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Hist, of Scume
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36938
HISTORY
O F
MATHEMATICKS,
Containing an Account of the Progrefs of MATHEMATICKS from
the Origin of that Science to the preſent Time; wherein are ex-
hibited the principal mathematical Diſcoveries, the Diſputes
they have given Riſe to, with an intereſting Detail of the Lives
of the moſt eminent MATHEMATICIANS.
7
CHA P. I.
Preliminary Difcourfe on the Nature, Divifions, and Utility of Mathematicks.
I.
HE Ancients are generally thought to have called this Science, Mathefis,
THE Mathemata, by which is meant the Sciences, on Account of the
luminous Certitude that characterifes it above all the other Branches of
human Knowledge.
This Etymoligy does fo much Honour to this Science, that we could
wiſh to eſtabliſh it on a more folid Foundation; it muſt be confeffed, that
it is not fupported by the Teftimony of any Author of Antiquity (a), and
without Doubt is the Conjecture of fome modern Panegyrift. Proclus, who
gives fuch great Encomiums to Mathematicks (b), and who relates with
fuch Care the Sentiments of his Predeceffors concerning their Nature, Di-
vifions, &c. fays nothing of this Etymoligy, but gives a metaphyſical De-
rivation of it, more fubtle than folid (c), which we therefore omit.
Some among the Moderns, not fatisfied with this Etymoligy, have given
another, which they thus account for: That at the Time Mathematicks
fu) Scapula in his Greek Dictionary, at the
word parlava, cites Philon, or the author of
the book De Mundo, in proof of this ety-
moligy, but no fuch thing is found there.
(b) Comm. in 1 Euc. 1, v.
(c) Ibid c. 15.
II
OF
HISTORY
took its Rife, and for fome Ages after, they were the only Sciences taught
in the Schools of Philofophers. Rhetorick, Dialectick, Grammar, Mora-
lity, which in fubfequent Ages had fo great a Share in the Improvement of
the Mind, being then unknown; Mathematicks and Natural Philoſophy
alone occupied the human Understanding; the latter being always preceded
by the Study of the former, as being the Avenue and Introduction to it.
it is well known that in the School of Pythagoras, the Claſs of Mathe-
maticians preceded, that of Natural' Philofophers, that Plato, in Times much
pofterior, excluded from his Phyfical and Metaphyfical Lectures, thoſe who
were not initiated in Geometry: it was, in fine, what gave Occaſion to
the Speech of Xenocrates to one, who, ignorant of Geometry and Arithme-
tick, attended his Lectures, withdraw, faid the Philofopher to him harſhly
Anfas Philofophiæ non habes (d).
>
Hence it appears that the Study of Mathematicks was the first purſued
in the Schools of Philofophers; and from this Priority of Time, not from
its fuperior Perfection, fay the Authors we mentioned (e), has this Branch
of Knowledge been called Mathefis, that is, Science, or rather Inftru&tion.
II.
As to the Nature of Mathematicks, they may be defined, the Science of
the Relations of Magnitude and Number, that all Things fufceptible of Aug-
mentation or Diminution can have to each other.
Thus Geometry, for Example, confiders the Relations of the different
Parts of Extenfion; fince to meafure, which is the principal Object of Geo-
metry, is nothing more than to determine the Ratio of a certain Portion of
Extenfion to another taken as a fixed Meafure. Aftronomy is wholly employ-
ed in difcovering the Order and Pofition of the Stars, that is, their Dif
tances from each other, in computing the Times in which they peform their
Revolutions, in foretelling their Conjunctions, Oppofitions, &c. Me-
chanicks, in compairing the Weights and Motions of Bodies with one ano-
ther, in calculating the Powers acting in oppofite Directions, &c. In all
thofe Confiderations, there are Relations of Magnitude, and to thoſe alone
Mathematicks are confined. If the Mind attempts to reafon on the Nature
of the Stars, of Extenfion, of Motion, or on the Caufe of Gravity, &c. it
refers thoſe Reſearches to Phyficks.
III.
Mathematicks are naturally divided into two Claffes; one comprehending
pure or abſtract, the other, mixed Mathematicks. The first confiders the
Properties of Quantities after an abftra& Manner, precifely as it is ſuſcep-
tible of Augmentation or Diminution; and as the Mind diftinguiſhes two
(d) Diog. Laert. in Acnecrat.
t
?
(e) Ramus Proem in math, Barrow lect.
math. lect, 1.
MATHEMATIC K S.
III
Sorts of Magnitude, one confifting in Number or Multitude, the other in
Space or Extenfion, thence arife the two principal Branches of the firft Di-
vifion, viz. Arithmetick and Geometry. Numbers are the Object of the
former; Extenfion, its Relations, its Meafure, that of the latter.
Mixed Mathematicks confift of certain Parts of Phyficks, to which ab-
ſtract Mathematicks may be applied: thus in Opticks, the Effects and Pro-
perties of Light are investigated by Means of certain Principles, which re-
duce their Confideration to pure Geometry. It is first affumed as a Prin-
ciple, that the Rays of Light are propagated in ftraight Lines, if no Ob-
ftacle is oppofed to their Paffage; that they are reflected making the Angles
of Reflection equal to thofe of Incidence; that paffing thro' one Medium
into another of different Denfity, they are deflected from their firft Di-
rection, obferving however in their Deflections a certain geometrical Law.
Thefe Principles once eſtabliſhed, the Mathematician examines no further
the Nature of Light, or of the Mediums it paffes through, or that reflect
it. He confiders the Rays of Light as ftraight Lines, the reflecting or re-
fracting Mediums as purely Mathematical Surfaces, the Form of which is
alone attended to. Thus he determines the Path of the Rays of Light on
Mirrors, through Optical Glaffes, their Effects on the Sight, &c. It can-
not be denied, but that thofe Reſearches are, properly fpeaking, Phyfical;
but as they are intimately connected with and dependant upon abftract Ma-
thematicks, from which they derive their Certainty, they are in fome Senfe
raiſed thereby to the Rank of Mathematicks, of which they form the fe-
cond Divifion, In this Refpect, they hold a middle Place between En-
quiries purely Phyfical, which are commonly involved in Obfcurity, and
pure Mathematicks, which fhine with unclouded Perfpicuity. They can-
not have a greater Degree of abfolute Certainty, than the Principle upon
which they are founded, and in this View they are Phyfical: on the other
Hand, they have an hypothetical Evidence equal to that of abftra&t Ma-
thematicks, that is, their Principle being fuppofed true, they are not lefs
certain than this latter. They have even the Advantage of having a Kind
of metaphyfical Certainty, tho' their Principle is not exiftent in Nature,
provided that Principle is not repugnant to Reaſon. What Archimedes has
demonftrated concerning the Ratio of Weights that are in Equilibrio at the
Extremity of a Balance, is equally true whether the Directions of heavy
Bodies are parallel or converge to a Point; in the latter Cafe only, the
Theory of Archimedes can be applied to Weights that gravitate on the Sur
face of the Earth, but it is likewiſe to thoſe which gravitate, or are con-
ceived to gravitate, in parallel Lines, which is not metaphyfically impof-
fible, and by Means of this Principle purely hypothetical, and that does not
take Place in the prefent Order of the Univerfe. The Mathematician of
Syracufa, fquared the Parabola. The Phyfico-mathematical Diſcoveries of
Newton, on the Form of the Orbits, that the Planets fhould defcribe, ac
་
❤
IV
HISTORY
cording to the different Laws of Attraction, would not be lefs true, tho' it
had been demonftrated that this Attraction did not exist, they would then
ſtand in the fame Rank with the Properties of a Triangle or Circle, if nonet
fuch really exiſted in Nature.
From what has been faid concerning mixed Mathematicks, it follows, that
their different Branches cannot be fixed and determined, as thoſe of abftract
Mathematicks. In Proportion às experimental Philofophy acquiring new
Riches, has afcertained certain Facts that may ferve as firſt Principles, mix-
ed Mathematicks have been extended. This the illuftrious Chancellor
Bacon obferved, with that Sagacity wherewith he forefaw the future Im-
provement of Natural Knowledge: Prout Phyfica, fays he (f), majora in
dies Incrementa capiet, et nova Axiomata educet, eo mathematica nova Opera in
multis Indigebit, et plures demum fient Mathematica mixte.
It is therefore no way furprifing, that mixed Mathematicks made fo flow
a Progreſs among the Ancients, whilft abftract Mathematicks advanced fo
rapidly by a Variety of important Discoveries. The human Mind need
only deſcend into itſelf to improve in pure Mathematicks, but to advance in
the other, a quite contrary Method is to be parfued; it requires a Series of
Experiments and Obfervations: in this the ancient Philofophers were de-
ficient; in general they neglected Obfervation too much, they laid too great
a Streſs on Reafoning and Metaphyficks. Excited by an impatient, and,
after all, a very excufable Curiofity, they wanted to explain Nature before
they were acquainted with her firft Operations: thus the Structure they had
raiſed, like that erected by ignorant Architects on a weak Foundation, focr
fell to the Ground.
The fucceflive Rife of the different Branches of Mathematicks, confirm
the foregoing Remark: The Pythagoreans divided this Science into four Parts
only, two of the abftra&t and two of the mixed Mathematicks; the two
latter were Muſick and Aftronomy. Already the Obfervations of Pythagoras
upon Sound, with thofe made from Time to Time on the celestial Pheno-
mena, joined to fome Hypothefes contrived to explain and calculate the
Motions of the Stars, afforded an Opportunity of employing pure Mathe-
maticks in Phyſical Enquiries. The Extent of thofe Sciences was not much
greater in the Platonick School; their Divifion into Geometry, Stereometry,
Arithmetick, Mufick, and Aftronomy (g), was injudicious, and comprehend-
ed no more than that of the Pythagorean School: in Effect, the two firft
Divifions are only a Subdivifion of Geometry. However, pure Mathema-
ticks were confiderably improved by the Platonicks; but as thofe Philofo-
phers were too much addicted to Contemplation, they were lefs fuccefsful
in phyfical Inquiries It does not appear that they eſtabliſhed one Fact that
(f) De Augmento Scient. lib. 3. cap. 6
(g) Plat. Dial. 7 de Rep. Theon de Smyrn
in Loca Math. Platonis.
MATHEMATICK
V
S.
could ferve as a leading Principle to a new Science, if we except perhaps the
rectilinear Propagation of Light, and the Equality of the Angles of Inci-
dence and Reflection.
However that may be, it feems Opticks and Mechanicks were not ranked
among the mathematical Sciences 'till a long Time after; and that about
the Time of Ariftotle, when they had at length traced out fome of the Laws
of the Propagation of Light, of Vifion, and the Equilibrium of Bodies,
the Queſtions of this Philofopher concerning Mechanicks, fome of his Pro-
blems, the Treatife of Opticks attributed to Euclid, feem to have been the
firſt Rudiments of thefe Sciences. The general Syftem of Mathematicks
was then compofed of fix Parts, Geometry and Arithmetick, Mufick and
Aftronomy, Opticks and Mechanicks; no other Branches were known to
the Ancients,
The Moderns, by cultivating natural Philofophy with Succefs, have re-
duced a great Number of other Subjects to Geometry, fcarcely known to
the Ancients: Opticks, among them, confifted only of a very fimple Theory
of the Illumination of Bodies; Catopricks, or the Science of reflected
Light; and a few Principles of Perfpective. The Science of Vifion or
direct Opticks, they were ignorant of; Dioptricks likewife as yet were un-
known. It is not much above a. Century and a half ago fince the Prin-
ciple on which they are entirely eſtabliſhed has been difcovered, as alfo that
which ferves as a Foundation to direct Opticks; thofe two Branches of a
Science ſo far extended at prefent, owe even their firſt Outlines to modern
Discoveries.
It is alfo butlately fince Mechanicks emerged from that feeble State in which
they were tranfmitted to us by the Ancients. Confined then to the Science
of Equilibrium, they only included what we now call Staticks and Hydro-
ftaticks, in which the Equilibrium of Bodies. is only confidered. At pre-
fent, Mechanicks are the Science of Motion in general; and what that
Science in former Ages totally confifted in, is now only a ſmall Part of it.
Is Motion impeded by a contrary Refiſtance, which, without deftroying the
Tendency, annihilates the Effect and produces an Equilibrium? This is the
Mechanicks of the Ancients. Do we confider the actual Motion in Bo-
dies, the Phenomena refulting from their Concurfe and Collifions, the
Paths they deſcribe, and the Velocities with which they move when acted
upon by a Combination of different Powers, the Refiftance of Fluids to Bodies
that move in them, &c. theſe are the Subject of Dynamicks. Thus all the Parts
of Mathematicks, without changing their Names, comprize, at this Day,
Objects more vaft and extenfive; and each of them have fent forth a Num-
ber of Scions, which cultivated with Care by the Moderns, foon furpaffed
the Stock from whence they took their Riſe,
VI
HISTORY Y OF
1
t
IV.
A principle Part of our Plan being to unfold the whole Syftem of Ma
thematicks, and to give a clear Idea of its different Branches, in order to
exhibit the Progrefs of the human Mind in this confiderable Part of natural
Knowledge, it will not appear unneceffary to trace its metaphyfical Generation.
Bodies are endued with feveral Properties, as Extenfion, Impenetrability,
c. but of all thefe Properties, that which feems to hold the first Rank,
and that without which the others could not exift, and which is equally
obferved by thoſe the leaſt accuſtomed to reflect as by the moſt fagacious, is
Extenfion it does not require much Skill in Metaphyficks to form this Idea,
to diſtinguiſh its different Species; tho' phyfically infeparable. The leaft in-
formed know well how to diſtinguiſh in a Globe of any Size, Matter or
Colour, what conftitutes it a Globe, and not a Cube, or a Pyramid.-
Should you ſpeak of the Extent of a Plane, the Mind naturally lays afide the
Idea of Depth, and only annexes to it the Idea of Length and Breadth.
:
Is the Diſtance of two Objects confidered, the Mind attends only
to the Length; it even goes further, and excludes every Idea of Extenſion
from the two Terms of this Diſtance. Such is the Origin of mathema-
tical Points, Lines, and Surfaces, the Subject of fo many groundleſs Ob-
jections made by Perfons, who, ignorant of Metaphyficks or Partifans of
Scepticiſm, have ftrove to raiſe Doubts concerning the Solidity of Mathe-
maticks.
Bodies therefore confidered only with Relation to their Extenfion, will be
the Limit to which the Mind can arrive, when by a natural Impulſe it
analyſes the Objects of its Reſearch. Thus, Extenfion and the Figure that
bounds it, will be neceffarily the firſt Confiderations that Men will attend to,
when they examine the Nature of Bodies that furround them. They will
begin by comparing them under thoſe two Points of View, which alone can
be the Subject of Confideration, in confequence of the Abſtraction that
lays afide all the other Qualities that may ſerve as a Foundation to any Com-
pariſon fuch is the metaphyfical Origin of Geometry.
The Idea of Multitude or Number, is not lefs natural to Man than that
of Extenfion. Surrounded by diſtinct Objects, more or lefs Numerous,
this Idea is every Inftant fuggefted to us by our Senfes; befides, at the fame ·
Time that the Mind gets the Idea of Space, conceives it divided into Por-
tions of different Forms, and compares them with one another, it acquires
the Idea of Number; Kence arifes the Divifion of Quantity into difcrete
and continued. Quantity confidered as divided into Parts, more or leſs in
Number, is the Object of Arithmetick; confidered as extended and inclofed
by Boundaries, is the Object of Geometry, fome Divifions of which we
hall proceed to point out.
Among the different Dimenfions of Bodies, there are fome more fimple
MATHEMATICKS.
V-II
than others: ftraight Lines are more fimple than Curves, and of thoſe the
Circle is the leaft complex. In like manner, plain Surtaces, bounded
by ftraight or circular Lines, Solids bounded by thofe Surfaces, are the moſt
fimple of their Kind. Thefe Subje&s, therefore, of Confideration, fhould
have paved the Way to more difficult Refearches; they are the Object of
elementary Geometry. Tranfcendental Geometry is the Part of this Sci-
ence by far the most extenfive, which treats of curvilinear Figures of a
more elevated and abftruſe Nature, fuch as the Conic Sections, and an in-
finite Number of others, to the Theory of which the former ſerves as an
Introduction. Figures may be confidered as Spaces endued with certain
Properties; or the Spaces may be analyzed and refolved, if we may fo ex-
prefs ourſelves, into the infinitely fmall Elements of which they are com-
poſed; whence arifes the Divifion of tranfcendental Geometry into Finite
and Infinitefimal. The Speculations of the Ancients and Moderns on
the Theory of Curves, furniſh an Example of the former; their Re-
fearches refpe&ting the Meafure of Curves, Refearches which commonly
proceed only by confidering the Law according to which their Elements in-
creaſe or decreaſe, form the latter.
The Mind, after being fome Time occupied in Reſearches purely geo-
metrical, Reſearches fo much the more pleafing, becauſe always accom-
panied by a pure and luminous Evidence, is foon forced, either by Necef-
fity or Curiofity, to return to the natural World. The Motions of Bodies,
and their mutual Efforts, occafioned by their impenetrability, are the firſt
Objects that claim his Attention, and thus give Rife to the moſt important
and uſeful Branch of mixed Mathematicks, to wit, Mechanicks.
We conceive in Bodies confidered as moveable, either a fimple Tendency
to Motion, a Tendency counteracted by contrary Efforts, or the Motion it-
felf. From the first Confideration arifes Staticks, which is divided into
Staticks, properly fo called, when it relates to Solids, and Hydroftaticks
when it relates to Fluids. If Bodies are confidered in Motion, that Science
is called Dynamicks, which is divided in the fame Manner as the former,
into Dynamicks and Hydrodynamicks. From Dynamicks a Variety of Theo-
ries arife, as the Laws of the Motion and Collifion of Bodies, the Theory
of central Forces, Baliftic, Theory of Ofcillations, &c. Many Sciences
are only a particular Application of Dynamicks, ſuch as the Theory of the
Direction and Motion of Water, Navigation, or the naval Science, in as
much as it is the Art of conducting a Ship by the Aid of mechanick Powers
which put it in Motion, as the Oars, the Sails impelled by the Wind, the
Rudder, &c.
متنوع
Next to the Knowledge of thefe Sciences requifite to fupply our Wants,
Aftronomy is the moſt pleaſing. The Motions of the heavenly Bodies are
fo regular, that they muſt have ever drawn the Attention of Men capable
of Reflection. The human Mind was foon prompted to inveſtigate the
B
VIII
OF
HISTORY
Caufe and the different Relations of the heavenly Motions. I call, with
Kepler (h), Spherical Aftronomy, that whofe Object is the Phenomena ariſing
from that ſeeming true Suppofition, that the Earth is in the Centre of a
Sphere, in whofe Surface the Stars are placed; this is the firſt Branch of
Aftronomy: the fecond is Theoretical Aftronomy, wherein it is propofed
to trace the different Relations of Pofition, Diſtance, and Velocity, of the
heavenly Bodies, that is to fay, to diſcover the real Form of the Univerſe.
From Aftronomy arife fome fubordinate Sciences, fuch as mathematical
Geography, wherein is determined the Figure of the Earth, and the Pofition
of the principal Places by Obfervation. Navigation, or the Art of con-
ducting a Ship across the Seas by the Obfervation of the Stars alone.
Gnomonicks, or the Art of dividing Time, and marking its Divifions by
Help of the heavenly Bodies, but particularly by the Shadow projected by
Bodies expoſed to the Sun. Chronology, or that Part of the Science con-
cerning Time, which confifts in regulating the Method of counting it, by
making the civil Period coincide as near as poffible with thoſe of the Sun and
Moon.
The Phenomena of the propagation of Light, that is to fay, of the Mo-
tion whereby it is tranfmitted from the luminous to illuminated Bodies, or
from thoſe to our Eyes, have given Riſe to Opticks. The firſt Obſerva-
tion upon the Rays of Light, is, that they are propagated in ſtraight Lines
as long as they remain in the fame Medium. After this Manner we gene-
rally perceive Objects, and the Senfation we thence receive is variouſly
modified according to the Circumftances of their Diſtances, their Pofi-
tions, &c. thofe Confiderations form what we call direct Opticks. It
would be natural to rank under this, Head, Perspective, which is no more.
than the Art of reprefenting on a Surface thoſe Degradations of Form and
Magnitude obferved in Objects that furround us, and all its Rules are folely
founded on the Principle of the rectilinear propagation of Light.
But Light is only propagated in ftraight Lines, when its Motion is not
impeded by any Obftacle; if it meets an opake Body in its Paffage, it is
reficeted, and if the Surface of the Body be fmooth, the Rays of Light
move on, making the Angle of Reflection, equal to that of Incidence; if
the Body it meets is tranfparent, and more or less denſe than the firſt Me-
dium, paffing into this Body, it moves in a Direction more or leſs O-
blique than before, which is called Refraction. From the former Obſer-
vation, the various phenomena of Mirrors are deduced, from the latter, thoſe
of Glaffes and Inftruments for remedying the Defects of Sight. The two
Sciences that are taken up with thefe Objects are called Catoptricks and
Dioptricks.
Acaufticks are nearly with regard to Sound, what Opticks are with re-
gard to Light, but is not near fo rich as the former in certain and incon-
(h) Epitome Aftron. Copern. p. 14.
MATHEMATICK S.
IX
teſtable Diſcoveries, the Reafon of which appears from the Difficulty of
reducing its Principle to the Simplicity of a Suppofition purely mathema-
tical. This Principle is that of the Vibrations of the elaftic Particles of the
Air, which, it is obvious, is complicated with feveral phyfical Difficulties.
We may refer to this gencral Divifion, Mufick, that enchanting Art of
raviſhing the Ear by the Harmony and Succeffion of Sounds; it is founded on
a Principle partly diſcovered formerly by Pythagoras, partly in our Days by
Rameau. Not that we pretend, that by the Aid of mathematical Rules alone
agreeable Muſick may be compofed. No without Doubt, Harmony mathe-
matically exact, may not be very pleafing: To Genius, to Tafle, it be-
longs to ſelect the various Tones beft fuited to the Subject propofed;
and the Muſicians who have treated this Art mathematically, have pretend-
ed no more, than to affign the Reaſons of certain Phenomena obferved
either in Melody or Harmony.
To avoid Prolixity, we fhall only point out the other Parts of Mathe-
maticks, the Confideration of the Relations of the Gravity, Elafticity,
Denſity of the Air, and other Fluids endued with thefe Properties,
has been called by fome Moderns, Pneumaticks. Calculation applied
for determining the Probability of Events, has produced the Art of
Conjecturing, of which the Doctrine of Chances is one of the principal
Branches. Pure Geometry, applied to the Art of cutting Stones in fuch a
Form, as from their Union certain Works of Architecture may refult,
compofe what is called Stone-cutting; in general, the Symmetry neceffary
to be obferved, whether in rearing Edifices for Defence as well as Orna-
ment, is altogether owing to the mathematical Sciences.
V.
We have already obferved more than once, that all the Parts of mixed
Mathematicks are intimately connected with abftract Mathematicks, of
which they are only particular Applications. This is an Obfervation pro-
per to be infifted upon, for the Advantage of thoſe who, defirous of ac-
quiring a folid and extenfive Knowledge of thefe Sciences, might miſtake
the Road to attain it, or would be defirous to know it: With this View, we
ſhall endeavour to point out clearly their Connection and mutual Dependance.
Every Queſtion in mixed Mathematicks is reducible to a Problem of
pure Geometry, by ftripping it of fome phyfical Circumſtances immaterial
to its Solution, as will appear by the following Example. In Gnomonicks,
as it is well known, it is required to find the Pofition of the Shadow pro-
jected at the different Hours of the Day, by a Style parallel to the Axis of
the World, on a Surface whofe Pofition is given. A flight Knowledge of the
Sphere is fufficient to fhew, that the Hours are determined by the Pofition
of the Sun in the twelve horary Circles that divide his diurnal Revolution
into 24 equal Parts, and that theſe Circles interfect each other in the fame
Line; it is further obfervable, that the Style fixed in a proper Pofition, that
M4
O F
HISTORY
is to ſay, parallel to this Axis, fenfibly coincides with it, and would fo in
effect if we were at the Centre of the Earth; and our Diſtance from it,
when compared with that of the Sun, is fo inconfiderable, that this Suppo-
fition may be allowed. Laftly, it is manifeft that the Shadow of the folid
Axis fixed in the common Interſection of all the horary Planes, is in the
fame Plane with the Sun and this Axis, the Shadow therefore projected by
this Axis, is only the horary Plane produced; hence the Problem for deter-
mining the Pofition of this Shadow, is reduced to the following: A certain
Number of Planes that cut each other in the fame Line and in the fame Angles
every Way, being given; to find their Interfection with a Surface whofe Form
and Pofition are alſo given.
1
Now it is eaſy to perceive, that this is merely a geometrical Problem; and
whilſt he who is unacquainted with, or indifferently fkilled in Geometry,
takes great Pains to learn the practical Rules of Gnomonicks and their Rea-
fons, the intelligent Geometrician finds in Himſelf thofe Reſources; he
folves the Queſtion, and invents and frames practical Methods of Solution.
The fame may be faid of Perfpective; this Branch of Opticks confifts
in a Problem very eafy to a Geometrician: it is required to determine on a
Plane, whofe Pofition is given, the Interſection of the different Lines con-
ceived drawn from the Eye to the Outlines of the Figure, which is ſuppoſed
to be placed behind the Plane. Little ſkill in Geometry fuffices to refolve this
Problem in its full Extent, whilft he that has made no Progreſs in it, is ſtopped
every Inftant, meets continually Difficulties which he is unable to folve;
and we fhall not heſitate to pronounce, that Geometry is the univerfal Key
of Mathematicks: He alone can afpire to penetrate into thofe Sciences,
who is Maſter of the firft; any other will ever remain confined in a nar-
row Sphere, and in a State of Mediocrity.
VI.
Mathematicks were always held in great Efteem by the moſt celebrated
Philofophers of Antiquity. We find, in Effect, that all thoſe who were
eminent for their Learning and the Purity of their Morals, cultivated thoſe
Sciences: I fay, all thofe eminent for their Learning and the Purity of their
Morals, being well aware that a fophift Pratogaras, a voluptuous Ariftip-
pus, an epicurean Zenon of Sidon, and fome others of the fame Stamp, have
endeavoured to decry them; but the moft refpe&table Characters have
rendered them the Juftice they deferve, fuch as Thales, Pythagoras, Demo-
critus, Anaxagoras, and all the Philofophers of the Ionian and Italick Schools;
in fine, Plato, Xerocrates, Ariftotle, &c. It is well known that the former
were indefatigable in promoting thofe Sciences in Greece; that Plato was one
of the moſt eminent Geometricians of his Time, and that his Works are
full of honourable Teftimonies in Favour of Mathematicks. Xenocrates,
one of his Succeffors, entertained no lefs high an Opinion of them; wit-
T
MATHEMATICK S.
XI
*
nefs his Anſwer quoted above (i). The Principle of the Peripatetick School
in his metaphifical Works frequently makes uſe of Examples taken from
Geometry; which clearly evince that he confidered the geometrical Me-
thod as the fittest to be employed in the Inveſtigation of Truth: befides,
it is well known that he wrote upon ſeveral mathematical Subjects. We
find among the Antients only Socrates, whofe Opinion can with any Shew
of Reaſon be oppoſed to this general Suffrage in Favour of Mathematicks.
This Sage, we must own, difapproved of too great a Curiofity to pener
trate into thofe Sciences. When we know, fays he, as much Geometry as
is neceſſary to meaſure our Land, as much Aftronomy as is requifite to
point out the Hours and regulate Time, to guide us in our Journies, either
by Sea or Land, we ſhould not affect to know more (k).
We fhall make fome Remarks on thofe Expreffions of Socrates, in order
to obviate the Confequences that fome might be apt to draw from them.
Firſt then, does not this Philofopher make large Conceffions, and even
more than in Appearance he intended, by allowing us to cultivate Mathe-
maticks as far as the Exigencies of Society may require? If the Circumſtan-
ces of the Times in which he lived, rendered the Ufe of Mathematicks
very confined, it is not fo now; we no longer navigate thro' a narrow Sea,
as they did at that Time; never more fecure from the Dangers of Navi-
gation, than when out of Sight of the Coafts, we fteer thro' the Ocean,
having, during a confiderable Time, no other Intercourfe but with the
Stars: the Knowledge of the pofition of all thofe celeftial Bodies is there-
fore neceffary. It is requifite that the Geography of our Earth ſhould be
correct; this only can be effected by perfecting and increafing the aftrono-
mical Methods. If fuch Pains are now taken to improve the Theory of the
Moon, and fo great an Apparatus of Obfervations and Calculations em-
ployed for this Purpofe, let it not be thought this is done merely to gratify
Curiofity; tho' even that might be easily juftified: It is with a View to
procure to Navigators a certain and perfect Method of diſcovering, at all
Times, the Place of their Stuation. Thus we fee a profound Knowledge
in Aftronomy becomes neceffary, even according to the Judgment of Socra-
We choſe Aſtronomy for an Example, becauſe the Utility of this
Branch of Mathematicks being lefs generally known, it might perhaps be
confidered as a vain and ufelefs Science. What great Advantages do not
accrue to Mankind from cultivating Mechanicks, Opticks, &c.
tes.
But we are principally to confider the Motives that induced Socrates to
hold Mathematicks in fo little Efteem. This Philofopher devoting himself
entirely to the Study of Morality, was
ferve a juft Medium) that the fole Study
to make him better and more virtuous.
(1) Art. 1.
of Opinion (fo difficult it is to ob-
of Man ſhould be that which ferves
We grant, it is the firſt and moſt
(k) Diog. in Surat. Xenoph. b. 1x, de dię.
fac. Sect.
XII
OF
HISTORY
{
effential Study; that, without moral Virtue, the moſt eminent Qualities de-
ferve little Regard: yet muft we not alfo allow, that it is too rigid to con-
fine the human Mind to that Study alone. If it be neceffary to furniſh ſome
Aliment to a Curiofity, too natural to Man, that to gratify it, fhould be look-
ed upon as criminal, what can fuit it better than the Study of Mathema-
ticks. Thefe Sciences, in effect, incapable of miſguiding the Heart, whilſt
at the fame Time they enlighten the Underſtanding, anfwer beft this Pur-
pole. Socrates, tho' his extreme Severity inspired him with little Efleem
for them, however acknowledged they were highly, uſeful in fome Refpects.
If we believe Plato, he regarded them as very fit to ftrengthen the Facul-
ties of the Mind: "Have you not obferved, faid he (1), that thoſe who
"naturally count well, are endued with an Underſtanding capable of making
a rapid Progress in all Arts; and that thoſe who are flow and dull, be-
"come, after being exerciſed in Arithmetick, more quick and ready of
Apprehenfion." And elſewhere (m), he feems to acknowledge the Uti-
lity of the Mathematicks in all Arts, which greatly foftens his harsh Judge-
ment, or, at leaſt, renders it inconclufive: For it is inconteftible, that
Branches of Knowledge ufeful to Society, fhould be the Occupation of ſome
Men endued with Talents and Genius fufficient to improve them; and that
it were to be wiſhed, that all could contribute by their Labours. In fine,
it muſt be allowed, that a Study calculated to render the Underſtanding more
ready to conceive, more capable of exercising the Faculties of Reaſoning
and Reflection, ought to form a confiderable Part of the Education of all
thoſe who are intended for a Way of Life that requires the Exertion of thoſe
Faculties. The Teſtimony of Socrates, therefore, is no way unfavourable
to Mathematicks.
We might collect in all Ages a Train of Suffrages no lefs honourable
to thefe Sciences, than thofe of the Philofophers of Antiquity. If there
were found even in thoſe Days of Darkneſs that fo long reigned in
the Weft, fome worthy of a more enlightened Age, who foared far
above their Cotemporaries, we obferve that they cultivated Mathematicks.
Such were the illuftrious Boetius, Caffiodorus in the fixth Century; the
venerable Beda and Alcuin his Diſciple, Preceptor to Charles the Fifth in
the eighth Century; Ferbert in the tenth; Albertus Magnus, Roger Bacon,
and fome others in the thirteenth Century: Thofe perfonages fo much
the more to be admired becauſe they were able to make their Way through
the ignorance and barbariſm of their Age, thofe perfonages I fay held
Mathematicks in Eſteem, and culitvated them with Ardour, witnefs Roger
Bacon, in whofe Writings we find the Seeds of fo many important Dif-
coveries; witnefs Ferbert, who, thirsting after the Knowledge of thofe
Sciences, fled from his Convent to feek among the Arabians the Aſſiſtance
which he could not find among Chriftians.
(1) In Phedro & in lib. v11. de Repub.
(m) In Phul.
MATHEMATICK
XIII
S.
Let us now pafs to the Moderns; we fhall find that the moſt eminent
Philoſophers who have flourished fince the Revival of Letters cultivated
the Mathematical Sciences; fuch was the illuſtrious Chancellor Bacon, that
great Genius, who at a Time that Learning only began to dawn, traced
the Road to be purſued for its Improvement. Mathematicks appeared to
him indifpenfably neceffary for the Restoration and Advancement of Phy-
ficks, the Study of which he fo warmly recommends. Who does not know
that Gallileo, Torrilicelli, Defcartes, Pafcal, &c. held the firft Rank among the
Mathematicians of their Time, and that they have enriched Phyficks with
ſeveral uſeful Difcoveries. Boyle, the principal Reftorer of experimental
Philofophy, has often regretted (o) his not having cultivated Mathematicks;
however, it cannot be faid that he was totally unacquainted with them,
as fufficiently appears by his Works. But he was fenfible that a more
profound Knowledge of thofe Sciences would have been of fignal Service
to him. The fame Genius to whom we are indebted for the nobleft Im-
provements in Geometry, the great Newton, is Author of the moſt ſublime
phyfical Diſcoveries. It was referved for the first of Mathematicians to
analyze Light, and to difcover and irrefragably demonftrate the Confti-
tution of the World, and the Laws according to which it is maintained
and preſerved. The moſt illuſtrious Metaphyſicians have likewife added
their Suffrages to thoſe we have already collected. Mallebranche thought
he could not propoſe a better Example of the Manner of proceeding in
the Inveſtigation of Truth, than the geometrical Method (o). I fhall fi-
niſh by the Teſtimony of Locke (p), I have mentioned, fays he, Mathema-
ticks as a Way to fettle in the Mind an Habit of reaſoning clofely and in Train,
not that I think it necessary that all Men ſhould be deep Mathematicians, but
that having got the Way of Reasoning, which that Study neceſſarily brings the
Mind to, they might be able to transfer it to other Parts of Knowledge as they
fball have Occafion.
To avoid Prolixity, I omlt feveral other Paffages of Locke favourable to
this Science. If the Authority of great Men is of any Weight, what
Names might we oppofe to the Enemies of Mathematicks, and to thoſe
Writers who from Time to Time have attacked them, fo little verſed in
thofe Sciences, that from their firfl fetting out they fall into the moſt grofs
Miſtakes. The famous Bayle (q), who from a Propenfity to Scepticiſm,
was induced to fay, that even Mathematicks had a weak Side, acknow-
ledged however, that none but an able Mathematician could oppoſe thoſe
Sciences with Succefs, but we confidently affirm, that this Attack would
in no Reſpect be difadvantageous to Mathematicks; and that nothing would
(1) In Confid circa utilit. Phil. experim.
Exercit. VI.
(o) Rech, de la Verité, liv. 6. chap. 5. &
P Jim.
(p) Of the Conduct of the Under. § 6, 7,
&c.
(4) Critical Dictionary, art. of Zenon de
Sidon.
XIV
OF
HISTORY
contribute more to make the Enemies of Mathematicks retract, than
prefcund Study of the Truths it contains.
The Annals of Philofophy and of the human Mind furniſh a Number of
Traits honorable to Mathematicks. The phyfical Diſcoveries we are now
in Poffeffion of have for the moſt Part been made by Mathematicians, as
we have ſhewn by the Examples of Defcartes, Pafchal, Gallilclo, Newton,
&c. on the contrary, if fome ufeful Doctrines have met with Oppofition,
it aroſe principally from Perfons unacquainted with mathematical Learning.
The mechanical Diſcoveries of Gallileo, the Gravity of the Air were op-
pofed only by Men who fhewed they were deftitute of this folid Branch of
Knowledge. Who are thoſe in the prefent Age who attack the mechanical
and optical Diſcoveries of Newton but Men for the moſt Part ignorant of
this Science? If we now take a View of the literary Societies, where a
Number of Mathematicians commonly take the Lead, we ſhall find the
fain Opinions in Phyficks adopted a long Time before they have made their
Way into the Unive fities, where in general Mathematicks are much ne-
glected, they do not gain Admittance there 'till very late, and even then
rather under the Denomination of popular Opinions than after a rational
Difcuffion. The Phyficks of Defcartes were difcuffed in the Academies
at the very Beginning of their Inſtitution, whilſt Ariftotle was implicitly
received for more than forty Years after in the moſt learned Univerſities.
Theſe illuftrious Societies rejected the Opinion of the French Philofopher
reſpecting the Collifion of Bodies, the ebbing and flowing of the Sea,
Colours, &c. They rejected his Vortices in Proportion as undoubted Ex-
periments and new phyſical Phenomena demonſtrated their inconfiftency.
}
#
VII.
After fuch refpectable Teftimonies, fo well attefted Facts that depofe in
Favour of Mathematicks, it perhaps would be unneceffary to attend to the
vain Declamations of its Enemies (r); however, as there are fome ca-
(*) In this Note wehall take Notice only
of fuch as have endeavoured to ridicule them,
or who have condemned them thro' Motives
that do not deférve a ferious Anſwer. A Sun
Dial being fhewn to Epicurus, to prove the
Utility of Mathematicks, Admirable Inven-
tion, faid he, not to miss the Hour of dining.
Verdier Vauprivas (in his Biblioth.) thinks
Euclid void of common Senfe: This is par-
donable in a Man whofe Head was crammed
ather with the Titles of Books, than with
real Knowledge.
→
I omit mentioning a Multitude of other Au-
thors who have declaimed against the pretend-
ed. Vanity and Uncertainty of the Sciences,
the greater Part excite the Laughter of the
Mathematicians, by the Manner they treat
Mathematicks; deferving no other Anſwer,
but an Exhortation to make themſelves ac-
quainted with their firft Principles, before
they attempt to fpeak of them. Of this Kind
are feveral Pieces inferted in the Literary
Journals. There are others who have regard-
ed Mathematicks as dangerous: The Under-
ftanding of Picus de la Mirandula appears to
have been greatly on the Declinc, when he
affeited that Mathematicks were incompatible
with Thcology, becauſe they accuftom the
Mind to demonftrative Reasoning. Peter
Hobbes, tho' praife-worthy in other Re-
fpects, when convicted of the grofs Errors he
had committed in attempting to fquare the
Circle, maintained Mathematicks to be an
illufory Science, The Particulars of this
Content may be feen in the Philofophical Poiret, in a Book intitled De Vera falfa et
Trankajons,
fuperficiaria Eruditione 1694 Leips. treats it as
MATHEMATIC K S.
X V
pable of miſleading thoſe who are not thoroughly acquainted with the Nature
of thofe Sciences, it will not be improper to difcufs them and expoſe their
Weakness.
Two Sects among the Ancients were the declared Enemies of Mathe-
maticks, the Pyrrhonics and the Epicureans. We fhall firft examine the Mo-::
tives of the Averfion of the former for this Science. This Sect, as it is
well known, only ftudied to raiſe Doubts against all Branches of human
Knowledge; it was but reaſonable therefore to expect that their firſt At-
tacks would be levelled againſt Mathematicks. Sextus Empyricus has tranf-
mitted to us the Arguments of his Sect in his famous Book against the
Mathematicians, under which Denomination are compriſed all thoſe who
make Profeffion of any Kind of Learning whatſoever, whom he attacks,
one after the other, and the Mathematicians ftrictly fo called, in the
III, IV, V, and vi Books.
To anſwer thefe Obje&ions, it would be almoft fufficient to obferve,
how ridiculous is a Doctrine that pretends, that there is no Demonftration,
no Means of attaining the leaſt Certainty, that the Evidence of Reaſon is
of leſs Weight than that of our Senfes fo liable to Deception; that, in
fine, attempts to annihilate all Science founded upon Reaſon. We do not
propofe here to refute in Form this Method of philofophifing, or to affert
the conteſted Prerogatives of human Reaſon; there is Nobody that is ca-
pable of the leaſt Reflection, who would not be able to anſwer thoſe vain
Subtilities. What Man in his Senfes would not laugh to hear Empiricus un-
dertaking to prove againſt the Geometricians, that there is neither Body nor
Extenfion; againſt the Arithmeticians, that even Number does not exiſt ;
againſt the Muſicians, that there are no founds? Paradoxes fo ridiculous in
their Nature, that barely to mention them, is to refute them.
The Objections ſtarted by the Pyrrhonics againſt Mathematicks that de-
ferve any Attention, are thofe that regard the Nature of the Objects they
are employed about, and particularly Geometry; to thofe a general An-
fwer has been given by Men of Eminence. The Objects of Mathematicks,
fay they, are fo Metaphyfical, that it is no Way furpriſing they are liable
to Difficulties; but it is a Rule obferved in the Reſearch of Truth, that
Objections, though they were unfurmountable, cannot invalidate a Doctrine
fupported by Demonftration; and this is the Cafe with Mathematicks; the
Doubts that are raiſed againſt them arifing from our imperfect Knowledge
of the Nature of Bodies, Extenfion and Motion, cannot affect Conclufions
deduced from evident Principles.
à Study that rather weakens than frengthens
the intellectual Powers; he difapproves of it
particularly becauſe it diverts the Mind from
the Contemplation of the Divinity. This de-
vout Philofopher perhaps did not ftand in need
C
of being refuted; he has been however re-
futed by the Count d'Herbſtein,in a Differtation
intitled Mathemata adv. umbratiles P. Poireti
impetus propugnata, 1709, in 8vo.
33.
1
XVI
HISTORY
OF
We shall not, however, confine ourſelves to this Kind of Defence; and
we fhall difcufs fome of thofe fo much boafted-Objections of the Scepticks,
or of the Enemies of Mathematicks.
The Objects of Mathematicks, fay they, have no Reality, and cannot
exift; Lines without Breadth, Surfaces without Depth, a mathematical
Point, that is, without Length, Breadth, or Thickness, are mere Chimeras:
The fame may be faid of the Figures whofe Properties are demonſtrated in
Geometry; there cannot be a perfect Circle, a perfect Sphere, &c. whence
they conclude, that the Objects of this Science are purely chimerical. They
enforce this Objection with feveral Arguments: If, fay they, from the
Center of a Circle, Lines be drawn to every Point of the Circumference,
they will fill up the whole Area of this Circle, and confequently the Cir-
cumference of every Circle concentric to the former, being cut by thoſe
Rays in as many Points will be equal to it, becauſe it will contain the fame
Number of Points. If a perfe& Sphere be fuppofed placed upon a perfect
Plane, their Contact will be a Point without Extent, a true mathematical
Point; but when this Sphere rouls upon the Plane, it will defcribe a Line
by the continual Application of its Surface to the Plane, and in this Man-
ner will be generated a Line compofed of Points without Extenfion, that is,
Extension formed of Parts that have no Extenfion, which is abfurd.
it appears, that the Suppofition of a perfect Circle, of a perfect Sphere,
&c. involves palpable Contradictions. Again, fay they, if thro" every
Point of the Ray of a Circle, concentric Circles be deſcribed, they will all
touch, and will fill up the Area, new Abfurdity, which confifts in this, that
Surfaces may be made up of Lines, or the Geometricians will be obliged to
allow, that Lines have Breadth, which is fufficient to overturn all their
Demonſtrations. It will be unneceffary to produce more Objections of this
Nature, becaufe, for the moft Part, they are only the fame Idea reprefent-
ed in a different Manner, and that the Solution of fome of them may ferve as
an Anſwer to all the reft.
Hence
To folve theſe Difficulties, it would be almoft fufficient to obferve, that
the Mathematicians never pretended that there were Bodies extended in
Length and Breadth without having Solidity. That there are others that
have only Length without having any other Dimenſion, they only refolve
Extenfion into its Parts, attending to fome of them, and abftracting from
all the reft. All Bodies have Length, Breadth, and Thickneſs; but we
may confider the Length and Breadth, without attending to the Thickneſs:
Hence ariſes the Idea of a Surface, and this Idea refolved again by a new
Degree of Abſtraction, produces the Idea of Length, thus the Surface is
the Term of the Bulk of the Body, and confequently has no Thickneſs ;
the Line is the Term of a bounded Surface, and the Point the Term of a
Line.
1
MATHEMATICK
XVII
S.
It follows from hence, that Bodies, Surfaces, Lines, are not made up of
Surfaces, Lines, Points, for the Term of an Extenfion cannot be confider-
ed as one of its conftituent Parts, whence the Hypotheſis on which the firſt
and laft Objection is founded, is not admiſſable. Whatever Number of Lines
are drawn from the Center of a Circle to its Circumference, or from the
Vertex of a Triangle to its Bafe, they will never compofe a Surface, they
will only be the Terms of the Divifions of this Surface into Parts, as the
Points of the Circumference are only the Terms of the Portions of this Cir-
cumference, for it is thofe Portions that compofe it, and not their Extremi-
ties: When, therefore, it is pretended that there are as many Points in a
little Line as in a great one, nothing elſe can be meant by it, than that one
can be divided into as many Parts as the other, confequently there will be
the fame Number of Terms of Divifions in each; but we can conclude
nothing with Refpect to their Size, which depends on that of the Portions.
into which they have been divided. The pretended Abfurdity that the
laft Objection feemed to point out, is no lefs groundleſs: all thofe con-
centric Circumferences do not fill up the Surface of the Circle, they only
divide it into circular Zones, of which they are the Boundaries.
It is of no Confequence whether a perfect Sphere or a perfect Plane exifts
or not, thofe Figures are only the intellectual Limits of material Magni-
tudes confidered by the Geometricians. What they demonftrate with re-
gard to thofe Limits, is more fenfibly true with refpect to material Bodies,
as they approach the nearer to them: Allowing therefore that the Truths of
Geometry are only Hypothetical, that is, for Example, if a perfect Globe
and Cylinder exifted, they would be to each other in fuch a Ratio, they
would be very far from being ill grounded; it would be neceffary to demon-
ftrate, that a perfect Sphere is the two Thirds of its circumfcribed Cylin-
der, to diſcover that the fame Ratio fubfiſts ſenſibly between material Be-
dies that approach to thoſe Figures, as far as we are able to judge by our
Senfes.
With refpe&t to mixed Mathematicks their Certitude depends partly up-
on Geometry partly upon the Truth of the Hypothefis affumed for their
Bafis. Hence by pleading the Caufe of that Science we have defended
mixed Mathematicks, at leaſt as far as regards the Confequences deduced
from the Fact they prefuppofe. As to this Fact or Principle, fince it is
founded upon Obfervation or inconteftible Experiments, it would be carry-
ing Scepticiſm further than the Scepticks themselves to difallow it; for
thofe Philofophers did not conteft the Truth of Facts and Experiments.
Empiricus who refufed to acknowledge the Truth of the Axioms of Geo-
metry allowed that Part of judicial Aftronomy which confifts in foretelling
the Viciffitudes of the Seafons, becauſe he believed it was founded upon
aftronomical Obfervations.
}
XVIII
HISTORY OF
The Invectives of Ariftippus againſt Mathematicks, the contemptuous
Opinion that Epicurus and his Followers affected to entertain of them can
have but little Weight with thoſe who are acquainted with thofe Charac-
ters. It is no Way furprifing to find a Science that requires a clofe Appli-
cation of the Mind condemned by a voluptuary fuch as the former; the
Pleaſures they afford, Pleaſures purely mental, are quite different from
thoſe in which they made our fupreme Felicity confift. (s) With regard
to Epicurus, to whom it would be unjust to impute fo fenfual a Syftem of
Morality, other Motives induced him to reject the mathematical Sciences;
it was becauſe his Opinions were incompatible with the Truths they con-
tain. In Effect, what Mathematician could he have perfuaded that the
real Magnitude of the Sun is the fame with the apparent, or even leſs
that the Eclipfes of the Sun and Moon, the fetting of the Stars, are oc-
cafioned by a total Extinction of their Light; that they are newly lighted
up at their Riſing, &c. Such was the phyfical Syftem of Epicurus, a
Syſtem well worthy of one who flighted Mathematicks.
*
;
Hence it is
that Cicero ridicules him in feveral Places, among others (t), where he
fays, he can be eafily induced to believe, without Epicurus fwearing it,
when he afferts, that he never received any Inftructions from a Maſter,
but that he would have done much better to have had one, and to have
learned Geometry rather than to have decried it: Finally he adds, that this
falutary Advice would have faved him from much Ridicule (u).
It is obfervable that the greater Part of the Scholaftick Philofophers op-
poſed the Study of Mathematicks thro' the fame Motives, as likewife in
our Days fome pretended Philofophers, who efpouſe theſe Syſtems, by
Means of which every Thing is explained in general and nothing parti-
cularly and with Preciſion. Thofe Sciences expofe the Weakness of the
Phyficks of the former, and Geometry deſtroys the phyſical Romances
of the latter.
I fhall fay nothing concerning thofe Edicts iffued by the Emperors
againſt the Mathematicians. It is well known, that the Aftrologers, were
diſtinguiſhed by that Name, who flocked to Rome during feveral Ages,
even down to the Time of St. Auguftine, who wrote a Homily on the Ře-
(s) Diog. Laert. in Ariftippo.
(1) De finib. Box. et Mal. lib. 1. § 7.
It is
(u) It appears by another Paffage of Cicero,
(Acad. Queft.B.2. Jthat Epicurushad gained over
a certain Polyneus, reputed a good Mathema-
tician, who afterwards maintained, that Geo-
metry was a mere Tiffue of Falfehoods.
very poffible that this Polyneus might have
paffed for an able Mathematician, though very
little verfed in them; it might alfo happen,
that an able Mathematician might have been
mifled, To this Example we may add that of
Chevalier de Mere, who, in a Letter to Paf-
chal, fet up for a first-rate Mathematician,
(Bayle Dict. Art. Zenon de Sidon Let. du Che-
val. de Mere, Num. 19.) treats as falfe the
Demonſtrations of Pafchal and of Geometry.
Thofe Arguments prove nothing againſt Ma-
thematicks: either more Examples fhould be
produced of eminent Mathematicians that de-
ferted this Science after having founded it,
or Objections equal in Force and Evidence to
the Principles on which Geometry is founded.
1
$
XIX
MATHEMATICK S.
conciliation of one of thofe pretended Mathematicians with the Church; (x)
but the Men of Underſtanding, the Philofophers, the Emperors themſelves
who profcribed the Mathematicians through the Empire, knew how to diftin-
guifh the real Profeffors of this Science from the Impoftors who ufurped
their Name, conferred Marks of diftinction on the former, whilft they de-
creed Puniſhments againſt the latter. There is a Decree of the Emperors
Theodofius and Valentinian, (y) conferring the honorable Titles of Spectabi-
les and Clariffimi on the Profeffors of Geometry. The Emperors Diocle-
tian and Maximien declared by a Reſcript that the Cultivation of Mathema-
ticks was an Object of public Concern; Artem Geometria difcere atque
exercere publice intereft.
VIII.
It remains now to anfwer Objections of another Nature; theſe do not
regard the Certainty of Mathematicks, but the Rank they fhouid hold
among the human Sciences. It is common in theſe Days for Men of Li-
terature, upon every Occafion, to undervalue thefe Sciences, and to ex-
tenuate the Merit of thoſe who excel in them. According to theſe Cri-
ticks, Mathematicks flouriſhed along with the Schoolmen in the Ages the
moft deftitute of Tafte, Science, and Delicacy. The greateſt Mathema-
ticians, fay they, elſewhere have been always the moſt aged or the moſt
laborious. We can clearly difcern the Motive that induced them to
ſpeak in this Manner. It is evidently in order to exclude all Genius from
Mathematicks, and reduce them to the Level of the fcholaftic Puerilities,
According to Scaliger, and others who have retailed his Opinions, Genius
is not requifite to fucceed in thofe Sciences, and confequently thoſe who
devote themſelves to the Study of them need not expect a diftinguiſhed
Place in the Annals of Literature.
Theſe Reproaches or Invectives rather, will not furpriſe thoſe who are
acquainted with the human Heart; it arifes ftom that Failing to which the
greater Part of Mankind are addicted, of over-rating their own Purfuits,
and contemning thoſe of others. With regard to Mathmetaticks, there is
a further Reaſon for this Conduct. Thofe Sciences being of a rude and diffi-
cult Acceſs, and requiring much Pains and Study to become thoroughly ac-
quainted with their fundamental Principles, the greater Part of thoſe who
endeavour to decry them with fo much Malignity, are animated with a Sort
of Defpight for their having wanted Talents to purfue fuch Enquiries,
It was, for Example, Vanity mixed with Envy, that excited Scaliger to
ſpeak with ſo much Contempt of the Mathematicks. Jofeph Scaliger, full
of that Self-fufficiency which made him fall into fo many Errors, was de-
firous of acquiring a Reputation even among Mathematicians; far from en-
tertaining at that Time fo contemptible an Opinion of them, he attempt-
(x) In Pfalm. 1. LXI. p. 32. Ed. Frot (y) L. 2. Cod. de excufat Artif.
1556.
W
XX
OF
HISTORY
ed the Solution of all thofe Problems that hitherto had baffled all their
Skill; fuch as the Quadrature of the Circle, the Trifection of an Angle,
the Duplication of the Cube, &c. he at length diſcloſed (z) his pretended
Diſcoveries with much Pomp. He propofed alfo a new Method of re-
forming the Calendar, which he oppofed to that of Gregory XIII; but all
thofe Novelties, far from being applauded by the Mathematicians, met with
that Reception which a Tiffue of grofs Parallogifms, propofed with the
greateſt Affurance, deferved: An univerfal Cry was raifed against Scaliger,
and Clavius among others convicted him of the Miſtakes he had committed.
From that moment, all thoſe who cultivated Mathematicks with Succefs,
were only dull heavy men; and the Jefuit Geometrician, his principal Ad-
verfary, was loaded with Abuſe in Proportion to the Affront he had re-
ceived. They deſerve no other Anſwer than this ſhort Hiſtory.
*
Thoſe who applaud fuch groundleſs Imputations, fhew themſelves either
very ignorant of Facts, or have very little Candour. Were then Pythago-
ras, Plato, and all the eminent Mathematicians among the Ancients, were
Descartes and Newton among the Moderns, dull and heavy Men? It would
be the Excefs of Injustice, to treat thus the celebrated Mathematicians of our
Times. Whoever can tafte the preliminary Difcourfe of the Encyclopedy,
a Difcourfe wherein the Talents of an able Writer appear confpicuous;
whoever, I ſay, can perceive the Beauty of this Difcourfe, will not heſitate
to rank the Author among the firſt Men that grace the Republick of
Letters. This is, however, the Production of one of our first-rate Ma-
thematicians, who, with the fame Pen that he calculated the Action of
Fluids, the Irregularities of the Moon's Motion, wrote this truly fublime
Piece. There is another whofe Name has been rendered famous by one of
the greateſt Operations that was ever attempted, by various mathematical
and phyfical Diſcoveries, and who had the Talent of adorning the drieft
philofophical Subjects. It would be eafy to cite a Number of others, in
whom profound Meditation has not dulled the Vivacity of Imagination.
If there are found Mathematicians of a different Caft, they are either not
of diftinguiſhed Abilities, or it may be a Defect contracted by Solitude, fo
apt to extinguiſh all Brilliancy and Livelinefs of Imagination. Great Ge-
niufes of every Species have experienced this Fáte in different Ages, but
particularly in thoſe Days when Men of Learning were confined folely to
their Books, and never paffed the Limits of the Science to which they de-
voted themſelves. If fome Mathematicians were then entirely Strangers to
polite Literature, how few were the Profeffors of Belles Letters who were
acquainted with the firft Elements of the Sphere? I fay, fome Mathema-
ticians, for it would be eaſy to prove, by a Multitude of Inftances, that
mofl of them were verfed in feveral Branches of polite Literature: But if
(≈) Cyclometria.
MATHEMATICK S.
XXI
there were even more that lived in a Kind of literary Barbariſm, this was
common to them with many others; however, they are much reformed as to
this Point, in this Age. It would be eaſy to find actually, Men of Letters, par-
ticularly among the Poets, who know not the Reaſon why the Days are longer
in Summer than in Winter. Should a Phenomenon, fo regularly and fre-
quently occurring, leſs excite the Admiration and Curiofity of the human
Mind, than the fublime Beauties of Poetry and Eloquence?
;
Thoſe who ſpeak of Mathematicians with fuch Contempt, doubtless have
miſtook the Compilers of voluminous Works for the ableſt Mathematicians
and as Authors do not compile large Volumes in the Prime of Life, or with-
out an immenſe Deal of Pains, they conclude from thence, that the oldeſt
and moſt laborious were the moſt eminent Mathematicians. This Miſtake
is only pardonable in a Stranger to mathematical Learning: Had theſe Cri-
ticks been lefs fo, they would have thought otherwife. Mathematicians
have always difcovered more Genius in fome Pages of Vieta, Kepler, Co-
pernicus, Tycho Brahe, than in the voluminous Writings of Clavius, Renal-
dini, Guarini, &c. Defcartes, while yet at the Flower of his Age, in-
ftructed all the Mathematicians of his Time, by publiſhing his Geometry,
written in a very concife Manner, The vaft Improvements Geometry has
received within this laft Century, are almoft entirely owing to young Ma-
thematicians: De Fermat was as young as Defcartes when he contended with
him, and laid the Foundations of the Infinitefimal Calculus; Wallis was very
young at the Time he grafted his Diſcoveries upon thofe of Defcartes and
Fermat; Newton had ſcarce attained his 23d Year, when he was the first
Mathematician in Europe, fince at that Age he had difcovered feveral of
his fublime analytical Methods, and among others the Principles of the di-
rect and inverſe Method of Fluxions; a few Years after, he analyzed Light,
and at the Age of 28, publiſhed his profound Theory of Opticks; his im-
mortal Treatife Principia Mathematica Philofophiæ Naturalis is partly the Pro-
duction of his Youth, he had then laid the Foundation of that immenſe
and admirable Structure; the Lives of many ordinary Men would ſcarce
be fufficient to collect and digeft the infinite Number of Materials he has
employed, and which he drew from Geometry and the moſt fubtle Me-
chanicks; however, he had fcarce completed the Half of his Career, when
at the repeated Solicitations of the Learned, he publiſhed this Work. Leb-.
nitz, propofing Cartels to the Geometricians, or anfwering them, was but
very young: This great Man's profound Knowledge in Antiquities, in Hif-
tory, in Politicks, and Jurifprudence, his Tafte for the moft refined Meta-
phyficks, are univerfally known; had it not been for these various Studies
which equally fhared his Attention during his Life, probably his youthful
Days would have been diftinguifhed, as well as thofe of Newton, by im-
portant Discoveries. What fhall I fay of the illuftrious Brothers, James and
John Bernoully, who, following in the fame Path Newton and Lebnitz, were,
XXII
OF
HISTORY
}
after them, the ableft and youngeſt Mathematicians in Europe? In fine, we
may venture to affirm, that there has not been a Mathematician of Repu-
tation who has not diftinguiſhed himſelf in his Youth by fome Work of
Genius.
We may conclude from thofe Traits, of which it would be eafy to augment
the Number, that the firft Accufation of the Enemies of Mathematicks is
deftitute of Foundation; nor is the fecond more equitable. Thofe Mathe-.
maticians that flouriſhed in the Ages of Ignorance and Barbarifm, were not
ſuch as thoſe Criticks would reprefent them; independant of their Num-
ber being very small, while the Seminaries of Learning fwarmed with
Schoolmen; independant that the greateſt Part of even thofe oppoſed the falſe
Taſte that prevailed in the Schools, could the moft eminent among them.
enter into Competition with thoſe Geniuſes that Greece in her flourishing
Days produced, with thoſe that appeared in Europe fince the Revival of
Letters. Their Knowledge, confined to the elementary Parts of thoſe Sci-
ences, to underſtand Euclid's Elements perfectly was confidered as an extra-.
ordinary Effort of Genius; the Geometricians of a more diftinguiſhed
Character, fuch as Archimedes, Apollonius, &c. were ſcarce known to them. But
let us admit for a Moment, that theſe dark Ages produced fome eminent
Mathematicians, why ſhould the Fecundity of Nature, that from Time to
Time produces great Geniuſes, fhould it be fufpended? Thofe Men were fo
much the more praife-worthy, becauſe they were able to make their Way in
fpite of the Ignorance and Prejudice that prevailed in thofe Times; and no-
thing is more honorable to Mathematicks, than that the great Geniuſes in
all Ages were verſed in thofe Sciences: We might conclude from thence,
that no Study is more proper to give that Strength and Vigour to the Mind,
which enables it to triumph over the Obftacles of Prejudice and Ignorance.
Beſides, might it not be asked, in what Age did Homer, Hefiod, &c. live?
Was it not when Greece was plunged in Barbarifm? Was it not in an igno-
rant Age that Dante, Petrarch, fhone forth in Italy? How many Poels of
Merit, how many Men of found Literature, flouriſhed in the 16th Century,
fo little productive of eminent Mathematicians every where but in Italy,
where Arts and Sciences were cultivated with unremitted Zeal? The Ob-
jection, therefore, of the Enemies of Mathematicks, proves nothing.
Upon a more candid Examination it will appear, that for the moft Part
Men famous for polite Literature, and thofe eminent for Mathematicks,
lived in the fame Age. The eminent Mathematicians that Italy produced at
the Revival of Letters in Europe, were cotemporary with Ariosto and Taffe.
The fame Age that produced in France a Defcartes, a Pafchal, a Fermat,
and a Marquis de L'Hopital, produced alfo Corneille, Moliere, Racine; in
England, Wallis, Newton, and Halley, were cotemporary with Milton, Ad-
difon, and Pope.
MATHEMATIC
XXIII
K S.
The Ancients, more equitable, feem to have been fenfible of this Truth,
when they affigned to one of their Mufes the Employment of prefiding over
the Study of the Heavens. As this Study, by the Noblenefs of its Subject,
claims the moſt diſtinguiſhed Rank, they had it chiefly in View when they
created that allegorical Being; but the Attributes (a) they gave it, apper-
tain to Mathematicks in general: In effect, the Compafs and Square are the
Symbols of Geometry, and evince that they had more extenfive Views
than one would at firſt be apt to imagine; befides, it is only by the mutual
Aids they afford, that we can attain to the fublime Knowledge of the Con-
ftitution and Laws of the Univerfe; they are therefore of the Number of
thoſe Sciences over which this Divinity prefides. The Mufe Urania, there-
fore, not only guides the Aftronomer thro' the Heavens, but alſo infpires
the Geometrician and the Mechanician, who alſo have a Seat on Parnaffus;
it being but juft, that thofe who explore the Myfteries of Nature with fo
much Sagacity, fhould afcend with thoſe who paint Nature with ſo many
Charms.
I might here avail myfelf of the Teftimony of Cicero, Not of Cicero ex-
alting his own Profeffion, by fhewing how few had attained Perfection in it;
but of Cicero the Philofopher, weighing the Sciences in the Balance of Rea-
fon. What Encomiums does he not give to Phyficks and Mathematicks (b):
Quid dulcius otio Litterato; iis dico litteris quibus infinitatem rerum ac Nature
et in hoc ipfo Mundo, Cœlum, Maria, terras cognofcimus. He makes Wiſdom
partly to confift in contemplating and unravelling thofe Mysteries: He ex-
claims (c), What Riches, what Crowns, can be preferred to the Pleaſures
tafted by a Pythagoras, a Democritus, an Anaxagoras, in contemplating the
furpriſing Spectacle of the Univerſe! In another Place (d), he even ven-
tures to call the Genius of Archimedes, divine, for having been able to imi-
tate, in a frail Machine, the ftupenduous Structure of the Univerfe. The
Sagacity of Aftronomers appeared to him fo great, that from thence he de-
duces one of his principal Proofs of the Exiſtence of a Soul, Portion, or
Image, of the Divine Being.
IX.
It now remains to fhew the Utility arifing from the Study of Mathema-
ticks (e). I fhall confine myſelf here to the Advantages refulting from that
(a) See the Frontispiece.
(b) Tufcul. Quæft. lib. v. verf. fin.
Ibid, lib. v. verf. med.
(d) Ibid, lib. 1. verf. med.
with Paffages and Texts taken from Mathe-
maticks; in his Opinion, the Conic Sections.
afford excellent Materials for Similes for the
Ufe of the Pulpit.
(e) Several Authors, thro' a miſtaken Zeal
for Mathematicks, have extolled their Utility
in a very puerile Manner. We think it ne-
ceffary to obferve this, leaft fuch injudicious
Pretenfions, if not taken Notice of, might
expoſe thoſe truly valuable Sciences to ridi-
cule. F. Merfennus (Harm. Univ. tom 2. lib.
VIII. Syn. Mat. Pref. § 13.) does not fcruple
to exhort the Orators to adorn their Difcourfes_the
D
Others have made ridiculous Applications of
the mathematical Truths to Queſtions in The-
ology, Metaphyficks, and Morality.
Pythagoreans formerly thewed the Example, by
Analogies of Figures and Numbers they
The
n
گھر
XXIV
HISTORY
OF
Science, in fupplying our Wants and adminiftring to our Pleaſures; how
this Study ſtrengthens and improves the Faculties of the Mind, we have al-
ready hinted. It were to be wiſhed, as Mr. Locke fays, that all thoſe
pretended to find in all Nature; but the Mo-
derns have far outdone the Pythagoreans in
thofe Conceits, particularly J. Caramuel de
Lobkowitz, Author of the Book entitled Ma-
thefis Audax, Rationalis, Naturalis, Superna- 4to:
Naturalis, Superna-
turalis, &c. Lov. 1644, 4to. All the Subtili-
ties of Metaphyficks, all the Mysteries of re-
vealed Religion, are explained by mathema-
tical Reafoning, the Application of which is
truly ridiculous. He examines whether God
could create Angels whofe Degree of Perfec-
tion is incommenfurable, if the Motion of the
Earth be poffible admitting the Rapt of St.
Paul, what Species of Triangle form the Tri-titude of trifling Queſtions, for the Sake of
nity, &c.
Caramuel had Imitators in one
Michael Berns, Author of a Treatife wrote
in High Dutch; in Gafpard Schmidt, who alfo
found all the Mysteries and Precepts of Re-
ligion in Mathematicks. His Work, a mere
Rhapfody, is entitled Aftrologia Cathetica.
Voffius does not give any great Proofs of his
Difcernment in his Book De Scientiis Mathe-
maticis, c. 7. when he difcuffes the Utility of
the Mathematicks, he finds them fit for every
Purpofe; for Poetry, Grammar, Economicks,
Theology, &c. The Reafons he alleges are
really curious: The Art of Combinations,
fays he, will teach the Poet that the Verfe
Rex, Lex, Sol, Dux, Fons, Lux, Mons, Spes,
Pax, Petra, Chriftus, will admit of 3628800
Variations. The Gramarian will learn, that
a Dictionary of the Size of Calepin, would
fcarce be fufficient to contain the different
Words arising from the Combination of 16
Letters. The Economift will learn from
Mathematicks, that a Pea, in 12 Years would
yield fo plentiful a Crop, that fold at a mo-
derate Rate, would amount to more than
1000000000000 Crowns. But the honeft Vof-
fius was ignorant of the firft Rudiments of
Trade, fince a Commodity fo abundaut would
be of no Value. The Divine, in fine, will
find there Subject fufficient to calm the Ap-
prehenfions of thoſe who might fear they
would not find Room in Paradife: They would
learn from Mathematicks, that tho' the World
fhould fubfift 12000 Years longer, and there
fhould be twenty thoufand Millions faved, the
Empyreal Heavens are fo vaft, that God might.
aflign to each of them a Space exceeding the
Extent of feveral Kingdom's on the Surface of
the Earth.
We shall alfo take Notice of feveral Books
+
published by injudicious Perfons, with a View
of fhewing the Utility of Mathematicks for
explaining the Holy Scripture. Thefe are the
Titles: Andreæ Arnoldi Mathefis Sacra, 1676,
4to. Altorf. Sam. Reyheri, Mathefis Mofaica,
1679. Krift. Sturmii, Math, ad. S. Script. in-
terp. Applicata, Norib. 1710. Wideburgi, Spe-
cimina Mathejeos Biblica. J. Schmidt. Mathe-
fis Biblica, 8vo. 1736. There is no Doubt,
that fome Knowledge of Arithmetick and
Geometry is neceffary to explain certain Facts
in Holy Writ; but it is quite ridiculous to rake
together, as thoſe Authors have done, a Mul-
applying to them the first Principles of Arith-
metick and Geometry, for fuch are for the
moft Part the Queſtions found in thofe Books
as the Calculation of the Sand, mentioned in
Genefis x111, 6; that of the Size of Goliath ;
the Weight of Abfalom's Hair; and, of the
Crown of the King of the Ammonites, &c.
Voffius has taken Care to extract fome of the
moft frivolous, to encourage Divines to ſtudy
Mathematicks.
A
We
Among the Abuſes of thofe Sciences, we
may reckon the pretended Application of them
to Metaphyficks and Phyfick. There are
fome Authors who have imagined, that when
they had digefted their Conceits in the Form
of Theorems, Problems, Corollaries, they had
raifed. them to the Rank of mathematical
Truths. Of late, feveral Works have ap-
peared. where the most contefted Points in Me-
taphyficks are treated after the Manner of the
Geometricians, whofe Authors, after having
heaped up a Number of quod erat Demonftran-
dums, Scholia and Corollaries have believed
their Opinions, have thereby acquired the
Certainty of a geométrical Theorem.
fhall here obferve, that it is not from the Form
of their Demonftrations, that Mathematicks
derive their Certitude; they owe it chiefly to
the Simplicity and Evidence of their Prin-
ciples; to the clear and inconteftible Connec-
tion of the Propofitions deduced one from the
other. Sr. P. de Crofa feems to have intend-
ed to expofe to Ridicule this improper Ufe of
the Form of geometrical Realoning, in a
Treatife on the Spirituality and Immortality of
the Soul: The Arguments he propofes are
quite humorous, and all in the geometrical
Style, which are a perpetual Satire on the
Metaphyficians we have ſpoken of.
3
MATHEMATICK S.
XXV
}
who are deftined for a Way of Life that requires the Exertion of the intel-
lectual Faculties, would apply themſelves to the Study of Mathematicks, we
ſhould fee fewer precipitate Conclufions, fewer falfe Arguments, boaſted for
Demonftrations; in fine, fewer Perfons feduced by the fpecious Appearances
of Truth. But we have already fufficiently infifted on this Article.
It muſt be allowed, that the Aid of Geometry and Arithmetick is abfo-
lutely requifite in Society, and in an infinite Variety of Cafes, in Econo-
micks, Jurifprudence, &c. As the Property of each individual muſt be
aſcertained in Number, Meaſure, and Weight; in general a Knowledge of
the Elements only of thefe Sciences fuffices, but there are Circumſtances
in which the Aid of the moſt profound Knowledge of thofe Sciences is
requifite: It is of Importance to a State, to private Companies, &c. that
eſtabliſh Annuities on Lives, Infurances, &c. that authoriſes certain Games
of Chance, as Lotteries, to know the Advantages and Diſadvantages, and
to State a certain Equality. Upon thofe Questions, Mathematicians are
always confulted; and the Inſpection alone of the Books wrote on thoſe
Subjects, fufficiently evince that they are of a higher Inveſtigation than
thoſe of common Arithmetick.
this Problem: A Distemper being given, to af
fign the Remedy for it. In all Probability, Pit-
carn refolved it badly; for notwithſtanding his
Solution, the Art of Healing is found every
Day defective in the Treatment of the best
known Diftempers.
tent to aflift the Lawyers in difcuffing thefe
Queſtions, the Motive is laudable; but if their
Authors intended thereby to fhew the Univer-
fallity of Mathematicks, by applying them to
a thouſand frivolous Queſtions that have any
Connection with Jurifprudence, we may claís
fuch Compofitions with Mathefis Biblica, Mo-
faica, St.
Phyfick furnish feveral Examples of the A-
bufe of Mathematicks. It is true, they may
ferve to explain fome mechanical Effects ob-
fervable in the human Body: Borelli's Trea-
tiſe De Motu Animalium, is, in this Refpect,
very valuable; but, to pretend to apply Cal-
culation to the combined Motions of Fluids Queſtions fometimes occur in Jurifprudence,.
and Solids, in a Machine the moſt complicated that require fome Skill in Arithmetick, and
exifting, is an Attempt, we may venture to fome Knowledge in Geometry; this has given
declare, chimerical. See Maupertuis' xiv Rife to the following Works: N. Vogt, Arith.
Letter on this Subject. It is entertaining, at Juridita. N. Polackii, Mathefis Forenfis. IF
leaſt to Mathematicians, to fee fome Phyfiolo-thofe Works have been compofed with an In-
gifts refolve, by a few Strokes of the Pen,
Problems, whole Solutions, thoſe who are
moſt profoundly killed in Mechanicks and
Geometry, would not attempt. The Differ-
tation of Mr. Bernoully, De Motu Mufculorum,
ought to be only confidered as an ingenious
Effay of his Skill on an hypothetical Pro-
blem, whofe Solution is modified by a thou-
fand Circumſtances. Here follow the Titles
of fome Medico-mathematical Books: N.
Stroem. Ratioc. Mechanic. in Medecina ufus
Vindicatus, L. Bat. 1707, 8vo. N. Gaukes De
Med. ad. Math. Certitud. evehenda, 1712, 8vo.
Archibaldi Pitcarnii, Elementa Medicine Phy-
fico-Math. Lond. 1717, 8vo. No Man appears
to have abufed Mathematicks more than this
baft Phyfician: He afferts what is truly ridicu-
lous; that by their Afffftance he had found
out a Method of curing the Dilorders incident
to the Eyes; lie even goes fo far as to propofe
·
A Mathematician (the Count d'Herbeſtein)
has published a Differtation on this Queſtion:
An Studium Geometriæ, rempublicam admini-
tranti adminicule an Obftaculo (Prague). I know
not how he refolves this Queſtion: I conjec-
ture, however, he concludes that Sovereigns.
should only chufe Mathematicians for their
Minifters of State. We could not expect lefs
from a Country and an Age, fo productive of
frivolous Compofitions on the Utility of Mas
thematicks.
XXVI
OF
HISTORY
It is by Mechanicks and the ingenious Combination of its different Pow-
ers, that human Induſtry is enabled to remove and tranſport Weights fo
fuperior to our natural Strength; to make Water ferve as a moving Power
to feveral Machines, to raiſe it to the Tops of Mountains, and from thence
diffuſe it as Occafion may require. Archimedes defended his Country for a
long Time by his mechanical Inventions; and almoſt all the Engines em-
ployed by the Ancients in War, were invented or perfected in thoſe Ages
when Mathematicks flouriſhed in Greece; which proves that they were of
great Uſe in perfecting that Part of the military Art.
The Advantages refulting from Aftronomy, cannot be conteſted by thofe
who will attend to the following Facts: This Science is the Soul of Geo-
graphy, of Navigation, and of Chronology. It muſt be allowed, that it
is of fome Importance to Man to know the Form, the Extent, the exact
Situations of the different Parts of the Globe he inhabits. How can this
Knowledge be attained, but by the Affiftance of Aftronomy? It is eaſy to
perceive how infufficient for this Purpoſe the moſt exa& Journals of Tra-
vellers are, at leaſt for determining the Situation of Places very remote from
each other. Beſides, in how few Cafes can this Method be employed?
And if it was the only one, we fhould not as yet have known the narrow
Limits of the Places that environ us. By the Affiſtance of Aſtronomy, the
moft diftant Countries, tho' feparated by innavigable Seas, Deferts, and
barbarous Nations, &c. hold a Sort of Correfpondence by the Interpo-
fition of the Heavens alóne.
Commerce, that Source of Wealth and Power to a Nation, is in a great
Meaſure indebted to Mathematicks for the extenſive Manner it is carried on
at this Day. In effect, this Science has had a greater Share than is common-
ly imagined, in the Diſcovery of thoſe Countries whence fuch Riches flow
in upon us. When Infant Don John of Portugal, who principally promoted
the Discovery of the Indies, propofed carrying this Scheme into Execution, he
employed the moſt eminent Mathematicians to contrive Inftruments for ob-
ferving, and to invent proper Methods for keeping a Reckoning at Sea; by
theſe Means, he induced Men to enter into his Views, and emboldened them
to brave the Dangers of unknown Seas: Such was the Origin of Nautical
Aftronomy. This Prince, who was well fkilled in Mathematicks himſelf,
was the Inventor of, the Charts employed in that Voyage; probably fo great
an Enterprize would have never been carried into Execution, without theſe
Circumftances.
If we now are able to traverſe the Ocean with ſo much Security and Skill,
it is owing to Mathematicks which has furniſhed the Means. To Mer-
cator, an Aſtronomer and Geographer of the Low Countries, we are indebt-
ed for the Invention of Charts by increafing Latitude, efteemed the beſt
by intelligent Navigators. And, without Doubt, it is from Aftronomers
that this Art will receive its laft Degree of Perfection, when the Motions
MATHEMATICK S
XXVII
of the Moon will be fufficiently known as to enable them to determine its
Place every Inftant with Accuracy.
The Chronologers have always employed the celeftial Phenomena as a
Means of verifying the Date: of certain fundamental Epochas; We find no
Order in the Chronologies of ancient Kingdoms, arifing from their igno-
rance of the celeſtial Periods. The Certainty of Hiftory, as far as it de-
pends on the affigning to Events their proper Places, is entirely owing to
Aftronomy. A well regulated Calendar feems to be the most important
Production of this Science. What Pains did not the ancient Greeks, Per-
fians, the modern Europeans, take to give to their Calendar a permanent and
perfect Form, which they attained only in Proportion as they cultivated
Aftronomy. I fhould not omit mentioning, that this Study has freed us
from thofe Terrors fo difgraceful to human Reaſon, that ufed to affright
whole Nations ignorant of the Cauſe of certain unufual Phenomena. We
recall to Mind, with Pity, the Story of that weak Prince, who, upon fee-
ing an Eclipfe of the Sun, ordered his Son's Hair to be cut off, as on a Day
of publick Calamity. The Ignorance of Nicias, who commanded the Athe-
nian Fleet in the War of Sicily, was the Cauſe of the fignal Defeat they
received on this Occafion: Nicias, terrified by an Eclipfe, was afraid to fet
Sail when it was Time, in Order to raiſe the Siege of Syracufa : The next
Day the Wind proved contrary, and prevented his Departure, and he was
taken Priſoner with his whole Army. It is not long fince, that the Ap-
pearance of a Comet infpired fuperftitious Terrors; Aftronomy alone could
calm them, by unfolding the Caufes of that alarming Phenomenon. The
confiderable Progreſs of thofe Sciences, has occafioned the Fall of judicial
Aftrology: That delufive Art, fprung from the Abuſe of Aftronomy whilſt
yet in its Infancy, has loft all Credit, except with a few weak Minds,
fince Aftronomy has been fo vaftly improved. The different Branches of
that Science, have thrown great Light on the general Syftem of the
World, an Object worthy of the Contemplation of rational Beings who
enjoy that wonderful Spectacle; fuch will doubtlefs be the Opinion of all
thofe whofe Eyes are not entirely turned to the Earth, and who recall to
Mind thoſe fine Verfes of Ovid:
Pronaque cum fpectent Animalia cætera Terram,
Os Homini fublime dedit, Cœlumque tueri
Juffit, et erectos ad Sydera tollere Vultus.
X.
To enter into a Detail of the Advantages refulting from the Cultivation
of the other Branches of mixed Mathematicks, would be affecting a uſeleſs
Prolixity; they are fufficiently obvious to diſpenſe us from infifting any
further on that Head, we fhall therefore confine ourſelves to ſome Obferva-
tions on geometrical Speculations, the End and the Utility of which might
be queſtioned. It must be allowed, that there are a great Number that are
merely intellectual Curiofities, and of no apparent Utility; but when wẹ
XXVIII
OF
HISTORY
confider that they are the only inconteftible Truths which the human
Mind can, by pure Reafoning, attain to, we fhall ceafe to regard them as
frivolous. In effect, Man being compofed of two Parts, the one Intellectu-
al, whofe Nature is to reflect and to inveſtigate the Properties of Objects;
the other formed to perceive and enjoy thofe fame Objects. It must be al-
lowed, that if their fenfible Properties ſhould be ftudied with a View of
fupplying our Wants and adminiftring to our Pleaſures, thofe that are purely
intellectual, are particularly fuited to our intelle&ual Part. Befides, to
what a narrow Compafs would human Sciences be reduced, if all thoſe
were excluded, from cultivating which, no apparent Advantage accrue to
Mankind, fhortly Ignorance would prevail, and bring back all the Evils at-
tending the rude and barbarous Ages.
We may further obferve, that thoſe purely theoretical Truths, whoſe
Ufe is not obvious, may probably be productive of fome which future Ages
may diſcover, but particularly may pave the Way to more important
Truths. What an Apparatus of Geometry does not fome Queſtions in Me-
chanicks and Aftronomy require? Among the latter, for Example, the Mo-
tion of the Moon, from the compleat Refolution of which we ſhall probably
derive the inestimable Advantage of a perfect Navigation. The Fate of
mixed Mathematicks is clofely connected with that of abftract Mathema-
ticks: Every Truth contained in the latter, is of Importance to the former.
I ſhall conclude by a Reflection. A Philofopher aſked, what would be
the Employment of Men, if they were exempt from Paffions and from the
Wants their Nature fubject them to? Doubtless, the Enquiry after Truth
and the Contemplation of the Phenomena of Nature, would be their only
Purfuits thro' a Life equally tranquil and happy. Well, then, thofe Objects,
ſo noble, the fole Occupation of perfect Beings, thofe Objects, I fay, are
thoſe of the Mathematician. Abſtract mathematical Enquiries, and their
Application to the Study of Nature, enter therefore into the Plan of the
Inftructions given in the Drawing School eſtabliſhed by the DUBLIN So-
CIETY, purſuant to their Refolution of the fourth of February, 1768; to
enable Youth to become Proficients in the different Branches of that Art,
and to purfue with Succefs, geographical, nautical, mechanical, commer-
cial, and military Studies.
W!
XI.
Introduction.
ISE Regulations relative to the Education of Youth in England,
Scotland, and other Parts of Europe. Fatal Confequences refulting
to this Country from the Neglect of this important Object. How far the
Drawing School established by the DUBLIN SOCLETY put on a proper
Footing has fupplied this Defect.
Plans of Inftruction put in Execution in this School.
Plan of a Courfe of pure Mathematicks: Utility of this Science: Me-
thod of teaching it: Is divided into Arithmetick numeral, and fpecious; into
MATHEMATICK S.
XXIX
Geometry, Elementary, Tranfcendental, and Sublime. Conclufion. Au-
thors who have fnrnifhed the Materials of this Courſe.
Inftructions relative to young Noblemen and Gentlemen of Fortune.
Plan of the Syftem of the Phyfical World. Utility of the Study of the
Syftem of the World. Advantages refulting from the Knowledge of the
Syftem of the World. Public Schools erected in England, Scotland, &c. for
inſtructing young Nobleman and Gentlemen of Fortune, in what regards the
Syftem of the World. Method of teaching the Difcoveries relative to the
System of the World. Progrefs of the Diſcoveries relative to the Syftem of
the World. Principal Phenomena of the Syftem of the World. Theory
of the principal Planets. Theory of the Figure of the Earth. Theory of
the Preceffion of the Equinoxes. Theory of the Tides. Theory of the
Refraction of Light. Theory of the Moon. Theory of the Comets.
Conclufion. Thofe Theories taken from the Principia, not as interpolated
and anatomized by Pemberton, Clark, M'Laurin, &c. but from the Original.
The Demonftrations being compleated by ſupplying the Steps, which were
defignedly omitted by the illuftrious Author. Enumeration of the Improve-
ments which thefe Diſcoveries have received from the united Efforts of the
firſt Mathematicians in Europe.
Plan of the Art of making Experiments, and that of employing them.
Neceffity there was of furniſhing the School with a compleat Collection of
the beſt executed Machines, adapted for experimental Enquiries, and of in-
ftructing Youth in the Management and Ufe of thofe Machines. I Clafs.
Machines for making Experiments on the Gravity, Motion, and Equilibrium
of folid Bodies. II Clafs. Machines for making Experiments on the Gra-
vity, Motion, and Equilibrium of fluid Bodies. III Clafs. Machines for
making Experiments on the Air. IV Clafs. Machines for making Experi-
ments on Fire. V Clafs. Machines for making Experiments on Light and
Colours. VI Clafs. Machines for making electrical and magnetic Experi-
ments. VII Clafs. Machines for making Experiments and Obfervations
in Cofmography. VIII Clafs. Machines for making Experiments and
Obfervations in Meteorology. Conclufion. The Conftruction and Uſe of
thofe Machines in experimental Enquiries propoſed to be defcribed.
Plan of the Syftem of the Moral World. Origin of Civil Society.
The different Forms of Government. Particulars in which all Forms of
Government agree. Particular Circumftances which fhould modify the
different Forms of Government. The Relations of which the different
Forms of Government are fufceptible. The Laws refulting from the Na-
ture, Circumſtances, and Relations of the different Forms of Government.
Conclufion. That the moral Philofopher does not employ his Time in
Speculations and Subtilities foreign to common Life.
