 0, U1, U2, Ua, &c. become also equal to nothing, none of
the quantities V, VI, Vs, &c. can become infinite in the supposition of i = 0. Hence by taking a value for i
sufficiently small, the second terms of the right sides of each of these equations may be made less than the first
terms in any proportion whatever; the signs of the quantities Uſ i + i Vo, U", i + i Vl, &c. will therefore be
the same as those of U, i, U," i., U.'i, &c. But we have supposed that U.’, UM, U.', &c. had all the same sign,
hence this is also the case with Uſ i, U," i, U, i, &c. and consequently with U1, Us– U1, Us–Us, &c. Therefore,
finally, the quantities U1, Us . . . . U, will have the same sign as the differential coefficient U", if i or b be positive,
and a contrary sign if b be negative.
(96.) We shall suppose now that in the series of Taylor a particular value has been substituted for r, but we
shall continue to represent the developement by
d at dº at h" dº nº h3
w = u + i k + z -- &c.
dº T. 2 " dra I. 2.3
The value of the series will then vary only with the value of h.
We have proved before that generally
d" u' d" w!
da" $ºmº d A”
these differential coefficients are functions of h, and vary accordingly with the value of that variable. Let m be
d" uſ d" w!
the least, and M the greatest value of drº T ſhº corresponding to the values of h between the limits h = 0,
Jº
and h = any constant quantity, so that
d” iſ d" ºf
M – , and — 7m,
iſ, *āh;
are functions of h which will remain positive for any value of h between these limits. These quantities are
respectively the first differential coefficients of
and consequently, of
dºn-1 w! d”-l 2. d"-l w/ dºn-l 7.
M h — (# T d ...) and HF – gi- m h,
n = 1
d a "-" does not contain h. But these new expressions vanish for h = 0, for then uſ = u, and we have
since
834 I N T E G R A L C A L C U L U. S.
Integral
Calculus.
S-N-"
dºn-l 7/ dº-1 tº/
besides — = —-.
d h”-1 d a "-1
as their first differential coefficients respectively, that is to say, both positive, for all the values of h between the
assigned limits. Again, they may be considered as the differential coefficients of
h2 d"-" tº d”-2 iſ d"-" tº d”-2 1/ d"-" ºf dº-i tº hº
* --> nº T n-2 sº 5-H h , all 770,
} , 2 d h”-2 d ºr d a "
. . Part II.
Therefore by the theorem demonstrated (95), these expressions are of the same sign ºl.
sa-mem===m ammº Gº- h — -
d h"-2 d p"-2 d rº-1 1 .. 2
After observing that these new expressions become nothing when h = 0, we shall conclude as before that they
remain positive for all the values of h between the assigned limits.
Proceeding with the same reasoning, we shall be able to prove that the two following quantities are both positive
between the same limits,
M h" w" — w d u. h. d” iſ h? d"-" at hº-1 and
1. 2. 3. . . . m. d ºr d a " I . 2 da"-" l. 2.... n – l /*
f d w dº tº hº d"-" at hº-l m h"
w" — u — – h – º – — . . . . . . . . . — — .
da, da" 1 .. 2 da"T" l. 2. . . . m — 1 l. 2. . . . m.
Let us now substitute for u' its value, given by the series of Taylor, and we shall find
M h” d" u h” d"-H at An-Fl
e--—-s-s-—-----es-s — — —- — &c. and
I.3.3. In 㺠I.3.3. In d a "+" l. 2. 3. . . . m + 1 C, all
H." d” }." dº-Fi - #74-1
-- 77? < QM, # + &c.
l. 2. 3. . . . m da" 1. 2. 3. . . . m ' d'a"+" l. 2. 3. . . . m + 1
M h” ºn h" tº gº g &
Therefore H-, and F-------—, are the limits between which are included the whole of that part of
1. 2. 3. . . . m. 1. 2. 3. . . . m.
the series of Taylor beginning with the (n + 1)" term ; or, in other words, remembering what M and m are
intended to represent, we conclude that when the series of Taylor is limited to the n first terms, the part
d” aſ d” ºt' s Tº e h”
neglected has for limits the greatest and least values of ... ſº º jirº multiplied by I. 2. 3. . . . m.
(97.) It is easy to infer from the preceding investigation, that, in the series of Taylor, a value may always be
assigned to h which will make any term greater than the sum of all those which follow it. For
d" it },” dº-Fi at hº-Fl
da" 1. 2. . . . m ' d a "+" l. 2. . . . m + 1
a g d" uſ 9 s h" & g \º &
being included between the greatest and the least value of dº' multiplied by T.I.' it is obvious that there
º g tº g º ºs 71,
+ &c.