Inftructions relative to Engineers, Gentlemen of the Artillery, and in general to
all Land Officers.
Plan of the Military Art. Neceffity there was of erecting a Military
XXX
HISTORY OF, &c.
School. Studies there purfued. Mathematicks, Mechanicks, Dynamicks,
Military Architecture, Baliſtic, Pneumaticks, Hydraulicks, Hydraulick
Architecture, Draughting, Attack and Defence, Geography, Hiftory,
Tacticks, Order of the Studies, practical Operations, public Examinations:
Conclufion. Pointing out the Advantages the young Officer will reap from
thofe Studies.
Inftructions relative to thofe intended for Trade..
Plan of the Mercantile Arts. Dignity of the Trader. Diſadvantages in
Point of Education he laboured under. The Neceffity there was of erect-
ing a Mercantile School. Studies there purfued. Mathematicks, Drawing,
Geography, Hiſtory, Navigation, moral Philofophy, Book-keeping, Com-
pofition, practical Negociations. Conclufion. Recapitulation of the Ad-
vantages which the young Trader, and the Public in general, will reap from
this Inftitution
Inftructions relative to Ship-builders, Sea Officers, and in general to all thofe
concerned in the Bufinefs of the Sea.
Plan of the Naval Art. The Neceffity there was of erecting a Marine
School, where are taught naval Architecture, mechanical Navigation, the
Art of Piloting, the different Branches of Drawing.
Inftructions relative to Architects, Painters, Sculptors, Engravers, Clock
Makers, &c. and in general to all Artifts and Manufacturers.
Neceffity there was of erecting a School of Mechanic Arts, where Artiſts
receive the Inftructions in Geometry, Perfpective, Phyficks, &c. which
fuit their refpective Profeffions, and contribute to improve their Tafte and
their Talents.
•Soboter acuees
!
[ 1 ]
:
ELEMENT S
O F
NUMERAL ARITHMETICK.
CHA P. I.
Of Numbers, of the general Principles of numeral Arithmetick, and of the
Operations performed upon fimple Numbers.
COMP
COMPUTATION is either performed by Numbers, or by cer-
tain Signs and Symbols which have been contrived for this purpoſe and
found convenient; whence the Science of computing is divided into nume-
ral and fpecious Arithmetick.
I.
By Number is meant a Collection or Affemblage of Units, and by an what is
Unit,an arbitrary Quantity, which is affumed to be a Meaſure of other Quan- underſtood
tities of the fame Kind.
by number.
upon for the
It has been found neceffary for the convenience of Commerce, to em-
ploy different Kinds of Units. In Money the Units authoriſed by Law, The diffe-
are the Pound sterling, the Shilling, the Penny, and Farthing. In mea-rent kinds of
furing Lengths and Diftances, the Units employed are the Fathom, the units agreed
Foot, the Inch and the Line, and ſeveral other Meaſures that have been convenience
introduced and confirmed by Cuſtom. In meaſuring of Areas and Surfaces, of trade.
the fquare Foot, the fquare Inch, the fquare Perch, the fquare Acre are
taken for Units: and when there is Occaſion of employing different Units
in order to render the Computation more eaſy; it is ufual to affume for
Units different Meaſures which have all the fame Length but different
Breadths, from whence they take their different Denominations. In the
Menfuration of Solids, the Units employ'd are the cubick Fathom, the
cubick Foot, the cubick Inch, and different Solids which have all for Bafe
a fquare Fathom but different Heights from which they take their different
Denominations, not to mention a Number of other Meaſures fuch as the
Gallon, the Bushel, &c. which vary in the different Parts of the World.
In fine, every Species of Things have their particular Unit whofe Value
has been fixed and authoriſed by Cuſtom or by Law.
The Units we have fpoke of, and all others. to whatever Kind or Species Numbers are
of Things they relate, are called concrete or applicate Units, and a Collec- either con-
tion or Affemblage of thofe Units is called a concrete Number. There is a- ftract,
crete or ab
2 A
157
ELEMENTS OF
The forma-
tion of num-
bers.
nother Unit which denotes no Kind of Things in particular, and is aplicable
to every Kind or Species of Things; this Unit is called abftract or abfolute
Unit and is expreffed by the Word One or once; and a Collection or Allem-
blage of thofe Units is called an abſtract Number.
II.
By adding an Unit to another Unit, a Number is formed which is called
Two; by adding to this Number a new Unit there refults the Number
which is called Three; and by continuing in this manner to add new
Units to the Numbers already formed, there will refult the following
Numbers which are called Four, Five, Six, Seven, Eight, Nine, &c. As
an Unit may be continually added to the Numbers that have been formed, let
them be ever fo great; it is manifeft that the Series of Numbers has no Limits,
If therefore each Number was to be expreffed by a Word or particular
ency of ex- Character, the Number of Words and Characters to be employed would be
preffing each
number by infinite, and the Life of Man would ſcarce be fufficient to learn to reckon
a partiular up to Fifty Thoufand, which is a ſmall Number not only among the
Numbers poffible, but even among thoſe which are in daily Ufe: but the
Computiſts obliged to employ great Numbers have found out the Art
of counting by means of a very few Words and Characters repeated feveral
times; this Art is called Numeration.
Inconveni-
word or
character.
Characters
employed to
repreſent
numbers.
How the
Arithmeti-
III.
Thow there is an Infinity of different Numbers, the Computifts have
found means of expreffing them all with ten Characters differently com
bined and repeated, viz.
O
Cypher,
I
2 3 4 5 6 7
One, Two, Three, Four, Five, Six, Seven
8 9
Eight, Nine.
It is eafy to perceive that with thoſe ten Characters we can reckon from
Nothing to Nine inclufively, without having recourfe to any new Arti-
fice, but not further, if we have no other Unit but of the Species to be
numbered. For Example, if the Units a Number confifts of are Feet,
we can reckon from Nothing to nine Feet, but we cannot reckon beyond.
9 Feet, if we had no other Unit but the Foot.
The Computits, therefore, befides the Unit of the Species to be rec-
koned, which is called principle Unit, for Example, befides the Foot,
which in the foregoing Example is the principle Unit, have imagined o-
thers which are called collective Units, which may be reckoned alſo from
Nothing to Nine inclufively; and by help of thofe collective Units, all
poffible Numbers arifing from the Repetition of the principle Unit may
be expreffed, in the manner we are going to explain.
IV.
When the Number of principle Units do not exceed Nine, and confe-
tal charac- quently may be expreffed by one Character, this character is called Num-
ters are dif- ber of Units of the first Degree,and the Place it ftands in is called the firſt Place.
NUMERAL ARITHMETICK
3
Of ten Units of the Firſt degree is formed a collective Unit called Ten pofed to re-
or an Unit of the fecond Degree, and may be reckoned from Nothing to
Nine, by means of the Characters 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
To diſtinguiſh the Character which repreſents a Number of thofe new
Units of the ſecond Degree, from that which reprefents a Number of Units
of the first Degree; it is wrote down in the ſecond Place to the left Hand
of the Figure, which repreſents a Number of Units of the firſt Degree,
and which ſtands in the firſt Place, this firſt Place being always filled up
with a Character reprefenting the Number of Units of the firft Degree, as
will appear by the following Examples.
་
1° When the Number of Units to be reckoned confifts only of Tens
or of Units of the fecond Degree, and do not exceed Nine, the Character
(0) which reprefents no Number, is fet down in the firt Place, and
by denoting that there are no Units of the firſt Degree remove the Cha-
racter which reprefents a Number of Units of the fecond Degree into the
fecond Place, and determines it to reprefent a Number of this Kind of
collective Units rather than any other.
Thus 10 repreſents Ten, 20 reprefents two Tens, or, Twenty, 30 three
Tens, or, Thirty, 40 four Tens, or, Forty, 50 five Tens, or, Fifty, 60 fix
Tens, or, Sixty, 70 feven Tens, or, Seventy, 80 eight Tens, or, Eighty, 90
nine Tens, or, Ninety.
preſent all
poffible
numbers.
ty nine are
two charac-
ters.
2º Since by writing down fome one of the ten Characters 0, 1, 2, 3, All numbers
4, 5, 6, 7, 8, 9, in the fecond Place we can reckon a Number of Tens from noth-
from Nothing to Nine, and by fetting down fome one of the fame Cha- ing to nine-
racters in the first Place we can reckon a Number of fimple Units or of the reprefented
first Degree from Nothing to Nine; it is ealy to perceive that we can by by help of
help of the two Characters which ſtand in the two first Places reckon
from Nothing to Ninety nine.
The Characters which ftand in the fecond Place, preferve the Names Each charac
of the Numbers of Tens they reprefent, and thoſe which ſtand in the firft ter retains
Place retain the Names of the Numbers of fimple Units they reprefent. of the num-
The following Numbers however are to be excepted, 11, 12, 13, 14, 15, ber it repres
16, 17, 18, 19, which are read and wrote as follows, Eleven, Twelve, fents.
Thirteen, Fourteen, Fiveteen, Sixteen, Seventeen, Eighteen, Nineteen,
and not Ten-one, Ten-two, Ten-three, Ten-four, Ten-five, Ten-fix,Ten-
feven, Ten-eight, Ten-nine, as the general Rule would require.
V.
the name
All numbers
nine
As ten Units of the first Degree form an Unit of the ſecond Degree, in
like manner ten Units of the fecond Degree, form an Unit of the third from noth-
Degree, which is called a Hundred, and the Character which reprefents ing to
a Number of thoſe new Units is fet down in the third Place to the left Hand. hundred and
ninety nine
Thoſe new Units of Hundreds may be reckoned from Nothing to Nine, are repre-
with the fame Characters 0, 1, 2, 3, 4, 5, 6, 7, 8, 9: and as we may fented by
ELEMENTS OF
1
蟋
​help of three reckon alfo with the fame Characters the Tens from Nothing to Nine, and
characters.
How anum-
ber repre-
fented by
three cha-
racters is
read.
All numbers
poffible are
repreſented
by help of
ten charac-
ters.
Every three
gures are
ods, and
have the
the fimple Units from Nothing to Nine; it is manifeft that we can by
help of the three Characters which ſtand in the three firſt Places reckan
from Nothing to Nine Hundred and Ninety Nine.
The different Numbers of Hundreds have no particular Names, they
are ſpecified by the name Hundred, preceded by the Word which expreffes
their Number; thus, 100 is named one. Hundred, 200 two Hundred, 300
three Hundred, 400 four Hundred, 500 five Hundred, 600 fix Hun-
dred, 700 ſeven Hundred, 800 eight Hundred, 900 nine Hundred.
;
The Reading of a Number reprefented by three Figures, is performed
by pronouncing firſt the Number of Hundreds reprefented by the Figure in
the third Place and afterwards the Names of the Numbers repreſented
by the two Figures which ſtand in the fecond and firſt Places; for In-
ſtance 364 is pronounced three Hundred and Sixty Four, 576 five Hundred
and Seventy Six, 193 one Hundred and Ninety Three, 916 nine Hundred
and Sixteeen, 807 eight Hundred and Seven, 290 two Hundred and Ninety.
VI.
Of ten Units of the third Degree called Hundreds, is formed an Unit of
the fourth Degree, which is called a Thoufand. Thoſe new collective
Units may be reckoned as the other collective Units already mentioned,
from Nothing to Nine, and the Figure which repreſents a Number of them
is always fet down in the fourth Place to the Left of the Hundreds.
repre-
By continuing in this Manner to form new collective Units each con-
fifting of ten Units of the Degree that immediately précedes; there will
refult a Progreffion of Units decuple of each other, by means of which it
is eaſy to perceive that all poffible Numbers however great may be
fented, employing only the ten Characters 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
The three Figures which occupy the three firſt Places, have parti-
places of Fi- cular Names, that which ftands in the firſt Place is called fimple U-
called peri- nits, that in the fecond, Tens, and that in the third Hundreds, par-
ticular Names might in like manner be given to the Units of higher De-
fame names, grees, but as thoſe new Names would burthen the Memory to no purpoſe,
and it would be too difficult to pronounce a great Number of Words in a
certain Order; the fame Names of Units, Tens, Hundreds are given to
the three following Figures which ſtand in the fourth, fifth and fixth
Places, and to every three fucceeding Figures: So that if the Figures
which repreſent a great Number, be parted or diftinguiſhed into Periods of
three Figures in each Period, beginning at the right Hand; each Period
will contain a Number of Units in its firft Place to the right Hand, a Num-
ber of Tens in its fecond Place; and a Number of Hundreds in its third
Place, except the laft Period which will not always conſiſt of three Fi
gures, but fome times only of One or of Two; viz. a Number of Units,
or a Number of Units with Tens,
NUMERAL ARITHMETICK.
3
A
To diſtinguiſh thofe Periods one from the other; a particular Name is Names of
given to the firſt Figure of each of them, from which the Period alfo the different
periods
takes its Denomination.
The firft Figure of the firft Period, is called fimple Units; and the first
Period which alſo contains the Tens and Hundreds of fimple Units is call-
ed the Period of fimple Units.
The firſt Figure of the fecond Period is called Thouſands, and this
fecond Period which contains alfo the Tens and Hundreds of Thouſands,
is called the Period of Thouſands.
The firft Figure of the third Period, is called Millions, and this Period
which contains alfo the Tens and Hundreds of Millions, is called the Period
of Millions
After the Period of Millions comes that of Billions, of Trillions, af
Quadrillions, of Quintillions, of Sextillions, of Septillions, of Octillions,
of Nonillions, of Decillions, &c. each of which contain their Units, Tens,
and Hundreds.
of figures.
A
To read therefore any propofed Number, we have no more to do, than Rule for
to divide it by Commas into Periods of three Characters each, beginning reading any
at the right Hand, and read one after another, beginning at the left Hand, the propofed
Numbers repreſented by each Period, expreffing the Names of each Period.
VII.
number.
fand that
earth
In order to fhew at one View the Names of the different Periods of a The number
great Number reprefented by Figures, with the Names of the Figures of of grains of
each Period, we ſhall take for Example the Number of Grains of Sand that would form
Archimedes found would form a Globe as big as the Earth, fuppof- a globe as
ing that ter Grains of Sand were in Length equal to the Diameter of a Seed big as the
of Coriander: that Ten Seeds of Coriander were in Length an Inch; that pofed as a
twelve Inches, make a Foot; five Feet, a geometrical Pace; three thou- example.
fand Paces, a League; that the Diameter of the Earth is two Thousand
eight Hundred and Sixty four Leagues; and that the Circumference of a
Circle is equal to three times its Diameter, together with the feventh
Part of this Diameter.
This Number of Grains of Sand, with the Names of
the Names of the Places
of each
Period is as follows.
its Periods and
pro-
}
Simple Units {
~ Units
+Tens
Hundreds
Millions {
Thousands {
Units
Tens
Hundreds
~ Units
Tens
Hundreds
Units
Tens
Hundreds
& Units
Tens
Sextillions {
Quintillions {
Quadrillions {
Trillions {
Billions {
71.764.5 46.80 9.2 70.8 57.1 4 2.8 5 7.1 4 2.8 5 7 · 14 2
Hundreds
∞ Hundreds
in Tens
Units
Tens
Units
& Hundreds
∞ Hundreds
O Tens
Units
+Units
Tens
Hundreds.
Nonillions
Qatillions {
Septillions { & Taits
Tens
Hundreds
- Tens
Units
6
ELEMENTS OF
Of decimal
parts.
Numbers
fet down
after the
are decimal
parts.
This prodigious Number is thus pronounced: feventy one Nonillions,
feven hundred and fixty four Octillions, five hundred and forty fix Sep-
tillions, eight hundred and nine Sextillions, two hundred and feventy.
Quintillions, eight hundred and fifty feven Quadrillions, one hundred and
forty two Trillions, eight hundred and fifty feven Billions, one hundred
and forty two Millions, eight hundred and fifty feven Thouſands, one
hundred and forty Two Units or Grains of Sand.
VIII.
Having fufficiently Explained how from the various Combinations and
Repetitions of the ten Characters, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, all
poffible Numbers confifting of fimple Units of every Denomination may
be repreſented: we fhall proceed to explain how by means of the fame
Characters all Numbers confifting of Units that are only the tenth the
hundreth or thoufanth Part, &c. of a fimple Unit confidered as the
principle Unit are repreſented.
Since ten Units of any particular Order form an Unit of an higher Or-
der, belonging to the next Place to the Left; it is manifeft that if we
divide an Unit of any degree into ten equal Parts, thofe Parts will be ten
Units of a lower Degree, belonging to the next Place to the right Hand.
For Example, if we divide a Hundred which is an Unit of the third Degree,
belonging to the third Place, into ten equal Parts; there will refult Tens
or Units of the ſecond Degree, belonging to the fecond Place: and if we
divide again one of thofe new Units into ten equal Parts, there will refult
fimple Units, or Units of the first Degree, which ſhould ſtand in the firſt
Place. Whence by dividing continually the Units into ten equal Parts ;
there continually refults new Units fubdecuple or ten times lefs, and the
Figures which reprefent a Number of them, ftand in places more and more
removed towards the right Hand, until we come to the fimple Units which
ſtand in the firſt Place; but there is no reaſon why this Divifion may not
be continued further on,
TX.
The Figures ftanding in Places continually removed towards the right
Hand, repreſenting a Number of Units fubdecuple of thofe of the preced-
ing Figures: if a Comma be prefixed to the Figure reprefenting the prin-
units place ciple Units, to diſtinguiſh the Place of thoſe Units, and if any Number of
Figures be wrote after the Comma towards the right Hand; as in this
Number 576, 347892; it is eafy to perceive that the Figures 576 to the
left Hand of the feparating Point, reprefents a Number of principle Units:
and that the other Figures 347892 to the right Hand, reckon Units fo
nuch the more fubdecuple of the principle Unit, as they are more remov-
ed to the right Hand from the feparating point. For the Character 3 which
ſtands next the Comma to the right Hand, or next the Character 6 which
reprefents fimple Units, will reprefent Units fubdecuple or ten times lefs
45
1
NUMERAL ARITHMETICK.
7
than thoſe repreſented by the Character 6, whence the Units of this Num-
ber 3 will be the tenth Part of a fimple Unit, and are called Tenths. In
3 to the
like manner the Character 4 which ſtands next to the Character
right Hand: repreſents a number of Units that are the tenth Parts of thoſe
of the Figure 3, and confequently will be the tenth Parts of a Tenth, or
the hundreth. Parts of the principle Unit, and for this reafon are called
Hundreths. In fine the Units reprefented by the following Characters
7892 to the right Hand, becoming continually fubdecuple, will be the
Thoufandth, the ten Thouſandth, the hundred Thoufandth, the Millionth,
Part, &c. of the principle Unit, and for this Reafon are called Thousandths,
ten Thouſandths, "bundred Thousandths, Millionths, as in the following
Example.
All the Figures which are to the right Hand
of the fimple Units, whofe Place is diftinguiſh-
ed by a Comma, are called decimal Figures,
and their Units are called decimal Parts or deci-
mal Fractions..
X.
Hundreds
6
Decimal Figures
Tenths
Hundreths
Thouſandths
Ten Thouſandths
~ Millionths
Hundred Thouſandths,
Tens
Units
It is now eaſy to perceive how Numbers whofe expreffion contain deci-
mal Figures are read. Let for Example 576,347892 be propoſed to be read.
?
decimal
decimal £<
gures.
1º This Number may be confidered as confifting of two Parts, the First manner
first part repreſented by the Figures 576 to the left Hand of the feparat- of expreffing
ing Point, reckon fimple Units, and confequently may be wrote thus, 576 numbers
fimple Units, as to the fecond Part reprefented by the Figures 347892 to the containing
right Hand of the feparating Point, the first Figure 2' expreffes Millionths,
and Ten of thoſe Units forming an Unit of the following Figure 9 to the left
Hand, this Figure 9 expreffes Ninety Millionths, in like manner ten Units
fimilar to thoſe of the 9, that is ten Tens or one hundred Millionths form
an Unit of the Figure 8 which follows to the left Hand, confequently this
Figure 8 expreffes eight hundred Millionths, and proceeding in this man-
ner it will appear that the ſecond Part of the propofed Number 347892
reckon Millionths, confequently may be wrote thus 347892 Millionths,
wherefore the entire Number reprefented by (576,347892) may be
read as follows, five Hundred and Seventy Jix Units, and three Hundred
and Forty feven Thouſand eight Hundred and Ninety two Millionths.
2. Since the Units of the propofed Number 576, 34789re continually Second man-
decuples of one another, proceeding from the right Hand to the Left, and ner of ex-
preffing
the Units of the loweſt Degree are Millionths; it is eafy to perceive that numbers
the entire Number may be confidered as confifting of Millionths only; con- containing
fequently may be read thus; five Hundred and feventy fix Millions three Hun- decimal Fi-
dred and forty feven Thouſands eight Hundred and ninety two Millionths,
gures.
8
ELEMENTS OF
In what a
taining deci-
one that has
none.
From whence it appears that a Number that contains decimal Figuress
number con- differs from one that has none, only becauſe they confiſt of different Units-
mal figures The Number which has no decimal Figures confifts of abfolute Units, or
differs from of Units of the Species to be reckoned. The Number which has decimal
Figures confiſts of Units which are only Parts of abfolute Unit or of the
Unit of the Species to be reckoned; and it is the Place of the Comma
which determines what Parts of the whole Unit the propoſed Number con-
fifts of, for Example, if there are One, Two or Three
decimal Figures after the Comma, the Number will confift of Tenths or
Hundreths or Thouſandths. or Millionths of the principle Unit.
Third man-
ner of ex-
preffing
numbers
or Six
3º Laſtly when Numbers contain decimal Figures, the Part to the left
Hand of the feparating Point may be firſt read, and afterwards the Values
of all the decimal Figures expreffed one after another, with the Names of
their particular Units, for Example, the Number 576,347892 may be read
decimal - thus, five Hundred and Seventy fix integer Units, and three Tenths, four
Hundreths, feven Thousandths, eight ten. Thousandths, nine hundred Thou-
fandths, and two Millionths.
containing
gures.
How num-
X I..
A Number very often has neither principle Units nor collective Units,,
bers confift- but only decimal Figures and as the Value of decimal' Figures can only be
ing only of
decimal
known from the Places they ſtand in; the Place of the fimple Units is diítin-
guiſhed by a Cypher to which a Comma is prefixed denoting that
repreſented, there are no fimple Units, and that all the Figures to the right Hand are
decimal Figures.
parts are
of the com-
It likewiſe often happens, that a Number not only has no principle
Units, but alſo has no Tenths, nor Hundredths, nor Thoufandths, &c. and
has decimal Figures of an inferior Denomination. In this Cafe in order to
affign to each decimal Figure its proper Place, not only the Place of the
fimple Units is filled up with a Cypher to which a Comma is prefixed, but
alfo Cyphers are wrote in the Places of the decimal Figures that are want-
ing; fo that a Number which has neither Units nor Tenths, nor Hun-
dreths, nor Thoufandths, will have four Cyphers to the right Hand..
For Example, (0,000289) will reprefent 289 Millionths..
XII.
Cyphers an-
Since the Comma prefixed to a Figure determines it to reprefent fimple
nexed to the Units; we may annex as many Cyphers as we pleaſe to the right Hand of
right hand
the Comma, without altering the Number to the right Hand of which
ma or fepa- thofe Cyphers have been annexed. For Example, the number 389 will
rating point not be altered by writing it thus (389,000) and will always reprefent
do not alter three Hundred and Eighty nine fimple Units; becauſe the Cyphers annexed
after the Comma, denote that there are no Tenths, no Hundreths, no
Thouſandths, nor any Kind of decimal Parts.
of a number
its value.
NUMERAL ARITHMETICK
9
By annexing Cyphers to the right Hand of the Comma of a Number,
the Denomination of the principle Unit may be changed, or it may be re-
duced into ſmaller Units. For Example, if it was required to reduce the
Units of the Number 389 into Hundreths; by writing it thus (389,00) it
will reprefent 38900 Hundreths. If it be required to reduce the Units of
this Number into ten Thouſandths by writing it thus (389,0000) it will
repreſent 3890000 ten Thaufandths, and fo of any other Transformation.
XII.
ons perform-
duced to ad-
After having fhewn how to read and write Numbers, we now proceed The diffe-
to explain the Operations performed upon them, which are reduced to the rent operati-
Four following. Addition, Subtraction, Multiplication and Divifion; all ed on num-
the other more complicated Operations being only different Combinations bers are re-
of thoſe. We may perform thofe Operations either upon Numbers dition fub-
that have only one principle Unit, as the Pound sterling, the Foot, or any traction
other Unit affumed at Will, or eſtabliſhed by Cuftom, which are called multiplicati-
fimple Numbers; or upon Numbers which have ſeveral principle Units, fion.
fuch as 5 Pounds 6 Shillings and 8 Pence, 3 Fathoms 4. Feet 9 Inches, &c.
which are called compound Numbers. We fhall treat of the Addition,
Subtraction, Multiplication and Divifion of fimple Numbers in this
Chapter.
XIII.
Addition is an Operation by which a Number is found equal to the Sum
of feveral Numbers.
on and divi-
added ſhould.
confiſt of
As all the Units of the fame Number ſhould be of the fame Species, The num-
it is eaſy to perceive that the Numbers to be added together in order to form bers to be
but one, fhould confift of Units of the fame Species, or that may be reduc-
ed to Units of the fame Species. For Example, if it was required to add units of the
together 20 Pounds fterling, 50 Horfes, and 60 Feet, three Numbers con- fame fpecies,
fifting of Units of different Species, and not reducible to the fame Species-
of Units, we are not fimply to add together 20, 50, and 60, their Units
being different, but fet them down feparately with their particular Deno-
minations. In like manner if it was required to add together 2 Fathoms, 5 Feet,
and 8 Inches, we are not fimply to add together 2, 5, and 8, their Units
being different. However as thofe different Units are reducible to the ſame
Species, fince 2 Fathoms are equal to 144 Inches, 5 Feet are equal to 60
Inches, and that a Number may be found equal to the three Numbers 144
Inches, 60 Inches, 8 Inches, we can add together 2 Fathoms, 5 Feet, 8-
Inches, fo that their Sum fhall form but one Number, by reducing all
the Units to the fame Species
XIV.
To perform this Operation more commodiouſly, the Computifts place Proceſs of
the Numbers to be added together under one another, in ſuch a Manner addition,
2 B
10
ELEMENTS OF
Firft exam-
ple of the
that the Figures of the fame Degree ftand in the fame vertical Column
and draw a Line under all thofe Numbers, to feparate them from that which is
to reprefent their Sum.
They afterwards add together all the Figures of the loweſt Degree,
which are in the firſt Column to the right Hand. If the Sum of thoſe Fi-
gures are lefs than 10, and confequently may be reprefented by one Fi-
gure; they fet down this Figure in this firſt Column, under the Line; but
if this. Sum exceeds 9 and cannot be reprefented but by feveral Figures;
they write the firft Figure of Units of this Sum under the Line, and the
other being of the fame Species of thoſe of the ſecond Column, is retained
to be added to them.
They perform the fame Operation upon all the Columns until they
come to the laft which contains the Figures of the higheſt Degree; and ad-
ding the Figures of this laſt Column with that or thofe that have been re-
tained from the preceding Column; they fet down the Sum in this Column
under the Line.
This Operation being performed; all the Figures under the Line repre-
fent the Sum of all the Numbers propofed to be added together.
X V.
Let the two Numbers 3456 and 4231 be given to be added together.
Thoſe two Numbers being difpofed as in the Margin, to
addition of determine one after another the Figures of their Sum, I pro-
integers.
ceed thus
1° I add together the two Figures 6 and 1 which form the
firſt Column: and as their Sum is Seven, which may be re-
preſented by the one Figure 7: I fet down 7 under the Line,
in this firſt Column,
4th Column
. +
3d Column
2nd Column 10 M
ift Column O
3
4
5
4
23
I
&
7 6 8 7
2º I afterwards add the Figures 5 and 3 of the fecond De-
gree, which form the fecond Column: and as their Sum is Eight, which
may be repreſented by the one Figure 8; I fet down 8 under the Line, in
this fecond Column,
3º In like Manner I add together the Figures 4 and 2 of the third Co-
lumn and as their Sum may be reprefented by the one Figure 6; I fet
down 6 under the Line, in this third Column.
4° Finally I add together the Figures 3 and 4 of the fourth and laſt Co-
lumn: and their Sum being reprefented by the one Figure 7; I write 7 un-
der the Line, in this third Column,
The Operation being finished; I find under the Line the four Figures
7687, which reprefents the Number feven Thoufand fix Hundred and
eighty Seven, to which amounts the two Numbers 3456 and
were propofed to be added together,
4231 which
NUMERAL ARITHMETICK.
II
XVI.
ample of the
Let the four Numbers 5874,9956,459,15 be given to be added together. Second ex-
Theſe Numbers being difpofed as in the Margin, to deter-
mine one after another, the Figures of this Sum, I proceed
as follows.
5874 addition of
9956 integers.
459
I 5
16304
1º I add together the Numbers repreſented by the Figures
which form the firſt Column, faying: 4 and 6 is 10, and 9 is
19, and 5 is 24. As this Sum 24 is reprefented by two Fi-
gures, viz. 4 which reprefents 4 Units, and by 2 which re-
prefents 2 Tens, I write 4 in the firft Column under the Line, and I retain
the 2 or rather the two Tens to add them to the four Numbers of Tens
which form the ſecond Column.
is 14,
2º.'I next add the Numbers of the fecond Column, whofe Figures confift
of Units of the ſecond Degree, faying: 2 I retained from the firit Column
and 7 is and 5 is
9,
and 5 is 19, and 1 is 20. As this Sum 20
Units of the fecond Degree, is reprefented by two Figures, viz. 2 which
denotes that there are two Tens of the Units of the ſecond Degree, which
form 2 Units of the third Column or of the third Degree, and o which ·
denotes that there are no Units of the fecond Degree that were to be
added; I write o under the fecond Column, and I retain 2 to add it with
the Figures of the third Column.
4 1s
3º. In like manner. I add together the Figures of the third Column
which repreſent Numbers confifting of Units of the third Degree, faying:
2 I retained from the fecond Column and 8 is 10, and 9 is 19, and
23. As this Number 23 is repreſented by two Figures, viz. by 3 which
denotes three Units of the Column which were added, and by 2 which
repreſents two Tens of Units of the fame Column, or two Units of the
fourth Column, I fet down 3 under the Line in the third Column, and I
retain 2 to add it with the fourth Column which follows.
4°. Finally I add together the Figures of the fourth Column with what
I retained, faying 2 that I retained and 5 is 7, and 9 is 16. This Sum con-
fifting of Units fimilar to thofe of the fourth Column, and of an Unit of
a higher Degree, I write 6 under the Line in the fourth Column. As to
the Unit of a higher Degree, I would have retained it, if there was another
Column of Figures to be added, but as there is not, I advance the 1 to
the left Hand of the 5, that is I fet down fimply 16.
XVII.
Example
numbers
Let the four Numbers (5874; 4631,752; 6872,44; 9797,5) the
three laft of which contain decimal Figures, be given to be added together. of the ad
I ſet down all thofe Numbers one under another, fo that the Figures of dition of
the fimple Units are in the fame Column, and all the other Figures of containing
the fame Denomination are likewiſe in the fame Column one under ano- decimal fi
ther. Having difpofed the Figures in this Manner.
gures
1
12
ELEMENTS OF
decimal
numbers
performed
by reducing
nits of the
1º. I add the Figures of the loweſt Degree, which
are Thouſandths and as there are but 2 Thouſandths, I fet
down 2 under the Line in the Column of Thousandths.
2º. I add together the Figures 5 and 4 of the follow-
ing Column which is that of the Hundreths; and as
their Sum 9 is repreſented by one Figure, I fet down
this Figure 9 in this Column under the Line.
5874
4631,752
68.72,44
9797,5
27175,692
3º. I add together the Figures of the Column of Tenths, ſaying 7 and
4 is II, and 5 is 16; and as this Sum 16 confifts of 6 Units of the De-
gree of this Colnmn, and of a Ten which makes an Unit of the follow-
ing Column; I fet down 6 under this Column and I retain I to add it to
the following Column.
4°. Adding the Unit I retained to the Figures of the Column of the
Simple or principle Units, I find 15, that is, 5 Units of this Column, and I
Ten which makes an Unit of the following Column; whence I write 5 un-
der this Column, and I retain 1 for the following Column.
I
50. Adding in like Manner the Unit I retained to the Column of Tens,
I find 27, that is 7, Units of this Column, and 2 Tens which make two
Units of the following Column: whence I write 7 under this Column
and I retain 2 for the following Column.
I perform the fame Operation upon the other Columns, and I find
(27175,692) for the Sum of the four propofed Numbers.
XVII.
The Numbers (5874; 4631,752; 6872,44; 9797,5) which were
Addition of given to be added together in the laft Example, might be changed, or re-
duced ſo as to have principal Units of the fame Denomination: and as one
of thoſe Numbers (4631,752) repreſents 4631 fimple Units and 752
Thouſandths, or 4631752 Thousandths; the other Numbers may be reduc-
ed fo as to have Thousandths for their principal Units: then the Number
5874 which reprefents 5874 fimple Units," will become 5874000 Thou-
Jandths, the Number (6872,44) will become 6872440 Thouſandths, and
the Number (9797,5) will become 9797500 Thousandths. All the propof-
ed Numbers being reduced fo as to confift of Units of the fame Denomina-
tion, they may be added as Numbers that have no decimal Figures, and
their Sum will be 27175692 Thousandths: to fupprefs afterwards the Word
Thouſandths,it fuffices to fet down the fame Sum thus27175,692 (Art. x).
fame denc-
mination.
Quantities
to be fub-
tracted one
from the o-
ther fhould
be of the
fame ſpecies,
XIX.
The Operation by which one Quantity is deducted or taken from ano-
ther, is called Subtraction.
The Quantity therefore to be deducted muſt be contained in the Quantity
from which it is to be deducted; and confequently thoſe two Quantities
muſt be of the fame Species, or reducible to the fame Species,
NUMERAL ARITHMETICK.
13
'To perform this Operation more commodiouſly, the Computifts fet down Procefs of
the Number to be deducted under the Number from which it is to be fubtraction.
deducted, placing the Units of the one under the Units of the other, the
Tens under the Tens, the Hundreds under the Hundreds, &c, and the
decimal Figures of the fame Species under one another. The two Num-
bers being thus diſpoſed, a Line is drawn to feparate them from their dif-
ference fought.
Then the Subtraction is performed by Parts, each lower Figure is deducted
from the Figure that ſtands over it, beginning at the right Hand, and pro-
ceeding from the Figures of the loweſt Degree to thoſe which are of a
higher Degree. But in this Operation, there occur three Cafes; either
the lower Figure is lefs than the Figure that ftands over it, or equal or
Greater.
If the lower Figure is less than the Figure that ftands over it, the For-
mer may be eaſily deducted from the Latter, and there will be a Remainder
which is fet down under the Line,
If the lower Figure is equal to the Figure that ftands cver it, the
former may alſo be deducted from the latter, and as there is no Remainder,
a Cypher is fet down under the Line for the remainder.
Finally if the lower Figure is greater than the Figure that ftands over
it, the Former cannot be deducted from the Latter without a Preparation
which confifts in borrowing an Unit from the Figure of an higher Degree
and adding it to the Figure which is too Small: then the Figure from
which an Unit is borrowed is accounted one Lefs than it is, and this Unit
being carried to a Place one Degree lower, is accounted as Ten, confe-
quently the Figure which is too Small being increaſed by Ten, the Figure
immediately under it which cannot exceed 9 may be deducted from it.
XX.
Let it be required to deduct 324 from 466,
Having fet down the Number to be deducted under the Number
from which it is to be deducted, placing the Figures of the fame
-Denomination under one another, and having drawn a Line as in
the Margin,to determine the Figures of the Remainder one af-
ter another, I proceed as follows.
if from 6 be taken
1°. Beginning at the place of Units, I fay
4 there will remain 2, which I fet down under the Line in the
Units Place.
4.66
324
1 4 2
2º. Then proceeding to the following Rank, I fay: If from 6 be taken
2, there will remain 4 which I fet down in the fecond Rank.
3°. Lastly, being come to the laft Rank, in which are the Figures of the
higheſt Degree, I fay: if from 4 be taken 3, there will remain I which I
write under the Line in the third Rank, and the Operation being finished,
the Remainder of the Subtraction will therefore be 142.
Firft exam-
ple of the
fubtraction
of integers.
14
ELEMENTS OF
Second ex-
ample of
the fubtrac-
tion of inte-
gers.
Third exam-
ple of the
fubtraction
of integers.
XXI.
Let it be required to deduct 51 from 552.
Having placed the Numbers as in the Margin. 1° Beginning by
the Figures of the loweſt Degree, Ifay; if from 2 be taken 1, there 5 5 2
will remain 1 which I write under the Line.
I
whence I fet down a
2º. Proceeding to the following rank, I fay: if from 5 be taken
5, there will nothing remain ; whence I fet down a Cypher un-
der the Line to denote that the Figures reprefenting Tens being
Subtracted one from the other, there is no Remainder.
5 I
501
3º. Laftly proceeding to the third Rank in which there is no Figure to
be deducted, I ſay: if from 5 be taken Nothing, there will remain 5;
whence I fet down 5 for the Remainder under the Line in this third Rank.
The Operation being finiſhed, the Remainder required will be 501.
XXII.
Let it be required to deduct 599 from 758.
Having difpofed the Figures of the fame Denomination under
one another, as in the Margin, I proceed as follows.
1º. Beginning by the Figures of the loweſt Order, becauſe 9
cannot be deducted from 8, I take from the following Figure (5)
of the upper Number an Unit, to add its Value to 8 which is too
Small and as this Unit carried to the Place of 8 becomes 10,
from 18 I deduct 9, and there remains 9, which I fet down
under the Line in the firſt Place.
7 5 8
599
I 59
2º. Proceeding to the following Rank in which 5 is reduced to 4, on
account of the Unit that has been taken from it, and which is denoted by
a Point fet over it, becauſe 9 cannot be deducted from 4, I take an Unit from
the following Figure (7) of the upper Number to add its Value to the 4
from which 9 is to be deducted, and as this Unit of the third Degree
carried to an inferior Rank becomes 10, from 14 I deduct 9, and there re-
mains 5 which I fet down under the Line.
3º. Proceeding to the Rank of Hundreds in which 7 is reduced to 6, on
account of the Unit which has been taken from it, and which is de-
noted by a Point fet over it; I fay if from 6 be taken 5 there will remain
I which I fet down under the Line; whence the Remainder required will
be 159.
XXIII.
It often happens that the Figure from which an Unit is to be borrowed
is a Cypher, which reprefenting nothing, can lend nothing; in this Caſe
the Computiſts borrow from the Figure to the left Hand of the Cypher or
Cyphers, if there are feveral; and leaving 9 above each of thoſe Cyphers,
referve only Ten to add it to the Figure from which the one immediately
under it is to be Subtracted,
i
NUMERAL ARITHMETICK.
is
XXIV.
وو
Fourth ex-
700 4 ample of
5 4 8 6 the fubtrac
1 5 1 8
Let it be required to deduct 5486 from 7004.
Having difpofed the Figures as in the Margin, I proceed as fol-
lows, becauſe 6 cannot be taken from 4 and I cannot borrow from
the Figure which is next to 4, to the left Hand, nor from the fol-
lowing, becauſe they are two Cyphers; I borrow an Unit from
the 7 which is next to thofe Cyphers. But as an Unit borrowed
from the 7 is equavalent to 100 Tens, and only one Ten is required to
add it to the 4 from which 6 is to be deducted, I leave above the Cyphers
which are in the third and fecond Places the two Figures 99 which repre-
fent 99 Tens: and adding the Ten to 4 I deduct 6 from 14, and there re-
mains 8 which I fet down under the Line for the firft Figure of the Re-
mainder fought.
The upper Number being thus prepared, the 7 from which I has been
borrowed is reduced to 6, and the two Cyphers over which the 9 have
been left, are now to be confidered as 9. To continue therefore the Subtrac-
tion, I deduct 8 from 9, there remains I which I write under the Line.
I afterwards deduct 4 from 9, and there remains 5 which I fet down like-
wife under the Line. Laſtly I deduct 5 from 6, and there' remains L
which I alfo fet down under the Line.
The Operation being finiſhed the Remainder required will be 1518.
XXV.
tion of inte-
gers,
numbers is
When there are decimal Figures in the Number to be deducted or in How the
the Number from which it is to be deducted, or in both; the Computits fubtraction
firſt diſpoſe under one another the Figures of the fame Degree; and when of decimal
one of the two Numbers has not as many decimal Figures as the other; performed.
they annex Cyphers to the right Hand of that which has the Leaft, to make
the decimal Figures equal in both; which will not change the Value of this
Number, fince the Comma preferves to each Figure the Place it had before
the Cyphers were annexed.
The two Numbers being thus prepared and their Units of the loweſt
Degree being of the fame Kind, the Subtraction is performed as if the
Numbers had no decimal Figures.
XXVI.
Let it be required to deduct 8716,257 from 230009, 3.
As thofe two Numbers have not the fame Quantity
of decimal Figures, the Second having two more than
8716,257
Example
of the ſub-
traction of
the firft; I annex two Cyphers to the right Hand of 230009,300 decimal
the Firſt, and the two propoſed Numbers being
transformed into theſe fet down in the Margin: whofe
Units of the loweſt Degree are Thousandths, I perform
the Subtraction as on Numbers thathave no decimal
Figures.
221293,043
16
ELEMENTS OF
How the
XXVII.
When there are Figures in the Number to be deducted greater than the
foregoing correfponding ones in the Number from which it is to be deducted, and
operation of
fubtraction when placed under one another, to perform the Subtraction, Units
may be con- are borrowed from the upper Figures of higher Degrees, to add their Va-
tracted..
An exam-
lues to the Figures which are too Small; inſtead of taking from the upper
Figures the Units that have been borrowed before the Figures that are
underneath are deducted from them, it is eafy to perceive that the Unit
borrowed from an upper Figure may be added to the Figure under it; and
fo from the upper Figure be deducted at once the Unit that was borrowed:
and the Figure under it.
畔
​XXVIII.
Let it be required to deduct 5486 from 7004.
Having placed the Numbers as in the Margin, to find all the
Figures of the Remainder, I proceed as follows
ple of fub-
traction per-
1º. Becauſe 6 cannot be taken from 4, I add ten to 4 and de-
formed by
the contract- duct 6 from 14, fetting down the Remainder 8 under the Line..
ed method.
1
Subtra&icm
7004
5486
1 5 1 8
2º The Ten added to the 4 ſhould have been borrowed from the Cy-.
pher which follows the 4. Confequently as this Unit fhould be taken
from this Cypher, as alfo the Figure 8 which ſtands under it, I add the
Unit borrowed to the 8, to deduct their Sum 9 from the Cypher but as
this cannot be done, I borrow an Unit of a higher Degree, which being
carried to the Place of the Cypher is equivalent to Fo; and deduct 9 from
10, and fet down the Remainder under the Line.
:
3º. As the Unit borrowed fhould be deducted from the fecond Cypher,
as alſo the 4 which flands under it, I add the Unit borrowed to the 4, to
deduct their Sum 5 from the Cypher which is above it ; but as this can-
not be done; I borrow an Unit of a higher Degree, which being carried
to the Place of the Cypher is equivalent to 10, and dedu&t 5 from 10,.
and fet down the Remainder 5 under the Line.
4°. Laftly as the Unit borrowed ſhould be taken from the 7, as alfo
the 5 which is under it, I add this Unit to the 5, and deducting their Sum
6 from 7, I fet down the Remainder 1 under the Line.
The Operation being finiſhed, the difference required will be 1518..
XXIX.
To diſcover whether any Error has been committed in computing, a
employed to particular Operation is employed, which is called the Proof.
prove additi-
on.
4
In Addition, as the Sum or total Amount ſhould contain all the Num--
bers which are added together, if from the Sum be deducted all the Parts
which were added, there fhould be no Remainder.. Addition therefore
may be proved, by deducting from the Sum all the Parts of the Numbers.
added together, the Operation being acurately performed, if after the
Subtraction there is no Remainder.
*
1
NUMERAL
1-7
ARITHMETICK.
The Order to be obſerved in this Operation, will appear in
the following Example: in which the three Numbers 473,
567,924 were added together as in the Margin, and their
Sum found to be 1964.
If this Sum 1964 is exact and contains all the Parts of the
Numbers added together; it is manifeſt that if all the Hundreds
all the Tens and all the Units of the Numbers added together,
be deducted from it, there fhould be no Remainder.
47 3
567
924
I 9 6 k
I I Ø
The Subtraction might be performed beginning at the Units
Place as ufual; but as the Units, Tens and Hundreds would then
be collected in the fame Order they were added together to find their Sum,
and confequently this Operation would become liable to the fame Errors
that might have been committed in performing the Addition. The Com-
putifts perform the Subtraction beginning by the Figures of the higheſt
Degree.
To deduct, therefore 1° the Hundreds of the Numbers added together,
from the Sum of thofe Numbers, I add thofe Hundreds, faying: 4 and
5 is 9, and 9 is 18; and deducting thofe 18 Hundreds from the 19 Hun-
dreds of the Sum, I fet down the Remainder I under the 9, after having
barred the two Figures 19.
2º. To dedu& the Tens, I add them together, faying: 7 and 6 is 13,
and 2 is 15; and deducting thoſe 15 Tens from the 16 Tens that remain
in the Sum, I fet down the Remainder 1 under the 6, after having barred
the two Figures 16.
I
3º. Laſtly I add the Units, faying: 3 and 7 is 10, and 4 is 14: and
deducting thoſe 14 Units from the 14 Units which are ftill in the Sum,
and there being no Remainder: I fet down a Cypher under the 4, after hav-
ing barred 14.
As there is no Remainder left after this Subtraction, I conclude, that the
Addition of the three Numbers 473,567,924, has been acurately per-
formed.
XXX.
Process of
this operati-
on explained
by an exam-
ple.
prove fub-
In Subtraction as the Quantity deducted is contained in the Quantity
from which it is deducted, and the Remainder is alfo contained in this Addition
Quantity; fo that the Quantity deducted and the Remainder are all the employed to
Parts of the Quantity out of which Subtraction is made: if the Quan- traction.
tity Subtracted be added to the Remainder; their Surn will be equal to the
Quantity out of which Subtraction has been made. Subtraction therefore
may be proved by adding the Number fubtracted to the Remainder, and
their Sum, if the Operation has been acurately performed, will be equal to
the Number out of which Subtraction has been made.
2 C
18
ELEMENTS OF
t
5 48 6
Let, for Example, 5485 be deducted from 004, and the Re-
Procefs of mainder be found to be 1518, as in theMargin, adding 1518 to
7 0 0 4
this operati-
on explained 5486, Ifind their Sum 7004 to be equal to the Number out of
in an exam- which Subtraction has been made, from whence I conclude that
the Subtraction of 5486 from 7004 has been acurately per-
formed.
ple.
The particu-
lar denomi-
nations given
to the num-
beit con-
cerned in
multiplicati-
OB.
11.
XXXI.
I 5 1 8
7 0 0 4
Multiplication is an Operation by which a Quantity is repeated a certain
Number of Times.
Two Numbers therefore are required to perform a Multiplication,
1° The Number to be repeated, which is called the Multiplicand.
2º The Number that denotes by the Number of its Units how often the
Multiplicand is to be repeated, and which is called the Multiplicator.
The Multiplicand and the Multiplicator are alfo called the Factors of the
Multiplication; and the Number which contains the Multiplicand as often
as the Multiplicator contains Unity, is called the Product.
For Example, if it was required to multiply 8 by 4, thofe twp Num-
bers 8 and 4 will be the two Factors of the Multiplication; the firſt (8)
which is to be repeated will be the Multiplicand; the fecond (4) which de-
notes that the Multiplicand is to be repeated 4Times,will be the Multiplicator;
laftly the Number 32 found by repeating 8 four Times, will be the Product
XXXII.
To denote that two Numbers are to be multiplied one by the other, the
The fignx Computifts place between them the Note x which fignifies Multiplied by.
is employed Thus 8 x 4 denotes that 8 is to be multiplied by 4; and as the Produ&
arifing from this Multiplication is 32,it may be faid that 8 × 4 is equal to 32.
If it was required to multiply again this Product 8 x 4 or 32 by a new
Multiplicator 2; the Computifts would write down 8 x 4 × 2 or 3 2× 2,
whoſe Product is 64; that is, they fet down one after another all the Fac-
tors that enter into the Product, feparating them by the Note X.
to denote
that two
quantities
are to be
multiplied
one by the
other.
Multiplication may be performed by reducing it to Addition, for Example,
the Multiplication of 32 by 4, might be performed by fetting down the Mul-
Inconveni- tiplicand 32 four Times in the Order that Quantities are fet down which are
ency of re- to be added; the Sum of this Addition containing 32 four Times, will be
ducing mul- the Product of 32 multiplied by 4. But as this Method of performing
tiplication to
addition. Multiplication could only be applied when the Multiplicator confifts of
very few Units; being impracticable when the Multiplicator is a great
Number. The Computifts were under the Neceffity of employing parti-
cular Rules to abridge the Operations.
•
XXXIII.
The Product therefore of a Multiplication will confift of Units of the
fame Species as thofe of the Multiplicand; for this Product reſulting from
the repeated Addition of the Multiplicand, can have no other Units than it,
NUMERAL ARITHMETIC K.
19
will
32
The product
of two fi-
gures multi-
From whence it follows, 1° that if the Units of the loweft Order
of the Multiplicand, are fimple Units, or Tens, or Hundreds &c. and the
Multiplicator confifts only of fimple Units, the Units of the loweſt Order
of the Product will be alfo fimple Units, or Tens, or Hundreds &c.
whence the Produ&t ſhould have at the right Hand as many Places of Fi-
gures as there are at the right Hand of the Multiplicand. For Example,
if 8 or 80 or 800 be multiplied by 4, which will be effected by repeating
the fignificative Figure 8 of the Multiplicand four Times, the Product
have to the right Hand as many Cyphers or Places as there are after the
Figure 8 which has been multiplied, and will be 32 or 320 or 3200.
2° If the Units of the Multiplicator are collective Units, as Tens, or Hun- one another
dreds or Thouſands &c. each of thofe collective Units will denote that the as many
Multiplicand is to be repeated ten Times, or a hundred Times, or a thou- places to the
right hand
fand Times: fo that the Product will be ten Times, a hundred Times, or a as there are
thoufand Times greater than if the Multiplicand was multi; lied by a. Num- places to the
ber of fimple Units; and confequently this Product will have alfo to the right hand in
the multipli-
right and as many Places as there are to the right Hand of the Figure em- cand and
ployed for Multiplicator.
multiplica-
For Example, if 8 be multiplied by 4, or by 40, or by 400, or by
4000, &c. which will be effected by repeating 8 four Times; the Product
32 will have in all thofe Cafes as many Cyphers or Places to the right
Hand as there are to the right Hand of the Multiplicator 4, and confequent-
ly will be 32, or 320, or 3200, or 32000.
From whence we may conclude that the Product of two Figures mul-
tiplied one by the other ſhould have to the right Hand as many Cyphers or
Places, as there are Cyphers or Places to the right Hand both of the Figure
multiplied, and of the Figure employed as Multiplicator.
For Example if 800 be multiplied by 4, or by 40, or by 400 or by
4000, &c; which will be effected by repeating the fignificative Figure 8 of
the Multiplicand as often as there areUnits in the fignificative Figure 4 of the
Multiplicator; the Product 32 will have to the right Hand the two Cyphers
which are to the right Hand of 8, and all the Cyphers which are to the
right Hand of the Figure 4, employed for Multiplicator, and will confe-
quently be 3200, or 32000, or 320000, or 3200000, &c.
XXXIV.
A Number let whatever be the Nature of its Units, may be repeated
any propoſed Number of Times; whence the Multiplicand of a Multiplica-
tion may be an abſtract Number, or a concrete Number confifting of any
particular Species of Units.
plied into
fhould have
tor.
always an
It is not fo of the Multiplicator which fhould denote how often the The multi-
Multiplicand is to be repeated. Each of its Units whether Simple or Collec- plicator is
tive ſhould only repreſent a Number of Times; whence it ſhould neceffarily be abſtract
confidered as an abftra&t or abfolute Number,and never as a concrete Number.
number.
20
ELEMENTS OF
In whatever
order the
factors of
a multipli-
cation are
difpofed the
However it is ſometimes propofed to multiply a concrete Number by a-
nother concrete Number. For Example, if an oak Board coft 5 Shillings and
the Price of 20 Boards be required at that Rate, it is propofed to multiply 5
Shillings by 20 Boards, but it is manifeft that this Propofition is contrary to
the Rules of Multiplication and that 5 Shillings is not to be multiplied by
the concrete Number 20 Baards, but by the abfolute Number 20 Times.
Since to obtain the Price of 20 Boards at the Rate of 5-Shillings per Board,
it ſuffices to repeat 5 Shillings 20 Times.
XXXV.
The Multiplicand remaining the fame, if the Multiplicator becomes dou-
ble, or triple, or quadruple, &c. of what it was; the Product will alſo be-
come double, or tr ple, or quadruple, &c. of what it was; becauſe the new
Product will contain the fame Multiplicand Twice, Thrice, or four Times
&c. oftner than the former Product contained it.
The Multiplicator remaining the fame, if the Multiplicand becomes
prodnat will double, er triple, or quadruple, &c. of what it was; the Product will
be always alſo become double or triple or quadruple &c. of what it was ; becauſe
the fame.
the new Product will contain a Multiplicand double, or triple, or quadru-
ple, &c. as often as the first Product contained the fimple Multiplicand;
and it is manifeft that a Whole ſhould become double, triple, or quadruple,
when its component Parts become double, or triple, or quadruple, &c. of
what they were,
1
From whence it follows 1° that the Product of two Factors, fuch as
7 X 5, will be multiplied by 2, or by 3, or by 4 &c. by multiplying one
of its Factors, either the Multiplicand 7, or the Multiplicator 5, by the
new Multiplicator 2, or 3, or 4, &c. becauſe by thus multiplying one of the
two Factors, their Product is rendered double, triple, or quadruple, &c.
of what it was.
It follows alfo from thence, that if any two Numbers, for Example, 3
and 7 are to be multiplied one by the other, there will always refult the
fame Product whether we multiply 3 by 7, or 7 by 3, and conſequently
whether we write 3 X 7 or 7 X 3.
For every Number may be confidered as a Product of itlelf into Unity.
Thus IX 7 is the fame Thing as 7, and confequently the Product of IX 7
multiplied by 3, is the fame Thing, as the Product of 7 multiplied by 3.
But to multiply 1 X7 by 3 it fuffices to triple the Multiplicand 1, which
will give 3 X 7; and to denote that 7 is to be multiplied by 3, it is fet
down thus 7 X 3.
Wherefore 3 X 7 is the fame Thing as 7 X 3; that is, when two Num-
bers are to be multiplied one by the other, it is indifferent which of them
is taken for Multiplicand or Multiplicator.
From whence it follows 3º, that if there are three Numbers fuch as
2,3,7 to be multiplied together, their Product will always be the fame in
NUMERAL ARITHMETICK.
21
whatever Order they are multiplied, fo that they may be fet down the fix
different Ways following 2 X3 X 7,3 X 2 X 7,3 × 7×2, 7 × 3 × 2,
7 X 2 X 3, 2 X 7 X 3. For fince the two Produ&s 3× 7, 7 × 3, are e-
qual; if they be multiplied by the fame Number 2, the two refulting Pro-
ducts will be alfo equal: and as the two Products 3 x 7, 7X 3, may be.
multiplied by 2, either by multiplying the Factor 3, or by multiplying
the Factor 7, and that the new Factor 2 may be fet down either before or
after the Figure to be multiplied, there will reſult ſix different Arrange-
ments of the three Factors 2, 3, 7. If there were a greater Number of Factors
to be multiplied together, whatever way they are ranged, it is eaſy to per-
ceive that the Product will be always the fame.
XXXVI.
To find the Produ& of two Numbers reprefented each by one Figure,
that is of two Numbers, each of which is less than 10. The Computifts
employ two different Methods. The first is a Table which Pythagoras is
faid to have invented, containing the Products of all the Numbers that may be
expreffed by one Figure.
This Table which con-
fifts of nine Rows of nine
equalSquares,is conſtruct-
ed thus. In the first ho-
rofontal Row is fet down
the Series of natural
Numbers from 1 to 9. In
the firſt Square of the
fecond horofontal Row is
fet down 2, and in the
remaining Squares of this
Row are fet down the
Numbers found by con-
tinually adding 2. In the
firft Square of the third
horofontal Row is fet
down 3, and in the re-
maining Squares of this
Row are fet down the
I
นา
14 16 18
2
3
4 5
6
7 8 9
2 4
68
IO
I 2
3
6
9
I 2
15
18
8
I 2 16
20
4
QO
5
ΙΟ
6
N
21
24 27
24 28 32 36
15 20 25 30 35 40 45
| 18|24| 30 | 36 | 42 | 48
4248 54
283542
35 42 49 56 | 63
12
7
14
2.I
| 28
8
16|
2432
32 40 48 5664 72
918 27 36 45
5463281
Numbers found by continually adding 3. In like Manner in the firſt
Square of each of the following Rows is fet down 4,5,6,7,8,9, and
in the remaining Squares of each Row, the Numbers found by continually
adding the Number which is fet down in the firſt Square of this Row.
To make Ufe of this Table in the Multiplication of Numbers less than
10, the Multiplicand is to be looked out for in the firſt horofontal Row,
and then deſcending to the horofontal Row that begins by a Number equal
First method
of finding
the product
oftwo auni-
bers lefs than
10 by the
table of
Pythagoras,
22
ELEMENTS OF
Second me-
thod for
finding the
product of
two num-
bers lefs than
10.
*Proceſs of
plication of
cator is re-
to the Multiplicator, the Product required will be found. For Example, if it
was required to multiply 8 by 7: the Figure 8 is to be looked out for
in the firſt horofontal Row, and then defcending to the horofontal
Row, that begins by 7, the Product of 8 into 7 required, will be found to
be 56.
XXXVIL
The fecond Method employed by the Computiſts to multiply two
Numbers leſs t'an 10, one by the other, confiíts in working with the Fin-
gers; but for this Purpofe it is neceffary to know how to multiply two
Numbers leſs than 6 one by the other. The Manner of performing this
Operation is as follows.
The Computift fhuts both his Hands, and attributing a Number to each
Hand, he lifts up as many Fingers in each, as there are Units from the
Number that is attributed to it,to Ten; then multiplying the Number of Fin-
gers lifted up in one Hand, by the Number of Fingers lifted up in the
other, he adds to the Product as many Tens as there are Fingers down in
both Hands.
Let it be required, for Example, to multiply 8 by 7. attributing 8 to
the right Hand, I lift up two Fingers, becauſe there are two Units from
8 to 10, and there remain three Fingers down in this Hand; attributing 7
to the left Hand, I lift up three fingers, becauſe there are three Units
from 7 to 10, and there remain two Fingers down in this Hand. As
there are three Fingers lifted up in one Hand, & two Fingers lifted up in
the other, I multiply 3 by 2, which gives 6 Units for the Product; and be-
cauſe there are five Fingers down in all, viz Two in .one Hand, and Three
in the other, I add 5 Tens or 50 to the 6 Units already found, and there
refults 56 for the Product of 8 into 7.
XXXVIII.
To multiply any Number which has no decimal Parts, by another
the multi- which has none alfo, and that is reprefented by one Figure. The Com-
whole num-putiſts fet down the Multiplicator under the Multiplicand, and drawing a
bers when Line to feparate the two Factors of the Multiplication, from the Product
the multipli- which is fet down under it; they multiply each Figure of the Multiplicand
prefented by one after another, beginning with the Units Place, by the Multiplicator.
one fignifi- When each Product can be expreffed by one Figure, it is fet down under
cativefigure. the Figure multiplied. But when fome Products cannot be expreffed but by
two Figures, the First, which is of the loweſt Degree is fet down under the
Figure multiplied, and the other is retained and added to the following
Produc. All the Figures of the Multiplicand being thus multiplied, and
all the particular Products fet down, all the Figures under the Line will
reprefent the Product of the Multiplication required.
.
NUMERAL ARITHMETICK
23
XXXIX.
Let it be required to multiply 964 by 4.
Having fet down thofe Numbers as in the Margin, to find
the Figures of this Product one after another.
Firſt exam-
964 ple of the
4 multiplica-.
tion of inte
3856 gers.
1° Begining at the Units Place of the Multiplicand, I fay:
four times 4 is 16; and becauſe this firſt Produ&t is repreſented
by two Figures the Firft of which (6) repreſents fimple Units, and thé
fecond (1) a Ten or an Unit of the fecond Degree; I fet down 6 in the
Units Place under the Figure multiplied, and I retain 1 to add it to the
following Produ&t whofe Units will be of the fecond Degree.
2° Proceeding to the fecond Figure (6) of the Multiplicand which re-
prefents fix Tens or fix Units of the fecond Degree; I fay: 4 times 6 is
24, and I that I retained from the foregoing Product makes 25, or rather
25 Tens or Units of the fecond Degree, and I fet down the first Figure (5)
in the fecond Place; retaining the fecond Figure (2) which reprefents two
Tens of Tens, or two Units of the third Degree, to add it to the following
product whofe Units will be alfo of the third Degree.
3° Proceeding to the next Figure of the Multiplicand, I ſay: 4 times 9 is
36, and 2 that I retained of the foregoing Product make 38, that is 38 Hun-
dreds or Units of the third Degree, and I fet down the firſt Figure (8) in
the third Place, as to the fecond Figure (3) which reprefents three Units of
the fourth Degree, I would have retained it to add it to the following Pro-
duct, if there remained a Figure to be multiplied; but as the whole is multi-
plied, I fet down this Figure 3 in the fourth Place to the left Hand of the
Figure 8.
All the Figures of the Multiplicand 964 being Multiplied by 4, and the
Figures of the particular Products being fet down one after another, in the
Manner we have explained; the Number 3856 under the Line is the Pro-
duct required.
XL.
Let it be required to multiply 964 by 60.
Having fet down thefe Numbers as in the Margin. I ob-
ferve that if the Number 964 was to be multiplied by 6, the
Product would be 5784 but the Multiplicator 60, being de-
cuple of 6, the Product required fhould be decuple of the
Product 5784 arifing from the Multiplication by 6.
964
60
Second ex-
ample of the
multiplica-
tion of inte-
gers,
5.7840
But according to the Rules of Numeration if each Figure of a Num-
ber be removed one Place towards the left Hand, which is effected by
fetting down a Cypher to the right Hand of this Number; it will be ren-
dered decuple of what it was. Wherefore fetting down a Cypher to the
right Hand of the Number 5784, we will have 57840 for the Product re-
quired of 964 multiplied by 60.
24
ELEMENTS OF
XLI.
Let it be required to multiply 964 by 200.
Multiplying 964 by 2 Units, I find 1928 for the Product
but the Multiplicator 200 being centuple of 2: the Product
plication of required fhould be centuple of 1928 and confequently the
Third ex-
ample of
the multi-
integers.
Procefs of
the multipli-
;.
964
200
0.
fimple Units of this Number ſhould be, Hundreds or Units of 1 9 2 8 0 0
the third Degree, as thofe of the Multiplicator 2 Hundreds.
Wherefore the Number 1928, ſhould have two Cyphers or two Places to
the right Hand, to express the Product of 964 by 200.
It will appear in like Manner that if the Multiplicator be reprefented
by a Figure whofe Units are of the fourth or fifth Degree, the Multipli-
cand is to be multiplied by this Figure, as if it reprefented only Units of
the firſt Degree, and as many Cyphers are to be fet down to the right Hand
of the Product as there are Places to the right Hand of the Multiplicator.
XLII.
When it is required to multiply a Number which has no decimal Parts,
cation of in- by another which has none alfo, and that is reprefented by feveral Figures.
tegers when it is eaſy to perceive that the whole Multiplicand muſt be multiplied by
the multipli- each Figure of the Multiplicator; whence there will refult, as many parti-
prefented by cular Products as there are fignificative Figures in the Multiplicator, and all
feveral fig thofe Products being added together: their Sum will be the Product required.
cator is re-
nificative fa
gures.
ple of the
XLIII.
Let it be required to multiply 964 by 264
Since the Multiplicator 264 confifts of four Units, fix
Tens, and two Hundreds, that is of 4, of 60, and of 200:
First exam- the Multiplicand 964 muſt be multiplied by thoſe three
multiplica- Numbers 4, 60, and 200. Having therefore fet down the
tion of inte- Multiplicator under the Multiplicand as in the Margin.
gers when
they are
1° I multiply every Figure of the Multiplicand one after
both repre- another by the first Figure 4, which reprefents 4 Units,
fented by fe- fetting down the Figures of the Product 3856 one after another
veral figni-
keative fi- according as I find them.
gures.
.
964
264
3856
5.7840
192800
254496
2º I multiply by the fecond Figure (6) of the Multiplicator, that is,
by 60, and becauſe the Units of the Product should be Tens, I fet down
a Cypher in the Units Place under the firſt Product, multiplying afterwards
every Figure of the Multiplicand by 6, I find 57840 for the fecond Product.
3º I Multiply by the third Figure (2), that is, by 200, and becauſe the
Units of the Product ſhould be Hundreds; I fill up with two Cyphers the
Places of the Units and Tens; multiplying afterwards by 2 every Figure
of the Multiplicand one after another, I find 192800 for the third Product.
Laftly I add together thofe three Products, and there refults 254496 for
the Product required of 964 multiplyed by 264.
1.
NUMERAL ARITHMETICK
25
XLIV.
Second ex-
43216 ample of the
30050 Multiplica-
tion of inti
2160800 gers repre-
Let it be required to multiply 43216 by 30050.
The Multiplicator confifting of two fignificative
Figures only 5 and 3, the firſt denoting 50 or 5
Tens, and the fecond 30000 or 3 Tens of Thou-
fands, there are no more than two particular Pro-
ducts to be found. Having fet down the Multiplica-
tor under the Multiplicand as in the Margin, and ob-
ſerving that the firft Multiplication by 5 tens, will
produce Tens; I fet down a Cypher in the Units
Place and multiply the Multiplicand 43216 by 5; Saying: 5 times.
6 is 30, writing down the first Character (0) in the fecond Place, and
retaining 3. I multiply the other Figures of the Multiplicand by 5, and
there refults 2160800 for the firſt particular Product.
A
?
fented by le-
veral fignifi-.
cative fi-
296-4
8
1298640800 gures.
The fecond Multiplication by 3 Tens of Thouſands will produce Tens
of Thoufands, confequently will have four Places of Figures to the right
Hand. I therefore fet down four Cyphers in the four firſt Places, or leave,
them Void: and multiply the Multiplicand 43216 by 3; and there reſults
129648 for the Second particular Product.
Laftly I add together thoſe two particular products, whofe Sum
1298640800 will be the whole Product of 43216 multiplied by 30050..
XLV.
o,
1
700800 Third exam-
ple of the
40600 multiplica-
tion of inte-
420480000 gers repre-
2803200
fented by fe-
veral fignifi-
28452480000 cative fi-
Let it be required to Multiply 700800 by 40600.-
1° Becauſe the firft fignificative Figure (6) of the
Multiplicator has two Cyphers, or two Places to the
right Hand of it, and confequently ſhould produce
Units of the third Degree, I fet down two Cyphers
in the two firſt Places; then faying: 6 times o is 03
I fet down a Cypher in the third Place, and proceeding.
to the Multiplication of the fecond Figure of the
Multiplicand, I fay 6 times o is o, ſetting down a
Cypher alfo in the fourth Place. Then I fay: 6 times & is 48, fetting
down 8 in the fifth Place, and retaining 4. I afterwards fay: 6 times o is o
and 4 I retained is 4, which I fet down in the fixth Place; then I fay: 6
times o is o, and fet down a Cypher in the feventh Place. Laftly I fay 6
times 7 is 42, and fet down 2 in the eighth Place, and 4 in the ninth Place,
and there reſults 420480000 for the first particular Product.
܂
2º Becauſe the ſecond fignificative Figure 4 tens of Thouſands of the
Multiplicator will produce Units of the fifth Degree, I fet down four Cy-
phers in the four firft Places of the fecond Product, or leave them void,
and then multiply 700800 by 4, and there refults 28032000000 for the
fecond particular Product.
Laſtly I add together the two particular Products and their Sum
28452480000 will be the whole Product required.
gures.
2 D
26
ELEMENTS OF
the multi-
numbers
XLVI.
If it be required to multiply a Number which contains decimal Parts,
Procefs of by another Number that alſo contains decimal Parts or that does not contain
plication of any. The Multiplicand is to be multiplied by the Multiplicator, as if nei-
ther of them contained decimal Parts and in the Product fo many Places
containing of decimal Figures to the right Hand are to be feparated with a Comma, as
there are decimal Parts both in the Multiplicand and Multiplicator, as will
appear by the following Examples.
decimal
parts.
First exam-
XLVII.
Let it be required to multiply 74,964 by 264
Since the Multiplicand (74,964) expreſſes 74964 Thou-
ple of the fandtbs, the Units of the Product will be thoufandths; and
multiplica- this Product will be 19790496 Thousandths; becauſe the
tion of num- Units of the Product are always of the fame Species as
ing decimal thofe of the Multiplicand, when the Multiplicator con-
parts. tains no decimal Parts (Art xxxIII.)
bers contain-
Second ex-
ample of
the multipli-
cation of
numbers
74,964
264
299,856
4497,84
I 4992,8
19790,496
In order therefore to fhew that the Units of the Pro-
duct 19790496 are Thousandths, the fame as thofe of the
Multiplicand. I feparate with a Comma three decimal
Figures in the Product, or in general I ſeparate with a Comma, as many
cimal Figures in the Product, as there are in the Multiplicand.
de-
To multiply therefore a Number which contains decimal Figures by
another that has none; the Multiplication is to be performed as if the Mul-
tiplicand had no decimal Parts, and afterwards as many decimal Figures are to
be feparated with a Comma in the Product, as there are in the Multiplicand.
As the Multiplicator has no decimal Parts, thofe in the Multiplicand
are all that are both in the Multiplicand and Multiplicator, whence the
Rule of the foregoing Article is applicable to this firſt Example.
XLVIII.
Let it be required to multiply 74964 by 2,64
Performing the Multiplication as if the Multiplicator
was a whole Number; I find for Product 19790496 fim-
ple Units. But the propofed Multiplicator (2,64) which
expreffes 264 Hundreths, being only the hundreth Part of
containing 264 Units: confequently the Product fhould be only the
hundreth Part of the Product 19790496 arifing from the
Multiplication by 264; the Units therefore of this Pro-
duct muſt be transformed into Hundreths; which is effect-
ed by ſeparating with a Comma as many decimal Figures in
this Product, as there are in the Multiplicator.
decimal
parts.
74964
2,64
2998,56
44978,4
149928
197904,96
As the Multiplicand has no decimal Parts thofe in the Multiplieator
are all that are in both the Multiplicand and Multiplicator; fo that the
general Rule is alſo applicable to this ſecond Example,
NUMERAL ARITHMETICK.
27
XLIX.
Let it be required to Multiply 74,964 by 2,64
Multiplying the Multiplicand (74,964) which con-
tains three decimal Figures, by the Multiplicator confi-
dered as a Number of fimple Units; that is, Multiplying
(74,964) by 264; I find (197,90496) for the Product.
74,964
2,64
Third ex-
ample of
the multi-
2,99856 numbers
44,9784
149,928
197,90496
But this Product is centuple of the One required; be-
cauſe it arifes from the Multiplication by 264 which
is centuple of the given Multiplicator (2,64): whence
this Product fhould be rendered a hundred Times lefs:
confequently the feparating Point muſt be ftill removed
fo many Places towards the left Hand as there are Places after the Comma
in the Multiplicator. And as there are already in the Product as many
Places to the right hand of the Coinma, as there are in the Multiplicand;
there will neceffarily be in the Product as many Places to the right hand of
the Comma, that is, as many decimal Figures, as there are in both the
Multiplicand and Multiplicator. Confequently (197,90496) will be the
Product of (74,964) into (2,64) agreeable to the general Rule. (Art. XLVI)
L.
0,125
plication of
containing
decimal
parts.
Fourth ex-
ample of the
multiplica-
tion of num-
bers contain-
0,0625
625
ing decimal
250
750
Let it be required to multiply 0,125 by 0,0625.
Having performed the Multiplication as if the two
Factors had no decimal Figures, and added together all
the particular Products; finding that there are not as
many Figures in the Product as there are decimal Places
in both the Multiplicand and Multiplicator: there being
ſeven decimal Places in both the Factors, and only five
Figures in the Product; I fet down two Cyphers to the
left Hand of this Product in order to have feven decimal
Figures, to which I prefix a Comma that they may be
reputed as fuch: and fetting down a Cypher to the left Hand of the Comma,
to ſupply the Place of the Units; there refults (0,0078125) for the Product
required of (0,125) multiplied by (0,0625).
LI.
0,0078125
parts.
tion are pro-
To explain how the Operations in Multiplication are proved, let us fup-
pofe that 4872 was multiplied by 863, and the Product found to be How the o-
4204536. To prove it, the Figures 4,8,7,2 of the Multiplicand confider- perations in
ed as reprefenting fimple Units are added together, and all the gs con- multiplica-
tained in their Sum being taken away, the Remainder 3 is fet down. In ved explain-
like manner, the Figures 8,6,3 of the Multiplicator are added together, ed by an ex-
and the 9s in their Sum being taken away, the Remainder 8 is fet down, ample,
Then the Remainder 3 of the Multiplicand, is multiplied by the Remain-
der 8 of the Multiplicator; and the two Figures of the Product 24 being
added together, if their Sum (6) not exceeding 9, is equal to the
28
ÉLEMENTS OF
The parti-.
minations
Remainder of the Sum of all the Figures 4,2,0,4,5,3,6 of the
Product, after all the 9s are taken away; there is a Prefumption that the
Multiplication has been accurately performed, if not, it is a Proof that fome
Error has been committed in the Operation.
For it is eafy to obferve, that if the 9s contained in a Number expref-
fed by a fignificative Figure followed by feveral Cyphers be taken away,
the Remainder will be repreſented by a Figure equal to the fignificative Fi-
gure of this Number.
elor
For Example, if all the gs contained in the Numbers 4000, 800, 70 be
taken away, the Remainders will be found to be 4,8, 7.1
So that if the 9s be taken away from any Number, fuch as 4872, which
is equivalent to 4000 more 800 more 70 more 2, the Remainder will
be
4 more 8 more 7 more 2; and if the 9s contained in thoſe four Fi-
gures whofe Sum is 21 be taken away; there will remain 2 more 1, or 3...
From whence it follows that if all the Figures of a Number be confider-
ed as repreſenting fimple Units, and from their Sum all the 9s are taken
away; the Remainder will be equal to what remains when the gs are taken
away from the given Number.
J
Now it is easy to perceive that when two Numbers fuch as 4872 and 863
are propoſed to be multiplied one by the other, that all the gs may be caft
out of their Product, by cafting out of the Factors all the 9s (becauſe the
Multiplication of thofe 9s by any Number, can only produce 9s, which
by the Suppofition are to be taken away); and by multiplying their Remain-
ders 3 and 8 one by the other, whofe Product 24 will be reduced to 6, after
the 9s are taken away.
LIL
To divide one Number by another, is to inveſtigate a third Number
which multiplied by the Second, gives a Product equal to the firft Number.
The Number which is propofed to be divided is called the Dividend ;
cular deno- that Number by which the Dividend is to be divided is called the Divifor;
given to the and the Number which refults from the Divifion is called the Quotient.
numbers "To denote that two Numbers are to be divided one by the other, the
concerned in Dividend is placed above a ſmall Line, and the Divifor under it. For
divifion. Example, to denote that 32 Feet is to be divided by 8 Feet, it is ſet down thus
32 Feet
8 Feet
LIII.
From whence it follows that the Divifor or Quotient muſt be an abſtract
Number, and that one of them must confiſt of Units of the fame Species
of thofe of the Dividend, otherwife the Divifor and Quotient multiplyed
one by the other would not produce a Quantity equal to the Dividend;
now acording as the Divifor is an abſtract Number, or a Number of the fame
Species with the Dividend we may form two diftinct Notionsof Divifion.
NUMERAL ARITHMETICK.
1° when the Divifor is a Number of the fame Species with the Dividend,
the Quotient that will refult will be neceffarily an abftract Number, denoting
how often the Divifor ſhould be repeated to produce the Dividend. In this:
cafe Divifion may be faid to be an Operation by which is diſcovered bow
often the Divifor is contained in the Dividend: and this bow often that is found,,
expreſſed in Latin by the Word Qoties, is called the Quotient.
For Example if it be propofed to divide 32 Feet by 8 Feet, or to find a
third Number by which 8 Feet being multiplied will produce the firft Num-
ber 32 Feet; the Number 4 that will be found will be an abfolute,
Number, denoting that 8 Feet is to be repeated 4 times to produce 32
Feet, and confequently that 8 Feet are contained 4 times in the Dividend Different
32 Feet, whence the Divifion of the Dividend 32 Feet by the Diviſor notions that
8 Feet, is reduced to find how often the Divifor 8 Feet, is contained in the may be for-
Dividend 32 Feet; and the abfolute Number 4 which is found, and denotes med of divi-
4 times, will be properly fpeaking the Quotient of this Divifion.
fion accor-
ding as the
'dividend are
2º When the Divifor confifts of abfolute Units, it denotes how often divifor and
the Quotient fought ſhould be repeated, to produce the Dividend. So that of the fame
to obtain this Quotient, the Dividend muſt be divided into as many equal or of diffe-
Parts as the Divifor contains abfolute Units. In this ſecond Cafe, it may rent ſpecies.
be faid that Diviſion is an Operation by which the Dividend is divided into as
many equal Parts, as there are abfolute Units in the Divifor, to obtain one of
thofe Parts which will neceſſarily be of the fame Species with the Dividend.
For Example, if it was propofed to divide 32 Feet by the abfolute Num-
ber 4, or to find a Number which repeated four Times will produce 32
Feet; it is manifeft that the Number which will be found will be the
fourth Part of 32 Feet, and confequently will be a Number of Feet. So that
the Divifion is reduced to divide 32 Feet into four equal Parts, to obtain
one of thoſe Parts which will be 8 Feet: but thofe 8 Feet when found
will be improperly ftyled the Quotient, becauſe not denoting 8 Times, it
does not correſpond to the Word Quoties, which fignifies bow often.
Q
3º when the Dividend and Divifor are both abfolute Numbers, the
Quotient will be alſo an abſolute Number; and either of the foregoing de-
finitions may be applied to Diviſion.
For Example, if it be propofed to divide the abfolute Number 32, by
the abſolute Number 8; according to the firſt Definition, the Diviſion will
be reduced to find how often the Divifor 8 is contained in the Dividend 32
which is of the fame fpecies with it. And the Number 4 that will reſult and
which will denote 4 Times will be properly fpeaking the Quotient of
vided by 8.
32
di-
But the Divifor 8 being an abfolute Number, may alſo denote that the
eighth Part of the Dividend 32 is to be taken; agreeable to the fecond De-
finition.
306
}
ELEMENTS OF
LIV.
It is eafy to perceive that the Number of Units of the Quotient, will be
the fame wether the Dividend and Divifor confift of Units of the fame
Species, or wether the Divifor is an abſtract Number, and that the Quoti
ents will only differ becauſe the Nature of their Units is different.
"}
For Example, if thoſe two Divifions be propofed, 32 Feet to be divided by
In the ope-
ration of di- & Feet, and 32 Feet to be divided by 8; both one and the other Quotient
will confift of 4 Units, and will differ only becauſe the Units of the first
will be abfolute Units, and the Units of the ſecond will be Feet.
vifion the
number of
units of the
dividend and
The Computifts therefore when a Divifion is propofed to be performed,
divifor is on- confider only the Numbers of Units of the Dividend and Divifor, without
ly confidered
and not their attending to the Nature of thofe Units; and feek for Quotient an abfolute
fpecies. Number which denotes how often the Number of Units of the Divifor
contained in the Number of Units of the Dividend, as if the Dividend and.
Divifor confifted of Units of the fame Species. Determining afterwards,
the Species of thofe Units, by obferving, that thofe Units fhould be ab-
ſtract ones, when the Dividend and Divilor are really of the fame Species,
and that they are of the fame Nature with thofe of the Dividend, when
the Divifor is an abfolute Number,
IV.
I
If the Dividend of a Divifion be multiplied by any Number, without
altering its Divifor; the Quotient of the new Divifion will be equal to the
Quotient of the former multiplied by the fame Number.
+
8
For Example, if 32 is to be divided by 8, which is expreſſed thus 32
the Quotient will be 4: and if the Dividend 32 be multiplied by 2 or 3, &c.
without altering the Divifor 8; thre will refult thofe new divifions 64 96
8' 8
&c. their Quotients 8, 12, &c. being equal to the former Quotient 4 multipli-
ed by 2 or by 3, &c. for it is manifeft that a conſtant Divifor is contained
twice oftener in a Dividend double, three times oftener in a Dividend
triple, &c.
And reciprocally if the Dividend of a Divifion be divided by any Num-
ber, without altering its Divifor, the Quotient of the new Divifion will.be
equal to the Quotient of the former, divided by the fame Number.
1
96
For Example, if 96 is to be divided by 8, that is, 26, whofe Quotient
is 12, and if the Dividend 96 be divided by 2 or by 3 &c; there will
refult thoſe new Divifions 48, 32 &c. their Quotients 6, 4.&c. being
8 8
e-
qual to the former Quotient 12 divided by 2 or by 3 &c. for by dividing
the Dividend by 2 or by 3 &c it is rendered twice or thrice &c. less than
it was; fo that the conftant divifor 8 fhould be contained twice or thrice,
B
NUMERAL ARITHMETICK.
3r
&c. lefs than before; confequently the Quotient fhould be twice, thrice
&c. less than the former Quotient, that is, equal to the former Quotient
divided by 2 or 3, &c.
LVI.
If the Divifor of a Divifion be multiplied by any Number, without
altering the Dividend; the Quotient of the new Divifion will be equal to
that of the former divided by the fame Number by which the Diviſor
was multiplied. For Example, if 96 is to be divided by 8, that is, 20
whofe Quotient is 12, and that without altering the Dividend 96, the Di-
vifor 8 be multiplied by 2 or by 3, &c. there will refult thofe new divifi-
96 96
&c. their Quotients 6, 4, &c. being equal to the former
16' 24' >
Quotient 12 divided by the Numbers 2, 3, &c. by which the Divifor was
multiplied: for it is manifeft that a Divifor become double or triple, &c. is
contained twice or thrice lefs, &c. in the fame Dividend: fo that the Quoti,
ent ſhould be twice or thrice &c. leſs than it would be if the Diviſor had not
been multiplied by 2 or by 3 &c.
ens
And reciprocally if the Divifor of a Diviſion be divided by any Number
without altering the Dividend, the Quotient of the new Divifion will be e-
qual to that of the former, multiplied by the Number which divided, the
Divifor,
96 whofe Quotient is
24
For Example if 96 is to be Divided by 24, that is
4; and that without altering the Dividend 96, the Divifor 24, is divided by
2 or by 3, &c. there will refult thofe new Divifions 2; 25 &c. their
12' 8.
Quotients 8, 12, &c. being equal to the former Quotient 4 multiplied by the
Numbers 2, 3, &c. by which the Divifor was divided; becaufe by divi-
ding a Divifor by 2 or by 3, &c. it. is rendered Twice or Thrice, &c. lefs;
and it is manifeft that a Divifor Twice or Thrice, &c. lefs, is contained
Twice or Thrice &c. oftener in the fame Dividend.
LVII.
From whence it follows 1° that if the Dividend, and Divifor of a
Divifion be multiplied by the fame Quantity; the Quotient of the Divifion The multi-
of the new Dividend by the New Divifor, will be the fame as the Quotient plication or
divifion of
32 the dividend
8 and divifor
by the fame
of the first Divifion; for Example if 32 is to be divided by 8; that is
whofe Quotient is 4, and the Dividend and Divifor be multiplied by 2 or
3 or by a fame Quantity: there will refult thofe new Divifions 64, 25 &c, not alter
by number does
16 24. 3.4.
which give the fame Quotient 4 as the former Divifion. For by multiplying
the
32
ELEMENTS OF
ded by a
the Dividend and Divifor of a Divifion by a fame Quantity, the Quotient
of the firſt Divifton is at the fame Time multiplied and divided by this
fame Quantity, (Art. LV. & LVI.) and its Value confequently is not changed.
2º If the Dividend and Divifor of a Diviſion is divided by a fame Quan-
tity, the Quotient of the new Divifion will be ftill the fame as that of the
former Division; becauſe by this Operation the Quotient of the former
Divifion is divided and multiplied by the fame Quantity. (Art. LV,LVI.)
For Example, if 96 is to be divided by 24, that is
4, and the Dividend 96 and Divifor 24 be divided by
fame Quantity, there will refult thoſe new Divifions
give the fame Quotient 4 as the former Divifion 96
LVIII.
24.