ſº
must exist an intermediate value of this quantity, which being multiplied by T2. Tº will be precisely equal to
. 3. . . . 7t
d"u h" + dºn-Fi 75, .* j,”--l + &c.
da" 1.2 . . . . m d p"+ 1 . 2. . . . m + 1.
Let U, be this value, then we shall have exactly
d it d” iſ h? d”-1 1, Hº-1 U, h"
! — --- ºm-wº-
70, = u + i h ºf I. T. g . . . . . . . . + j = T2.-- [+H.
ſº * d” u Hº-1 w ſº # tº a gº
To find the value of h which will make d a "7" i. 2 Tv — 1 greater than the remainder of the series, it is
therefore sufficient to find that which will make that term greater than H.-E. and we shall clearly satisfy
this condition by taking
h ºn d”" w
< U, dr"-"
Thus to obtain the required value of h it will not be necessary to know the value of Ua, but simply any
d” aſ
quantity greater than it, for instance the greatest value of H.
When we are at liberty to take any value for the quantity h, and that none of the differential coefficients
become infinite for the particular value of w, the series of Taylor may always be rendered a converging series, since
each term may be made greater than all those which follow it.
In the series we have found (94) for ſº X dr, h has a determinate value equal to a - b, and therefore what
has just been stated cannot be applied; but we shall always be able to determine the limits of the error made
by taking only a limited number of terms of the series.
(98.) Before proceeding to investigate other series for the value of ſ*X d ar, we must briefly state that with
respect to the developement of a function of two or more variables, limits of the series may also be determined
I N T E G R A L C A L C U L U. S. 835
Integral Let u = f(x, y), w = f(x + h, y + k). For h and k put h t and k t, them u' may be considered as a func-, * *
Calculus, tion of t, and if we substitute instead of it t + a, the two following developements, which must be equal to each S-
S-N- other, are obtained:
d w a dº u a” dº w as
f == * =ººl tº-º-º-mºmº-º-º-
u'- u + i + + HE TE → ; ; Tiga-F &c.
| a /d at d u a” (dº u , , d” at d? /
= mº - dºmesºs - a -º- º =-a-t-ºss gºmºsº | | *s-s 2 — h k + -- k” —- &c.
º, wit (#####)++,(# *- High k + iy, k)+&c
If we take only n terms of the first, the sum of the remaining terms will have for limits the greatest and
n .../
#: multiplied by Taº In the same supposition, the limits of the remaining terms of
smallest value of
the second developement will therefore be the greatest and the least value of
d" w d”-1 at d” at
tº-dºº- h” n- I tº e º 'º ",
day" + n = h k –– jº"
multiplied byi gº m’ Making in this last result a = 1, we find the limits of the developement of
j (1 + h, y + k). º
(99.) Let us return now to the definite integral ſ," X da', if we suppose a - b = n i, n being an integer, by
taking it sufficiently large, i may be made less than any assigned quantity. Let Y, Yi, Y, &c. be the values of the
indefinite integral, corresponding to a = b, - b + i = b -- 2 i, &c., and Y', Yi', Ya', &c., Y/, Yı", Y,", &c.
Y”, Y/", Y,", &c. the values of X, ax d x.
- d a d a "
same manmer as in (94), the following equations:
ſº P s // 72 * *
Y, -Y +Y i + Y” H + X"Hi + &c.
&c. corresponding to the same values. Then we shall get, in the
73 &
La., H &c.
Y, = Y, -i-Y,' i + Y,"H. --Y,”
, , , , * 73
Y. = Y,+y/i-Y"H; + Y"Hi + &c.
p vº A ſº 72 ºpy 33
Y. == Yº- + YA-1 * + Y. I T2 —-Y., T.2, 3 + &c.
If we add all these equations, suppress the terms which would be common to the two sides of the sum, and
place Y on the left side, we find
v.-v-f-x a- o'+x+x+... + Y.-)
+ i. (Y" + Y,"-- Y," + . . . . -- Y.-). . . . . . . . (e)
+H (Y"+y,"+y"+ ... + Yº)
+ &c.
Instead of substituting successively for a the values, b, b + i, b + 2 i, &c., we might have followed an
inverted order, beginning with b + n i, b + (n − 1) i, &c.. . . . down to b. We find in this manner
Y. – Y = ſ.” X d w = i (Y/ -- Y,' +... . . . . . . . . . + Y,'
- ?? / |
—fi (y,” + y,"+ y^+. tº E → * * * * Y."). . . . . . . . (f)
S3
+ º (Y," + Y.” -- Y," + . . . . . . . . Y.”)
— &c.
The two series (e) and (f) are formed by the addition of a limited number of series, in each of which any
term may be made greater than the sum of all the following ones, by taking i sufficiently small. (97.) Hence
it is easy to infer that these two series will have the same property, and consequently, that they may be con-
sidered as converging series.
(100.) Another important consequence, relative to the equations (e) and (f), may be derived from the
theorem demonstrated (97.) It results from this proposition, not only that a value may always be assigned to i
which will make any term greater than the sum of all those which follow, but also, that by taking for i values
less and less than this, the remainder of the series may be rendered less than any assigned quantity, however
small. Hence, in the equations (e) and (f) the quantities in the left side are, at the same time, the sums of
836 I N T E G R A L C A L C U L U. S.
Integral the series; and with respect to the decreasing values of i, the limits of any limited number of their terms, Part II.
Calculus. beginning with the first. Therefore the definite integral ſ.” X dr, with respect to decreasing values of i, is S-a-’
S-a-' the limit of the first term of either of these series. But the quantities i y', i y", i y”, &c., of which the first
terms are composed, are the values of X d a corresponding to a = b, E b + i, = b + 2 i, &c. . . . and da = i,
consequently the definite integral of a differential expression Xd ar, taken between two limits a = b + m i and b,
may be considered, with respect to increasing values of n, or decreasing values of i, as the limit of the sum of
the values assumed by that differential when i is substituted for dar, and a successively replaced by the terms of
either of the two following arithmetical progressions: -
b, b -- i, b + 2 i. . . . . . b -- (n — ?) i.
b+ i, b + 2 i, b + 3 i...... b -- m i.
(101.) When it is known or assumed that for a particular value a = c the integral of a differential expression
vanishes, this value is said to be the origin of the integral. In that case the integral may be considered as the
definite integral between the limits w and c ; it may be represented by ſ... X dr, and all that has been said
hitherto, with respect to definite integrals, applies to it.
(102.) When instead of the first differential coefficient, it is that of a higher order that is known, the deter-
mination of the primitive function will require several integrations, and as many arbitrary constants will be
introduced.
Let X be the given differential coefficient, which we shall suppose to be of the n° order, and let y represent
the primitive function, then
d" y
++ = X.
Multiplying both sides by dr, and integrating, we get
d" 3/ dº-i Q/ * –
If we multiply again by d ar, and integrate, and we find
. dº-1 3/ d”-9 3/
ſº =# = ſtarſ xas; +o, + .
The same operation being repeated n times will give the value of y with n arbitrary constants.
A symmetrical form may be given to the value of y, by means of the integration by parts. We first observe
that d" y = X da", and consequently
d"-'y = ſix da", d"-ºy = ſ.ſ Xda' or ſex dr", d"-s y = ſ.ſº X da" = ſº X dr",
and 3/ = ſº X d ºp”.
We shall now examine the transformations which may be made upon
ſex d tº, ſº X drº, &c.
We find, according to the rules of integration by parts, and recollecting that da' is constant,
ſº X dr = ſ.ſ X da' = ſārſ X da = r ſ X da — ſa X dr;
by means of this value we shall have
ſex da' = ſil r ſ drſ X dr = ſ.r drſ X da – ſ d a ſa X dw.
But ſw drſ X de-, eſ: dº--ſex dr.
and ſda ſ r. X da = a ſm X da — ſwº X dr.
These values being substituted, we shall have after reduction
fºx de = Gºſz dº – 2 ſix d. 4 ſex dº).
By the same means, we shall find the value of ſ: X d arº, &c. Thus we have
fx da: = ſix dr,
fºx de – (a ſix da – ſº X da),
fºx de = f's Gºſz dº – 2 r ſax dº 4 ſex da),
I
ſix da = l. 2.3
&c. &c.
The law of these values is obvious, and we may, without difficulty, form, by analogy, the developement of
f. X da"; and we shall find -
(tº ſix dº – 3 tº ſº X dr H-3 a ſwº X dº -- ſº X dr),
I N T E G R A L C A L C U L U. S. 837
I I l * , - 2 fe
§. ſº X dr" = 1.2.3 ("-" ſix dr – (m — 1) r"-" ſº X da + (m 1) (m. 2) *-* frºx da – Part II
N-V-7 e fºº º . . . . n – 1 1. 2 \-> 2-’
- &c. ... + ſa"-" X dw),
the last term being + when n – l is an even number, and — when it is odd.
To prove that this value is exact, it will be sufficient to show that the law, according to which it has been
formed, is true, it being admitted that ſ * X. daº" will subsist for J’ * X da". For since it has been verified for
ſ 2 X daº, ſ * X da", it will, of course, be true for all the following orders.