96
24.
whofe Quotient is
2 or by 3 or by any
48
32 8
32 &c. which
The quoti-
When a Dividend is to be divided by a Divifor compofed of feveral
ent of a di- Factors multiplied together; inſtead of dividing the Dividend by the com-
vidend divi- pofit Divifor, it may be divided by a Factor of the Divifor, and the Quo-
compofit di- tient of this Diviſion may be divided by a fecond Factor of the Diviſor,and the «
vifor is equal new Quotient may be divided by a new Factor of the Divifor, and fo on
to he quo until the Divifion is made by all the Factors: the laft Divifion being perfor-
tient arifing
from the med, the laft Quotient will be the Quotient of the Divifion of the Divi-
continual .dend by the compofit Divifor.
division by
the compo-
nent parts
W
For Example, if 840 is to be divided by the compound Divifor
of the divi- 3x5x7; which may be expreffed thus
for,
840
; inſtead of multiply-
3×5×7
ing the three Factors 3,5,7 one by the other, and dividing the Dividend
840 by their Produ& 105; the Dividend 840 may be firft divided by 3,
and the Quotient 280 that refults by 5, and the Quotient 56 arifing from
this laſt Diviſion, by 7, which will give 8 for the laſt Quotient, the fame.
that refults by dividing 840 by the Product 105 of the three Factors 3,5,7-
840
For
repreſenting the Divifion of 840 by 3×5×7, if the
3 X5 X 7
Dividend 840 and Divifor 3X 5X7 be divided by the fame Number 3, the
280
new Divifion
will give the fame Quotient (Art. LVII.), and if the
5 X 7
new Dividend 280 and Divifor 5 X 7 be again divided by the Number
5; the
Divifion that remains to be performed will ſtill give the fame Quoti-
ent (Art. LVII.) wherefore by dividing 840 by 3, and the Quotient of this
divifion by 5, & dividing again the new Quotient by 7, there will refult the
fame Quotient as will arife by dividing 840 by the compound Divifor
3 × 5 × 7.
56
7
NUMERAL ARITHMETICK.
33
LIX.
The Quotient arifing from the Diviſion of any Number lefs than 90, by
another Number lefs than 10, and the Remainder of fuch a Divifion is The divifion
of any num-
found by Help of the Table of Pythagoras, after the following Manner.
ber leſs than
The Divifor being looked out for at the Top of the Table; and then defcend- 90 by ano-
ing vertically under it until the Dividend is found, or a lefs Number that ap-
proaches the neareſt to it; the Number oppofite to it in the firft Column ed by help
to the left Hand, is the Quotient of the Divifion required.
Let it be propofed, for Example, to divide 72 by 8. I look out for
8 at the Top of the Table, and defcending vertically to 72; oppoſite to
which in the firſt Column, I find the Number 9, which is the Quotient
required.
Let it be propofed to divide 78 by 9. I look out for 9 at the Top of.
the Table, and defcending vertically under 9, and not finding 78; but 72
being lefs and opproaching the neareſt to it; oppofite to which in the firſt
Column, I find the Number 8 for the Quotient of 72 divided by 9; but
as it is 78 and not 72 that is propoſed to be divided by 9, there will be a
Remainder 6 which cannot be divided.
Let it be propoſed to divide 64 by 7. I look out for 7 at the Top of the.
Table, and defcending vertically under 7, and not finding 64, but 63
approaching the neareſt to it ; oppofite to which in the firft Column, I find the
Number 9 for the Quotient of 63 divided by 7; but as it is 64 and not 63
which is propoſed to be divided by 7, there will be a Remainder 1 which.
cannot be divided.
LX.
لا
therless than
10 perform-
of the table
of pythago
ras.
number by
-one figure.
To divide any Number by another Number reprefented by one Fi- Procefs of
gure, the Computifts divide all the Parts of the Dividend by the the method
Divifor, beginning the Divifion by the Figures whofe Units are of the of dividing a
higheſt Degree. Suppofing, for Example, that the Dividend confifts of another re-
three Figures, of Simple Units, of Tens and of Hundreds; the Computifts prefented by
begin by dividing the Part of the Hundreds which can be divided into as
many equal Parts as there are Units in the Divifor. Then reducing the
Remainder of the Hundreds into Tens, and adding them to the Tens
of the Dividend, they divide the Part of the Tens that can be divided
into as many equal Parts as there are Units in the Divifor. Laſtly redu-
cing the Remainder of the Tens into fimple Units, and adding thoſe
to the Simple Units of the Dividend they divide the whole by the
Divifor. So that there are as many particular Divifions to be perfor-
med, and conſequently as many particular Quotients to be found, as
there are different Parts of the Dividend to be divided.
Thoſe Operations that have been announced in general, will be ren-
dered obvious, by their application to the following Examples.
ย
"
2 E
34
ELEMENTS OF
divifion of
LXI.
Let it be required to divide 952 by 4:
Having fet down the Diviſor to the right Hand of the
Firft exam- Dividend, feparating them by a ſmall vertical Line or
ple of the Crotchet; and drawing a Line under the Divifor to fe-
a number by parate it from the Quotient, under which is to be fet
another re- down the Figures of the Quotient according as they are
prefented by found; as in the Margin.
one figure.
I begin by dividing the 9 Hundreds by 4; faying:
the fourth Part of 9 Hundreds is 2 Hundreds ; or more
fimply, the fourth Part of 9 is 2; which I fet down in
the Quotient in a Place which will be that of the Hun-
dreds.
To determine the Part of the 9 Hundreds, I divided and
to find the Remainder of the Divifion; I multiply the Di-
952
{
8
15
1 2
32
32
00
4
235
vifor 4 by the Quotient 2 Hundreds," and fet down the ProduЯ 8 Hun
dreds under the 9 Hundreds; then deducting 8 Hundreds from 9 Hundreds,
the Remainder 1 Hundred will be the Part of the Hundreds which has
not been divided by 4.
Setting down to the right Hand of the Remainder of the Hundreds,
the 5 Tens of the Dividend, there refults 15 Tens to be divided by 4; Í
therefore take the fourth Part of thofe 15 Tens, which can only be 3 Tens
with a Remainder.
I therefore fet down 3 in the Quotient in the Place of the Tens, that is,
to the right Hand of the 2 Hundreds which have been fet down for the
Quotient of the first Divifion.
To determine the Part of the 15 Tens that have not been divided. I
multiply the Divifor 4 by the Quotient 3 Tens, and fet down the Frodu& 12
Tens under the 15 Tens; then deducting 12 Tens from 15 Tens, the
Remainder 3 Tens will be the Part of the 15 Tens which has not been
compriſed in the Divifion.
Setting down to the right Hand of the Remainder 3 Tens, the 2 Units
of the Dividend, there refults 32 fimple Units to be divided by the fame
Divifor 4. I therefore take the fourth Part of 32, which is 8, and fet it
down in the Quotient in the Place of fimple Units, that is, to the right
Hand of the 3 Tens.
To diſcover whether any Remainder is left after this laft Divifion, I
multiply the Divifor 4 by the Quotient 8, and fetting down the Produc
32 under the 32 Units that were to be divided, I deduct one from the o-
ther, and there being no Remainder left, and the diviſion being ended,
I conclude that the Divifor 4 is contained 238 Times exactly in 952.
NUMERAL ARITHMETICK.
3
LXII.
Let it be required to divide 7264 by 9.
Having difpofed the Dividend and Divifor as in the
Margin; I divide one after another all the Parts of
the Dividend, beginning by the Thouſands, and pro-
ceeding afterwards to the Hundreds, and then to the
Tens, and Units,
1° As the first Figure 7 cannot be divided by 9,
I join it to the following Figure 2, and divide 72 Hun-
dreds by 9, faying: the ninth Part of 72 Hundreds is
8 Hundreds, which I fet down in the Quotient in a
Place which will be that of the Hundreds.
7264
9
807
7 2
064
63
I
9
To find the Remainder of this first Diviſion, I multiply the Divifor
by the Quotient 8; and fetting down the Product 72 under the 72 which Second ex-
ample of the
was to be divided, I deduct one from the other; and as there is no Re- divifion of a
mainder, I fet down a Cypher underneath to denote that the Divifion of 72
Hundreds by 9, gives exactly 8 Hundreds for Quotient.
2° I bring down to the right Hand of the Cypher, the 6 Tens of the Divi-
dend to divide them alfo by 9; and as that is not poffible, I fet down a Cy-
pher in the Quotient, in the Place of the Tens, as well to denote that there
are no Tens in the Quotient, as to conferve to the firft Figure of the
Quotient the Place of Hundreds: the 6 that has been fet down and that
could not be divided, will remain for the following Divifion.
3º I bring down the 4 Units of the Dividend to the right Hand of the 6
Tens, which makes 64 Units which I divide by 9; and as 64 contains 9
feven Times, I fet down 7 in the Quotient in the Unite Place: and the
Divifion will be ended, with reſpect to the Quotient required; fince all the
Figures that it confifts of have been found. To determine the Remainder
of the Divifion, I multiply the Divifor 9 by the laſt Quotient 7; and fet-
ting down the Product 63 under 64, I deduct it from 64; and find 1 to be
the Remainder of the Diviſion.
Wherefore the Dividend 7264 being divided by the Divifor 9, gives
807 for the Quotient, with a Remainder 1 which has not been divided; fo
that 807 is the exa& Quotient of 7263 divided by 9.
I
LXIII.
number by
another re-
prefented by
one figure.
another.
In General, when any Number is propoſed to be divided by another, the General me
Computiſts first determine the Number of Places of the Quotient, by thod for di
pointing off fo many Places from the right Hand of the Dividend, as are eviding one
qual to, or not exceeding the Product of the Divifor into any of the Nine number by
fignificative Figures; the remaining Places, in the Dividend more one, being
equal to the Number of Places in the Quotient. To find afterwards
the Figures correfponding to thofe Places, beginning by the Figure whofe
Units are of the higheſt Degree, they divide the firft Figure of the Places
36
ELEMENTS OF
{
pointed off in the Dividend, which is called the firſt Member of the Divifi-
on by the first Figure of the Divifor, when this Member and the Divifor
confift of the fame Number of Figures, or the two firft Figures of this
Member when it has one more; whereby is difcovered the Number of
Times the Divifor is contained in this Member, which is repreſented by
the firſt Quotient Figure fought. They then fubtract the Product of the
Quotient into the Divifor from this Member, and to the Remainder affix
the next Place in the Dividend, which will be the fecond Member of the
divifion; with which they proceed as before, and if the Divifor is not once.
contained in any Member, they increaſe the Quotient with a Cypher before
any new Place is taken down to the right Hand of this Member
LXIV.
Let it be required to divide 32361 by 469.
Having fet down the Divifor to the right Hand of
The fore-
the Dividend as in the Margin, I obferve that as the
going me-
thod explain- Dividend contains Tens of Thouſands, and the Divi-
ed by an ex- for Hundreds, and that one Hundred is contained in
ample.
Ten Thouſand a Hundred Times, the Quotient may
contain Hundreds, but no Units of a higher Degree,
It is Queftion therefore to determine how many Hun-
dred Times, how many Tens of Times, how many U-
nits of 'Times, the Divifor is contained in the Dividend.
32361
2814
469
69
4221
4221
0.000
To determine how many Hundred Times the Dividend contains the
Divifor, I point off fo many Places from the right Hand of the Dividend as
there are in the Divifor, that is, I Point off the Part 323 of the Dividend,
which really is 32300; ſetting aſide for a moment the two laft Figures 61,
and divide 32300 by 469, to diſcover how many Hundred Times 469 is
contained in 32300; to effect which, I obferve that it fuffices to divide
323 by 469, confidering the Quotient Figure not as repreſenting fimple
Units, but Hundreds. But as 323 can not be divided by 469, I conclude
that the Quotient will not contain Hundreds. It would have contained
Hundreds if inſtead of 323 there had been 523, or in general a Number
equal or greater than 469: for then the Quotient would contain at leaſt 1
Unit of the third Degree, or 1 Hundred. The Units therefore of the
higheſt Degree that the Quotient can contain will be Tens, but it will
neceffarily contain Tens, becauſe the Dividend having two Figures more
than the Diviſor, is neceffarily ten Times greater than the Divifor, in effect
469 repeated ten Times gives 4690 which has a Figure less than 32361
I confider therefore how often 32360 contains 469, fetting afide for a mo-
ment the Number 1, or what amounts to the fame Thing, I confider how
often 3236 contains 469, remembring that the Quotient Figure will repre-
fent a Number of Tens; but 3236 can not contain 469 oftener than the
Number 32, (the two firft Figures of the Dividend) contains the firſt Fi-
NUMERAL ARITHMETICK.
37
gure 4 of the Divifor: for 32 contains 4 eight Times, and if, for Example, 9
is fet down in the Quotient inſtead of 8; multiplying 469 by 9 there
would refult a Number greater than 3236 for 4 times 9 giving 36, the two
first Figures of the Number equal to 9 Times 469 would be greater than
the two first Figures 32 of the Number 3236; fo that it fuffices to di-
vide by the firſt Figure of the Divifor the firft Figure of the Member of
the Divifion, when this Member confifts of as many Figures as the Divi-
for, or the two firſt Figures when it contains one more.
дав
However we are not to conclude that this Operation will never give too
much, but it is certain it will never give too little, and it is for this Reaſon
that the Computifts divide the firſt Figures of the Member of the Diviſion
by the firſt Figure of the Divifor. When the Diviſion gives too great a
Quotient, as in the preſent Cafe, where 8 is too great, and even 7, I di-
miniſh fucceffively the Quotient until I find one that is not two great,
which is 6 Tens, which I fet down in the Quotient. I afterwards mul-
tiply the Divifor 469 by 6 Tens, fubtracting the Product 2814, which
is really 28140 from 3236 which is really. 32350. To the Remainder 424
which is really 4220, I add the 1 I fet afide, which will be the fecond
Member of the Divifion, and divide 4221 by 469: proceeding as before, I
find it is contained 9 Times, which I fet down in the Quotient in the Units
Place, that is, to the right Hand of 6, and multiplying the Dividend 469
by the Quotient 9, and fubtracting the Product from this Member of the
Diviſion, and there being no Remainder, I conclude that 32361 is exactly di-
vided by 469, and that 69 is the Quotient of this Divifion
LXV.
the number
of places
confiſt of.
When as many Places has been pointed off in the Dividend to the General rule
right Hand, as there are in the Divifor, or one more if neceſſary, it is eaſy to for finding
perceive that the Quotient will confift of as many Figures more one, as
will remain in the Dividend. For Example, let 523032 be propofed to that a quo-
be divided by 469, having pointed off 523 which confiits of as many tient of a di-
Places as 469, there will remain three Figures 032, I fay the Quotient fhould. vifion will
contain three Figures more One, or Four: for it is manifeft that 523000
is a Thouſand Times greater than 523, which is greater than 469;
and that 523032 is lefs than 469 repeated Ten Thoufand times, becauſe
4690000 confifts of a Figure more, therefore the Quotient ſhould contain
Thouſands, and not Ten Thoufands, and therefore fhould confift of four
Places, neither more nor lefs. If the Dividend was 1523032, pointing
off 1523, which has a Figure more than the Divifor 469, it will appear
in like Manner that the Quotient ſhould confift of four Places, neither
more nor lefs; whence it is that a Cypher is fometimes fet down in the
Quotient and fometimes feveral.
ར་
Why in each
It remains to explain why in the Quotient, more than 9, is never ſet
down at once, which will eaſily appear when it is obſerved that a Mem- operation of
38
ELEMENTS OF
1
greater than
divifion a ber of the divifion is always leſs than ten Times the Divifor; for a Mem-
number ber of the Diviſion confifts of as many Figures as the Divifor, or of one
9 is not fet more : In the first Cafe it is manifeſt that it is less than the Divifor re-
down in the peated ten Times, in the fecond Cafe, if a Figure be taken from the
quotient. Member of the divifion it will be less than the Divifor, therefore the
Member of the divifion with this Figure reſtored is less than ten Times
the Divifor.
Second ex-
divifion of
LXVI.
45711.12
4485
861 I
Let it be required to divide 4571112 by 897.
Having difpofed the Dividend and Divifor as
ample of the in the Margin, and obferving that the three Fi
integers re- gures of the Divifor, form a Number greater than
prefented by the three firft Figures of the Dividend to the left
feveral fig- Hand; I Point off 4571 for the firft Member
of the divifion, and as there remain three Fi-
gures 112 in the Dividend, the Quotient will
confift of four Places of Figures.
nificative fi-
gures.
I
8073
5382
5382
0000
5897
15096
To determine the Numbers correfponding to
thofe Places, beginning by that whofe Units are
of the higheſt Degree, I confider how often the
Divifor 897 is contained in the firft Member of
the divifion 4571, and find 5Thoufands for the firſt Figure of the Quotient.
To find the fecond Figure of the Quotient, I multiply the Divifor
897 by the Quotient 5, and fubtract the Product 4485 from 4571, and to
the Remainder 86 I affix the 1 Hundred of the Dividend, fetting a Point
over it to denote that it has been brought down; which will be the fecond
Member of the Diviſion, and divide 861 by 897, and as that can not be
done, I fet down a Cypher in the Quotient, as well to denote that it con-
tains no Hundreds, as to preferve to the firft Quotient Figure found its Place.
To diſcover the third Figure of the Quotient, I affix to 861 the 1 Ten of
the Dividend, (fetting a point over it) which will be the third Member of
the divifion, and confider how often 897 is contained in 8611, dividing
86 by 5, and I find 9 for the third Figure of the Quotient.
1
+
•I
To determine the fourth and laft Figure of the Quotient, I multiply
the Divifor 897 by the Quotient 9, fubtract the Product 8073 from 8611,
and to the Remainder 538 I affix the Units of the Dividend (fetting a
Point over it) which will be the fourth Member of the Divifion, and con
fider how often 897 is contained in 5382, dividing 53 by 8, and. I find
6 for the Fourth and laft Figure of the Quotient, and multiplying the
Divifor-469 by the Quotient 6, and fubtracting the Product 5382 from this
Member of the divifion, and there being no Remainder, I conclude that
4571112 is exactly divided by 897, and that 5096 is the Quotient of
this Divifion,
7
NUMERAL ARITHMETICK.
39
LX VII.
239200
- 208
Let it be required to divide 239200 by 52
The Dividend and Divifor being difpofed as
in the Margin, the Divifor 52 not being contain-
ed in the two first Figures of the Dividend to
the left Hand; I Point off the three Figures 239,
and there being three Figures 200 remaining in
the Dividend, I conclude that the Quotient will
confift of four Places of Figures.
312
312
000
{1
52
4600
To determine the Numbers correfponding to thofe
Places, beginning by that whofe Units are of the
higheſt Degree; I confider how often the Divifor 52 is contained in the Third ex-
first Member of the Divifion 239 Thouſands, and find 4 Thouſands for ample of the
the Quotient with a Remainder 31 Thoufands.
divifion of
integers re-
To determine the fecond Figure of the Quotient, I bring down the 2 Hun- prefented by
dreds of the Dividend to the Remainder 31; and dividing 312 Hundreds leveral fig-
by 52, find 6 Hundreds for the Quotient without a Remainder.
To continue the Divifion, according to the Rules already explained, the
two Cyphers of the Dividend fhould be fucceffively taken down; but as each
of thole affixed to the Remainder o, being divided by 52 will give o for Quo-
tient; I fet down two Cyphers to the right Hand of theQuotient already found.
Whence the Quotient of the Divifion of 239200 by 52, will be 4600.
LX VIII.
nificative fi-
gures
decimal
The general Rule for dividing Numbers containing decimal Parts, con- Procefs of
fifts in reducing the Units of the Dividend, and thofe of the Divifor to the divifion
the fame Denomination; by fetting down after the feparating Point, or of numbers
after the Decimals of the Term which has lefs decimal Figures than the containing
other, as many Cyphers as are neceffary that the Dividend and Divifor parts.
may have the fame Number of Characters after their Commas. The Divi-
dend and Divifor being thus prepared, they are to be divided one by the
other, without attending to the Commas which may be fuppreffed, and
which are conferved only that the Values of the Dividend and Divifor may
not be changed.
For Example, if (172,8) was to be divided by (1,44) that is, 1728
Tenths, by 144 Hundreths; as the Divifor has a decimal Figure more
than the Dividend, I fet down a Cypher to the right Hand of the Divi-
dend. This Dividend being transformed into (172,80) which denote
17280 Hundreths, will confift of Units of the fame Species, with thoſe
of the Divifor (1,44) which denotes 144 Hundreths.
The Dividend and Divifor being thus prepared and transformed into
(172,80) and (1,44) will not have changed their Value. The Quotient
therefore of (172,80) divided by (1,44) will be the fame as that of
(172,8) divided by (1,44).
40
ELEMENTS OF
Divifion of
Suppreffing the feparating Points both in the Dividend (172,80) and in
the Divifor (1,44), all the Figures of the Dividend and Divifor will be ad-
vanced two Places, and thereby will be rendered centuple of what they
were, that is, they will be multiplied both one and the other by roo. The
Quotient therefore will be the fame (Art. LVII.) as that of (172,80) divi-
ded by (1,44).
If therefore, to the right Hand of the Dividend and Divifor there be
annexed Cyphers to make the Number of decimal Figures equal in both,
and the new Dividend be divided by the new Divifor, after fuppreffing
the feparating Points, the Quotient fought will be obtained.
LXIX.
When the Units of the Dividend and Divifor are reduced to the fame
Denomination, by annexing to each the fame Number of decimal Fi-
gures; and the Divifion is performed as if the Dividend and Divifor had no
decimal Parts; the Quotient arifing will confift of fimple Units; but it
feldom happens that the Divifor is contained in the Dividend a certain.
Number of Times without a Remainder: and in this Cafe, it may be
found neceffary to reduce into decimals the Part of the Units Time.
that the Divifor is contained in the Dividend. Thoſe different Cafes.
will be explained in the following Articles.
LXX.
To divide a Number which has decimal Parts, by a Divifor that has
numbers con- none; the Dividend is to be divided by the Divifor as if the Dividend
taining deci- had no decimal Parts: and when the Quotient is found; fo many decimal Fi-
mal parts by
gures are to be ſeparated with a Comma, as there are in the Dividend : ´as
have none. will appear by the following Example.
others that
Example.
LXXI.
Let it be required to divide (79,58) by 23
79,58
{
5.23
3,46.
69
105
92
138
I divide (79,58), by 23 as if it was propofed to di-
vide 7958 by 23: and having found 346 for the Quo-
tient; I feparate with a Comma two decimal Fi-
gures in this Quotient; and there refults (3,46) for
the Quotient of (79,58) by 23. Which will eafily
appear, when it is obferved that the Number (79, 58)
propoſed to be divided, denotes 7958 Hundreths; and
the Divifor 23 denotes that the twenty third Part of
this Dividend 7958 Hundreths is to be taken. Now,
fince a Part of a Number whofe Units are Hundreths,
will alfo confift of Hundreths: the Quotient 346 found
by dividing 7958 Hundreths, by 23, can confift of no
other Units than Hundreths; confequently fhould exprefs 346 Hundreths;
whence each. Figure of this Quotient fhould be removed two Places
towards the right Hand which is effected, by feparating with a Comma
two decimal Figures,
;
138
NUMERAL ARITHMETICK,
41
LXXII.
To divide any Number by a Divifor that has no decimal Parts, and Method of
carry on the Divifion until the Quotient ſhall not differ from the exact approxima-
ting to the
Quotient, by a decimal Unit of any propoſed Order. Cyphers must be Quotient of
annexed to the Dividend, until the Place of the Decimals of the loweſt a Divifion
Order that the Quotient ſhould contain is filled up. And the Dividend when the
thus prepared being divided by the Divifor, as if it did not contain any tains no de-
decimal Parts; fo many Decimals muſt be ſeparated by a Comma in the cimal Parts.
Quotient, as there are in the Dividend prepared, as will appear by the
following Example.
LXXIII.
Divifor con-
Let it be propofed to divide (103,2) by 33, and to find a Quotient that
ſhall not differ from the exact Quotient by a thousandth Part of an Unit. The forego-
As the propofed Dividend contains in Decimals, only Tenths; I annex ing Method
to it two Cyphers to fill up the Place of the Hundreths, and of the explained by
Thouſandibs; becauſe the Error of the Quotient
-an Example.
must be less than I Thousandth, the Divifion
ſhould be carried on to Thoufandths.
The Dividend being thus prepared, I divide
(103,200) by 33; without attending to the fepa-
rating Point, and I find 3127 for the Quotient
which fhould repreſent Thouſandths; as the Di-
vidend is compofed of Thoufandths whence this
Quotient fhould be (3,127).
As it is not poffible to put an Unit more in the
Quotient 3,127, without rendering it too great;
and that this Unit can only be 1 Thouſandth; it
is manifeft that the Quotient (3,127) found, tho'
not exact, yet does not differ from the real Quo-
tient by a thoufandth Part of an Unit.
LXXIV.
33
103,200
(3,127
99
42
33
90
66
240
231
9
Divifor is
To divide any Number by a Divifor greater than the Dividend, and Method of
carry on the Divifion until the Quotient ſhall not differ from the exact approxima-
Quotient by a decimal Unit of any propofed Order: If the Divifor ting to the
Quotient of
contains no decimal Parts, Cyphers must be annexed to the Dividend, a Divifion
until the Place of the Decimals of the loweſt Order that the Quotient when the
ſhould contain, is filled up. But if the Divifor contains decimal Parts;
greater than
its feparating Point muſt be fuppreffed, and that of the Dividend muft the Divi-
be removed ſo many Places to the Right Hand as there were decimal dend,
Figures in the Divifor: then Cyphers must be annexed to the Dividend,
that it may contain as many decimal Places as it is propofed the
Quotient fhould contain. The Dividend and the Divifor being thus pre-
pared, they muſt be divided one by the other, as if they contained no
decimal Parts.
2 F
42
ELEMENTS OF
1
Firſt Ex-
ample.
1
Second ex-
ample.
As the Quotient fhould contain as many decimal Places as the Divi-
dend; and it may happen that there will not be as many Figures in this
Quotient as there are Decimals in the Dividend, in this Cafe, Cyphers
must be prefixed to it for the Defect.
LXXV.
Let it be propofed to divide the Number 2 by 189, and carry on the Di-
vifion until the Quotient shall not differ from the exact Quotient, by the
hundred-thouſandth Part of an Unit.
2,00000
I 89.
ΙΙΟΟ
945
189
Having placed a Comma to the right Hand of the Number 2 to be
divided; I fet down five Cyphers to the right Hand of this Comma,
that the Place of hundred thousandths may be fil-
led up. I divide the Dividend (2,00000) thus
prepared, by 189, as if this Dividend did
not contain decimal Parts, and I find 1058 for
the Quotient, which ſhould repreſent hundred
thousandths, as the Dividend is compofed of
bundred thousandths; whence its Figure 8 to the
right Hand ſhould ſtand in the Place of Hun-
dred Thousandths, and confequently in the fifth
Place to the right Hand of the feparating Point.
I therefore fet down a Cypher to the left Hand of the Quotient 1058
prefixing a Comma to it, that the Quotient may confift of five decimal
Figures as the Dividend, and there refults (0,01058) for the Quotient.
LXXVI.
1550
1512
38
1
Let it be propofed to divide (0,025) by (1,89), and to carry on the Divi-
fion until the Quotient shall not differ from the exact Quotient by the Mil-
lionth Part of an Unit.
Having fuppreffed the feparating Point of the
Divifor, I remove the Comma of the Divi-
dend two Places towards the right Hand, be-
cauſe there are only two decimal Figures in
the Divifor. The Dividend and Divifor will-
thereby be multiplied by 100, (Art. 1x) and the
Value of the Quotient will not be altered (Art..
LVII). The Divifion therefore will be reduced
to that of (002,5 or 2,5) by 189. As the Divi-
fion is to be carried on to Millionths, which are
Decimals of the fixth Order, which ſtand in
the fixth Place to the right Hand of the Com-
ma; and that the Dividend (2,5) has already a
decimal Figure, I annex five Cyphers to this
Dividend, and divide 2,500000 by 189.
1
The Divifion being performed, there refults
13227 for the Quotient: and as the Dividend
I
189.
2,500000
1 89
00013227
610
567
430
378
520
378
1420
1323.
97
}
43
NUMERAL ARITHMETICK.
confifts of Millionths, this Quotient will alfo reckon Millionths; ſo that
its laft Figure 7 fhould ſtand in the fixth Place to the right Hand after
the Comma. I therefore fet down a Cypher to the left Hand of this
Quotient, prefixing a Comma to it, and there refults (0,013227) for
the Quotient, of (2,500000) divided by 189, or for that of (0,025)
divided by (1,89).
As the Dividend (2,500000) which denotes 2500000 Millionths is not
exactly divisible by 189; after the Divifion there is a Remainder 97
Millionths, which cannot be divided by 189, unless it fhould be pro-
pofed that the Quotient fhould contain Decimals of an inferior Order
to Millionths.
If the Divifion fhould be carried on to Hundred Millionths, which
are Decimals of the eighth Order, two more Cyphers ſhould be annexed
to the Dividend, that is (2,50000000) fhould be divided by 189 and
there would refult for Quotient 1322751 Hundred Millionths, or
(0,01322751) with a Remainder 61 Hundred Millionths.
In fine, if the Divifion was carried on indefinitely, there would
refult for Quotient (0,01 322751 322751 322751 &c.), the fame Figures
322751 being continually repeated.
LXXVII.
fion that
cannot be
When it is required to exprefs in decimal Parts, the Quotient of a A Quotient
Divifion that cannot be obtained without leaving a Remainder, after a of a Divi-
certain Number of Figures of this imperfect Quotient is found, the ſame
Figure or Period of Figures will continually recur as in the foregoing obtained
Example, and in the following.
without
leaving a
Remainder
1º İf 1 be divided by 3, there will reſult (0,3333 &c.), that is, the
Quotient will confift of three Tenths, three Hundreths, three Thou- is expreffed
fandths, and fo on continually ad infinitum.
by a decimal
Series, con-
20 If I be divided by 6, the Quotient will be found to be 0,16666 &c. fifting of e-
that is, the first Figure of the Quotient will be 1 Tenth and all the
other decimal Figures will be 6s.
I
of Figures
fucceeding
3° If I be divided by 7, there will refult (0,142857 142857 &c.) that each other
is, after the fix first decimal Figures 142857 of the Quotient are found, ad infinitum
the fame Figures will recur for the fix following ones, and fo on conti-
nually ad infinitum.
40 If I be divided by 24, the Quotient will be found to be
(0,041666 &c.) that is, the three first decimal Figures will be 041,
which repreſent 41 Thouſandths, and all the decimal Figures following,
ad infinitum, will be 6s.
When the fame Figures recur in this Manner in a Quotient, two
Periods only of thofe circulating Figures are fet down, with an &c.
annexed to them, to denote that thofe Periods will continually recur,
ad infinitum.
44
ELEMENTS OF
Property of
which di-
bers lefs
than itself.
LXXVIIL
Every Number lefs than 9 will give for Quotient an infinite Series
the Digit 9 of Decimal Figures, the fame as that of the Dividend: For every Num-
vides Num- ber lefs than 9 cannot be divided by 9, until it is reduced into Tenths,
and then will be equivalent to as many Tens of Tenths, as it confifts
of Units; now each ten Tenths will give 1 Tenth for Quotient with a
Remainder 1 Tenth. Wherefore all the Tens of Tenths of which the
Dividend is compofed, will give as many Tenths for the Quotient and
Remainder as there are Units in the Dividend; confequently the first de-
cimal Figure of the Quotient, and the first decimal Figure remaining will
be the fame as that of the Dividend which is fuppofed lefs than 9.
Method of
"confifting of
Since the remaining Figure of the first Divifion is equal to that of
the Dividend, and is to be divided by 9, it muſt be reduced into Hun-
dreths, and confequently will give as many Hundreths for Quotient
and Remainder as the Dividend contains Units, and fo on ad infinitum.
For Example, if 7 is propofed to be divided by 9, I reduce the Dividend
into 7 Tens of Tenths. Now each ten Tenths being divided by 9 will give
1 Tenth for the Quotient, and there will remain Tenth. Wherefore
the 7 Tens of Tenths being divided by 9, will give 7 Tenths for the
Quotient, and will alfo give 7 Tenths for the Remainder.
Since thofe 7 Tenths remaining after the first Divifion cannot be
divided by 9, I reduce them into 7 Tens of Hundredths, and as each
ten Hundreths divided by 9, will give 1 Hundreth for the Quotient with
a Remainder 1 Hundreth, the 7 Tens of Hundredths will give 7 Hun-
dreths for the Quotient, and 7 Hundredths for a Remainder, the fame
is to be faid of the Thoufandths, &c. that is, each decimal Figure of
the Quotient and of the Remainder, will be the fame as the Figure of
the Dividend.
LXXIX.
Whence is derived an abridged Method of multiplying, when the
multiplying Multiplicator is a Number reprefented by the fame Figure feveral
by Numbers Times repeated. Let it be propofed for Example, to multiply 3787 by
equal Peri- 5555. I multiply 3787 by the repeating Figure
ods of Fi- 5, annexing as many Cyphers to the Right
ceeding each Hand of the Product, as the Multiplicator con-
gures fuc-
other.
3787
50000
9)189350000
2 1 0 3 8 8 8 8
2103
fifts of Places. I then divide the Number
that refults 189350000 by 9, neglecting the
Remainder of the Divifion, and from the Quo-
tient 21038888 I fubftract the Number repre-
fented by as many Figures of this Quotient to
the right Hand, as there are Places of Figures in the Multiplicator,
the Remainder will be the Product required.
210367 85
For the particular Products of this Multiplication are 18935X1,
NUMERAL ARITHMETICK.
45
18935×10, 18935X100, 18935×1000, the Sum of which, or the
Product required, is equal to 18935×1111, 18935X10000
but
and
I
9
or
21038888, 8888 &c. is equal to 18935X1111 more 18935X1111 &c.
18935XI
or 2103,8888 &c. is equal to 1893X,1111 &c. where-
9
fore deducting 2103,8888 &c. from 21038888,8888 &c. that is
18935×,1111 &c. from 18935XIIII more 18935X,1111 &c. the Re-
mainder 21036785 will be equal to 18935×1111 the Product required
of 3787 into 5555.
I
LXXX.
Every Number lefs than 99, divided by 99, will give for Quotient
Property pe-
an infinite Series of decimal Periods of two Figures equal to thofe of culiar to all
the Dividend. For every Number lefs than 99, cannot be divided by Divifors lefs
by an Unit
99, until it is reduced into as many Hundreds of Hundreths as it confifts than the
of Units; now each hundred of Hundreths being divided by 99, will Terms of
give 1 Hundreth for Quotient, with a Remainder 1 Hundreth; where- the progref-
fion 10, 100
fore all the Units of the Dividend will give for Quotient a Number of 1000
Hundreths, expreffed by the fame Figures as the Dividend, and there 10,000
will remain the fame Number of Hundreths, and fo in like Manner of which divide
the other Figures of the Quotient.
1000,
Numbers
less than
For Example, if it was required to divide 42 by 99, I reduce the Di- themselves.
vidend 42 into 42 Hundreds of Hundreths, and each hundred Hundreths
divided by 99, giving 1 Hundreth for Quotient, with a Remainder 1
Hundreth, the 42 Hundreds of Hundreths will give 42 Hundreths for the
Quotient, with a Remainder 42 Hundreths; fo that the Number of
Hundreths of the Quotient, and the Number of Hundreths of the Re-
mainder will be expreffed by the fame Figures as the Dividend 42.
--
As the 42 Hundreths remaining, cannot be divided by 99, I reduce
them into 42 Hundreds of Hundreths of Hundreths, that is into 42
Hundreds of Ten-Thouſandths, but each Hundred of Ten-thousandths being
divided by 99, will give 1 Ten-thoufandth for Quotient, with a Re-.
mainder 1 Ten thouſandth; whence 42 Hundreds of Ten-thousandths --
will give 42 Ten-thousandths for the Quotient, with a Remainder 42
Ten-thoufandths; wherefore the Number of Ten-thousandths,
of the Quotient and the Number of Ten-tboufandths of the
Remainder, will be expreffed by the fame Figures as thofe of
the Dividend 42, and it will appear in like Manner that all
the other Figures of the Quotient will be equal two by two to thoſe of
the Dividend, which are fuppofed lefs than 99; whence dividing 42 by
29, there will refult for Quotient (0,42 42 42 &c.). If the Number to
be divided by 99, was reprefented by one Figure, for Example, if 5.
40
ELEMENTS OF
Method of
finding from
whence fim-
or o5, was to be divided by 99, I transform 5 into 500
500 Hundredths,
which I divide by 99, and there refults for Quotient (0,05) that is 5
Hundreth's with a Remainder (0,05), I transform this Remainder into
500 Ten-thousandths, which I divide by 99, and there refults 5 Ten-
Thouſandths or (0,0005) with a Remainder 5 Ten-Thousandths or
(0,0005); fo that the Quotient will be compofed of fimilar decimal Pe-
riods, each confifting of the two Characters 05 equal to thoſe of the
Dividend, that is, 5 or 05 being divided by 99, will give for Quotient
(0,05 05 &c.). In fine, any Number divided by another Number greater
than it, all whofe Figures are 9, will give for Quotient an infinite
Series of decimal Periods, confifting of - as many Figures as
the Divifor, and each of thofe Periods having the fame fignificative
Figures as the Dividend; fo that if the Dividend has lefs Figures than
the Divifor, there will be places in each Period filled up with Cyphers,
fet down to the left Hand of the fignificative Figure or Figures, equal
to thoſe of the Dividend.
LXXXI.
Reciprocally the Sum of an infinite Series of decimal Periods, con-
fifting of the fame Figures, is equal to the Quotient of one Period di-
ple and com- vided by a number, confifting of as many 9s as there are Figures in the
-pound re- Period.
petends are
derived.
}
For Example, the Series (0,333 &c.) each of whofe Periods confifts
of one Figure 3, is equal to the Quotient of the Divifion of 3 by 9, or
of I by 39
The Series (0,23 23 23 &c.) each of whofe Periods (23) conſiſts of
two Figures, is the Quotient of the Divifion of 23 by 99,
The Series (0,087 087 087, &c.) each of whofe Periods (087) confifts
of three Figures, is the Quotient of the Divifion of 087 or of 87 by
999,
The Series 0,001 OCI 001 &c.) each of whofe Periods (001) confifts
of three Figures, is the Quotient of the Divifion of oor or of 1 by
999, and ſo on.
´LXXXII.
•
•
According as the Comma or feparating Point of any propofed decimal
Method of Number, is advanced one Place towards the left Hand, its Value is
finding from
whence fim- thereby rendered ten Times lefs. For Example, if in the Series
ple andcom- (0,298 298 &c.) the Comma be advanced one Place to the left Hand,
pound repe- there will refult (0,0298 298, &c.) whofe Value is ten Times lefs than
ded by Cy- that of (0,298 298 &c.) if the Comma be 'again advanced one Place
phers are de- to the left Hand; there will refult (0,00298 298 &c.) whofe Value is
ten Times less than that of (0,0298 298 &c.) or a hundred Times lefs
than (0,298 298 &c.) and fo on.
tends prece-
rived.
But the Series (0,298 298 &c.) compofed of equal Periods, the first
beginning immediately after the Comma, is the Quotient of the Divi-
fion of 298 by 999.
NUMERAL ARITHMETICK.
47
Wherefore the Series (0,0298 298.298 &c.) whofe Value is ten Times
lefs than that of the Former, is the Quotient of the Divifion of 298
298
by 9990, that is equal to and the Series (0,00298 298 &c.) whofe
9990
Value is ten Times lefs than the Former, is the Quotient of the Divi-
and ſo on; that is, when
298
fion of 298 by 99900, or is equal to
99900
a Series of decimal Periods does not immediately begin after the fepa-
rating Point, it reprefents the Quotient of a Divifion, the Dividend of
which, is equal to one Period; and the Divifor confifts not only of as
many 9s as there are Figures in the Period, but alſo of as many Cyphers,
as there are Places between the Comma and the firft fignificative Figure
of the firſt Period.
LXXXIII.
in- finding from
the whence fim-
&c. pound repe-
tends preced
ple and com-
Beſides thofe decimal Series that confift only of equal Periods, there Method of
are others, that independant of an infinite Series of equal Periods,
clude a certain Number of decimal Figures, after which begin
equal Periods. Such are the following, 0,1666 &c. 0,08 333
0,004 629 629 &c.
The first of thofe Series 0,1666 &c. confifts of o, I and an infinity ed by acer-
of Periods of 6s.
tain number
of decimal
The fecond 0,08333 &c. confifts of a Part 0,08 and an infi- figures are
nity of 3s.
The third 0,004 629 629, is compofed of a Part 0,004 and an infinity
of Periods 629, and ſo on.
To diſcover from whence thofe Quotients are derived; for Example,
to diſcover from whence the Series 0,004 629 &c. is derived, I feparate the
infinite Series of Periods from the decimal Figures that precede them,
in order to form two decimal Numbers of the propofed one, and there
reſults 0,004 and 0,000629 629 &c. contained in the given Number
0,004 629 629 &c.
to
Now the first Part (0,004) which denotes 4 Thoufandths, is equal
4
1000
The fecond Part 0,000629 629 &c.
an infinite Series of equal Periods, is equal to
which confifts only of
629
999000
; Wherefore the
Sum of the two Parts 0,004 and 0,000 629 629 &c. or the propofed deci-
mal Series 0,004 629 629 &c. is equal to the Sum of the two Divifions
4
629
1000' 999000 ·
Multiplying the Dividend and Divifor of the first Di-
derived.
ELEMENTS OF
vifion by 999, there refults
3996
999000
and
629
999000
and their Sum
4625
I
or
999000 216
(found by dividing its two Terms by the Dividend
4625) will be the Divifion from whence the propofed decimal Series
0,004 629 629 &c. is derived.
LXXXIV.
To abridge the foregoing Operation of Diviſion, the Computiſts inſtead
of fetting down the Products of the Multiplication of the Divifor into
the Figures of the Quotient, deduct from the Dividend the Figures of
thoſe Products according as they find them. Let for Example 7958 be
propoſed to be divided by 23, placing the Divifor at the right Hand of
the Dividend, as in the Margin, and beginning the
Diviſion by the Figures of the higheſt Denomination,
I first divide 79. Hundreds by the Divifor, and there re-
fults 3 Hundreds for the firſt Quotient Figure.
7958
23
346
105
138
000
To obtain the Remainder of this Divifion, I multi-
ply the Divifor 23 by the Quotient, and deduct the
Figures of the Product according as I find them from
Method of the Dividual 79. Saying: 3 Times 3 is 9, which I de-
contracting duct from the 9 of the Dividend, and there being no
the Operati-
ons of Divi- Remainder, I fet down a Cypher under the 9, then faying :3 Times 2 is
6, which I dedu&t from the Figure 7 of the Dividend, and fet down
the Remainder under the 7, whence from the Divifion of 79 Hun-
dreds by 23, there refults for Quotient 3 Hundreds, with a Remainder
to Hundreds, which could not be divided by 23.
fion.
4
To find the fecond Figure of the Quotient I bring down the 5 tens of
the Dividend to the right Hand of the 10 Hundreds that remain after
the firſt Diviſion, and divide 105 Tens by 23, letting down the Quotient
Tens at the right Hand of the 3 Hundreds. To obtain the Remainder
of this fecond Divifion of Tens, I multiply 23 by 4, and dedu& the Fi-
gures of the Product according as I find them, from the fecond Member
of the Diviſion 105, faying: 4 Times 3 is 12, which is to be deducted
from 5, but as this cannot be done, I borrow 1 Ten, or an Unit of a fu-
perior Degree, and adding it to 5, I deduct 12 from 15, and fet down
the Remainder 3 under the 5, then ſaying: 4 Times 2 is 8 and 1 I bor-
rowed, which is to be deducted, is 9, and deducting 9 from 10, I fet
down the Remainder I underneath, fo that the Remainder of the fecond
Division will be 13.
I
To find the third Figure of the Quotient, I bring down the 8 Units of
the Dividend to the right Hand of the 13 Tens remaining of the forego-
NUMERAL ARITHMETICK.
49
ing Divifion, and there refults 138 Units to be divided by 23; which
gives 6 Units for the Quotient, which I fet down to the right Hand of
the two Figures 34 already found. To obtain the Remainder of the
Divifion, I multiply the Divifor 23 by 6; and fubtract from this
Member of the Divifion 138, the Figures of the Produ& according as
I find them. And there being no Remainder left, I fet down a Cypher
underneath.
The Divifion being ended, I find 346 for the exact Quotient of 7958
divided by 23.
LXXXV.
The foregoing Operation is performed by fome Computifts in a man-
ner ſomewhat different, in each particular Diviſion, the Divifor being fet
down under the Dividend, and the Remainder above it.
10
7958 (3
Spanish Me-
thod of per-
forming Di
vifion ex-
For Example, if it was propofed to divide 7958 by 23, having point-
ed off as many Figures to the left Hand of the Dividend,
as will contain the Divifor: And as the two Figures 79
of the Dividend contains the Divifor 23, I fet down
23 under 79, and confider how often it is contained in
79, or how often 2 is contained in 7; and as it is
contained 3 Times, I fet down 3 in the Quotient.
2z
To difcover the Remainder of this Divifion, I multiply the Diviſor
23 by the Quotient 3; and fubtract the Figures of the Product ac-
cording as I find them from the fuperior Figures, and fet down the Re-
mainders over thoſe fuperior Figures. Saying: 3 Times 3 is 9, which
I deduct from the fuperior Figure 9; and as there is no Remainder left,
I fet down a Cypher over this 9 which I barr, as alſo the Figure 3
multiplied. Then faying: 3 Times 2 is 6, which I fubtract from the
fuperior Figure 7, and fet down the Remainder I over the 7, and
barr this 7 and the Figure 2 multiplied.
By this firſt Operation, the 79 Hundreds of the Dividend will be divided
by 23; the Quotient will be 3 Hundreds, and there will remain 10
Hundreds that could not be divided by 23.
I
203
7958 (34
233
To continue on the Divifion, I annex the 5 Tens of the Dividend to
the 10 Hundreds that remain after the first Divifion ;
and fet down anew the Divifor 23 under the Dividend
105 Tens, fo that the 3 will stand under the 5, and the
2 under the Cypher; that is, I remove the Figures of
the Divifor one Place towards the right Hand. I then
confider how often 23 is contained in 105; and fet down the Quotient
4 to the right Hand of the Figure 3 first found.
2
4 Times
3
To obtain the Remainder of this fecond Divifion, I fay: 4
is 12; and as this particular Product cannot be deducted from the fupe-
plained by
an Example.
2 G
50
ELEMENTS OF
rior Figure 5, I borrow 1 Ten to add it to 5, and deduct 12 from 15;
and fet down the Remainder 3 over the 5, barring this 5 and the 3
multiplied. Then I fay, 4 Times 2 is 8 and 1 I borrowed is 9, which
I deduct from. 10; fetting down the Remainder I over the Cypher,
after having barred 10 and the 2 multiplied.
By this fecond Operation, the 105. Tens will be divided by 23; the
Quotient will be 4 Tens, and there will be a Remainder 13 Tens which
could not be divided by 23.
I:
X0Z.
To compleat the Divifion, I annex the 8 Units to the Remainder 13
Tens; and fet down the Divifor 23 under the new Dividend 138: So
that the 3 will ſtand under the 8, that is, I remove
the Figures of the Diviſor one Place towards the right
Hand. And finding that 23 is contained 6 Times in
138, I fet down 6 in the Quotient to the right Hand of
34, and multiplying the Divifor 23 by 6, I fubtract the
igures of the Product according as I find them from
thoſe above them, as in the two foregoing Operations,
and after the Subtraction, there being no Remainder left, I conclude that
346 will be the exact Quotient of 7958 divided by 23.
LXXXVI.
7988 (346
2833
22
To render the foregoing Method of performing Divifion more com-
modious, the Computifts have found Means of avoiding the Trouble of
borrowing when the Product of the Divifor into the Quotient is fub-
tracted from the Members of the Divifion. Let it be required, for Ex-
French Me- ample, to divide 84162 by 98, as the Divifor 98 is not
thod of per- contained in the two firft Figures 84 to the left Hand of
forming Di- the Dividend, I point off the three Figures 841 for the
plained by firſt Member of the Divifion, and ſetting down the Divi-
an Example. for 98 under 41, I confider how often 98 is contained in
841 Hundreds, and fet down the Quotient 8 Hundreds, in
a Place which will be that of the Hundreds.
vifion, ex-
5
* 27
$4x62 (8
98
To obtain the Remainder of this firſt Diviſion, I multiply fucceffively
the two Figures of 98 by 8,- beginning by the Figure 9 of the higheſt
Degree; and deduct the Figures of the Product, according as I find
them from thofe immediately over them. Saying: 8 Times 9 is 72,
which I deduct from 84, fetting down the Remainder 12 over it, after
having barred 84 and the Figure 9 multiplied. Then I fay, 8 Times 8
is 64, which I deduct from 121, and fet down the Remainder 57 over
it, after having barred 121 and the Figure 8 multiplied.
By this first Operation, the 841 Hundreds will be divided by 98; the
Quotient will be 8, and there will remain 57 Hundreds that could not
be divided by 98.
NUMERAL ARITHMETICK.
51
18
82
2276
$4x62 (85
988
To continue on the Divifion, I remove the Divifor 98 one Place to
the right Hand, fetting it down under 76, in order to di-
vide the 57 Hundreds remaining with the 6 Tens, that
is, in order to divide 576 Tens by 98. The Diviſor 98
being contained 5 Times in the ſecond Member of the
Divifion 576, I fet down 5 in the Quotient at the right
Hand or the 8 already fet down, and multiply 98 by 5, be-
ginning by the Figure of the higheſt Degree; and deduct
the Figures of the Produ&t according as I find them from
the ſuperior Ones. Saying: 5 Times 9 is 45, which I dedu&t from
57; and fet down the Remainder 12 over 57, after having barred 57
and the Figure 9 multiplied. Then faying: 5 Times 8 is 40, which
I deduct from 126, and ſet down the Remainder 86 over 126, or rather
over 26, after having barred 126 and the Figure 8 multiplied.
9
By this fecond Operation, the Number 576 Tens will be divided by
98; the Quotient will be 5 Tens, and there will remain 86 Tens that
could not be divided by 98.
I
287
To compleat the Divifion, I remove the Divifor ftill one Place to-
wards the right Hand, in order to divide the 86 Tens remaining with
the 2 Units, that is, in order to divide 862 Units by
98. Finding that 98 is contained 8 Times in the Mem-
ber of the Divifion 862, I fet down 8 in the Quotient
to the right Hand of 85; and fearch for the Remain-
der of this Diviſion, Saying: 8 Times 9 is 72, which
I deduct from the fuperior Figures 86, and fet down
the Remainder 14 over thefe Figures, after having
barred 86 and the Figure 9 multiplied. Then laying
8 Times 8 is 64, which I deduct from 142; and
fet down the Remainder 78 Units over 42.
5 24
22768
$4162 (858
98 8 8
99
Wherefore the Quotient arifing from the Divifion of 84162 by 98 will
be 858, and there will remain 78 Units, which cannot be divided by 98.
LXXXVII.
in Divifion
Since to divide a Number by another, is to find a third Number which
multiplied by the fecond, will give a Product equal to the firft Number, Proof of the
Divifion may be proved, by confidering it as a Multiplication of which Operations
the Divifor is the Multiplicand, the Quotient the Multiplicator, and made by caf
the Dividend the Product, that is, by cafting out the 9's out of the Di-ting out the
viſor and Quotient, and multiplying the Remainders one by the other, af-nines, as in
Multiplica
ter having added to the Product the Figures of the Remainder of the tion
Diviſion, when there is one; and cafting out the 9's, if the Remainder is
equal to what remains when the 9's are caſt out of the Dividend; there is
a Prefumption that the Divifion has been accurately performed, other-
wife it is certain that fome Error has been committed in the Operation.
32
ELEMENTS OF
Example.
1
1
For Example, if 5478989 was divided by 375, and the Quotient found
to be 14610 with a Remainder 239, I caft the 9's out of the Divifor
and Quotient, and multiplying the Remainders 6 and 3 one by the other,
and adding to their Product 18 the Remainder of the Divifion 239, and
cafting the 9's out of their Sum 257; the Remainder 5 being equal to
what remains when the 9's are caft out of the Dividend 5478989, there
is a Prefumption that the Operation has been accurately performed.
LXXXVII.
To prevent the Multiplicity of Errors and to diſcover them in the
Courſe of the Divifion, the Computiſts verify each Figure of the Quo-
tient according as they are found.
964
254496
2376
264
1689
For Example, if 254496 was to be divided by 264, beginning the Di-
vifion, by the Figures of the higheſt Degree, I first divide 2544 Hun-
Method of dreds by 264, and find 9 Hundreds for the Quoti-
verifying
each figure
ent, I therefore fet down 9 in the Quotient in a
of the quo-
Place which will be that of the Hundreds; to difco-
tient accord ver whether the Figure 9 placed in the Quotient is
ing as they
are found, exact, or if the Divifor 264 is contained 9 Times
in the Member of the Divifion 2544; I multiply
264 by 9; fetting down the Product 2376, under
the Member of the Divifion2 544, and fubtracting this
Produ&t 2376 from 2544 I find 168 for theRemainder.
1584
1056
1056
0000
To diſcover whether this firft Remainder 168
Hunreds is exact; there are two Things to be confi-
dered. 1°. We are to examine whether 2376 is the
exact Product of 264 into 9; by taking the ninth Part of this Product:
as this ninth Part ſhould be equal to the Divifor 264. 2°. We are to
examine whether the Subtraction has been accurately performed; by
adding the Remainder 168 to the Product 2376: as this Sum fhould
be equal to the Number 2544 from which it is fubtracted.
To côtinue on the Divifion, I bring down the 9 Tens of the Dividend
to the right Hand of the 168 Hundreds, and there Refults 1689 Tens to
be divided by 264; which gives 6 Tens for the Quotient. I then multi-
ply 264 by 6, and deduct the Product 1584 from 1689; and there re-
mains 105. I prove this fecond Divifion as the former. 1°. By taking
the fixth Part of 1584, which is equal to the Divifor 264, and proves
the Product 1584 to be exact. 29. By adding 105 to 1584 whofe Sum
is equal to 1689, the fecond Member of the Divifion, and proves that
the Remainder 105 which is less than the Divifor 5264 is exact.
The Operations employed to obtain the 4 Units of the Quotient will
be proved in like Manner.
NUMERAL ARITHMETICK.
53
CHA P. II.
Of Fractions and their various Reductions, &c. Of the Operations per-
formed upon compound or applicate Numbers, and the Menfuration of
Surfaces and Solids.
T
HE Numbers we have hitherto treated of are compofed of feveral
integer Units, which for this Reaſon are called integer or whole Numbers, but
it often happens that the Unit that has been adopted, or eſtabliſhed
by Cuſtom, is too great to be contained once or ſeveral Times exactly in
the Magnitude which is propoſed to be meaſured. In this Cafe the
Computifts form fmaller Units that may meaſure exactly the propoſed
Magnitude, that is, which may be contained in it once, or a certain
Number of Times, exactly.
I.
To obtain Units fuitable to the Magnitude which is propofed to be
meaſured; the Computiſts divide the principal Unit that has been adopt- Origin of
ed into feveral equal Parts, which in general are called fractional Units, Fractions,
and that receive their particular Denominations from the Number of
Parts into which the principal Unit has been divided. For Example, if
the principal Unit be divided into 2, or into 3, or into 4, or into 5 equal
Parts, each Part is called 1 Half, or 1 Third, or I Fourth, or I Fifth ;
and thofe Parts are fractional Units.
A fractional Unit or a Collection of feveral equal fractional Units, is
called a Fraction or broken Number. There are therefore two Numbers:
required to reprefent a broken Number; viz. a Number to denote the
Species of the fractional Unit, that is, to fhew into how many equal
Parts the principal Unit has been divided, and another Number to denote
how many Times thoſe new Units are repeated.
To diftinguish thofe two Numbers, the Computiſts place ope above
the other, with a Line drawn betwixt them, placing under the Line Notation of
the Number which denotes into how many equal Parts the principal Fractions.
Unit has been divided, and which confequently denotes the Species of
the fractional Unit; and placing above the Line, the Number which de-
notes how often the fractional Unit is taken.
7
For Example, is a Fraction or broken Number, the inferior Num-
8
I
ber (8) denotes that the principal Unit has been divided into 8 equal Parts,
and that confequently each Part is the 1 Eighth of the principal Unit;
and the fuperior Number (7) denotes that there are 7 of thoſe new
Units; ſo that the Fraction denotes 7 Eighths of the Unit or Quanti
7
पु
ty that has been taken for Unit.
54
ELEMENTS OF
The
different
fpecies of
fractions.
How a
whole Num-
7
·
7
As the fractional Unit receives its Denomination from the inferior
Number of the Fraction, it is called the Denominator, and becauſe the
fuperior Number denotes the Number of thoſe new Units, it is called
the Numerator. Thus in the Fraction the inferior Number (8)
which repreſents Eighths, is the Denominator, and the fuperior Number
(7) which denotes that there are 7 of thoſe Units called Eighths, is the
Numerator.
8
The Numerator and Denominator of a Fraction are called the two
Terms of this Fraction; the Numerator is called the firft Term, and the
Denominator is called the fecond Term:
'FI.
The Computifts diftinguiſh two Sorts of broken Numbers; abftract
broken Numbers, and concrete or applicate broken Numbers. The broken
Numbers, fuch as
which denote 1 Fourth, 2 Fifths,
1
2
5
2
,
...4
5 6
5 Sixths, that are not applied to number any Species of Things are
called abſtract or abfolute broken Numbers. Broken Numbers that are
applied to Number the Parts of any Thing, are called concrete or appli-
cate broken Numbers. For Example, Pound, Foot, Hour,
I
4
2
5
-5
6
which denote one fourth of a Pound, two Fifths of a Foot, five Sixths of
an Hour, are concrete or applicate broken Numbers. It is manifeft that all
broken Numbers, whether abſtract or concrete, may be conſidered as
Species of integer Numbers, having for principal Units their particular
fractional Units.
III.
*
A Fraction may be alfo confidered as the Quotient of a Divifion, of
which the Numerator is the Dividend, and the Denominator the Divi-
for. For Example, the
Fraction may be confidered as the Quoti-
7
8
1
tient of 7 divided by 8. For to divide 7 by 8 is to take the eighth
Part of 7, but to take the eighth Part of 7 the eighth Part of
each of the Units which compofe 7 must be taken: and as each
Unit will give 1 Eighth, for its eighth Part; 7 Units will give 7
Eighths, that is, the Fraction for their eighth Part.
Whence the Fraction
~7
8
7
is the Quotient of 7 divided by 8.
8
As a Number is not changed when multiplied and divided by the fame
ber is chang- Quantity; being rendered fo much leſs by Divifion, as it is increaſed by
Multiplication, it is manifeft that an Integer may be changed into a Fracti-
on; by multiplying it by any Number for to make a Numerator, and
giving it this fame Number for a Denominator.
ed into a
fraction.
NUMERAL ARITHMETICK.
55
whofe nu-
merator and
Since a Fraction is equal to the Quotient of the Divifion of its Nu- A fraction
merator by its Denominator; it is manifeft that it is equal to an integer
Unit, when its Numerator is equal to its Denominator, for the Denomi- denomina-
nator will be contained once in its Numerator.
IV.
Since by multiplying or by dividing the Dividend and Diviſor of a
Divifion, by a fame Quantity, the Value of the Quotient is not changed
if the Numerator and Denominator of a Fraction be multiplied or divi-
ded by the fame Quantity; the Value of the Fraction will not be chan-
ged.
4
5
tor are e-
qual, is an
unit.
The value
of a fraction
If, for Example, the Fraction be given, and the Numerator 4
and the Denominator 5 be multiplied by 3 or by 4, there will refult a
new Fraction 12 or of the fame Value as the firſt. And recipro- ged by mul
15
16
20
cally, if the Numerator and Denominator of a Fraction
5
12
IS
is not chan
tiplying its
numerator
be divided, and deno-
by a fame Number for Example,by 3, there will refult a new Fraction
which will be equal to the firſt 12.
15
5
By dividing the Numerator and Denominator of a Fraction by a fame
Quantity it is rendered more fimple, and fo much the more fo, as the
Quantity by which it is divided is greater; and when the two Terms
of a Fraction are divided by their greateſt common Divifor, the Fracti- -
on which reſults whofe two Terms can no more be divided by a fame
Quantity, is faid to be reduced to its loweft. Terms..
V.
minator by
the fame
quantity,
their lowest..
terms.
To reduce a Fraction to its loweſt Terms, without altering its Value, Method of
the greater Term muſt be divided by the leffer; and if there is no Re- reducing
mainder, the leffer Term will be manifeftly the greateſt common Divi- fractions to
for of the two Terms of the Fraction. If there is a Remainder, the leſ-
fer Term must be divided by this Remainder, and if there is no Remain-
der, the Remainder of the firſt Divifion will be the greateſt common
Divifor of the two Terms of the Fraction.
•
If this ſecond Divifion leaves a Remainder, the firft Remainder muſt
be divided by the fecond, and the fecond by the third, and fo on conti-
nually, dividing the laft Divifor by its Remainder, till there is no Remain-
der left And then the laſt Divifor will be the greateſt common Divi-
for of the two Terms of the Fraction, and by dividing the two Terms by
this laft Divifor, the Fraction will be reduced to its lowest Terms. But
if it should happen that the laſt Remainder fhould be Unit, then is the
Fraction already expreffed by its moft fimple Terms.
56
ELEMENTS OF
fractions to
VI..
2016
5796
Let it be propofed, for Example, to reduce the Fraction -to its low-
eft Terms.
The forego 1°. I divide the greateſt Term 5796 by the leffer 2016. The Di-
ing method vifion being performed, there will remain 1764. 2°. I divide 2016 by
of reducing the Remainder 1764; without attending to the Quotient, but only to
their loweſt the Remainder 252. 3°. I divide the firft Remainder 1764, by the
terms, ap fecond Remainder 252; and as there is no Remainder left after the Divi-
plied to an
example.
fion, I conclude that the laft Divifor 252, will be the greateſt common
Divifor of the two Terms of the Fraction
2016
5796
and dividing the two
8
Terms of this Fraction by 252, there refults the new Fraction 8
23
whoſe Terms cannot be rendered more fimple, and which will have the
fame Value as the propofed Fraction 2016
VII.
5796
Grounds of The ground of this Operation is as follows: 1º By dividing the Deno-
2016
, the Quotient will be found to be 2
5796
Whence the Denominator 5796 is compoſed
1764, and the Fraction
the forego minator by 2016 of the Fraction
ing method.
with a Remainder 1764.
of two Parts 4032 and
2016
5796
may be reduced
2016
to this Form
wherefore any Number that is the
2×2016 more 1764’
greateft common Divifor of the two Terms of the Fraction
2016
>
5796
will
be the greatest common Divilor of 2X2016 and of 1764, which are
the two Parts of the Denominator, and confequently ſhould be the great-
eft common Diviſor of 2016 and of 1764.
2º By dividing 2016 by 1764 it will be found to be contained once
with a Remainder 252, that is, 2016 is compofed of two Parts, of
1764 and of 252. Whence the Number which will be the greateſt
Diviſor of 2016 and of 1764 will be alfo a Diviſor of 252. But 252
is the greatest Divifor of 252, and divides 1764 exactly: It will there-
fore alſo divide 2016 which is the Sum of 252 and of 1764, and will be
alſo a Diviſor of twice 2016, that is, of 2×2016. The Number 252.
being a Divifor of 2X2016 and of 1764, will be alfo a Divifor of their
Sum 5796, and will confequently divide the Numerator and Denomina-
2016
tor of the Fraction
Moreover, 252 is the greatest common Di-
5796*
viſor of 2016 and of 5796; fince the common Divifor of thoſe two
Numbers ſhould be a Divifor of 252.
7
57
NUMERAL ARITHMETICK.
{
VIII.
terms.
There is another Method of reducing a Fraction to its lowest Terms, Another
more eaſy, and upon feveral Occafions, more commodious than the method of
reducing
foregoing. 1. If the Numerator and Denominator of a Fraction are fractions to
even Numbers, they may be both divided by 2, till one of thoſe two their lowest
Terms becomes an odd Number. 2°. If the two Terms end by 5, they
may be divided by 5, till one of them ceaſes to end by 5. 3°. When the
two Terms of the Fraction are odd Numbers, and do not end by 5,
Tryal may be made to divide them by 3, till one of them ceafes to be
divifible by 3; afterwards Tryal may be made to divide the two new
Terms by 7, then by 11, afterwards by 13, 17, 19, 23; and fo on by
all the Numbers that have no other Divifors but themſelves and Unity.
In fine, when one of the Terms is no more divſible, or when the two
Terms can be no more divided by a fame Quantity, the Fraction will
be reduced to its lowest Terms.
IX.
Let it be propofed, for Example, to reduce the Fraction to its low.
eft Terms.
2016
3780
1008
I divide the two Terms which are even by 2, and there reſults 1890 The fore-
I divide again by 2 the two new Terms, which are alfo even Numbers, going me-
thod appli-
and there refults
I then divide thofe two Terms by 3, and there ed to an ex-
ample.
refults
168
315
504
945
Dividing again by 3, there reſults
56
105
Dividing after-
8
wards by 7, I find. And this Fraction will at Length be the
15
Fraction reduced becauſe its Numerator 8, cannot be divided but by 2,
or by a Multiple of 2, and its Denominator 15 cannot be divided by. 2. ¸
X.
If the two Terms of the Fraction have Cyphers to their right Hand, The fore-
it is manifeſt that an equal Number may be fuppreffed in thoſe two Terms; going me-
thod appli
becauſe each Term will be divided by 10 each Time a Cypher is fup-ed to ane.
preffed.
Thus to reduce the Fraction 3150 to its lowest Terms, I first divide
11400
the two Terms by 10, by fupprefling in each of them the Cypher that
occupies the Place of the fimple Units, and there refults
315
1140°
Then
becauſe the new Numerator ends by 5, and is confequently divisible by
5, as alſo the new Denominator which ends by a Cypher, I divide by
3, and there reſults I afterwards divide by 3, and there refults
63
228
2. H
21
76
ther ex-
ample.
58
ELEMENTS OF
And this Fraction
21
Method of
any two
mination,
76
1
will be reduced to its loweſt Terms; ſince the Nu-
merator 21 is diviſible only by 7 or by 3, and the Denominator 76 is
divifible neither by one or the other.
XI.
To reduce two Fractions to the fame Denominator without altering
reducing the Value of thoſe Fractions; having reduced them to their lowest
fractions to Terms (if they are not fo already) the Numerator and Denominator of
the famede the first Fraction must be multiplied by the Denominator of the Second,
and the Numerator and Denominator of the ſecond by the Denominator
of the first, whence will refult two other Fractions of the fame Value
with the two first; (Art. iv.) having the fame Denominator, fince the De-
nominator of each of them will be the Product of the Denominators of
the two firft Fractions.
Application
of this me-
"
XII.
Let it be propofed, for Example, to reduce to the fame Denomination the
·Fractions — and 5.
2
.3
7
1º. I multiply the two Terms 2 and 3 of the firſt Fraction, by the
3
thod to an Denominator 7 of the fecond, and there refults a new Fraction 14
example.
21
equal to the firſt 2
3
cond Fraction 5
20. I multiply the two Terms 5 and 7 of the fe-
by the Denominator 3 of the firft, and there refults
General
A
7
15
a new Fraction equal to the ſecond.
21
7
5
By this Means the two propofed Fractions and without hav-
2
3
7
ing changed their Value will be reduced to the two Fractions 14 and
.15
-21
21
which have each for Denominator the Product of the Denomiators
3 and 7 of the propoſed Fractions.
XII.
To reduce any Number of Fractions to the fame Denominator; being
method of reduced to their loweft Terms, 10. All the Denominators of the Frac-
reducing tions must be multiplied into one another, and the Product will be the
any number
of fractons Denominator that all the Fractions reduced to the fame Denomination
fhould have.
NUMERAL ARITHMETICK.
59
1
20. To obtain the Numerator of the first of the Fractions that are to the fame
denomina-
propoſed to be reduced to the fame Denomination, the Denominators of tor.
all the Fractions, except that of the firſt, muſt be multiplied into one
another, and the Product that arifes multiplied by the Numerator of
the firft will be the Numerator of this firft Fraction reduced.
To obtain the Numerator of the fecond Fraction reduced, all the
Denominators except that of the fecond, must be multiplied into one
another, and the Product that refults multiplied by the Numerator of the
ſecond will be the Numerator of this fecond Fraction reduced; and in
like Manner the Numerators of the other Fractions will be found.
XIV.
Let it be propofed, for Example, to reduce to the fame Denomination the
four Fractions —, —,,,
Ι
2
6
2 3 5 7
Application n
I multiply into one another all their Denominators 2, 3, 5, 7; and
the Product 210, will be the new Denominator, which ſhould be com of this Me
mon to all the Fractions.
105.
210
thod to an
To obtain the Numerator of the firft of the new Fractions; I multi- Example..
ply into one another all the Denominators, except the firſt (2), that is,
I multiply together the three Denominators, 3, 5, 7; and the Product
that arifes 105 by the Numerator 1 of the firft Fraction, and the Product
105,
will be the Numerator of the firſt Fraction, which will become
To obtain the Numerator of the fecond new Fraction; I multiply into
one another all the Denominators except the fecond (3), that is, I multiply
together the three Denominators 2, 5, 7, which produce 70; I afterwards
multiply 70 by the Numerator 2 of the fecond propofed Fraction, and
the Product 140 will be the Numerator of the fecond new Fraction
140 In like Manner I find the Numerators of the two other Fractions
210.
I
168 180
which will be
210 210
Whence the four Fractions 1 2
2. 3
105 140 168 180
4
}
5
77
9
reduced to the fame Denomination will be
XV.
210 210 210 210
T
By the foregoing Method any Number of Fractions may be reduced
to the fame Denominator without altering their Value, but thofe Frac- Inconveni
tions will not always be reduced to as low Terms as they might be, and
retain ftill a common Denominator.
1º. If among the Fractions reduced to their loweſt Terms, fuch as the
following
I
2
6.
3 5 7
; there are not ſeveral whofe Denomina-
cyto
ency
which this
foregoing
Method of
of reducing s
бо
ELEMENTS OF
*
fractions be tors have a common Divifor: When thofe Fractions will be reduced to
the fame de-
nomination the fame Denomination; the new Fractions
is liable.
105 140 168 180
210 210' 210
210
,
that refult cannot be reduced to lower Terms and retain ſtill a common
Denominator.
2º. If among the Fractions which are fuppofed reduced to their low
Method of eft Terms, there are feveral whofe Denominators have common Divi-
remedying
this inconve-fors: When all theſe Fractions are reduced to the fame Denomination,
niency ac- they may be reduced to lower Terms, and retain ftill a common Deno-
cording to
minator, by dividing their Numerators and the common Denominator
by the common Divifor fo many Times lefs one as there are Denominators
to which thoſe Divifors are common in the firſt Fractions.
For Example, if thoſe four Fractions were given —, ——,
2
the two
different
cafes that
may occur.
I
I
4
I
I
7
reduced to their loweſt Terms, among which are three whofe Denomi-
Example of
the art cafe nators 2, 4, 6, are diviſible by 2: When all thoſe Fractions are reduced
168 84 56 48
336 336
to the fame Denomination, and become
336
3369
the Numerator of each of them and the common Denominator may be
divided twice fucceffively by 2, which will reduce them to the follow-
42 21 I 4 12
ing ones
84 84 84 84
,
XVI.
When the Divifors of the Denominators are compofite Numbers, if
their Factors are common to a greater Number of Denominators,
they should be employed, as Diviſors, preferably to thoſe compofite Di-
Example of viſors.
the fecond
mfc.
I
I
1
I
If thofe four Fractions
•
were propofed, which
2
4
12
I
I
I
I
>
may be reduced to this Form.
2 2X2 2X3 2X2X3
Though
the Denominators 2×2, 2X2X3 of two of them are divisible by the
compofite Divifor 2X2; I do not make Ufe of this compofite Divifor,
becauſe the ſimple Divifor 2 is common to a greater Number of Deno-
minators than the compofite one 2X2: And I obferve,
1. That the four Denominators 2, 2X2, 2X3 2X2X3 are divifible
by 2, and may be reduced to 1, 2, 3, 2X3.
2º. That of thoſe four Terms reduced by Divifion, there are two, viz.
2 and 2×3 ſtill diviſible by 2, fo that thoſe four Terms are reducible
to I, 1, 3, 3.
3°. Laftly, that thofe four new Terms, ftill include two, viz. 3 and
3, which are divifible by 3.
NUMERAL ARITHMETICK.
6I
1
Whence the four propoſed Fractions when redu-
I
I
2
등​,
1
12'
ced to the fame Denominator, and are transformed into
288 144
2
576' 576'
96 48
may be reduced to lower Terms and ftill retain a com-
576 576
mon Denominator, by dividing their Numerators and Denominators,
three Times fucceffively by 2, or once only by 8, then once by 2, and
6 3
laftly once by 3, which will reduce thofe Fractions to 12 12'
2
I which cannot be reduced to lower Terms, and retain a com-
mon Denominator.
12
12
XVII.
The Operations performed upon Fractions will often produce other Reduction
Fractions whofe Numerators will be greater than their Denominators. of improper
And fince a Fraction is equal to a whole Unit when its two Terms are fractions
equal, thofe Fractions will contain as many integer Units as the De-
nominators are contained in their Numerators.
Whence to find the Number of integer Units contained in a Fraction;
the Numerator must be divided by the Denominator, and the Quotient of
this Divifion will be the Number of integer Units contained in the Frac-
tion. And the Remainder of the Divifion, if there be any, will be the
Numerator of a Fraction having the Diviſor for Denominator.
18
4
For Example, being a propofed Fraction; I divide the Nu-
Example
merator 18 by the Denominator 4, and there reſults for Quotient 4 in-
teger Units with a Remainder 2; and this Remainder being divided by
the Denominator 4 will give the Fraction which is reduced to
I
2
2
4
I
fo that the Fraction will be changed into 4 Units and —•
18
4
XVIII.
2
The foregoing Reductions of Fractions are neceffary to prepare them
for Addition and Subtraction, for we can really add together, or fubtract
one from the other only Quantities which conſiſt of Units of the fame
Species. Whence Fractions cannot be added together or fubtra&ed
one from the other, but when their fractional Units are the fame;
and then they must have the fame Denominator.
62
ELEMENTS OF
XIX.
1º. If the Fractions which are propoſed to be added have the fame Dno-
Addition of minator; their Sum will be obtained by forming a new Fraction of the
the fame de-fame Denominator, and having for Numerator, the Sum of their Nume-
nomination. rators.
fractions of
}
For Example, if the Fractions
3
7
6
"
7
7
which have the fame
Denominator 7, were to be added together; I confider them as con-
crete Numbers, whofe principal Units are Sevenths; and I fay 3 Sevenths
and 4 Sevenths is 7 Sevenths which joined to 6 Sevenths make 13 Sevenths,
Example. which I fet down thus 3: That is, I add together the three Numera-
13
7
tors 3, 4, 6, and fubfcribe under their Sum 13, the Denominator 7
common to all the Fractions added together; and there refults for
the Sum required.
13
7
2º. If the Fractions to be added together have not the fame Deno-
Addition of minator: They must be reduced to the fame Denomination, and thofe
fractions of new Fractions added, as in the former Caſe.
different de-
nominations
For Example, if the Fractions,
-
· 3
2 4 6
5' 7
were propoſed to
140 168 180
Example.
>
105
be added together; I transform them into
210 210 210 2109
which have the fame Denominator. I then add together their Nume-
rators 105, 140, 168, 180, and fubfcribing under their Sum 593 the
Denominator 210, there will refult the Fraction equal to the Sum
105 140 168 180
of the Fractions
propofed ones.
>
210 210 210
210
593
210
which are equal to the four
XX.
From the Addition of feveral Fractions, there often refults a Frac-
tion whofe Numerator is greater than the Denominator. Such a Frac-
tion being greater than the principal Unit, fhould be reduced to the
integers it contains, and to a Fraction that it may contain befides.
For Example, having found that the Sum of the three Fractions
3 4 is, whofe Numerator 13 contains the Denominator 7
>
7 7 7
7
once with a Remainder 6, whence this Fraction
6
13
7
may be divided in-
to thoſe two others
Zand
91 and confequently will be equivalent to 1
7
7
6
and
3
63
NUMERAL ARITHMETICK.
XXI.
of the fame
To fubtract one Fraction from another, 1°. If they have the fame De- Subtraction
nominator, it ſuffices to deduct the Number of fractional Units of the of fractions
one from the Number of the fractional Units of the other, fince their denomina-
fractional Units are equal. Now the Numerators of the Fractions ex-tion.
preſs the Number of fractional Units they contain; wherefore the Re-
mainder of the Subtraction will be obtained, by deducting the Numera-
tor of one from the Numerator of the other, and fubfcribing under the
Remainder the Denominator common to the two Fractions; becauſe the
Units remaining after the Subtraction fhould be of the fame Species as
thoſe of the Number from which the Subtraction has been made.
For Example, if it was propoſed to deduct from I fubtract 2
2
9
8
9
from 8, and under the Remainder 6, I fubfcribe the common Denomi-
nator 9; and there refults for the Remainder of the Subtraction the
Fraction which may be reduced to the Fraction 2
6
9
3
20. If the propofed Fractions have different Denominators, they muft Subtraction
be reduced to the fame Denomination, and then deducted one from the of fractions
other as in the foregoing Cafe. For Example, if it was propoſed to of different
fubtract from =; I firſt reduce thoſe two Fractions to the ſame
2
3
6
7
14
21
18
21
و
Denomination, whereby they will be transformed into and then
fubtracting the Numerator 14 from the Numerator 18, I fubfcribe under
the Remainder 4 the Denominator 21 ; and there reſults for the Re-
4
mainder of the Subtraction.
XXII.
21
The Multiplications and Divifions of Fractions by Fractions, being
Operations compofed of the Multiplication and Divifion of Fractions
by whole Numbers, before we proceed to treat of the Multiplication and
Divifion of Fractions by Fractions, it will be convenient to. explain the
Multiplication and Divifion of Fractions by whole Numbers
XXIII.
denomina-
tions.
whole num-
To multiply a Fraction by a whole Number, for Example, by 2, or Multiplica-
by 3, or by 4, &c. is to repeat it twice or three Times, or four Times, tion of frac-
&c. or in general as often as the Multiplicator contains Unity: confe- tions by
quently it is to make a Fraction twice or three Times or four Times, &c. bers.
greater than the propoſed Fraction. Now this Operation may be per-
formed two different Ways, either by operating on the Numerator only,
or by operating on the Denominator only.
}
64
First me
thod.
ELEMENTS OF
10. If it be propofed to operate only on the Numerator, the Nume-
rator of the propoſed Fraction muſt be multiplied by the whole Number
which is to ferve as Multiplicator; and fubfcribing under this Product
the Denominator of the propofed Fraction, there will refult a new Frac-
tion, which will be the Product required; as will appear by the follow-
ing Example.
2
If it was propoſed to multiply the Fraction by 4; I confider the
9
Example of Multiplicand 2 as a concrete Number (2 Ninths) which is to be repeat-
this first
Method.
Second me
thod.
9
ed 4 Times; whence I fay 4 Times 2 Ninths is 8 Ninths, which I fet
down thus : that is, I multiply the Numerator 2 by 4, and ſubſcribe
8
9
under the Product 8 the fame Denominator 9; which will give 8
for the Product required.
9
2º. If it be propoſed only to operate on the Denominator, this Deno-
minator muſt be divided by the propoſed Multiplicator; and fubfcribing
the Quotient under the Numerator of the Fraction to be multipled;
the new Fraction that will refult from this Operation, will be the Pro-
du& required, as will appear by the following Example,
5
Let it be propoſed to multiply by 2: I divide the Denominator 12
12
5
Example of by 2, and there refults a new Fraction for the Product ariſing from
this fecond
Method.
5
12
the Multiplication of by 2.
6
For the principal Unit being divided into twice more Parts in the Frac-
5
6
tion than in the Fraction
12
5
944
the fractional Units of the Frac-
5
tion will be double of thofe of the Fraction
6
5
And as thoſe frac
12
5
tions confiſt of the fame Number of Parts, it is manifeſt that will be
double of 12.
6
As it is always poffible to multiply one Number by another, and that
one Number cannot always be divided by another without a Remain-
der; it will always be poffible to perform the Multiplication of a Frac-
tion by a whole Number, by multiplying its Numerator by this whole
Number, but not by dividing its Denominator by this whole Number.
•
NUMERAL ARITHMETICK.
65
XXIV.
Fractions
To divide a Fraction by a whole Number, for Example, by 2, or by Divifion of
35 or by 4, &c. is to form a new Fraction which will be twice, or 3
by whole
Times, or 4 Times, &c. lefs than the Fraction propofed to be divided. Numbers.
Now this Operation may be performed two different Ways; by ope-
rating on the Numerator only, or by operating only on the Denominator.
1" If it be propofed to operate only on the Numerator: the Numerator First Me
of the propofed Fraction must be divided by the whole Number which thod
is to ſerve as Divifor; and fubfcribing under the Quotient the Denomi- explained
nator of the propofed Fraction, there will refult a new Fraction which by an Ex-
will be the Qotient required. As will appear by the following Example.
Let it be propoſed to divide the Fraction by 4, I confider the
Dividend 8
9
8
9.
as a concrete Number which reprefents 8 Ninths; and
to divide it by 4, I lay the fourth Part of 8 Ninths is manifeftly 2 Ninths
Wherefore the Fraction found by di-
which I fet down thus
viding the Numerator
8
2
8
9..
9
by 4, is manifeftly the Quotient of the Frac-
tion divided by the whole Number 45-
9
ample.
thod
2º. If it be propoſed to operate only on the Denominator; the
Denominator of the Fraction must be multiplied by the propofed Second Me-
Divifor and fubfcribing the Product under the Numerator of the explained
propoſed Fraction; the new Fraction that refults will be the by an Ex-
Quotient required.
:
→
-
6
7.
For Example, if it was propofed to divide the Fraction
— by 4;
without medling with its Numerator 6, I multiply only its Denomina-
tor 7 by 4; and there refults the Fraction for the Quotient of the-
6
Fraction — divided by 4.
་
7
6
28
For the principal Unit being divided into four Times more Parts in
6
the Fraction than in the Fraction, the fractional Units of the
6
28
Fraction will be only the fourth Parts of thofe of the Fraction
6
7
28
and as thoſe two Fractions confift of the fame Number of Parts,
6
that, 28
whofe Parts are four Times lefs will be contained four Times
in the other
6
윽​,
that is, as often as there are Units in the Divifor 4.
ample.
66
ELEMENTS OF.
f
The fecond
As it is always poffible to multiply the Denominator of a Fraction by
Method not a whole Number, and it often happens that its Numerator can not be di-
always prac- vided exactly without a Remainder: it is manifeit that a Fraction can al-
ticable. ways be divided by a whole Number, by multiplying its Denominator
by this whole Number, but not by dividing its Numerator by this whole
Number.
Multiplica
tion by
Fractions.
XIV.
1
To multiply any Magnitude by a Fraction, is to multiply this Quan-
tity by its Numerator, and afterwards divide it by its Denominator,
For let it be propofed, for Example, to multiply any propoſed Magnitude
2
by, if inſtead of multiplying it by the Fraction —, it had been
multiplied by the Numerator; the Product arifing would be triple of
the one required, fince the propofed Quantity has been multiplied by
a Number triple of the given Multiplicator: confequently the third
Part of this Product muſt be taken, that is, it must be divided by the
Denominator 3, in order to reduce it to its juſt Value.
2
3
Whence it follows that the Multiplication by a Fraction, whofe Nu-
merator is Unity, is really a Divifion by the Denominator of this Frac-
tion. For Example, the Multiplications by,,, &c. are really
Divifions by the Denominators 2, 3, 4 &c. of thefe Fractions.
For to multiply by
→
.1
·I
or by or by, we muſt firſt multiply
3
by Unity, which will not alter the Quantity multiplied; and after-
wards divide by 2, or by 3, or by 4.
XXVI.
The Multiplication of a Fraction by a Fraction may be performed
Four diffe- four different Ways, three of which are confined to particular Cafes,
rent Me- but the fourth is applicable to every Cafe.
thods of
multiplying
1. A Fraction may be multipled by a Fraction, by operating only on
a Fraction the Numerator of the Multiplicand, and confequently without altering
by a Fracti- the Denominator of this Fraction.
en.
2º. A Fraction may be multiplied by a Fraction, by operating only
on the Denominator of the multiplicand, and confequently without alter-
ing the Numerator of this Fraction.
3. A Fraction may be multiplied by a Fraction, by operating on
the two Terms of the Multiplicand, by Means of Divifion.
4°. A Fraction may be multiplied by a Fraction by operating on the
two Terms of the firft, by Means of Multiplication.
We shall explain thofe different Methods of multiplying a Fraction
by a Fraction, in the following Articles.
NUMERAL ARITHMETICK
67
}
XXVII.
ed by an
To multiply a Fraction by a Fraction, by operating only on the Nu- Firft Me-
merator of the Fraction which is confidered as the Multiplicand; the thod explain
Numerator of the Multiplicand must be multiplied by the Numerator Example.
of the Multiplicator; and this Product being divided by the Denomi-
nator of the Multiplicator, and under the Quotient the Denominator of
the Multiplicand being fubfcribed; there will refult a Fraction equal to
the Product required, as will appear by the following Example.
9
9
Let it be required to multiply by; the Fraction being con-
10
3
10
fidered as a concrete Number 9 Tenths, I first multiply it by the Nume-
rator 2 of the Multiplicator, faying: twice 9 Tenths is 18 Tenths or
18
10
3
but by multiplying by 2, I multiplied by a Number 3 Times too-
2 -which is the
great; fince it ſhould have been only multiplied by
3
third Part of 2. Wherefore the Produ& 18 Tenths is alfo three Times
too great; I therefore take the third Part of it, or divide it by 3; and
there refults 6 Tenths or 2
9
for the real Product of multiplied by.
6
10
XXVIII. -
10
2
3
To multiply a Fraction by a Fraction by operating only on the De-
nominator of the Fraction which is confidered as the Multiplicand: the Second
Denominator of the Multiplicand must be divided by the Numerator of Method
the Multiplicator; and the Quotient multiplied by the Denominator of explained
the Multiplicator; and this Produ& being fubfcribed under the Nume- by an Ex-
ample,
rator of the Multiplicand; there will refult a Fraction equal to the Pro-
duct of the two Fractions which were to be multiplied one by the other;
as will appear by the following Example.
IQ
;
3
Let it be propoſed to multiply the Fraction by by dividing the
Denominator 10 of the firſt Fraction by the Numerator 2 of the fecond,
the first Fraction will be multiplied by 2; (Art. xx111.) conſequently
9
10
the Fraction that refults will be triple of the one required, fince the
5
Multiplicand, has been multiplied by
has been multiplied by 2 which is triple of
10
the Fraction by which it ſhould have been multiplied; this Pro-
3
68
ELEMENTS OF
duct therefore
ſhould be divided by 3 which is the Denominator of
5
-
the Fraction. But by multiplying the Denominator of the Pro-
5
・3
by 3, this Product will be divided by 3 (Art. XXIV.) Wherefore
duct
the Fraction that refults
9.
$5
multiplied by.
will be the real Product of the Fraction
?
10
XXIX.
Third
Method
explained
by an
Example.
Fourth
Method
explained
by an
1
{
*
To multiply 2 Fraction by a Fraction, by operating on the two
Terms of the Fraction which is confidered as the Multiplicand, by
Means of Divifion. The Denominator of the first must be divided
by the Numerator of the fecond, and the Numerator of the first by the
Denominator of the fecond; and from thoſe two Operations there will
refult a Fraction equal to the Product required.
9
10 3
Let it be propoſed to multiply by: I first divide the Denomi-
nator 10 of the firft Fraction by the Numerator 2 of the Second, and
by this Operation the Fraction will be multiplied by 2. (Art. xx111;)
✔
9
ΤΟ
but the Produ&will be triple of the one required; fince the Num-
-5
ber 2 by which I multiplied, is triple of by which I ſhould have multi-
5
3
plied. This Produ&t 2 which I confider as a concrete Number 9
Fifths, fhould therefore be divided by 3; whence I fay the third Part of
9 Fifths is 3 Fifths or 3, and this Fraction will be the ProduЯ re-
quired.
XXX.
5
To multiply a Fraction by a Fraction, by operating on the two Terms
of the Fraction confidered as the Multiplicand, by Means of Multiplica-
tion, the Numerator of the first Fraction must be multiplied by that of
Example. fecond, and the Denominator of the firft by that of the fecond: and
the Fraction that reſults having for Numerator the Product of the two
Numerators, and for Denominator the Produ& of the two Denomina-
tors, will be the Product required.
Let it be required to multiply by: I confider the Multipli-
NUMERAL ARITHMETICK.
ég
cand
9
10
as a concrete Number 9 Tenths, and multiply it firft by the
Numerator 2 of the Multiplicator, faying: twice Tenths is 18
18
9
Tenths or 15; this Product will be triple of the one required, becauſe
the Number 2 by which I multiplied, is triple of the Fraction
18
3
by which I ſhould have multiplied: the Product therefore must be
ΙΟ
2
3
divided by the Denominator 3 of the Multiplicator to reduce it to
its juft Value: which is effected by multiplying the Denominator 10 by
3, (Art. xxiv.) which will give the Fraction for the Product required.
XXXI
18
30
As a whole Number may be confidered as a Fraction of which it is
the Numerator, and Unity the Denominator: when a whole Number The Multi
is propofed to be multiplied by a Fraction, this Operation may be re- plication of
duced to the Multiplication of a Fraction by a Fraction. For Éxample, Number by
if it be propoſed to multiply 3 by; this Multiplication may be re-is the fame
3
duced to that of by
2 and 2.
5
5
4
; and there will refult for Product 12
5
5
a whole
a Fraction
as the Mul-
tiplication
or of a Fractic
Whence the Multiplication of a whole Number by a Fraction, for
Example, of 3 by is the fame as the Multiplication of a Fraction by
4
3× is equal to 4×3 or
X ·+x3.
a whole Number 3.
For 3x
—
or×
XXXII
I
5
5
To divide by a Fraction, is to divide by its Numerator, and after-
wards to multiply by its Denominator. For let any Quantity (for Ex-
ample) be propoſed to be divided by the Fraction 3
4
Let it first be di-
on by a
whole Num
ber.
Io what
conffts the
by
Divifion by
vided by the Numerator 3 of this Fraction: this Divifior being 4 Times
too great, fince it was propofed to divide only by the fourth Part of 3, a Fraction
will give a Quotient 4 Times too little; confequently this Quotient muſt
be multiplied by the Denominator 4 of the fame Fraction, to reduce it
to its juſt Value.
}
"
70
NUMERAL ARITHMETIC K.
From whence it follows 12 that the Divifion by a Fraction, for Ex-
ample, by 2, is reduced to a Multiplication by the converfe Fraction
4,
39
4
for the divifion by the Fraction, is performed by dividing by 3
The Divis- and afterwards multiplying by 4: now the Multiplication by the con-
on by a
Fraction verfe Fraction is performed by precifely the fame Operations; that
reduced to
& Multipli
On,
4
3.
is, (Art. xxv.) by multiplying by 4 and dividing by 3.
It follows 2°. that the divifion by a Fraction whofe Numerator is
Unity, is really a Multiplication by the Denominator of this Fraction.
For to divide by the Fractions, &c. is to multiply by the
converfe Fractions, 2, 4, &c. that is, by the whole Numbers
2, 3, 4, &c.
XXXIII.
The Divifion of a Fraction by a Fraction may be performed two dif
ferent Ways. 1. Since the Divifion by a Fraction is reduced to a Mul-
tiplication by the converfe Fraction; when a Fraction is propoſed to be
divided by a Fraction, for Example, if 2
9 was propoſed to be divided
20
Firſt Me. by, it ſuffices to invert the Terms of the Divifor 3, to obtain its
thod of.
4.
9
dividing a converfe 4: and afterwards multiply the Dividend by this con-
Fraction by
1 Fraction.
3.
20
verfe of the Divifor, and the Product that arifes 36
4
3.
9
or
will:
60
15.
3
be the Quotient of the Fraction-divided by the Fraction
20
·
2º. If the Dividend and Divifor have the fame Denominator or are
reduced to the fame Denominator; the Divifion may be performed by
dividing the Numerator of the Dividend by the Numerator of the Di--
vifor; or elfe by forming a new Fraction having for Numerator the Nu-
merator of the firft Fraction, and for Denominator the Numerator of
Method of the fecond Fraction, the Fraction refulting from this Operation being the
Fraction by
Quotient of the Divifion of the Dividend by the Divifor.
a Fraction.
Second-
dividing a
រ
8:
9.
Let 19 the Fraction be propoſed to be divided by the Fraction;
I confider thefe two Fractions as concrete Numbers 8 Ninths, 41
whoſe fractional Units are Ninths; whereby the Divifion of
Ninths
8
by
୨
9
ELEMENTS OF
78
will be reduced to find how often the Dividend 8 Ninths, contains the
Divifor 4 Ninths, which is effected by dividing the Number 8 of the frac-
tional Units of the Dividend by the Number 4 of the fracional Units of the
Divifor, that is, by dividing the Numerator 8 of the Dividend
8
9'
by the
Numerator 4 of the Divifor ; which will give the whole Number 2
for the Quotient required.
4
9
9
8
9
or the
2º. If it had been propoſed to divide the Fraction by 5,
concrete Number 4 Ninths by the concrete Number of the lame Spe-
cies 8 Ninths; the Number 4 of the Fractional Units of the Dividend
must be divided by the Number 8 of the Fractional Units of the
Diviſor, which will give for Quotient the Fraction having for Nu-
merator the Numerator of the Dividend, and for Denominator the
Numerator of the Divifor
8
XXXIV.
The diffe
vitions of
and the
As the principal Unit is divided into feveral Fractional Units, fo a
Fraction confidered as a collective Unit may be divided into ſeveral equal
Parts, which are called Fractional Units of a Fraction; and a Quantity
confifting of one or feveral Fractional Units is called a Fraction of a
Fraction. For Example, if the Fraction confidered as an Unit be
divided into three equal Parts; each Part which is only one third of the rent Subdi
Fraction, will be a fractional Unit of a Fraction, and the Quan- Fractions
tity confifting of one or feveral of thofe fractional Units is called
Fraction of a Fraction.
To exprefs a Fraction of a Fraction, two Fractions are required fe-
parated by the Word of. For Example, to repreſent the two Thirds of
the Fraction, it is fet down thus
which expreffes 2
Thirds of Sixths. The Denominator 3 of the first Fraction denot-
ing into how many Parts the fecond Fraction is divided; and the Nume-
rator of the firſt Fraction denoting how many of thoſe Parts of the Frac
tion are taken.
2
3
of 동​,
a Manner of
expreffing
them.
As a Fraction may be divided into ſeveral equal Parts, one or ſeveral
of which form a Fraction of a Fraction, in like Manner a Fraction of a
Fraction may be divided into feveral equal Parts, one or feveral of which
form a Fraction of a Fraction of a Fraction; and ſo on ad Infinitum,
}
92.
ELEMENTS
OF
All thoſe different Species of Fractions of Fractions are fet down one
after the other, being feparated by the Article of. For Example, if the
2 of the Fraction be taken, it is expreffed thus, the two Thirds of
3
2
five ſixths, and ſet down thus of, and if the three Fourths of
3
this Fraction be taken, it is expreffed thus, the three Fourths of two
A Fraction Thirds of five Sixths, and fet down thus
of a
Ai.
on is equa
to the Pro-
duct arifing
from the
Multiplica-
tion of the
XXXV.
2
of
of 동​.
3
To determine the Value of a Fraction of a Fraction, let us take for
Example the Fraction of a
swo Practi- the one Third of five Sixths.
ons by
2
Fraction of, which repreſents twice
3
Now the one Third of
is obtained
is
one Third of
which it is by dividing this Fraction by 3: confequently twice the
expreſſed. ༨ will be obtained by dividing it by 3, and multiplying the Quotient by 2; -
A Fraction:
of a Fracti- that is, by multiplying one by the other, the two Denominators and the
the famways two Numerators of the Fractions
is
whatever
2
and; wherefore a Fraction of
3
Way the a Fraction is equal to the Product arifing from the Multiplication of the
two Fractions by which it is expreffed.
Fractions
by which it
is expreífed
Since a Fraction of a Fraction is the Product of the two Fractions by
are arrang- which it is expreſſed, and this Product will be always the fame in what-
2
ever Order thofe Fractions are multiplied, it is manifeft that
ed.
A Fraction
of a Fracti is equal to of 2.
on of a
Fraction,
&c. is
equal to a
Fraction
having for
3.
of y le
3
6
XXXVI.
Since a Fraction of a Fraction of a Fraction, E. g.
3
2 of 2 of 2
4
3
3
or
of 18
and
4
Numerator is equal to the Fraction of a Fraction 3 of 2X5
the Produc
of the Nu
3
10
meratork of that 2-of-18 is equal `to
all the Frac
tions by
which it is
and for deno
Fraction of a Fraction
4 3×6
3X2X5 it follows that the Fraction of a
4x3x6
of is equal to the Fraction
3 2
of of
4 3
3X2X5
4X2x6
expreffed, and in general that a Fraction of a Fraction of a Fraction, &c. is equal
minator the to a Fraction having for Numerator the Product of the Numerators of
Product of all the Fractions by which it is expreffed, and for Denominator the =
their Deno
minators.
Product of their Denominators.
NUMERAL ARITHMETICK.
-73
XXXVII.
If there was only one Kind of Unit for all Magnitudes of the fame
Neceffity of
Species; thofe Magnitudes could not be expreffed by Numbers with employing
fufficient Accuracy, unles the Unit was very fmall; and in this Cafe the Units of
Numbers requifite to exprefs the most ordinary Magnitudes would be ex- nominations
ceffive, and confequently very inconvenient in Trade.
different De
for meafur-
Kind of
In Computations of Money, for Example, if the Farthing was al- ing the fame
ways affumed for Unit, the Numbers reprefenting the different Sums Magnitudes.
would be four Times greater than if the Penny was taken for Unit, and
would be 48 Times greater than if the Shilling was taken for Unit, and
960 Times greater than if the Pound Sterling was taken for Unit.
If the Pound Sterling was affumed for Unit, and no other was em-
employed; all the poffible Sums could not be reckoned without omit-
ting fuch as are lefs than the Pound Sterling, which though too confi-
derable to be rejected; could not be attended to without dividing the
Pound Sterling into different Parts, which would require a particular
Knowledge of Fractions.
1
XXXVIII.
To remedy thoſe two Inconveniencies, the Computifts employ Units The diffe-
of different Degrees of Magnitude for meaſuring the fame Species of rent Kinds
Quantities
of Units em
ployed in
ures.
In Computations of Money; they commonly employ four Sorts of Units, Money
viz. the Pound Sterling, the Shilling, Penny and Farthing, they alſo em- Weights
ploy for Units the Crown, the Guinea, and all the Coins current in Trade, and Mea-
In Computations of Weights; the Units employed are the Pound, the
Qunce, the Penny Weight, the Dram, the Scruple, the Grain; and in heavy
Weights, the Stone, the hundred Weight, the Tun, &c.
In Computations of Lineal Meafure; the Units employed are the
Fathom, the Yard, the Foot, the Inch, the Line, the Ell, the Pole, &c.
In Computations of Time; the Units employed are the Year, the
Month, the Day, the Hour, the Minute, the Second, &c. the Week, which
confifts of 7 Days, is alfo employed for Unit, and an Age which
confifts of 100 Years.
In Computations of Angles and of Arcs of Circles; the Units em-
ployed are the Degree, the Minute, the Second, &c. the Signs each of
which contain 30 Degrees are alfo employed for Units. In fine, for cal-
culating each Species of Quantities, the Computifts employ for Untis
fome of the known Parts of the fame Kind of Quantities.
XXXIX.
Value of
Units em-
ployed in
The Value of the different Units, above mentioned, and the Charac- the different
ters by which they are diftinguiſhed from each other are as follows.
IN MONEY, 1. denotes Pounds Sterling, s. Shillings, d. Pence, and Money
qrs. Farthings.
Weights
2 K
74:
ELEMENTS OF
and Mea-
fures, and
the Charac-
ters by
which they
are diſtin-
guiſhed
from each
other.
1 Pound Sterling is equal to 20 Shillings, 1 Shilling to 12 Pence,
and I Penny to 4 Farthings.
IN WEIGHTS lb. denotes Pounds, oz. Ounces, dwt. Penny Weights,
Scruples, 3. Drams, cut. hundred Weight.
IN TROY WEIGHT ufed for weighing Gold, Silver, Seeds, Liquors,
Bread, Medicines, &c. I Pound is equal to 12 Ounces, 1 Ounce to 20
Penny Weight, and I Penny Weight to 24 Grains.
I
-
In compounding of Medicines, &c. 1 Pound is divided into 12 Ounces,
1 Ounce into 8 Drams, and I Dram into 3 Scruples, and 1 Scruple
into 20 Grains.
IN AVOIR-DU-POISE WEIGHT, ufed for weighing courſe and
heavy Goods, fuch as grocery Wares, Pitch, Tar, Rofin, Wax, Tal-
low, Flax, Hemp, &c. Copper, Tin, Iron, Lead, Steel, &c. 1 Tun is equal
to 20 hundred Weight, I hundred Weight to 8 Stone, I Stone to 14
Pounds, and 1 Pound to 16 Ounces.
I
IN LINEAL MEASURE, F. denotes Fathoms, Y. Yards, f. Feet I.
Inches, L. Lines, E. Ells.
1 Fathom is equal to 2 Yards,
Inches, I Inch to 12 Lines, and
1-4 of a Yard. -
Yard to 3 Feet, 1 Foot to 12
Line to 12 Points, Ell is equal to
XL.
In treating of the Addition of fimple Numbers, we defined Addition™
to be an Operation by which a Number is found equal to the Sum of
fereral other Numbers. We proved afterwards that the Numbers to be
added ſhould confift of Units of the fame Species, and that the Units of
the Sum refulting from their Addition are of the fame Species. All which
is likewife applicable to compound Numbers: for though the Units of
all their Parts are not abfolutely the fame, they are reducible to fimilar
Units; that is, a certain Number of Units of a lower Denomination, form
an Unit of another Part of a higher Denomination.
XLI.
1
Compound or applicate Numbers that are propofed to be added to--
Procefs of gether, muſt be ſet down one under the other; fo that all the Parts
Addition of whofe Units are fimilar, may ftand in the fame Column; and all the Figures
bers, of the fame Degree that are in decuple Proportion, may alſo ſtand under -
mixt Num-
one another. Then a Line being drawn under them, to ſeparate them
from their Sum; all the Parts whofe Units are of the loweſt Denomi-
nation muſt be firft added: and if their Sum contains one or feveral
Units of the next fuperior Denomination; they must be retained to add
them to thofe of the next Column; and what remains fet down under
'the Column that was added. The fame Operation being performed
upon each Column, the Number under the Line will be the Sum re-
quired of the compound Quantities which were propoſed to be added.
r
NUMERAL ARITHMETICK.
75
XLII.
To illuſtrate thofe Rules, we ſhall proceed to apply them to ſeveral
Examples; in Money, affuming for Units the Pound Sterling, the Shilling,
Penny and Farthing; in Weights, affuming for Units the Pound, the
Ounce, Penny Weight and Grain; in lineal Menfuration, affurning for
Units the Yard, the Foot, the Inch, and the Line. Thofe Examples well
underſtood will be fufficient to fhew how thofe Rules are to be applied,
for finding the Sum of any other Kind of compound Numbers.
XLIII.
Let it be propofed to find the Sum of the four following mix'd Numbers
3871. 125. 8d. 3qrs. 759/. 19s. Lid. 2qrs. 8961. 175. 10d. Iqrs.
45631, 19s. 9d. 3qrs.
Having fet down thofe Numbers as in the
Margin, 1°. I begin by adding the Farthings,
as being the Units of the loweſt Denomination,
faying 3 Farthings and 2qrs. is 5grs, and 1qrs.
is 6qrs, and 3qrs. is 9qrs, and becauſe
9 qrs.con-
tains twice 4qrs, which is 2d. and 1qrs, over.
I
fet down Iqrs. under the Line in the Column
1.
d. qrs.
387 12 8
3
Addition of
759 19 II
2
17. 10
I
9 3
896
4563. 19
6608 10 4 .I
4
of Farthings, and retain 2d. to add them to the folowing Column.
2º. As the tens of Pence form no particular Species of Units, and that
12 d. is equal to 1 Shilling; I add together not only all the Pence, but
alſo the tens of Pence; to find the Sum of the Pence. Saying: 2 Pence
I retained from the Column of Farthings and 8d. is. 10d. and 11d. is 21d.
and rod. is 31d. and 9 is 40d. and as 40d. contains three Times 12 Pence
or 3 Shillings and 4d. over; I fet down 4 Pence in the Column of Pence,
and I retain the 3 Shillings to add them to the following Column.
3º. Since 2 tens of Shillings is equal to 1 Pound Sterling, and that
there are Units of Shillings which may amount to fome tens of Shillings;
I add the Column of Shillings at twice, beginning by the Units place
of the Shillings. And I find 30s, or 3 tens of Shillings, I therefore ſet
down a Cypher in the Units place of Shillings, and retain 3 which I
add to the tens of Shillings, whofe Sum is 7; and as 2 tens of Shillings
is equal to one Pound Sterling, the 7 tens of Shillings will be equal to
31. 10s. I therefore fet down in the Column of tens of Shillings, and
retain 31. Sterling to add them to the Pounds.
4°. Proceeding to the Column of fimple Pounds, I fay 3 Pounds I
retained from the Column of tens of Shillings, and 77. is iol. and 97. is
197. and 67. is 251. and 37. is 281. which contains 2 tens of Pounds and
87. over: I therefore fet down 81. in the Units Place of the Pounds, and
I retain 2 tens to add them to thofe of the following Column. And
performing the Remainder of the Operation as explained in Addition of
fimple Numbers, I find 66087. 10s. 4d. 1grs. for the Sum required.
Pounds Shil
lings and
Pence
explained by
an Example.
76
ELEMENTS OF
Addition of
Pounds,
Ounces,
Penny
and Grains
explained by
XLIV
Let it be propofed to find the Sum of the four following mixt Numbers
47216. 1102. 19dwt. 22gr. 1438/b. 100%. 14diut. 21gr.
18det. 10gr. 3481b. 10oz. 17dwt. 9gr.
lb.
472
1438
2568
348
25681b. 110%.
f
oz. dwt. gr..
II 19 22
ΙΟ
14 21'
I f
18 10
IO
17 9
14.
4829 9 ΙΟ
Having fet down the thoſe Numbers as in the
Margin, I begin by adding the Grains, as being
the Units of the loweft Degree. Saying 22gr.
Weights and 21gr. is 43gr, and 10gr. is 53gr. and 9 is
62gr, which contains twice 24gr. or 2 Penny
20 Example. Weights and 14 Grains over. I therefore fet
down 14gr. in the Column of Grains, retaining .the
2- Penny Weights to add them to the Column of Penny Weights.
Proceeding to the Column of Penny Weights, I find 68 dot to which
adding the 2 Penny Weights retained from the Column of the Grains,
there refults 70dwt, which contains three Times 20 Penny Weights, or
3 Ounces with 10dwt, over. I therefore fet down 10 dwt, in the Column
of Penny Weights, and retain 3 for the Column of Ounces,
Inches and
Lines ex
J
Proceeding to the Column of Ounces, and adding to it the 3 Ounces
I retained, I find 450%. which contains 3 Times 12 Ounces, or 3 Pounds
with 9oz. over; I therefore fet down goz. in the Column of Ounces,
retain the 30z, to to add them to the Column of Pounds,
and
Proceeding to the Column of fimple Pounds, and adding to it the 3.
Pounds that I retained from the Column of Ounces, I find 29/6. that
is, 2 tens of Pounds and 9 Pounds over, which I fet down in the Column
of fimple Pounds, and retain the 2 tens to add them to the Column of
tens of Pounds, and performing the Remainder of the Operation as ex-
plained in Addition of fimple Numbers, I find 4829lb. 9oz. 10dwt. 14gr.
for the Sum required..
XLV.
Vi fi
32 2.
144 I 9
67 2 10
78
I. L.
6 10
9
8.
2 II I.I
I
3
2
Let it be propoſed to find the Sum of the four following mixt Numbers,
321. 2f. 61. 10L. 144. 1f. 91. 9L. 67Y. 2f. 101. 8L. 78r. 2f. 111. 11LA
Addition of Having placed thofe Numbers as in the Mar--
Yards, Feet gin, I begin by adding the Lines, as being the
Units of the loweſt Degree. Saying: 10L. and
plained by L. is 19L. and 8L is 27 L. and 11L. is 38L.
2nExample. which contains 3 Times 12 Lines, or 3 Inches
with 2 Lines over, I therefore ſet down 2L. in
the Column of Lines,, and retain 34 to add them -
to the Column of Inches.
Proceeding to the Column of Inches, I fay: 31. that I retained from
the Column of Lines, and 61. is 91. and 91. is 181. and 101. is 287. and
III. is 391, which contains 3 Times 12 Inches, or 3 Feet, with 3 Inches
over, I therefore fet down 3 Inches in the Column of Inches, and-retain
3f. to add them to the Column of Feet.-
334
NUMERAL ARITHMETICK
77
фото
Proceeding to the Column of Feet, I fay: 3f. that I retained and 2f
is 5f. and if is 6f. and 2f. is 8f. and 2f. is 1of. which contains 3
Times 3 Feet, or 3 Yards and 1f. over. I therefore ſet down 1.f. in the
Column of Feet, and retain 3 Yards to add them to the Column of
Yards.
Proceeding to the Addition of the Column of Yards, I fay 3 Yards
that I retained and 27, is 5 and 47. is 97. and 7r. is 167. and 87.
is 24. which contains 2 tens of Yards, and 4 Yards over. I therefore
fet down 4 Yards in the Column of Yards, and retain 2 tens of Yards
to add them to the Column of the tens of Yards that follow, compleat-
ing the Addition, after the manner explained in the Addition of fimple
Numbers, I find 324F. 1f. 31. 2L. for the Sum required.
XLVI.
་
In treating of the Subtraction of fimple Numbers, we defined Sub-
traction to be an Operation by which one Quantity is deducted from
another, whence we concluded that the Quantity to be deducted fhould"
be contained in the Quantity from which the Subtraction is required to be
made; and conſequently that thofe two Quantities fhould be both of the
fame Species, or reducible to the fame Species, it is the fame with re-
fpect of applicate Numbers; one Quantity cannot be Subtracted from
another if they are not both of the fame Species, or reducible to the
fame Species. For Example, a Quantity made up of different Weights,
cannot be deducted from another Quantity made up of Yards and Parts
of a Yard; but Yards, or known Parts of a Yard, may be deducted ·
from another Quantity confifting only of Yards or Parts of a Yard.
XLVII.
-
tion of mixt
To fubtract one compound Number from another compound or
fimple Number, the Parts of the Quantity to be fubtracted muſt be dif- The procefs
pofed under the fimilar Parts of the Quantity from which the Subtrac- of Subtrac-
tion is required to be made, obferving to place the Units, Tens, Hundreds, Numbers.
&c. of one Species, under the Units, Tens, Hundreds, &c. of the fame
Species, then drawing a Line under thofe Numbers to feparate them from
the Remainder, each Part of the Quantity to be fubtracted muſt be de-
ducted from each corresponding Part of the other Quantity that ſtands
over it, beginning by the Parts of the lowest Denomination, and pro-
ceeding to the next fuperior Denomination, and fo on through the
whole till all the Parts of the Quantity to be deducted are fubtracted.
If the Number of Parts to be deducted, is lefs than the Number of
the correfponding Parts in the other Quantity that ſtands over it, there
will be no Difficulty in fubtracting them, and the Remainder muſt be fet
down under the Line in the fame Column with thofe Parts. But if in
the inferior Number that is to be deducted, there are Parts whofe Num-
ber is greater than that of the correfponding. Parts of the upper Num-
78
ELEMENTS OF
。
Subtraction
of Pounds,
Shillings,
and Pence.
First Ex-
ample.
ber, an Unit must be borrowed from the following Parts that are
of the next fuperior Denomination, in the upper Number, and this
Unit reduced into Parts of the fame Species with thofe from which the
Subtraction is required to be made, added to them, whereby they will
be fufficiently increaſed, fo as the Number of Parts may be deducted
from them, that could not be deducted before. The fame Operation
muſt be performed for all the Species of Parts in the Quantity to be de-
ducted, which are in greater Number than in the correfponding Parts of
the Quantity from which the Subtraction is required to be made, and
Care taken to deduct the Unit borrowed from the Number from which
it has been borrowed.
XLVIII.
1.
199
98
S.
d. qrs.
14
I 6 2
10 3
6
7
3
Let it be propofed to fubtract 981. 14s. 10d. 34rs. from 1997. Is. 6d. 2qrs.
Having placed the Numbers as in the Margin,
to obtain the Remainder of the Subtraction, I first
fubtract the Farthings, which are the Parts of the
loweſt Denomination, but as the 3 Farthings to
be fubtracted are not contained in the 2 corref-
ponding Farthings of the upper Number, I bor-
row I Penny from the Number of Pence in the upper Number, and
reducing this Penny into 4 Farthings, I add them to the 2 Farthings from
which the Subtraction is required to be made, and deduct 3 Farthings
from 6 Farthings, and fet down 3 Farthings for the Remainder of the
Farthings.
100
Proceeding to the Column of Pence, I have 10 Pence to deduct from
5 Pence, (becauſe the Unit borrowed from 6 to carry it to the Column
of Farthings has reduced this Figure to 5) I therefore borrow I Shilling
from the upper Number of Shillings, and reducing this Shilling into 12
Pence, I deduct 10 Pence from 17 Pence, and fet down 7 Pence for
the Remainder of the Pence.
Then proceeding to the Column of Shillings, I have 14 Shillings to
fubtract from nothing, becaufe, the Shilling in the upper Number was
borrowed and employed in the Pence: I therefore borrow 1 Pound from
the Pounds of the upper Number; and reducing this Pound into
20 Shillings I deduct 14 Shillings from 20 Shillings, and fet down 6 Shil-
lings for the Remainder of the Shillings.
Proceeding to the Units Place of the Pounds, I fubtract 8 Pounds
from 8 Pounds, (becauſe the Unit which was borrowed from the Figure
9 to carry it to the Column of Shillings has reduced this Figure to 8)
and there remains nothing. In fine, continuing the Subtraction as in fimple
Numbers, I find 100%. 65. 7d. 3grs. for the Remainder required.
NUMERAL ARITHMETICK.
79
XLIX.
1.
40
S.
d.
9 19
О
8
16
ΙΙ
3 9
9
30
Let it be propofed to fubtra& 9l. 16s. 11d. from 401. os. 8d.
Having placed thofe Numbers as in the Margin, and
as the 11 Pence to be fubtracted cannot be deducted
from 8 Pence which ftand over it, and that nothing can
be borrowed from the Shillings, or from the Units of
the Pounds, I borrow one ten of the Pounds, leaving 9
Pounds in the Rank of the Pounds, and 19 Shillings in
the Rank of Shillings, and adding the Shilling remaining reduced into
Pence to the 8 Pence, I deduct 11 Pence from 20 Pence, and fet down
9 Pence for the Remainder of the Pence, and the Operation is there by
reduced to deduct 97. 16s. from 39. 19s. I therefore dedu& 16s. from
19s. and fet down 3. for the Remainder of the Shillings, then proceeding to
the Pounds I 'fubtra&to Pounds from 39 Pounds, and fet down 30 Pounds
for the Remainder of the Pounds, and there refults 30%. 3s. 9d. for the
Remainder required.
L.
Let it be propofed to fubtract 8lb. 1002. 7dwt. 22gr. from 161b. 802:
7dwt. 17gr.
Having placed thofe Numbers as in the Margin,
I begin the Subtraction by the Grains, which are the
Weights of the loweſt Denomination; and as 22
Grains cannot be deducted from 17 gr. I borrow Idwt.
or 24gr. which adding to the 17gr. I deduct 22gr.
from 41gr. and fet down 19gr. for the Remainder of
the Grains.
lb. oz. dwt. gr.
16 8 7 17
7 22
Second Ex
ample.
Subtraction
of Pounds,
Ounces,
Penny
Weights
8 10
and Grains.
7 7 9
19 19
Proceeding to the Column of Penny Weights, as 7 Penny Weights
cannot be deducted from 6 Penny Weights, to which the Penny Weights
of the upper Number are reduced, becauſe idwt, has been borrowed from
them, I borrow 1 Ounce or 20dwt. and adding them to the 6dwt. I
deduct 7dwt, from 26dwt. and fet down 19dwt. for the Remainder of the
Penny Weights.
Then proceeding to the Column of Ounces, as 10oz. cannot be de-
ducted from 70z: to which the Ounces of the upper Number are reduc-
ed, becauſe 1oz. has been borrowed from them, I borrow I Pound or
120%. and adding them to the 70%. I deduct 100%, from 19oz. and fet down
goz. for the Remainder of the Ounces.
Laſtly proceeding to the Column of the Pounds, I fubtract 8 Pounds
from 15 Pounds, becaúfe 1lb. has been borrowed from the 6 Pounds, and
the Subtraction being finished, there refults 7b. 9oz. 19dwt. 19gr. for the
Remainder required.
·80
ELEMENTS OF
LI.
r
Let it be propofed to fubtract 97r. 2f. 111. from 100r. 2ƒ. 0Ï.
Subtraction Having difpofed thofe Numbers as in the Margin,
and as 11. cannot be deducted from ol. I borrow an
Unit from the Feet, and dedu&t 11. from 121, and fet
down II for the Remainder of the Inches.
100
97
f. 1.
ΙΙ
2 2
2 2
I
of Yards,
Feet, and
Inches, ex
plained by
an Example. Proceeding to the Column of Feet, as 2 Feet cannot
be deducted from I Foot, to which the Feet of the up-
per Number are reduced, becauſe 1 Foot has been borrowed from them,
i borrow Yard, or 3 Feet, and adding it to the 1 Foot, I deduct 2
Feet from 4 Feet, and fet down 2 Feet for the Remainder of the Feet,
and compleating the Operation according to the foregoing Rules, I find
2r. 2f. 11. for the Difference required.
and Subtrac
I
LII.
A Queſtion
In the foregoing Articles we have explained all the little Difficulties
for Practice that occur in the Subtraction of mixt Numbers, and that might embar-
in Addition rafs Beginners, either in paffing from one Column to another, or in bor-
tion of mixt rowing, when the Number of Parts to be fubtracted, is greater than
Numbers. that from which the Subtraction is required to be made. For the
further Exercife of the young Practitioner the following Queſtion is pro-
pofed, in which Addition and Subtraction are equaly concerned.
Process of
ration of
mixt Num
bers.
A Merchant in balancing his Books, finds he hath in ready Money,
1561. 175. in Goods, 1749/. 19s. 6d. his Stock in a Company Trade was
4997. 19s. 6d. 2qrs. due to him in open Accounts 29771. 19s. 7d. 39rs.
In Confignments 479. 19s. 7d. He owes to A. 14561. 18s. 7d. 2qrs.
To B. 991. 19s. Id. To C. 497. 17s. 10d. and to the Bank 4901.
What is his nett Stock? Anſwer, 3319/. 18s. 10d. 3grs.
LIII.
The Multiplication of applicate Numbers, is performed by multiply-
the Multipli ing the Multiplicand by all the Parts of the Multiplicator: but as the
Multiplicator when it is a mixt Number confifts of Parts less than the
principal Unit, and that each of thofe principal Units denote that the
Multiplicand muſt be taken once, the Parts of the Multiplicator which
are less than the principal Unit, will denote that Parts of the Multipli-
cand muſt be taken for the Product. To render the Rules of the Multi-
plication of applicate Numbers more intelligible, we fhall proceed to ex-
plain them in their Application to feveral Examples.
Multiplication being an Operation by which the Mutiplicand is re-
peated a certain Number of Times, expreffed by the Number of prin-
cipal Units in the Multiplicator; the Multiplicator fhould be confiderec
as an abfolute Number, even when it is applied to reckon any particular
Species of Units, and the Units of the Product fhould confequently be
of the fame Species as thofe of the Multiplicand.
NUMERAL ARITHMETICK.
81.
LIV.
Let it be propofed to multiply 5181. 14s. 8d. by 744
Having placed thofe Numbers as in the
Margin. To Multiply regularly all the Parts
of the propofed Multiplicand, by all the Parts
of the Multiplicator; I begin by multiplying
the whole Multiplicand by 74, multiplying af-
terwards the fame Multiplicand by the Frac-
tion. But the Multiplicand 5187. 145. 8d. con-
fifting of three Parts, I multiply thofe three
Parts one after the other by 74.
4.
1º. I multiply 5181. by 74, and fet down
the particular Products 2072 Units of Pounds
and 3626 tens of Pounds, as in the Margin.
70
The Multi
&d. plication of
Pounds,
Shillings,
and Pence
explained by
an Exampl.
518%. 145.
74 룸
​4
4
2072/.
3626
IOS.
37
45.
14
Iбs.
¿d.
2
9
4d.
4
T29
13 8
385157. 195. oď.
+
20. To multiply 14. by 74, I divide 145. into Parts that may be con-
tained each a certain Number of Times in the Pound, which is the prin-
cipal Unit of the Multiplicand. Thofe Parts will be 10s, and 4s. which
I multiply feparately by 74.
Σ
of
To multiply 10s. by 74 I obferve that 10s. is of a Pound, and that
1 Pound Multiplied by 74 will give 741, whence I conclude that the
a Pound or 10s. fhould only give for Product the of 74. that is 371.
To multiply 4. by 74, I obſerve that 45. being of a Pound, and I
Pound multiplied into 74 giving 741. the of 17, or 4. multiplied by 74
fhould only give the of 741. which is 14. 16. for the of 74/. is 147.
and there remains 41. or 80s. the of which is 16s.
3.
30
3º. To multiply 8 Pence by 74, I obferve that 8 Pence being the
of a Pound, and that confequently 8 Pence multiplied into 74 fhould
only produce the 3 of 74/. whence we may take for the Product the
of the Multiplicator 74, confidered as a Number of Pounds.
36
30
But instead of taking the Part of 74. it will be more commodious
to make ufe of the Produ& 14. 16s. found for 4. fince 8 Pence is the
fixth Part of 4s. the Product of Sd. into 74 will be the fixth Part of the
Product 14. 16s. found by multiplying 45. by 74; whence the Opera-
tion is reduced to take the fixth Part of 14/. 16s. which will be 21. 95. 4d.
becauſe the fixth Part of 14/. is 27. and there remains 21. or 40s, which
adding to 16s. I take the fixth Part of 56. which is 9s. and there re-
mains 25. or 24d. the fixth Part of which is 4 Pence.
Having multiplied the Multiplicand 5187. 14s. 8d. by 74 it remains to
multiply it by the Fraction, to effect which I obferve that if
5187. 14. 8d. was multiplied by 1, the Product would be the whole Multi-
plicand 5181. 14s. 8d. the Product therefore arifing from the Multipli-
cation of the fame Number by, will be the of 5181. 145. 8d. which
I find to be 1291. 13. 8d. becauſe the 4 of 518 is 129/, with a Remain-
2 L
82
ELEMENTS OF
4
der 21. or 40s. which added to 145. makes 54s. the of which is 13. with
a Remainder 25. or 24d. which added to 8d makes 32d. the of which is 8d.
All the Parts of the Multiplicand being thus multiplied by all the
Parts of the Multiplicator, and all the particular Products fet down, I add
them together and their Sum 385157. 19s. will be the Produ& required
of the mixt Number. 5181. 14. 8d. multiplied into 74
be taken fór
LV.
The Rules to be obſerved in the Multiplication of Money fufficiently
appear from the foregoing Example. The only Difficulty that can occur
in this Operation, is to find what Parts of the Multiplicator confidered as
a Number of Pounds are to be taken for the different Numbers of Shil-
lings and Pence in the Multiplicand, which, for the Afiftance of the
young Practitioner, we ſhall proceed to explain.
Table for When the Number of Shillings or Pence, or of Shillings and Pence.
finding what
Parts of the together, are contained exactly a certain Number of Times in 20s. that
Multiplice- is, in a Pound, the Shillings or Pence, or the Shillings and Pence to-
tor fhould gether, are faid to be aliquot Parts of a Pound. So that the aliquot Parts
the different of a Pound is always a Fraction of a Pound having for Numerator
Numbers of Unity, and for Denominator the Number of Times that the Shillings.
Shillings,
or Pence, or the Shillings and Pence together, are contained in a Pound.
Pence, and
Farthings in
10s. is the of a Pound, whence for 10s. the of the Multiplicator,
2
the Multipli conſidered as a Number of Pounds is taken.
cand.
I
4
'
I
I
20
40
13
ΤΖ
ठ
of
N4
TO
20
40
12
ठ
of
55. Is the of a Pound, whence for 5s. the 4 of the Multiplicator is taken.
For 4s, which is the of a Pound, the of the Multiplicator is taken.
For 25. which is the of a Pound, the of the Multiplicator is taken.
For Is. which is the of a Pound, the of the Multiplicator is taken.
For 6s. 8d. which is the of a Pound the of the Multiplicator is taken.
For 35. 4d. which is the of a Pound, the of the Multiplicator is taken.
For Is. 8d. which is the of a Pound, the of the Multiplicator is taken,
For 2s. 6d. which is the of a Pound, the of the Multiplicator is taken.
For 6d. which is the a Pound, the the Multiplicator is taken.
For 3d. which is the of a Pound, the
of a Pound, the of the Multiplicator is taken.
For 8d. which is the of a Pound, the of the Multiplicator is taken.
For 4d. which is the of a Pound, the of the Multiplicator is taken.
For 2d. which is the
of the Multiplicator is taken.
For Id. which is the 4 of a Pound, the 4 of the Multiplicator is taken.
When a Number of Shillings is not exactly contained a certain Num-
ber of Times in a Pound, it is called an aliquant Part of a Pound. A
Number of Pence which is not contained exactly a certain Number of
Times in a Pound, is alfo called an aliquant Part of a Pound.
ठ 이
​30
60
I
of a Pound the
120
240
BO
35
1
I20
I
240
·
When the Number of Shillings or Pence in the Multiplicand is not
an aliquot Part of a Pound, but only an aliquant Part, the Computiſts
divide it into two or three Parts, each of which are aliquot Parts of a
Pound.
NUMERAL ARITHMETICK.
83
LVI.
one Opera
The Multiplication of Numbers confifting of Pounds, Shillings, Pence,
and Farthings, according to the foregoing Method, being often Times Method of
tedious and perplexed, we fhall proceed to explain the more compendi- finding at
ous Rules, the Computifts have found for remedying this Inconveniency. tion the
The Pounds and Shillings of the Product, arifing from the Multipli- Pounds and
cation of the Shillings of the Multiplicand by a whole Number, is ob- Shillings
arifing from
tained, at one Operation, by the following Rule. Multiply the Num- the Multipli
ber of Shillings by the Figure in the Units Place of the Multiplicator, cation of a
and if this Product be leſs than 20, fet it down in the Column of Shil- Shillings
lings; but if this Product is greater than 20, retain 17. for every twenty into a whole
Shillings it contains, and fet, down the Remainder in the Column of Number.
Shillings. Then multiply one half of the Number of Shillings by all the
other Figures of the Multiplicator, removing each Figure of this Pro-
duct one Place towards the right Hand, and adding to the firſt Part of
the Product the Number of Pounds that were retained.
LVII.
Let it be propofed to multiply ol. 1s. by 457.
Having fet down thofe Numbers as in the Margin.
1. To obtain the Shillings of the Product, I multiply the
Figure 1s. by the Figure 7 of the Units of the Multipli-
cator; and as the Product 75. is less than 20s. I fet it down
in the Column of Shillings.
Number of
First Ex-,
ol.
Is. ample.
457
221. 175.
2º. To obtain the Pounds of the Product, I multiply,
which is the one half of the Number of Shillings, by the remaining
Figures 45 of the Multiplicator, that is, I take the one half of 45, re-
moving each Figure of the Product one Place towards the right Hand,
which will be 22/. 10s placing the first Part 227. in the Rank of the
Pounds, and for the 10s. placing I ten to the left Hand of 75. already
fet down. By this Operation, I find 22/. 175. for the Product of Is.
multiplied into 457.
LVIII.
01 185.
457
Let it be propofed to multiply ol. 18s. by 457.
Having placed thofe Numbers as in the Margin, 1°. I
multiply 18s. by the Figure 7 of the Units of the Multi-
plicator. But as it would be too difficult to perform this
Multiplication at one Operation; I multiply firft 8s. by 7, 4117. 6s.
which gives 56s. I fet down the Figure 6 in the Rank of
the Shillings, and retain the 5 tens of Shillings, I then multiply the tens
of Shillings by 7, and there refults for Product 7 tens of Shillings, which
added to the 5 tens retained, makes 12 tens of Shillings or 61. there be-
ing therefore nothing to be fet down in the Rank of the tens of Shillings,
I retain 61. to add them to the Pounds.
Second Ex-
ample.
مانية
ELEMENTS OF
Third Ex-
ample.
*Method of
› finding at
one Opera
-tion the
Pounds,
2º. I multiply the one half of the Number of Shillings, that is, gr.
by the two remaining Figures 45 of the Multiplicator, removing the
Figures of the Product one Place towards the Right Hand, and adding
to this Produ&t the 61. that was retained, there refults 411. So that
4111. 6s. will be the whole Product of 18s. multiplied into 457.
LIX.
Let it be propofed to multiply ol. 17s. by 457.
Having fet down thofe Numbers as in the Margin, 1º. I -
multiply 175. by the Figure 7 of the Units of the Multipli-
cator, which gives 51. 19s. I fet down the Part 195. in the
Rank of Shillings, and retain the Part 57.
ol. 175.
457
365 19
22
ΙΟ
3881. 9.
2º. I multiply by the two remaining Figures 45 of the
Multiplicator, the one Half of the Number of Shillings, that
is 8; but as this Operation cannot beperformed but at
twice, I first multiply 8 by 45, removing the Figures of the
Product one Place towards the right Hand, which with the 51. retained
gives 365. I afterwards multiply by 45, that is, I take the of 45
confidered as a Number of Pounds which is 227. 10.
By this Operation I find the Produ&t of 175. multiplied by 457, in
two Parts, the Sum of which is 3887. 9s.
•
.LX.
The Pounds, Shillings, Pence, and Farthings of the Product ariſing
from the Multiplication of the Number of Pence or Farthings of the
Multiplicand, which is an aliquot Part of a Shilling, by a whole Num-
ber, is obtained at one Operation by the following Rule. Divide the
Shillings, Figure in the Units Place of the Multiplicator, confidered as a Number
Pence, and of Shillings, by the Number of Times that the Pence or Farthings
arifing from are contained in a Shilling, and fet down the Quotient in the Rank of
the Multipli the Shillings Pence and Farthings, if there be any; then divide the re-
Number of maining Figures of the Multiplicator by double the Number of Times
Pence or Far that the Number of Pence or Farthings of the Multiplicand is contained
things into a in a Shilling; and the Quotient being removed one Rank towards the
whole Num right Hand, will exprefs the Pounds, and the Parts of the Pounds if
Farthings
cation of a
ber.
Firft Ex-
ample.
of
there be any.
EXI.
ol. OS
457
3d.
Let it be propoſed to multiply ol. os. 3d. by 457.
Having difpofed thofe Numbers as in the Margin, I
obferve that 3d. being the one Fourth of a Shilling,
the one Fourth of the Figure in the Units Place of the
Multiplicator confidered as a Number of Shillings, and
the one half of the one Fourth, or the one Eighth of
the remaining Figures of the Multiplicator, muſt be
taken, that is, the Figure 7 must be divided by 4, and the remaining
Figures by 8. But as in Divifion, the Figures of the higheſt Degree
57. 145. 3d.
NUMERAL ARITHMETICK.
85
must be first divided, in order that the Remainders may be reduced and
added to the following Figures; I begin by taking the one Eighth of the
Figures of the Multiplicator which precede that in the Units Place; re-
moving the Figures of the Quotient one Rank towards the right Hand,
to repreſent the Number of Pounds. I therefore take the one Fighth
Part of 45 tens, which is 5 tens and as this Quotient mufl be removed
one Place towards the right Hand, I fet it down in the Rank of the
Pounds, the Remainder 5 tens being added to the Following Figure 7
makes 57, the one fourth Part of which is 144, which I fet down in the
Rank of the Shillings and Pence, under this Form 14. 3d. whence
there refults 57. 145. 3d, for the Product required of 3d. multiplied into
457.
LXII.
ol.
457
os. 8d.
Second E
15. 45. 8d. ample.
Let it be proposed to multiply ol. os. Ed. by 457.
Having placed thofe Numbers as in the Margin, I
obferve that the 8d. in the Multiplicand, being the
two Thirds of a Shilling; the two Thirds of the
Figure 7 Units of the Multiplicator, or the one Third
of 14 double of this Figure 7, and the one Third only
of the remaining Figures of the Multiplicator, must be
taken. But as this Operation is a Divifion, we fhould begin by the
Figures of the higheſt Degree, and take the one third of them, removing
the Quotient one Place towards the Right Hand, I therefore take the
one Third of 45 tens, which is 15 tens and as thofe 15 tens which
confiſt of 1 hundred and 5 Tens, fhould be removed one Place towards the
right Hand; I fet down I in the Rank of the tens, an 5 in the Rank
of the Units of the Pound. I afterwards take the two thirds of 7 Units,
or the one Third of 14 Units; which gives 43, which I fet down in the
Rank of the Shillings and Pence, under this Form 45. 8d. whence
157. 4. 8d. will be the Product of 8d. multiplied into 457.
I
LXIII.
:
ol. os. IId.
457
Let it be propofed to multiply ol. Os. 11d. by 457
Having fet down thofe Number as in the Mar-
gin, and obferving that the 11d. of the Multipli-
card is not an aliquot Part of a Shilling; I di-
vide id. into two Parts which are aliquot Parts,
or at leaſt commodious Fractions of a Shilling.
Thoſe two Parts will be 8d. and 3d. which I mul-
tiply ſeparately by 457.
I
1. Multiplying 8d. by 457, I find as in the
fecond Example 157. 4s. 8d. for the Product.
8d. 15
3d. 5
00
4
14
3
20/. 185. 11d.
2º. Multiplying 3d. by 457, I find as in the firſt Example 57. 145. 3d.
3º. Adding together thole two Products, there reſults 201. 18s. Iid.
for the Product of 11 Pence multiplied into 457.
Third Ex-
ample
$6
ELEMENTS OF
Fourth Ex-
ample
going Me-
taining
1
LXIV.
Let it be propoſed to multiply ol. os. Od. 3qrs. by 457.
01.
457
Os.
Having placed thoſe Numbers as in the Mar-
gin, and obſerving that 3qrs. is the of a Shil-
ling, I divide 45 by 32, and place the Quotient
I in the Rank of the Units of the Pounds, the Re-
mainder 13, which with the Figure 7, makes 1375.
I divide by 16, and fet down the Quotient
8s. 6d. 3qrs. in the Rank of Shillings, Pence and Farthings.
LXV.-
When a mixt Number, confifting of
The fore- Pounds, Shillings, and Pence and Far-
thod appli- things, as 1897. 18s. 11d. 2qrs. is pro-
ed to anEx- pofed to be multiplied by a whole Num-
ample, con- ber, for Example by 457; first the
Pounds, whole Number 189 must be multipli-
Shillings, ed into 457, according to the Rules of
Pence, and
the Multiplication of fimple Numbers;
Farthings..
then the 18. muſt be multiplied into
457, according to the Rule for multi-
plying Shillings, and afterwards the
11d. 2qrs. by 457, according to the
Rule for multiplying Pence and Far-
things.
od. 3ars.
11. 8s. 6d. 3q¶rs.
1897. 18s. 11d. 2grs.
457 3
4
1323
50°
945
400
756
185.
4II
Es.
8d.
15
4
8d.
3d.
5
14.
3
2qrs.
19
O
2qrs.
3
3.
4
2
冉
​113 19
86920%. 35. 4d. oqrs.
If the Multiplicator contained alfo a Fraction, for Example if the Mul-
tiplicator was 457; after the whole Multiplicand 1897. 18s. 11d. 2qrs.
is multipled by 457, it must be multiplied again by, that is 3 Times.
the one Fifth of the Multiplicand must be taken; and when all the Parts
of the Product are found, they must be added together, to obtain the
whole Product of the Multiplication.
LXVI.
The Grounds of the foregoing Rule for multiplying mixt Numbers,
confifting of Pounds, Shillings, Pence, and Farthings, will fufficiently
the forego appear from the following Confiderations.
Grounds of
ing Method
1. by multiplying the Number of Shillings by the laft Figure of the
for multiply Multiplicator, there will manifeftly refult a Number of Shillings; where-
ing Shillings fore the Product arifing from this Multiplication fhould be fet down in
the Rank of Shillings, when it does not exceed 20s. and when it exceeds
20s. one Pound fhould be retained for each 20s. and the Remainder fet
down in the Rank of Shillings.
by a whole
Number.
1
NUMERAL ARITHMETICK.
87
The Rule next directs to multiply all the Figures of the Multiplica-
tor, by one half of the Number of Shillings, and to remove each Figure
of the Product one Place towards the right Hand. Now,
1º. The Product that refults by multiplying by one half of the
Number of Shillings will be only the one half of the Product arifing
from the Multiplication by all the Shillings. 2°. by removing each
Figure of this Product one Place towards the right Hand, the tenth Part
only of this Produ& is taken, and confequently only the one tenth Part
of the one half of the Product refulting from the Multiplication of the
Number of Shillings of the Multiplicand by the Multiplicator; but the
tenth Part of the one half of this Product, that is, the twentieth Part
of this Product is equal to the Number of Pounds it contains. Where-
fore by multiplying all the Figures of the Multiplicator, except that in
the Units Place, by the one half of the Number of Shillings, and removing
each Figure of the Product one Place towards the right Hand; there
will refult a Quantity equal to the Number of Pounds contained in the
Product arifing from the Multiplication of the Multiplicator into the
Number of Shillings of the Multiplicand.
we by a whole
one
Number.
2º. If the Multiplicand contains but Is. according to this Rule, the Grounds of
Figure in the Units Place of the Multiplicator confidered as reprefenting, the forego
Shillings, together with the one half of the remaining Figures of the Mul- ing Method
for Multiply
tiplicator confidered as reprefenting Pounds, will be the Product of Is. ing Pence
multiplied into the Multiplicator; and as for the one half, the one or Farthings
fourth, the one third, the one fixth, the one twelfth, &c. of Is.
ſhould take but the one half, the one fourth, the one third, the
fixth, the one twelfth, &c. of what ſhould be taken for 1s. it is manifeſt
that for the one half, the one fourth, the one third, the one fixth, the
one twelfth &c. of 1s. we fhould take but the one half, the one fourth,
the one third, the one fixth, the one twelfth, &c. of the Figure in the
Units Place of the Multiplicator, confidered as reprefenting Shillings,
together with the one half of the one half, or of the one fourth, or of
the one third, or of the one fixth, or of the one twelfth, &c. of the re-
maining Figures in the Multiplicator, confidered as reprefenting Pounds,
removing each Figure of the Produ&t one Place towards the right Hand;
that is, we ſhould divide the Figure in the Units Place of the Multipli-
cator by the Number of times that the Pence or Farthings of the Mul-
tiplicand are contained in a Shilling, and divide the remaining Figures
of this Multiplicator confidered as reprefenting Pounds by double the
Number of Times that the Pence of the Multiplicand is contained in a
Shilling, removing each Figure of the Product one Place towards the
right Hand; which is precifely what the Rule preſcribes.
88
ELEMENTS OF
LX VII.
Let it be propofed to multiply 24lb. 7oz. 6dwt. by 51
Having difpofed thofe Numbers as in
Multipli- the Margin. 1º. I multiply 24lb. by
cation of
Weights ex 51, and fet down the two particular Pro-
plained by ducts, viz. 24 and 120, as in the Mar-
an Example. gin.
50
24lb.
70%. 6dwt.
51
I
2416.
120
402.
17
20%.
8
6oz.
10%.
4
3
ΙΟ
4dwt.
5
2
4dwt.
2dwt.
002.
125516.
6dwt.
2. to multiply 70%. by 51 I divide
the 70%. into the 3 aliquot Parts of a
Pound, 40%. 20z. 10z. and as the Product
of 1lb. multiplied by 51 is a Number of
Pounds equal to the Multiplicator 51,
the Product of 40z, which is but the one
third of a Pound, will be only one third
of the Multiplicator 51. I therefore for 40%. take the one third of 5176.
which is 1776. For 20z. which is the one fixth of a Pound, or the one
half of 40%. I take the one fixth of the Multiplicator confidered as 51
Pounds, or the one half of the Product 17/6. found for 4 Ounces which
is 8lb. 6oz. for 1 Ounce which is the one twelfth of a Pound, or the one
fourth of 4 Ounces, or the one half of 2 Ounces, I take the one
twelfth of the Multiplicator confidered as 51 Pounds, or the one fourth of
the Product 1776. found for 4 Ounces, or the one half of the Produ& 8lb.
60z. found for 2 Ounces, which is 4lb. 30z.
3º. having found the Product of 1 Ounce multiplied by 51, I divide the
6dwt. which remains to be multiplied, into the aliquot Parts of an
Ounce, and thofe aliquot Parts will be 4dwt, and 2dwt.
For 4dwt. which is the one fifth of an Ounce, I take the one fifth of the
Produ& 4lb. 30%. found for 1 Ounce; which is 100%. 4dwt.
For 2dwt, I take the one half of the foregoing Produ& 100%. 4dwt.
found for 4dwt, which is 5oz. 2 dwt.
Having thus found all the Parts of the Product, I add them together;
and there reſults 1255lb. ooz. 6dwt. for the Product required.
LXVIII.
Hitherto we have treated of Arithmetical Multiplication, we ſhall
Multiplica
tion of Lines now proceed to treat of Geometrical Multiplication, fo called becauſe
it relates to Extenfion, which is the Object of Geometry.
and Sur
faces.
As every Quantity is meaſured by fome other Quantity of the fame
Kind, the Computifts are under the Neceffity of employing three dif
ferent Species of Meaſuring Units, for meafuring the three different
Species of Extenfion: Lineal meaſuring Units for meaſuring Distances
and Lines; Superficial meaſuring Units for meaſuring Surfaces; and
Solid meaſuring Units for meaſuring Solids..
NUMERAL ARITHMETICK.
89
LXIX.
The Number which fhews how often the meaſuring Unit is contain- Figure ‹.
ed in the Extention meaſured, is called the Content of the Extention
fo meaſured. Let the Extention to be meaſured be a Rectangle ABCD,
whoſe Baſe BC and Altitude A B have been meaſured, for Example,
with a lineal Yard, fimply called a Yard; let B G, GI, IL, LN,
NP, PC, exprefs the Yards contained in the Bafe B C, and let A Q,
QR, RS, ST, T B, repreſent the Yards contained in the Breadth or
Altitude A B. If through the Points G, I, L, N, P of the Bafe, be drawn
GF, IH, LK, NM, PO, parallel to the Altitude A B; the Rectangle
A B C D will be divided into as many Rectangles ABGF, FGIH,
HILK, K L N M, MN PO, OPCD, as there are Yards in the
Baſe B C.
a
is found by
› Meaſures in
Altitude.
Each of thoſe Rectangles having one Yard in Breadth, and as many
Yards in Length as there are Yards in the Altitude AB of the Rec- The Area of
tangle, will manifeftly contain as many fquare Yards as there are lineal Rectangle
Yards in the Altitude AB. Whence to obtain the Number of ſquare Yards multiplying
contained in the Rectangle A B C D, the Number of Rectangles ABGF, the lineal
FGIH, HILK, &c. which ſtand on the Bafe B C, or the Number of the Bafe by
Lineal Yards BG, GI, &c. contained in the Bafe BC, muſt be re- thofe in the
peated as often as there are fquare Yards in each of thofe Rectangles,
that is, as often as there are lineal Yards in the Altitude AB. But to
repeat the Number of lineal Yards that are in the Bafe BC of the Rec-
tangle, as often as there are lineal Yards in the Altitude A B of this
Rectangle, is to multiply the Number of lineal Yards in the Baſe
BC, by the Number of lineal Yards in the Altitude A B. Whence the
Number of fquare Yards contained in a Rectangle is found, by multi-
plying the Number of lineal Yards in its Baſe by the Number of lineal
Yards in its Altitude.
For Example if the Length of the Bafe BC of the Rectangle ABC D
be fix Yards, and its Altitude AB 5 Yards, the Rectangle ABCD Example,
will contain 6 Rectangles ABGF, FGIH, HILK, KLNM,
M N PO, O P CD, each of which are 5 Yards long, and 1 Yard broad,
and confequently contains 5 fquare Yards. Whence the Rectangle will
contain 6 Times 5 or 30 fquare Yards.
LXX.
It is manifeft that if the Bafe BC and Altitude A B of the Rectangle
was meaſured in Feet, it would contain as many Rectangles 1 Foot Another
Example
broad as there are Feet in the Bafe A B, and that each Rectangle would
contain as many fquare Feet as there are lineal Feet in the Altitude A B
of the Rectangle ABCD. Whence the Number of ſquare Feet contained
in the Rectangle ABCD will be found, by multiplying the Num-
2 M
90
OF
ELEMENTS
Figure 2.
The conti
ber of lineal Feet contained in the Bafe B C, by the Number of Feet
contained in the Altitude A B.
The fquare Yard, for Example, having 3 Feet in Length and 3 Feet
in Breadth, its Superficies will contain 3 Times 3 or 9 fquare Feet.
If the Square ABCD reprefented a fquare Foot, its Bafe BC would
contain 12 lineal Inches, and its Altitude AB 12 lineal Inches. Whence
its Superficies would contain 12 Times 12 or 144 fquare Inches. In like
Manner it will appear that a fquare Inch whofe Bafe and Altitude are
divided each into 12 Lines, contains 12 Times 12 or 144 fquare lines:
and fo of any other Meafure that is employed.
LXXI.
It often happens that the fame Meaſures are not employed for mea-
guous Sides furing the Bafe and the Altitude of the Rectangle. In this Cafe, the
of the fnper fuperficial Meafures contained in the Superficies of the Rectangle, are not
fquare Meaſures, but Meaſures whofe Length is the Meafure employed
tained in any for meaſuring the Bafe, and whofe Breadth is the Meafure employed
Figure are for meaſuring the Altitude.
ficial Mea-
fures con
the lineal
Meaſures of
ons..
Example..
For Example, if it was propofed to determine the Number of Bricks
its Dimenfi lying flat which are contained in a Rectangle: Since a Brick is 8 Inches
long and 4 broad; the Length B C, fhould be meafured with a Mea-
fure of 8 Inches, and the Altitude A B with a Meaſure of 4 Inches, and
the Product arifing from the Multiplication of the Number of Meafures
of 8 Inches contained in the Bafe B C, into the Number of Meaſures of
Inches contained in the Altitude AB, will be the Number of Bricks,
or of the fuperficial Meafures 8 Inches long and 4 broad, contained in
Area of the Rectangle ABCD.
The fuperfi
cial Mea
fures are
divided and
4
LXXII.
When the Length and Breadth of the Rectangle is measured by the
Yard, and is not contained in them a certain Number of Times with-
out a Remainder; the Computifts meafure this Remainder in Parts of
a Yard, viz. in Feet, Inches, Lines, &e. and as the Products of thofe
fubdivided Meaſures do not always produce fquare Yards without a Remainder,
into Parts they eftimate this Remainder in Parts of a ſquare Yard.
to thoſe of
analagous. Though the moft regular Parts of a fquare Yard, are fquare Feet,
the Lineal fquare Inches, and fquare Lines; thofe however are not the Parts that
Meaſures. the Computifts commonly employ, they find it more convenient to di-
vide the fquare Yard into Parts analogous to thofe of the lineal Yard:
Example. and as the lineal Yard is divided into 3 lineal Feet, and the lineal
Foot into 12 lineal Inches, and the lineal Inch into 12 lineal Lines, fo
they divide the fquare Yard into 3 Rectangles, each of which is 1 Foot
broad and I Yard long, expreffed thus, Foot-Yard, or Yard-Foot; on
account of their two Dimenfions; they divide the Rectangle Yard-Foot
into twelve equal Parts each of which is 1 Yard long and 1 Inch broad,
NUMERAL ARITHMETICK.
91
expreffed thus Yard-Inch; in fine they divide each of thofe new Rec-
tangles into 12 equal Parts each 1 Yard long and 1 Line broad, ex-
preffed thus Yard-Line on Account of their two Dimensions. And fo
on of the other Meaſures employed.
LXXIII.
When the Sides of a Rectangle are measured by the Foot, and is not Another
contained in them a certain Number of Times without a Remainder; Example.
the Computifts meaſure this Remainder in Inches and Lines; and as
the Product of the two dimensions meaſured will not always produce an
exact Number of fquare Feet without a Remainder, they have recourfe
to the Parts of a fquare Foot to meafure this Remainder.
The moſt regular Parts of the fquare Foot with Refpect to the Di-
vifion of the Foot into Inches and Lines, are the fquare Inch, and
fquare Line. But as the fquare Foot contains 144 fquare Inches; and
the fquare Inch 144 fquare Lines; and that the lineal Foot is only di-
vided into 12 Inches, and the lineal Inch into 12 Lines: The Computifts
chufe rather to divide the fquare Foot into 12 equal Parts, each of
which is Foot long and 1 Inch broad, expreffed thus Foot-Inch on
account of their two Dimenſions. They divide alfo the Foot-Inch as
the lineal Inch, into 12 equal Parts, each of which is Foot long and
Line broad expreffed thus Foot-Line.
LXXIV.
From what precedes it is eafy to conclude, that two Numbers of equal
lineal Meaſures, multiplied into one another, produce a Number of fquare
fuperficial Meaſures, whofe Sides are the lineal Meafures contained in the trical Multi
Multiplicand and Multiplicator. For Example, a Number of lineal plication of
Yards multiplied into a Number of lineal Yards produces a Number of a Line by &
fquare Yards; a Number of lineal Feet multiplied into a Number of li- duced to
neal Feet produces a number of fquare Feet: And ſo on.
Line is re-
arithmeti
cal Multipli
cand the
tors.
It is alfo manifeft that a Number of equal lineal Meaſures, multi-cation, by
plied into a Number of other equal lineal Meaſures, different from the afcribing to
firft, produces a Number of fuperficial Meaſures whofe contiguous Sides
are the two Sorts of Meaſures of the Multiplicand and Multiplicator. two Dimen
For Example, a Number of lineal Yards multiplied into a Number of fions affect-
lineal Feet will produce a Number of fuperficial Meafures each of which ing the Fac
is 1 Yard long and 1 Foot broad; a Number of lineal Yards multiplied
into a Number of lineal Inches or Lines, will produce a Number of
fuperficia! Meaſures each of which will be r Yard long and 1 Inch or
Line broad; And fo on. The geometrical Multiplication therefore of
a Line by a Line will be reduced to common Multiplication, by giving
to the Units or Meaſures of the Multiplicand the two Dimenfions which
affect thoſe of the Multiplicand and Multiplicator, and conſidering the
propofed Multiplicator as an abftra&t Number.
1
92
ELEMENTS OF
LXX V.
The Value of the different Units relative to the fquare Yard, with
the Characters whereby they are diftinguished from each other, are as
follow.
rr Square Yard Irr
off
If f
ff
Square Foot
r" Yard-Second Ir" 12" or 3LL
iff
14411
r
Yard-Tierce Ir"
-
II Square Inch
ALI
III
I 44LL
LL Square Line
rf Yard-Foot
iff
12fI
fl
Foot-Inch
If I
12fLor 12II
Irr
3rf
fL
Foot-Line
fL
12for ill
Irf
12rlor 3ƒƒƒ'
Foot-Prime
YI Yard-Inch
YL Yard-Line
IrI
12 Y Lorfff"
If
12f" or 12LL
Foot-Second
If
m
ILL
ITL
12r or 311
III
I2IL
r'
Yard-Prime IT'
12r" or II
IL
Inch-Line
I/L
12ľ or 12LL
I'
Inch-Prime
Il'
ILL
LXXVI.
Method of
finding the
Area of a
Rectangle
whofe Di-
Let it be propofed to find the Area of a Rectangle, whofe Length is
58r. 2f. 81. and Breadth 8r. if. 61.
Though either of the Factors may indif-
criminately be taken for Multiplicator, yet
that which contains the leaft Number of
fed in Yards Yards is commonly affumed for Multipli-
menfions
are expref-
cator.
and Parts of
a Yard ex-
When the Number of Yards of the
plained by
an Example. Multiplicator is expreffed by one Figure,
582.
2f.
81.
8r
if. 6I.
471rr. orf. 4r1.
19
I
IO
8rL.
9
2
5 4
as in this Example; the Computifts first 500rr. 1rf. 871. OYL.
multiply each Part of the Multiplicand,
beginning by the Parts of the loweſt Denomination, by the Number of
Yards of the Multiplicator, afterwards taking for the Product arifing
from the Multiplication of the Multiplicand into the other Parts of the
Multiplicator, the fame Parts of the Multiplicand that the Parts of the
Multiplicator are of a Yard. According to thofe Rules.
1º. I multiply 8 Inches by 8 Yards, which produces 64 Fard-Inch.
As this Number of Yard-Inch contains 5 Yard-Feet and 4 Yard-Inch
over, I therefore fet down 4 Yard-Inch in the Column of Inches and
retain 5 f. to add them to the Yard-Foot of the Product.
I next multiply 2f. by 87 which produces 161f. to which adding the
51f. I retained, there refults 21f. as this Number of Yard-Foot "con-
tains 7rr. I fet down o in the Column of Feet, and retain 7rr. to add
them to the Square Yards of the Product.
Laftly multiplying 587. by 87, and adding to the Product the 7r.
NUMERAL ARITHMETICK.
93
I retained; there refults 471rr. orf. 4rI. for the Produa of 58r. 2f. 81.
multiplied by 8r.
2º. To multiply the fame Multiplicand 587. 2f. 81. by the Part 1f.
of the Multiplicator; I obferve that if this Multiplicand was multipli-
ed by 1 Yard, it would produce 58rr. 27. 871, that is, the Product
would be equal to the Multiplicand, with this difference only that each
Part would acquire a Dimenfion of 1 Yard: Whence multiplying by 1 Foot
which is only the one third of a Yard, the Product will be the one third
of the Multiplicand, each Part having a fecond Dimenfion of 1 Yard.
Now this one third will be 19rr. 1rf. 10rI. 8YL. becaufe the one
third of 58. is 19rr. and 1r. over which is equal to 327. which
added to 2ƒ makes 5rf. the one third of which is If. with 21f. or
2471. over, which added to 8YI makes 32rI the one third of which is
Iori. and 2rI, or 24YL. over, the one third of which is 87L.
I
3º. To multiply the Multiplicand by the 6 Inches in the Multipli-
cator, I obſerve that 6 Inches is the one Half of 1 Foot, and confe-
quently ſhould give the one Half of the Product 19rr. 1rf. 10r1, 8rL.
which is 9rr. 2rf. 5YI. 4rL. becauſe the one half of 197r. is 9rr.
with a Remainder of Irr, or 3f. which added to 1f. makes 47.
the one Half of which is 21f. and the one half of 10rI. 8YL. is 5r1.4rL.
Adding together thofe three particular Products there refults 500ry..
Irf. 8r7. for the Product required.
LXXVII.
Let it be propoſed to find the Area of a Rectangle 68Y. 1f. long and
572 2f. 81. broad.
57
2
81.
68r. If
45
Irf.
orf. 4rı.
0
IOYT 8YL
15
19
The Part 68 of the Multiplicator by which
the Multiplicand is firft to be multiplied, con-
fifting of feveral Figures; all the Parts of the 456.
Multiplicand cannot be multiplied after a direct 342
Manner by this firft Part of the Multiplicator,
as in the foregoing Example; but after the
Part 572 of the Multiplicand is multiplied by
68r. to obtain the Product of the remaining
Parts of the Multiplicand multipled into 68, the 3955
fame Parts of 68 must be taken that the Parts 2 Feet 8 Inches of the Mul-
tiplicand are of a Yard; and the Sum of the feveral particular Pro-
ducts that refult will be the Produ&t of 57r. 2f. 81. into 68 Yards. Af-
térwards the fame Multiplicand 57. 2f. 81. muſt be multiplied by 2
Feet, after the Manner explained in the foregoing Example.
2rf 2r1 8YL
1º. Multiplying the Part 57, of the Multiplicand by 687, I find the
two particular Products 456 Y, and 3420rr.
Application
of the fore
going Me
thod to
another
Example.
་
94
ELEMENTS OF
reducing
20. to multiply 2 Feet by 68 1ards, I obferve that I Fard multiplied
into 68 will give 68 Jquare Yards, that is, a Number of fquare Yards
equal to the Multiplicator. Whence 2 Feet which are only the two
Thirds of a Yard, fhould only give the two Thirds of 68 Square Yards:
and each third Part being 22YY, 2rf: I fet down twice 22r. 2rf or
·45XX, 1Xƒ.
3º. to multiply 8 Inches by 68 Yards, I obferve that 8 Inches are the
one third of 2 Feet, which produced 45. 1f. Whence for 8 Inches
I take the one third of 45r. If. which is 15r1. orƒ. 4rI.
}
+
It remains to multiply the fame Multiplicand by 1 Foot, for which I
take the 1 third of 57YF. 2¥f. 871. which is 19rr. off. 1ori. 8rL.
Adding together all thofe particular Products, there refults 3955YY
2rf. 2ri. 8¥£. for the Product required.
LXXVIII.
+
When the Area of a Superficies is found in fquare Yards, and in Parts
Method of of the fquare Yard divided into 3 and fubdivided continually into 12
the Parts of equal Parts, as in the foregoing Examples. Thofe Parts may be re-
fuperficial duced into fquare Meafures, fuch as fquare Feet, fquare Inches, fquare
Meaſures Lines, after the following Manner.
analogous
to thoſe of
the lineal
Meaſures;
into ſquare
rº. The Number of f. being multiplied by 3, will produce fquare
Feet.
2º. Each Yard-Inch being 36 Inches long and 1 Inch broad, contains
Meafures. 36 Square Inches, or the one fourth of a fquare Foot. Whence the one
fourth of the Number of Yard-Inch, will be fquare Feet, and every
If. that remains will be 36 quare Inches.
}
3°. Each Yard-Line being the twelfth Part of a Yard-Inch, and
the Yard-Inch being equal to 36 fquare Inches, confquently the Yard-Line
is equal to 3 Square Inches: Whence the Number of Yard-Line being
multiplyed by 3 will produce Square Inches.
49. each Unit of the Meaſures affected by the Mark Y being equal
to the twelfth Part of a Yard-Line, which is equal to 3 Square Inches,
it will be the one fourth of a Square Inch, or 36 Square Lines; whence
the one fourth of the Number of ' will be fquare Inches, and every
Ir' that remains will be equal to 36 quare Lines.
1
I
5. Each " is equal to 3LL, becaufe 1" is the one twelfth of 12',
which is equal to 36 Square Lines. Whence the Number of Y" multi-
plied by 3 will produce Square Lines.
69. It will appear in like Manner, that each is equal to LL:
Wherefore the one fourth of the Number of r" will be Square Lines:
It is the fame with refpect of the other Parts continually 12. Times leſs
which may be reduced into quare Primes, which are the one hundred and
forty fourth Parts of a Square Line.
When the Parts of the loweft Denomination in the two Factors of
the Multiplication are Lines, and the principal Parts are Yards, the
NUMERAL ARITHMETICK.
95
Parts of the loweſt Denomination in the Product will be r¹, ·4ˆ of
which make a fquare Line: becaufe when the two Factors of the Mul-
tiplication are reduced into Lines, the Product can only confift of fquare
Lines.
LXXIX.
Let it be propofed to reduce into Jquare Feet, Jquare Inches, and quare
Lines, the Meafures that are less than the quare Yard in this Product.
120851rr 2rf 5r1 grl of 42" 82".
The firft Part
being compofed 120851YY 2Y 5YI 9YL oY 4Y" 8Y""
of Square Yards.
which
are the
principal Mea
fures, are not to
be altered. To re-
duce the other
3
3
6ff 361
I 27
12LL
2
3 HC
120851YY 7ff 63II
14LL
under the YI, 3 under
under the Y"", and
Parts, it fuffices to fet down 3 under the Yf,
the YL. under the Y', 3 under the Y", and
multiply each Part of the Product by the Number wrote under it, viz.
the first, third and fifth Parts after the fquare Yards, by 3; and the fe-
cond, fourth, and fixth Parts after the fquare Yards, by 4; obferving that
the firſt and ſecond Parts after the Yards that are multiplied by 3 and 4,
produce Square Feet; that the third and fourth, multiplied by 3 and by
1, produce fquare Inches; and that the fifth and fixth Parts, multiplied
by 3 and by, produce fquare Lines.
I
Whence I fay: 3 Times 2Yf is 6ff, which I fet down under the
Line. Then I fay, the of 5YI is iff and 36II: I therefore fet
down rff under 6ff, and 36II in the following Column.
4
I then multiply 9YL by 3, which produces 2711 which I fet down
in the Column of fquare Inches under the 3611; and multiplyiny oY' by
4, which produces nothing more for the quare Inches.
I
Laftly I multiply 4Y" by 3, which produces 12LL; and 8Y" by
which produces 21.L.
Adding together the new fquare Meaſures of the fame Species, I find
that the propofed Product 120851 YY 2Yf 5YI 9YL OY' 4Y""8Y"
is reduced to fquare Meaſures 120851YY 7ff 631 14LL.
LXXX.
The forego-
ing Method
applied to an
Example.
When a Parallelogram ADCB is a Rhomboid, if from the Extremi- Fig. 4.
ties A and B of one its Sides, there be drawn two Perpendiculars AE,
BF to the oppofite Side the rectangular Parallelogram. AEFB will be The Con-
equal to the Rhomboid ADCB. (Euc. Prop. xxxv. B. 1) And as the Parallelo-
Number of the fquare Meaſures contained in the Superfices of the rectan- gram is
tent of any
96
EMELENTS OF
the Baſe into
}
found by gular Parallelogram AEFB, is found by multiplying the Number of
multiplying Meaſures in the Bafe EF by the Number of Meafures in the Side AE;
alfo the Number of fquare Meafures contained in the Rhomboid ADCB
will be found, by multiplying the Number of Meaſures in EF or in AB
or in DC, by the Number of Meaſures contained in the Perpendicular AE.
Half the
Altitude.
Fig. 5.
LXXXI.
Since a Triangle DAC is the one half of a Parallelogram ADCB of
the fame Baſe DC and Altitude AE (Euc. Prop. xx1, B. 1); and that
The Con-
tent of any
the Number of fquare Meafures contained in the Parallelogram ADCB
Triangle is found by multiplying the Number of Meaſures in the Bafe DC by
found by the Number of Meafures in its Altitude AE; it is manifeft that the
multiplying
the Bafe in- Number of fquare Meaſures conrained in the Triangle DAC will be
to Half the found, by drawing a Perpendicular AE from its Vertex to its Bafe DC
produced if neceffary, and multiplying the Number of Meaſures in this
Baſe DC by one half of the Number of Meafures contained in the Al-
titude AE.
Altitude.
Fig. 6.
Another
Method of
finding the
Area of a
Triangle.
LXXXII.
The Area of any Triangle ABC may be alfo found, thus: add to-
gether the three Sides of the Triangle, take the one Half of the Sum;
from this Sum fubtract fucceffively the three Sides of the Triangle;
multiply the three Remainders into one another: multiply the Product
again by the one Half of the Sum of the three Sides, and the Square
Root of the Product will be the Content of the Triangle.
Let, for Example, the Length of the three Sides AB, BC, AC of
the Triangle ABC, be 4 Yards, 15 Yards and 13 Yards. I take the one
Half of the Sum of thoſe three Sides which is 16 Yards, I then fubtract
fuceffively from this Half Sum, each of the three Sides, that is, I fub-
tract firft 4 Yards from 16 Yards, and there remains 12 Yards, then
15 Yards from 16Y, and there remains 1 Yard, and laftly 13Y
from 16Y and there remains 3 Yards. I afterwards multiply thofe
four Numbers 16, 12, 1, 3 into one another, and extracting the Square
Root of the Produ& 576, I find 24 for the Number of fquare Yards
contained in the Triangle ABC.
LXXXIII.
To demonftrate this Rule, from the Angle A of the Triangle ABC
Grounds of let a Perpendicular AE be drawn to the oppofite Side BC; then
the forego-
ing Method, AC²+2BCXBF=AB²+EC² (Fuc. Prop. XII. B. 11.) Confequently
BE
2
2
2
AB +BC —AC
BC²¬AC
fults BE
2
2BC 2
Squaring each Member ofthis Equation, there re-
2 2
(AB +BC-AC ),
2
(2BC2)
2
·AC ³)
AE²=AB¾— (AB²+BC² —Að ²
(2 BC' ) ²
2
2
confequently AB-BE” or
multiply ingeach Member by (2BC)²,
NUMERAL ARITHMETICK.
97
there reſults AE¹× (2BC) ²=AB³x(2BC)² —(AB²+BC²—AC²)². Ex-
racting the Square Root of each Member, and dividing them by 4, there
AEXBC
refults
2
2
=-=-√ [ABˆ× (2BC)²—(AB²+BC³—AC³)³]= Area
4
2
2 1
of Triangle ABC, but AB x (2BC)-(AB+BC-AC) is equal to
[AB × 2BC + AB²+BC-AC X ABX2BC-AB—BC+AC³]
I
Whence the Superficies or the Area of the Triangle ABC is equal to
{√/[(AB×2BC+AB‍+BC”—AC¾×(AE×2BC—AB¾—-BC¾+ac)]
4
2
2
2
2
2
but ABX2BC+AB² +BC-AC or AB²+BC²+2AB XBC¬AC₁s the
Product of AB+BC + AC multiplied into AB+BC AC. And
ABX 2BC-AB-BC +AC, or AC —AB —BC +2ABXSC is the
-AB-BC²+2A5XSC
Product of AC+AB—BC multiplied into AC-AB+BC, Whence
the Superficies or the Area of the Triangle ABC is equal to
✔[AB+-BC+AC)X(AB+BC—AC)X(AC+AB—BC)×(AC+BC—AB]
AB÷BC—AC AC÷AB-BCAC+BC—AB]
equal [AB+BC+ACK
24
AB+BC-AC
2
X
AB+BC+AC
But
2
A B+BC+AC
BC, and
2
2
2
AC, and
2
AC+AB-BC
2
AC+BC-AB AB+BC+AC
2
=
2
AB.
Whence the Area of the Triangle ABC may be exprefled thus
AB+BC+AC(AB+BC÷AC
X
2
2
AB+BC+ACBC)X(AB+BC+AC __AB)]
AC)X(A)
LXXXIV.
2
2
The Content of any quadrilateral Figure ABCD, having two parallel
Sides AB, CD, is found by multiplying the Sum of thofe Sides by Half Fig. 7:
their perpendicular Diſtance CE. For if the Diagonal AC be drawn,
the Triangle ACB will be equal to CEXAB, and the Content of the Rules for
finding the
Triangle DAC will be equal to CEXCD, and therefore the Content Content of a
of the whole Quadrilateral ABCD will be equal to
+ CEXAB+ CEXCD=1 CEX(AB+CD).
LXXXV.
quadrilateral
Figure
whoſe op-
pofite
Sides are
parallel,
From the Manner of finding the Area of a Triangle, the Area of any Sale,
right-lined plane Figure, as ABCDE, may be determined, by divid
ing the whole into Triangles and finding the Content of each Triangle. Figure S.
Thus let the dividing Lines AC and AD, be 20 and 16 Inches, and the
Perpendiculars BF, DG, EH falling thereon, 8, 12 and 16 reſpectively;
then the Content of the Triangle ABC being So, that of ACD 120,
2 N
98
ELEMENTS OF
1
Method of
Figure by
dividing it
into
Triangles.
and that of ADE 80, it is evident that the Content of the whole Figure
finding the will be the Sum of all thofe, or 280 fquare Inches. But, when the given
Area of any Lines are expreffed by Fractions or very large Numbers, the Work
right lined
will be fome what fhortened by finding the Content of every two Tri-
angles, having the fame Baſe, at one Operation; that is, by first adding
the two Perpendiculars together, and then multiplying half their Sum by
the common Bafe of the two Triangles. Thus, in the laſt Example,
the half Sum of the two Perpendiculars BF and DG being 10, if this
Number be therefore multiplied by 20, the Meaſure of the common
Bafe AC, the Product which is 200 will be the Content of the Trapezium
ABCDA, to which 80 the Content of the Triangle ADE being added,
the Sum will be 280, the fame as before.
Fig. 9.
Role for
LXXXVI.
The Content of a regular Polygon ABCDEF, that is, one whofe Sides
and Angles are all equal, is found by multiplying half the Sum of its
finding the
Area of any
Sides by the Length of the Line KG drawn from the Middle of
regular Po- AB to the Center of the Polygon.
lygon.
Grounds of
the forego-
ing Rule for
finding the
gon.
1;
any Side
For, from the Center K let there be drawn Rays to all the Angles of
the Polygon, which will divide it into as many equal Triangles as it has
Sides, and which will have all the fame Altitude. (Euc. Prop. XIV. B. IM.)
Now if upon a fame ſtraight Line AH, the Baſes AB, BC, CD, &c. of all
thofe Triangles be extended, and upon AH be conftructed the Triangle
AKH having for Altitude KG, this Triangle AKH will be equal to the
Area of any
Sum of the Triangles which compofe the regular Polygon ABCDEF,
regular Poly confequently the Superficies of any regular Polygon is equal to a Tri-
angle AKH whofe Bafe is equal to the Contour of the Polygon, and
whofe Altitude is the Line drawn from the Middle of any Side to the
Center of the Polygon, but the Superficies of a Triangle AKH is equal
to one Half of the Product of its Bafe AH into its Altitude KG, where-
fore the Superficies of any regular Polygon ABCDEF is equal to the one
Half of the Product arising from the Multiplication of its Contour into
the Line drawn from the Middle of any Side to the Center of the
Polygon.
The Area
of a Circle
found by
LXXXVII.
By increafing the Number of Sides of a regular Polygon infcribed in
a Circle, its Contour approaches continually nearer and nearer to the
Periphery of the Circle, which is its Limit, and the Line drawn from the
multiplying Middle of any Side to the Center of the Polygon, approaches nearer and
its Periphery
by Half its nearer to the Ray, which is its Limit, wherefore fince the Superficies of a
regular Polygon, what ever the Number of Sides may be, is equal to a
Triangle whofe Baſe is the Contour of this Polygon, and Altitude the
Line drawn from the Middle of any Side to the Center of the Polygon,
it is evident that the Superficies of a Circle is equal to that of a Triangle
Radius.