We have ſ” X dr" = ſ d a ſ” X dr", substituting for ſ” X dr" the above developement, integrating by
parts each term, and uniting under the same coefficient similar integrals we find
H 71, m (n − 1)
f* x d ? - T. ( ſx dº – ; , ſex dº -- ***, * w"-" ſwº X dw......
I - 70, n (n − 1) -
n = 1 º unime –--------- . . . . . . . . 17. c
+ m a.ſa. xas) - Ta-r ( I + 1.2 + n)ſ. X d a
The coefficient of ſ a" X da is equal to (1 – 1)" + 1 = -El, and therefore the assumed law is verified for
ſºr x dºt.
There is another developement of ſ” X da", no term of which requires to be integrated. By applying to
J Xa" dw the integration by parts, we shall easily find
ſ X r" dr = X arº-Fl d X tº-H2 + d? X tº-Hs &c. -- c
T m + 1 dr (n + 1) (n − 2) " dº (n-ET) (n + 2) (n +3) C. -H. C.
If in this series we suppose successively m = 0, = 1, = 2,.... = n – 1, and substitute the values of ſ X da,
ſº X da; ſwº X da, &c. which will arise, in the value found above for ſº X d ar", we shall have, after
reduction,
* …+. " ("If D +,
ſ" x d fº = - X p" d X l + d? X 2 da X
* = I.T. - I. T.I.TI d r* 1.2. . . . . . m + 2 cl as
n (n + 1) (n + 2) rººs
l. 2. 3
l. 2. 3. . . . m + 3 -- &c.
To complete this developement, it is necessary to add to it the terms containing the arbitrary constants, which
are clearly
C ac"-1 C. tº 2 C, r"-a
tºmºmºmºmºsºms-- *-**-w —--—-4 - &c.
I.T. t I.E.L. 2 # 1.2.0 L 3 —H &c
or simply C r" + C, a "-" + C, a "-" —H &c. including the denominators in the constants C, C, C, &c.
(103.) We proceed now to the integration of differentials containing more than one variable.
With respect to functions of more than one variable, two cases may occur. In the first it may be required to
find the value of the primitive function, when one of the partial differential coefficients is given ; and in the
second to determine the primitive function when the complete differential is known.
The first case presents no greater difficulty than the integration of the differential coefficient of a function of
one variable. If it be the partial differential coefficient with respect to a that is given, then all the other variables
must be considered as constant, and the integration is to be performed by means of the preceding rules ; but
instead of adding an arbitrary constant, it will be an arbitrary function of all the other variables that will be added.
(104.) Let us next examine the second case. We have seen (35) that the differential of a function of several
variables, of three, for instance, is of the form
X d r + Y d y + Z dz,
X, Y, Z being respectively the partial differential coefficients of the function with respect to r, to y, and to z.
If in the given differential it happen that X contains neither y nor z, Y neither a nor 2, Z neither a nor y, then
the integration will present no difficulty, for we shall have obviously - -
ſ (X dw H-Y d y + Zd z) = ſ. X d r + ſ Y dy--ſ Z d 2 + c.
(105.) When the variables are mixed in the quantities X, Y, Z, &c. this method of integration cannot be
applied. To begin with the simplest case, let M da + N d y be the differential of a function of two variables, in
which M and N are each functions of the two variables a and y. If u represent the primitive function, then
d
** = M, and d w = N ;
dr dy
W OL. I. 5 Q
838 I N T E G R A L C A L C U L U. S.
Integral
Calculus.
\-y-Z
e dº at dº nº tº a 9
and because it has been proved (35) that da dy tºmmºns dy dº' the two quantities M and N must be such that
d M d N
dy T d a
Unless this condition be satisfied, M. da + N dy cannot be the result of the differentiation of a function of two
d M d N
variables. When the equation dy = -ſ. obtains, we shall find the integral in the following manner. Since
d
+ = M, we have u = 'ſ M da + Y, the integration being performed with respect to the variable a alone,
d
and Y being an arbitrary function of y. To determine the value of Y, we observe that * must be equal to
N, hence we must have
d'ſ M dw +* Y
dy dy T ‘’’
- d d Y - d
or, if we represent f M d a by v, #; +- dy = N, and consequently Y =/(N T j ;) d y + c.
Therefore we shall have
w = ſm as 4./(N- #; d y + c.
We may derive from this result the condition of integrability, already determined. It is obvious, that M
d -
being the partial differential coefficient of u with respect to w, N — #; must be independent of a, therefore
its differential coefficient with respect to that variable must be equal to nothing; that is to say, we must have
d d v
d N d? Q, d N. d? v da, d v
— — — — —- = 0 — = —-—- sº —- in or — = heref
da, d y d a , or T.I. d y day d y , but v being equal to ſ M dr, d a M, therefore
d N d M * * * * tº a Q g
dº. T dy” which is the condition previously found.
(106.) Differentials of functions of more than two variables may be integrated by generalizing the rules
already given. It will be sufficient to consider the case of a function of three variables; and then it will be
easy to extend the same method to any other number.
Let M da + N d y + P d 2 be the proposed differential, M, N, P being functions of w, y, and 2, and let u
represent the primitive function. Then
d u d u d at
d a = M, d y d c
= P,
and consequently unless we have
d M dN d M. d P C. N. d 2
dy T d a ' d'z T d a d 2 T dy’
M d a + N d y + Pd 2 cannot be the differential of a function of three variables. But if these conditions are
fulfilled. then the integral may readily be obtained. In that hypothesis each of the three quantities M. d a +
N d y, M. d a + P d z, N d y + Pd z, represents a complete differential of u, corresponding respectively to the
supposition of 2, y, ac, being considered as constant. Any one of them may therefore be integrated by the pre-
ceding rule. Let v be the integral of M d a + N dy, for instance, we shall have
ſ (M da -- N d y + P d z) = v - Z,
Z being a function of 2 alone, which must be determined by the condition that the partial differential coefficient
d d
of v - Z, with respect to 2 shall be equal to P, that is P = # + #. From this last equation we find
2. 2
d 2 d v d v
− = P — — . - sºma gº-seº g
d z # and Z ſ(p +: )d 2 + 6
Hence it is necessary, in order to be able to integrate the proposed differential, that P – #: should contain
neither a nor y, which condition we shall express by making the differential coefficient of P – # with
2
respect to either of these variables equal to nothing; thus we must have
d P d2 v 0 d d P d2 v
-º-º: * *-º-º-º-º-me tº U, I\ — — — sº
da: da d 2 a. d y d y dz
Part II.
\-v-”
I N T E G R A L C A LC U L U.S. 839
Integral
Calculus,
N-V-Z
But dº v_ _ _d M nd dº v_ _ d N
Ul III: = Hi-, and TT = H .
d P G M d P d N
therefore 'dar T TG 2 ° and dy – i.
These two equations, together with the supposition we have already made that M da + N d y was a complete
d M d N
differential ; or, which is the same thing, that 7- = -i-, are precisely the expression of the conditions
$/ ſº
which should be fulfilled in order that M da -- N d y + P dz might be the differential of a function of three
variables; therefore, when they are satisfied, the integral may always be found.
After having proved that the conditions of integrability are fulfilled, the value of the primitive function may
be obtained by integrating with respect to a alone the term M da ; with respect to y the terms of N dy,
which do not contain a ; and, finally, with respect to z the terms of z, which contain neither a nor y.
It is obvious, now, that the number of conditions of integrability relative to differentials of n variables is
**** , and that to obtain the integral of such a differential, it will be sufficient to determine, first, the
integral of the differential, considering one of the variables as a constant, and to add to it an arbitrary function
of that variable, which will be determined by the method already used. -
(107.) The differentials of an order higher than the first, may be considered, with respect to functions of
several variables, as well as with respect to functions of one variable, as the first differentials of the differentials
of the order immediately preceding. Hence when the differential of any order m is given, and that it is proposed
to pass from it to the differential of the order n – 1, the conditions of integrability we have found above, and
the methods of integration which have been explained, may still be used. However, they require some modifi-
cations, which will be sufficiently elucidated by the following remarks upon a differential of the second order of
a function of two variables.
Let Q d tº + R da d y + S d y? be the proposed differential. We must observe, first, that the term R d a dy
is the aggregate of two terms, the one resulting from a differentiation with respect to a, and the other from a
differentiation with respect to y. In order to put the proposed differential under the form M d a + N dy, we
shall assume R = R' + R', and we shall be able then to write it in the following manner,
(Q d a + R' dy) d a -- (R" da + S dy) dy.
The condition expressing that this is the differential of a differential of the first order will be
d : (Q d r + R'd y) ... d (R"d r + S d y),
d y tºmºs da, 9
d Q d R/ d R' d S
#; as + -ā- a y = -ā- as ++.
and because a and y are variables independent of each other, and consequently da and d y are in the same
case, this equation will give •
or developing
d y,
d Q d R" d R/ d S
d y T d a ’ d y T d a
fº f
d R' _d R_ _ d R. Substituting this value, we find
da, da, d a
d Q d R. d R' d R' d S
dy T dr d a ' d'y T d a
d2 R! &
If we differentiate the first equation with respect to w, and the second with respect to y, dad y will be in
But R” = R – R', and consequently
*º-e
both equations, and by eliminating it, we shall have
d? Q –– dº S dº R.
d y” daº T d a d y'
Such is the condition to be fulfilled, in order that Q d'aº –– R. da d y + S d y” should be the differential of a
differential of the first order. Similar means would lead to the conditions relative to higher orders.