NUMERAL
99
ARITHMETICK
whofe Bafe is the Circumference of the Circle, and Altitude the Ray of
the fame Circle.
If therefore a ſtraight Line could be found exactly equal to the Cir-
cumference of a Circle whofe Radius is given, a re&tilineal Figure might
be made equal to the Area of this Circle, and what is called the
Quadrature of the Circle would be found. But though this Problem has
not been folved, and is thought never will, yet feveral Ways have been
invented by which the Value of the Circumference may be obtained to
any affigned Degree of Exactness, whereof the following is one of the
moſt fimple.
LXXXVIII.
Method of
finding the
If from the Extremity A of any Arch AB, there be drawn a Ray CA,
and an indefinit Tangent AD; and from the other Extremity B of the
fame Arch there be drawn a Line BE perpendicular to the Ray CA, and
through the Points B and C, the Line BC which produced will meet the Fig. 10
Tangent in D: BE is called the Sine of the Arch AB, AD its Tangent,
CD its Secant, AE its Verfe Sine.
The Ray CF being fuppofed perpendicular to the Ray CA, that is, AF Ratio of the
being a Quadrant of a Circle, and confequently the Arch BF being the Diameter to
Complement of the Arch BA, if from the Extremity B of the Arch BF the Circum-
the Line BG be drawn perpendicular to the Ray CF, and from the
other Extremity F of the fame Arch be drawn a Tangent FI meeting
the Secant CI in I, BG or its equal CE Sine of the Arch BF, is called the
Co-fine of the Arch AB, FI its Co-tangent, CI its Co-fecant, and GF its
Verfe co-fine.
ference.
Theſe Notions being premifed, let the Radius of the Circle be di-
vided into ten Billions of equal Parts, or let CA=10000000000, and let
the Arch ALB be 30°, Confequently its Sine BE, which is equal to Half
the Radius, will be 5000000000. Whence is deduced the Sine AK of
AL, half of the Arch ALB, that is, of 159. for EC/BC-BE=
8660254037, 84438, and CA-CE—AE=1339745962, 15562, and Fig. 11.
2
2
AB=BE +AF =26794919243112264579,77747602confequently AB
or AK - 6698729810778066144, 94436901, and of Courfe AK or
S. 15° 2588190451, 0252.
4
In like Manner the Sine of 7° 30' will be found to be equal to
1305261922, 2005, the Sine of 3° 45′ to be 654031292, 3014, the Sine
of 1° 52′ 30″ to be 327190828, 2177, the Sine of 56' 15" to be equal to
163617316, 2649, the Sine of 28' 7" 30" to be 81811396, 0394, the
Sine of 14' 3" 45"to be 40906040, 2624, the Sine of 7' 1" 52" 30"H
100
OF
ELEMENTS
Circle by
Means of
the forego-
V
VI
to be 10453062, 911641, the Sine of 3′ 30″ 56" 15"" to be equal to
10226536, 8034, the Sine of 1' 45" 28" 7 to be 5113269, 0701,
the Sine of 52" 44" 3" 45" to be 2556634, 6186, the Sine of 26" 22~
1*52 30 to be 1278317, 3198, the Sine of 13" II" .0""""
be 639158, 66118383; the Sine of 6" 35" 30""""
319579, 3307512: the Sine of 3" 17" 45" 14"
159789, 665396; and confequently the Sine of 1" 38"
52 30v will be found to be 79894, 8327005.
56 15 to
VI
30" 28"
3
7V
VI
3011
45 VII
to be
to be
V
5.2""
37 IV
Now I" 38" 52" 37 IV' 52" 30" being reprefented by the Arch AL,
half of the Arch AB, or of 3" 17" 45"" 14" 3" 45", if the Chord BF
be drawn, which an account of the Angle FAB being a right Angle, will be
perpendicular to AB, and confequently parallel to the Secant CH the
two Triangles FEB, CAH will be therefore equiangular, and of Courfe
FE: BE=CA : AH (Euc. Prop. Iv. B. vi.) but FE=EC÷CF¬R+
Coſ, AB=10000000000+9999999998, 723363=19999999998, 723363,
BE= Sin. AB=159789, 665396 and CB=10000000000.
Conſequently AH or T. AL or T. 1" 38" 52" 37°
159789,665396X10000000000
= 79894, 832703.
37V IV 520"
30v-
19999999998, 723363
From whence it follows 1. that fince the Sin. I" 38" 52""" 37" Iv!
52" 30", viz 79894, 8327005 is less than its Arch, and this Arch
being found by dividing the Arch 300 by 65536 or 16 Times fucceffively
by 2, and being confequently contained 65536 Times in 30°, or 6 Times
65536, that is, 393216 Times in the Semi-Circumference, it follows
I fay, that 79894, 8327005X393216, or 31415926535, 159808 will be
lefs than the Semi-Circumference, whofe Radius is expreffed by
10000000000. 2º. Since the T. 1" 38" 52" 37v IV 52" 30", viz.
79894, 832703 is greater than its Arch, 79894, 832703X393216, or
31415926536, 339456 is greater than the Semi-circumference whofe
Radius is expreffed by 10000000000, whence the Periphery of a
Circle whofe Radius is expreffed by i is equal to 6, 283185308 very
nearly.
LXXXIX.
From the Periphery thus found, the Area of the Circle will alfo be
Method of known, being equal to the Rectangle of the whole Periphery and half
finding the
Area of any the Radius, that is, equal to 6, 283185308X, 5 or 3, 141592654.
Therefore fince the Peripheries of Circles are as their Diameters (Euc
cor. III. Prop. xxxiii. B. vI.) and the Circles themſelves are as the Squares
ing Propor- of thofe Diameters (Euc. Prop. II. B. XII.), it follows that as 2 to
tion of the 6, 283185308, or 1 to 3, 141592654:: the Diameter of any Circle to
the Circum its Periphery, and as 4 to 3, 141592654 or as I to 0, 785398163: the
ference, Square of the Diameter to the Area very nearly.
Diameter to
་
NUMERAL ARITHMETICK,
ΙΟΙ
A
XC.
found more
by the Pro-
But when the Cafe propofed does not require any great Degree of The Area
Accuracy then thoſe of Archimedes may be uſed, viz. 7: 22 Diam: Cir- of a Circle
cum. and 14: II
14: 11 fquare Diam: Area, which Proportions differ but expeditiouf-
little from thoſe above as will appear from the following Example, ly but lefs
wherein the Diameter of a Circle being 28, its Circumference and Area accurately
are required, here according to the firft Proportions I multiply 28 by portions of
3, 141592654 for the Circumference, and the Square of 28 or 784 by Archimedes
0, 785398163 for the Area, and there refults 87, 964 &c. and 615, 75, &c.
refpectively. But according to the Proportions of Archimedes the Cir-
cumference willbe found equal to 88 and the Area 616, which differ very
little from the former. After the fame Manner the Area of any other
Circle will be found.
XĊI.
Area of a
Since a Sector DKBCAD, or DKCAD, or DKA, contained by the Fig. 12.
three fourths, or the one half or the one fourth, &c. of the Circumfe-
rence, is equal to the three fourths or to the one half or to the one Rule for
fourth, &c. of the Circle, and confequently is equal to a Triangle finding the
whofe Bafe is the three fourths or the one half or the one fourth, &c. Segment of
of a Line equal to the Circumference of this Circle, and Altitude the a Circle.
Ray of the Circle; it is manifeft that the Content of a Sector will be
found, by multiplying its Arch by one half of its Ray.
If, for Example, the Arch AD of the Sector be fuppofed to be 60
Degrees, and the Ray AK 14 Yards, the Diameter of the Circle of the
Sector will be 28 Yards; Whence the Circumference of the Circle will
be found to be 88 Yards, and 60 Degrees being the one fixth of 360 De-
grees, or of the whole Circumference, the Number of Yards contained
in the Arch AD will be 14 Y 2f, confequently the Content of the Sector
AKD will be 14Y 2f X7Y or 102 YY 2Yf.
XCII.
Since the Superficies of any Sector AKDFL is determinable, as alſo
Method of
the Area of the Triangle AKD contained by the two Rays AK, DK
finding the
and the Chord AD, it is manifeft that if the Area of the Sector Area of a
AKDFL, and the Area of the Triangle AKD be feparately found, and Sector of a
the Area of the Triangle be deducted from that of the Sector, the Re-
mainder will be the Area of the Segment ADFL.
If, for Example, the Ray AK be 21 Yards, the Chord AD 30, and
the Perpendicular KE 14Y 2f 21, and the Angle AKD 100 Degrees.
I first determine the Superficies of the Sector AKDFL, which I find to
be 385 fquare Yards. I afterwards determine the Superficies of the
Triangle AKD, by multiplying AD by KE and taking the one half of
the Product, which will be 220YY 2Yf 6YI, then fubtracting from
385 YY the Area of the Sector AKD! L, 220YY 2Yf 6YI the Area
of the Triangle AKD, the Remainder 164YY oYf 6YI will be the
Area of the Segment ADFL,
Circle,
102
ELEMENTS OF
XCIII.
From the Manner of computing the Areas of plane Figures, the
Convex Superficies of folid Bodies may alfo be determined, thus.
1
The Convex Superficies of a Prifm AQPCB, is found by multiply-
Fig. 13.and ing the Periphery of a Section RSTVXR perpendicular to one of the
Sides AM, by the length of this Side.
14.
Method of
finding the
convex Su-
perficies of
a Prifm.
Grounds
of this
Method.
That this may appear, let a Rectangle mr be conftructed, whofe Bafe
ar is equal to the Periphery of the Section RSTVXR, and whofe Alti-
tude am is equal to the Side AM of the Prifm; and let the Bafe ar of the
Rectangle mr be divided into Parts af, ft, tu, ux, xr equal to the Sides.
RS, ST, TV, VX, XR of the Section RSTVXR, and through the
Points of Divifion let there be drawn Parallels to the Side am; I fay that
the Parts F, G, H, I, K into which the Rectangle mr is divided, will be
equal to the Parallelograms AN, BO, CP, DQ, EM which contain the
Priſm.
For fince the Side AM is perpendicular to the Section RSTVXR, all
its Parallels BN, CO, DP, EQ will be alfo perpendicular to the ſame
Section, (Euc. Prop. VIII. B. XI.) from whence it follows 1°. that the
ſtraight Line RS will be perpendicular to the two Parallels AM, BN,
confequently the Parallelogram AN will be equal to the Product of AM
into RS, or to the Rectangle F whofe Altitude am and Bafe af are
equal to AN, RS.
Fig. 15 and
.16.
2º. That the Straight Line ST will be perpendicular to the two Pa-
rallels BN, CO; whence the Parallelogram BO will be equal to the
Product of ST multiplied by BN, or by its equal AM, and confequently
equal to the Rectangle G, whofe Bafe st and Altitude am are equal to
ST, AM.
In like Manner it will appear, that the other Parallelograms CP, DQ,
EM are equal to the Rectangles H, I, K; whence the Sum of the Pa-
rallelograms that contain the Prifm, or the Superficies of the Prifm, is
equal to the Sum of the Rectangles F, G, H, I, K, that is, to the
Rectangle whofe Bafe is the Periphery RSTVXR of the Section, and
whofe Altitude is the Side AM of the Prifm.
f
From whence it follows that the convex Superficies of an erect Priſm,
will be found, by multiplying the Periphery of the Bafe by the Altitude
of the Solid, for in an erect Prifm, the Section perpendicular to any of
its Sides, is parallel to the Bafe, and confequently equal to it.
XCIV.
Since the foregoing Conclufions hold univerfally whatever the Num-
ber of Faces of the Prifm may be, and as the Prifm by increafing the
Rule for de Number of its Faces approaches continually nearer and nearer to the
termining infcribed Cylinder, which is its Limit, it is evident that the convex Su-
ficies of a perficies of a Cylinder is equal to a Rectangle mr, whofe Baſe is a ſtraight
Cylinder Line equal to the Periphery of the Section perpendicular to one of its
the Super
NUMERAL ARITHMETICK.
103
Sides AM, and whofe Altitude is a Line equal to this Side: and if the deduced
Cylinder be erect, its convex Superficies is equal to a Rectangle whofe from the
contiguous Sides are equal to the Altitude and Periphery of the Baſe of foregoing
this Cylinder.
XCV.
Methed.
Method of
The upper Superficies of a regular Pyramid SABCDE is found by Fig. 17 and
multiplying the Periphery of its Bafe by half the Length of the Line 18.
SG drawn from the Vertex to the Middle of any Side AB of this Baſe.
For the Pyramid SABCDE being a regular one, its Bafe ABCDE
finding the
Superficies
will be a regular Polygon, and confequently all its Sides AB, BC, CD, of a regular
&c. will be equal, and the ftraight Line SF drawn from the Vertex to Pyramid.
the Center F of the Circle defcribed about this Polygon will be perpen-
dicular to the Plane of this Polygon, confequently all the ftraight Lines
SA, SB, SC, &c. drawn from the Vertex of the Pyramid to the Angles
of its Bafe, will be equal. (Euc. Prop. IV. B 1.) Wherefore all the Tri- this Me
anglès ASB, BSC, CSD, &c. whofe Vertices coincide at the Vertex of thod.
the Pyramid will be equal, and ifofceles Triangles, and their Altiudes
will be equal to the Line SG drawn from the Vertex of the Pyramid
to the Middle of any Side of the Bafe, as AB, and confequently per-
pendicular to this Side.
Wherefore the Sum of all the Triangles ASB, BSC, CSD, &c.
which compofe the upper Superficies of the regular Pyamid SABCDE,
is equal to the Triangle ZHI, having for Bafe the fum of all their Baſes,
and whofe Altitude ZH is equal to the ftraight Line SG drawn from the
Vertex of the Pyramid to the Middle of any Side AB of the Baſe.
XCVI.
Grounds of
The Superficies of any Fruftum ABCDEMNOPQ of a Pyramid, is Rule for
found by multiply ng the Sum of the Peripheries of the two Ends finding the
ABCDE, MNOPQ, by Half the Length of the Line drawn through of a Fruf-
Superficies
the Middle of any two correfponding Sides AB, MN.
tum of a
Pyramid.
Grounds of
For the two Ends of the Fruftum MNOPQ and ABCDE being pa-
rallel, the Sides MN, NO, OP, &c. of the one will be parallel to the
Sides AB, BC, CD, &c. of the other, wherefore AB: MN=SB: SN this Rule.
and SB: SN-BC: NO (Euc. Prop. II. B vI.) Confequently AB: MN-
BC: NO, and alternately AB : BC-MN: NO; but the End
ABCDE being a regular Polygon, AB BC, confequently MN=NO.
By a fimilar reafoning it will appear that NO-OP, &c. and the Lines
SA, SB, SC, SD, &c. which are all equal, will be cut proportionally,
confequently SM, SN, SO, SP, &c. will be alfo equal, wherefore the
Triangles MSN, NSO, OSP which compofe the Superficies of the Py-
ramid will have their Sides equal each to each, and confequently all
their Altitudes will be equal to that ST of the Triangle MSN; whence
the Sum of all the Triangles MSN, NSO, OSP, &c. or the upper
- 104
OF
ELEMENTS
Superficies of the Pyramid, is equal to the Trangle ZKL, whofe Baſe KL
is equal to the fum of the Bales of all thofe Triangles, and whofe Alti-
tude ZK is equal to the Line ST; but the Superficies of the Pyramid
SABCDE is equal to the Triangle Z I; wherefore the Superficies of
the Fruftum contained between the parallel Bafes ABCDE MNOPQ,
is equal to the Trapezium HILK, whofe two parallel Sides are equal
How the Su to the Peripheries of the two Ends of the Fruftum, wherefore &c.
uperficies
of an irre-
gular Pyra
mid is
determined.
Fig. 19.
As to the Superficies of an irregular Pyramid, there is no other Rule
for finding its Content, but to compute feparately the Area of each
of the Triangles which meet at the Vertex of this Pyramid.
XCVII.
Since the foregoing Conclufions hold univerfally whatever the Num-
ber of Sides of the Pyramid may be, and as the Pyramid by increaſing
the Number of its Sides approaches continually to the infcribed Cone
which is its Limit, it is evident that the convex Superficies of a Cone is
termining equal to the Product of the Circumference of the Baſe of the Cone into
the Super half the flant Side thereof, and that the convex Superficies of any
Fruftum of this Solid is equal to the Product of the Peripheries into
half the Slant-ſide of the Fruftum.
Rule for de
ficies of a
Cone.
XCVIII.
The Superficies of a Sphere, is found by multiplying the Periphery of
Method of
the greateſt or generating Circle, by its Diameter; or by multiplying
finding the
Superficies the Square of the Diameter by 3, 1416.
Fig. 20.
of a Sphere. Let a Circle RQSq be infcribed in a regular Polygon ABCDE &c. of an
even Number of Sides, from the Center O to the Point of contact Q
of any Side BC, let the Radius OQ be drawn; alfo draw BbM, QPq,
CcL &c. perpendicular to AF, and BN perpendicular to CL, and let the
Polygon and its infcribed Circle be ſuppoſed to revolve about the pro-
duced Diameter AF.
this Me-
thod.
Becauſe the Solid generated by the Plane be CB is the Fruftum of a
Grounds of Cone, the convex Superficies thercof generated by BC, is equal to the Rec-
tangle under BC and the Sum of the two Peripheries defcribed by Bb
and Cc: but the Sum of theſe two, as Qq is an arithmetical mean be-
tween them, is equal to twice the Periphery Qq, and therefore the
convex Superficies of the faid Fruftum is equal to BCX2 Periph. Qq
BCX Periph. Qq, but becauſe of the fimilar Triangles OPQ,
BNC, we have BC : BN(bc)=OQ: PQ=Periph. RQSq: Periph. Qq
and confequently BCX Periph. Qq=bcx Periph. RQSq Superficies
generated by BC. By the very fame Argument it will appear that the
Superficies generated by any other Side CD is COX Periph. RQSq:
Whence it is manifeft that the Superficies of the whole Solid is
=Ab+bc+c0+, &c.X Periph. RQSq, and as the whole Superficies is
NUMERAL ARITHMETICK.
IOS
equal to AFX Periph. RQSq, let the Number of Sides of the gene-
rating Polygon be what it will, and as the faid Superficies, by increafing
the Number of its Sides, approaches nearer and nearer continually
to the Superficies of the infcribed Sphere, which is its Limit; it is evi-
dent that the Superficies of the Sphere itſelf is likewife equal to a Re&t-
angle under its Axis RS and Periphery R QS q; wherefore, &c.
XCIX.
Rule for de
The convex Superficies of any Segment of a Sphere BALDB, is found Fig. 21.
by multiplying the Square of twice the Altitude LB of the Segment, toge- termining
ther with the Square of the Diameter of the Bafe of the Segment, by the Superfi
0,7854; as being equal to the Superficies of the Circle, having for Ray cies of a Seg
the Altitude LB of the Segment, together with the Superficies of the ment of a
Sphere
Baſe of the Segment..
of
For it is evident from the foregoing Article, that the convex Superficies Crounds of
any Segment BALDB of a Sphere, is equal to the Product of the Al- this Rule
titude multiplied into the Periphery of the greateſt Circle ABDGA of
the Sphere, but if the Chords AB, AG, be drawn, the Triangles
ABL, BAG being Equiangular, LB: AB AB: BG, and dividing
=
the Confequents of this Proportion by 2, then LB : = AB: BC:
AB
2
But AB BC= Circum. AB: Circum. BC: Wherefore LB :
2
AB
2
AB
2
AB
Circum.AB: Circum. BC, whence Circum. ABX = LBXCircum.BC.
But the Produ& LBXCircum. BC, reprefenting the convex Super-
ficies of the Segment BALDB, & Circum. ABX repreſenting the
Superficies of a Circle whofe Ray is AB, which on Account of the Trian-
gle ALB, being right angled in L, is equal to the Sum of the Circles,
whoſe Rays are the Sides AL, LB of this Triangle, it is manifeft that
the convex Superficies of the Segment BALDB is equal to the Super-
ficies of the Circle, whofe Ray is the Altitude LB of the Segment, toge-
ther with the Superficies of the Bafe of the Segment,
C.
be
Fig. 22.
Having explained how the Superficies of folid Bodies are determined,
it remains to fhew how the Contents of the Solids themſelves may
found. Let the Solid to be meaſured be a Parallelepiped A O, whoſe
Length AB, Breadth B C, and Height A M, have been meaſured, for
Example, with the lineal Yard, and let this Solid be cut by Planes D P,
EQ, FR, GS, HT, IV, &c. parallel to its Bafe, into as many equal
Solids one Yard high, as there are Yards in its Height A M.
Each Solid as MV contained between two parallel Planes MO, IV, tent of a Pa
The Con
› rallepiped
106
OF
ELEMENTS
its found by being one Yard high, will contain as many cubical Yards as there are
multiplying
the Area offquare Yards in the Superficies of the Bafe MO; becauſe a cubical
its Bafe by Yard may be placed upon each of the fquare Yards in the Bafe MO,
its Altitude. and that the cubical Yards that occupy all the fquare Meaſures of the
Bafe, will be exactly contained between the two parallel Planes M O,
I V, diftant one Yard from each other.
Example
Another
But the Number of fquare Yards contained in the Rectangle M O, is
equal to the Product of its Length M N multiplied by its Breadth NO.
Wherefore the Number of cubical Yards contained in each of the So-
lids one Yard high, into which the Parellelepiped AO is divided, is
equal to the Product of its Length MN multiplied by its
Breadth NO; and as the Parallelepiped A O contains as many Solids
one Yard high as there are Yards in its Altitude A M; it follows that
thel Number of cubical Yards contained in the Parallelepiped A O
wil be found by multiplying the Product of its Length MN and Breadth
NO, meaſured in lineal Yards by the Number of Yards contained in its
Altitude A M.
CI.
Let, for Example, the Length of the Bafe AB be 6 Yards; the
Breadth B C 5 Yards; and the Height A M of the Solid 3 Yards; the
Number of cubical Yards contained in the Paralellepiped A O, will be *
equal to the Product of the three Numbers 6, 5, 3; and confequently
will be go cubical Yards.
CII.
If the Length AB of the Bafe, the Breadth BC, and the Height AM
Example. of the Parallelepiped AO were meaſured in lineal Feet, Inches, or
Lines, it is manifeft that the Content of the Paralellepiped would be
expreffed in cubical Feet, cubical Inches, or cubical Lines,
A cubical Yard, being a Parallelepiped 6 Feet long, 6 Feet broad,
and 6 Feet high, the cubical Yard will contain 6 × 6 × 6 or 216 cubical
Feet.
A cubical Foot being a Paralellepiped 12 Inches long, 12 Inches
broad, and 12 Inches high, the cubical Foot will contain 12 X 12 X 12,
or 1728 cubick Inches:
In like Manner it will appearthat a cubical Inch which is 12 Lines long,
12 Lines broad, and 12 Lines high, contains 1728 cubical Lines, and fo
of the other Meaſures that are employed.
CIII.
T
When the Dimenfions of a Parallelepiped are meaſured by the
Yard, and is not contained in them a certain Number of Times without
a Remainder, the Computiſts meaſure the Remainder in Parts of a Yard,
viz. in Feet, Inches, Lines, and as the Product of thoſe three Dimen
NUMERAL ARITHMETICK.
107
Meaſures
are divided
vided into
Parts anala
fions is not always an exa& Number of cubical Yards without a Remain- The folid
der, they eftimate this Remainder in Parts of the cubical Yard.
Though the cubical Foot, the cubical Inch, and cubical Line, are and fubdi
the moſt regular Parts of the cubical Yard, thofe however are not the
Parts that the Computifts commonly employ in eftimating the Parts of gous to
Solids that are leſs than the cubical Yard, after the greateſt thofe of the
Part of the Solid has been computed in cubical Yards, to render the
Computation more eafy, they divide the cubical Yard into Parts analo-
gous to thofe of the lineal Yard.
They divide therefore the cubical Yard into 3 equal Parallelepipeds
having each a fquare Yard for Bafe, that is 1 Yard long, 1 Yard broad,
and I Foot high, which on account of their three Dimenſions is expref-
fed thus Yard-Yard-Foot,
They divide the Yard-Yard-Foot, or the third Part of the cubical Yard,
into 12 equal Parts, having each for Bafe a fquare Yard, that is,
1 Yard long, 1 Yard broad, and I Inch high, expreffed thus
Yard-Yard-Inch.
They divide alfo the Yard-Yard-Inch, or the twelfth Part of the
third Part of the cubical Yard into 12 equal Parts, having
each a fquare Yard for Bafe and I Line for Height, which on account.
of their three Dimenfions is expreffed thus, Turd-Yard-Line; and fo of
the other Meaſures that are employed.
CIV.
lineal Mea
fures,
Example
When the Dimenfions of a Parallelepiped are meaſured by the Foot,
and is not contained in them a certain Number of Times without a Re- Another
mainder, the Computifts meaſure this Remainder in Inches and Lines. Example.
In this Cafe the Parallelepiped contains a certain Number of cubical
Feet with a Remainder, which the Computifts eſtimate in Parts of a cu-
bical Foot. To this End they divide the cubical Foot into 12. equal Parts,
and thofe Parts which are 1 Foot long, 1 Foot broad, and 1 Foot high,
are expreffed thus, Foot-Foot-Inch, on account of their three Dimenſions
they fubdivide afterwards the Foot-Foot-Inch into 12 equal Parts, each
I Foot long, 1 Foot broad, and 1Line high, expreffed thus, Foot-Foot-Line,
to diſtinguiſh them by their three Dimenfions.
In fine, whatever Meaſures are employed to meafure a Solid, it is
ufual to affume cubical Meaſures for the principal Meaſures of the Solid,
and to measure the Part which is lefs than the principal folid Meaſure
with leffer Meaſures, formed by fubdividing the principal cubical Meaſure
into as many equal Parts as are contained in the lineal Meaſure; ſo that all
the folid Meaſures refulting from thofe Divifions, have two Dimenſions
equal to thoſe of the principal Meaſure,
¡
108
}
ELEMENTS OF
J
}
J
CV.
TheLength, From what precedes it is eafy to conclude that a Number of equal
Breadth, and fuperficial Meaſures of any Length and Breadth, multiplied by a Num-
the folid ber of lineal Meafures of any Length, produces a Number of folid Mea-
Meaſures fures, having for Bafes the Meafures of the Multiplicand, and for Height
the Meaſures of the Multiplicator.
contained in
any Solid,
are the line
For Example, if a Number of fquare Yards 1 Yard long, and 1 Yard
al Meaſures broad, be multiplied by a Number of lineal Yards, the Product will be
compofed of a Number of folid Meafures, 1 Yard long, 1 Yard broad,
and 1 Yard high, that is, of cubical Yards.
of its Di
menfions.
Ifff
If a Number of fquare Yards be multiplied by a Number of lineal Feet,
or of Inches, or of Lines, the Produ&t will contain a Number of
folid Meaſures, 1 Yard long, 1 Yard broad, and 1 Foot or 1 Inch or
Line high.
If a Number of fuperficial Meaſures that have two different Dimenfions,
be multiplied by a Number of lineal Meaſures that differ alfo from thoſe
Dimenfions, for Example, if a Number of fuperficial Meaſures 8 Inches
long, and 4 Inches broad, be multiplied by a Number of lineal Meaſures
of two Inches, there will refult for Product, a Number of folid Meaſures,
each 8 Inches long, 4 Inches broad, and 2 Inches high, and fo on.
CVI.
The Value of the different Units relative to the cubical Yard,. and the
Characters whereby they are diftinguished from each other, are as
follow.
rryCubical Yard
CubicalFoot
III Cubical Inch
LLL Cubical Line
I rrr 27fff
1fff 1728III
₁ III 1728 LLL
rrr 3rrƒ
irrƒ 12rri or 9fff
irri 12rrl fff
12YYL
I 1rrl 12rr'
YYL
rrf Yard-Yard-Foot
YYI Yard-Yard-Inch
YYL Yard-Yard-Line
Yard-Yard-Prime
Yard-Yard-Second Irr" 12rr""
Yard-Yard-Tierce I XX!!!
rri
Yr
rn
Irr' 12rr"
Υγ 12rr¹v
Yr¹v Yard-Yard-Quart ¡YY'v 12YYv
I
Yr Yard-Yard-Quint Tv 12rY
rv
12/II
■ III
12ff 144LLL
12ff" 12LLL
ffL Foot-Foot-Line IffL 12ff
ff Foot-Foot-Prime iff 1 2ff"
ff Foot-Foot-Second iff"
ff Foot-Foot-Tierce ff"
ff Foot-Foot-Quart ff
fff IIL Inch-Inch-Line
911II1 Inch-Inch-Prime
¡III 12IIL
1 LLL
IIL 12' 144LİL
III II" Inch-Inch-Second II"
7zIII
9LLL Yff Yard-Foot-Foot
LLL YI Yard-Foot-Inch
rrv
LLLYfI
I
Irry'
LLLYL Yard-Foot-Line
rry Yard-Yard-Sext
ffl Foot-Foot-Inch
fff 12ff!
iffI 12ffL
YII Yard-Inch Inch
YIL Yard-Inch-Line
144III|YLL Yard-Line-Line
III' [2]["
1 2 LLL
X
1 LLL
I
irff
4rri
4rri 3fff
I r fl
4 r r l
4rrl #fff
irfl
4rr! 36111
36111
3III
IrII rr'
I YIL 4rY"
1 YLL 4Y7"||
I
NUMERAL ARITHMETICK.
109
CVII.
Let it be propofed to find the Content of a Parallelepiped the Area of
whofe Baſe is 3957rr 2rf 8r1, and its Altitude 22Y 1ƒ 61.
Having difpofed thofe Numbers as
in the Margin, I first multiply the Num-
ber of YY of theMultiplicand, by22Y,
and afterwards take for the Products of
the other Parts of the Multiplicand,
multiplied by 22Y, the fame Parts of
23YYY, that the Parts 2 Yf 8YI of
the Multiplicand are of a fquare Yard.
1º. Multiplying the Part 3957Y Y
of the Multiplicand by 22Y, I
find thoſe two particular Products
to be 7914YYY and 79140YYY.
7914rr
7914
3957rr 2rf
22r If
8YI
61
14
2rrf
4
2
8rri
О
ΙΟ
8rYL
I
I I
4
1319
Method of
finding the
Content of
a Parallele
piped, whofe
Bafe is ex
preffed in
fquare Yards
and Parts of
a fquare
Yard, and
Altitude in
Yards and
659
890521rr 1r 6rri orYL Parts of a
ÖYYL Yard.
2º. To multiply 2 Yf by22 Y, I obſerve that 1YY multiplied by 22 Y,
will produce 22ŸŸY, whence 2 Yf which are only the two thirds of YY
fhould give for Product the two-thirds of 22YYY, and each third Part
being 7YYY 1YYf, I fet down twice 7YYYYYf or 14YYY 2YYf.
3º. To multiply the 8YI of the Multiplicand by 22Y, I obferve that
8YI are the one third of 2Yf, which produced 14YYY 2YYf, whence
for 8YI I take the of 14YYY 2YYf, which is 4YYY 2YYƒ 8YYI,
4°. If the Multiplicand was to be multiplied by IY, the Product
would be 3957YYY 2YYf 8YYI, but as it is only to be multiplied
by Foot, or the of a Yard, the Product will be the of
3957YYY 2YYf 8YYI, viz. 1319YYY OYYf roYYI 8YYL.
5°. As 61 is only the one half of I f, it ſhould give for Product the
one half of the Product 1319YYY OYYf 11YYI SYYL, which is
659YYY IYYf 10YYI 4YYL.
3
Adding together all thofe particular Products, there refults 89052YYY
1YYf 6YYL for the Content required of the Parallelepiped.
CVIII.
81
Let it be propoſed to find the folid Content of a Parallelepiped 871 2f 81 long,
681 2f broad, and 32r if 61 high.
As the Length is to be mul-
tiplied by the Breadth, and that
Product by the Height of the
Parallelepiped, I multiply
87 Y 2f 81, by 68Y 2f, and
the Product 6035 YY OYf
IYI 4YL by 32Y if 61,
from whence there refults
197138YYY 2YYf 1YYI
871 2f
682 2f
6035rr orf iri 4rL
6035rr orf ri
322 If 61
4YL
OYL
Method of
finding the
Content of a
Parallelepi
ped, whofe
Dimenfions
are expreff
ed in Yards
and Parts of
197138rrr 2rrƒ ırrı 4rYL OYL & Yard.
4 YYL for the folid Content required of the Parallelepiped.
110
ELEMENTS OF
!
CIX.
When the folid Content of a Body is found in cubical Yards, and in
Method of other folid Meaſures that have all a fquare Yard for Bafe, and for
reducing the
parts be the Height the Parts into which the lineal Yard is divided, thofe Meaſures
lid Meaſures may be reduced into cubical Meaſures, fuch as cubical Feet, cubical Inch-
analogous to es, and cubical Lines, after the following Manner.
thoſe of the
lineal Mea
fures into
cubical
Meaſures.
I
1º. Each ſquare Yard containing nine fquare Feet, and 1YYf being
the Product of 1 fquare Yard or of 9 fquare Feet multiplied by 1 Foot,
which produces 9 cubical Feet, it follows that the Number of YYf being
multiplied by 9, will produce cubical Feet.
2º. Each YYI being the one-twelfth Part of 1YYf, or of 9 cubical,
Feet, it follows that IYYI is the of a cubical Foot; whence the Num-
ber of YYI being multiplied by, will produce cubical Feet, and if
there remains 1 or 2 or 3 Units, they will be equal to 432 or 864 or
1296 cubical Inches.
16
3°. Each YYL being the one-twelfth Part of IYYI or of the of a
ž
cubical Foot, it follows that 1YYL is the of a cubical Foot, or 108
cubical Inches, whence the one-fixteenth Part of the Number of 1YYL
will be cubical Feet, and every 1YYL that remains will be equal to
108 cubical Inches.
4°. Each YY' being the one twelfth Part of 1YYI, or of 108 cubical-
Inches, it follows that 1YY' is equal to 9 cubical Inches, confequently the
Number of YY' being multiplied by 9, will produce cubical Inches.
5°. Each YY" being the one twelfth Part of 1 YYY', or of 9 cubical
Inches, it follows that 1YY" is the of a cubical Inch, whence the Num-
ber of YY" being multiplied by will produce cubical Inches, and if
there remains 1 or 2 or 3 Units they will be equal to 432, 864, 1296
cubical Lines.
I
of a
6°. Each YY" being the one twelfth Part of 1YY", or of
cubical Inch, it follows that 1YY" is the Part of a cubical Inch, or 108
cubical Lines, whence the one fixteenth Part of the Number of 1YY"
will be cubical Inches, and every IYY" that remains will be equal to 108
cubical Lines.
7°. Each YY" being equal to the twelfth Part of 1YY", or of 108
cubical Lines, it follows that 1YY" is equal to 9 cubical Lines, whence.
the Number of 1YY" being multiplied by 9, will produce cubical Lines.
8°. Each YYv being the twelfth Part of 1YY or of 9 cubical
Lines it follows that 1YYv is equal to the of a cubick Line, where-
fore the number of YY being multiplied by will produce cubical
Lines.
V
9º. Laſtly, 1YY being the twelfth Part of
cubical Line, it follows that I YY will be the
1YY", or of of a
of a cubical Line, and
+
NUMERAL ARITHMETICK.
III
confequently the number of YY being divided by 16, will produce
cubical Lines.
I
When the Parts of the lowest Denomination, in the three mixt Num-
bers that are propofed to be multiplied into one another, are Lines, the
of the Number of 1 YY will always be an exact Number of cubical
Lines, and the Number of IYYV will be always divisible by 16; be-
cauſe the three Numbers when reduced into Lines, and multiplied into
one another, can only produce cubical Lines.
CX.
Let it be propofed to reduce into cubical Feet, cubical Inches, cubical Lines,
the Numbers of folid Meafures in the following Product, orrf orri
2rYL 3rr' 5rr" 5rr" 2r"""" 4rrv 48rr' that have all for Baje a
Square Yard, and for Height the Parts into which the lineal Yard is divided.
Το re-
duce thofe orrf orri ₂rrь 3rr 5rr" 5rr" 2rrrrv 482v' The forego
Meaſures it 9
fuffices, to
fet down
under the
three first,
the Num-
bers 9, 4,
I
, which
will
pro-
3
4
2rYL
I
178
Tъ
9
216/11
27
3
24711F
V
I
16
9
94
I
To
1296LLL
540
18
3
3
132LLL
Method
applied to an
Example.
3
duce cubical Feet, likewiſe under the three following the fame Numbers
9,416, which will produce cubical Inches, and under the three laſt
the fame Numbers 9,,, likewife, which will produce cubical Lines,
and multiply each Number of Meafures by the Number wrote under it.
As the three firſt cannot produce an Unit, the Reduction will not pro-
duce cubical Feet, but becauſe there remains 2 Yard-Yard-Lines,
which multiplied by or divided by 16, will produee the of a Foot,
or the of 1728 cubical Inches, I fet down 216 in the Rank of cubical
Inches.
ह
Multiplying the three following Numbers 3YY', 5YY", 5YY", which
fhould produce cubical Inches by the Numbers 9, 2, 3, wrote under
them, I find for the first Product 27 II, for the fecond 3 III, 1296LLL
and for the third 540LLL.
Multiplying the three laſt Numbers of the Meaſures that ſhould pro-
duce cubical Lines, by 9, the firft Produ&t will be 18LLL, the
fecond 3LLL, and the third will be 3LLL.
Having reduced all the Parts of the propofed Product, into cubical
Meaſures, I add them together, and there reſults offf 247III 132LLL.
1
112
ELEMENTS OF
Fig. 23.
BG
2
CXI.
From the Content of a Parallelepiped thus known that of a Prifm is
alfo known. Let it be propofed for Example to find the Content of a
triangular Priſm ABCDEF, becauſe this Priſm is the Half of its Parallele-
piped (Euc. Prop. XXVIII B. XI) it will be equal to a Parallelepiped of
equal Bafe and Altitude. Let therefore in its Baf e ABC, and from one
of its Angles B, a Perpendicular BG be let fall upon the Side AC oppofite
to this Angle; then ACX will reprefent a rectangular Parallelo-
gram MNO whofe contiguous Sides MN, NO are equal to the Side
AC and to one half of the Altitude BG of the Baſe ABC of the Priſm
tent of any ABCDEF. Let a Perpendicular DI be drawn from any Point D of the
found by upper Baſe of the Priſm to the Baſe ABC: then AC X X DI will re-
multiplying
the Area of preſent the folid Parallelepiped MR of equal Baſe and Altitude with the
its Bafe by triangular Prifm ABCDEF.
The Con
Priſm is
its Height.
Fig. 24.
The Con
tent of a
Cylinder
BG
2
As all Polygons may be divided into Triangles, and that each Triangle
may be transformed into a rectangular Parallelogram, it is manifeſt that
every Priſm is equal to a Parallelepiped of equal Bafe and Altitude;
therefore the Content of any Prifm will be found by multiplying the
Area of the Baſe, (found by the Rules for Surperficies) by the Height of
the Priſm, and the Product will be the required Content.
CXII.
After the very fame Manner the Content of a Cylinder is found (it
being alfo equal to a Parallelepiped of equal Baſe and Altitude): There-
found by fore, if for Example, the Diameter of the Baſe be ſuppoſed to be 20
multiplying Inches, and the Height of the Cylinder 15 Inches, then the Content of
the Area of the Solid will be 4712, 4 cubical Inches, very nearly. For the Area of the
its Altitude, circular Bafe being 314,16, this multiplied by15, gives 4712,4, as before
its Baſe by
Fig. 25.
tent of a
3
CXIII.
Hence alfo the Solidity of a Pyramid or Cone is likewife known; every
The Con fuch Solid being of a Prifm or Cylinder of equal Bafe and Altitude
Pyramid or (Euc. Prop. VII. Cor. II. Prop. x. B x11). Therefore the Content of a
Cone, found Pyramid orCone is found by multiplying the Area of the Baſe by of
by multiply the Altitude.
ing the
Meaſure of
its Bafe into
I third of its
Altitude.
CXIV.
I
3
The Content of a Fruftum of a Pyramid is found by multiplying the
Area of the two Ends, and a geometrical Mean between them by of the
Height of the Fruftum.
The Area of the two Ends of the Fruftum is found by the Rules for
Superficies, but as thoſe two Ends are fimilar Figures, and confequently
NUMERAL ARITHMETICK
113
proportional to the Squares of their homologous Lines, (Euc. Prop. xx. Fig. 26.
Cor. II. B. VI.) when one of the two Ends of this Fruftum is found,
the other may be determined by the following Proportion.
Rule for
finding the
As the Square of any Line in one Bafe, whofe Length is known, is to the folid Con
Square of the homologous Line in the other Bafe; fo is the Area of the first Bafe tent of
to the Area of the fecond Bafe.
Fruftum of
a Pyramid
When the Area of the two Baſes of the Fruftum is found, the Area of or Cone
the Baſe, which is a geometrical Mean between them, is determined by
the following Proportion.
As a Line in one of the Bafes, is to the homologous Line in the other Baje;
Jo is the Area of one of the Bafes to the Area of the geometrical Mean between
them.
CX V.
the forego
That this may appear let ABCDEF be the Fruftum of a triangular
Pyramid, from an Angle A of the upper Bafe of the Fruftum, let there Grounds of
be drawn in the lateral Faces, BD, CD of this Fruftum, two Diagonals ing Rule for
AE, AF, and let the Fruftum be cut by a Plane EAF, paffing through determining
the Solidity
thoſe Diagonals. It is manifeft that this Fruftum will be divided into of a Fruf
two Pyramids AEDF, ABCFE, the firft AEDF having for Bafe, the tum of a
Baſe EDF of the Fruftum, as alfo the fame Altitude.
Let the fecond quadrangular Pyramid ABCFE, be cut by a Plane CAE
paffing thro' AC, AE, which will divide it into two Pyramids ABCE,
ACFE; the first ABCE having for Baſe the Triangle BAC, and its Ver-
tex in E, confequently the fame Altitude as the Fruftum.
To find the Expreffion of the third Pyramid ACFE, I compare it
with the Pyramid ABCE, and obferving that the Vertices of thoſe two
Pyramids coincide in A, and that their Bafes BCE, CFE, are in the
fame Plane BCFE, I conclude that they have the fame Altitude, and
confequently that they are proportional to their triangular Bales BCE,
CFE.
will
Now the two oppofite Bafes BAC, EDF, of the Fruftum of the Py-
ramid, being parallel; BC and EF, will be parallel; whence the Trian-
gles BCE, CFE, contained between thofe two Parallels BC, EF,
have the fame Altitude, and will be proportional to their Baſes BC, EF
and confequently the Pyramids ABCE, ACFE, will be proportional to
the fame Lines BC, EF.
But instead of taking the Point A for the common Vertex, and the
two Triangles BCE, CFE for the Bafes of the two Pyramids, ABCE,
ACFE; let the Point E and the Triangle ABC, be confidered as the Ver-
tex and Baſe of the Pyramid ABCE, and let the Pyramid ACFE, be fup-
poſed to be reduced to another Pyramid of the fame Height as the Fruf-
Pyramid or
Cone.
2 P
}
114
ELEMENTS
=
OF
X
tum; the two Pyramids ABCE, ACFE having the fame Altitude, will
be proportional to their new Baſes, BAC & X: and as thofe Pyramids
have been demonftrated to be proportional to the two ftraight Lines
BC, EF, their new Bafes will be alfo proportional to the Lines.
BC, EF, or BAC: X BC: EF, but the
but the two oppofite.
Bafes EDF, BAC,
ВАС, of the Fruftum being fimilar Figures.
EDF: BAC EFXEF: B CX BC; wherefore by the Compofition
of Ratios EDF : X = EFX EF X BC : BC × BC X EF-EF: BC,.
and invertendo, X: EDF BC: EF; wherefore BAC: X= X: EDF..
From whence we may conclude that the Fruftum of a Pyramid or
Cone is equal to the Sum of three Pyramids, or of three Cones, of the
fame Altitude as the Fruftum, two of which have for Bafes the two Ends.
of the Fruftum, and the third having for Bafe a geometrical. Mean be-
tween the two Ends of the Fruftum. Confequently its folid Content is
equal to one third of the Product, arifing from the Multiplication of the.
Altitude of the Fruftum into a Bafe compofed of the two oppofite Bafes:
of this Fruftum, and a geometrical Mean between them...
2
3
CXVI.
The folid Content of a Sphere is found by multiplying the Area of its
Rule for greateſt or generating Circle by of its Diameter; or becauſe the Area of
fuch a Circle is to the Square of the Diameter, as 0,7854 to Unity;
multiply the Cube of the Diameter by ,5236 which is of 0,7584.
and the Product will be the Content of the Sphere.
finding the
folid Con
tent of a
Sphere.
Grounds of
the forego
ing Rule.
Fig. 27.
Thus if the Diameter of a Sphere be 20, the Cube thereof will be 8000,
which multiplied by the Fraction ,5236, gives 4188,8 for the Solidity
of the Sphere very nearly.
CXVII.
23
=
To demonftrate this Rule, let AB be the Axis, about which a Sphere-
and Cylinder are generated by the Rotation of a Semi-circle AGB, and
a Rectangle ADCB; let HL be any right Line perpendicular to AB,
meeting the Periphery of the Semi-circle in K, and from the Center O let
OK and OD (interfecting HL in I) be drawn becauſe AD=OA, therefore
is HI OH, and confequently HI (OH OK—HK') = HĽ—HK;
whence becauſe all Circles are as the Squares. of their Radii
(Euc. Prop. 2. B. XII.) it is evident that the Circle deſcribed by HI, or
the Section of the Cone generated by the Triangle AOD is equal to the
Difference of the Cireles defcribed by HL & HK, that is equal to the
Anulus deſcribed by KL, or the Section of the Solid, which remains
when the Sphere is taken out of the Cylinder; wherefore ſeeing the
Sections are every where equal, the Solids themſelves must be alfo equal
or the Cone EOD generated by the Triangle AOD, equal to the Ex-
NUMERAL
115
ARITHMETICK.´
}
cefs of the Cylinder, GDEg, above the Hemiſphere GAg, whence, as
the Cone or Excefs is of the Cylinder, the Hemiſphere must confequent-
ly be the other two thirds, and fo the whole Sphere equal to of its
circumfcribing Cylinder CDEF
CXVIII.
2
the folid
Content of a
The Content of a Segment of a Sphere is found by multiplying the Rule for
Square of twice the Height or Thickness of the Segment by the Ray of the determining
Sphere lefs by of the faid Height, and that Product again by 0,7854.
For to obtain the Content of a Segment BAD of a Sphere, we muft Segment of
fubtract the Cone CAD, from the fpherical Sector CABD.
3
a Sphere.
But 1º. The ſpherical Sector CABD is equal to a Cone, having the
Radius BC of the Sphere for Altitude, and the fpherical Superficies of the
Segment BAD for Bafe, confequently is equal to the Sum of two Cones, Fig. 28.
both having the Ray BC of the Sphere for Altitude, and whoſe Baſes
have for Rays the two ftraight Lines AL, BL, that is, Sector
Cir. AL X BC Cir. BLX BC
CABD =
3
+
3
2º. 'The Cone CAD to be fubtracted from the Sector, having AL for
Cir. AL x LC
Ray and LC for Altitude, this Cone CAD=
Grounds of
the forego
3
confequent-ing Rule.
Cir. ALX (BC-LC)
+
ly Sector CABD-Cone BAD or Segment BAD=
Cir. BLXB C Cir. A LXBL, Cir. B LXB C.
3
+
3
3
3.
But Cir.BL: Cir.AL-BL: AL and BL: AL-AL: LG and BL AL
BL:LG. Wherefore Cir. BL: Cir. AL-BL: LG; confequently
Cir. ALXBL Cir. BLXLG
Cir. ALXBL = Cir. BLXLG and
Hence the ſpherical Segment BAD =
But
3
3
Cir. BLXLG Cir. BLX BC
+
Cir. BLX LG Cir. BLXBC
3
+
3
3
Cir. BL x (LG + BC)
BLX (LGBC)
Cir. BLX (3 BC — BL) = Cir. B L X (B C – BL).
3
3
3
3
But Cir. BL being the Area of a Circle whofe Ray is the Thickness of
the Fruftum BAD, it is manifeft that Cir. BL X (BCBL) is the
folid Content of a Cylinder, whofe Ray is BL, and Altitude is Equal to
the Ray BC of the Sphere, lefs the one third of the Height BL of the
Segment. Wherefore, &c.
CXIX.
Having fully explained the Method of multiplying mixed Numbers, and
fhewn how Superficies and Solids are produced by the Multiplication of
116.
}
ELEMENTS
OF
}
Method of
dividing a
ber by a
fimple one.
Numbers, fubllituted for Lines, we now proceed to explain the Method
of dividing mixed Numbers, and to fhew how Superfices and Solids are
are refolved by Divifion.
The Divifor given to divide a mixed Number by, may be either a fim-
ple or compound Number. When the Divifor given is a fimple Number,
the Divifion of mixed Numbers does not differ from that of fimple
mixed Num Numbers, and is performed by dividing each Part of the mixed Number
by the given fimple Divifor, beginning the Divifion by the Parts of the
higheſt Denomination. For Example, if the mixed Number that is pro-
pofed to be divided, confifts of Pounds, Shillings, Pence, and Farthings,
the Pounds are first to be divided, then the Shillings, after adding to them
the Value of the Pounds that could not be divided; afterwards the Pence,
after adding to them the Value of the Shillings that could not be divided;
and lastly the Farthings are to be divided, after adding to them the
Pence that could not be divided.
dividing a
ber by ano
ther mixed.
When the given Divifor is a compound Number, the Computifts re-
Method of duce it to a fimple Number, by multiplying it by fuch Numbers as will
mixed Num make all the Parts, whofe Units are lefs than the principal Unit, difap-
pear; and, that the Quotient may be the fame as would refult if the
Dividend was divided by the compound Divifor, they multiply the Di-
vidend by the fame Numbers that the Divifor was multiplied by, to re-
duce it to a fimple Number; the Reafon of this Operation is, that a Di-
vidend and Divifor being multiplied by the fame Number, gives the fame
Quotient as they would give if they had not been mutiplied.
Number.
exx.
When the Divifor is an abſtract Number, the Units of the Quotient
How to dif are of the fame Species as thofe of the Dividend, becauſe the abftra&
tinguish
Divifor denotes, by the Number of its Units, that the Dividend ſhould
when the
Quotient be divided into a certain Number of equal Parts, and it is manifeſt that
fhould be the Parts of the Dividend are of the fame Species as this Dividend.
concrete
or abſtract
Number.
a
Dimenfions
of the Units
When the Divifor is a concrete Number, its Units should be always of
the fame Species as thofe of the Dividend, except the Dividend be a Num-
ber of fuperficial or folid Meaſures, for in this Cafe the Divifor may be a
concrete Number of Meafures that have one or two Dimenſions lefs
than thoſe of the Dividend.
If the Dividend and Divifor confift of the fame Species of Units,
the Quotient is always an abſtract Number, fince it ſhould expreſs how
often the Divifor is contained in the Dividend.
If the Dividend contains a Number of fquare Meaſures, and the Di-
of the Que vifor contains a Number of Meaſures that are the Sides of thoſe ſquare
tient, when Meaſures, the Quotient will be a Number of Meaſures that will be the
NUMERAL ARITHMETICK.
117
Dividend
Sides of the fame fquare Meaſures; and in general, when the Dividend Units of the
confifts of a Number of Meaſures of a certain Number of Dimenfions, and Divifor
and that the Units of the Divifor have fome of the Dimenfions of the confift of a
Units of the Dividend, the Units of the Quotient always will have the certain
Dimenſions of the Units of the Dividend, that the Units of the Divifor Dimenfions.
have not: All which will be explained in the following Examples.
CXXI.
Let it be propoſed to divide the mixed Number 38386 £. 6s. 10d.½ by the
abſtract Number 74.
Having fet down the
Divifor to the Right Hand
of the Dividend, and drawn
a Line, under which the
Figures of the Quotient
are to be fet down, accord-
ing as they are found.
1º. I divide the Part
38386 £. of the Dividend
by the given Divifor 74,
and I find 518 £. for the
Quotient, with a Remain-
der 54 £. which cannot be
divided by 74 under the
Form of Pounds.
2º. This first Divifion
being performed, I re-
duce the 54 £. remaining
into Shillings, by multi-
plying it by 20, which
will produce 1080 s. to
which adding the 6 s. in
the Dividend, there re-
38386 £. 6 s. 10 d. 1
370
138
74
646
592
54 £. 6 s. or 1086 s.
20
- 74
74
Number of
The Divifi
518 £. 14s. 8.d. 1 on of
4 Pounds,
Shillings,
and Pence,
by an ab
ftract Num
ber, explain
ed by an Ex
ample.
346
296
50s. 10d.orб10 d.
12
592
18d. £or 74 grs.
74
00
fults 1086s. to be divided by the given Divifor 74, the Quotient will be
found to be 14s. with a Remainder of 50s. that cannot be divided by 74.
3º. Having fet down this fecond Quotient 14. to the Right Hand of
518. already found, I reduce the 50s. remaining into Pence, by
multiplying it by 12, which with the 10 d. in the Dividend, will produce
610 Pence, to be divided by 74, the Quotient will be found to be 8 d.
with a Remainder 18 d. which reduced into Farthings, and adding to the
Product, the in the Dividend, there refults 74 Farthings, which di-
vided by 74 gives for the Quotient without a Remainder; whence
38386 £. 6s. 10d. divided by the abſtract Number 74, will give
for the Quotient required.
518 £. 14 s. 8 d.
I
4
118
ELEMENTS OF
CXXII.
Let it be propofed to divide the mixed Number 1267 lb. 8 oz. 5 dwt. 15 gr
by 51 4.
i
1267 lb. 8oz.
5 dwt.
15 gr.
3r. { 51 %
The Divi
fion of
Weights
5070 lb.
9oz.
2 dwt.
12 gr. S 205
by an ab
}
410
{
ftract Num
24lb.8 oz. 16 dwt. 12 gr.
A
ber explain
ed by an
970
Example.
820
150lb. 9oz. or 1809oz.
1640
12
169 az. 2 dwi ver*3382 dwt.
}
20
205
*8332
-1230
102 dwt. 12 gr. or 2460 gr.
24.
205
410
410
000
The Divifor 51 being a mixed Number, I multiply it by 4, to make
the Fraction difappear, and there refults 205 for a new Divifor qua-
druple of the one propofed; and that the Quotient may be the fame as
if the Dividend was divided by the propoſed Divifor 514, I multiply the
Dividend alfo by 4, which will produce 5070 lb. 9oz. 2 dwt. 12 gr. for a
new Dividend, and having placed the new Divifor to the right Hand of
the new Dividend.
1º. I divide the firft Part 5070 lb. by 205 and I find 24 lb. for the
Quotient, with a Remainder 150 lb. which cannot be divided by 205.
2º. I reduce the Remainder 150 lb. into Ounces, by multiplying it by
12, and adding to the Product the 9 Ounces in the Dividend, there refults
1809 Ounces for a new Dividend, which I divide by 205, and I find 8 oz.
for the Quotient, with a Remainder 169, that cannot be divided by
205.
NUMERAL ARITHMETIC K.
119
3º. Ireduce the Remainder 169 oz. into Pennyweights, by multiplying it
by 20, and adding to the Product the 2 dwt. in the Dividend, there re-
ſults 3382 dwt, for a new Dividend, which I divide by 205, and I find
16 dwt, for the Quotient, with a Remainder 102 dwt. which cannot be
divided by 205.
4°. I reduce the Remainder 102 dwt. inte Grains, by multiplying it
by 24, and adding to the Product the 12 gr. in the Dividend, there re-
fults 2460 gr. for a laſt Dividend, to be divided as the reft by 205, and I
find 12 for the Quotient, without a Remainder, whence the Quotient
required will be 24 lb. 8 oz. 16 dwt. 12gr.
CXXIII.
38515. £. 19 s.
{ 518 £.
14 s. 8 d. The Divif
115547 £. 17 s. {1556 £. 45.
{}
5:77739 £. 5 s. { 7781 £.
74
I
Let it be propoſed to divide 38515 £. 19s. by 518 £. 14 s. 8 d.
The Divifor being a mixed
Number, I multiply it by fuch-
Numbers as will make the
Shillings and Pence difappear;
to find thofe Numbers I ob-
ferve that 8 d. is the one third
of 2 s. confequently by multi-
plying the Dividend and the
Divifor by 3, there will reſult
a new Dividend 115547. 17s.
and a new Diviſor 1556 £. 4 s.
in which there are no Pence. I
obferve further that the 45.
in the new Divifor being the
one fifth of 1. by multiply-
ing the Dividend and Divifor by 5, there will refult a new Dividend
577739 £. 5 s. and a new Divifor 7781 £ in which there are neither
Shillings or Pence.
54467
33069
31124
1945 £ 5.
The Dividend and Divifor being thus prepared, I divide one by the
other. 1. Dividing 577739 £. the first Part of the prepared Dividend,
by the prepared Divifor 7781. I find the abftract Number 74 for the
Quotient, which denotes that the Divifor 7781 £. is contained 74 Times
in the Dividend 577739 £. and there remains 1945 £. 5. that cannot
be divided by 7781. but as this Remainder is precifely the one fourth
of the Divifor 7781. I fet down in the Quotient; whence 74 is the
exact Quotient required.
CXXIV.
I
When the Dividend and Divifor confift of Units of the fame Species,
as in the foregoing Example, the Diviſion may be performed after the
on of Pounds
Shillings,
and Pence,
by Pounds,
Shillings,
and Pence,
explained
by an Ex
ample.
120
OF
ELEMENTS
"
Another
Method of
dividing
following Manner: Reduce the Dividend and Divifor to the loweſt De-
nomination they contain: Thus the loweft Denomination in the
mixed Num foregoing Example being Pence, reduce the Dividend and Divifor into
bers of the Pence, by multiplying the Pounds by 240, and the Shillings by 12, and
fame Spe
then divide the Dividend reduced to one Term, by the Divifor reduced
alfo to one Term, and the Quotient will be an abftra&t Number, expref-
fing how often the Divifor is contained in the Dividend.
cies.
Divifion of
fquare
Yards and
Parts of a
fquare Yard
by lineal
Yards, and
Parts of a
lineal Yard,
explained
by an Ex.
ample.
CXXV.
Let it be propofed to divide 512rr 1rƒ 8r1 1YL by 57r 2f 81.
512rr irf 8r1 1YL {57X 2f 81
1537rr 2rf ori 3YL{173r 2f
521r
4613rr orf orI 9YL
82
4168
»rz{gr_of_of_gt
445rr orf or 13351ƒ
2f 61
f
3
1042
12
293f oFI or 3516ri
3126
390ri gri or 4689 YL
12
4689
0000
To make the 8 Inches in the Divifor diſappear, I multiply it by 3; I
likewiſe multiply the Dividend by 3, and there refults 1537YY 2Yf
OYI 3YL for a new Dividend, and 173Y 2f for a new Divifor.
To make the 2 Feet in the new Divifor diſappear, I multiply like-
wife the new Dividend and Divifor by 3, and there refults
4613YY oYf, oYI 9YL for the prepared Dividend, and 521Y for
the prepared Divifor; the Quotient of which will be the fame as that of
the propofed Dividend, divided by the propofed Divifor.
As a Number of Yards, Feet, and Inches, multiplied by a Number of
Yards, produce a Number of fquare Yards and Parts of a fquare Yard,
NUMERAL ARITHMETIC-K.
121
divided into 3, and fubdivided continually into 12 equal Parts, and that
the Extention produced by Multiplication is refolved by Diviſion
; it is
manifeft that 4613 YY oYf oYI 9YL, divided by 521Y, will give
for Quotient a mixed Number, compoſed of Yards, and Parts of a Yard.
Dividing therefore the Part 4613 YY, of the Dividend by the
red Divifor 521 Y, the Quotient will be 8Y, with a Remainder 445 YY,
which cannot be divided by the Divifor 521 Y.
prepa-
To continue on the Divifion, I reduce into Yard-Foot the 445 fquare
Yards remaining, by multiplying it by 3, which will produce 1335 Yf,
to be divided by 521 Y, and the Quotient will be 2 Feet with a Remain-
der 293 Yf; becauſe a Number of Yard-Foot, arifing from the Multipli--
cation of a Number of Feet into a Number of Yards, when divided by
a Number of Yards, will give a Number of Feet for Quotient.
To continue on the Divifion, I reduce into Yard-Inches the Re-
mainder 293 Yf, of the foregoing Divifion, by multiplying it by 12,
which will produce 3516YI, to be divided by 521 Y, and the Quotient
will be 61, with a Remainder 390YI; becaufe a Number of Yard-Inch
divided by a Number of Yards, fhould give a Number of Inches for
Quotient,
To continue on the Divifion, I reduce into Yard-Line the Remain-
der 390 YI, by multiplying it by 12, and adding to the Product the
9YL in the Dividend, there refults 4689 YL, to be divided by 521 Y,
and the Quotient will be 9L, and there being no Remainder, the
Quotient required will be 8Y 2f 61 9L.
CXXVI.
A
In the foregoing Example, the Yard being the principal Dimenſion Divifion of
of the Meaſures of the Dividend, we were under the Neceffity of fquare Feet
reducing the mixt Divifor into Yards; and we found for the Quotient a fquare
a Number compofed of Yards, Feet, Inches, &c.
and Parts of
Foot by
lineal Feet
and Parts of
Foot, ex
plained by
6off ifI 10fL { 2f 61 61 an Example
1443ff 8f1.
611
23 81
6L
If the Foot was the principal Dimenſion of the Parts of the Dividend,
the Divifor muſt be reduced into Feet, and the Quotient would be aa lineal
Number compofed of Feet, Inches, Lines, &c.
For Example, if it was pro-
poſed to divide 6off ifI 1ofL,.
by 2f 61 6L. To reduce the
Divifor to a Number of Feet,
I multiply the Dividend and Di-
vifor by 24 and there refults
1443ff 8fI to be divided by
61f, and performing the Divi-
fion, as in the Margin, I find
the Quotient to be 23f 81.
122
223
183
40ff or 488f1
12
488
000
2 a
122
ELEMENTS OF
1
CXXVII.
Let it be propofed to divide 90372rrr orrf 4rrI rYL by
22r 2f 61.
qrrı
9037½rrr_orrƒ·4rri_xrrL{ 22r 2ƒ 61.
180744rrr orrƒ 8rrI 2rYL { 45r 2f
Division of
542232rrr 2rrf orri 6rYL
cubical
411
Yards and
YL {
1372
Parts of a
3957rr 2rf 8r1 6YL
cubical
1312
Yard by li
Deal Yards
1233
and Parts
of a lineal
793
Yard, ex
plained by
685
an Example
1082
959
1
123 Irr 2rif or 371rŸƒ
3
274
97 rŸƒ orri or 1164rri
12
7
1096
68YYI 6YYL or 822YYL
12
822
'000
To make the 2 f 61 in the Divifor difappear, I multiply the Divi-
dend and Divifor fucceffively by 2 and by 6, and there refults
542232YYY 2YYf oYYI 6YYL for a prepared Dividend, and
137Y for a prepared Divifor.
As a Number of fquare Yards and Parts of a fquare Yard divided
into 3, and ſubdivided continually into 12 equal Parts, multiplied by a
Number of lineal Yards, produce a Number of cubical Yards, cubical
Feet, cubical Inches, &c. and that the Extenfion produced by Multipli-
cation, is reſolved by Divifion, it is evident that 542232YYY 2YYf
oYYI 6YYL; divided by 137Y, will give for Quotient a Number
of ſquare Yards, fquare Feet, fquare Inches, &c.
}
NUMERAL ARITHMETICK.
123
Dividing therefore the Part 542232YYY, by 137Y, the Quotient
will be 3957 YY, with a Remainder 123YYY, which I reduce into
Yard-Yard-Feet, and adding to the Product the 2YYf in the Dividend,
there refults 371YYf, which I divide by 137Y, and the Quotient will
be 2Yf, with a Remainder 97YYf. Reducing this Remainder into
Yard-Yard-Inch, and continuing on the Divifion, I find the Quotient
required to be 3957YY 2Yf 8YI, 6YL.
CXXVIII.
Let it be propofed to divide 2747rII 1fII 4III by 7211.
The Dividend and Divifor of a Divifion, being divided by the fame.
Quantity, the Quotient arifing from the Divifion of the new Divi-
dend, by the new Divifor, will be the fame as that of the firſt Divi-
dend, divided by the first Divifor. Now the propofed Dividend
2747 YII 1fII 4III, being the Product of 2747Y If 41 into II, it
may be divided by III, and confequently reduced to 2747Y f 41,
and the Divifor 7211, being the Product of III, multiplied by the Divifion
abſtract Number 72, may be alfo divided by III, and confequenrly of a Num
reduced to the abfolute Number 72; wherefore the Quotient of Yard-Inch-
the Divifion will be the fame, whether the propofed Dividend be di- Inch, Feet-
vided by the propofed Divifor, or the abridged Dividend 2747Y If 41, Inch-Inch,
be divided by the abridged Divifor 72; and as this new Dividend and Inch by a
Divifor are more fimple than the former, they ſhould be employed
the Divifion.
1º. Dividing 2747Y
by 72, there refults
38 Y for Quotient, with
a Remainder 11Y.
720
2747
216
if
If of
4f
38r of 51 8L 8'
ber of
? Inch-Inch-
in Number of
Inch-Inch,
explained
by an Ex
ample.
I
2º. I reduce the Re-
mainder Y into Feet,
by multiplying it by 3,
and adding to it the If
in the Dividend, there
refults 34f, which can-
not be divided by 72; I,
therefore, fet down of
in the Quotient, and
reduce the 34 Feet into
Inches, by multiplying
it by 12, and adding to
it the 4 Inches in the
587
576
112 if or 34ƒ or 412/
360
3
12
521 or 624L
12
5.76
48 L or 576'
-12
576
000
Dividend, there refults 412 Inches, which divided by 72, gives 51 for
Quotient, with a Remainder 521, which being reduced into Lines, and
124
ELEMENTS OF
divided by 72, gives 8L 8' for the Quotient, and there being no
Remainder left, the Quotient required will be 38 Y of 51 8L 8'.
CXXIX.
Let it be propoſed to find the Side of a Square, whoſe ſuperficial Content
is 24rr orf 2rI 3rL.
Extraction 1º. I extra&
oftheSquare
Root of the Square Root
compound of 24ŸY, and I
Numbers, find 4Y for the
explained by
an Example first Part of the
Root, with a Re-
mainder 8 YY.
24rr orf 6YI 3YL
16
8
5
42
2f 91
8r 2f
24
2rr 2rf 8r1
4
2
2
8
2 2
2rr irf 2rI
$6
2rr irƒ 2rI 3YL
O
O
5rr 2rf 4r1
gr If 91
O
2rr ilƒ 2ri 3rL
2º. To find the
Number of Feet
of the Root, I re-
duce this 8YY
into Yard-Feet,
and adding to them
the Yf in the propofed Number, there refults 24 Yf, which I divide by
Double of 4 Y, or by 8 Y, and fet down the Quotient in the Root, and to
the right Hand of 8 Y; cancelling the 24 Yf that I have no further Occa-
fion for, then multiplying 8Y 2f by 2f, and the Product 5 YY 2Yf 4YI
fubtracted from 8YY oYf 6YI, leaves 2YY 1Yf 2YI for a Remain-
der. Continuing on the Operation, as in the Margin, I find 4Y 2f 9I
for the Side required of the Square.
CXXX.
Let it be propoſed to find the Side of a Cube, whofe folid Content is
118rrr 2rrƒ˜ árri årrl 9rr'.
1º. I extract
او
4r 2f ol
the Cube Root
118rrr 2rrƒ 6rYI 8rYL 91r'
of 118 YYY,
48rr
64
and I find sŸ
65rr irf
Extraction for the first Part
54
164
of the Cube of the Root,
Root of
compound with a Remain- 101rry Irrf 10rr 81YL
Numbers, der 54YYY.
2º. To find
explained by
an Example
Feet of the
the Number of
17rrr orrf grrl orrl 9rr!
620
Root, I reduce 118rrr 2rrf 6rrl 8rrl 9rr'
this Remain-
der into Yard-
О
О
О
О
O
Yard-Feet, and continuing on the Operation, as in the Margin, I find
4Y 2f 9I for the Side required of the Cube.
NUMERAL ARITHMETICK.
125
CHA P. III.
Of Proportion, and the principal Rules of Arithmetick which depend thereon.
THE
I.
HE Compariſon of one Quantity with another, is called What is
Ratio.
meant by a
Ratio
Difference
When two Quantities are compared, if it be confidered how much
the one is greater than the other, and what is their Difference; this
Difference is called their Arithmetical Ratio. For Example, comparing
12 with 3, if it be confidered how much 12 is greater than 3, or that
3 is exceeded by 12 by 9 Units; thofe 9 Units which is the Difference
between 12 and 3, or 3 and 12, is the Arithmetical Ratio of 12 to 3. between an
But when two Quantities are compared, it be confidered how many arithmetical
Times the one is contained in the other, this Number of Times is called Ratio and
their geometrical Ratio. For Example, comparing 12 with 3, if it be a geometri
confidered that 12 contains 4 Times 3, or that 3 is contained 4 Times
in 12, this Number 4 is called the geometrical Ratio of 12 to 3.
From theſe two Definitions of Ratios, it follows that only Quantities
of the fame Kind can have a Ratio to one another.
cal Ratio.
1º. The Arithmetical Ratio being the Difference of two Quantities, Quantities
this Ratio or Difference cannot be obtained, but by fubtracting the of the fame
leffer from the greater; whence the leffer Quantity fhould be a Part of
the greater, and confequently of the fame Kind with it.
Kind can
only have
a Ratio to
2º. The geometrical Ratio of two Quantities, being the Number of one another
Times that one contains the other, fuppofes manifeftly that the leffer is a
Part of the greater, and confequently of the fame Species with it.
II.
metical Ra
From what precedes it is alfo eafy to conclude, that the Arithmetical The arith
Ratio of two Quantities is a Quantity of the fame Species with thofe tio is of the
which are compared; becauſe the arithmetical Ratio being the Differ- fame Spe
ence of the two Quantities compared, or the Excefs of the greatest the Quanti
above the leaſt, is neceffarily a Part of the greateft, and is confequently ties compa-
of the fame Kind with it.
cies with
red.
The geome
It is not fo with refpe& to the geometrical Ratio. This Ratio
is always an abſtract Number, becauſe it repreſents a Number of Times, trical Ratio
that is, the Number of Times, that one of the two Magnitudes compar- is always
ed, contains the other.
III.
Since the geometrical Ratio of two Quantities is the Number of Times
that one contains the other; and that this Number of Times is found
an abftra&
Number.
126
OF
ELEMENTS
trical Ratio
The geome by dividing one by the other; it is manifeft that the geometrical Ratio
is expreffed of two Quantities, is the Quotient arifing from the Divifion of one of
the Quo the Quantities by the other. For Example, the Ratio fubfifting between
tient of the 12 and 3, is the Quotient of the Divifion of 12 by 3.
Divifion of
the Terms
When two Quantities are compared, for Example, 12 and 3, the
of the Ratio Quantity 12 expreffed or fet down the firſt, is called the Antecedent, and
one by the and the other 3 is called the Confequent; if 3 was compared with 12,
the Number 3 exprefled the firft, is the Antecedent of the Ratio, and
the other Number 12 is the Confequent.
other.
Becauſe the geometrical Ratio of two Quantities is the Quotient of
the Diviſion of one by the other, the two Terms of a Ratio may be fet
down after the Manner of a Fraction, that is, the Antecedent may be
fet down above a fmall Line, and the Confequent under it, for Example,
I 2
39
the Ratio of 12 to 3, may be fet down thus, which denotes 12 divided
by 3, or rather the Quotient of 12 divided by 3, and the Ratio of 3 to
12 is fet down thus which denotes the Quotient of 3 divided by 12.
12
IV.
Two equal Ratios for Example. The Ratio of 2 to 3, and that of
4 to 6, form a geometrical Proportion. Whence a geometrical Proportion
confifts of 4 Terms; the first of which contains the Second as many
Two equal
Ratios form Times as the Third contains the Fourth; or of four Terms, of which
a Propor- the First is contained in the Second, as often as the Third is contained
tion.
in the Fourth.
finding the
fourth
2
To reprefent a geometrical Proportion, for Example, that, compoſed
of the two equal Ratios and, the Computiſts ſet it down thus,
2:34:6; which denotes that 2 is to 3 as 4 is to 6, or that 2 is con-
tained in 3 as 4 is contained in 6.
3
The firft and fourth Terms of a geometrical Proportion, are called
the Extreams, and the fecond and third are called the Mean Terms.
V.
The fourth Term of a geometrical Proportion is found by multiply-
Method of ing the third Term by the Quotient of the fecond divided by the firft.
To make this appear, we fhall diftinguith two Cafes; either the first
Term is contained in the fecond, or the fecond is contained in the firſt.
In the firft Cafe, it is manifeft that the fourth Term of a geometrical
Proportion will be found by multiplying the third Term by the Number
of Times it is contained in the fourth; but from the Nature of Propor-
tion, the third Term is contained in the fourth as many Times as the
Term of a
Proportion.
NUMERAL ARITHMETICK.
[27
first is contained in the fecond, and this Number of Times that the firſt
Term is contained in the fecond, is equal to the Quotient of the fecond
Term divided by the firft; wherefore the fourth Term of a geometrical
Proportion is found, by multiplying its third Term by the Quotient of
the ſecond Term divided by the firſt.
For Example, if a geometrical Proportion begins by thofe three
Terms 2:37, and that the fourth Term is required. I divide the
fecond Term 3 by the first Term 2; and there reſults for Quotient the
Fraction, which expreffes the Number of Times that the firit Term
2 is contained in the fecond 3, or that the third Term 7 is contained in
the fourth fought. Whence, by multiplying 7 by, the Product
21 or 10 will be the fourth Term required, and the whole Proportion
will be 2:3 = 7:10 1/1.
I
I
Grounds of
In the fecond Cafe it is manifeft that the fourth Term of a Proportion
will be found, by dividing the third Term by the Number of Times that
the fourth is contained in it. But from the Nature of Proportion, the
fourth Term is contained in the third as many Times as the fecond is
contained in the firft; and this Number of Times is equal to the Quo-
tient of the Divifion of the first Term by the fecond. Wherefore the
fourth Term of a geometrical Proportion will be obtained, by divid- this Methot
ing the third Term by the Quotient arifing from the Divifion of the first
Term by the ſecond, that is by a Fraction, having the firft Term for
Numerator, and the fecond for Denominator. But to divide by a Frac-
tion, having for Numerator the firft Term, and for Denominator the
fecond Term, is to multiply it by the converfe Fraction, having for
Numerator the fecond Term, and for Denominator the first, and which
is confequently the Quotient arifing from the Divifion of the fecond
Term by the firft. Wherefore the fourth Term of a geometrical Pro-
portion will be obtained by multiplying its third Term by the Quotient
of the ſecond Term divided by the first, as in the firſt Cafe.
For Example, if a geometrical Proportion begins by thofe three
Terms 12:8 20:, and the fourth Term be required, I divide the firſt
Term 12 by the fecond 8; and there refults for Quotient the Frac-
tion, which will exprefs the Number of Times that the fecond
Term 8 is contained in the first 12, or that the fourth Term fought is
contained in the third 20. Whence, by dividing 20 by 2, the fourth
Term required will be obtained. But to divide 20 by the Fraction
is to multiply 20 by the converfe Fraction. Wherefore, the fourth
Term of a Proportion, whoſe three first Terms are 12:820:, will be
found, by multiplying the third Term 20 by the Fraction, which is
the Quotient of the ſecond Term divided by the first; and this fourth
Term being 13, the whole Proportion will be 12: 8 = 20: 13 3.
11
128
ELEMENTS OF
VI.
To multiply a Number by a Fraction, is to multiply by the Nume-
The fourth rator of the Fraction, and divide the Product by the Denominator of
Term of a the fame Fraction; fince therefore the fourth Term of a Proportion.
Proportion
found by is found, by multiplying its third Term by a Fraction, having the
multiplying the fecond Term for Numerator, and the firft for Denominator; this
and fecond fourth Term will be obtained, by multiplying the third by the fecond,
together, and by dividing the Product by the firft Term; that is, the fourth
and dividing Term will be equal to the Product of the mean Terms of the Propor-
by the firſt tion divided by the firſt Term.
the third
the Produc
Term.
The Tranf-
the mean
Terms of a
Proportion
do not alter
For Example, if a geometrical Proportion begins by thofe three
Terms 2:37: and the fourth Term be required; the third Term 7
must be multiplied by the Fraction; that is 7 must be multiplied by 3,
and the Produ& divided by 2, which will give 2 or 10 for the fourth
Term required, whence the Proportion will be 2:37: 101
VII.
If there be three Terms of a geometrical Proportion given, for Ex-
pofition of ample, thoſe three Numbers 2: 37, the third 7 may be put in the
Place of the fecond 3, and the fecond 3 in the Place of the third 7,
thus 2: 7
3, without altering the fourth Term required.
For the fourth Term is equal to the Product of the Means,
divided by the first, and in thoſe two Arrangements 2:37:
and 2: 73, the Means being the fame, as alfo the firſt Term, the
Quotient of the Divifion of the Product of the Means by the firſt Term
will be the fame.
the fourth
Term.
ometrical
VIII.
Since the fourth Term of a geometrical Proportion, for Example, of
In every ge the following 2: 34: 6, is found by multiplying the third Term by
Proportion the fecond, and dividing the Product by the firft; it follows that the
the Product Product of the Extreams of a geometrical Proportion is equal to the
Product of the Means of the fame Proportion, that is 6X2 and 4X3 are
equal to the equal Products, as will appear when it is obferved, that the fourth
Product of Term confidered as the Product of the Means divided by the firſt Term,
of the
Means is
the Ex
tremes.
being multiplied by the firft Term, will produce the Product of the
Means; becauſe the Diviſion of the Product of the Means by the firſt
Term is deſtroyed by the Multiplication by the firſt Term. Wherefore
the Product of the Extreams of a geometrical Proportion is equal to the
Produ&t of the Means of the fame Proportion.
IX.
The Product of the Extreams of a geometrical Proportion being equal
to the Product of the Means, if thofe two Produ&s be divided by an
extreme or by a mean Term, it will appear that each extream Term of
NUMERAL
129
ARITHMETICK.
Terms of a
a Proportion is equal to the Product of the Means divided by the other Any three
Extream, and that each mean- Term is equal to the Product of the Ex- Proportion
treams divided by the other Mean. Wherefore when three Terms of a being given,
geometrical Proportion are given, and the Order in which they are dif- how to find
poſed in the Proportion is known, the Term which is wanting in this that is want
Proportion may be found.
ing.
X.
the Term
When three Terms of a geometrical Proportion are given; the Ope- What is
ration to be performed to find the Term that is wanting in this Propor- meant by
tion, is called the Rule of Three; it is alfo called the Rule of Proportion, the Rule of
Three, or
an I by fome Computifts The Golden Rule, on account of its great Utility Golden
in Trade.
Having fhewn how the Term which is wanting in a Proportion of
which three Terms are given may be difcovered; we ſhall now proceed
to give ſeveral Examples of this Operation, and to explain how the giv-
en Terms are to be confidered.
XI.
Rule.
The Computiſts diftinguish two Species of Rules of Three, the Rule of The diffe
Three direct, and the Rule of Three inverfe, which are both either Simple rent Species
of Compound. Whence there are four Species of Rules of Three; the of Rules of
Simple Rule of Three direct, and the Simple Rule of Three inverſe, the
Compound Rule of Three direct, and the Compound Rule of Three inverse.
Three.
The fimple Rule of Three direct, is that whofe three given Terms The Simple
are the three firft Terms of a geometrical Proportion. The Object Rule of
therefore of this Rule is to find the fourth Term of a geometrical Pro- Three Di
portion, of which the three first Terms are given.
rect.
verfe.
The fimple Rule of Three inverſe, is that in which three Terms are The Simple
given, of which two are the Extreams of a geometrical Proportion, and Rule of
the other a mean Term of the fame Proportion; fo that the Object of this Three In
Rule is to find a mean Term of a Proportion of which three Terms are
given. But becauſe the Computifts do not put the unknown Term
fought in its Place, but fet down the three given Numbers one after the
other, as if thoſe three Numbers were the three firſt Terms of a Propor-
tion; the last Ratio of the Proportion is inverted, when the unknown
Term required is really a mean Term of the Proportion; and it is for
this Reaſon that the Operation to be performed to find the mean Term
fought, is called Rule of Three inverfe.
The Com
The Compound Rule of Three is that in which more than three pound Rule
Terms are given. But all thofe given Terms may be always reduced of Three
to three, and the Term fought is always the fourth Term, or a Factor of
Direct and
Inverfe
2 R
130
OF
ELEMENTS
7
The Simple
Rule of
Three Di
rect explain
ed by an
Example.
Of the three-
Terms
given in a
Rule of
Three the
the fourth Term of a Proportion, when the Rule is dire&t, and is al-
ways a mean Term, or a Factor of a mean Term, when the Rule is inverte.
XII.
It has been faid that the Object cf a Simple Rule of Three direct, is to
find the fourth Term of a geometrical Proportion of which the three
firſt Terms are given. For Example, in this Queſtion,
If 37 Pieces of Cloth coft 148 £. what will 30 Pieces of the fame Cloth be
worth?
It is manifeft that the Price of 30 Pieces of Cloth, which is the
Anſwer to this Queftion, is the fourth Term of a geometrical Proportion,
of which 37 Pieces, 30 Pieces and 148. are the three firft Terms; be-
cauſe it is manifeſt that 37 Pieces of Cloth fhould contain 30 Pieces of
the fame Cloth, as the Price 148 £. of 37 Pieces, contains the Price re-
quired of 30 Pieces:
The like may be ſaid of this other Queſtion, which is the Converſe of
the foregoing one.
If 148 £. will buy 37 Pieces of Cloth, how many Pieces of Cloth may I buy.
for 120 £?
It is manifeft that the Number of Pieces of Cloth, which is the
Anfwer to this Queſtion, is the fourth Term of a geometrical Proportion,
of which the three firft Terms are 148£. 120£. and 37 Pieces of Cloth;
becauſe the two Numbers of Pieces of the fame Cloth fhould be propor-
tional to the two Sums 148 £. and 120 . which is to be paid for them,
that is,
148 £. to be paid for 37 Pieces of Cloth, is to 120 £. to be paid for the
Number fought of Pieces of the fame Cloth; as 37 Pieces of Cloth, to the
Number of Pieces required of the fame Cloth.
XIII.
To perform thofe Rules of Three, that is, to difcover the fourth
Terms of thofe two Proportions, whoſe three firſt Terms are given,
we have feen, (Art vI.) that the fecond Term must be multiplied
by the Third, or the Third by the Second, and that Produ&t divided by
the firft Term; but here there occurs a Difficulty which however is
eafily folved.
In the Operation to be performed to find the fourth Terms of the
Proportions propofed for Examples, we find concrete Numbers to be
multiplied one by the other, which is contrary to the Rules of Multipli-
cation, it having been proved that the Multiplicator is always an abſtract
two which Number.
are of the
But if it be confidered that the two firft Terms whoſe Units are of
cies fhould the fame Species, affect the fourth Term, but by the Number of Times
fame Spe
NUMERAL ARITHMETICK.
131
ed as ab
that one contains the other, fince to obtain the fourth Term, it fuffices be confider
to multiply the third by the Quotient of the fecond, divided by the first, ftract Num
and that this Quotient is an abftra&t Number fimilar to that refulting bers.
from the Diviſion of an abftra&t Number by another abſtract Number,
it will appear, that thofe two concrete Terms may be confidered as ab-
fra& Numbers.
In Effect, fince 37 Pieces contains 30 Pieces after the fame Manner
that 37 abſtract Units contains 30 abftra&t Units, and that 148. con-
tains 120. after the fame Manner that 148 abftra&t Units contains 1 20
abſtract Units; it is manifeft that the two Proportions propofed, as Ex-
amples may be reduced to the two following, whofe two firft Terms
are abstract Numbers.
37 is to 30 as 148 £. to a fourth Term required,
And 148 is to 120 as 37 Pieces of Cloth is to a fourth Term required.
whence of the three Terms given in a Rule of Three, the two which
are of the fame Species fhould be conſidered as abftra&t Numbers.
XIV.
The fourth
is of the
With Refpect to the Units of the fourth Term which is the one re- Term re
quired; it is manifeft that thofe Units are of the fame Species with thofe quired of a
of the mean Term which is not confidered as an abſtract Number. For Proportion.
in order to find the fourth Term, the concrete mean Term must be mul- fameSpecies
tiplied by the other abftra&t mean Term, and that Product divided by the with that
firſt, which is an abftra& Number, and it has been proved that the Na. which is not
ture of the Units of a concrete Number is not altered whether it be mul- as an ab
tiplied or divided by an abſtract Number.
From what has been faid it appears, that if the three firft Terms of a
Rule of Three be abftra&t Numbers, the laſt Term required will be alfo
an abftra& Number.
XV.
In order that the two firft Terms of each Proportion might be of the
fame Species, and form a geometrical Ratio, we were under the Necef-
fity of inverting the Order of the fecond and third Terms of the Rules of
Three propofed as Examples. But it is manifeft that this Inverfion
does not alter the Value of the fourth Term required, (Art. vII.) as it
is therefore unneceffary for difcovering the fourth Term, the Com-
putifts fet down the Terms of the Rules of Three, in the Order they are
expreffed, even when the two firft Terms are not of the fame Species,
and in this Cafe the firſt and third Terms, which are of the fame Species,
are confidered as abftra& Numbers.
ftract Num
ber.
"
132
ELEMENTS OF
$
Firft Exam
XVI.
If 37 Pieces of Cloth coft 148 1. what will 30 Pieces of Cloth be worth?
To perform this Rule of Three, I confider the
ple of the first and third Terms as abftra& Numbers, that is
as if the Queftion had been propofed thus:
Simple Rule
of Three
Direct.
If 37 cost 148 £. what will 30 be worth?
I multiply therefore 148. by 30, which pro-
duces 4440. which I divide by the first Term
37, and there refults 120 £. for the Quotient, or
for the Value required of the 30 Pieces ofCloth.
XVII.
1 4 8 £.
30
37
4 4 4 0 %. {3206.
37
74
74
ОО
I
If 1481. will buy 37 Pieces of Cloth, how many Pieces of the Jame Cloth
will 120 1. buy?
I confider the firft and third Terms,
which are of the fame Species, as abſtract
Numbers, and as if the Queſtion was pro-
pofed thus:
If 148 Units will buy 37 Pieces of Cloth,
bow many Pieces of the fame Cloth will 120
Units buy?
Second Ex
ample of
the Simple
I multiply the fecond Term 37 Pieces of
Rule of
Three Di- Cloth, by the third 120, which will pro-
duce 4440 Pieces of Cloth; and I divide
rect..
Third Ex
37
37 Pieces
I 20
740
148
4440 Pieces
444
30 Pieces
О
this Product by the first Term 148, which gives 30 Pieces of Cloth,
for the Anſwer to the propofed Queſtion.
XVIII.
201
30 -/-
65. 9d.
If 71. 135. 4d. gained 201. 6s. 9d what will 30l. 135. 4d. gain?
I confider the Pounds of the first and
third Terms as abftract Numbers, and
ample of the their Shillings and Pence as Fractions of
Simple Rule abftra&t Units; fo that 135. 4d. being
the two-thirds of a Pound, I operate as
if the Queſtion had been propoſed thus,
If 7 gain 201. 6s. 9d. what will 303
gain?
of Three
Direct.
The Queſtion being thus reduced to
more fimple Terms, I multiply the fe-
cond Term 201. 6s. 9d. by the third
30, and there refults 623. 135. 8d.
for the Product, which I divide by the
first Term 7, and I find 817. 75. for
the Quotient, that is, for the Anſwer to
the propofed Queſtion.
6231. 135. 8d. } 73
23
18717. IS.
184
2817. 75.
31
23
84. 15. or 161 J.
20
161
NUMERAL ARITHMETICK.
133
XIX,
fes the first
It often happens that the Shillings and Pence of the firft and third la what Ca-
Terms are not easily reduced into Fractions; in this Cafe, it will be and third
more ready to reduce thofe Terms into Pence; for Example, in the Terms of 2
propofed Question I might have reduced the first Term 7. 135. 4 d. Rule of
and the third 30. 13. 4 d. into Pence, whereby the two Terms ſhould be
will be transformed into 1840 d. and 7360 d. and the propofed Queſtion reduced to
i to the following:
Three
the loweſt
Denomina
tion they
If 1840 Pence gained 201. 6 s. 9 d. what will 7360 Pence gain?
Confidering the first and third Terms in this Queſtion as abftra& contain.
Numbers, I multiply the fecond 20l. 6s. 9d. by the third 7360, which
produces 1496841. and divide this Product by the first Term 1840, and
I find, as before, 817. 7. for the Quotient, and for the Anſwer to the
propoſed Queſtion.
The three firft Terms of the Proportion might have been reduced
into Pence, and after the fourth Term was found in Pence, it might
have been reduced into Pounds and Shillings; but this Method of ope-
rating occafions two ufelefs Reductions.
XX.
Three be a
concrete
Since to find the fourth Term of a Rule of Three, the fecond Term If the fift
muſt be multiplied by the third, and their Product divided by the firft, Term of a
and that a Number divided by Unity gives a Quotient equal to this Rule of
Number, it is manifeft, that if Unity be the firft Term of a Rule of
Three, the fourth Term required will be found by multiplying the fe- Unit it is
cond Term by the third, or the third by the fecond: However, it is to not to be
be obferved, that when the firft Term is a concrete Unit, it is not to as ufelds.
fuppreffed
be fuppreffed as uſeleſs; becauſe the concrete Unit of the firſt Term
ferves to determine the Nature of the Units of the Product of the two
other Terms, by indicating which of the mean Terms of the Propor-
tion fhould be confidered as an abſtract Number, viz. that which is of
the fame Species with it. For Example, let the two following Rules of
Three be propofed.
If Yard of Silk coft 41. 10s. 6d. what will 27 Yards coft at that
Rate?
If I I. will buy 27 Yards of Galloon, bow many Yards will 41. 10s. 6d. buy?
Unity being the firft Term, and the two other given Terms being the
fame in thoſe two Rules of Three, viz. 27 Yards, and 4/. 10s. 6d.
the fourth Terms of each of thofe Proportions will be found by multi-
plying 27 Yards by 47. 10s. 6d. but the fourth Terms of thofe Rules of
Three, tho' produced by the Multiplication of the fame Terms, do not
confift of the fame Units; becauſe the first and third concrete Terms
Example.
134
OF
ELEMENTS
How a Rule
of Three In
formed.
气
​of the first Rule, confifting of Yards, and the fame Terms of the fe-
cond Rule confifting of Pounds, they are rendered abftra&t Numbers by
fuppreffing the Denomination of Yards in the first Rule, and the Deno-
mination of Pounds in the ſecond. In the first Rule therefore the Pro-
duct of 27 Yards into 47. 10s. 6d. is reduced to 4. 10s. 6d. repeat-
ed 27 Times, which gives 1221. 35. 6d. for the fourth Term of
this Rule; but in the fecond, the Product of 27 Yards into 41. 10s. 6d.
is reduced to 27 Yards repeated a Number of Times, expreſſed by
4 and, or by 4 23, which gives 122 Yards for the fourth Term
of this fecond Rule. Wherefore, when Unity is the firft Term of a
Rule of Three, and that confequently the fourth Term is found by mul-
tiplying the ſecond by the third, the Unity that conftitutes the firſt Term
is not to be fuppreffed as ufelefs, fince it is the Species of this Unit
which determines that of the fourth Term.
7
丁
​Whence if it be propofed to multiply a concrete Number by another
concrete Number; for Example, 27Y by 4. 10s. 6d. Since from the
Multiplication of thoſe two Numbers there refults Products confifting of
Units of different Species, as appears from the foregoing Rules of Three;
it is manifeſt that the Units of the Product of a fimilar Multiplication
cannot be determined, and confequently the Multiplicand and Multipli-
cator of a Multiplication cannot be both concrete Numbers.
XXI.
It has been faid that a Rule of Three is inverſe, when of the three
verfe is per given Terms there are two that are the Extremes of a Proportion, fo
that the Term required is a mean Term of the fame Proportion. And
as the Terms of the Rule of Three are expreffed one after the other; in
order to obtain this mean Term, the first Terin muſt be multiplied by
the ſecond which in Reality is the fourth Term of the Proportion, and
that Product divided by the third Term, as will appear by the following
Example.
Example.
XXII.
If 30 Men can do a Piece of Work in 40 Days, bow many Days muſt 10
Men require to do the Jame Work?
As 30 Men and 10 Men are to do the fame Work, fo much more
Time is required as the Number of Men is lefs; that is, a Number of
Men twice lefs, would require twice as much Time; a Number of Men
three times lefs, would require three times more Time, &c. wherefore
30, which is the first Number of Men, will be to the fecond 10, as the
unknown Time, required by the fecond Number of Men, will be to
the Time, 40 Days, employed by the firft Number of Men.
The Proportion being thus expreffed, the Time required will be the
third Term; the Value therefore of this Term will be obtained by mul-
NUMERAL ARITHMETICK.
135
tiplying into one another the Extremes 30 and 40 Days, and dividing
the Product by the mean Term 10 which is given, that is, the Time re-
quired will be 30 X 40 Days
ΙΟ
>
or 120 Days.
But the two Terms 30 and 40 Days that have been multiplied, are
the firſt and ſecond Terms of the propofed Rule of Three Inverſe, whoſe
three Terms, 30 Men, 40 Days, 10 Men, are fet down one after the
other; wherefore the unknown Term of a Rule of Three Inverfe is
found by multiplying the firſt and ſecond Terms together, and dividing
the Product by the third Term.
General
Rule of
The foregoing Queſtion, and all others of the fame Kind, may be fo
ſtated, that its Terms will be the three firft Terms of a geometrical Rule for
Proportion, whence we are difpenfed from confidering two Species of performing
Rules of Three. The Rule for placing the Terms is as follows; firft a fingle
fet down the Quantity that is of the fame Kind with the Quantity Three, ci-
fought, then confider, from the Nature of the Queftion, whether that ther direct
which is given is greater or lefs than that which is fought; if it is or inverfe.
greater, then place the greatest of the other two Quantities on the left
Hand; but if it is lefs, place the least of the other two Quantities on
the left Hand, and the other on the right: Then ſhall the Terms be
in due Order, as will appear by the following Example.
XXIII.
If a Penny white Loaf ought to weigh S Ounces, Troy Weight, when
Wheat is fold for 6s. 6d. the Bushel, what muſt it weigh when Wheat is fold Example,
for 4s. the Bufbel &
Becauſe it is a Number of Ounces that is fought,
I firſt fet down 8, the Number of Ounces that are
given; I easily fee that the Number that is given
is lefs than the Number that is fought, therefore I
place 4.on the left Hand, and 6 on the right, and
6X8
fay 4: 8 oz. =6/12:
or 13 oz. confequently
4
6 1/2 X 8
4
, or 13 oz. is the Anfwer required to the
propofed Queſtion.
XXIV.
4:80%. = 6 1/
16 /1/20
48
4
4
52) 13.02.
4
12
12
A compound Rule of Three is when the Queftion propoſed contains “
more than three known Quantities; and though it confifts of more than
three Terms, it is ſtill called Rule of Three, becauſe it may be reduced
to a Rule of three Terms refulting from the Multiplication of thoſe ex-
preffed in the Queſtion.
136
ELEMENTS OF
How a com
To reduce a compound Rule of Three to three Terms, the Com-
pound Rule
of Three is putifts confider it as confifting of two Cauſes, and of two Effects; fet
reduced to down under one another all the Terms which relate to the first Caufe;
a fimple
Rule of
Three.
ple.
they alſo fet down under one another all thofe that compofe the firſt
Effect; they difpofe in like Manner the Terms which belong to the fe-
cond Caufe, as alfo thofe which compofe the fecond Effect: Obferving
to diſpoſe alternatively the Caufes and their Effects, and to begin by the
firſt Cauſe, or by the firſt Effect, according as the Term fought belongs
to the fecond Effect, or to the fecond Caufe. They afterwards multi-
ply together all the Quantities that compofe each Term, whence re-
fults a fimple Rule of Three, whoſe three firſt Terms are known Quan-
tities, fo that the fourth Term may be found (Art. XIII.)
But after the Terms of the compound Rule of Three are thus redu-
ced, there may happen two Cafes; either the Quantity fought will be
the fourth Term of the Proportion, or only a Factor of this fourth Term.
In the firſt Cafe, the Refolution of the Rule of Three (Art. x111.) will
manifeftly give the Quantity required. In the fecond Cafe, when the
fourth Term is found, it muſt be divided by the given Factors that enter
into its Compofition, to obtain the Quantity required. All which will
be explained in the following Examples.
XXV.
If 20 Perſons ſpend 991. in 15 Days, how much will 60 Perfons pend
in 25 Days?
In this Queſtion there are five Terms given to find a fixth Term,
The forego which is the unknown Expence of 60 Perfons in 25 Days. Of thofe fix
ing Method Terms, there are two (20 Perfons and 15 Days) which are the Cauſe of a
explained
first Effect, or of the first Expence 997. and two others (60 Perfons and
by an Exam
25 Days) which are the Caufe of a fecond Effect, or of a fecond un-
known Expence. I therefore fet down under one another, for a firſt
Term, 20 Perſons and 15 Days which compoſe the firſt Cauſe; I then
fet down for a ſecond Term the first Effect 99l. I afterwards fet down
under one another, for a third Term, the two Quantities 60 Perfons
and 25 Days which compofe the fecond Cauſe, and the fourth Term
will be the unknown Expence of 60 Perfons in 25 Days.
2d Effect.
Ift Caule
20 Perlons
15 Days
ift Effect.
2d Caufe.
} 991.
bo Perions
25 Days
} Unknown.
300
B 495!.
991. = 1500
As the Caufes are proportional to their Effects, thofe four Terms, of
which the laſt is unknown, compoſe a geometrical Proportion; whence
NUMERAL
[37
ARITHMETICK.
the fourth Term unknown, viz. the Expence of 60 Perfons in 25 Days,
will be found by multiplying the fecond Term by the third, and divi-
ing the Product by the firft.
But before this Rule of Three can be performed, we muſt find what
the firſt and third Terms of the Proportion are reduced to. To this
End I obſerve that 20 Perfons will ſpend in 15 Days as much as 15 times
20 Perſons in a Day; and that 60 Perfons will fpend in 25 Days as
much as 25 times 60 Perfons in a Day: So that the first and third
Terms are reduced to thoſe two Products, 15 times 20 Perfons, and 15
times 60 Perfons; wherefore the propofed Queſtion will be folved by
this Rule of Three.
If 15 times 20 Perfons ſpend 991. how much will 25 times 60 Perſons
Spend?
The Rule being thus reduced, the fourth Term required will be found
to be 495. that is, the Expence of 60 Perſons in 25 Days.
XXVI.
If 20 Perfons Spend 991. in 15 Days, in how many Days will 60 Perfons
Spend 4951.
Second Ex
ample of
the com
In this Example, the Cauſe of the firft Expence 997. is compofed of
20 Perfons and of 15 Days, and the Caufe of the fecond Expence 495/
is compoſed of 60 Perfons and of a certain unknown Number of Days. pound Rule
The unknown Number of Days fought being in the Caufe of the fe- of Three
cond Expence, this Number of Days cannot be found in the fourth
Term of the Proportion, except the Caufe of the Expence be fet down
after the Effect; whence I fet down the given Terms of the Queſtion,
as follows:
2 Caufe.
Unknown Number
Ist Effect.
991.
Ift Caufe.
20 Perfons.
15 Days.
2d Effect.
60 Perfons.
4951
99
:
35 Days
495
of Days.
4th Term.
As the two Quantities which produce each compound Term ſhould
be multiplied one by the other, and that the fourth Term is compound-
ed of 60 Perfons, and of the Number of Days required; this fourth
Term muſt be inveſtigated, and, when found, divided by 60, to obtain
the unknown Number of Days required.
To diſcover this fourth Term, I confider all the other Terms, except
15 Days, as abſtract Numbers; and after having multiplied 15 Days
by 20, which produces 300 Days for the fecond Term, I multiply this
direct,
2 S
138
OF
ELEMENTS
Third Ex
fecond Term 300 Days by the third 495, and there refults 148500 Days
for the Product. In fine, this Product being divided by the first Term
99, I find 1500 Days for the Quotient, and for, the fourth Term of the
Proportion; and as this Number 1500 Days is made up of the required
Number of Days multiplied by 60, I divide 1500 Days by 60, and there
reſults 25 Days for the Number of Days required.
XXVII.
If 60 Men, working 8 Hours a Day, can dig in 12 Days a Ditch 60 Yards
ample of long, 5 Feet broad, and 7 Feet deep; what will be the Length of a Ditch 4
Feet broad and 6 Feet deep, that 50 Men will dig in 15 Days in the fame
Pound Rule Soil, working 6 Hours a Day?
the com
of Three di
rect.
1
1º. It is manifeft that 60 Men, 12 Days during which they work,
and 8 Hours they are employed each Day, compofe by their Multiplica-
tion the Cauſe of the firſt Ditch; becauſe 60 Men will do 12 times
more Work in 12 Days than they would do in one Day, 60 Men ſhould
be multiplied by 12; and as they would perform 8 times more Work
by working 8 Hours a Day, than they would perform by working
1 Hour a Day; the first Product of 60 Men multiplied by 12 fhould be
again multiplied by 8.
2. The Ditch 60 Yards long, 5 Feet broad, and 7 Feet deep, being
confidered as a Parallelepiped, is an Effect compounded of thoſe three
Dimenſions, 60 Yards, 5 Feet, and 7 Feet.
3º. Fifty Men, 15 Days during which they work, and 6 Hours they
are employed each Day, compofe by their Multiplication the Caufe of
the fecond Ditch.
4º. In fine, the fecond Ditch, compofed of the Multiplication of its
unknown Length by its Breadth and by its Depth, is the Effect of the
fecond Caufe.
I therefore difpofe under one another the Factors of each Caufe; I
range likewife under one another the Factors of each Effect, as follows.
Ift. Cauſe.
60 Men
12 Days
8 Hours
5760
Ist. Effect.
2d. Caufe.
60 Y
50 Men
5 F
15 Days
7 F
6 Hours,
1
2d. Effect.
Length required.
4 F
6 F
2100 Y = 4500
4th Term.
I afterwards multiply together the Factors of the three firft Terms,
confidering all thofe Factors as abftra&t Numbers, except that, 60 Yards
of the first Effect, in order to obtain the Length of the Ditch in Yards;
laftly, I'multiply the fecond Term reduced 2100Y by the third Term
NUMERAL ARITHMETICK.
139
reduced 4500, which produces 9450000, and dividing this Product by th
firſt Term reduced 5760, there refults 1640 Y 1f 10l 6L for the fourth
Term. But this fourth Term is the Length of the Ditch multiplied into
the Product 4 X 6, or 24 of its Breadth and Depth; wherefore dividing
1640 if to 6L by 24, there refults 68Y if of 11L 3Y' for the
Length required of the Ditch.
I
XXVIII.
Three Men are employed to make a Ditch; the first could make the Ditch
in II Days, the Jecond in 22 Days, and the third in 33 Days; it is required
in what Time thofe three Men together would make the Ditch?
33
I
22
I
22
33
II
pound Rule
Here it is plain that the first Workman will dig the of the Ditch Fourth Ex
in I Day, the ſecond Workman will dig of the fame Ditch in I Day, ample of
and the third Workman will dig of the Ditch alſo in I Day, where- the com
fore the three Workmen will dig, 2 and 3 of the Ditch in a Day; of Three
adding together thoſe three Fractions, or Parts of the Ditch, after hav- direct.
ing reduced them to the fame Denominator, I find their Sum to be
12 or, that is, the three Workmen together will dig of the Ditch
in a Day.
I I
726
As thofe Workmen the longer they are employed the more Work
they will perform, the Quantities of Work will be directly proportional
to the Times in which they are performed; wherefore affuming the
Ditch for Unity, I fay,
6
I
As % of the Ditch is to 1 which represents the whole Ditch; fo is 1 Day,
Time employed to dig of the Ditch, to the Number of Days that the three
Workmen will employ to dig the whole Diteb.
Wherefore to find the Number of Days that the three Workmen will
require to dig the Ditch, I multiply the third Term 1 Day by the ſecond
I repreſenting the Ditch, which will produce 1 Day, and divide this
Product 1 Day by the firft Term, that is, by the Fraction, or multi-
ply it by 6, which will produce 6 Days for the Time required.
XXIX.
The Compound Rule of Three Inverſe does not differ from the fim- The Com
ple Rule of Three Inverſe, except that the given Terms of the latter pound Rule
are fimple, and thofe of the former arife from the Multiplication of feve-
ral others; as will appear by the following Example.
of Three
Inverfe ex
plained by
If 30 Men can finish a Piece of Work in 40 Days, when they work 8 an Example
Hours a Day, in how many Days will 10 Men finiſh the fame Piece of Work,
when they work 6 Hours each Day?
As the 30 Men and the 10 Men are to do the fame Piece of Work,
the 10 Men will require more Time in Proportion as their Number is
140
OF
ELEMENTS
OF
Direct Me
lefs, and the Number of Hours they work each Day is lefs; whence
there refults the following Proportion.
As 30 Men working 8 Hours, are to to Men working 6 Hours; Jo is the
unknown Number of Days employed by the 10 Men, to the 40 Days employed by
the 30 Men.
The firſt Term of this Proportion is reduced to 8 times 30 Men, or
to 240 Men, and the fecond is reduced to 6 times 10 Men, or to 60
Men; becauſe 30 Men working 8 Hours, perform the fame Work that
8 times 30 Men working Hour; and that 10 Men working 6 Hours,
perform the fame Work that 6 times 10 Men working 1 Hour; whence
the Proportion is reduced to the following one.
240 Men, are to 60 Men; as the unknown Number of Days, is to 40 Days.
Whence the Number of Days required will be obtained by multiply-
ing 40 Days by 240, and dividing the Product 9600 by 60; which gives
160 Days for the Anſwer to the propofed Queſtion.
XXX.
Any Queſtion that falls under the compound Rule of Three inverſe
thod of folv may be folved, by confidering in it two Cauſes and two Effects, and
ing Quef comparing direaly the Caufes with their Effects.
tions in the
compound
Rule of
In the propofed Queſtion: If 30 Men, working tight Hours each
Day, can do a Piece of Work in 40 Days; in how many Days will 10 Men
Three Iq do the Jame Piece of Work, when they work 6 Hours each Day?
verfe.
Rule of
Conjunc
tion,
3
It is eaſy to obferve that 30 Men, 8 Hours, and 40 Days, compofe by
their Multiplication the firft Caufe of the Piece of Work; and that 10
Men, 6 Hours, and the Number of Days fought, compole alfo by their
Multiplication the Caufe of the fame Work which thofe 10 Men fhould
perform. Now the two Effects, or the two Pieces of Work being
equal, their Caufes are equal; wherefore all the Numbers given being
confidered as abftra&t Numbers, except that of the Days, 30X8X40 Days,
or 9600 Days, are equal to 10 times 6, or 60 times, the Number of
Days fought; whence the fixtieth Part of 9600 Days, that is, 160
Days, will be the Number of Days required. Wherefore when all the
Terms of a compound Rule of Three Inverfe can be reduced to two
Cauſes which produce a fame Effect, you may obferve the following
Rule: Multiply together all the Numbers that compofe the first Cauſe,
and divide the Produ& by the Product of the Terms that are given in
the fecond Cauſe, and the Quotient will be the Term required.
XXXI.
When it is propoſed to join feveral Statings in the Rule of Proportion
into one, and by the Relation that the feveral Antecedents have to their
Confequents, to difcover the Proportion between the firft Antecedent
7
NUMERAL ARITHMETICK.
141
and the laſt Confequent, this Operation is called the Rule of Conjunc-
tion.
To perform a Rule of Conjunction, the Computifts range the Ante-
cedents in the left Hand Column, and the Confequents in the right
Hand one; fo that the first Antecedent and laft Confequent, whoſe An-
tecedent is fought, is of the fame Species, as alfo the fecond Confequent
and third Antecedent: This Order being continued throughout the
whole. They then divide the Product arifing from the Multiplication of
all the Confequents into one another, by the Product of all the Antece-
dents multiplied into one another, and the Quotient refulting from this
Divifion gives the Antecedent required, as will appear by the following
Examples.
lb.
XXXII,
16.
70 of Lyons.
100 of Rouen.
100 of Toulouſe.
Suppoſe 100lb. of Venice weighs 70lb. of Lyons, and 120lb. of Lyons 100lb.
of Rouen, and 80lb. of Rouen 100lb. of Touloufe, and 100lb. of Toulouſe 7416.
of Geneva, how many Pounds of Geneva will equiponderate 100lb. of Venice?
Let the Pound
Weight of Venice
be expreffed by 1V,
that of Lyons by IL,
that of Rouen by 1R,
that of Toulouſe by
IT, that of Gene-
va by IG, becauſe
100XIV 70XIL,
100 of Venice
120 of Lyons
80 of Rouen
100 of Toulouſe
How many of Geneva
70X100X100X74X100
100X120X80X100
R
11
74 of Geneva.
100 of Venice.
53 23 lb.
2.4
120×1L=100X1R, 80X1R=100X1T,100×1T=74X1G,it follows, that
1V: 1L=70:100,1L1R=100:120, 1R:1T=100:80, IT: 1G=74:100.
Wherefore multiplying in order all the Antecedents by the Antece-
dents, and the Confequents by the Confequents,
IVX LXIRXIT: 1LXIRXITXIG=70X100X100X74: 100X120X80X100,
IVXILXIRXIT 70X100X100X74
ILXIRXITXIG
confequently
IV
ced to
70X74
=
IG
120X80
=
which is redu-
100X120X80X100'
7X74
12X80
7X37
259
Whence
6X80
Application
of the Rule
of Conjunc
tion to an
Example.
9
480*
IV IG 259: 480, which denotes that the Pound Weight of
Venice is to the Pound Weight of Geneva as 259 is to 480, or that
the Pound Weight of Venice is only the Parts of the Pound Weight
of Geneva. I now fay, if a Pound Weight of Venice is reduced to
288 of a Pound Weight of Geneva, what will 100 lb. of Venice be re-
259X100
23
duced to ? or I :
lb. that is 53
<= 53
480
24
25
259
480
100:
will equiponderate roo lb. of Venice.
23 16.
24
142
OF
ELEMENTS
Another
Example.
Varieties in
XXXIII.
If 1 French Crown be equivalent to 80 Pence of Holland, 415 Pence of Hol-
land to 240 Pence English, 240 Pence English to 420 Pence of Hambourg,
64 Pence of Hambourg to Florin of Frankfort, how many Florins of Frank-
fort are equal to 166 French Crowns ?
Let the French
Crown be expreffed
by IF, the Penny of
Holland by IH, that
of England by 1A,
that of Hambourg
by 1h, the Florin of
Frankfort by If, then,
I French Crown
415 Pence of Hol.
240 Pence English
64 Pence of Hamb.
How many F. of Frank.
R
80 X 240 X 420 X 166
415X240X64
=
80 Pence of Hol.
240 Pence Engl.
420 Pence of Ham.
I Florin of Frank.
166 French Crowns.
210 F. of Frank.
IF: 1H=80: 1, 1H: 1A=240:415, 1A: 1h=420: 240 1h: 1f=1:64,
IFXIHXIAX1h_80X2 0X420 IF
hence
IHXI AX1hXI
Ι
1050
830
=
or
80X420 8X10X105X4
415X240X64 If 415X64 415X8X4X2
105 Whence IF af = 105 : 83, and confequently
83
1FX83 If X 105, that is, 83 French Crowns are equivalent to 105
Florins of Frankfort; I now fay, if 83 French Crowns are equivalent
to 105 Florins of Frankfort, what will 166 French Crowns be equiva-
105X166
lent to, or 83: 105 = 166:
=210, that is, 210 Florins of
83
Frankfort are equivalent to 166 French Crowns.
XXXIV.
Having fully explained the different Rules of Proportion, it remains
Proportion. to fhew their Ufe in the common Affairs of Life and Commerce, which,
for greater Perfpicuity, the Computifts range under diftinct Heads.
Things to
which the
XXXIV.
Intereſt is an Allowance made by the Borrower to the Lender for the
Loan of Money, or any other Goods. In Queſtions relating to Intereſt
there are four Things confidered. 1°. The Sum lent, which is called
the Principal. 2°. The common Standard at which the Intereſt is fix-
Queſtions ed, which by common Conſent is at ſo much for the Forbearance of 100l.
for 1 Year, and is called the Rate. 3°. The Time during which the
Principal is lent out. 4°. The Amount of the Principal and Intereſt at
the Expiration of the ftipulated Time, which is fimply called the
Amount. Any three of thefe Things being given, we ſhall ſhew how
the other may be found by the Rules of Proportion.
in Intereſt.
relate.
I
J
NUMERAL ARITHMETICK.
143
XXXVI.
Rate, and
CASE I. The Principal, Rate, and Time being given, to find the From the
Intereft. Multiply the Principal by the Rate, and that Product by the Principal,
Time, the laſt Product divided by 100 is the Intereft required, as will Time given,
appear by the following Example.
§
to find the
Let the Intereft, for Example, of 4261, 5s. 9d. lent out at 4 per Cent, Interest.
for 6 Years be required.
To folve this Queſtion I ſay, if 100l. in 1 Year gives 41. what will
4261. 5s. 9d. give in 6 Years? wherefore by the compound Rule of
Three, 100XI: 41% = 426,2875X6: 1291. 95. 8d. 4.
XXXVII.
№
Example.
From the
CASE II. The Amount, Rate, and Time being given, to find the Amount,
Principal. Say, as the Amount of 100l. at the Rate and Time given, Rate, and
Time given
is to fool. fo is the given Amount to the Principal required.
to find an
Let it be required to find what Sum in ready Money is equivalent to Principal.
3998/. 12s. 10 d. due 3 Years and 145 Days bence.
To folve this Question, I first find the Amount of 100l. in 3 Years.
and 145 Days, at 3 per Cent. per Annum, ſaying 1: 3,4 = 3%. : § 10,2/.
and adding 10,2/. to1ool. I fay110,2: 100=3998,64271.: ₪ 3628, 532391.
XXXVIII.
Example.
CASE III. The Amount, Principal, and Time being given, to find From the
the Rate of Intereft. Say, as the Principal multiplied by the Time, is Amount,
to the whole Intereft, fo is rool. to the Rate per Gent. required.
Principal,
Let it be required to find at what Rate of Intereft will 500l, become given, to
5371. 10s. in 1 Year and 6 Months?
and Time
find the
Rate of La
To folve this Queſtion, I ſay if 500l. in 1 Year and 6 Months gives tereft.
37%. Ios. what will 100l. in 1 Year give, whence
500×1,5 : 100 = 37%, 10s. : № 5%
I
XXXIX.
Example.
From the
CASE IV. The Principal, Amount, and Rate of Intereft being Principal,
given, to find the Time. Say, as the Intereſt of the Principal for I Amount,.
Year at the given Rate is to Year, fo is the whole Intereſt to the and Rate
Time required.
given, to
find the
Let it be required to find in what Time 500l. will become 5371. 10s. at Time.
5 per Cent. per Annum?
To folve this Queftion, I fay if 100l. in 1 Year gives 57. in what
Time will 5001. give 371. 10s. whence 5: 100x1Y= 37,5 : *X 500 ;
wherefore the Time required x equal 37,5X100
Months,
5X500
equal 1 Year and 6
Example..
-
1
144
Things to
which the
Queſtions
in Diſcount
relate.
From the
Sum due,
Rate and
time given
preſent
Worth.
ELEMENTS OF
XL.
Diſcount is an Allowance made for the Paying of Money before it
falls due.
In Queſtions relative to Difcount there are four Things to be confi-
dered. 1°. The Sum due. 2º. The Rate per Cent. per Annum of the
Diſcount. 3°. The Time that the Payment is anticipated. 4°. The
preſent Worth. Any three of thefe Things being given, the fourth will
be found by the Rules of Proportion.
XLI.
CASE I. The Sum due, Rate and Time being given, to find the
prefent Worth. Say, as the Amount of 1ool, at the Rate and Time
given, is to 1ool. fo is the Sum due to its prefent Worth; becauſe the
prefent Worth is the Principal which laid out at Intereft at the given
Rate of the Diſcount, as long as the Sum is paid before it is due, will
amount to the Debt.
A Bill, for Example, of 2000l. which has 17 Months to run, is prefented
Example. to the Bank to be discounted, Intereft at 5 per Cent. bow much Money is the
Holder to receive?
To folve this Queftion, I find the Amount of 100/. for 17 Months, which
1285
85
is
1. and adding it to 100l. I ſay : 100= 2000 1: ₪ 18677.
R
12
12
XLII.
181
257°
A young Gentleman going to travel, depofits with a Banker 1500l. who fur-
nifbes him with Bills of Exchange to the Amount of that Sum, en Condition of
his paying 3 per Cent. of the Money be fhall receive. The Value of the Bills
is required.
To folve this Queſtion, I fay, if 1037. are reduced to 100l. what will
1500/. be reduced to? Whence 103: 100 1500l.: 14567. 6s. 2d. ½,
which is the Value required of the Bills.
XLIII.
#
CASE II. The Sum due, prefent Worth and Rate of Intereſt being
From the given, to find the Time. Say, as the Intereſt of the prefent Worth for
1 Year at the given Rate is to 1 Year, fo is the whole Diſcount to the
Time required.
Sum due,
preſent
Worth and
Rate given,
A Gentleman who contracted a Debt of 13441. that becomes due a certain
to find the Time bence, diſcharges it by paying 1200l. in ready Money; bow many Years
did he anticipate the Payment, fuppofing the Difcount to be 3 per Cent. per
Annum ?
Time.
Example.
To folve this Queſtion I fay, if 100l. in 1 Year produces 37. in what
Time would 1200l. produce 144/. whence 3: 100XIY= 144: xX1200;
wherefore the Time required x equal 100 X
Years.
144
3X1200
equal
144 or 4
36'
NUMERA.L
145
ARITHMETICK.
{
噼
​XLIV..
Sum due,
CASE III. 'The Sum due, prefent Worth and Time being given, to From the
find the Rate of Difcount. Say, as the prefent Worth multiplied by the preſent
Time is to the whole Difcount, fo is 100l. to the Rate per Ċent. Worth and
Time given,
A Gentleman who contracted a Debt of 2000l. payable in two Years, dif- to the
charges it at the End of 7 Months by paying 1867 1371. ready Money, what Rate.
was the Rate per Cent. per Annum of the Discount? 51. per Cent.
XLV.
Tare and
Tret.
In weighing feveral Commodities, the Weight of the Package is in- What is
cluded in the Weight of the Goods, and the whole upon that Account meant by
is called grofs Weight; the Allowance for which is regulated either by
Cuſtom, or by fome exprefs Stipulation between the Buyer and Seller,
and goes under the following Denominations.
1º. Tare, which is an Allowance for the Weight of the Cafk, Cheft,
Box, &c. in which the Goods are packed, allowed to be either fo much
per Bag, Barrel, Cheft, &c. or fo much per Cut....
2º. Tret, which is an Allowance for Duft contracted by keeping, Waſte
by Freight, Carriage, &c. When a Deduction is made for the Allowance:
of Tare from the grofs Weight, the Remainder is called Nett, unleſs
Tret is likewiſe allowed, and then it is called Subtle, out of which the
Tret is deducted, and the laft Remainder is called-nett Weight.
XLVI.
I
1
When the Allowance is made at fo much per Cwt. the Nett is found Rüle for de
by the following Proportion: As 112lb. is to the Difference betwixt the ducting the
given Tare or Tret, and 112/b. fo is the given Grofs to the Nett required, Tare.
What is the Net of 410Cwt. 2qrs. 12lb. at 20 per Cwt. ?.
To folve this Queftion, I fubtract 20 from 112, the Remainder is 92,
and then fay, if 112/b. is reduced to 92lbs what will 410Cwt. 29rs. 12lb.
be reduced to? H337Cwt. 1gr. 4lb.
When Tare and Tret are both allowed, to find the Nett. Say, as Rüle for de
112X112 is to the Product of the Difference betwixt the given Tare, ducting
and 112 multiplied into the Difference of the given Tret and 112, fo is Tare and
the given Grofs to the Nett required.
Tare being allowed at 4lb. to 112lb. and Tret at 5lb. to 112lb. what is
the nett Weight in 87lb. grofs. ?
To folve this Queftion, I fay as 112lb. is to 1086. (viz. 112 lefs 4)
fo is 871b, to the Subtle. I then fay, as 112lb. to 107lb. (viz. 112 lefs 5)
fo is the Subtle to the Nett. Multiplying therefore the Antecedents
of the Ratios of thoſe two Proportions together, as alfo their Confequents ·
together, 112 X 112: 108 X 10787 X Subtle : Nett X Subtle, that
is, 112 X 112 : 108 X 107 = 87: Nett.
Tret at one
Operation,
explained by
an Exampler
2. T
146
ELEMENTS OF
Rule of
Mixtures.
Firft Cafe.
tities and
XLVII.
It is ufual in Trade to mix feveral Sorts of Wares together for the
Convenience of Sale. The Operation to be performed, to proportion
the Price of the Mixture to the ſeveral Prices of the Simples, or to find
the Quantity of each Ingredient that will proportion the Mixture to a
certain Price, is called the Rule of Mixtures.
XLVIII.
When the Quantities as well as the Prices of the Ingredients are
given, to find the Rate of the Mixture. Multiply the Value of an Unit
The Quan of each Ingredient by the Number of Units of that Ingredient, whence
there will refult as many particular Products as there are Sorts of Wares
the Ingredi- to be mixed; divide the Sum of thofe Products by the Sum of the In-
ents given, gredients, the Quotient will give the Rate of the Mixture, as will ap-
pear by the following Example,
Prices of
to find the
Rate.
Firft Exam
ple.
Second Ex
ample.
1
Third Ex
ample.
XLIX.
A Farmer mixes three Sorts of Grain of different Prices, viz. 10 Sacks of
Wheat at 125. per Sack, 8 Sacks of Wheat at 14s. per Sack, and 6 Sacks of
Qats at 8s. per Sack; what is a Sack of the Mixture worth?
To folve this Queſtion, I ſay, 19. If 1 Sack coft 12. what will
10 Sacks coſt? R 61. 2°. If I Sack coft 14s. what will 8 Sacks coſt?
R 57. 125. 3º. If I Sack coft 8s. what will 6 Sacks coft? y 21. 85.
Whence the 24 Sacks of Corn mixed together are worth 14. I now
fay, if 24 Sacks of Corn mixed together are worth 14%. what will 1
Sack of the Mixture be worth? R 11s. 8d.
L.
If 278 Gallons of Rum, at 11s. 6d. per Gallen, were mixed with 174 Gal-
lons, at 9s. 3d. per Gallon, what would a Gallon of the Mixture be worth?
Multiplying 11s. 6d. by 278, I find the Price of 278 Gallons, at 1 15. 6d.
per Gallon, to be 159/. 175. Multiplying 9s. 6d. by 174, I find the
Price of 174 Gallons, at 9s. 6d. per Gallon, to be 82. 135.
Confequently the 278 Gallons and 174 Gallons mixed together, or
the 452 Gallons of Rum, are worth 242/. 10s.
Wherefore dividing 242/. 10s. the whole Price of the 452 Gallons of
Rum, by 452, the Quotient 10s. 8 d. will be the Price of a Gallon of
the Mixture.
LI.
A Goldsmith melted down together 60lb. of Silver of different Standards,
viz. 32lb. of 110z. fine, 2016. of 110%. 12dwt. fine, 8lb. of 100%. 10dwt.
fine. The Standard of the Mixture is required.
If Silver is pure and free from Mixture, that is, if the 12 Parts into
which the Pound of Silver is divided are fine, the Silver is faid to be
NUMERAL ARITHMETICK.
147
120%, fine. If the lb. is compofed of 11 Parts of pure Silver, and of I of
another Metal, the Silver is faid to be 110z. fine. If the lb. of Silver is
compoſed of 100z. and 10dwt. of pure Silver, and of 10%. 10dwt. of ano-
ther Metal, the Silver is faid to be 100%. 10dwt. fine, and fo on. Theſe
Notions being premifed, the Queftion is folved thus:
As there are 32lb. of 110%. fine, 2016. of 110%. 12dwt. fine, 81b. of
100%. 10dwt. fine. Multiplying 110%. by 32, there refults 35202. Mul-
tiplying 110z. 12dw, by 20, there refults 232 oz. Laftly, multiplying
100%. 10dwt. by 8, there refults 840z.
The 60lb. therefore will contain in all 668oz. fine; whence dividing
thofe 6680%. fine by 60, the Quotient 11oz. 2dwt. 16gr. will be the
Quantity of fine Silver contained in a lb. of the Mixture, and confe-
quently will be the Standard of this Mixture.
LII.
The particular Rates of two Simples to be mixed, and the Rate of Second Café
the Mixture being given, to find how much of each Ingredient must be The Rates
taken to compofe the Mixture.
of two In
and that of
find the
Form two Fractions that fhall have for common Denominator the gredients
Difference of the Rates of the two Ingredients; the one having for Nu- the Mixture
merator the Difference between the mean Rate, and the loweſt Rate will given, te
expreſs the Portion to be taken of an Unit of the Ingredient of the higheſt
Quantity of
Rate; and the other having for Numerator the Difference between the each Ingre
mean Rate, and the higheſt Rate will exprefs the Portion to be taken dient.
of an Unit of the Ingredient of the lowest Rate, as will appear by the
ollow in Example.
LIII.
How much Wheat, at 13s. the Sack, and Rye at 10s. the Sack, will com-
pofe a Mixture that may be fold for 125, the Sack?
To folve this Queſtion, let the three Sacks, that of 10s. the lowest First Exam
Rate, that of 135. the higheſt Rate, and that of the Mixture to be fold ple.
at the mean Rate 125. be conceived to be divided into a Number of
equal Parts.
The Difference 25. between the mean Rate 12. and the lowest Rate
ros. being double of the Difference Is. between the higheſt Rate and the
mean Rate; it is manifeft that every Part that is taken of the Corn of
the loweſt Rate to compofe the Mixture, will diminish the mean Rate
twice more than each Part of the Corn of the higheſt Rate will increaſe
it, we ſhould therefore take two Parts of the Corn of the higheſt Rate
for every one that we take of the loweft Rate, in order that the mean
Price may not be altered, that is, of the three Parts that are taken in all
to compoſe a Sack of Corn of the mean Rate, two fhould be of the Corn
*
¿
£48
ELEMENTS
OF
-Second Ex
ample.
of the highest. Price, and one of the Corn of the loweft Price; whence
the Sack of the Mixture to be fold for 125. will confift of of the Sack
43
at 13. and of the Sack at 10s.
Now it is eaſy to perceive that the Denominator 3, common to the
two Fractions and, which exprefs the two Portions of the Sack of
the Mixture, arofe from the Difference 3 in the Prices of the two Sorts
of Corn, and that the Numerators 2 and I of thofe Fractions are the
Differences between the mean Rate and the particular Rates.
LIV.
Let it be propoſed to mix two different Subftances, a cubical Foot of one of
which weighs 650lb. and of the other 480lb. Jo as a cubical Foot of the Mix-
ture ſhall weigh 500lb.
The heaviest cubical Foot weighing 650lb. the lighteſt 480%. and the
cubical Foot of the mean Weight weighing 500lb. the Difference be-
tween the lighteſt and the heaviest will be 170lb. the Difference between
the mean Weight and the heaviest 150lb. the Difference between the
mean Weight and the lighteſt 20lb. whence two Fractions having 170lb.
or fimply 170 for common Denominator, and 20lb. and 150lb. or fimply
20 and 150 for their Numerators, will exprefs the Portions to be taken
of the two given cubical Feet to compofe the cubical Foot required,
weighing 500lb. that is,
1º.
20
170
or will be the Part to be taken of the cubical Foot weigh-
ing 650lb.
20.1% or
ing 480lb.
T7
I S
7
will be the Part to be taken of the cubical Foot weigh-
LV.
The foregoing Queſtion is of Ufe in Gunnery. The Metal employed
Ufe of the in the Conſtruction of Cannon is a Mixture of Copper and Block-Tin.
foregoing
Example in Some Founders, to every 100lb. of Copper, add 11lb. of Block-Tin;
Gunnery. others, to every 100lb. of Copper, add 12/6. of Block-Tin. This laft
Proportion being fuppofed to be the beſt, when old Pieces of Cannon,
compoſed of thoſe two Metals, are to be melted down, it will be necef-
fary to determine the Quantity of each Metal they contain, in order to
difcover whether thofe Metals have been mixed in the Proportion,
which Experience has proved to be the beſt.
It is eafy to perceive, that if the Weight of a Quantity of Copper, as
alfo of a Quantity of Block-Tin of the fame Bulk with the Mixture,
could be determined, the Quantities of each Metal in the Mixture
would be easily found.
A heavy Body immerſed in Water lofes a Part of its Weight equal to
the Weight of the Quantity of Water' that it puts out of its Place;
whence Bodies that lofe, when weighed in Water, equal Parts of their
NUMERAL ARITHMETICK.
149
Weight, put equal Quantities of Water out of their Place, and confe-
quently are of the fame Bulk, agreeable to this Principle.
Let a Portion of the Mixture, weighing for Example 80lb. be
taken; likewiſe a Piece of Copper and one of Block-Tin, each of the
fame Weight, and let the Portion of the Mixture, when weighed in
Water, lofe 9 lb. Copper lofing in Water the 9th Part of its Weight,
and Block-Tin the 7th Part, the Piece of Copper will lofe 8. and the
Piece of Block-Tin 11 lb.
I
J
Now to find the Quantity of Copper and Block Tin feparately that
compoſe the Mixture, it fuffices to find how much Water the Copper
and Block-Tin fhould put out of its Place feparately, fo that the Mix-
ture fhall put 9 lb. out of its Place.
ठ
To find which, I take the Difference between 11 lb. and 8 lb. which
is 2 1/4 or 33,
2詩 ​160, the Difference between 23 and 83, which is or 3, and
the Difference between 3 and 11, which is 2; wherefore the Frac-
tions having for common Denominator 160, and for Numerators 28 and
132, will expreſs the Parts of the 9 lb. of Water, which the Copper
and Block-Tin in the propofed Mixture put out of its Place; wherefore
the Quantity of Copper in the propoſed Mixture is 661b. and of Block
Tin igl.
उ
Having diſcovered after this Manner the Quantity of each Metal in
the Portion of the Mixture, it will be eafy to find how much of each the
whole Mixture contains. Let the Piece, for Example, be a 24 Pounder,
weighing 5100lb. to find the Quantity of Copper in this Piese. I fay,
if 80lb. of the Mixture contain 661b. of Copper, how much will 5100lb.
contain? 4207 16. In like Manner the Quantity of Block Tin will
be found to be 892, that is, 21. of Block-Tin for every 100lb. of
Copper; but as the Proportion of the Quantity of Copper and Tin ſhould
be as 100 12, I ſay, if 12lb. requires 100lb. what will 892 lb. require?
R 7437 lb. Confequently there ſhould be added 323016. of Copper to
the propoſed Mixture, that it may be of the Quality required.
LVI.
The Total of the Compound of two Simples, with the total Value of
that Compofition, and the Value of an Unit of each Simple being given,
to find the Quantity of each fimple Ingredient in the Compofition.
To find the Quantity of the higher priced Simple, 1. multiply the
leffer Price of the Unit by the total Quantity of the Compofition, de-
du&t the Produ&t from the total Value of the Compofition, and divide
the Remainder by the Difference in Value of an Unit of the two Simples
given, and the Quotient is the Quantity of the higher priced Simple.
Third Cafe.
A Compoud
two Sim
Value,
ples, its to
tal
of
and that of
Į
150
an Unit of
ELEMENTS OF
2º. To find the Quantity of the lowest priced Simple, multiply the
each Ingre greateſt Price of the Unit by the total Quantity of the Compofition, de-
dient given,
to find the duct the Product from the total Value of the Compofition, and divide the
Quantity of Remainder by the Difference in Value of an Unit of the two Simples
given, and the Quotient is the Quantity of the lowest priced Simple; as
will appear by the following Example.
each.
Example.
1
A Com
pound of
any Num
ber of Sim
LVII.
Suppose there are 20 Ounces of Gold melted into one Mafs, confifting of
Gold at 4l. per oz. and Gold at 41. 5s. per oz. the Value of the whole being 821.
it is required to find how much of each was taken to make the Compofition ? ·
If the 2002. confifted of Gold at 47. per oz. the total Value of it would
be 80l. and confequently 27. lefs than the propofed Value 821. This
Product 80%. must therefore be increafed by 27. without increafing the
Number of Ounces. Now it is manifeft that it is this Increaſe of 21.
that is found by the Rule, when it directs to multiply the leffer Price.
of the Unit by the total Quantity of the Compofition, and dedu& the
Product from the total Value."
As 41. 5s. exceeds 41. by of a Pound, every Ounce at 4. 55. that is
taken in the Room of an Ounce at 41. will increaſe the Product 80l. by
1. without increafing the Number of Ounces; wherefore, in order to
increaſe the Product 80/ of 200z. at 41. per oz. by 21. we must take in
the Room of the Gold at 41. per oz. as many Ounces at 41. 55. as 11. is
contained in 27. that is, we muſt take 8oz. of the Gold at 4/. 5s. per oz.
Now it is this Number 8oz. that is found by the Rule, when it directs
to divide the Remainder by the Difference in Value of an Unit of the
two Simples given.; wherefore there were 120z. of Gold at 41. per oz.
and 8oz. at 41. 5. in the propofed Mixture.
2º. If the 20 Ounces confifted of Gold at 41. 5s. per oz. the total Va-
lue of it would be 857. and confequently 37. more than the given Value
827. This Product 851. must therefore be diminiſhed by 37. without in-
creaſing the Number of Ounces; to effect which, we must take as ma-
ny Ounces of Gold at 41. per oz. as 4. is contained in 31. the Difference in
Value of an Unit of the Ingredients, that is, we must take 120%. of the
Gold at 41. per oz, and confequently 8oz. of the Gold at 41. 5s. per oz.
LVIII.
The Total of the Compound of any Number of Simples, with the to-
tal Value of that Compofition, and the Value of an Unit of each Simple
being given, to find the Quantity of each Ingredient in the Compofition.
ples, its to Multiply the lowest Price of the Unit by the total Quantity of the Compo-
tal Value, fition, and deduct this Product from the total Value of the Compofition;
and that of divide the Remainder into as many Parts lefs one as there are Ingredients,
each Ingre divifible by the Differences between the lowest Price of the Unit, and all
an Unit of
NUMERAL ARITHMETICK.
151
the other Prices Thofe Parts being divided by thofe Differences, the dient given,
Quotient will exprefs the Quantities of each Ingredient, except the loweſt
priced one.
LIX.
How many Pounds of Gun-Powder at 2s. Is. and at 6d. per lb. muft be ta-
ken to compound a Mixture of 221b. worth 30s.?
to find the
Quantity of
each.
If the 22/6. confifted of Powder at 6d. per lb. the total Value of it Example.'
would be 132d. and confequently 228d. lefs than the propoſed Value
360d. whence this Product 132d. muſt be increaſed by 228d. without in-
creafing the Number of Pounds of the Mixture. Now this Augmenta-
tion of 228d. found by dedu&ting the Produ& 132d. from the propoſed
Value 360d. cannot be made but by taking a Number of lb. of Powder at
25. and at Is. per 16. in the Room of a fimilar Number of 15. of Powder
at 6d. per lb.
1º. Each lb. of Powder at 23, taken in the Room of a lb. at 6d. will
give an Augmentation of 18d. equal to the Difference in Value of an
Unit of the highest rated and loweft rated Powder; whence the Part
of the Augmentation produced by taking a Number of Pounds of Pow-
der at 25. per lb. in the Room of a like Number of Pounds at 6d. per 16.
will be a Number of Pence multiple of 18d. and confequently diviſible
by 18d. Difference in Value of an Unit of the higheſt priced and loweſt
priced Powder.
2º. Every lb. of Powder at 12d. taken in the Room of a 1b. at 6ď.
will produce an Augmentation of 6d. which is the Difference in Value
of an Unit of the mean priced and lowest priced Powder ; whence
the Part of the Augmentation, produced by fubftituting a Number of
lb. of Powder at Is. per lb. in the Room of a like Number of lb. at 6d.
per
1b, will be a Number of Pence multiple of 6d. and confequently di-
viſible by 6d. Difference in the Value of an Unit of the mean priced and
lowest priced Powder.
We must therefore divide the Augmentation 228d. into two Parts,
one of them divifible by the Difference 18d. in the Value of an Unit of
the higheſt and lowest priced Powder, and the other divifible by the
Difference 6d, in the Value of an Unit of the mean priced and loweſt
priced Powder.
The Parts of 228d. diviſible by 18d. and the corresponding Parts of
228d. divifible by 6d. are
18, 36, 54, 72, 90, 108, 126, 144, 162, 180, 198, 216,
210, 192, 174, 156, 138, 120, 102, 84, 66, 48, 30, 12,
each lb. of Powder at 2s. fubftituted in the Room of a 1b, at 6d. produ-
cing 18d. in the firſt Parts of 228d, and each 1b, at 1s. fubftituted in the
*
152
OF
ELEMENTS
How the
Profit or
Lofs made
upon any
Article in
Tráde is
ekimated.
Room of a 1b. at 6d. producing 6d. in the fecond Parts of 228d, it is ma-
nifeft that if the first Parts be divided by 18, the Quotients will be the
different Numbers of lb. of Powder at 25. that produce the first Parts,
or that can compofe the Mixture; and that by dividing the fecond Parts
by 6d. there will refult the different Numbers of lb. of Powder at Is.
per lb. that can compoſe the Mixture. Thofe Divifions being perform-
ed, the Number of lb. of Powder at 25. and at 1s. per lb. will be as
follows:
1
lb. at 2s. 1, 2, 3, 4, 5, 6,
4, 5, 6, 7, 8, 9, 10, гI, 12,
lb. at Is. 35, 32, 29, 26, 23, 20, 17, 14, 11, 8, 5, 2,
1
As the Compofition fhould confift but of 22lb. it is manifeft that the
ſeven first Numbers of 16, at 25. per lb. which with the correſponding
Numbers of lb. at 1s. make more than 2276. fhould be rejected; and
that the eighth Number 876, at 25. which with the correfponding Number
14b. at 1s. makes precifely 22/6. fhould be alfo rejected with its Fellow
14b. fince the Mixture fhould contain a Number of b. at 6d. whence
there will remain but four different Numbers of 1b, of Powder at 25.
with four correfponding Numbers of lb. at Is. per b. and as the whole
Compofition is limited to 221b, if from 22 be deducted each Sum made
up of a Number of lb. at 25. together with the correfponding Number of
16, at 1s. each Remainder will be the correſponding Number of 15. at 6d
per lb. confequently the Particulars of the Mixture fhould be glb.. at 25.
11lb. at Is. and 2lb, at 6d. per lb. Or, 10lb. at 2s. 815. at Is. and
41b. at 6d. per lb. Or, 11lb. at 25. 51b, at rs. and 6lb, at 6d, per lb. Or,
12lk. at 25. 2lb. at Is. and 81b.. at 6d. per lb.
LX.
In buying and felling Goods the Merchant fhould be able, by compa
ring both together, on the different Articles in which he deals, to make
a true Estimate of his Trade, thereby to judge with Certainty what
Articles turn out more or less to his Account. To effectuate which it
was found neceffary to have fome common Standard, by which the Gain
or Lofs made, or propofed to be made upon any Commodity or Article
in Trade, fhould be tried and expreffed, and this by univerfal Confent
feems to have been fixed to the Centum or Hundred; fo that when it is
faid that the Gain is at 10 per Cent. it is to be underſtood that when
100%. 100s. &c. have been laid out in purchaſing Goods, 110/. 110s. &c.
have been recovered by the Sales; and in the fame Manner if 100l.
were laid out in the Purchaſe of Goods, and but 90l. received back, it
is faid that 107. per Cent, was loft by fuch Goods..
1
NUMERAL ARITHMETICK
153
LXI.
The buying and felling Prices being given, to find the Rate per Cent, Caſe I.
of the Gain or Lofs. Say, as the buying Price is to the Difference be- The buying
tween the buying and felling Price, fo is 100 to the Gain or Lofs and felling
Prices given
per Cent.
to find the
A Merchant bought Cloth at 15s. 6d. per Yard, and fold it again for 18s. Rate.
what did he gain per Cent.?
To folve this Queſtion I ſay, if 15s. 6d. gained 2s. 6d. what will 100 Example.
gain? 16 per Cent.
A Merchant bought Cloth at 10s. per Yard, and fold it again for 9s. per Second Ex
Yard; what did he lofe per Cent.?
I fay, 10 : 1 = : 100: R 10 per Cent.
A Merchant bought 5 Puncheons of Rum for 2451. 19s. 6d.
for 2951. 195. 6d. what did be gain per Cent?
I fay, if 245,9751. gain 50l. what will 100 gain?
Cent.
ample.
and fold them Third Ex
3220
20 330 per
ample.
A Merchant bought a Puncheon of Rum for 661. 135. 4d. it run 150 Gal- Fourth Ex
lons, which he retailed at 12s. 6d. per Gallon; whether did he gain or loje, ample.
how much, and at what per Cent.?
I fay, as 661. 13. 4d. the buying Price, is to 27%. Is. 8d. the Diffe-
rence between the buying and felling Price, fo is 100l. to R 40%, the
Rate required per Cent.
LXI.
The buying Price, and the propofed Rate per Cent. Profit being given,
to find the advanced Price. Say, as 100 is to the Amount of 100 at the
Rate given, fo is the buying Price to the advanced Price required.
A Merchant bought Cloth at 15s. per Yard; how may be charge it per
Yard to gain 25 per Cent.?
Cafe II.
The buying
Price and
the Rate per
Cent. Profit
given, to
find the ad
vancedPrice
I fay, as 100 is to 125, fo is 15s. to 18s. 9d, the advanced Price.
A Merchant bought 400 Spindles of Tarn for 411. Carriage and other Second Ex
Charges came to 17s. how may be retail it per Spindle to clear 30 per ample.
Cent, ?
I fay 100: 130 41,85%. : 54,4051. the advanced Price of the 400
Spindles, which is at the Rate of 25. 8d§ d. per Spindle.
A Merchant bought 80 Puncheons of Rum, containing 7548 Gallans, Third Ex
at 10s. 6d. per Gallon. Lighterage, Porterage, and other Charges came to ample.
251. 75. There was of Leakage 25 Gallons. How may be fell the Remain-
der per Gallon to clear 12 ½ per Cent. upon the whole ?
To folve this Queſtion, I firſt inveſtigate the Price of the 7548 Gal-
lons, at 10s. 6d. per Gallon, to which adding the Charges, I find the
2 U
154
ELEMENTS OF
ple.
first Coft to be 3988,057. 1 then ſay, 100: 112,5=3988,057.: 4485,431257.
the advanced Price of the 7523 Gallons that remain after the Leakage
is deducted, which is at the Rate of 11s.. 11d. per Gallon.
Fifth Exam A Merchant bought 17Gut. 3qrs. of Sugar, at 60s. and 14Cwt. 2qrs. 14lb.
at 70s. which he mixed together and propofes to fell it at 30 per Cent, ad-.
vance, how must be value I Gwt. of the Mixture?
Cafe III.
The Rate
given and
To folve this Queſtion, I firſt inveſtigate the Prices of 17Cwt. 3qrs.
of Sugar, at 60s. per Cwt. and of 14Cwt. 2qrs. 14lb. at 70s. per Cwt.
and I find the firft Coft of the Mixture 32,375 Cwt. to be 104,4375.
I then fay, 100: 130 104,437: 135,768751. the advanced Price of
the Mixture, which is at the Rate of 41. 35. 10 d. per Cwt.
. LXIII.
at
If the Rate proportional to another upon a new advanced Price is re-
quired. Say, as the loweſt advanced Price is to the Amount of 100,
another the given Rate; fo is the higheſt advanced Price to the Amount of 100
portional to at the required Rate.
pro
Firft Exam
it required. A Merchant fold Tobacco. at 7d. per lb. upon which he gained 10 per Cent.
Markets rofe to 8d. what did that advanced Price bring per Cent. ?
I fay, 7: §d.
110 = 8d.: 125 d. 25 per Cent. -
ple.
Second Ex
ample.
Cafe. IV.
The advan
ced Price
given, to
find the
A Merchant fold Cloth at 18s. per Yard, upon which he had 15 per Cent.
How much per Cent. had his Neighbour, who fold his Cloth of the fame Coft at
19s. 6d. ?
24 per Cent.
R 24
2
LXIV.
If it be required to find the prime Coft. Say, as the Amount of 100
at the Rate given, is to 100; fo is the advanced Price to the prime
Coft required.
A Merchant fold a Pipe of Wine for 431. 15s. by which be bad 20 per Cent.
prime Coft. what was the prime Coft?
First Exam
ple.
Second Ex-
ample..
What is
meant by
Barter.
Method of
folving
Queſtions
I fay, 120: 100=43,751. R 361. 95. 28.
A Merchant fold 50 Pieces of Scotch Lawns, 10 Yards each, for 1351. by
which he gained 15 per Cent. what did it cost him per Yard?
*
I fay, 115 100 1357.: 117,3917. the prime Coft of the 50 Pieces,,
or 500
Yards of Lawn, which is at the Rate of 4s. 8 d. per Yard.
44
LXV.
Barter is the Commutation of one Commodity for another, and teach-
eth fo to proportion the Quantities to be exchanged, according to the
Conditions of the Barter, that neither Party may ſuſtain Lofs.
To folve Queſtions relating to Barter at one Operation. Say, as the
Rate of the Quantity required per Yard, lb. Cwt. &c. is to the Rate of
the affigned Quantity, fo is the affigned Quantity to the Quantity re-
one Opera quired; as will appear by the following Example.
in Barter at
tion.
First Exam
ple.
How many Yards of Shalloon may a Merchant barter for 80 Yards of broad
Cloth, rating the broad Cloth at 15s. 6d. per Yard, and the Shalloon at 25.2
NUMERAL ARITHMETICK.
155
its
I fay, as 1 Yard of broad Cloth is to 15s. 6d. fo is 80 Yards to x
Value. I then fay, as 2s. is to 1 Yard of Shalloon, fo is x the Value of
the broad Cloth to y, the Number of Yards of Shalloon required, Con-
fequently. 1X2: 15XI=80YXx: yxx, or 2: 15-80Y: y equal 620 Y.
How many
Yards of Irish Linen, at 2s. 3d. per Yard, may a Linen-Draper Second Ex
have in Barter for 80 Pieces of Holland, of 20 Yards each, at 3s. 6d. per ample.
Yard?
I fay, 2 80X20 Y= 3 R 2488 Yards.
How many Pieces of Indian Chints may a Merchant have in Barter for 86 Third Ex
Pieces of broad Cloth, rating the former at 251. 10s, per Piece, and the latter ample.
at 151. 15s. per Piece? I fay, 25: 86 Pieces
15 53
How many lb. of Almonds may a Grocer have in Barter for
Raifins, reckoning the Almonds at 7, and the Raifins at 3
I fay 7: 3791b. 3: R 189 lb.
LXVI.
½
2
7
Pieces.
37916. of Fourth Ex
per lb? ample.
Exchange is the Commutation of the Money of one Country for that
of another, by Means of a Bill or Writ, commonly called a Bill of Exchange.
When the Purchaſes and Payments of one Country in another exa&- From
ly balance the Purchaſes and Payments of the latter in the former, the whence ari-
Exchange on both Sides are at Par; that is, one who gives Money in one
Country will receive as much in the other in Weight and Standard.
But if one Country fupplies another with more than it takes from it,
there will be a Balance against the latter, which it muft neceffarily pay;
in order to which, the Demand for the Money, or Bills of Exchange
of the former, becomes greater in the latter than the Quantity to fup-
ply that Demand, and confequently puts the Price of their Money above
Par, and that of the other below Par, and this conftitutes what is called
the Courſe or current Rate of Exchange.
LXVII.
fes the
Courſe of
Exchange.
Manner of
In Ireland, Accounts are kept in Pounds, Shillings, and Pence, Iriſh.
Britain exchanges with Ireland upon the rool the Par of which in Iriſh exchanging
Money is 1087. 6s. 8d. fo that the Shilling Sterling is worth 13d. Irish. between
The Courſe of Exchange runs from 5 to 12 per Cent, according as the Ireland.
Balance of Trade is in Favour of Ireland, or againſt it.
ठ
If Iriſh Money is required; as, for Example, how much in Dublin for
3571. 18. in London, when the Exchange is at 10 per Cent. ?
Say, if 100l. English gives 1101. Iriſh, according to the Courfe,
what will 3571. 185. Engliſh give? R 3951. 18s. 6d.
If English Money is required; as, for Example, how much in Lon-
don for 395% 18s. 6d. in Dublin, Exchange at 10 per Cent. ?
Say, if 110 Irish, the Courfe of Exchange equivalent to 100l
Engliſh, gives 1ool, English, what will 3957. 18s. 6d. give?
357% 18. the English Money required.
*
England and
Reduction
of English
Money to
its equiva-
lent Irish.
Reduction
of Iriſh Mo-
ney to its
equivalent
English,
1
1
156
with the
ELEMENTS
OF
Manner of In America and the Weft-Indies, as in other Parts of the British Do-
exchanging minions, Accounts are kept in Pounds, Shillings and Pence, divided as
British Plan in Britain, and their Money, for Diftinction's Sake, is called Currency.
The Method of computing the Exchanges is the fame as with Ireland.
tations,
Exchange
applied to
LXVIII.
When Britain exchanges upon the 100%. the higher the Exchange is,
Drawing the Advantage to Britain in remitting is the greater, and in drawing
and Remit- the lefs, as will appear by the following Examples.
ting,
Manner of
between
When the Exchange was at 12 per Cent. Britain remitted to Ireland 50001.
for how much Sterling ought Britain to draw for a Reimburſement, when the
Exchange falls to 6 per Cent. ?
To folve this Queftion, I first fay, if 100l. English amounts to II 2.
Irish, what will 5000l. English amount in Ireland to? R 5600l. I next
fay, if 106d. Irish gives 100l. English, what will 5600l. Irish give?
R 52831. os. 4 d. English; whence Britain gains by this Tranfaction
2831. Os. 4žd. which is above 5 per Cent.
When the Exchange with Ireland rofe to 12 per Cent. Ireland drew on Lon-
don for 5000l. English, how much Irish must be remitted to London to diſcharge
the Debt, when the Exchange falls to 6 per Cent.?
To folve this Queſtion, I firſt fay, if rool. English amount to 1127.
Irish, what will 5000l. English amount to in Ireland? 5600l. I af-
terwards fay, if 100l. English amounts to 106/. Irish, what will 5000l.
Engliſh amount to in Ireland? 5300l. whence Ireland gains by this
Tranfaction 300l. which is 6 per Cent.
LXIX.
In France Accounts are kept in Livres, Sols, and Deniers, divided as
exchanging the British Pound. The Manner of exchanging between London and
Paris, &c. is to give fo many Pence Sterling for the Crown or Ecu of
3 Livres, or of 60 Sols Tournois. The Par of Exchange, according to
the prefent Currency of the Coins in England and France, is 29,149 Pence
Engliſh per Crown, or per 60 Sols Tournois. According to this Par,
their Coins may be eſtimated as follows.
England and
France.
Eftimation
of the
French
Coins.
Reduction
3.9
I 3.
32
1º. A Denier d. 2°. A Liard of three Deniers d. 3°. A Dardene
of 2 Liards d. 4°. A Sol of 2 Dardenes 32d. 5º. A Frank of 20 Sols of
1 Livre gåd. 6º. A Crown of Exchange or 60 Sols 25. 54d. 7º. A double
Crown of 6 Livres 4s. 101d. 8°. A Louis d'or of 8 Crowns 19s. 6d.
English Money.
To compute the Exchanges with France. If French Money is re-
of English quired, as, for Example, how many Livres in Paris for 5661. 135. 44.
Money to
its equiva- in London, when the Exchange is at 31d. per Ecu?
lent French.
Say, if 31d. the Courfe of Exchange, gives 3 Livres, what will
5661. 13s. 4d. English give? 13161 Liv. 5Sol. 93 Den.
NUMERAL ARITHMETICK.
157
If English Money be required, as, for Example, how much in Lon-
don for 13161 Liv. 5 Sol, 9 Den, in Paris, the Exchange being at 31d.
per Ecu?
Say, if 3 Livres gives 31d. what will 13161 Liv. 5 Sol. 9 Den.
give? 5661. 135. 4d.
R
LXX.
Manner of
In Genoa Accounts are kept in Piaftres, or Pezzos, which are di-
exchanging
vided into Soldi and Denari, as the Britiſh Pound. Some of the Mer- between
chants keep their Accounts in Lires or Liras, Soldi and Denari, divided England and
as before. This Money is only the of the Value of the other.
The Manner of exchanging between London and Genoa, is to give fo
many Pence Sterling for the Pezzo or Piaftre, according to the Courſe
of Exchange which runs between 45 and 50d. per Piaftre. The Par of
the Piaftre is 54d. Sterling, and the Par of the Lire 10 d.
9
Genda.
of Genoa.
According to which their Coins may be cftimated as follows: 1. A Eſtimation
Denari zod. 2°. A Soldi or 12 Denari d. 3°. A Chevalet or 4 Sol- of the Coins-
di 1d. 4º. A Teftoon or 30 Soldi 1s., 1 d. 5º. A Genouine or 6
Teftoons 6s. 9d. 6º. A Pistole 155. 7°. A Spaniſh Piſtole 17s. 11d.
If Genoiſh Money is required, as, for Example, how much in Genoa
for 1710l. 16s. 4d. in London, the Exchange at 47 d. Sterling per Reduction
Pezzo?
of Sterling
to its equi-
how valent Mo-
Say, if 47 d. Sterling, the Rate of Exchange, gives I Pezzo,
many Pezzos will 1710l. 16s. 4d. give? R 8644Pez. 2Sol. 6Den.
If Sterling be required, as, for Example, how much in London
8644Pez. 2Sol. 6Den. in Genoa, the Exchange at 471d. per Pezzo?
Say, if 1 Pez. gives 47d. what will 8644 Pez. 2Sol. 6 Den.
R 1710. 16s. 4d. Sterling.
I
LXXI.
for
give?
ney of Ge-
noa.
Reduction
of Money
of Genoa to
its equiva-
·lent Ster-
ling.
Manner of
In Leghorn Accounts are kept in Piaftres, Soldi, and Denari, divided
as at Genoa. Some likewife keep their Accounts in Liras or Lires, di- exchanging
vided as the Piaftre; but this Money is only of the Money of Ex-between
change. The Par with London is 45. 4d. but the Courfe runs from 45 to London and
50d. Sterling only per Piaſtre.
Leghorn.
The Coins of Leghorn are eftimated as follows. 1°. A Denari. Eftimation
2º. A Quatrini or 4 Denari 3d. 3°. A Soldi or 3 Quatrini d. 4°. A of the Coins
of Leghorn.
Craca, or Grain of 5 Quatrini 3d. 5º. A Julio, or Paulo of 8 Grains 3d.
6º. A Piaſtre of Exchange 4s. 4d. 7. A Ducat of 150 Soldi 5. 5d.
8°, A Piſtole of 21 Lires 155. 6d.
Reduction
When Money of Leghorn is required, as, for Example, how much of Sterling
in Leghorn for 4651. 19s. 6d. Sterling, the Exchange at 46d.
Piastre?
per talent Mo-
to its equi-
ney of Legh
158
ELEMENTS OF.
1
Reduction
1
Say, if 46d. gives 1 Piaftre, what will 465/. 19s. 6d. Sterling give?
R 2431 Piaft. 3Sol. 5&Den.
When Sterling is required, as, for Example, how much in London
of Leghorn for 2431 Piaft. 3Sol. 5 Den. Exchange at 46d. per Piaftre?
Money to
its equiva-
Say, if i Piaft. gives 46d. what will 2431 Piaft. 3Sol. 52Den. give?
lent Sterling N 4651. 19s. 6d.
Manner of
between
N
LXXII.
The Accounts of the Bank of Venice are kept in Livres, Sols and
exchanging Deniers Gros. The Livre is equal to 10 Ducats Bank, or 240 Grofs,
England and (the Ducat being equal to 24 Grofs) 100 whereof make 120 Ducats
Venice. current Money, fo that the Difference betwixt Bank and current Mc-
ney is an Agio of 20 per Cent.
Eftimation
London exchanges with Venice, by giving an uncertain Number of
Pence Sterling for the Ducat Banco. The Par of a Ducat Banco is
45. 4d. Sterling, and the Courſe between 45 and 50d.
The Venetian Coins are as follow: 1. A Picoli
66
181
163
I
7 5
5
d. 2°. A Soldi
of the Vene- or 12 Picoli d. 3. A Jule or 18 Soldi 5 d. 4°. A Teſtcon or
tian Coins. 3 Jules Is. 5 d. 5. A Ducat current or 124 Soldi 35. 4d. 6°. A
Chequin or 17 Lires 9s. 2d. Lire Money is divided as the Britiſh
Pound, and I Ducat Banco is worth 7 Lires.
Reduction
4.
5
When Venetian Money is required, as, for Example, how much in
of Sterling Venice for 541. 185. Sterling in London, the Exchange at 45 d. per
to its equi- Ducat Banco ?
-valent Ven.
•
Money.
A
Say, if 45 d. the Rate of Exchange, gives 1 Ducat Banco, how
many Ducats will 5417. 18s. Sterling give? 2850Duc. 10Sol. 10Den.
When Sterling is required, as, for Example, how much in London
of Ven. Mo- for 2850 Duc. 10 Sol. 10 Den. in Venice, Exchange at 45d. per
Ducat?
Reduction
ney to its
equivalent
Sterling.
Manner-of
between
England and
Portugal.
14
73
4-
Say, if I Ducat gives 45 d. what will 285cDuc. 10Sol. 104 Den.
5417. 18s. Sterling.
give?
LXXIII.
In Liſbon, and in general throughout the Portugueſe Dominions,
exchanging Accounts are kept in Milreas and Reas, reckoning 1000 of the latter to
one of the former. The Exchange betwixt London and Portugal is ra-
ted at an uncertain Number of Pence Sterling for a Milrea, according to
the Courſe of Exchange, which runs betwixt 5. and 5s. 8d. per Milrea.
The Par of a Milrea is 5s. 7 d. according to which the Gold Monies of
Portugal are eſtimated as follow:
of the Coins
1º. The Piece of 25 600 double Joannes 71. 4. 2º. Of 24 ditto
Eftimation 61. 15. 3°. Ditto of 12800 fingle Joannes 3. 125. 4°. Of 12
of Portugal. ditto 31. 75. 6d. 59. Ditto of 6 400 half Johannes 1. 16s, 6°. Ditto
of 4800 Moidore ftamped 17. 75. 70. Ditto of 3200 quarter
NUMERAL
159
ARITHMETICK.
Joannes 185. 8°. Ditto of 2
of 1
400 half Moidore 13s. 6d. 9°. Ditto
Coo Joannes 9s. 10°. Ditto of 1
ठ
I
·
200 quarter Moidore 65. gd.
11º. Ditto of o 800 Joannes or Teftoon Piece 4s. 6d.
w
'The Silver Monies are as follow: 1. The Crufado of 400 Reas not
ftamped 25. 3d. 2º. Ditto of 480 Reas ftamped in 1643, 25. 8 3 d. ‍3º. The
12 Vintin Piece of 240 Reas 15, 6d. 4°. The 5 ditto of 100 Reas 9d.
5. The 2 Vintin, ditto of 50 Reas 4 d.
The Copper Coins as follow: The Vintin Piece of 20 Reas 1žd. the
half and quarter ditto..
Reduction-
of Sterling
to its equi-
When Portugueze Money is required, as, for Example, how much in Re
Liſbon for 5781. 19s. 6d. in London, Exchange at 5s. 3d. per Milrea?
Say, if 5. 3d gives 1 Milrea, how many Milreas will 5781. 1gs. 6d,
give? R 2205 Mil. 619 Reas.
valent Por-
tuguiſe Mo-
ney.
Reduction
of Portu-
When Sterling is required, as, for Example, how much in London
for 2205 Mil. 619 Reas in Lisbon, Exchange at 5s. 3d. per Milrea?
Say, if 1 Milrea gives 5. 3d. what will 2205 Mil. 619 Reas give? guile Money
R 5781. 19s. 6d.
LXXIV.
in
to its equi-
valent Ster-
ling.
Manner of
exchanging
between
In Spain the foreign Bankers, or Remitters, keep their Accounts
Piaftres, Rials, and Maravedies, old Plate, reckoning 34 Maravedies to
a Rial, and 8 Rials to a Piaftre,
The Shopkeepers of Madrid, the Cuſtomhouſe, and other Dealers England
within the Kingdom, keep their Accounts in Rials and Maravedies,
and Spain.
Vellon or Copper Money. Some Merchants, particularly in Valentia,
Alicant, &c. keep their Accounts in Piaftres, Sols and Denier, divided
as the French Livre. The Exchange betwixt London and Spain is rated
at an uncertain Number of Pence Sterling per Piaftre, or Piece of 8, ac-
cording to the Courfe of Exchange, which varies from 35 to 40 Pence.
per Piaftre. The Par of the Piaftre is 3s. 7d. Sterling, according to
which the Spaniſh Silver and Copper Coins are eſtimated as follows:
-
4.3
27
nifh Coins..
1º. A Maravedie d. 2. A Quartil equal 2 Maravedies. Eftimation
3º. A Rial Plate equal 17 Quartils or 34 Maravedies 5 d. 4°. A Pif- of the Spa-
trine equal 2 Rials Plate ros. d. 5°. A Dollar old Plate of Seville equal
10 Rials 45. 6d. 6º. A Dollar of new Plate equal 8 Rials Plate 3s. d.
70. Mexico ditto 4s. 6d. 8. Pillar ditto 4s. 6d. 9º. Peru ditto old
Plate 4s. 5d. 109. A croſs Dollar 4s. 4 d.
The Gold Coins are Piſtoles, Half Piftoles, &c. Double Pistoles,
Quadruples. The Piftole is worth 4 Dollars or 175. 11d.. Sterling..
When Spanish Money is required, as, for Example, how much in
Cadiz for 5761. 12. 2d in London, Exchange at 37 d. per Piaftre?
Say, if 37 d. gives 1 Piaftre, how many Piaftres will 5761. 12. 2d.
give? 3653 Piaft: 6 Ri. 7 Mer.
*
Reduction
of Sterling
valent Spa-
to its equi-
Biſh Money.
160
OF
ELEMENTS
Reduction
of Spanish
Money to
its equiva-
lent Sterling
Manner of
When English Money is required, as, for Example, how much in
London for 3653 Piaft. 6 Ri. 7 Mer. Exchange at 37 d. per Piáſtre?
Say, if 1 Piaſtre gives 37 d. what will 3651 Piaft. 6 Ri. 7 Mer. give?
5761. 124. 22d.
LXXV.
In Holland Accounts are kept in Pounds, Shillings, and Pence Fle-
exchanging miſh, divided as the Britiſh Pound; but more generally in Guilders or
between Florins, Stivers, and Phinnings, reckoning 16 Phinnings to a Stiver,
England and
Holland. and 20 Stivers to a Guilder or Florin, and 6 Guilders or Florins to a
Pound Flemish.
Britain exchanges with Holland upon the Pound Sterling, for which
the latter gives an uncertain Number of Shillings and Pence, or Grotes
Flemiſh, according to the Courſe of Exchange, which runs from 30 to
to 40s. Flemish per 20s. Sterling. The Par of a Pound Sterling is
1. 16s. 6d. Flemish; but a Guinea paffes in Holland for 12 Guilders.
According to this Eftimation their Coins may be reckoned, as follows:
1º. A Duke d. 2°. A Stiver 1 d. 3°. A Schilling 6d. 4°. A
I zod.
of the Dutch Guilder Is. 9d. 5º. A Zeland Dollar 2s. 7žd. 6º. A Rix-Dollar
45. 4 d. 7°. A Dry Guilder 5s. 3d. 8°. A Ducat 9s. 2 d.
If Dutch Money is required, as, for Example, how much in Amfter-
of Sterling dam for 270l. 8s. 2d. Sterling, when the Exchange is at 35s. 6d. Fle-
to its equi- mifh per Pound Sterling ?
Eftimation
Coins.
Reduction
valent
Dutch Mo
ney.
Reduction
of Dutch
2 1
Say, if 1. gives 35s. 6d. Flemish, what will 270k. 8s. 2d. give ?
479. 19s. 5d. Flemish.
If Sterling is required, as, for Example, how much in London for
4791. 19s. 6d. Flemish, the Exchange being at 35s. 6d. per Pound
Money to
its equiva- Sterling?
lent Sterling
between
Say, if 17. 15s. 6d. Flemiſh gives 11. Sterling, what will 4797. 195. ád.
Flemish give? ₪ 2701. 8s. 2d. Sterling.
LXXVI.
تجھے
Manner of In Hambourg Accounts are kept in Rix-Dollars, Sols, and Deniers.
exchanging Lubs, or in Marks, Sols, and Deniers Lubs. The Rix-Dollar is worth
England and 3 Marks or 48 Sols Lubs, the Livre Gros or Pound Flemiſh is equal to
Hambourg. to 7 Marks Lubs, or 20 Sols Gros, or 120 Sols Lubs. The Mark
Lubs are divided fometimes into 32 Gros, but more generally into 16
Schillings Lubs, and each of thefe into 12 Phennings.
The Exchanges betwixt London and Hambourg are rated at an un-
certain Number of Schillings and Grotes Flemish per Pound Sterling,
according to the Courfe of Exchange. The Par of their Rix-Dollar is
reckoned at 4s. 6d. Sterling; fo that the Par of 17. Sterling is 13 Marks
5 Schillings Lubs, 35s. 6d. Flemish; according to which Eftimation
their Coins may be eſtimated, as follows:
1
NUMERAL ARITHMETICK
161
128
1". A Tryling of a Phenning d. 2°. A Sexling of a Phen-
ning d. 3. A Phenning of a Schilling Lubs d. 49. One Schil- Eftimation
ling Lubs of a Mark 1 d. 5º. The Dollar equal 2 Marks 3. of the Co
60. The Rix-Dollar equal 3 Marks 4s. 6d. 79. The Ducat of 6 bourg.
Marks 9s. 4 d.
I
When Hambourg Money is required, as, for Example, how much in
Hambourg for 500l. Sterling, the Exchange at 35s. 6d. Flem. Banco
per Pound Sterling?
Say, if I gives 35s. 6d. Flemish, what will 500l. give?
R 6656 Marks, 4 Schillings Lubs.
Coins
Ham-
Reduction
of Sterling
to its equi
valent Ham
bourg Mo
ney.
of Hamb.
How much in London for 6656 Marks, 4 Schill. Lub. in Hambourg, Reduction
the Exchange at 35s. 6d. Flem. Banco per Pound Sterling?
Say, if 35s. 6d. Flem. gives 17. Sterling, what will 6656 Marks, its equiva-
4 Schillings give? 500l. Sterling.
Money to
lent Ster-
The foregoing are the moſt remarkable Places of Exchange in Europe ling.
with which Britain hath Occafion to negotiate; and it is prefumed
that thofe Examples are fufficient, to fhew how Sterling is reduced to its
equivalent Value in the Money of Account in any other Country, and
the contrary, the Courfe of Exchange being given.
LXXVII.
Being commiffioned to remit a certain Sum of Money to a Place, at Method of
a certain Rate of Exchange, and at the fame Time to draw for the Re- arbitrating
imburſement upon fome other Place, at a certain Rate of Exchange; Exchanges,
to determine whether the Advantage in performing the one Part of the by an Ex-
› explained
Commiffion will be fufficient to compenfate for the Lofs that may arife ample.
from the other.
Say, as the Rate of Exchange affigned for remitting, is to the Rate
of Exchange affigned for drawing, fo is the real Rate at which the Re-
mittance can be made, to the Rate at which the Draught ought to be
made to be at Par with the Rate of the Remittance; which being com-
pared with the Rate of Bills upon that Place where the Draught is to
be made, will fhew whether the Order is to be obeyed or not, as will
appear by the following Example.
LXXVIII.
A Factor at London receives an Order to remit to Venice 1000 Ducats, at
4s. Sterling per Ducat, and, for this Purpoſe, to put himſelf in Caſh by draw-
ing on Spain at 35. 2d. per Piaftre. When this Order came to Hand, Bills for
Venice were at god. at what Price muft London draw upon Spain, to compen-
fate the Advance on the Remittance to Venice by the Rife of the Exchange ?
2 X
162
ELEMENTS OF
What is
meant by
Stock,
The differ
To folve this Queftion, let the Number of Piaftres, at 38d. per Piaftre,
that the Factor is to draw from Spain, to put himfelf in Caſh to make
the Remittance of 1000 Ducats, at 48d, per Ducat, be expreffed by x;
then 1000 X 48d. equal x X 38d. or 48: x = 38: 1000. But fince he
is obliged to remit at 50d, per Ducat, and yet not draw on Spain for
a greater Number of Piaftres, the Piaftres must be rated higher than
at. 38d. to anfwer this Remittance. Let this Rate be expreffed
by y, then 1000 X 50d. = x Xy, or x: 50 = 1000: y; wherefore
1000:y;
48Xx: 50Xx= 38×1000: yX1000, or 48: 50'= 38d. y equal 39 7zd.
3,
LXXIX.
By the Word Stock is meant any Sum of Money lent to Government,
or to publick Companies, on Condition of receiving a certain Intereft till
the Money is repaid. Every Stock, or Fund of a Company, being raiſed
for a particular Purpofe, and limited to a certain Sum, when the Fund
is compleated, no Stock can be bought of the Company, though Shares
already purchafed may be transferred from one Perfon to another;
whence there is frequently a great Difproportion between the original
Value of the Shares, and what is given for them when transferred; for
if there are more Buyers than Sellers, a Perfon who is indifferent about
felling, will not part with his Share without a confiderable Profit; and
on the contrary, if many are difpofed to fell, and few Purchaſes appear,
the Value of Stocks will naturally fall in Proportion to the Impatience of
thoſe who want to turn their Stock into Specie.