When the above condition is satisfied, the first integral of Q d a " + R. da d y + S dy” is readily obtained.
We know that it must be of the form Ud a + V dy, therefore the term Q daº must be the differential of U d a
taken with respect to r, and consequently U = ſq d w. In the same manner V must be equal to f S d y.
Thus ſ (Q dº + R. da d y + S dy") = da ſ Q d a + dy'ſ S dy.
We ha w to verify that this integral is exact, when d.º.º. d° S di R. For that ust
We IMO ıry Is Integral is exact, d y” ... - Tº dºſ' Or at we must prove
that its complete differential is equal to Q drº + R da d y + S d y”. By differentiating, we obtain Q d r*-ī-
Part II.
3 Q 2
840 I N T E G R A L C A L C U L U.S.
Integral *. - d
d. S d y” + d a d y ( ..ſ Q d a +- d. ſ S d y ) It will be sufficient to show, therefore, that - Part II
d . d d : ſ Sd
R = *...**Q d a + *...** Q
Differentiating this equation first with respect to a and then with respect to y, we shall find
d’ R _ dº Q + (l? S
d a d y T dy” d as '
which is precisely the condition of integrability. -
It may be also observed, that since the first integral of Q d a " + R da d y + S dy” is dr ſoda + dy'ſ S dy,
the conditions to be fulfilled, in order that Q d tº + R da d y + S dy” should be the second differential of a
function of a and y, are
dº Q d’s d" R and d.ſº dº – d.ſs dy
d y” d tº T d a d y ' d y T da º
Or differentiating the last equation twice, first with respect to w, and afterwards with respect to y, the two
conditions will become
- d° Q 1 d2 R. d? Q gº dź S
dy; T 2 dad y dy; T dº
These conditions are verified, for instance, for the differential y” da' + 4 + y da d y + tº dy”, and we find
by means of the preceding rules, that the integral is a y” without the constants.
(108.) We have proved (41) that if n be the sum of the exponents in each term of a homogeneous function
at of the variables w, y, z, &c. then .
d u d w d u
gº w -- ây 9 + d 2 + &c.
This theorem may sometimes facilitate the integration of the complete differential of a function of several
variables. It follows from the rules given for the differentiation of algebraical functions, that the differentials of
homogeneous functions are themselves homogeneous. Hence if a given differential M da -- N d y + &c. be
homogeneous, we may infer that the integral is in the same case. If, therefore, M da + N d y + &c. fulfil
the conditions of integrability, if u represent the integral, and m the sum of the exponents in each of its terms,
we shall have gy
7, 7A, st
d d
7m u = M a + N2) -- &c. since M = **, N = +, &c.
da, d y
This value of m w proves also that m = n + 1, n being the degree of the functions M, N, &c., consequently
Ma -- N y + &c.
m + 1 tº
This method of integration cannot be used when n = — 1, since then the denominator of the value of w
becomes nothing.
The relations which have been found (41) between a homogeneous function of several variables and the
partial differential coefficients of orders higher than the first, might also be used, in some cases, to find the
integrals of differentials of higher orders.
We shall now apply the rules which have been given for the integration of differentials of functions of several
variables to a few examples.
Example 1. Let the differential be (a" + a y + y”) da + (a” — a y + y”) dy.
Here M = a + a y + y”, N = a – a y + y”, therefore
w = ſ. M da + N d y + &c. =
d M d N
d y Jº 9, d. 2a – y
tnese two quantities are not equal, therefore (a + a y + y”) da + (tº — a y + y”) d y is not the differential of a
function of two variables.
Eacample 2. (a w -- b y + c) da + (b a + e y + f) d y.
hi d M d N
In this case a y T da T
We shall have the primitive function by integrating first (a r + b y + c) da, considering y as a constant, and
adding to it the integrals of the terms of (ba-H e y + f) d y which do not contain y. We shall find
2 2
ſa, + by + C) dz--(ºr +ey+ſ) dy=#|+ bºy 4. cz + 3 +fy + c.
d a dy y d a r dy
Example 3. au =#-F#-º
dº? y”
º
I N T E G R A L C A L C U L U.S. 841
Integral • – 1 3/ amº l gº - Part II.