The prefent Government Funds are, 1°. Three per Cent. reduced
ent Govern- Bank Annuities. 2. Three per Cent, confolidated ditto. 3º. Three
ment Funds, per Cent. ditto, 1726. 4°. Ditto, 1751. 59. Three and a Half per
and Stocks Cent. ditto, 1756. 6°. Ditto, ditto, 1758. 7°. Four per Cent. ditto,
nies. 1760. 8°. Long Annuities, 1761. 9°. Four per Cent. Subfcription,
1762. 10. Long Annuities, 1762.
of Compa-
Computa
tions in
Stocks of Companies are, 1°. Bank Stock.
3º. India Stock. 4°. South Sea Annuities.
6. India Bonds.
LXXX.
2°. South Sea Stock.
5°. India Annuities.
The Price of a Quantity of Stock bought or fold is obtained by mul-
tiplying the Quantity by the Rate per Cent. and dividing the Produc
Stock-job- by 100; as will appear by the following Examples.
bing.
What must be paid for 135 Annuities 3, at 87 per Cent.?
To folve this Question I fay, if 100 is reduced to 87, what will 135
be reduced to? R 1187. 2s. 6d.
What is the Value of 971. 10s. Bank Stock, at 120 per Cent. ?
To folve this Question I fay, if 100 becomes 120, what will 97%. 10s,
become? R 117%
NUMERAL ARITHMETICK.
163
LXXXI.
A TABLE, exhibiting at one View the intrinfick Value per Cent. of Table, ex-
hibiting the
the feveral publick Funds, and the Proportion they bear to each other, intrinuck
by which any Perfon may know which it will be most advantageous to Value of
purchaſe, and what Proportion fuch Purchaſes bear to the Value of land- the diffe-
rent publick
ed Eſtates, and Life Annuities.
Funds.
3 per
C.
3/4 4 1/2
Years Ann. Inter
3 per C. 3 4
556 Par. 5 per C. 3 per
at 60 70 80 90 100
61171
3
2
120 20 5 O о
82 92102112123 2024 17
176
63 73 84 94105 1151126 21 4 15
6475 86 96 107118129 214 13 0
66 77 88 99 100 121 132 22 4 10 10
6778 90101112123135 224 8 10
698092103115 1261138 23 4
23 4
70829410421171201412324
T
6 11
5 }
72 84 96108 120 132 144 24 4 3 4
7385 98110 120 134147 2424 I 7
75 87 100 1124125137150 25 4 0 0
7689102114127140153253 18 5
78 91 104117 130 143 156| 26 |3 16 11
791922106119|13221453159 263 15
81 94108 121|135 |148|162 27 3 14
Years Ann. Inter.
6 Pur. 5 per C.
at 829611012331372151165 273 12 C
84 98 112126 140 154 168 28
3 11
85 991141281421563171 283 102
87 1011161302145 1592 74 29 3 9 с
I
881103118132147162177|29|3
7
4
2
7
7
5
6
90 105 120 135 150 165 180 30 13 6 8
911061221371521673183 303 5
93108124139155 170186 31 3
941101 261411572173189 313 3
96 112 128144 160 176 192 32 3
97113130146|1621|1782|195 32|3 I 6
99 11
1151132|148|165 181198 33 3 0 7
100117134150167184201 332 19 8
5 102 119 1 3153 170 187 204 34 2 18 10
10312031381554|172|1893|207 342 18
the forego
To illuftrate the foregoing Table by an Example, let it be fuppofed
that 3 per Cent. Annuities may be purchaſed at 84 per Cent. and India The Ufe of
Stock at 140. In the Column of 3 per Cent. find 84, and oppofite there-ing Table 3
to, in the Column of Intereft, it will be found to produce 37. 11s. 4d. In explained
the Column of 6 per Cent. the Dividend on India Stock ftands 168, which by an Ex-
ple.
ſhews, that when 3 per Cent. give 84 per Cent. India Stock ought to give
1681. fo that to purchaſe India Stock is, in this Cafe, by 28 per Cent, a
better Bargain than 3 per Cent. Annuities.
LXXXII.
When it is propofed to divide a given Number into Parts proportional
to thoſe of another Number divided any how, the Operation to be per-
formed to folve this Queſtion is called a Rule of Company; becauſe it is
uſually employed to divide Gains or Loffes among Merchants in Com-
pany.
Rule of
Company.
164
ELEMENTS OF
General
It appears from this Definition, that each of the Parts of the Number
Rule for to be divided will be found by the following Operation of the Rule of
performing Three.
a Rule of
Company.
Application
As the Number already divided, is to one of its Parts; fo is the Number to
be divided, to one of its Parts correfponding to the Part that has been taken for
Second Term
A Rule of Company may be either fimple or compound. It is fim-
ple when the Rules of Three on which its Solution depends are fimple,
and is compound when thofe Rules are compound. As the different
Rules of Three have been fufficiently explained, a few Examples will
be fufficient to fhew how every Rule of Company, either fimple or
compound, is to be performed.
LXXXIII.
Three Merchants, A, B, C, were concerned in a Store of Corn, whereof
of the fore A bad 3000 Sacks, B 2500 Sacks, and C 4500 Sacks. The Store was fold
going Rule
to an Exam off for 12700l. what is the Dividend of each?
ple.
A more
commedi
It is manifeft that in order to folve this Queftion, we muſt divide
12700/. into Parts proportional to the Shares 3000 Sacks, 2500 Sacks,
and 4500 Sacks of the Merchants in the Store; and that the whole
Number of Sacks is to their Produce, as each particular Number of
Sacks is to the Dividend of each Merchant.
I therefore add together the three Numbers of Sacks of Corn that
compoſe the Store, and there refults 10000 Sacks of Corn, which is
worth 12700/. I then fay,
1º. If 10000 Sacks are worth 127001. what will 3000 Sacks be worth?
B 3810%.
2º. If 10000 Sacks are worth 12700l. what will 2500 Sacks be worth?
R 31751.
39. If 10000 Sacks are worth 12700l. what will 4500l. Sacks be worth?
B 57151.
We might have found the Dividend of the third Merchant C without
employing the Rule of Three; for adding together the Dividends of
the two firft Merchants, 3810l. and 31757. and fubtracting their Sum
69851. from the Price of the whole Store 12700l. the Remainder 57157.
will be the Dividend of the third Merchant.
LXXXIV.
When it is required to find a great Number of proportional Parts, it
will be more commodious to inveſtigate the Part of the Number to be
ous Method divided that correfponds to an Unit of the Number already divided, and
of finding then multiply by this Part each Part of the Number already divided; by
ional Parts. this Means all the proportional Parts required will be obtained, without
employing as many Rules of Three as there are Numbers to be found.
the propor
NUMERAL ARITHMETICK.
165
LXXXV.
Suppofe a Bankrupt's Effects bould amount to 17391. 13s. 8d. what Di-
vidend thereof will fall to each of the following Creditors in Proportion to their
respective Sums? He owes to A 3131. 7s. 3d. to B 2901. 4s. 6d. to C 700l.
to D 4861. 135. 8d. to E 6ool, to F 500l to G 3811. 10s. and to H 4187.
To folve this Queſtion, I firſt find the Dividend for 17. by the follow-
ing Rule of Three.
As the total Debt 36891. 155. 5d. is to the whole Subje& 17397. 13s. 8žd.
fo is 1. to 0,4714875, the Dividend for 17.
Whence the Dividend of A will be 3137. 75. 3d. X,4714875, that is,
1477. 14s. 114d. The Dividend of B will be 2901. 45. 6d. X4714875,
that is, 136, 16s. 9 d. The Dividend of C will be 700l. X,4714875,
that is, 330l. os, 10d. The Dividend of D will be 4861. 135. 8d. X,4714875,
that is, 2297. gs. 32d. The Dividend of E will be 600l. X,4714875,
that is, 2827. 175. 10d. The Dividend of F will be 500l. X,4714875,
that is, 235/. 14. 10 d. The Dividend of G will be 3817. Ios. X,47148755-
that is, 1797. 175. 54d. And the Dividend of H will be 418/. X-4714875
that is, 1971 is. 8d.
LXXXVI.
:
pound Rule
A Merchant put 40000l. into Trade. At the End of 6 Month's a Second
Merchant takes a Share of 2500l. in the 40000l. and at the End of two other The com-
Months the first Merchant made over to a third a Part 3000l. of his Share of Compa-
that remained in the Capital 40000l. and at the End of 6 other Months there ay, explain-
was a nett Gain of 18000l. Now it was agreed by the Parties, that each ed by an
fhould have a Share of the Gain proportional to their Shares in the Capital,
and the Time of their Continuance in Trade; the Dividend of each Merchant
is required.
Since the Capital 40000l. has been 14 Months in Trade, and confe-
quently has gained during thofe 14 Months as much as 14 times 40000%.
would gain in one Month, 5600ool. may be confidered as the Cauſe of
the whole Gain.
The fecond Merchant having, at the End of 6 Months, taken a Share
of 2500/. in the 40000%. the 2500k of this Merchant have been 8 Months
in Trade, and confequently will have gained during 8 Months as much
as 8 times 2500l. or 20000l. would have gained in a Month; whence-
20000l. may be confidered as the Caufe of the Gain of the fecond
Merchant.
The third Merchant having taken, at the End of two other Months,
a Share of 3000l. in the 40000l, the 3000l. of this Merchant have been
6 Months in Trade, and confequently will have gained during 6
Months as much as 6 times 3000l. or 18000l. in one Month; whence
18000l, may be confidered as the Caufe of his Gain. But the Causes of
Example.
166
ELEMENTS OF
J
The diffe
rent Species
of Rules of
falfe Pofi-
tion.
the Gains are proportional to the Gains; whence the Gain of the fe--
cond Merchant will be found by the following Rule of Three.
As 560000l. reduced Caufe of the whole Gain, is to the whole Gain 18000l
fo is 20000l. reduced Caufe of the Gain of the Jecond Merchant, is to the Gain
of this fecond Merchant; which will be found to be 7017. 175. old.
In what the
fition con-
The Gain of the third Merchant, who had 3000l. in Trade during
6 Months, will be found by the following Proportion.
As 560000l. reduced Caufe of the total Gain, is to the total Gain 18000%
fo is 18000l. reduced Caufe of the Gain of the third Merchant, to the Gain
of this Merchant; which will be found to be 5787. 11s. 5d.
To obtain the Gain of the first Merchant, there is no Rule of Three
required; becauſe, the ſecond having gained 6421. 175. 1 d. and the
third 5781. 11. 5d. deducting their Sum 12211. 8s. 6d. from the to--
tal Gain, the Remainder will be the Gain of the firſt Merchant
16778% 11s. 57d.
LXXXVII.
The Rule of falſe Pofition reſembles the Rule of Company, being
employed to divide a given Number, or a Part of a given Number, into
Parts proportional to thofe of another Number taken at Will, as the
Nature of the Queſtion may require.
The Computifts diftinguiſh two Species of Rules of falle Pofition,
viz. the Rule of one falfe Pofition, and the Rule of two falfe Pofitions.
In the Rule of one falfe Pofition there is only one Suppofition made
of Parts, proportional to thofe into which the propoſed Number is to be
divided; but in the Rule of two falfe Suppofitions 'there are two Sup-
pofitions made, both falfe, whereby the true Parts of the propofed Num-
ber to be divided are difcovered.
LXXXVIII.
The Rule of one falſe Poſition confifting in affuming Parts propor-
Rule of fin- tional to thoſe into which the propoſed Number is to be divided, thoſe
gle falfe Po- fuppofed Parts form a falfe Pofition, as not being equal to the Parts into
which the propoſed Number ſhould be divided; but as thofe fuppofed
Parts are proportional to thoſe required, the Totality of thoſe fuppofed
Parts is to each of them in particular, as the Number given to be divi-
ded is to each of its Parts required.
fifts.
The Rule
of fingle
LXXXIX.
2
A Father dying, left his Wife with Child, and an Estate worth 20000..
to whom he bequeathed, if fhe had a Son, of his Eftate, and to the Son;
falle Pofi- but if he had a Daughter, to ber, and to ber Mother. It happened that
tion explain ſhe had both a Son and a Daughter; how fall the Eftate be divided to anſwer
ed by an Ex the Father's Intention ?
ple.
3
NUMERAL
167.
ARITHMETICK.
N
As the Father plainly defigned the Son to have double of the Mother's
Part, and the Mother double of the Daughter's Part, therefore for
every 1 the Daughter got the Mother muſt have 2, and the Son 4;
whence the Queſtion is reduced to divide 20000l. into Parts proportional
to 4, 2 and 1, ſaying, the Totality 7 of thoſe Parts is to each of them in
particular, as the Number 200ool, is to each of its Parts required. The
Operation being performed, the Dividend of the Son will be found to be
1428. the Dividend of the Mother 5714. and the Dividend of the
Daughter 2857 3/4
XC.
An Army employed three Days to pass thro' a Town: The of the Army
paffel the first Day, the the Second Day, and 6000 Men the third Day.
The Number of Men the Army confifted of is required.
3
3.
Let the whole Army be expreffed by 1, the Part that paffed thro' the Application
Town the first Day will be expreffed by, and the Part that paffed the of the Rule
of fingle
fecond Day will be expreffed by . Thoſe two Parts, reduced to the falle Pofi-
fame Denomination, will be 2 and 3, whofe Sum is g, which fubtracted tion to ano-
from the whole Army 1, leaves for the Part of the Army that paffed ple.
thro' the Town the third Day. Now as the Parts are proportional to
2
the wholes, I fay, as, the Number expreffing the Part of the Army
that paffed thro' the Town the third Day, is to I, the fuppofed Num-
ber expreffing the whole Army; fo is 6000 Men, the real Part of the
Army that paffed thro' the Town the third Day, to 36000, the real Num-
ber of Men that the Army conſiſted of.
XCI.
In a Rule of two falfe Pofitions, it is Queftion of dividing a Num-
ber into two Parts, and to divide again one of thofe Parts into Parts.
proportional to other fuppofed Parts. Thofe two Divifions require two
falfe Suppofitions, as will appear by the following Example.
ther Exam-
Let it be propofed to divide 1201. between three Perfons, fo that the firft The Rule
may have twice as much as the fecond and 31. over, and the third as much as of double
the two first and 41. over.
falfe Pofi-
tion ex-
If the Share of the firft was fuppofed to be 17. the Share of the fecond plained by.
would be 51. and the Share of the third rol. confequently the Totality of an Example.
the Shares would be 16. Here is the firft Suppofition, which is falſe,
not only becauſe the fuppofed Shares are not the real ones, but alſo be-
cauſe thoſe Shares are not proportional to thofe into which 120/. is to be
divided; for the two laft fuppofed Shares include each two other Parts,
one of which is relative to the firft Share 17. and the other is a determin-
ed Quantity.
The fecond Share 51. for Example, is compoſed of two Parts, 21, and 3%.
168
ELEMENTS, &c.
one of which ſhould be double of the first fuppofed Share, and will change
its Value proportionally to the Variations made in the firft Share 17.
whilft the fecond Part 31. is a determined Magnitude, which will not be
altered when the Value of the firft Share 17. is changed.
Each Share being thus confidered as compofed of two Parts, one of
which is relative to the firft fuppofed Share, and the other is a determin-
ed Quantity, we muſt examine which is the Portion of the Number 1201.
containing the Parts of the Shares proportional to the firft fuppofed Share,
and which is the Portion of the fame Number 120/. that contains the
determined Parts of thofe Shares, and when this lat Portion of 20l. is
found, fubtract it from 120/. in order to obtain the former.
To determine this fecond Portion of 1207. I make a fecond Suppofi-
tion, in which the determined Parts 31. and 4. do not enter, that is, I
confider the Queſtion as if it was propofed thus:
To divide 120l. among three Perfons, fo that the Jecond ſhall have twice as
much as the first, and the third as much as the two others.
I ſuppoſe as before the Share of the first to be 17. confequently that of
the ſecond will be 24 and that of the third will be 37.
This fecond Suppofition is alfo falfe, not only becauſe the fuppofed
Shares are not the real ones, but likewife becauſe they are not propor-
tional to the Shares required.
As the Shares taken in this fecond Suppofition do not contain the de-
termined Parts 31. and 47. their Totality 67. will not contain the Refult
of thoſe two determined Parts, as the Totality 161. of the Shares of the
firft falfe Poſition does.
Wherefore, if the Sum 61. of the three Shares of the fecond Suppofi-
tion be deducted from 167. the Sum of the three Shares of the first Sup-
pofition, the Remainder 10l, will be the Portion for which the deter-
mined Parts 31. and 47. enter into the Sum 120l. to be divided; whence
deducing 10l. from 120l. the Remainder 110l. will be the Portion that
contains the Parts of the Shares relative to the first Share; whence to
find the first Dividend,
I fay, as the Sum 61. of the three fuppofed Shares, is to the firſt Share
1. fo is 110l. to the firſt Dividend 181. 6s. 8d.
The fecond Dividend being equal to twice the firft, and 31. over,
will be 391. 135. 4d. and the third Dividend being equal to the two firſt,
and 4% over, will be 621.
END OF NUMERAL ARITHMETICK.
[ ]
ELEMENT S
O F
SPECIOUS
ARITHMETICK.
CHA P. I.
Of the analytick Methoi of expreffing Problems by Equations, and of the
Refolution of Equations of the first Degree.
AMONGST the different Problems which employed the firſt
Mathematicians, called Analyfts, I chufe the following, as the moſt pro-
per to fhew how they formed the Science ftyled fpecious Arithmetick.
I.
a Problem
thoſe which
To divide a Sum, for Example, £890 between three Perfons, in fuch a Example of
Manner that the firft may have £180 more than the fecond, and the fecond fimilar to
£115 more than the third.
It is thus I imagine a Perfon would have argued, who, without the the firft Ana
leaſt Tincture of fpecious Arithmetick, attempted to ſolve this Problem. have propo
It is manifeft that if one of the three Parts was known, the other ed to them-
two would be immediately difcovered; let us fuppofe, for Example, the felves.
third which is the leaſt to be known, we muſt add 115 to it, and this this problem
lyfts might
Solution of
Sum will be the Value of the fecond; to obtain afterwards the firft Part, fuch as
we must add £180 to this fecond, which comes to the fame as if we ad-
ded £180 more £115 or £295 to the third.
Let therefore this third Part be what it will, we know that this Part,
more itfelf together with £115, more itfelf again together with £295
ſhould make a Sum equal to £890.
From whence it follows that the Triple of the leaft Part, more £115
more £295, or more £410, is equal to £890.
But, if the 'Triple of the Part fought more £410 be equal to £890, this
Triple of the Part fought muſt be lefs than £890 by £410, therefore it is
equal to £480, therefore the leaft Part is equal to 160. The fecond
will confequently be £275, and the first or greateſt £450.
-
It is probable the first Analyfts argued in this Manner when they pro-
poſed to themſelves Queſtions of this Nature, without doubt in propor-
tion as they advanced in the Solution of a Problem, they burthened
their Memories with all the Arguments which had conducted them to
might be
found with-
out fpecious
Arithmetic.
3 A
2
ELEMENTS OF
Algebraic
the Point they had arrived at, and when the Problems were not more
complicated than the foregoing, it was no difficult Matter; but as foon
as their Reſearches prefented a greater Number of Ideas to be retained,
they were under the Neceffity of having recourfe to a more conciſe Me-
thod of expreffing themſelves, and of employing fome fimple Symbols,
by Means of which however advanced they were in the Solution of a
Problem, they might perceive at one View what they had done and
what remained for them to do. Now the Kind of Language they ima-
gined for this Purpofe, is called fpecious Arithmetick.
II.
To explain the Principles of this Science more clearly, we will re-
Method of fume the fame Queſtion, write down in Words the Arguments which
expreffing
the forego- the Analyft employs to folve his Problem, and in analytick Symbols what
ing Problem is requifite to affift his Memory.
The Sign+
The leaft or third Part, be it what it will, I denote by one Letter,
for Example by x.
The ſecond confequently will be x more 115, which I denote thus,
denotes ad-x+115 employing the Sign + which fignifies more to expreſs the Addi-
tion of the two Quantities between which it is placed.
dition.
The Sign=
As to the firſt Part or greateſt, fince it exceeds the fecond by 180, it
will be expreffed by x + 115 + 180.
Adding thofe three Parts we will have 3 x + 115 + 115 + 180, or
when reduced 3 x + 410.
=
But this Sum of the three Parts fhould be equal to 890, which I
defigns E- exprefs thus 3x+410 890, employing the Symbol which figni-
quality. fies equal to denote the Equality of the two Quantities between which it
is placed.
An equation
lity of two
The Queſtion therefore by this Computation is changed into another,
is the equa- where it is required to find a Quantity, the Triple of which being ad-
quantities. ded to 410 makes 890. To find the Refolution of fimilar Queſtions, is
To folve an what is underſtood by folving an Equation, the Equation in the prefent
equation is Cafe is 3x+410890 it is fo called, becauſe it indicates the Equa-
value of the lity of two Quantities, to folve this Equation, is to find the Value of
unknown the unknown Quantity x from this Condition that its Triple more 410
quantity it
makes 490.
to find the
includes.
Refolution
tion which
III.
To folve this Equation the Analyſt argues and writes down his Ar-
equa- guments as follows. The Equation to be folved 3x+410890, teaches
expreffes the us that we are to add 410 to 3 x to make up the Sum 890, wherefore
foregoing 3 x are less than 890 by 410, which I exprefs thus 3 x 890-410 em-
ploying the Sign which fignifies lefs to denote that the Quantity
denotes which it precedes fhould be fubtracted from that which it follows.
From this new Equation 3 x 890-410, we deduce, by fubtracting
in effect 410 from 890, this other Equation 3 x = 480.
Problem.
The ſymbol
fubtraction.
=
SPECIOUS ARITHMETICK.
3
=
But if three be equal to 480, one x will be the third Part of 480
er 160, which I write down thus, x430 160, and the Queftion is
folved, fince it fuffices to know one of the Parts to difcover the reft.
IV.
3
If we were defirous of folving the Queſtion by inveſtigating the great-
eft Part first, we might effect it in like Manner.
Another fo-
lution of the
foregoing
Let this firſt Part be expreffed by y, the fecond having 180 lefs, will Problem.
180, and the third having 115 lefs than the fecond, will be
y 180 115.
be y
Now the Sum of thoſe three Quantities is 3 y 180 180 115,
that is, 3 y 475.
But this Sum fhould be equal to 890.
We have therefore the Equation 3 y
475
890, by which we
learn that 3 y exceeds 890 by 475, fince we muſt ſubtract 475 from 3 y
to obtain 890. Wherefore 3y=890 + 475, or 3 y=1365. Where-
fore y or the greateſt Part 455, as found above.
V.
If in the Problem it was propoſed to divide a Sum greater or leſs than
the one employed, and that the Differences were Numbers different
from thoſe made ufe of, it is manifeſt it would be folved after the ſame
Manner. Let us fuppofe, for Example, the Problem to have been ex-
preffed thus.
Another ex-
ample of the
To divide 9600 between four Perfons, in fuch a Manner that the first
may bave 300 more than the fecond, and the ſecond 250 more than the third, foregoing
and the third 200 more than the fourth.
We would have argued after the following Manner: Let the fourth
Part be x, the third will be x + 200, the ſecond x + 200 + 250, and
the first x + 200 +250 + 300..
But the Sum of all thoſe Parts fhould be equal to 9600, we have
therefore the Equation 4x+1400 9600.
=
To folve this Equation, I obferve, as in the foregoing, that if to 4 %
1400 is to be added to make it equal to 9600, it muſt be equal to what
remains of 9600 when 1400 has been fubducted from it, which is wrote
down thus, 4 x = 9600 1400, or 4 x = 8200.
x =
But if four x be equal to 8200, one x will be equal to the fourth Part
of 8200, that is, 8200 2050, the leaft Part x being known, the
other Parts will be immediately difcovered, the third = 2250, the fe-
cond 2500, and the firſt 2800.
VI.
The Problem may be ftill varied, and yet ftill depend upon the fame
Principles; Let us fuppofe, for Example, that it was expreffed thus.
To divide 5500 into two Parts, fo that the firft may have a third more
than the fecond, together with 180.
Problem,
Third exam
ple of the
foregoing
Problem.
4
ELEMENTS OF.
It would be folved in the following Manner: Let the fecond Part be
x, the firſt will be x+x+180. Now as their Sum fhould be equal
to 5500, 2.x
we will have the Equation. 2. + 3x + 180 5500.
3
To folve this Equation, I first add 2 x & x together, and their Sum
will be x, becauſe two Integers is equal to fix thirds, and confequently
thoſe two Integers with one third make up feven thirds. Wherefore the
foregoing Equation is reduced to 3x + 180 5500, which, by the
fame Reafoning employed in the foregoing Examples will become
3 x = 5500· 180, or x=5320.
7
3
7
उ
Now if the third Part of 7x be 5320, the whole 7x will be three
The fign X
Times that Sum, which I expreſs thus, 7 x = 5320 X 3 employing the
denotes mul Sign X to denote the Multiplication of the two Quantities it feparates.
Afterwards, instead of 7x=5320 X 3, it fuffices to write 7x=
15960, which reſults on multiplying in Effect 5320 by 3.
tiplication.
A new pro-
fame nature
And by the Means of this new Equation we have x = 59602280,
Value of the ſecond Part.
The firſt Part will be readily found, fince we have only to add to this
Quantity 2280 its third Part 760 and 180 more, agreeable to the Con-
ditions, and we will have 3220 for this firſt Part.
Beginners may exerciſe themſelves in varying ftill more the Expreffions
of the foregoing Problem, and folving it in the different Cafes they may
imagine, the Readineſs they will acquire will recompence them fuffici-
ently for their Trouble. For their further Affiftance, I fhall propoſe
another Problem, in feveral Reſpects fimilar to the former.
VII.
Three Merchants form a Society, the first furniſhes £17000, the fecond
blem of the £13000, the third £10000; but as they have Need of a Perfon to tranfact the
with the Bufinefs attending their Commerce, he who furnished £10000, is fatisfied to
foregoing. take on bimfelf this Trouble, on Condition of being allowed 3 per Cent on the
whole Profit that will arife more than the reft: It happens that this Profit
amounts to £100000, it is required to determine the Share of each.
Let the Part of the first be x.
The ſecond having furnifhed lefs in the Ratio of 13 to 17 fhould have
a Sum lefs in this fame Ratio, that is, only 1 x.
1.
I 3
The third, ſuppoſing he received only a Sum proportioned to what he
furniſhed, would have the ten 17ths of the firft, but as he is to have be-
fides 3 per Cent upon the whole Profit, that is £3000 his Share will be
1 x + 3000.
10
17
be
1 3
17
And as the Sum of thofe three Parts fhould
x + 13 x + 19×+ 3000 = 100000, or x +
To difengage the unknown Quantity in this
that x + x + 19 x, or 17 x + 13 x + 18
1 3
17.
£rooooo, we will have
x + 19x=97000.
Equation, I obferve,
is the fame as 49x;
SPECIOUS ARITHMETICK.
5
wherefore 49x=97000, or 40 x 97000 X 17, or 40 x = 1649000,
or x=
1649000
=41225.
40
The Share of the first being found, that of the fecond expreffed by
13 x, will be 1 X 41225, that is 31525, that of the third expreffed by
19 x + 3000 will be 19 X 41225 + 3000=27250.
17
VIII.
The analy-
tick folution
problem
By thofe two Problems the Readers may form fome Notion of Speci
ous Arithmetick, they learn that in general the Solution of a Problem
conſiſts of two Parts, in the firft the Analyſt expreffes by a Letter as x confifts of
or y, &c. the unknown Quantity fought, or one of thoſe which when two parts.
known, ferves to determine the reft, he afterwards endeavours to arrive the problem
at an Equation including the unknown Quantity, which is effected by is expreffed
expreffing the fame Quantity two different Ways.
In the firſt
by an equa-
tion.
cond the
equation is
In the ſecond it is Queftion of bringing the unknown Quantity to one In the fe-
Side of the Equation, leaving only known Quantities on the other.
The firſt of thoſe two Parts is not eaſily reduced to precepts intelligible folved.
to Beginners, and perhaps can be learned only by Examples.
As to the ſecond Part, it may be explained with much greater Eafe, after
a general Manner.
IX.
degree are
In the Queſtions which we have already folved, we arrived at Equa- Equations
tions, in which the unknown Quantity was affected only by the Multi- of the firft
plication or Divifion of known Numbers; thefe Kinds of Equations are thofe in
called Equations of the first Degree, fuch are 2x-
10 = 56, which the
3 x + 15 = x x + 30, &c. and the Problems which lead to thoſe
Equations, are called Problems of the first Degree.
5
ΙΟ
They are fo called to diftinguish them from thofe in which the
known Quantity is raiſed to the Square, Cube, &c. which are faid to
of the fecond Degree if the unknown Quantity be fquared, &c.
un-
be
unknown
quantity is
only multi-
plied or di-
vided by
known quan
tities.
Let it be propofed, for Example, to find a Number, the Triple of
which being added to its Square, will be 65, the Problem to be folved
in this Cafe is of the ſecond Degree, and the Equation 3 x + x x =65,
(in which x x denotes the Square of x) which expreffes the Conditions
of this Problem is an Equation of the fecond Degree.
The first Analyfts could not have attained to the Refolution of thofe
Equations till they had been a long Time exercifed in folving Equations
of the firſt Degree. We fhall therefore proceed to inveftigate the Rules
which thofe require.
X.
To find them, let us firſt reſume the Equation 4x1400 9600
treated Art. v. which is compofed of three Terms, 4 x, 1400, 9600 The terms
(thus are called all the Parts of an Equation feparated from each other of an equa-
by the Signs+ or -) and let us obferve, that by the fame Reaſoning,
by which we deduced 4 ≈
x = 9600 1400, we may in all Kinds of rated by the
tion are its
parts lepe-
6
'ELEMENTS OF
I
3
figns+or Equations take any Term whatfoever preceded by the Sign+and carry
it over to the other Side of the Sign by giving it the Sign. If we had
for Example, '50+x=5x+30, we may carry the Term 10 x
over to the other Side, giving it the Sign and write down the Equa-
tion thus, 505 + 30-x; for we may fay, as in Art. v. that
fince is to be added to 50 to make it equal to the Quantity
530, therefore 50 muſt be lefs than 5x+30 by the Quantity x,
that is, it must be equal to 5 x + 30 LX.
Any term
may be car-
ried over
ΙΟ
3
50=5x+30
I
We may in like Manner as we found, Art. III. that the Equation
31-475 890 was changed into 3 y890 +475, we may, I fay, per-
33-475890
ceive that in general, the Terms which have the Sign on one Side of
the Sign of Equality, may be carried over to the other, by giving them
the Sign+. If we had, for Example, 32-6x=9x+119, we would
conclude that 32=6x+9×+ 119. For if 32 ſhould be diminiſhed by
6x to equal 9x+119, it muſt exceed this Quantity by 6 x, that is, it is
equal to 6x+9x+119.
XI.
Hence there refults this general Principle for all Equations, that we
may carry over what Terms we pleaſe from one Side of the Equation to
the other, provided we change their Signs. Now this Principle is of
fide of the infinite Ufe, in as much as it faves a great deal of Reaſoning.
from one
equation to
the other on
changing its
fign.
The mem-
XII.
By its Means we may change any Equation into another, in which
there will be on one Side of the Sign, that is, in one of the Members
of the Equation, the Terms affected by x, and on the other Side of the
equation are Sign, that is, in the other Member of the Equation the Terms which
its two parts are entirely known.
bers of an
ſeparated by
the fign=
Let, for Example, the Equation 8x + 30 x 250 be given;
I deduce from it 8x
3
5 x = 250
/ x=250-x, I deduce x
other Equation.
60
XIII.
94
+
30; in like Manner from
x=250-60, and fo of any
When after the neceffary Tranfpofitions we have carried over all the
Terms affected by x to one Side of the Equation, and the known Terms
to the other; what moſt naturally occurs, is to reduce each of the two
Members of the Equation to its moft fimple Expreffion. Let, for Ex-
ample, the Equation 8 x x = 250 30 be given; I reduce it to
- 3
x=220, by taking away in effect 30 from 250, and ½ x from 8 x₁
or from 24 x which is equal to it. The Equation 3x-4x=250-60
will be reduced to 12x190, becauſe x and x brought to the fame
Denomination become 22 x and 12 x whofe Difference is 1x, and taking
5
1x
away 60 from 250 there remains 190.
19
3
28
XIV.
7
3
5
£
By fimilar Reductions all Equations of the firft Degree, however com
1
SPECIOUS ARITHMETICK
pofed, will be changed into others, confifting only of two Terms, one
of which compofed of a certain Number of either integral or fractio
nal, and the other a. Term entirely known; fuch are the Equations
4x=8200, 7x=5320 and folved in the Articles vandvI...
$
4
Let us now refume what we have faid concerning thofe Equations, in
order to deduce from thence general Principles applicable to all other
Equations.
=
4
E
tor which
From the Equation 4 x 8200 we deduced x 8200 becauſe 4 Method of
being equal to 8200, it followed that ene x must be the fourth Part of making the
multiplica-
that Sum, from this Reaſoning, and from thoſe which may be framed in
like Manner for other Numbers of x, we deduce this general Principle, affects the
that the Multiplier which affects the unknown Quantity in one of the unknown
Members of the Equation may be taken away, by making it ferve as a appear.
Divifor to the other Member.
3
XV.
quantity dif
From the Equation x 5320 we deduced 7 x=3× 5320, obferv- Method of
ing that if the third Part of 7x was equal to 5320, the whole 7 x muft making the
be equal to three Times that Sum. From whence we deduce this general which af
Principle, that the Divifor which affects the unknown Quantity in a
Member of an Equation may be taken away, by making it ferve as a quantity dif-
Multiplier to the other Member.
XVI.
With thoſe Rules we are enabled to folve all Kinds of Equations of
the firſt Degree. To exercife Beginners here are fome Examples.
IS
-
=
7
$x
3
8
***
15
I
fects the
unknown
appear.
*
Examples of
gree folved
by the fore-
ciples.
§ x − 90 + 3 x = 82 is changed by Tranfpofition into equations of
{ x + 3 x − 1 × 90 82, and by Reduction becomes x-x8, the firft de
or 18 x 1 x = 8, or
948
8, or 8x8 X 15, or infine x 15.
In like Manner x+9=4x-10 by Tranfpofition is changed in- going prin-
to x 2 x = 10 + 9, or * x=19, or 21 *= 19, or x=399.
Likewife x40x60 x, by Tranípofition becomes
3/2 x − 1 x + 3x=100, which by reducing firft 3 & 4 to the fame Deno-
minator, becomes xx100, and by reducing & to the fame
Denominator, becomes 24% x=100, or x =
80
4
5
I
XVII.
18000
247
澧
​36
Inſtead of reducing all the Fractions to the fame Denominator, all
the Divifors of the Equation may be made to vaniſh one after the other,
by the following Method, which must have foon occurred to thoſe who
firft handled thofe Kinds of Equations.
2
Let the foregoing Example x-x+7x=100 be refumed, it is Method of
manifeſt that if the two Members of this Equation be multiplied by 9, making the
the two Products will be the fame; for equal Quantities multiplied by the fractions of
fame Number ſhould give the fame Product, we will have by this Mul- difappear.
tiplication - 2x + 3 x = 900, which on Account of x=2%,
18 x
63
18
an equation
1
$
ELEMENTS OF
}
Another
method
whereby
once.
../
2
is reduced to 2 x
63
2/2 x + 3/3 x900, in which the Divifor 9 has va-
nifhed, and it is eafy to perceive that it fhould neceffarily do fo; for of
any Quantity whatſoever multiplied by 9 fhould give 2 Integers of this.
Quantity. To make 4 vanifh in the fame Manner, I multiply all the
Terms of the Equation by 4, only obferving that with Refpect to the
Termx, the Multiplication by 4 is performed by taking away the De-
nominator 4. Hence there refults 8 x 9x+252 x = 3600, or
25.2 x-x=3600, which by multiplying the two Members by 5, will
become 252 x − 5 ×
5x=28000, or 247 x 18000, or x = 247
The general Principle which reſults from hence, is that to make a Di-
vifor of a Term vanifh, we have only to multiply all the other Terms
by the Diviſor, and take it away from the Term it affects.
5
We
may
}
XVIII.
ड
9
4.
1,8000
5
find another Method for taking away all the Diviſors at once,
by obferving that if we multiply all the Terms by the fame Number di-
vifible by each of thofe Divifors, each Term will be reduced. If we mul-
they are
made to dif- tiply, for Example, the Equation 3x-4x+3x= 100 by 180, which
appear all at is diviſible by 9, by 4, and by 5, we will have 360 180 x + 1260 x
18000, or 40 x — 45x+252x=18000, or 247 x 18000.
Now to find this Number which may be divisible by all the Diviſors,
we have only to multiply fucceffively thoſe Diviſors into each other. Let
the Diviſors of the Equation 3x + x = 160 x be propoſed to be
taken away. I first multiply 3 by 5, and afterwards their Product 15 by
7, which gives 105 for the Number divifible by 3, 5, 7; this Number
being found, I multiply the whole Equation by it, which gives me
235 x + 105 x = 16800 210 x, or 245 x + 21 x = 16800
3
5
7
·
7
30 x.
To render this Operation more comodious, inftead of forming the
Product 105 of the three Divifors, it fuffices to write it down thus,
3 × 5 × 7, and then we will have 7 × 3 × 5 × 7
7 X 5 X 3 X 2
✡
3.
7X3 X5
5
x =
* + ZX 3 × 5
160 X 3 X 5 × 7
X2x, where we fee by Inſpection,
that the Number 3 fhould vanish in the Numerator of the firft Fraction,
becauſe the Divifion by 3 fhould be deftroyed by the Multiplication by
35
it is the fame with Reſpect to 5 and 7, which are at the fame Time
in the Numerators and Denominators of the other Fractions.
By this Means we arrive at the Equation 7 X5 X 7x+7×3x=
160 × 3 × 5 × 7 — 2 × 3 × 5 ×, which after performing the Multipli-
cations indicated by the Signs X, becomes 245 x 21 x 16800-30%
delivered from Fractions.
XIX.
·
To follow in the moſt likely Manner poffible the Order of the Inven-
tors, we ſhall dwell no longer on the Method of difengaging the un-
SPECIOUS ARITHMETICK.
1
known Quantity, but return to the Manner of reducing Problems to
Equations. The Refolution of Equations might, independent of the Pro-
blems to which they relate, employ the firſt Analyſts when this Science
had been advanced to a certain Point, but it is to be prefumed that thoſe
who layed the Foundations of it, only examined Equations as they rela-
ted to Problems, theſe being the Conclufions to which Problems were to
be brought. Beſides, there fometimes occur Complications in Equations
which had never been thought of, if the Nature of the Problems fought
had not pointed them out.
We cannot explain more fully, the Manner of reducing Problems to
Equations, than has been already done, Art. vIII. but will proceed to
give ſeveral Examples in order to render this Reſearch familiar to Begin-
ners.
blem.
To pay a certain Number of Workmen at the Rate of £3 each, £8 is Third Pro-
wanting to the Perfon who employed them, but on giving them £2 each be
bas £3 left, it is required to determine bow much Money be had.
Let x exprefs the Number of Pounds which this Man poffeffes, where-
forex 8 will be the Sum fufficient to pay all the Workmen at the
Rate of £3 each; and fince the Number of Workmen ſhould be three
Times leſs than that which is expreffed by this Sum, it will be expreſſed
In fpecious
by the third Part of +8, which is denoted thus, *+; for in fpeci- as inaume-
x
x+8
3
ral
-
tick a bar is
employed to
indicate di-
ous as in numeral Arithmetick, a Line placed between two Quantities
indicates the Divifion of the fuperior Quantity by the inferior.
Moreover, fince there Remains £3 after £2 has been given to each vifion.
Man, x 3 will be the Sum fufficient to pay all the Workmen at this
rate. Wherefore 3 will denote the Number of Workmen, but
fince we have two Values of the fame Number, they must be equal,
the Problem is therefore reduced to the Refolution of the Equation
2
* + 8
3
2
To folve this Equation, I first make the Divifor 2 of the Member
3 diſappear, by multiplying the other Member by this fame Num-
2x+16
3
2
ber 2, which will change the Equation into x-3=
; for it
3
is x-3, and that the Double of
2
is manifeſt that the Double of
+8
will be
2x + 16
3
3
*
for the fame Reaſon that 2 x + 16 is double
of +8. I afterwards make the Divifor 3 vanish in the Equation
3 B
- ΤΟ
ELEMENTS OF
2x+16
3
3, by multiplying the fecond Member by 3 and taking
it away from the firft, which will give 2 x + 16=3x 9, or x 25.
If we would know how many Workmen there were, we muſt take
one of the two Expreffions
found for this Number,
x-3 or
2
x+8
3
≈
3 for Example. Now fince x = 25, -3 will be 22, and confe-
*
X
2
quently
2
3 will be 2211 Number of Workmen fought.
2
XX.
3
* +8
3
It is to be obſerved with Reſpect to the Equation 2
that it is not permitted in order to apply the Rule of Art. XI. to change
the Side and Sign of the Quantities 3 and 8, and write the Equa-
- tion thus
becauſe the Number 3 is not, properly
Another fo-
lution of the
fame prob
lem,
X 8
2
x + 3
3
2
fpeaking, a Term of the firſt Member, but only a Term of its Dividend
-3; the Quantity
Equation, as alſo
·x+8
3
2
3 being in Reality only one Term of the
3.
To apply therefore the Rule of Art. x1. we
muſt firſt take, as indicated by the Number 2 which is under the firſt
Bar, the Half of x 3 which will be ½ * 3; afterwards we muſt
take, on Account of 3 which is under the other Bar, the third Part of
+8 which will be x+: Then equalling thoſe two Quantities we
will have the Equation x-x+3 in which we may make what
Tranfpofitions we pleaſe.
उ
XXI.
The foregoing Problem may be alfo folved after the following Manner.
Let y exprefs the Number of Workmen, 3y will be the Sum of Mo-
ney which ſhould be diftributed among them at the Rate of £3 each.
But £8 is wanting to fatisfy them at this Price: Wherefore 3-8 is
the Sum of Money which their. Employer poffeffes.
;
On the other Hand 2 y will be the Sum fufficient to pay the Work-
men at the Rate of £2 each, and in this Cafe there will remain £3
Wherefore 2 y 3 is another Expreffion of the Sum which their Em-
ployer poffeffes.
8,
We must therefore equal the two Quantities 2 y 3 and 3y
or what amounts to the fame, we must folve the Equation
21+3=318 to obtain the Value of y. Which being done by the
foregoing Principles, I will be found for y, that is, for the Number of
Workmen required.
SPECIOUS ARITHMETICK.
II
XXII.
blem.
A Meſſenger being departed 9 Hours from a certain Place, travelling at Fourth pro-
the Rate of 5 Miles in 2 Hours, another Meſſenger is fent after him,
who travels at the Rate of 11 Miles in 3 Hours; it is required to find what
Number of Miles the fecond Meffenger will ride before be overtakes the firft.
Let x be this Number of Miles, it is manifeft that this Diſtance
fhould be equal to the Number of Miles which the firft Meffenger made
during his 9 Hours of Advance, together with what he made whilſt the
fecond Meffenger is on Rout. I inveftigate firſt the Number of Miles
travelled by the firſt Meffenger in 9 Hours, by the following Proportion
or Rule of Three.
+
As 2 Hours are to 5 Miles, fo 9 Hours are to a fourth Term, which,
according to the known Rules of Arithmetick, is obtained by multiply-
ing the fecond Term 5 of the Proportion by the third 9, and dividing
their Product by the first 2; and which confequently will be 45 Num-
ber of Miles made by the firft Meffenger in 9 Hours,
Manner of
But as the Analyfts always endeavour to exprefs their Operations in
the moſt concife Manner; they denote this Proportion thus; expreffing
Hours. Miles.
2
Hours.
9
Miles.
2
2
proportions
in fpecious
5
:
45 the Signs: Serving, one to compare arithmetick
2 to 5 and the other 9 to 45, and the Sign= ferving to denote the Equa-
lity which fubfifts between the Ratios of 2 to 5 and that of 9 to 45.
To find afterwards the Diſtance rode by the fame Meffenger during
the Time the fecond Meffenger travels the Number of Miles x, I in-
veftigate, first, the Time which the fecond Meffenger takes to travel the
Distance x, by Means of the following Proportion;
Miles. Hours.
TI
3
Miles.
Hours.
3
II
. Whereby, without embarraffing our-
felves about the Number of Miles expreffed by x, we learn that it fuf-
fices to multiply this Number by 3 and divide it by II, to obtain the
Number of Hours employed by the fecond Meffenger in travelling the
Number of Miles x.
Now without minding whether the Number of Hours expreffed by
x be known, or unknown, I make the following Proportion;
Hours.
2 33
Miles.
5
Hours.
3
IT
X
Miles.
플돌​x.
I
22
The fourth Term of which 15 x expreffes the Number of Miles tra-
velled by the firft Meffenger in the Timex, that is, before he has
been overtaken.
By this Means we have the fame Quantity expreffed two different
Ways; for the Diſtance rode by the fecond Meffenger is firit expreſſed
by x, fecondly it is the Sum of 45 Miles which the firſt Meſſenger
had travelled before the other had fet out, and of 1x Miles which this
fame Meffenger muſt have travelled, before he was overtaken.
2
22
12
ELEMENTS OF
Equalling therefore thofe two Expreffions we will have the Equation
x = 42 + 1 ½ x which by the foregoing Rule gives x 70 +3.
45
XXIII.
If the first Meffenger, befides the Advantage he has of having fet
out fooner, has alfo that of having fet out from a Place more advanced,
the Problem though more complicated, will be eaſily reduced to the fame
Principles.
Let the firſt Meffenger, for Example, going to Londonderry, fet out
from Cashel on Monday at 8 in the Afternoon, travelling at the Rate of
7 Miles in 3 Hours, and the fecond Meffenger purſuing the firſt ſet out
on Tueſday at 10 in the Morning from Cork, fuppo'ed to be 34 Miles
diftant from Cafbel, travelling at the Rate of 13 Miles in 4 Hours, it
is required to find what Number of Miles the fecond Meffenger will
ride before he overtakes the first.
To folve this Problem, we must take the Difference between 8 in the
Afternoon and 10 in the Morning, which is 14 Hours; and as the first
travels 7 Miles in 3 Hours, we will have by the following Rule of three;
Hours.
Miles.
Hours.
:
Miles.
98
3 : 7 = 14 23, which being added to 34 Miles af
Advance will give 34+ 28 or 20 Miles for the Diftance of the firſt
nce
Meffenger from Cork, when the ſecond fet out. I afterwards make as
3
3
Miles. Hours.
13 : 4
Miles.
x
Hours.
4
13x
above the following Proportion;
Number of Hours in which the fecond Meffenger travels the Diſtance x.
But during this fame Number of Hours, the first Meffenger would
have travelled a Number of Miles with will be found thus
Hours.
Miles.
3 :
7
==
Hours.
4
13
:
28
39
Miles
28
39
3
X. We will have therefore
the following Equation, x=2x+200, from whence is deduced, by
the Rules explained above x 236 +, Number of Miles travelled
by the fecond Meffenger before he overtakes the firft.
XXIV.
As foon as the first Analyfts had found out the Solution of any in-
tereſting Problem, they did not fail of making ſeveral Applications by
varying the Numbers given in thofe Problems. For Example, they would
have repeated ſeveral Times the foregoing Queftion, by varying the Ra-
tio of the Velocities of the Meffengers, and the Diftances of the Places
from each other they fet out from. In thofe different Applications, they
perceived that a Part of the Operation was repeated in each particular
Example of the fame Problem, which might be performed once for all
by inveſtigating a Solution which was not reſtrained to any particular
Number, but applicable to every Number given. To explain what
they imagined in this Refpect we will refume the foregoing Problem,
and treat it after as general a Manner as poffibly we can.
SPECIOUS ARITHMETICK.
13
folution of
Let the Diſtance of the Meffengers from each other be expreffed by General
a, when the Queſtion is brought to a Conclufion, we may make this the forego-
Leiter reprefent any Number of Miles we pleaſe.
Let the Number of Hours which the Departure of the first Meffen-
ger precedes that of the fecond be expreffed by the Letter b.
Let the Velocity of the firſt Meffenger be fuch that he travels the
Number of Miles c in the Number of Hours d.
Let the Velocity of the ſecond Meffenger be ſuch that he travels the
Number of Miles e in the Number of Hours f.
Finally, let the Number of Miles which the ſecond Meffenger tra-
vels before he overtakes the firſt be expreffed by x.
It is ufual in Specious Arithmetick to exprefs the known Quantities
by the first Letters of the Alphabet a, b, c, &c. and the Quantities
fought by the laſt s, t, u, x, &c.
ing problem
The first let
ters of the al
phabet ferve
to exprefs
the known
laft
To find now, according to the Method purſued in the foregoing Ex-
ample, the Number of Miles travelled by the firſt Meſſenger in the quantities
Number of Hours b, we muſt inveſtigate the fourth Term of a Propor- letters the
tion, of which the first Term is the Number of Hours d, the ſecond the unknown
Number of Miles c, the third the Number of Hours b, and it is mani-
feſt that this Operation will be performed, as in all other Rules of three,
by multiplying the ſecond and third Term,, one by the other, and dividing
their Product by the firſt Term.
ones.
fol-
As to the Manner of expreffing the Product of thoſe Terms which Letters
are no more as heretofore Numbers, but Letters adapted to exprefs any which fano
Numbers, what has appeared the moſt fimple, is to place the Letters ther with-
to be multiplied, befide each other; with Refpect to Divifion, we have out having
ſeen already that in fpecious as in numeral Arithmetick, a Bar is placed
between the Quantities to be divided.
By this Means the foregoing Proportion is wrote down
d:c=b: bc
d
b c
thus,
2
Having therefore to exprefs the Distance which the firſt Meffen-
ger travelled before the fecond fet out, if we add to it the Diſtance a
of the Places from whence they fet out, we will have for the Number
of Miles which the first Meffenger is advanced at the Time of the De--
bc
parture of the feconda +
To find afterwards the Distance travelled by the first Meffenger
whilst the other purfues him and rides Miles, I first inveſtigate, as
above, the Time in which the fecond Meffenger rides the Diſtance x,
by Means of the Proportione : fx: whofe firft Term is the
ef
fx
e
any fign be-
tween them
are to be con
fidered as
multiplied
by each
other.
14
ELEMENTS OF
}
{
#
Number of Miles e, the fecond, the Number of Hours f, the third the
fx
Number of Miles x, and the fourth the Time ſought.
e
Now, whatever be the Number of Hours
fx
e
which the ſecond Mef-
fenger employed to overtake the first, it is obvious if a Proportion be
made of which the three firft Terms are, 1°. the Number of Hours d;
e
fx
2º. the Number of Miles c; 3°. the Number, the fourth Term will
be the Diſtance travelled by the firſt Meſſenger in the fame Time that
the fecond travelled the Diſtance x.
fx
I write down this Proportion thus d: c=
fx
: CX
e
÷
e
d
Number
of Miles travelled by the firft Meffenger in the Time the fecond rides.
x Miles.
But the Number of Miles travelled by the first Meffenger together
with the Number of Miles a+ which he had advance, ſhould be
bc
d
equal to the Number of Miles travelled by the fecond.
ъс
We have therefore the Equation x = a+b+cxf. Now recol-
e
d
lecting the Operations on Fractions, I find that to multiply a Fraction as
by 4, we are to multiply the Numerator by this Number and write
or 24. in like Manner to multiply
.down
6 X 4
3
tiply c by fx which gives
cfx
for c X
fx
e
e
C
fx
by c we muſt mul-
e
It alſo appears that to di-
vide a Fraction as by any Number as 6, we are to multiply the De-
nominator 3 by this Number 6, and write downX6 or. In like
Manner to divide the Fraction
cfx
e
C
x
ed.
by d, we muft write down f
Having thus changed the foregoing Expreffion c
b c
c fx
cfx
finto the
e
de
Equation to be folved will be x = a++. Operation which
d
de
requires we ſhould begin as has been taught Art. XVIII. by multiplying
all the Terms, except the laſt, by the Divifor de in order to take it
away from this Term.
By this Operation we ſhall have de x = a de + b c d e + cfx,
-d
SPECIOUS ARITHMETICK.
15
d
or de x = a de + b c e +cfx, becauſe bede is the fame as b c e,
fince the Quantity bce remains the fame when multiplied and divided:
by d.
Carrying the Term cfx over to the first Member, we will have
dex cfxa de + bce.
In order to find in this Equation, I obferve that if the Numbers-
de and cf which exprefs how often x is contained in the Terms dex
and cfx were known, we would fubtract the fecond from the firſt, and
the Remainder which would exprefs the Quantity of x contained in the
firſt Member of the Equation, would' ferve as a Divifor to the fecond
Member, to obtain the Value of x. But without knowing the Num-
bers de and cf, it is manifeft that decf exprefs their Difference,
and confequently the Quantity of x which is contained in the firft
Member of the Equation de x cfx = ade+bce. Therefore the
Value of x will be what refults dividing the fecond Member by this
a de + boe
Number de — cf. wherefore x
by
a.f.
de, and this is the general
Solution of the foregoing Problem; for when the Quantities a, b, c, d, e, f
are known, we have no more to do than to make the Ufe of them in-
dicated by this general Value of x, that is, to multiply fucceffively
a, d, e, one by the other: To add to this Product that arifing from the
Multiplication of the Quantities b, c, e, one by the other, and to divide
the Sum of thofe Products, by the Number which is the Difference of
the Product of c into ƒ compared to the Product of d into e, and we will
have by this Operation any particular Solution we pleafe.
*
-
1
XXV.
Let us fuppofe, as in Art. XXIII the Diffance of the two Meffengers Example of
the forego-
from each other to be 34 Miles, that the firft Meffenger fets out ing folution
14 Hours fooner than the fecond, and travels 7 Miles in 3 Hours, in numbers,
and that the fecond rides 13 Miles in 4 Hours, we will have
a = 34, b = 14, c = 7, d = 3, e = 13, ƒ = 4, which will give
f
ade = 34 X 3 X 13, that is, 102 × 13 = 1326, bce = 14 × 7 × 13
X X
=1274, and confequently a de + b c e = 2600, de = 39, cƒ— 28, and
cf
of Courſe de cf= = II.
From whence we deduce x =
fame as found in Art. XXIII.
ade+bce
de-cf
2600 = 236+, the
II.
From this general Solution we may alfo deduce the firſt Cafe calculated
in Art. XXII. in which the two Meffengers were fuppofed to ſet out.
from the fame Place, the First 9 Hours fooner than the Latter, and
travelling at the Rate of 5 Miles in 2 Hours the fecond travelling
at the Rate of 11 Miles in 3 Hours, in this Cafe a=0, b = 9, c = 5; »
16
ELEMENTS OF
Fifth pro-
-blem.
Example in
d = 2, e = 11, ƒ= 3, and fubftituting thofe Values in the general
Formula or Value of x, we will have x = 9 × 5 × 11
= 495 = 70+ 3,
7
2X11-5X3
as found in Art. XXII, in the fame Manner we may make as many Ap-
plications as we pleaſe.
XXVI.
No fooner had the firft Analyſts found out the Method of rendering
the Solution of a Problem general by making Ufe of Letters inftead of
Numbers, but probably they ever after confidered Problems in their
greateſt Generality; it is therefore proper to accuftom Beginners to treat
them in this Manner. With this View we fhall folve the following
Problem.
A Workman can do a Piece of Work expreſſed by a in a Time expreſſed
by b; a fecond can do a Piece of Work c in a Time d; a third can do a
Piece of Work e in a Time f, it is required to determine in what Time thofe
three Workmen together can do a Piece of Work g.
Let x be the Time fought, the Work done by the firſt in this Time
will be found by the following Proportion: b: a = x :
ax.
b
The Work done by the ſecond in the fame Time will be found by
the following Proportion: dc = x : ~
CX
d
Finally we will have the Work done in the fame Time by the third
Workman, by Means of the following Proportion; f : e = x :
ex
CX
ax
'ex
f.
Wherefore **+*+ is the Work done by the three Work-
f
d
b
men together in the Time fought, but this Work ſhould be equal g, we
CX
have therefore the Equation++
f
ax
= g.
b
To folve it, I multiply according to the Rules of Art. xvIII. the
whole Equation by the Product fb d of the Divifors, and I have
ed bf x + cdbfx +
F
a x ƒ d b =
ax/
bdfg, which is reduced to
e d b x + fc b x + ad fx = b d fg, in which obſerving that edb+
fcb+adƒ expreffes the Number of x contained in the fecond Member,
I conclude that x =
bdfg
b de + b c f + adf
XXVII.
To make fome Application of this Problem, let us fuppofe that a
Numbers. Maſon can do 7 Feet fquare of a Wall in 5 Days, that a fecond Maſon
can do 10 Feet in 3 Days, and a third 11 Feet in 4 Days, it is required
+
SPECIOUS ARITHMETICK.
17:
in what Time thofe three Mafons together, would do 150 feet fquare of
the fame wall.
According to thofe Suppofitions we will have a 7; b = 5; c = 10;
d 3; e = 11; f = 4; 8 = 150; and confequently b d f g =
5X3 X4 X 150 9000, b de=5X3 X 11=165; b c f = 5 X 10X4
=
=200, a⋅d ƒ = 7×3×4=84; wherefore x =
= 20 + 449
number of Days in which the propofed Work will be finiſhed.
XXVIII
9000
449
20
Let us fuppofe it be required in what Time a Ciftern of 200 cubick Another
Feet will be filled by three Pipes, the first of which can fill 9 cubick Example,
Feet in 2 Days, the fecond 15 cubick Feet in 3 Days, and the third
2
19 cubick Feet in 5 Days; we will have a
Х
d = 3, or; e =
19, ƒ = 54, or ; g
ZI
ΙΟ
X
Wherefore, x=
5 ΤΟ
5
X
2
3
which becomes
950
2 X 3
+
2
3.
3
9, b = 2 ½, or ½ ; c = 15;
200.
21
4
× 19 + 2 × 15 ×
210000
2 X 3 X ÷
1575
+
X 200
21
X X +9x X
1890
2 X 4 3 X 4
10
3
21
4
To reduce this Quantity I multiply the Numerator and Denominator
of the firſt Fraction of the Divifor by 4; the Numerator and Denomi-
nator of the ſecond by 3; and the Numerator and Denominator of the
third by 2, which gives
210000
3800
2 X 3 X 4
+
2 X 3 X 4
4725
2 X 3 X 4
+
3780
2 X 3 X 4
210000
or
2 X 3 X 4
12305
210000
or
· or 17 +
12305
163
2461
Number of Days
2 X 3 X 4
fought in which the three Pipes running at the fame Time will fill the
propoſed Cistern.
XXIX.
thoſe rules
It appears from the two foregoing Problems that the Rules delivered The appli-
in Art. x &c. for folving numeral Equations of the first Degree are cation of
equally applicable to literal Equations; but it appears at the fame Time has produc-
that Beginners require to be directed in the Manner of employing them, ed feveral
we think ourſelf fo much the more obliged to affift them by a great of fpecious
Number of thoſe Applications, as it is probable that it is to them we arithmetick
operations :
3 C
1
18
First exam-
ELEMENTS OF
are indebted for a great Number of ufeful Operations of Specious Arith-
metick, which we ſhall difcover as it were accidentally.
Let it be propofed to folve the Equation 2 a c + a b
ple of the 3 a c + zax 5abdx.
refolution
of literal
equations.
1
Second ex-
a x =
I firft carry the Terms 3 a c and 5 a b over to the other Mem-
ber of the Equation having changed their Signs, and there refults
2ac+ab a x 3 ac + 5 a b = 2 a x
dx. . I carry likewife
the Terma x over to the other Side, obferving alfo to change its Sign,
which gives me 2 ac + ab 3ac + 5 a b = 2 a x — d x + ax. I
afterwards reduce this Equation, 1°. by adding a b to 5 a b, which gives
me 6 a b; 2°. by fubftituting ac for the Terms 2 ac and Зас
3º. by ſubſtituting 3 a x for 2 a xax; the propofed Equation there-
fore becomes 6 a b
dx which gives x
Let
ас 3, a x
XXX.
ab 2. a´x
ав
3.
ample of the the Terms 5 a b
refolution
of literal
equations.
3
b d — 2 a b — 5 a x + 7 b d
bd will become
6 a b
a.c
3 a
d
ас
d x ;
5 a b + 3 b d when carried
ас
5 a x dx will become
we will have therefore
5 a b + 3 b d, which is
a c by ſubſtituting 7 a x for
over to the ſecond Member, and the Terms
+5ax + dx when carried over to the first;
2 a x + 5 a x + d x = 2 a b + 7 b d
reduced to 7 a x + dx = 10 b d
10 b d — 3 a
зав
2 a x + 5ax, 10 bd for 7 b d + 3 b d,
Now difengageing x in this Equation we will have ≈ —
XXXI.
and
ས
3 a b for 2 a b
10bd- ac
5 à b.
зав
7at d
Reduction
In the Refolution of the two foregoing Equations we had Occafion to
of quanti-
ties to their reduce to a more fimple Expreffion different Terms of the fame Kind,
moft fimple fuch as 2 a c and 3 ac; 5 ab and a b, &c. As this Operation is fre-
quently neceffary in Equations which are to be folved and in the other
Parts of Specious Arithmetick, Beginners fhould render it familiar to
themſelves, to affift them thereto; Here follow fome Examples.
expreffion.
Affirmative
terms are
Let 15 abc-13 b c d 7 a b c + 29 b c d - 5 a bf + 9 a b c + 6 chi
be propoſed to be reduced.
b
6,
I first take the Terms 15 a b c, -7
ча abc and 9 a b c which are of the
fame Kind, I add the two Terms ab bc and 9 a
15
which are both
thoſe which one and the other Affirmative, that is, affected with the Sign +; I after-
are preced- wards fubtract from their Sum which is 24 a b c, the Term 7 abc be-
ed by +.
Negative cauſe it is negatif or preceded by the Sign, by this Means the three
thoſe which Terms 15 ab c - 7 a b c + 9 a b c are reduced to 17 ab c. In like Man-
inſtead of 29 b c d 13 b c d I ſubſtitute 16 b c d. As to the
Terms 5a bf and 6c bi which are alone of their Kinds, I write
them down as they are, hence the Quantity reduced will be
17 a b c + 16 b c d - 5 a b f + 6 cbi.
are preced-
ed by
ner,
$
SPECIOUS ARITHMETICK.
19
3
5
The Quantity abac + 2 ax-ad+7 ab + ax, will be
reduced to 2 a b + 4 1 a x — acad.
A
26
3
4. I
28
–
The Quantity 2 acd-5ach-3acd + 3 a cb 6 b fi will
become after being reduced-acd2ach-6bfi, which being en-
tirely negative, fhews that the Quantity which was to be reduced includes
more negative than affirmative Parts.
XXXII.
fame opera
It is proper to obferve here that the Reduction which has been em- The alge
ployed in the foregoing Examples, is precifely the fame Rule as that braick addi-
which is called Addition; for when two Quantities are propofed to be ad- tion is the
ded, it fuffices to write them down one after the other, and after- tion as the
wards to reduce them to their moft fimple Expreffion: If it was foregoing.
required, for Example, to add the Quantity 6 a b 2 a c 3 ad to
3ab + ac- 2 ad+bf, there is no more to be done than to re-
duce the Quantity 6 a b 2 a c 3ad + 3ab + ac = 2 ad + b ƒ,
which will confequently give 9 a bac-5 ad+bf for the Sum
of the two propofed Quantities.
formed there will refult
5.a c
If it was propofed to add the two Quantities 2 ac-zad + af and
ad 5 ac 2 af, no more is required than to reduce the Quantity
2 ac 3 ad + af + ad=
2 af. The Reduction being per-
Зас 2 ad af. It may appear at first
View furprizing that the Refult of an Addition fhould be a negative
Quantity; but this Difficulty will foon be folved when it is obferved
that the two Quantities 2 a c 3 ad af and ad јас 2 a f
must be both Negative, or at leaſt one of the two negative and greater
than the other.
2ac
This will more readily appear by fome Examples in Numbers. Let
us firſt ſuppoſe a = 2, c = 3, d = 4; f = 5, in this Cafe, inſtead of
3 ad + af we will have 12 24 +10 or fimply
or fimply
inftead of ad 5 ac 2 af we will have 8 30 20
their Sum will be 44, and it is no way furprizing that the Sum of
two negatif Quantities fhould be negatif.
Let us now fuppoſe a = 6, c
-2 ac
2, and
42 and
3 ad + a ƒ = 18 and a d
5, d.
5 ac
d = 3, f: 2, we will have
2 af = 156. Where-
fore the fecond Quantity being negative, and greater than the first the
Sum fhould be negative.
XXXIII.
It may perhaps be aſked, if a negative Quantity can be added to an af- in what
firmative one, or rather can a negative Quantity be faid to be added. To fenfe a neẻ
which I reply that this Expreffion is exact when the Words to add gative quan
tity can be
and to augment are not confounded. Let two Men, for Example, join faid to be
their Fortunes, be them what they will, I would fay they add their For- added.
tunes; let one have Debts and real Effects, if his Debts exceed his Ef-
20
ELEMENTS OF
From the
foregoing
operation
the algebra-
tion is de-
duced.
fects, he will poffefs but a negative Fortune, and the Junction of his
Fortune with that of the firft would diminish that of the firft; fo that the
Sum will be found, either lefs than what the firft poffeffes, or even en-
-tirely negative.
a x
a x
2 ax
XXXIV.
a x
The Reduction taught in the foregoing Articles, has given rife to
another Rule of Specious Arithmetick namely Subtraction; for when,
for Example, in the Equation 2 ac + ab
3ac+2ax— 5 a b
dx (Art. XXIX.) we carried over the Terms 3 a c 5 ab to the
ick fubtrac- other Side, after having changed their Signs, and we arrived at the
Equation 2 a c + a b
3 ac + 5 a b = 2 a x
ab
d x, or
a c + 6 ab
dx, I fay that the Quantity 3ac- 5 ab
was fubtracted from the Quantity 2 a cab a x and the Remainder
is a c6 ab-ax. For by making 3 a c-5 a b difappear in the
fecond Member of the Equation, a Subtraction of this Quantity was
´performed, but to preferve the Equality, a fimilar Subtraction must have
been made in the firft Member; therefore 2 ac+ab-ax-3ac+5ab
or a c + 6 a b
a x is what remains of the Quantity 2 acab
after 3 ac- 5 a
has been fubtracted from it.
Proceſs of
fubtraction.
Hence when there are two Quantities of which one is to be ſubtracted
from the other, we must change the Signs of the Quantity to be fub-
tracted, and write it down after the other, then reduce the Quantities of
the fame Kind, which independant of what has been faid may be demon-
ftrated after the following Manner.
b
3ac
Let 2 ac+a ax be the Quantity from which it is propoſed to
fubtract the Quantity 3 a c-5 ab. It is manifeſt that if 3 a c was on-
ly to be fubtracted from it, we ſhould write down 2 ac+ab
ax - зас,
but by fubtracting the Quantity 3 ac instead of 3 ac5 ab a Quantity
too great by 5 a b has been fubtracted: Wherefore we must add the
5 a b which has been taken away too much by fubtracting 3 a c. There-
fore we must write down 2 acab
3ac + 5 ab for the
Remainder of 2 ac + ab a x after 3 a c — 5 a b has been ſubtract-
ed from it.
a x
To exerciſe Beginners in this Rule which it is obvious fhould frequent-
ly occur I have added the following Examples..
If from 5 ab+10fg-3 ac+2 de we fubtract 2 a b-5fg+6ac+de,
there will remain 5 a b + 10fg 3ac + 2 de ļ --
2ab+5f8
-6acde, or 3 a b + 15 fg -9ac + de.
6a b+3a8
If from the Quantity 6 ae b + 3 a gb - 10 b c d we fubtract
8 ag b, there will remain 16 a e b
a b c
10 ae b
10 b c d + 1 I ag
b.
a b c
If from the Quantity 3 ac + ab + be we ſubtract ➡ac — 3 a by.
there will remain 4 a c + 4 ab + be.
SPECIOUS ARITHMETICK.
XXXV.
A quantity
is increaſed
tracted from·
If it appears furprizing that in this fubtraction the Remainder
4 ac + 4 ab + be is greater than the Quantity 3 ac + ab + be
from which it was propofed to ſubtract a c 3 a b, it can only arife when a ne--
from not making a proper Diftinction between fubtracting and diminish- gative quan
ing; for if we confider that to fubtract any Quantity, a, for Example, tity is fub-
from another b, is to determine how much b exceeds a, it will appear it.
very poffible that a Quantity may be increaſed by a Subtraction. Let it
be required to determine, for Example, how much one Man is richer
than another, if the latter has only Debts, it is manifeft that the Ex-
cefs of the Riches of the firſt will be what he poffeffes more a Sum
equal to the Debts of the other.
XXXVI.
Let it be propoſed to folve the Equation
CX
2 a
ас
26
• Third ex--
the refolu-
ample of
tion of lite--
to make firſt the Divifor 2 a difappear, I make it ferve according to
Art. xv. as a Multiplier to all the Terms of the Equation,
4 ad
3 c
and we will
a c X 2 a
z b
= 2 a Xx
4 ad X 2 a
3 с
but it is ma-
ral equati--
ons..
骂
​have c x
nifeft that inſtead of a c X 2 a we may fubftitute 2 a a c, becauſe the
Product of 2 a into a c fhould be double of the Produ&t of a into a c, and
the Product of a into a c is a ac. Likewife 2 a Xx will be 2 a x and
4 ad X 2 a will be 8 a ad; for the Product of a d into a is a ad and
the Product of 4 ad into 2 a ſhould be octuple of that of ad into a.
The Equation is therefore changed into c x
or c x
a-ac
= 2 a X
24
b
are one and the fame.
will beome b xc x
8 a ad
3 c
2a a c
=24x
26
8aad
becauſe
,
заас
a a c
or
30
26
b
8 a ad
3 с
x bor
Multiplying the Terms of this Equation by b, it
хъ
=20x x b
аас
8 a a b d
3 с
a a c
x
3 c
a a c = 2 a b x
b c x
into cbx x 3 c
3 bc.cx. Заасс 6 a b c x
into cbx, a ac and 2 a b x,
confequently the Products of 3
of the foregoing, that is, 3 b
1pofing we will have 3 b c c
which at Length gives x =
which will be again changed
= 2 a b x X. 3 c
8 a abd,
will be bcc.x,
8 a ab d, or
for the Products of c
a a cc, 2 a b c x, and
by the fame Quantities fhould be triple
6 a b c x =
c
c
c x, 3 a ac c,
6 a b c x;
x
-
3 a acc
Заасс
3 b c c
6 a b c
Заассасаси
bc
XXXVII.
8 a a b d
now tran-
8 a a b d
In the foregoing Example, the Multiplication of fome Quantities.
22
ELEMENTS OF
the top of a
A number which contained the fame Letters, produced the Repetition of thofe Let-
placed at ters in the Products: Now as the Analyits always endeavour to expreſs
letter to the themfelves in the most concife Manner, inftead of repeating a Letter
right de- feveral Times one after another they write it down but once, placing at
often it is the Top of it on the right Hand a Figure denoting the Number of Times
repeated by that this Letter fhould be repeated. Hence, inſtead of the foregoing
8 a a b d
notes how
multiplica-
tion, and in Expreffion x =
this cafe the
letter is faid
to be raiſed x =
to the
power
led expe-
nent.
заасс
b
3
C C
6 a b c
3
a2 c2
8 a² b d
3
b c²
ъс
6 a b c
they write
C
expreffed by When in an Operation the Analyft has Occafion for a a a, that is, for
this number the Product of a a by a or for a multiplied by itſelf twice, he puts down
which is cal fimply a³. In like Manner for cc cc, he writes ct. When a Letter is
thus repeated or rather confidered as repeated by Means of a Number, it
is faid to be raiſed to the Power expreffed by this Number, and that this
The num- Number is its Exponent, thus c or cccc which is the Product of c mul-
are to the tiplied three Times by itſelf is faid to be raiſed to the fourth Power,
left and on and 4 is its Exponent. Care is to be taken not to confound the Numbers
the fame which ferve as Eexponents with thoſe which are on the left Hand of the
line are cal- Letters and on the fame Line, which are called Coeficients; in 4a² c,
for Example, 4 is the Coeficient of the Term, 2 is the Exponent of a.
bers which
led coefici-
-ents.
Fourth ex-
ample of the
XXXVIII.
Let the Equation
2ab2x
3 c² d
+
refolution of to be folved, multiplying all its
literal equa-
5 ac² X3c² d
tions.
2ab² x +
62
=
6cd2
a²
5 ac²
3x, be propofed
62
Terms by the Divifor 3 c²d, we will have
6 c d² X 3 c² d
a²
3 x X 3 c² d.
To perform the Multiplications indicated by the Signs X, I obferve
first that a c² multiplied by c² d ſhould give for Product a c4 d, for if in-
ftead of a c² and of c²d we write acc and cc d, as may be done, their
Produ& will be accccd, that is, according to the foregoing Art. a c4d.
Having therefore a c4d for the Product of a c² into c²d, it is manifeſt
that 15 a c4d will be the Product of 5 a c² into 3 c2 d.
་
In like Manner, the Product of 6 c d² into 3 c²d will be found to be
18 c³ d³ and the Product of 3 x into 3 c² d to be 9 c² d x. Wherefore
the foregoing Equation will be changed into
2ab²x +
15ac4 d
62
18 c3 d3
24
a
Multiplying afterwards this new
18 b² c3 d3
2 a b¹ x + 15 ac² d =
a²
-9 c² d x.
Equation by b2 it will become
9 b² c² d x, and this laſt by
3
a², there will refult 2 a³ b4 x + 15 a³ c4 d = 18 b² c³ d³. — 9 b² a² c² dx,
a3
SPECIOUS ARITHMETICK.
23
15 a3 c4 ď
and by tranſpoſition 2 a3 64 x + 9 b² a² c² d x = 18 b² c³ d³ — 1 5 a³ c4 d,
18 b² c3 d3
from whence at length is deduced x =
2a3b4 +9a² bu cü d
XXXIX.
which con-
In the foregoing Examples, we had Occafion to multiply Quantities Incomplex
expreffed by a fimple Term fuch as 4 a d, 9 c² d, &c. which are com- quantities
monly called incomplex Quantities or Monomes, and we found at the are thofe
fame Time how this Operation was to be performed. The general Me- fift only of
thod which refults from the Reaſoning employed in thofe particular Ex- one term.
amples, is firſt to multiply the Coeficients; to add afterwards the Ex- plication
ponents of the fame Letters, and to write thofe which are different one deduced
after another. Hence according to this Rule, 3a4b3dX7a²bd² = 21a6b4d3; from the
foregoing
} a ² c d x }
{ a c³ b d = 13 a³ c4 b d² = { a³ c4 b d²; } a c² de X9 at fg = examples.
6 a5 c² d e f g.
2
ड
a3
Let it be propoſed to folve the Equation
Their multi
- 6 a
34, Fifth exam-
ple of the
refolution of
6 a b², literal equa
tions.
XL.
a²c
262
8b2 c x
+
4 cx
3 a
5 ab
C
10 a b3
3 a
C
30 a² b3
-18· a² b²,
62,
C
multiplying it by 2 621 have a²c +
then multiplying by 3 a, I have 3 a³ c + 8 b² c x =
and performing again the fame Operation to make the Divifor c vaniſh,
there reſults 3 a³ c² + 8 b² c² x
30 a² b3
8 a² b².c, from whence.
may
= 30 a² b3
1
is deduced x =
18 a² b² c
3 a3 c²
which
862 c²
alfo be.
wrote thus x -
30 a² b3
864 c²
18 a² b² c
862 c²
2
3 a3 c²
862 c²
; becauſe
8 b2c2 dividing the whole Quantity 30 a2 63
vides each of its Parts.
18 a² b² c — 3 a³ c² di-
a3
Now the Value of x thus wrote may be more fimply expreſſed by re-
ducing each Term. For 1°. inftead of 30 a² b3
"
15 n² b
4 c²
,
8b2ch may be put down
becauſe the Numerator may be confidered, as the Product of
2 b² into 15 a² b, and the Denominator as the Product of the fame Quan-
tity 2 62 into 4 c²; dividing therefore both
the fame Quantity 2 62, there refults 15 a² b
we may put down 9 e²
4 c
4 c²
one and the
2º. inſtead of
other by
18 a² b² c
864 c
C
; for the Numerator is the Product of 2 b² c
into 9 a², and the Denominator is the Product of the fame Quantity 2 b2c
a3 2
a3
into 4 c. Instead of 3 a³ c²
86 we may put down 3 43
862
Therefore the
+24
ELEMENTS OF
Value of x reduced is
15 a² b
4c2
9a2
3 a3
4 C
XLI.
і
862
Divifion of The Method to be purfued in general in all Operations of the fame
incomplex Nature as the foregoing, that is, in Divifions of incomplex Quantities
quantities is easily deduced from what has been faid, particularly after having feen.
the Multiplication of incomplex Quantities. This Method may be ex-
example. preffed thus.
deduced
from this
Sixth exam-
ple of rhe
refolution
. of literal
}
Divide firft the Coeficients if the Divifion be poffible, take away the
Letters which have the fame Exponents in the Numerators and Deno-
minators, divide afterwards the Letters which have different Exponents.
in the Denominator and Numerator, by fubtracting the leaft Exponents
from the greateft, and writing the Remainders in the Place of the great-
eft Exponents. As to different Letters there is no more to be done than
to copy them.
As this Operation frequently occurs, to render it familiar to Begin-
-ners here follow fome Examples.
9a5 d² b²
3 a4 c² d²
C
3ab2
c2
2
= 9 a b c³, 5 m² b4 c²
15 a b 3
1 8 a 4 b c d
14 a b²
1
a b c²
3
XLII.
9a3 c d
76
>
27 43 62 c5
3 a² b c²²
a²x
b-c
+ dc = b x
a c.
Let it be propofed to folve the Equation
To make the Divifor b c vaniſh, I multiply all the Terms by this Di-
equations. vifor, which gives a² x + (b − c ) x d c = (b x — a c) x (b — c),
X
where I obferved 1°. to put b - in the firſt Member between Paren-
Ufe of Pa- thefes left it might be imagined that only c was to be multiplied by dc;
fpecious a- 2° to put b x- a c and b
c in the ſecond Member between Paren-
rithmetick. thefes in Order that it may appear that thoſe two entire Quantities are
to be multiplied by each other.
rentheſes in
It is now queſtion to perform the Multiplications indicated by the
Signs X. Let it be propoſed firſt to multiply dc by bc, it is mani-
feſt that we muft multiply dc by b and from their Product fubtract the
Product of dc into c; for b
for bc being less than b by the Quantity
the Product of b cinto de fhould be lefs than that of b into dc,
by the Quantity c X d c. Therefore the Product of bc into dc is
ccd.
b d c
To find the Product of b X a c into b— c; I obſerve that if the two
Terms b x - ac be confidered as a fingle Quantity, the Product of this
Quantity into be ſhould be, the Quantity which the Product of
b.x
a c into b exceeds the Product of b x a c into c. The Quef-
SPECIOUS ARITHMETICK.
25
tion is therefore reduced to two Multiplications fimilar to the foregoing
one and to a Subtraction.
The first of thoſe two Multiplications, that of bx ac by b, will
give b² x
abc; the ſecond that of b xa c by c, will give bcx - ac².
It remains therefore to fubtract this laft Quantity from the first, which
will give, according to Art. xxxiv. 6²x — a b c — b c x + ac², and
this is the Product of b x
ac into b C.
So that the Equation
xcd= (bx — a c) x
— a b c - b c x+ac²,
c² d + a b c
b c d
X
b c d
૨૨
=
+ c d = b x — a c, or a² x + (b − c )
b-c
(b—c) is become a² x + b c d — c² d = b² x
which, by the ordinary Tranfpofitions, will give
a c² = b² x
b c x ax, and at length
c² d +
62 b c
a b c
22
ac²
XLII.
tities or mul
In this Example, we had Occafion to form a Rule of fpecious Arith- Multiplica-
metick, which we had not as yet employed, and which on feveral Occa- tion of com-
fions may be of Ufe. This Rule is called the Multiplication of Mul- pound quan
tinomials. A Multinomial or compound Quantity fignifies in general a tinomials de
Quantity compofed of feveral Terms. When the Number of Terms of duced from
the forego-
a Quantity are ſpecified, if it confifts of two it is called a Binomial; if
it confifts of three it is called a Trinomial, &c.
ing article.
To exercife Beginners in the Multiplication of thoſe Kinds of Quan- Example of
tities, here follow fome Examples: Let first the Product of the multi-
2 a³ c² 5 a² 6 + 6 a5 into 3 a b² - 4 b c d be required.
3
plication of
multinomi-
By reaſoning in the fame Manner as in the foregoing Article, it will als.
appear, that fince the Quantity 3 a b² - 4 b c d is less than 3 a b by
4bcd, the Product of this Quantity into 2 a3c2 — 5 a+ b + 6 a5
fhould be less than the Product of 3 a b² into 2 a³ c²
by the Product of 4b c d into 2 a³ c²
quence, I write the Product thus (2 a³ c²
— (2 a³ c² — 5 a4 b + 6 a5 ) × 4 b c d .
5
5 at b + 6a5
a+b+ 6 a5. In Confe-
4
5 a4 b + 6 a5) X 3 ab²
Performing now the two Multiplications indicated by the Signs X,
after the fame Manner as thofe of fimple Quantities, we will have
6 a4 b² c² — 15 a5 63 + 18 a6 b² for the firſt Produ&t (2 a³ c² 5
a46
+ 6 a³) × 3 a b². In like Manner we will have 8 a3 b c3 d 20 a 4 b² c d
+24 a5 b c d for the ſecond Produ& (2 a³ c² — 5 a4 b + 6 a5 ) × 4 b c d .
Then fubtracting the fecond from the firft, as indicated in the foregoing
Expreffion, we will have 6 a4 b2 c2 15 a5 b3 + 18 q6 b²
8 a3 b c 3 d
+ 20 a4 b² c d 24 a5 b c d for the Product of the two propofed
Quantities.
>
3 D
20
ELEMENTS OF
Fundamen-
1
XLIV.
If the Multiplier of the foregoing Quantity, contained befides the two
Terms 3 a b²-4bcd, another 5 a b c, for Example, it is mani-
feft that to obtain the whole Produ&t, we muſt ſubtract from the fore-
going Quantity, the Product of 2 a³ c² 5 a4 b + 6 a5 into 5 abc.
For the Multiplier 3 a b² - 4 b c d 5 a be being less than the Multi-
plier 3 a b² 4 b c d, by 5 a b c, the Product of it into 2 a³ c2 5
a4 b
+6 a5 fhould be lefs than the Product of 3 a b² 4 b c d into 2 a3 c²
- 5 a4 b + 6 a5 by 5 abc X (2 a³ c² 5 a4 b6a5). In like Man-
ner if there was another Term in the Multiplier, 3 a c² for Example,
with the Sign + the Product 3 ac² × (2 a³ c² — 5 a4 b + 6 a³) fhould
be added to the foregoing Products.
X
It appears in general that any Quantity being given to be multiplied,
tal principle with the Quantity which is to ferve as Multiplier, we are to form all
of multipli- the Products of the Multiplicand into each of the Terms of the Mul-
tiplier, and add or fubtract thofe Products, according as the Terms of
the Multiplier are affected with the Sign + or -.
cation:
Method to
cation.
To perform this Operation after a regular Manner, proceed thus.
XLV.
First write down the Multiplier under the Multiplicand, and draw a
be purfued Bar under the Multiplier. To form afterwards the firft Line of the
in multipli- Product which is to be fet down under this Bar, multiply the firſt Term
of the Multiplier by each of the Terms of the Multiplicand, obferving
to prefix to each of thofe Products, the Sign of the Term of the Mul-
tiplicand, if the firft Term of the Multiplier has no Sign, and conſe-
quently is confidered as having the Sign +.
To form afterwards the fecond Line which fhould be wrote under
the first, multiply the fecond Term of the Multiplier by all the Terms
of the Multiplicand, and if this fecond Term of the Multiplier has alfo
the Sign+, the Operation is abfolutely the fame as for the firſt Line,
but if it has the Sign, to each of the Products which compofe this
Line, prefix a Sign contrary to that which affects the Term of the Mul-
tiplicand to which it relates. All the other Lines of the Product being
formed after the fame Manner, by Means of the other Terms of the
Multiplier multiplied by all thofe of the Multiplicand, draw a Bar,
and add or reduce all thofe particular Products; the Quantity which re-
fults will be the Product required.
We ſuppoſed that the firft Term of the Multiplier had the Sign+,
if however it had the Sign, it is eafy to perceive that with Refpect
to this Term, as with Reſpect to the others affected by the Sign-,
Signs contrary to thofe of the Terms of the Multiplicand, are to be
prefixed to the Product of thoſe Terms.
SPECIOUS ARITHMETICK.
27
XLVI.
2 a b
4ac +
a d
Mult.
3 a b
6 a² 62
5ac + 2 a d
ad
by
I
12 a² b c +
3 a² b d
10 a² b c +
20 a² c²
[
5 a² c d
Product.
+ 4a² b d
8 a² c d +
2 a² ď²
622 62
22 a²² b c + 7 a² b d + 20 a² c²
1 3 a² c d + 2 a² ď²
Sum.
За
of the fore-
To explain this Method, we fhall apply it to an Example: Let it be
propoſed to multiply the two Quantities 2 a b 4 ac+ad and 3 a b Application
-5ac + 2 ad. The firft being chofen for Multiplicand, and the going me-
fecond for Multiplier, I write down this laft under the other, as above. thod to an
Which being done, I remark that the first Term of the Multiplier is example.
affirmative, and confequently all the Signs of the Terms of the first Line
of the Product ſhould be the fame as thoſe of the Multiplicand. In
Confequence of this Remark, I write down 6 a² 62 the Product of 3 a b
into 2 ab, without prefixing any Sign to it, which comes to the fame
as if it had been affected by the Sign +.