*Calculus. We have *m. y - . . iN = — - Fº \-, -º
&
S-N-"
- d M. d. N I l I %) £
--→ - — º — — — — := _ ºn tº d - – ſº-º-º- Y,
dºy d a w? gº ' ſ M dw (; gº £ 3/ ºr +
Ç 3/
N = + &E , (; +4+x) * + 1 + 2*
- - a y - d y T y” a y’
hence ** = 0, Y = c, and ſ * +*-*-*)=; +4+.
d y $y @ º? y” y a
Example 4. du = (3a*-ī- 2 a. a y) da + (a aº + 3 y”) d y.
In this example the functions M and N are homogeneous and of the same degree n = 2, and moreover, the
condition of integrability is satisfied, for
d M d N
dy T da
= 2 a r, therefore ſ(3 w” + 2 a w y) da + (a a”-- 3 y”) d y =
(3 a " + 2 a. a y) r + (a a” + 3 y”) y
- 3
= * + a rºy + y + c.
Erample 5. – 9: (Wit?... a. --→ *(*# *), dy-Lºy (*-H 9 a.
a'ample du– #####, dº-Fiji, Hy av-Kºź, dź.
The conditions of integrability are satisfied, and M, N, are homogeneous functions the degree of which is one,
therefore we shall have
wesºo tº tre+º- a y z
2 (a + y + 2)” J T ~ + y + 2
(109.) When a function of several variables w, y, &c. is differentiated in the supposition that r, y, &c. are
functions of other variables, there arise differential expressions containing a, y, &c. dr, d y, &c. d’a, dº y, &c.
d" ar, d" y, &c. Reciprocally, when such a differential expression occurs, it may be required to determine the
function from the differentiation of which it is supposed to have resulted.
We shall consider the case of two variables only. Let U be any function of w, y, d w, dy, dº r, dº y . . . . . . . .
d" ar, d" y, and let U, be its integral, that is a function of the same kind, but containing neither d" a nor d” y : if
we make
+ C,
d a = ºri, dº a = a, dº w = a, tº sº E & ſº e s is
d y = yi, d’y = y, dº y = y, . . . . . . .
U, will become a function of the variables a, y, all, 91. . . . an 1, yn-1, and its complete differential will be
U dº U
d U1 = ****** * * * * * * * g tº-
is a y + #ay, ++, dy,+ © tº e º ºs e º gº + i. 29.- .
but d'U, = U, substituting also for d ar, d al., &c. their values all, as, &c. we shall have
U = **-*****, +…++.
++,+,+,+ © s tº G e º e a +#y.
If we differentiate U successively with respect to each of the variables a, y, .... a, y, we shall have first with
respect to a
d” U, d" U. d” U, d? U,
d U d tº * + i. º. Hää.”-H........+ H+.
da: T dº U, dº U, d? U, d? Ul
* 3rd,' " ii. 9-Fāzā," tº . . . . . . . + did.”
and by inverting the order of differentiation in each term, we may easily see that the right side of the equation
842 I N T E G R A L C A L C U L U. S.
Y d U d U . tº o
.. is the complete differential of d U.U. therefore − = d !. Let us differentiate now with respect to wi. we
CUlill Se da: da: da:
shall find
d U, d” U, dº U, dº U. . d” U,
d" U dr + a...', tà.”- i.” T ſº e º e º & J & * dr, d.”
dr, T dº U, d” U, dº Ul d" U. . .
| täzij% + i. i. 9, t d ... i.,9. * ' ' ' ' ' ' ' " i. i.”
inverting the order of the differentiations in every term in which U, is differentiated twice, we shall have
d U d U, dº U -
- J + d H, and we shall find in the same manner
da, d a da,
d U d U, d U,
da, T da', + d da,
d U d U, d U.
da, da, + d H 4's
d U d U, d U,
da-, dra- + d dra-
But when we come to a, = d" r, which is not in U1, since that function is only of the (n − 1)* order, we shall
have simpl d U d Ul
3.V * T *mºmºsºm- = gº
simply a £n dan-,
By differentiating with respect to y, y, &c. we should obtain similar results. . We may without difficulty
d
eliminate U, from these equations; for that it will be sufficient to subtract from the equation # = d. #,
e e d U d U d U º tº
the differential of da. :-- dº: -j- d da,º, then add the differential of the second order of the following equation,
I l
subtract the differential of the third order of the next, &c. &c. We shall find
d U d U d U d U
— — d. — * —— — * — . = 3.
d a ãº, t "ar *H,--& 0.
and in the same manner -
U
d'U – a 30 + ºr 40 – d.d. U
d ya d y
ſº-º-º-º-º-º: &c. := 0.
d y d y, + &c
3
We should have had as many similar equations as there were variables, if instead of two we had supposed any
other number. These equations will be verified whenever U is the differential of a function U1 of an order less by
one than U. If, therefore, we wish to ascertain whether a differential function, of the mº" order, be the differen-
tial of another function of the (n − 1)" order, we shall assume da = a, dº a = a, . . . . d y = yi, dº y = y, &c
. . . . d z = z, d” z = zs, &c., and then, the function being represented by U, we shall form the quantities
d U d U d U d U d U d U
---, - . . . . . . -, - . . . . . . H., H+, &c. &c.,
d a d ar, d y d y, d z d zi
and we shall substitute them in the equations we have found ; if they are not satisfied, we may safely conclude
that the function U is not the differential of a function of the (n − 1)* order.
Let us take for example the function a dº y – y dº w, it will be changed into a y, - y as = U, and we shall
have
d U d U d U
# = y, Hi + 0 +z=-y
d U d U d U
† = - “... Tº = 0, ± = ,
which give the following equations,
ya – dº y = 0, - a, dº a = 0;
these being satisfied, we may infer that a dº y — y dº a is the differential of a function of the first order; and it
is, in fact, the differential of a dy — y dºc. -
When U is of an order superior to the first, then it may be required to determine whether U, be the differen-
tial of a function Us of an order less by one, or in other words to determine whether U be the second differential
Part II.
N-V-2
I N T E G R A L C A L C U L U.S. 843
s da,
&
Integral of a function U. of an order less by two, &c. The equations which express these conditions may easily be
Calculus. , formed by means of what precedes. If U, be the result of the differentiation of Us, then the equation
d Ul d U, d U. a d U.
mº sºmºsºm sºmºn &c. = 0,
da, a #4 &# d'Hi + &e
8
limited to the differential re-i, and others similar with respect to the other variables, must be satisfied. The
values of the differential coefficients may be readily found in terms of the differential coefficients of U, by means
of the relations found before ; if we substitute them in the equation above, we shall find
d U d U o is d U a d'U
#–2 a #4 sai-44;
4
+ &c.
and we shall have similar equations for each of the other variables; all these joined to the equations which
express that U is the differential of a function U, must be satisfied, in order that U should be a second differen-
tial of a function Us. Similar considerations will prove that U, in order to be a third differential of a function
Us, must satisfy, besides the preceding, the equation
d U d U d
— — 3 d + 6 as “.." – &c. = 0,
d as da, da,
4.
*.
and those alike applicable to the other variables. -
We have supposed, in order to be more general, none of the first differentials d ar, d y, &c. to be constant ; if
it were not the case, then the equations relative to the variables, the differentials of which are supposed to be
constant, should be suppressed. If we suppose d ar, for instance, to be constant, it is obvious that all the
differential coefficients taken with respect to w, would vanish.
Part II.
\-N-Z
TA B L E g
PRINCIPAL MATTERS
IN THE
TREATISE OF DIFFERENTIAL AND INTEGRAL CALCULUS
ALPHABETICALLY ARRANGED.
N. B. The Numbers in Parentheses refer to the Articles, and the others to the Pages.
CALculus, differential (1)
integral (57) -
Coefficient, differential, to obtain (15)..............................
— equal when two functions of the same
value are equal (19)
of their sum and difference (21), of their
products (22), of fractions (24)
exponential, logarithmic, and trigono-
- metrical (26) ..............................
Definitions and preliminary observations (1)
Differential expressions containing circular functions, inte-
gration of (86)
Equations, differential (44)
Functions of two variables, transformation and developement
of into a series of terms containing successive
powers of one of them (10)
containing two or more intermediate variables
(33) ...................................................
a s e s e a s as w = w e s is ºn a w is e º º sº s a s tº e s a sº e s s s s a s gº tº
e s = e º a s s = e s e s a e e s a e s a s a s is a w s s = a a º e º ºs e e s & E &
r Page
Functions, homogeneous, of several variables (41) ............... #.
implicit of two or more variables (51)............... 796
of one variable, to find the value of, when the first
differential coefficient is given (58) ............... 802
irrational (83) .......................................... 8 : 5
differential of trigonometrical lines (87)............ 825
Integral calculus (57) .................................. * - © & & & ſº & m m ºn º ºs e 802
definite (92) ............................................. 832
Integration of differentials containing one variable (58)......... 802
more than one variable (103)......... 837
by parts (86)......................................... ... . 822
Series employed to represent the value of an integral of a
differential expression (89) ........................... 828
Taylor's Theorem analytically expressed (12) .... ................ 773
Series to determine the limits of begiuming with any
term (96) ................................................ 833
Transformation of differential functions of trigonometrical
lines (87) ................... ................... ........ 825
THE END OF WOL. I. - ...
*',
LoNDoN : PRINTED BY w. CLowes, stamford STREET.
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