I afterwards put down for the Sign of the fecond Term of the
fame Line, becauſe it is the Sign of the fecond Term of the Multipli-
cand, and after this I write 12 a2 b c Product of 4 a c into 3 a b. I
prefix in like Manner the Sign of the third Term of the Multipli-
cand to the third Term of the first Line of the Product, which is
3 a² b d Product of ad into 3 a b. The first Line of the Product be-
ing thus finiſhed, I remark that the ſecond Term of the Multiplier has
the Sign, and confequently that the Signs of the Multiplicand muſt
be changed to form the Terms of the fecond Line of the Product.
Wherefore the firft Term of this fecond Line ſhould have the Sign
which I prefix to the Product 10 a² be of the two Terms 2 a b, 5 ac.
The fecond Term of the fame Line fhould have the Sign +, fince
the fecond Term of the Multiplicand has the Sign, I therefore pre-
fix the Sign to the Product 20 ac² of the two Terms 4 ac, 5 ac.
The third Term a d of the Multiplicand being preceded by the
Sign+, the third Term of the fecond Line will be affected by the Sign-
which I prefix to the Product 5 a² cd of the two Terms a d, 5 a c.
As to the third Line of the Product fought, becauſe the third Term
of the Multiplier has the Sign +, all the Signs of the Multiplicand are
to be preſerved, confequently the firſt Term, that is the Product of
2 ab into 2 ad will be 4 a2 b d preceded by the Sign+, the fecond,
that is, the Product of 4 ac into 2 ad will be 8 a² cd preceded by the
Sign and the third, that is, the Product of 2 a d into ad will be
2 a² d² preceded by the Sign +.
28
ELEMENTS OF
To render this Operation familiar to Beginners, here follow more
Examples.
Multiplicand.
Multiplier.
Product.
5 a3 b
2 a b3 + 4 a² c²
223 b
a 63 + 3 a² cz
10 26 32.
4 24 64 +
8 95 b c²
2 a2 66
6 a3 b3 c² +
5 24 64 +
+ 15 a5 b c²
9a4 b4 + 23 a5 b c² + 2.a² 66
2a3 202
2
363 2
2 a3 x² + 3.63 y2
Sum.
10 46 62
Multiplicand.
Multiplier.
Product.
4 26x4 6 a3 63 y²x²
2
+ 6 a3 b3 x² 332
9b6 y4
Sum
4
225x64
966 74
4 a3 63 c²
12 a4 c4
10 a3 b3 c² + 12 24 c4
5ab+3ac
5 ab + zac
25a262
15
6.2
a² b c + 5 a b c²
+ 15 a²bc + 9 a² c² — 3 a c³
3ac3
c4
6 a c3 c4
+
5 a b c²
+
3 ac3
25a² b²+9a2 c²
Multiplicand.
Multiplier..
3y+2.ay.
α a
yy
z a x + a a
a ay y + 2 a³ y
x x
x + a
a x
a4
x3
a x x
Product.
Sum.
—2 ay3-4 a ayya³y
p4 + 2 ay³ — Zaayy
y4
3.aayy + 3, a³µ — — a4
+axx
лах
a ax
y
Multiplicand.
Multiplier.
aa + 2.ac
b c
2 a
a
b
aab
Product.
z a b c + b b c
46
2a + 4
+4
46
4 a a
8 a b
a32aac
a b c
+
8 ab
- 16 b b
Sum.
n³ + zaαc
a ab
3 a b c + b b c
дас
4 a a
16 bb.
Multiplicand.
Multiplier.
Product.
4.a² bx
z a b x − b x y + a² x + 3:02:39:
2 ax
zab x²y + 2 a³ x² + 6 a³ x y
6 a² b x y + zab x y² 3 a3 x y
9a3 y 2
4 a² b x² - 2 a b x² + 2 a³ x² + 6 a³ xy-6 a² bx y + zab xy² - 3a3 xy-9a3y²
2
a
3 ay
SPECIOUS ARITHMETICK..
29*
XLVII.
Let it be propoſed to folve the Equation
ax — ac, I first make the Divifor de difappear by multiplying folution of
a x
-
ac by d
C
a b² + abd — a b x
Sixth exam-
d
C
ple of the re
(a x
acx + ac²,
a c) literal equa
tions.
which gives a b² + a b d ·
a b x
= a dx
a b x
a c d
× (d — c), or a b² + a b d
which, by carrying over all the Terms affected by x to one fide,
the known ones to the other, will become a b² + abd + ac
a c d
a b x + a d x acx, from whence is deduced
X
a b² + a b d a c d
a b + ad a c
a c²
and
a c²
the divifion.
A certain Relation which is perceived to fubfift between the Terms.
of the Dividend and. Divifor in this Expreffion, might induce us to think
that the Diviſion may be performed exactly, and confequently invites
us to attempt this Operation, which ſhould appear eaſy to execute, af-
ter having feen that of Multiplication of which it is the Converſe.
To difcover in effect if a b + a d a c can divide exactly a b Manner of
+ abd + ac d a c². I divide firſt one of the Terms of this laft performing
Quantity by one of thofe of the firft, for Example, a b by ab, and indicated in
write down the Quotient b, I afterwards multiply this Quotient b, or this exam-
rather this first Part of the Quotient fought, by the whole Divifor a b. ple.
+ ad a c, and fubtract the Product a b² + abd abc from the
Dividend, the Remainder a b² + abd + acd
azz a b d
+abc, or ac d ac² + abc, is ſtill to be divided by the fame Diviſor,
and its Quotient added to the foregoing b to form. the whole Quotient
fought.
a 62-
To perform this Diviſion I divide one of the Terms of the Quantity
acd
acabe which remains to be divided, by one of thoſe of
the Divifor. For Example, ac d by a d, which gives c for the Quotient;
I multiply this new Quotient by the whole Divifor ab + ad
ac and
I fubtract the Product abc + ac d ac² from the remaining Dividend
a c d
ac² + abc; and as the two Quantities are the fame, there re-
mains nothing to be divided, from whence I perceive that b + c is the
exact Quotient of a b² + a b d + a c d a c² divided by a b + ad- a c
and confequently the Value of x.
LVNI.
thod for di--
After having performed the foregoing Divifion, it is eafy to perceive General me.
what Method is to be purſued in other Examples. To reduce this Ope- viding com--
ration to a regular form, the Analyfts write down the Divifor at the pound quan
right Hand of the Dividend, feperating them by a Line, as in Diviſion tities.
of Numbers, chufing afterwards a Term in the Dividend divifible by
one of thofe of the Divifor, they write the Quotient of thofe two Terms.
=
30
Manner of
ELEMENTS OF
under the Divifor, prefixing to it the Sign+, if the two Terms which
are divided one by the other have the fame Signs, but prefixing to it the
Sign, if thofe two Terms have different Signs. This being done, they
multiply the Quotient by all the Terms of the Divifor, and write the
Product under the Dividend. But as the Ufe of this Product ſhould be
to fubtract it from the Dividend, they obferve when writing it under this
Dividend to prefix to each Term a Sign contrary to that which reſults
from the Multiplication.
The Product being thus wrote down, they draw a Line and perform
the Reduction with the Dividend, and the Quantity which remains, is
to be divided a-new by the fame Divifor. They chufe as before a Term
diviſible by one of thoſe of the Divifor, and write the Term which re-
ſults for Quotient beſide the firſt, obferving to prefix to it the Sign +,
or the Sign, according as the two Terms that have been divided one
by the other have the fame or different Signs. They multiply after-
wards this Term by all thofe of the Divifor, and write the Produc un-
der the Quantity to be divided, obferving as before to change the Signs
which refult from the Multiplication. Then drawing a Line and reduc-
ing, if all the Terms do not deftroy each other they write the Remainder
under this Line, and continue the Operation after the fame Manner un-
til all the Terms of the Dividend have vanished.
As in this Operation, it may be fome Times embarraffing to chuſe
avoiding among the Terms of the Dividend and the Divifor, thofe which ſhould
working by ferve to form the Terms of the Quotient. To remove all uncertainty
conjecture
in divifion. in this Choice: Here is what the Analyfts have imagined.
They first chufe at will a Letter which is found in the Dividend and
What is un- Divifor, and they difpofe the Terms of thofe two Quantities in fuch a
derstood by Manner that the Terms may ftand first in which this Letter has the
ordering a
greateſt Exponents, and thofe next in which this Letter has the next
quantity ac-
cording to a greateſt Exponent and fo on. Having therefore ordered the two propof-
ed Quantities according to the fame Letter (it is thus this Operation is
called) the Terms to be divided are no more determined by conjecture,
it is always the firft Terms of the Dividend and Divifor which are to
be choſen.
letter
When the firſt Term of the Quotient is formed by thofe two Terms
of the Divifor and Dividend, and the Product is wrote with different
Signs under the Dividend, if this Operation introduces Terms of a
different Kind from any of thofe in the Dividend; Care is to be taken.
in writing the Quantity which refults after the Reduction to place the
Terms, fo that the Quantity which remains to be divided be always
ordered according to the fame Letter as the Divifor.
SPECIOUS ARITHMETICK.
31
XLIX.
2a4 — 13 ba³ + 31 b² a² — 38 63 a +2464
2 a² — 3 ba + 462
2a4 +
3
b a3
4622
a² — 5 ba +662
10 b a3 + 27 biz az
38 b3 a +24 64
+106a3 - 15
64
b2 a² + 20 63 a
+1262a2 18 63 a + 24
12 62 2² + 18 b3 a 24 64
о
To illuftrate this Method, we will apply it to fome Examples.
-
Let Application
going me-
it be first propoſed to divide the Quantity 31 a² b² + 2 a4 + 24 64 of the fore-
38a b3
13 a3 b by the Quantity 3 ab + 2 a² + 4 b².
Having wrote down thofe Quantities in the Manner above, where they example.
are ordered according to the Letter a.
I divide the firſt Term 2 a4 of the Dividend by the first Term 2 a² of
the Diviſor, and I write the Quotient a² under the Divifor, without pre-
fixing any Sign to it, that is, I mak it affirmative, becauſe the Terms
2 a4 and 2 a² are preceded by the fame Signs. The Quotient a² being
fet down I multiply it by all the Terms of the Divifor, and as this Mul-
tiplication ſhould give me for firſt Term 2 a4 Product of a² into 2 a² with
the Sign+, I fet down this Term under the Dividend with the Sign-,
becauſe it is to be fubtracted.
In like Manner, becauſe the fecond Term 3 b a3 Product of a² into
3 ba fhould have the Sign by the Multiplication, I write under the
Dividend + 3ba3 fince it is to be fubtracted, finally becauſe the third
Term 4b2a2 Product of a² into 462 fhould have by the Multiplication
the Sign I write it under the Dividend with the Šign.
4.
This being done, I draw a Line and reduce; the Quantity which re-
mains is 10 b a³ + 27 62 a2 38 63 a + 24 64 which is to be di-
vided by the fame Dvifor 2 a² 3ba + 462. To perform this Divi-
fion, I divide the firft Term 10 b a3 of the Quantity to be divided by
the first Term 2 a2 of the Divifor, I write the Quotient 5 ba befide
a², and multiply this new Term of the Quotient by the Divifor, and
write the Terms of the Product with contrary Signs under the Quantity
to be divided. After the Reduction there remains 12 62 a²
1863 a
+24 64 to be divided.
The firſt Term 12 63 a² of this Quantity being divided by 2 a² of the
Divifor, gives + 6 b² for third Term of the Quotient, and as the Pro-
duct of this third Term into the Divifor deſtroys all thofe of the Quan-
tity to be divided, I conclude that the Divifion is ended, and that
a² — 5ba + 6 62 is the Quotient fought.
thod to an
32
ELEMENTS OF
Lo
404 96² c² + 663 c — 64
-4c4 +66-22262 c²
2c² 3 bc +62
2c² +36c-62
66c3
-{{ b² c² + 663 c →→
b2
84
6bc3 +9
62 c2
ር
3
63 с
2
62 c² +
3
33 с
64
+26² c²
3
63 c + 64
O
Another ex-
ample.
Let it be propofed to divide 6 63 c — 64 — 3cb
9c262 + 404 by .
+ b² + 2 c², I write thofe two Quantities in the Manner above after
having ordered them according to the Letter c.
C
Dividing afterwards the two firſt Terms, I have 2 c² for the firft.
Term of the Quotient, which being multiplyed by the Divifor, gives, af-
ter changing the Signs, the Quantity
4c4 +66c3
2 b2 c², which
being placed under the Dividend, gives for Remainder 6bc3 II b² c²
+663c 64, in which I obferved to place first the Term 6b c³ af-
fected by c3 introduced by the Multiplication, in order that the Quantity
may remain ordered according to c. Dividing afterwards this firſt Term
6bc3 by 2 c², I have 3 bc for Quotient with the Sign+. I multiply
this new Term of the Quotient by the Divifor, and write the Terms of
the Product under the Dividend, after having changed their Signs. After
the Reduction there remains -262 c² + 363 c C 64 to be divided.
The firſt Term of this Quantity being divided by that of the Divifor,
gives for third Term of the Quotient b² affected with the Sign, be-
cauſe the Terms 2 b² c² and 2 c² have different Signs, and as the Product
of this third Term into the Divifor, deſtroys all thofe of the Quantity
to be divided, I conclude that the Divifion is ended, and that 2c+3bc
b2 is the Quotient fought.
LI.
.
If in ordering the Dividend and the Divifor according to the ſame
Letter, there occur feveral Terms in which this Letter has the fame Ex-
ponent, we would fall into the fame Inconveniency which we propoſed
to avoid; unleſs thofe Terms be ordered again according to another Let-
ter, common to the two Quantities.
Let us ſuppoſe, for Example, that the Dividend being ordered accord-
ing to the Letter d, the firft Terms are, 3 ac² d3. c3 d3
3 a²c d3
+a³ d³, and thoſe of the Divifor, ordered according to the fame Letter
a² d² + c² d² - 2acd. Arranging theſe two Quantities thus a3 d3
3ca2 d3 + 3 c² a d³ c3 d³, a² d2 2 cad² + c² d, that is,
ordering them according to the Letter a, the Diviſion will be performed
without being liable to be embarraffed in chufing in the Dividend and
Divifor the Terms which fhould ferve to form thofe of the Quotient.
SPECIOUS ARITHMETICK.
33
Product.
-14
Reduction, 1,
32
½
To accuftom Beginners to be attentive to thofe Particularities in Di-
vifion, here follow more Examples.
Dividend,
3 — a² y² + 3a3 y — 1/2 at
J43a² a4
+2ay³ — a² y²
ауз
+2 ay³ — 4 a² y² + 3 a³ y — — α4
Divifor.
2.
zay + a²
32+2ay - 21/2 a²
44
Product.
-2 ay3+4 a² y² —
y2
2 a3 y
Reduction. 2.
1/2
a² y² +
Product.
+ // a² y²
—
a3 y — // a4
a³y + // a4
Reduction. 3..
O
Dividend.
Divifor.
{
36 +
a ay 4
- 2 b b y4
+ b 4 yy
25
aa
b b
ayy
2 a4 b b
a a 64
34 + za ayy
b by y
+ at
+ a a b b
Product,
{
Reduction, 1.
{
Product,
{
Reduction. 3.
1
Product
Rednction. 5.
{
35 + a a y4
bb y4
+2a ay 4 +
b by 4
a y y
—2nay4 + 2a4yy
+ 2 a² b²yy
b by4 + b4 yy
+ at yy
+ zaabbyy
Reduction. 2.
Product.
{
+ bb y4
Product.
{{
Reduction. 4.
{
aabbyy
b4 yy
+ at yy
+ aabbyy
a437
+ aabbyy
26
20466
a a b4
}
as
2a4bb
·a a b4
25
2a4bb
a a 64
+a6
111 ++|| I
a4 b b
a4 b b
a a b4
a abbyy + aA bb
+ aa b4
}
}
}
64yy
}
}
3 E
34
ELEMENTS OF
Problem in
LII.
In the Solution of the foregoing Problems, we had occafion to employ
only one unknown Quantity, becauſe in thofe Problems there was, pro-
perly ſpeaking, only one Quantity to be inveſtigated. But as we ad-
vance in the Science of fpecious Arithmetick, there occur Problems, to
folve which, it is requifite to employ feveral unknown Quantities, we
fhall therefore proceed to explain how they are to be managed.
There being given the Specifick Gravities of two Subftances which are
which two mixed together, the Bulk and Weight of the Mixture, to find the Quantity
of each of thofe Subftances which compofe the Mixture.
unknown
quantities
are employ-
ed.
Let the Number of cubick Inches contained in this Mixture, or in ge-
neral, its Bulk after whatever Manner it is meaſured, be expreffed by a,
and its Weight by b.
Let the Quantity of the firſt Subſtance contained in the Mixture, for
Example, the Number of cubick Inches of this Subftance be expreffed
by x.
Let the Weight of a cubick Inch of this Subftance, or in general, its
fpecifick Gravity be expreffed by c.
Let the Quantity of the fecond Subſtance be expreffed by y, and its
ſpecifick Gravity by d.
The Weight of the Quantity of the firft Subftance contained in the
Mixture will be expreffed by c x.
C
For if x expreffes the Number of cubick Inches of this Subſtance,
and the Weight of each cubick Inch, their whole. Weight will be the
Product of thoſe two Numbers. In like Manner the Weight of the
Quantity of the ſecond Subſtance will be expreffed by dy.
But thofe two Weights added together is equal to the Weight of the
Mixture, hence we have the Equation cx + dy = b.
x=
C
But this Equation is not fufficient to folve the Problem; for diſen-
gaging one of the unknown Quantities, for Example, x, we find
b-dy
, whereby x cannot be determined unleſs we ſuppoſe y
to be known. There is therefore fome other Operation to be performed:
to diſcovery. To arrive at which we must examine whether all the
Particulars expreſſed in the Queſtion have been attended to, or to ſpeak
in the Language of the Analyfts, if all the Conditions of the Problem
have been fulfilled; upon Examination it will appear that only one of
the Conditions has been expreffed, and the Condition implying
that the Quantity of the firſt Subſtance, added to the Quantity of the
fecond, fhould be equal to the Bulk of the Mixture has not been em-
ployed. Hence this fecond Condition furniſhes the Equation x+y=a,
which, as the firſt, determines the Value of x only by Means of that
of y, by giving xay.
SPECIOUS ARITHMETICK.
35
But if we cannot by either of thoſe two Equations, taken feperately,
find x independent of y, by employing them both together we are en-
abled to determine y. For fince each of thofe two Equations gives a
Value of x, we may equal theſe two Values, which gives the Equation
dy = a a - y, from which is deduced by the foregoing Methods
b
or
C
dyaccy, or a c-bcy—dy, or finally y =
ac
- b
y being known it eaſily appears that x which is expreffed by a y',
bdy is alfo known. We have therefore only to fubftitute in
which of thoſe two Quantities we pleaſe, in the firft a-y, for Exam-
ple, inſtead of y,
· b
a c
b
c = d
and we will have a-
a c
for the
c d
ढ
On examining the foregoing Value a -
Value of x.
acb it will eafily be
ас
C
d
perceived that it can be reduced; for if we put a to the fame Denomi-
nator as the Fraction
gives
a c ad
C
d
ac-b
C
we muſt multiply it by c-d, which
inſtead of a; we have no more to do than to fub-
tract from this Fraction the fecond. To this End ſubtra&-
ing their Numerators, and dividing the Remainder by the common
Denominator, we will have
C ---- -d
ас- ad
ac + b
or
C
d
bad
c-d
for
The Quantities required therefore of the first and fecond Subſtance
the reduced Value of x.
which compoſe the Mixture are expreffed, one by
a c
C
b
other by
a d
b
and the
confequently the Problem is folved.
LIII.
a c
C
b
If inſtead of ſubſtituting the Value of y in a -y, we
fubftituted it in
b-dx
C
b
C
a c
b
d
fult
C
Ъ
ad
lue from
C
and
b
C
dy
d
of
dy, which alſo expreſſes x. There would re-
which at first View feems to be a different Va-
But as we know that the two Values a
x are equal, thoſe two Values of x expreffed in
36
ELEMENTS OF
known Quantities muſt coincide. The Manner of reducing one to the-
other is as follows.
I first give the Denominator cd to the Letter b, by multiplying
c-
cd, that is, by fubftituting
b c
b d
it by c
for b, and then the
C
d
d x ( a c
b)
b
C
d
C
foregoing Quantity
b c - b d d x ( a c — b)
C
C
d
but instead of dx (ac
*
or
,
will be transformed into
bcbd dx (ac - b)
d c
CC
;
b) we may write a c d — b d, and as this
Quantity fhould be fubtracted from bc bd, the foregoing Quantity
b c — b d
c²
dx (a c
do
d c a
b c
b)
when reduced is
c²
2 — d c
which
by dividing the Numerator and Denominator by the fame Quantity 'c,
da the fame Value as before.
at length becomes
Ъ
d
LIV.
To apply this general Solution to an Example; let the Mixture be
Application fuppofed to be compofed of Gold and Silver, its Weight to be 30
of the fore- Ounces, its Bulk 3 cubick Inches, the Weight of a cubick Inch of
going folu-
Gold 12 Ounces, and that of a cubick Inch of Silver 6 & Ounces.
We will have a = 3, b = 30, c = 123, d = 63. Subftituting
thoſe Values in the two general Formulas x =
and y =
tion to an
example.
Problem in
which two
unknown
ac - b
they will become x = 1
and y =
ture will contain cubick Inches of Gold, and
Silver.
13
LV.
I 8
da
c-d
, that is, the Mix-
1 cubick Inches of
I 3
It is eaſy to perceive after what we have ſeen in the foregoing Pro-
blem, that as often as two unknown Quantities are employed in a Queſ-
tion, there muſt be two Equations to difengage them, and when there
are two Quantities required in a Problem, there must be two Conditions
given to determine them, in order to deduce from thofe two Conditions
two Equations. To explain the Method of employing thoſe Conditions
we have added the following Problem.
LVI.
Two Pipes, each of which run uniformly together, filled a Ciſtern a, the
one running during the Time b, the other during the Time c; the two fame
SPECIOUS ARITHMETICK.
37
are employ-
Pipes filled another Ciſtern d, the firſt running during the Time e, the fe- quantities
cond during the Time f. The Discharge of each Pipe is required:
Let x and y exprefs thofe Difcharges, that is, for Example, the
Number of Hogfheads that each of thoſe two Pipes furnifh in a Day,
fuppofing the Cisterns a and d to be meaſured in Hogfheads, whilft the
Times b, c, e, f are counted in Days.
The Quantity of Water furniſhed by the firſt Pipe, during the Time
b, will be expreffed by. bx; and the Quantity of Water furniſhed by
the fecond Pipe in the Time c, will be expreffed by cy. But thofe two
Quantities of Water by the firft Condition of the Problem fhould be
equal to the Ciſtern a, henee we have bx + cy = a.
In like Manner the Quantities of Water furnifhed by the fame Pipes
during the Times e, f will be expreffed by ex, fy, and confequently
the ſecond Condition will give e x + ƒ y · d:
We have no more to do now than to deduce from thoſe two Equati
ons.the Values of x and of y, which will be performed, as in the fore-
going Problem, by deducing a Value of x from each of thofe two Equa-
tions, and then putting thefe two Values equal to each other.
a
су
The first will be
b
e
Putting theſe
d-fy
the ſecond
df
cy
cey = b d
a e, or fi-
two Values equal to each other, there will arife
a су
Ъ
e
ed.
}
Subftituting this Value of y in one of the two
or a e -
cey
b d
b d
bfy, or b fy
nally y =
Values of x,
of
in
b d
a e
се
a
a e
b
су
a
- CX
(
-)
b
of-ce
or x =
b
for Example, there will refult x =
ax (bf-ce) — cx (bd — ae)
bx. (b.fce)
•
by giving the firft Term a the fame Denominator as the fecond, and by
multiplying the two Denominators, one by the other.
Performing afterwards the Multiplications indicated in this Value, and
a f − c d
reducing, there will arife x =
bf-ce
Wherefore when the particular Values of a, b, c, d, e, f, are afcer-
tained, we have only to fubftitute them in thofe two general Values of
x and of y, to obtain any particular Solution we pleaſe
ولا
it
Inftead of difengaging in the two foregoing Equations, and putting
the two Values which arife equal to each other, in order to obtain
is, manifeft that if we difengaged y, and afterwards put the two Values
which ariſe equal to each other, to obtain x, the Refult. would be the
fame.
3.8
ELEMENTS OF
LVII.
Example of
To make fome Application of this Problem, let us fuppofe that the
the forego firft Pipe running five Days, and the fecond four, filled a Ciftern
in numbers, which contains 330 Hogfheads. That afterwards the first Pipe run-
ning two Days, and the fecond three, filled a Refervoir containing 195
ing problem
Another ex-
ample.
Hogfheads.
In this Cafe a
confequently af
fequentlyx=
=
330, b = 5, c = 4, d= 195, e 2, f= 3, and
210, bf-ce = 7, bdae 315, con-
=
de
af
d c
b f
се
21030, and y =
=
b d
a e
bf - ce
35545. Hence the firft Pipe in this Example furniſhes 30 Hog-
fheads a Day, and the ſecond 45.
LVIII.
Let us now fuppofe that the firſt Pipe running during 3 Days, and
the ſecond during feven, filled a Ciſtern containing 190 Hogfheads, that
afterwards the first running 4 Days, and the fecond 6, they filled a Cif-
tern containing 120 Hogfheads.
In this Cafe a≈ 190, b = 3, c = 7, d = 120, e =
confequently a fdc = 300, bf - ce=
which will give x =
af
b f
d c
се
400
4, ƒ:
4, ƒ = 6, and
de = 400,
d.b a e
10, bd
300
10
and y =
bf - ce
IO
Singularity The firſt Time the Analyfts found fimilar Values, that is, negative
of the ex- Quantities divided by negative Quantities, and pofitive Quantities divid-
preffion ar-
rived at in ed by negative ones; without Doubt they were embarraffed to know
what they meant, and probably reſumed the Queſtion, in order to avoid
thofe Kinds of Divifion, and in the prefent Cafe would have pro-
ceeded thus.
this exam-
ple.
Method of
their mean-
"They would have reſumed the two general Equations bx + cya,
diſcovering and ex+ fyd, and fubftituting in thofe Equations for a, b, c, d, e, f,
the Values which thoſe Letters have in this Example, there would re-
fult 3x + 7y190 and 4x+6y= 120, deducing from thoſe two
Equations
190 73,
-7, and x = 2y, they would have
ing.
3
3
3
190
3
3
put them equal to each other, which would have given y =
3037, or zyy190 -
༡༡
30, or y = 40.
Subftituting afterwards this Value of y in 30-y Value of x, they
would have found x 30 60, that is, x = 30. In this Manner
they would have affured themſelves that the Quotient arifing from
400 divided by 10 is + 40, and that arifing from 300 divided by
IO is 30.
SPECIOUS ARITHMETICK.
39
LIX.
Theorems
They foon after eftabliſhed as general Principles that + divided by +, General
produces, that divided by produces, that divided by concerning
+
produces that divided by produces +, and the fame for Mul- the figns of
tiplication.
Thoſe Principles were the more eaſily diſcovered as they refulted from
the Obſervations which must neceffarily have been made on the Signs
found for the Terms of the Products and Quotients, in applying the
Rules given for the Multiplication and Divifion of compound Quantities,
but it is probable that the firſt Analyſts did not adopt them until after
they had verified them in feveral Examplès.
a
EX:
quotients or
products.
-binto
S
d
are not pre-
To affure ourſelves that the Multiplication of -- by fhould pro- Demonftra-
duce in the Product, let us fee what Affiftance we can draw from tion that
the general Method of Multiplications delivered Art. XLV. According is +bd
to this Method, we fee plainly that the Product of a Quantity, fuch as when thofe
into another c d fhould be a c —
b c
ad + bd, and con- quantities
fequently we ſee at the fame Time that the Term b d which arifes from ceded by
the Multiplication of binto d has the Sign+, whilft its Factors b and d others. ·
have the Sign It remains therefore to know if when two negative
Quantities, fuch asb and d are not preceded by any pofitive.
Quantities, their Product will be ſtill bd, the Truth of which will
eafily appear by obferving that the Method by which we found the
Product of a b into c- d to be ac
adbd was not
reftrained to any particular Value of a or c, and confequently took Place
when thofe Quantities were equal to nothing. But in this Cafe, the
Product a c
b c
ad bd is reduced to + b d, wherefore-
―bxd = + bd.
LXI.
.bc
-
The other
As to the other Cafes, that is, with Refpect to the Multiplication cafes are de
and Diviſion of by, they will be proved after the fame Manner.
LXII.
30 and Y. = + 40 we
monſtrated
in like man-
ner.
To return now to our laft Application of the foregoing Problem, it
is to be obſerved that after having found x =
fhould be ftill at a Lofs to know what this Value of x meant. The How the ne
Method the firſt Analyſts probably would have taken to fatisfy them- gative value
felves in this Point, would be to return back to the Conditions of the the problem
found,folves
Problem or which comes to the fame, they would reſume the two
Equations 3x + 7y 190, and 4x+6y=120, and try. how the
Values 30 and 40 of x and of y, anfwered thofe Equations, we
find first, that in this Cafe 3x90 and 7 y 280, confequently
3x+7=90
90 + 280, which in effect is equal to 190. In like
Manner 4x+6y is found to be 120 + 140 which is reduced to 120.
40
ELEMENTS OF
;
Unknown
quantities
=
Having therefore difcovered how the Values 30 and 40 of x
and of y anſwer the Equations 3x + 7y = 190 and 4x + 6y ≈ 120,
we perceive at the fame Time how they answer the Conditions of the
Problem; for fince the Ufe that have been made of the Quantities
3x and 4x, which exprefs the Quantities of Water diſcharged by the
first Pipe in the firſt and ſecond Operation, was to fubtract them from
7y and from 6y, which expreſs the Quantities of Water furniſhed in
the fame Operations by the fecond Pipe. The first Pipe muſt
be confidered in this Cafe as depriving the Cifterns of Water inſtead of
Furniſhing any, as it did in the other Exampie, and as it was ſuppoſed
in expreffing the Conditions of the Problem.
We have here an Example of the Generality of the analytick Art,
whereby we find in a Queſtion Cafes which we did not foreſee could be
included in it.
LXIII.
In almost every Queftion folved after a general Manner, the Ana-
becoming lyfts found Cafes fimilar to the foregoing, and they always concluded,
negative are that when the Value of the unknown Quantity became negative, the
Quantity expreffed by it ſhould be confidered as being of an oppofite
Kind from what it was fuppofed in expreffing the Conditions of the
of an oppo-
fite kind
from what
they were
fuppofed in
the expreffi-
on of the
Problem.
What has been faid with Refpe&t to unknown Quantities, is equally
applicable to known Quantities, that is, when a general Solution is ap-
problem, as plied to any particular Cafe, if any of the given Quantities a, b, &c. in
allo known the Problem are negative.
quantities.
Example of
knownquan
tities made
negative.
LXIV.
Let it be propofed, for Example, to find what ſhould be in the fore-
the ufe of going Problem the Diſcharges of two Pipes that the first furniſhing
Water during 3 Days, and the fecond during four Days, may fill a Cif-
tern containing 320 Hogfheads, and that the fecond Pipe afterwards
furniſhing Water during fix Days whilft the firſt diſcharges it during
3 Days, may fill a Cistern containing 180 Hogfheads.
We have only to put in the general Solution a
c = 4, d = 180, e = 3, ƒ = 6.
.a e
f
=
=
And there will refult de 720, af 1920, ce = 18,
− 960, d b = 540, and confequently af
-ce30, d b a e = 1500, which gives x =
e=
and y
d b
bf-ce
a e
= 50.
320, b = 3,
bƒ =— 12,
dc
1200, bf
af
de
bf-ce
=40,
From whence it appears that the Diſcharge of the firſt Pipe is 40
Hogfheads a Day either to carry away the Water as in the fecond Ope-
ration, or to furnish it as in the firft, and the Diſcharge of the fecond 50
Hogfheads a Day, which it furniſhes in both Operations.
'
41
SPECIOUS ARITHMETICK.
LXV.
To accuftom Beginners to the Manner of extending the Solutions of
Problems to thofe Cafes in which the given Quantities are of an oppo-
fite Kind from what they have been ſuppoſed to be at firſt, I ſhall give
another Example taken from the Problem of Art. xxiv. in which it is
required to know where two Meffengers will meet, and will endeavour
to deduce from the general Solution, the Solution of the following
Cafe.
which the
Two Meffengers at 50 Miles Diſtance from each other, one being, Another ex-
for Example, at Newry, the other at Dublin. The first fets out from ample in
Newry for Dublin at 8 in the Evening, running at the Rate of 4 Miles known quan
an Hour, the fecond fets out the fame Day from Dublin for Newry, at tities are
11 in the Morning, running 3 Miles an Hour, it is required to know made nega
at what Diſtance from Dublin they will meet.
Comparing this Cafe with the general Expreffion of the Problem,
it appears first, that the Letter c which denoted the Number of Miles
run by the first Meffenger in a given Time, fhould be negative, fince
in the general Solution the firſt Meffenger was fuppoſed to recede from,
and, in this Cafe, he goes to meet the fecond. It alfo appears that the
Letter b, which expreffes the Number of Hours that the firſt Meſſen-
ger fet out fooner than the fecond, fhould be alſo negative, becauſe in
this Cafe he fet out later than the fecond.
Hence we have only to put in the general Formula -
པ་
a de + b ce
de
- c
cf
=1, and we will have x =
a = 50, b = -
F
50 X 1
9,c=- 4, d = 1, e = 35
X 3 — 9 X — 4 × 3,
X
IX 3 + 4X I
150 + 108
258
=
3 + 4
7
= 36, by which we learn that when the
Meſſenger from Dublin has run 36 Miles, he will meet the Meffen-
ger from Newry.
7
LXVI.
tive.
foever of the
known quan
tities may
One of the principal Advantages of fpecious Arithmetick, and which Two equa
fhews the Utility of affuming at Will the Signs of the given Quantities tions what
in the general Solution of Problems, is the reducing to the Solution of first degree,
Equations expreffed indefinitely all thoſe in which the unknown Quan- including
tities are difpofed after the fame Manner, for Example, by Means of two un-
the two Equations bx + cya, and ex +fyd folved Article
LVI. any two Equations whatſoever of the firſt Degree may be folved, be reduced
provided they include only two unknown Quantities.
Let, for Example, the two Equations mnx = p² y — b b
m ny
p3 nnx be propoſed to be folved.
with the two first I write them thus m n x-
to the fore-
going ones.
p² y
b bg, and
To compare them
bbg, and
Example.
3 F
42
ELEMENTS OF
་
Another ex-
ample.
=
f
p3.
nnx + n my p³, then comparing them with the two Equations
bx + cy a and ex + fy d. The firft with the firft,
and the fecond with the fecond, we will have b = m n, c —— p², a=
b b g, e = n², ƒ = m n, d =
Which will give cd p³, a f =
= m² n², a e = b² n² g, bd
c d = m n b² g + p³, bf
b f
af
=
a e = m n p³ + b² 12² g.
Now fubftituting thofe Values in
a f - c d
bf - ce
p5
a e
bf-ce
b d
>
and } =
y
m n b²
g
2
} =
m² n² + p² m²
Let the Equations
+ (p + q) × y =
first this Form
=
-
m n b² g, ce= p² π²,
m np³, and confequently
ce = m² 12² + p² m², b d
the general Formulas x
we will have at length x=
m n p3 + b² n² g
m² n² + p² n22
3mpx
P
n q
Þ - q
3mpx
Þ - 9
9
LXVII.
p2y.
p + q
279
p. q
and m x
be propoſed to be folved, giving the
p2
p + q
y =
2 19
P 9
and com-
3 m p
paring it with the general Equation bx + cy = a, b =
C
2n9
p q
Þ - 9
, comparing the fecond with the
Equation ex+fy = d, we will have em, f = p + q, d =
a
p + q
n q
and thofe Values
being fubftituted in the Formula x =
p
9
19
af
c d
bf
,
gives x =
p- 9
2n9 × (p+q) +
$2
q n
X
p + q
ce
3 mp
p-q
x(p + q) +
$2
X. m
p + q
To reduce this Quantity, I first multiply the Numerator and Deno-
minator of the Fraction
changes it into
- 2 ng X ( 79)
p
9
(p − q) x (p + q)
Numerator of the Value of x becomes
by p+q, which
9,
by this Means the whole
Z
2
(p + q) X (p − q)
4)
or
[p² - 2 x (p + q) ²] × n q
or
q)
(p −9) X (p + q)
q n X
+p² + 4 p q +29²
(p −q)X(p + q)
I operate afterwards on the Denominator of the Value of x, reducing
its two Parts to the fame Denominator.
SPECIOUS ARITHMETICK.
43
2
Which gives 3 mp X (p + q) ² +
تمر
رو
p² x (p − q) × m
(p + q) X (p − q)
2
m p × [p² = p q + 3 x (p + q) ²], or
(p + q) × (p − q)
Thoſe two Operations change the
p² + 4 p q + 2 q²
q n X
mp x
(p + q)X(p − q)
4p² +509 +39²
(p + q) × (p −q)
X
;
or
m p × (4 p² + 5 p 9 + 39²)
X
(p + q) X (p − q)
foregoing Value of x into
but as the Numerator and De-
nominator of this Fraction are each divided by (p + q) X (p- },
I take away this Divifor, and the Value of x becomes
— q n X (p² + 4 p 9 +299)
mp × (4 p² + 589 +39²)
or finally x =
like Manner the
} =
nq p p
4 m p³ + 5 m p² q
fame Values of a,
pp +49 +299
4 p p + 5 9 + 399
2193
9
9 n
or
X
m p
4 p q q n
+ 3 m p q q
Subſtituting in
b,
c, &c. in the general Formula
b d
bf-ce
a e
we will have y:
} =
3 m p
Þ - 9
3 m'p
n q
X
+
214
X m
p9
× (p + q) +
pp
X 112
p + q
whoſe Numerator I reduce to this Form 3 m n p q + 2 m
n
q X p.
(p − q)
L
2 m n q
by multiplying the Numerator & Denominator of the Fraction
by p q.
I afterwards reduce this new Form and it becomes
m n q × (5 p − 2 q)
(B
درو
24
As to the Denominator of the Value of y, it being the fame as that
of x, it will be reduced after the fame Manner, and confequently we
رو
m n q
X
5 P 29
(p − q) 2
will have y:
mp x
4 PP + 5P 9 + 3 9 9
(p + q) X (p − q)
which by taking
away the common Divifors pq, and reducing, there reſults
q n X
5P-29
$
y
p x App +
5 p 9 + 399
which, by putting the Divifor
p + q
p+q at Top, and the Divifor pq at Bottom, according to the
Rules of the Divifion of Fractions, will at length become
q n X (p + q) × (5 p — 2 q)
P X (p − q) × (4 p² + 5 p q + 3 q²) '
ELEMENTS OF
44
or y
Another
manner of
folving the
2
2 q3 n + 5 p² q n + 3 p q² n
2 2
2 p² q² + 4p4 + p³ q 393
LXVIII.
If we had freed the Equations propofed in this Example from Frac-
tions, before we compared them with the general Formulas, they would
have been more readily folved.
Let the two Members of the Equation
foregoing
example.
3 m p x
or
ppy
p + q
3 mp x
Þ - q
ppy
p+9
29 n
2nq
p
4
be multiplied by pp — 9 9,
Product of the two Divifors pq, p + q, and we will have the
Equation (3 m pp + 3 m p q ) x x + ( p p q − p³) × y = — 2 n p q
2 199
In like Manner, let the two Members of the Equation m x + (p+ q)
× y =
n be multiplied by p—
q
р q
A:
xx + (pp - qq) × y = q ni
q, and we will have m p
mq
Now comparing thofe two new Equations with the two general For-
mulas b = 3 m p p + 3 m p q, C = p p q p³, a =
2 n q², e = mp m q, f
3
4
From whence is deduced c d = p² q² n
2 22 p² q² + 2 p q³ n + 2 n q², a e = 2 nm q³
b d = 3 m n p² q + 3 m n p q²; ce = 2 p³ q m --
b f = 3 m p³ q + 3 m p4
af
b f
b d
c d = 2
се
3
3 m p q 3 m p² q², hence
3 n p² q²
3
2 m p² q² + m p³ q + 4 m p²,
m n p q² + 5 m n p² q,
2 n p q
$2 q², d q n.
p³ q n, a
q3
af
2 n p³ q
p3
m p4
2 n m p² q,
m p² qz
n qʻ
94 + 2 π p
3 m p q³
q³
ñ p³™ q,
n p³ q
mp3 q. +
3 n p²
4 m p4
q np3 +
or x=
and
y
2 m p p q q
2 n m q3
2 m p p q q
3 p p q q n
4 m p4
—
2 n p 93
3 m p q3
2nq4
2 m p² 92
2
mp3 g + 3 m p q3
5 m n p²
4 m p4
q
3 m n p q q
mp3 q + 3 m p q³
a e
which gives x =
2 nm q³ + 3
2
q² + 2 p q 3 n + 2 n q4
મ
Compariſon
folutions.
LXIX.
Comparing actually thoſe two Values of and of y with thofe found
of the two in the foregoing Article, firft the Identity of the two Values of y is
foregoing
eafily perceived. As to the Values of x to diſcover how the former can
be the fame with the latter, we are to obſerve, that the Equality which
fhould fubfift between thoſe two Expreffions, neceffarily fuppofes that
the Numerator q n p³ + 3 n p² q² 2 12 p q3 2 n q4 of the lat-
ter contains the Numerator
former, after the fame Manner that
4
q n p²
n
4 p q² n 2 q3 n of the
the Denominator 2 m p² q²
- 4m p² - mp3 g + 3 m p q³ of the latter contains the Denomina-
SPECIOUS ARITHMETICK.
tor 4 p³ m q + 5 p² q m + 3 m p q² of the former, and dividing the
fecond Numerator by the firft, we find in effect the fame Quotient
pq, as refults by dividing the fecond Denominator by the firft, that
is, the Expreffion n p 3 + 3 p² q² n — 2 n p q3 — 2n94
2 2
2 m p² q 4 m p4
ed into q. X (— nq p²
p
9
p³
m p³ q + 3 m p q3
4 p q² n 2n93)
p = q x ( 4 m p3 g + 5.mp² q + 3 m p q²)
away p-q, becomes
nq p²
is chang-
2
or by taking
4 P. q² n 2n93
4 mp3 q + 5 m p² q + 3 mp q3
LXX.
It was eaſy to difcover the Method employed to reduce the more
complex Value of x to the moſt fimple, when both one and the other
of thoſe two Expreffions were known; but if the more complex Value
was only known, and it was propofed to reduce it, to its lowest Terms,
it would have been much more embarraffing, fince we would not know
by what Quantity to divide the Numerator and Denominator of the
Fraction. Now as it would be an Imperfection in the Solution of a Prob-
lem that a Quantity ſhould be reducible and not reduced, the Analyſts
have fought a Method for reducing Fractions to their leaft Terms, or
which comes to the fame, a Method for finding the greateſt common
Divifor of any two given Quantities.
To explain what they have imagined in this Reſpect, let us firft fup-
poſe thoſe two Quantities to be Numbers; let it be propofed, for Ex-
ample, to find the greatest common Divifor of the Numbers 637 and
143, or to reduce the Fraction, to its leaft Terms.
Dividing 637 by 143, the Quotient is 4 and there remains 65, that
is, the Fraction 33 is changed into 4+, hence the next Step
is to reduce the Fraction, or to find the greatest common Di-
viſor of the Numbers 143 and 65. For when this Number is found, it
is manifeſt that it will be the greateſt common Divifor of the Numbers
637 and 143, fince we can not reduce the Fraction to its loweſt
Terms, without reducing at the fame Time 4 + or to its leaft
Terms alfo.
6 5
65
773
632
65
143
The two. Numbers 143 and 65, on which we are now to operate,
being much fimpler than the two firſt 637, 143, it is eafy to perceive-
that the Difficulty is leffened, and that proceeding after the fame Man-
ner it will be ftill leffened. Inftead of the Fraction to be reduced,
I write 143, not that thofe Fractions are the fame; but becauſe one
cannot be reduced without the other being reduced alfo after the fame
Manner. To reduce 143, I divide 143 by 65, and the Quotient is 2,
the Remainder being 13. We have therefore no more to do according
to the fame Principle, than to find the greatest common Divifor of 13
and 65, becauſe the greatest common Divifor of thoſe two Numbers,
65
45
46
ELEMENTS OF
3
General me
bers.
will be alfo that of 143 and 65, fince the Fraction 43 is changed into
2 + 3/3/3.
5
Now the greatest common Divifor of 13 and 65 is 13 itfelf, fince it
divides 65 exa&ly. Wherefore 13 is alfo the greateſt common Divifor
of 143 and 65, and confequently of the propofed Numbers 637, 143.
In effect 637 is 49 X 13 and 143, 11 X 13, from whence is deduced
1434 an irreducible Fraction.
637
11
LXXI.
It is eaſy to perceive that the Method employed in the foregoing
thod for find Example, is applicable to any Numbers whatfoever. Let A and B ex-
ing the grea prefs in general any two Numbers, and let a be the Quotient arifing
teſt common
divifor of from the Divifion of the firft by the fecond, and C the Remainder, the
two num Queſtion is reduced to find the greateſt common Diviſor of B and C ;
then b being fuppofed to be the Quotient refulting from the Diviſion
of B by C, and D the Remainder, we have no more to do than to find
the greateſt common Divifor of C and D, that is, to divide C by D,
and if there is a Remainder to divide D by it, and to proceed thus con-
tinually dividing until we find two Numbers, the leaft of which is con-
tained exactly in the greateft, and this Number which is contained ex-
actly in the other, will be the greateſt common Divifor of the two first
Numbers A and B.
This Rule in its full Extent, is founded, as, in the foregoing Exam-
being changed into
ple, on this Principle, that the Fraction
야야
​C
a +
B
0.
cannot be reduced until
is reduced, and that
C
B
B
cannot be reduced but after the fame Manner that
and
В
D
С
that being changed into b + c
C
cannot be reduced until
C
3
D
C
is reduced, and fo on.
LXXII.
We now proceed to fhew what Alterations are to be made in this
Method to render it applicable to fpecious or algebraic Quantities, to
underſtand which, let us first take an Example.
63
Let it be propofed to find the greateſt common Divifor of the Quan-
tities 3 a3 3 ba a + b b a
b³, and 4 a a 5 ba + bb.
We are, according to the foregoing Method to divide the firft of thoſe
two Quantities by the fecond; but as the Divifion cannot be performed,
the firſt Term 3 a3 of the Dividend not containing exactly the firſt
Term of the Divifor, I multiply the firft Quantity by 4, obferving that
as 4 is not a Divifor of the fecond Quantity 4 a a5 ba + bb, the
SPECIOUS ARITHMETICK.
47
greateſt common Divifor of 12 a3 1 2 ba a + 4 b b a
4 a a
3 a3
4 b³, and
5 ba + b b muſt be alfo the greateſt common Divifor of
3 ba a + b b a
63 and 4 a a
5 ba + bb.
I divide then, according to the foregoing Rules, 12 a3 12 b a²
+46² a 4 63 by 4 a² 5 ba + bb; the Quotient is 3 a, and
there remains 3 ba a + b ba—4 63, by which, according to the fame
Rules we ſhould divide 4 a² 5 ba + bb; but as the Diviſion of
thoſe Quantities can not be performed without firft preparing them, I
obſerve 1º. that b being common to all the Terms of the firſt Quantity,
and not to thoſe of the fecond, it cannot be a Part of the greateſt com-
mon Diviſor of thoſe Quantities, wherefore I take it away from all the
Terms of this firft Quantity, and I affume in its Place 3a a + ba
+4 bb. I obferve 29. that if the fecond Quantity 4 a a5 ba
+bb be multiplied by 3, which is no Divifor of 3 a a+ba-
the Diviſion will fucceed; I divide therefore 12 a a 15 a b + 3 b²
by 3 aa + ba 4 b², the Quotient is 4, and there remains — 19 ab
+ 1962.
4
62
We have therefore no more to do than to find the greateſt common
Divifor of 3 a²+ba 4 b², and 19 a b + 19 62. As this Opera-
tion requires that we ſhould divide the firft of thoſe two Quantities by
the ſecond, and that it is neceffary in order to make the Divifion of the
two firſt Terms fucceed, to multiply the first by 19 b, which is an ex-
act Divifor of the fecond, I take away this Divifor from the fecond,,
whereby it is reduced to
a + b
But the greateſt common Diviſor of 3 a a + b a
15 a b
4bb and
a + b, is ab itſelf, fince the Divifion of thoſe two Quan-
tities is performed exactly. Wherefore a+b is the greatest com-
mon Divifor of 3a a + ba 4 b b and of 19 a b + 19 b b¸ .
wherefore it is alfo the greatest common Divifor of 12 a a.-
+ 3b b, and of 3a a + b a 462, and confequently of 4 aa-56.
+b², and 3 b a a + b b a 4 63, as alfo of 12 a3 iz ba a
+ 4bba 463, and 4.aa 5 b a + b b, and finally it is the
greatest common Divifor of the propofed Quantities 3 a3 3 ba a
+ b b a
63 and 4 a² -5b a + b².
LXXIII
ba
It is eaſy to perceive that the greatest common Divifor of any
other two Quantities may be found after the fame Manner. The only
Principle which we are obliged to add in this Enquiry to the Method
of Art. LXXI. is that any two Quantities A and B will preferve their
greatest common Divifor when one of thoſe two Quantities, for Exam-
ple, A, is multiplied or divided by a Quantity which has no commen
Divifor with B.
18
ELEMENTS OF
General me
thod for find
gebraic
The general Method for determining the greatest common Divifors,
ing the great may be expreffed thus. Let A and B be the two propofed Quantities,
eft common firft order thofe two Quantities according to the fame Letter, examine
divifor offpe afterwards what Quantity m is moft proper to multiply A by. in order
cious or al- that the Terms affected by the higheſt Power of the Letter according
quantities. to which it is ordered may be divifible by the Terms of B affected by
the higheſt Power of the fame Letter; if this multiplier m has no com-
mon Divifor with B, multiply A by it, but if it has a common Divifor
take away this common Divifor, both from m and B, and multiply
A by
which will form a new Quantity C, which affume in the
Place of A. In like Manner affume in the Place of B the Quantity D,
which reſults when B is divided by the Divifor n, common with m.
Firſt exam-
ple.
·979
•
m
n
Then divide C by D, and the Divifion being performed, if it be
exact, D will be the greatest common Divifor fought of A and B, but
if there be a Remainder E, perform upon D and E the fame Operation
as on A and B, and fo on until two Quantities are obtained, of which
one divides the other without a Remainder, and that Quantity will be
the greateſt common Divifor required.
It is proper to obferve, that if before you attempt this Operation,
you can difcover in one of the propofed Quantities A or B any Quan-
tity which is an exact Divifor of it and not of the other, it will be
convenient to take away this Diviſor.
To render the Application of this Method familiar to Beginners, here
follow more Examples.
LXXIV.
Let there be given the Quantities q n p3+3 n p² q² — 2 n p q³
93
2´n q4, and 2 m p² q²
92 4 m p4
m p³ q + 3 m p q³, of which
we found p q to be a common Divifor, only becauſe we knew be-
fore Hand that the firſt of thofe two Quantities, divided by the fecond
fhould give the fame Quotient as the Quantity
3
n q p²
2
4 p q² n
2 n q³, divided by 4 m p³ + 5 m p² q + 3 m p q²:
To reduce now thofe two Quantities by the foregoing Method, I
take away qn which is common to all the Terms of the first of thoſe
two Quantities, and is not contained in thoſe of the ſecond; I take
away likewiſe pm, which is common to all the Terms of the fecond
without being contained in thofe of the firft, and thereby the Operation
is reduced to find the greatest common Divifor of the Quantities (A)
4p3
p² q + 2 p q ² + 3 93 and (B) p³ + 3 p p q − 2 p q²
2q3.
Dividing by B the Quotient is 4, and there remains (C) 11 p² q
6 p q²
5 q³, as B fhould be multiplied by 11 q in order to render
its firft Term divifible by the firft Term of C, and that q is contained
SPECIOUS ARITHMETICK.
49
in all the Terms of C, I multiply B fimply by 11, and I divide C by 4,
hence the Queſtion is reduced to find the greateſt common Divifor of
the two Quantities (D) 11 p + 33 p² q — 22 p q² - 22 q³, and (E)
3,
бря 5.92.
11 p²
2
I divide the first by the fecond, the Quotient is p and there remains
(F) 39 p² q- 17 p q² 22 93. As (E) must be multiplied by 39 9
to render its first Term divifible by that of this new Quantity F, and
that 9 is common to all the Terms of F, I multiply E only by 39,
and I'divide its Product (G) 429 p² - 234 p q — 195 q² by (H) 39 p²
The Quotient is II, and the Remainder
22 92.
17 p q
(1) 47 p q +47 9².
2
22 q². And
To render H diviſible by I, we muſt multiply all its Terms by 47 9 ;
but this Quantity is a Divifor of H, wherefore I take it away from H,
and there remains q p for Divifor to 39 p² — 17 p q
as it divides it exactly, I conclude that qp is the greatest common
Divifor fought of the propofed Quantities.
accc
LXXV.
·
5
ba
46c
Let there be given the two Quantities ab + 2 a a 3 b b — 4 b c Second ex-
cc and 9 ac + 2.a a − 5 a b + 4 c c + 8 b c 12b b, rang- ample.
ing thoſe two Quantities according to the Dimenfions of a, we have
2 aa+ba — ca 366 48.0 ccc and 2 aa + 9 ca~---
12 b b + 8 b c + 4 cc, or (A) 2 aa + (b. — c) x a 3bb
-cc and (B) 2 a a + (9 c 5 b) x a 1 2 b b + 8 b c + 4 c c.
Dividing the firft by the fecond, the Quotient is 1, and there re-
mains (C) (6 b 10 c) X a + 9 b b + 1 2 b c 5 с с. To render
B divifible by this Quantity, I obferve that it must be multiplied by
3b5c, but before I perform this Operation, I try the Divifion of C
by 3 6 5c, which fucceeds, and gives (D) 2 a +36 + c for Quo-
b
tient; the Queſtion is therefore reduced to find the greateſt common
Divifor of B and D; but B is diviſible exactly by D; wherefore D, or
2a + 3 b + c is the greateſt common Divifor ſought of a b + 2 a a
3bb
4 bc- a c cc and 9 ac + 2 aa 5.ab + 4cc
12 b b. The first of thoſe Quantities being the Product of
2a+36 +c into a — b. c, the ſecond the Product of 2 a +36
+ c into a 4.b + 4c, and thofe two Quantities a
464c have no common Divifor.
•
+86
a
b c
·
LXXVI.
b
Let there be given the two Quantities (A) (dd — cc) x a² + c4 Third ex-
-ddcc, and (B) 4 d a² — (2 c c + 4 c d ) × a 2 c³ ranged ac- ample.
cc cd) Xa+
cording to the Dimenſions of a. I first change B into (C) 2 d a²
= (c.c + 2 cd) a + c3 by taking away from all its Terms the
Divifor 2, which is not common with A. I afterwards multiply A by
3 G
50
ELEMENTS OF
Another
greateſt com
2.d, in order to make the Divifion fucceed, the Quotient is dd-ce
and there remains (D) (d dc c) x (c c + 2 cd) xa (dd — cc)
X &³ + 2 d. c4 2 d³ c c.
d3
To make this Quantity ferve as a Divifor
to C, we must first multiply C by (d dc c) x (c c + 2 cd).
64
(d d
But before I perform this Multiplication, I examine whether
(d dcc) x (cc+2 cd) be not a Divifor or a Multiple of fome Di-
vifor of D. By inveſtigating the greateſt common Divifor of (dd-cc)
X (cc + 2 cd) and
- cc) x c³ + 2 d c4 2 d³ c², that is,
of d dc c
c4 + 26 d³ 2 c3 d and d d c³ + c5 + 2 d c4
2 d³ c²; and I find that the ſecond of thoſe Quantities is the Product
of the firſt into c, and confequently that the Quantity D is the Pro-
du&t of (d d c c) X (cc + 2 d c) into a - c; wherefore inſtead of
multiplying C by (d dcc) X (cc + 2 dc). I divide D by this
Quantity, and the Quotient is (E) a c, and as a -c divides C ex-
actly, I conclude, that ac is the greateft common Divifor fought.
LXXVII.
The greateſt common Divifor of two Quantities may fometimes be
manner of obtained without having recourfe to the general Method. For Example,
finding the the two foregoing Quantities (d dcc) x a a + c4d dcc and.
mon divifor 4 da a― (2 c c + 4 c d ) Xa+2 c3 being ordered according to d,.
of the quan- and confequently being reduced to this Form (a a- cc) x d d + c4.
tities in the a a cc and (4 a a 4 a c) x d + 2c3 2 c2 a, it is eaſy to per-
foregoing
example.
Other quan-
•
ceive that a acc. is a Diviſor of the firſt, and c a a Divifor of the
fecond. But a² 2 is divifible by c- a, wherefore ca is a Divifor
of the two propofed Quantities; I divide therefore both one and the
other by ca, and the Quotients are (c c d d) x (c + a); and
(cc-
4 ad + 2cc; which by Infpection are found to have no common
Divifor, confequently ca or ac is the greateſt common Divifor
of the propofed Quantities.
LXXVIII.
Let it be propofed to find the greateſt common Divifor of the two
tities whofe Quantities 6 a5 15 at b 4 a³ c² 10 a abcc and 9 a3 b-27 a abc
+ —
greateſt
common di-
vifor is
pendent of
the forego-
6 a b c c + 18 b c³, I firſt take away aa from all the Terms of the
first, and 3 b from all thofe of the fecond, whereby they are reduced-
found inde- to 6 a³· + 1.5 a² b.--. 4 acc- - 10 b c c and 3 a³ 9aac 2 acc
+6c3; but as b is not contained in any of the Terms of the ſecond
ing method. Quantity, I conclude, that if it has a common Divifor with the firſt, it
muſt have one with its two Parts 6 a³ 3 4 acc and 15 a² b. 10 b c c,
and that thoſe two Parts ſhould alſo have the fame common Divifor.
But it appears by Inſpection that 3 a a. — 2 cc is the common Divifor of
thoſe two Parts, wherefore it is the greateſt common Diviſor of the
propoſed Quantities, if they have one, and dividing, in effect, thoſe two
SPECIOUS ARITHMETICK.
51
Quantities by it, the Divifion fucceeds, confequently it is their greateſt
common Divifor.
LXXIX.
are three
quantities re
be three
From what precedes, it is eafy to conclude, that when two Quanti- When there
ties are required in a Problem, there must be two Equations given.
Likewife when there are three unknown Quantities there must be three quired in a
Equations in order to determine them, and in general there must be as problem,
many Equations as Quantities required. As to the Manner of difen- there muft
gaging the unknown Quantities involved in thofe Equations, it is pre- equations
cifely the fame as for Equations involving two unknown Quantities. given to find
For let three Equations be given, each involving three unknown Quan-
tities x, y, z; if a Value of x be deduced from each of thoſe Equations, How the va-
and thoſe different Values be put equal to one another, it is obvious unknown
there will refult two new Equations, involving only y and z, which will quantities
be refolved by the foregoing Method. If four Equations are given, in- involved in
volving four unknown Quantities, their Values may be found after the tions are
fame Manner, and fo on.
The Ufe of this Method will be more clearly underſtood by Help of
the following Problem, containing the greateſt Complication that Equa-
tions of the firſt Degree involving three unknown Quantities are ſuf-
ceptible of.
LXXX.
them.
lues of the
thofe equa-
found.
The Prices of three Magazines, each containing three Sorts of Grain, Problem in
and the Number of Meafures which each Magazine contains of those three which three
different Sorts of Grain being given; to determine the Price of a Mea- unknown
fure of each Sort of Grain.
quantities
are requir
Let a, b, c exprefs the Numbers of Meafures of each Sort of Grain ed.
contained in the firſt Magazine, and let m be the Price of this Maga-
zine.
Let d, e, f exprefs the Numbers of Meaſures of each Sort of the
fame Kinds of Grain, contained in the fecond Magazine, and let n be
the Price of this Magazine.
Let g, b, k exprefs the Numbers of Meafures of each Sort of the
fame Kinds of Grain contained in the third Magazine, and let p be the
Price of this Magazine.
And let x, y, z exprefs the Prices of a Meaſure of each Sort of Grain.
It is manifeft that the Price of the Quantity of the first Sort of Grain
contained in the Magazine m will be expreffed by a x, fince a is the
Number of Meaſures of this Sort of Grain, and the Price of a Mea-
fure; in like Manner the Price of the Quantity of the fecond Sort of
Grain contained in the fame Magazine, will be expreffed by by, and
the Price of the Quantity of the third Sort of Grain contained in the
fame Magazine, will be expreffed by cz. Wherefore fince the Price
52
ELEMENTS OF
m of this Magazine is equal to thoſe three Sums together, we will have
ax + by + c z = m.
•
Expreffing in like Manner the Conditions refpecting the two other
Magazines, we will have dx+ey+fzn and gx+by+kz=p.
It is now Queſtion to deduce from thofe Equations the Values of
x, y, z; with this View I first deduce the Value of x from the first
m by Cz
Equation, which is
and putting this Value of x
a
equal to that deduced from the fecond, we will have the Equation
ey -f, putting afterwards the fame
772
by
Value
a
CZ
n
d
fx
equal to that deduced from the third Equation,
m
by
CZ
a
by
C Z
a
we will have
В
by — k z
g
From the firft of thoſe two Equations I deduce a n
an
Rey - d by + afz
dc
c z, or y =
from the fecond I deduce ap
a p
gm + gcz
a b
g b
a
k
Z
d m =
a e
d m + dc z
d b
a fz
,
b
gy
- 68 29
Manner of
abridging
or y =
mg = a by tak˝z
Putting thoſe two Values of y equal, it is manifeft that we will have
an Equation involving no other unknown Quantity but z, and that folv-
ing this Equation the Value of z will be found. As the Calculations in
this Operation would be confiderable, I fhall fhew how they may be
avoided by employing fome Abbreviations which the first Analyfts who
were engaged in great Calculations eafily imagined.
LXXXI.
Thoſe Abbreviations confift in fubftituting new Letters in the Place
of ſeveral Terms compofed of known Quantities.
the calcula-
tions by par
ticular deno for a p
minations.
Inſtead of a n-
gm, D,
By thoſe new
A + B z
C
db, C,
Denominations the foregoing Equations will beccome
D + E z
B F z =
dm I fubftitute A, for dc
for
g
& c
- a f, B, for a e
—ak,
ak, E, for a b gb, F.
y =
and y =
F
which gives AF +
DC
A F
DC + CE %,
from
whence is deduced z =
B F
CE
a e p
- a b n + d b m
a e k
a b f + d b c
(a e
d b) p + (g b
— a b) n + (
d b k + g b f
(d b
d b p + g
+ g⋅ b n
gem
g e c
g
e) m
fubftituting after-
(a e — db) k + (gb — a b) ƒ + (d b − g e) c'
terwards this Value of z in one of the two foregoing Values of y, in,
SPECIOUS ARITHMETICK.
53
A + B F
BDC-BA F
CE
the firft, for Example, we will have y =
or
С
BD - A E
y
akn afp + dcp-dkm+gfm — gcn
BF CE aek-abfd b c − d b k + g b f − g e c
(ak — gc) n + (dc — a f) p + (g f − d k) m
(a e
= 86) n + (2 6 = ab); + (b = db) m
d b) k
(g, b
f (d
c
This being done, I fubftitute thoſe Values of y and z in one of the
foregoing Values of x, in
reſults x =
or x =
m
a
for Example, and there
m
C Z by
a
C
DC-
AF
Ъ
X (
-)
X
a
Ᏼ Ꭰ A E
BFCE
a
AE)
m X (B F — CE) —
(e k — f·b) m + (c b −
(a e — d b) k + (g b —
b
BF-CE
c X (D C A F) — b × (B D
aX (BF-CE)
'k) n + (b ƒ − c e) p
f
a b) ƒ + (d b − g e) c.
f
LXXXII.
To apply this Method to an Example, let us fuppofe that the firſt
Magazine contains 30 Meaſures of Rye, 20 of Barley, 10 of Wheat,
and that it cost £11 105.
That the fecond contains 15 Meaſures of Rye, 6 of Barley, and 12
of Wheat, and that it cost £6 18 s.
That the third Magazine contains 10 Meaſures of Rye, 5 of Barley,
and 4 of Wheat, and that it cost £3 15s. To determine the Prices of
a Meaſure of Rye, that of Barley, and that of Wheat, we must put
— 20, c = 10, m = 230, d = 15, e = 6, f. = 12,
a = 30,
b
k
4, p = 75.
42
n = 138, g = 10, b = 5,
Subftituting thofe Values in the Formulas, you will find x
y=3, and z = 5, confequently the Price of a Meaſure of Rye is 4
Shillings, that of a Meafure of Barley 3 Shillings, and that of a Mea-
fure of Wheat 5 Shillings.
LXXXIII.
three
quan-
As the Equations of the foregoing Problem are the moft general of lems of the
All prob
the first Degree involving three unknown Quantities, fince each in- firit degree.
volves three unknown Quantities combined with known Quantities, it in which
follows that every Problem of the firſt Degree involving three unknown tities are re-
Quantities, is included in the foregoing, as foon as it is expreffed ana- quired when:
lytically. To give an Example, let the following Problem be propofed.
There are three Ingots compofed of different Metals melted down to- are contain--
gether, of the first of which a Pound (Averdupois) contains, of Silver 7 Oun- ed in the
ces,, of Brafs 3 Ounces, and of Tin 6 Ounces; of the fecond a Pound con- problem.
reduced to
foregoing
54
ELEMENTS OF
In what caf-
tains, of Silver 12 Ounces, of Brafs 3 Ounces, of 'Tin 1 Ounce; and
a Pound of the third contains, of Silver 4 Ounces, of Braſs 7 Ounces,
and of Tin 5 Ounces: How much of each Ingot muſt be taken to make
a fourth, which fhall contain, of Silver 8 Ounces, of Brafs 34 Ounces,
and of Tin 4 Ounces.
Let x, y, z exprefs the Number of Ounces to be taken of each of
thofe Ingots.
༡
It is manifeft that x will exprefs the Quantity of Silver in the Por-
tion of the first Ingot, that 12y will exprefs what is contained in the
Portion of the fecond Ingot, and that that contained in the Portion
of the third.
4
As the Sum of thofe three Quantities fhould be 8 Ounces of Silver,
there refults the Equation x + 3y + + z = 8, or 7 x + 12y
+4% = 128.
ठ
3
ठ
In like Manner the Quantity of Brafs taken out of each of the three
Ingots, will be expreffed by xy, and the Sum of which
ठ x 16
3
ठ
>
hould be 3 Ounces, wherefore +1 +16% = 14, or 3 x
+31 + 7 z = 60.
6
6
I
I
S
5
The Quantity of Tin taken of each Ingot, will, in like Manner be
expreffed by x, y, z, the Sum of which ſhould be 4 Ounces,
wherefore 18x + 789 + 182 = 4 + 4, or 6x + y + 5 z = 68.
y
We have no more to do now than to refolve thofe three Equations,
which will be effected by putting in the foregoing Solution a = 7,
b = 12, c = 4, m = 128, d = 3, e = 3, ƒ = 7, n = 60, g
b=1, k = 5, p=68.
Subftituting thofe Values in the Formulas, you will find x 8,
5 and z =
3, that is, we must take 8 Ounces of the firft Ingot,
Ounces of the fecond, and 3 of the third to form the Ingot required.
J
5
LXXXIV.
f
6,
It is eaſy to perceive that if there are more Quantities required than
es problems Equations given, the Queſtion is not limited to determinate Quanti-
are indeter- ties, but is capable of a Number of Solutions, which however are con-
minate. fined within certain Limits, as will appear by the following Example,
There are three Ingots of Gold: The Mark * of the first contains 23
Firft exam- Carats of fine Gold, that of the fecond 21, and that of the third 18. How
ple. much of each Ingot must be taken to compofe a fourth, weighing 9 Marks,
each Mark containing 22 Carats of fine Gold.
Let x, y, z exprefs the Number of Marks to be taken of each of
the Ingots.
It is manifeft that 23 x will exprefs the Quantity of pure Gold in
the Portion of the first Ingot, 21 y that in the Portion of the fecond,
and 18 z that in the Portion of the third. Adding therefore thofe three
I
The Mark contains 8 Ounces. The Carat is the 24 of a Mark.
SPECIOUS ARITHMETICK.
5.5
Quantities, their Sum fhould contain 9 Times 22 Carats, confequently
23x + 2y + 18 z = 22 × 9. And as the fourth Ingot compofed
of thofe three Portions fhould weigh 9 Marks, x + y + z = 9.
All the Conditions of the Problem being expreffed, it is manifeſt that
the Queſtion is not limited to determinate Quantities, becauſe there are
three unknown Quantities and only two Equations, but it is evident that
if one of the three unknown Quantities be determined, the other two
will be alfo determined, and the Suppofitions which may be made and
the Limits within which they are confined, will be difcovered by folving
thoſe Equations; which will be effected by putting a = 23, 6 = 21,
C 18, m = 22 X 9, d I e = 1,
>
I n = 9.
>
For then the Equation a n
f
md = a ey b d y + a ƒ z
is reduced to an - m = (a
(a - b) x y + (a
or y =
(a
a n
Y
*****
m
a
(a—c) z
- b
a n
m
a
C
2
— c) × Zg.
f:
6
- c d z
where I obſerve, 1°. that the Product
=
c) z of the greateſt Difference a- cinto z cannot exceed the
Product an — m of the Number of Marks into the leaft Difference,
otherwife would become negative. I obferve, 2°. that z cannot be
fuppofed
becauſe would in this Cafe become o,
that is, the fourth Ingot would contain none of the fecond, which is
contrary to the Conditions of the Problem, hence the Limits within
which the Values of z are confined, are z < 1 + and ≈ > o.
o. The
Sign being employed by the Analyſts to denote the Inequality of the
two Quantities between which it is placed, the Point being always di-
rected towards the leaft Quantity.
It is manifeſt that all the Suppofitions for z included within thoſe Li-
mits will folve the Problem.
9
Let, for Example, z=1+
2
8
Ingot, we must take
Marks, then y = 9-5×9
2
1; which denotes that if we take 1 + Marks of the third
a Mark of the fecond, and confequently 6 of
+ 3 Marks of the third Ingot contains 28 ₫ Ca-
rats of fine. Gold, a Mark of the fecond contains 10 Carats, and
the firft. In effect 1
9
이
​5
2
6 Marks of the firft contains 158 Carats, and confequently the
three Portions together contain 198 Carats of pure Gold, the fame
Quantity that is contained in 9 Marks, each Mark conſiſting of 22 Ca-
rats of fine Gold.
If we ſuppoſe z = 1, then y = 2, which fhews that if we take r
Mark of the third Ingot we muſt take 2 of the fecond, and confequently
6 of the firſt. In effect i Mark of the third furniſhes 18 Carats, 2
Marks of the fecond will give 42, and 6 Marks of the firft makes 138,
36
$
ELEMENTS OF
ample.
which together amount to 198 Carats, which is precifely the. fame
Quantity, as is contained in 9 Marks, each Mark conſiſting of 22 Ca-
rats of fine Gold.
LXXXV.
To accuftem Beginners to the Method of limiting the Anſwers to
all Sorts of Queſtions of this Kind, here follows another Example.
Suppofe a Piece of Metal to be compoſed of Tin, Gold and Copper,
Second ex-melted down together, its weight to be 375 Ounces, its Bulk 80
cubick Inches, let the Weight of a cubick Inch of Tin be 4 Ounces,
that of Gold Ounces, and that of Copper 5 Ounces, it is re-
quired to determine how much of each of thoſe Metals are contained
in the propofed Maſs.
Let x, y, z exprefs the Quantities of each of thofe Metals.
As the Quantity of Tin added to that of Gold and Copper fhould
compoſe the whole Bulk, x + y + z = 80. And it being manifeft
that 4 expreffes the Weight of the Quantity of Tin in the Mix-
ture, II y that of the Gold, and 5z that of the Copper, and that
thoſe three Weights together thould be equal to the whole Weight of
the Mixture 4 x + 11 y + 5% 375 or taking away the Frac-
tions 35 +939 + 41 2 = 3000.
5
Thofe two Equations expreffing the Conditions of the Problem, it is
manifeft that it is not limited to determinate Quantities, but if one of
the unknown Quantities be determined, the two others will be deter-
mined alfo. In order therefore to folve thoſe two Equations, and there-
by diſcover what Suppofitions may be made, and within what Limits.
they are confined, I put a = 1, b = 1, c = 1, m = 80, d = 35,
e = 93, ƒ = 41, n = 3000, fubftituting thofe Values in the Equation
d m = (a e b d) y + (a fc d) x z, there refults
a n
58 y + 6 z = 200.
From whence it appears that z < 333, that is, that there cannot be
33 cubick Inches of Copper in the Mixture, for the Equation 58y+6z
200, becoming in this Cafe 58y + 6 × 33 200 will be changed
into 58y + 200 200, from whence is deduced 58 y = 200
3
200
o, that is, there would be no Gold in the Mixture, which is con-
trary to the Suppofition, but all the Suppofitions for the Value of z leſs
than 33 will anſwer the Conditions of the Problem.
=
Let us fuppofe z 4, confequently 6 z 24, and 58 y 176,
ory 325, which denotes that if the Mixture be fuppofed to contain
4 cubick Inches of Copper, it will contain 35 cubick Inches of Gold,
confequently 72 23 cubick Inches of Tin, fince thoſe three Quantities
fhould compofe 80 cubick Inches, and 4 X 72 29 + 11 § × 3 29
+ 5 1 × 4 = 375 Ounces.
28
7229
28
SPECIOUS ARITHMETICK.
57.
1 8 2
62
58 =
If we fuppofe z=3, then 6 z = 18, and the Equation 58y + 6 z
200 will be changed into 58y + 18 = 200, wherefore y
3, which denotes that if the Mixture is fuppofed to contain 3 cu-
bick Inches of Copper, it will contain 3 cubick Inches of Gold, and
confequently 73 25 cubick Inches of Tin. Thoſe three Quantities
together making 80 cubick Inches, and 4 X 73 255 + 11 § X 3 4
+ 5 1 × 3 = 375 Ounces.
29
LXXXVI.
29
which the
It is obvious that when the Number of Equations and unknown Inconveni-
Quantities is confiderable, the Calculation for finding their Values ac- ency to
cording to the foregoing Method would be very laborious, we ſhall foregoing
therefore proceed to explain what Means the Analyfts have found to re- method for
exterminat-
medy this Inconveniency.
ing un-
On examining the Formulas expreffing the Values of the unknown knownquan
Quantities in the general Solutions of the Problems of Art. LVI and tities is lia-
LXXX, it will appear, that the common Denominator of thoſe Values is
formed of all the Products that can be made of a Number of Coefici-
ents equal to the Number of Equations, and that are taken each from a ons which
different Equation, and are prefixed to a different unknown Quantity; to improve
thoſe which involve the Products of two fuch Coeficients having con- this method.
trary Signs.
For Example, in two Equations involving two unknown Quantities
ax + by = c and d x + ey f, all the Products that can be
made of two Coeficients taken each from a different Equation, and are
prefixed to a different unknown Quantity, are a e, bd, to which if con-
trary Signs be given, there will reſult a e
b d, which is the common
Denominator of the Values of x and y deduced from thofe Equations
Art. LVI. In like Manner in three Equations including three unknown
Quantities a x + by + c z = m, d x + ey +ƒz=n, and g x + by
ax dx
+kx= p, all the Products that can be made of three Coeficients ta-
ken each from a different Equation, and are prefixed to a different un-
known Quantity, are a e k, abf, d b c, d b k, g bf, gec, and if con-
trary Signs be given to thofe which involve the Products of two fuch
Coeficients, there will refult a e ka bf + db c — d b k + g b f
gec, which is the common Denominator of the Values of x, y and
z, deduced from thofe Equations Art. LXXX.
z
It alſo appears that the common Denominator is changed into the
Numerator of any of thofe Values, by fubftituting in this Denominator
in the Place of the Coeficients of the unknown Quantity in the given
Equations thofe which affect no unknown Quantity.
db,
For Example, if we fubftitute in the common Denominator ae
for the Coeficients e and b of y, in the Equations ax + by c, d x
+ey=f, thoſe f, c which affect no unknown Quantity, there will
=
ble.
Obfervati-
have ferved
3 H
5.8
ELEMENTS OF
I
General
rule for find
ing the va-
lues of any
number of
refult af d c, which is the Numerator of the Value of y, in like
Manner fubftituting in this Denominator for the Coeficients a and d of
x, thoſe c and f, which affect no unknown Quantity, there will reſult
се bf, which is the Numerator of the Value of x.
If we fubftitute in the common Denominator a ek a b f + d b c
— d b k + g b f-gec for the Coeficients k, f, c of z in the Equa-
tions ax + by + c z = m, d x + ey + f z = n, g x + b y + k z
p, thoſe p, n, m which affect no unknown Quantity, there will re-
fult a ep
a b n + d bm d b p + g b n
gem which is the
Numerator of the Value of z.
In like Manner fubftituting in this Denominator for the Coeficients
b,e, b of y, thoſe p, n, m, there will reſult a k n — afp + dcp
+ gf m gen, which is the Numerator of the Value of y.
Finally fubftituting in this Denominator for the Coeficients g, d, a of
x, thoſe p, n, m, there will refult e
fb m + c b n
+bfp cep, which is the Numerator of the Value of x.
k m
LXXXVII.
d k m
b k n
From theſe Obfervations the Analyfts have deduced this general Rule,
for exterminating unknown Quantities in Equations of the firſt Degree.
Let a, b, c, d, &c. be the Coeficients of thofe unknown Quantities
and A that which affects no unknown Quantity, in the firft Equation.
Let a', b', c', d', &c. be the Coeficients of the fame unknown Quan-
quantities tities, and A that which affects no unknown Quantity, in the fecond
required
Equation.
when as ma
ny fimple
equations
are given.
Let a", b", c", d", &c. be the Coeficients of the fame unknown
Quantities, and A" that which affects no unknown Quantity in the third,.
and fo on.
a
ba;
Form the two Permutations ab and ba and write down
with thoſe two Permutations and the Letter c, form all the poffible Per-
mutations, obferving to change the Sign as often as a will change its-
Place in ab, as likewife in ba; and there will refult
a b c a c b + c a b — bac + b c a
c b a:
With thoſe fix Permutations and the Letter d, form all the Permu
tations poffible, obferving to change the Sign as often as d will change.
its Place in the fame Term, and there will refult
abcd—abdc + adbc-dabc-acbd + acdb―adcb+dacb+cabd—cadb+cdab-dcab
-bacd+badc-bdac+dbac+bcad-bcda+bdca-dbca-cbad+cbda-cdba+dcba
and proceed in this Manner until all the Coeficients of the firſt Equa-
tion are exhauſted.
Afterwards preſerve the Letters which occupy the firſt Place, give
thoſe which occupy the fecond Place the fame Mark they have in the
fecond Equation, and thoſe which occupy the third Place the fame Mark
they have in the third Equation, and fo on. And this Refult will be the
common Denominator of the Values of the unknown Quantities.
SPECIOUS ARITHMETICK.
59
The common Denominator being thus formed, the Value of xe will
be obtained by giving to this Denominator the Numerator which is
found by changing in all its Terms a into A, and the Value of is the
}
Fraction which has the fame Denominator, and for Numerator the
Quantity which refults by changing b into A in all the Terms of the
Denominator, and in like Manner the Values of the other unknown
Quantities will be obtained.
Hence if there be two Equations and two unknown Quantities.
a x + by = A, d' x + b'y', the common Denominator will be
ba' or a b' — a' b.
a b'
C
- ab" c + a b' c
A",
A", the
b a' cl
If there be three Equations and three unknown Quantities ax + by
+cz = A, a' x + b' y + c' z = A', a'' x + b" y + c" z =
common Denominator will be a b'c'l
+bca" - cb' a", or a b'c"
a b c + a" bď
a" b' c, or (a b' — a' b) c" + (a" b — a b') c' + (a' b" — a'' b!) c.
If there be four Equations and four unknown Quantities a x + by
+cz+dt = A, a' x + b' y + c' z+d' t = A', a" x + b" y + c% ·
+ d't = A', a'"' x + b'il y + c'll z + d""t=A"", the common Deno-
minator will be found after ranging the Letters in alphabetic Order to be
-
a
+a" b'll c d'
+a" b' c'"' d
a b' cli d'!!
b' c''' d'"' + a bil c''' d' -- a bll c'll d
ta b'll c' d"
— a b'"' c'' d' + a b'll c'' d +ď b"c d'"'
d' - a" b'll c' d ・a' b c" d'" + a b c"" d"
+a" b c d'"
a
bil c' ¿!!!
· abc d'
a" b c" ď
- all b' c d''!
or [(a b'
+[lab
all' bi c d'
a b ) c' + (all b
d'
a"" b c d" ta"" bc" d'
a'" b" c
.all! b' c'' d
+ a"" b" c' d
a b''). c'
+ (a' b'l
Q" b' ) c ] d'"'
a
b') c'"' + (a
b'"
+[(a""' b - a
+[(a' B!!!
a
b''!) d'' + (a
b'") c'' + (a
b"
bil
all! b)
— a""' b )' c'
- a" b) c'" + (a" b'" -
B!!!
+ (a"" b'
a
b''') c
] d'
a""b" ) c } d'
''
a"" b') c"' + (a""'
b" — a"
LXXXVIII.
""") c' + (a" b' - a' b") c'''] d
Now it is eafy to obferve, 1 that the firſt Term of any one of thofe
Denominators, is formed of the foregoing Denominator multiplied by
the next Letter in the Order of the Alphabet, which it does not in-
clude, this Letter being affected by the Mark which immediately fol-
lows the higheſt of thofe in this fame Denominator.
2º. That the fecond Term is formed of the first, by changing the
higheſt Mark in it into that which is immediately below it, and that
which is immediately below the higheft into the higheft, as alfo by
changing the Signs.
3°. The third Term is formed of the firft, by changing. in it the
higheſt Mark into that of two Numbers below it, and that of two Num-
bers below the higheſt into the higheſt, as alſo by changing the Signs.
Obfervati
ons which
have ferved
this rule
to render
more fimple
60
OF
ELEMENTS
The fourth is formed of the first by changing the higheſt Mark
in it, into that which is three Marks below it, and reciprocally, as
alſo by changing the Signs.
For Example, the fecond Denominator has for firft Term (a b'—a' b) c'd
which is the firft Denominator multiplied by c", which is the Letter
which follows immediately the Letters a and b, and which has the
Mark ", which follows immediately the Mark', the higheſt of thoſe
which enter in the Expreffion a b' a' b.
1
The fecond Term of this fecond Denominator is (a" ba b!) c',
which is no other than (a b'a' b) c", in which the Signs have been
changed, and the higheſt Mark" into which is immediately below it,
and which is immediately below the higheſt into the higheſt ", and ſo on.
Hence the common Denominator of the Values of the unknown
Quantities in any Number of Equations is eafily determined, for Exam-
ple, if there be five Equations and five unknown Quantities, the com-
mon Denominator will be found by this Method to be
[(a b'
a' b ) ɗ" + (a"
b
a
b'l ) c'
+ (a' b" — a"
b') c ]d"
+ [(a b
a
b') c'"' + (a
b'" -- a"" b
) c'
+ (a'"' b'
· a' b"") c
]d"
elllo
+ [(all b
- a
b'") c'"
+ (a
b!! — a"
b ) c""
+ (a" b'"'
B!!!
a!" b"
) c
] d'
+ [(a' b'"' — a""'
b' )c"
+ (a'll
b"!
- a"!
b!!! ) c'
+ (a" b'
·
— a' b!
) c!" ] d
+[(a' b - a b') c''
+ (a
b"
— a"
b
) c'
+ (all bt - a'
b") c
] d'"""~
+[(a b' — a' b ) c!!!!
+[(a b'""' - all!! b) c"
+ [(a'""' b' - a
+ (a!!" b
- a
b'''') c²
+ (a' b!!!! — a'''' b') c
]d"
+ (a"
b
a
b") d'""' + (a"""' b"a"
b'""') c ] d'
e'll
b!!!!) c'"
+ (a"
b!!!! -a"""" b'! ) c' + (a' b'
— a"
-
b') c'""'] d
+ [(a b
+[(a b' — a'
- a
b') c''"' + (a
b'""" — a!""" "b
) c
).c'
+ (a"" b!
- a'
b'""') c ]d"
b ) c'"' + (all b
+ [(a b'll - all
+[(a'""' b' - a
b) c'!!!! + (a!!!! b
- a
b''') c!!!! + (a!!!! b!!!
bill
-
+[(a""" b
+[(a bill
+[(a b'll — a b ) c"
· a
b'') c!!
+ (a
b'
— a"
a!!!! b) c!!!
+ (all' bi
+ (a" b
a
a!!!! b''') c'i
+ (a!! b!!!
b) c!'!!! + (a" B!!!! —
B!!! ) c!!!! + (a!!!! B!! —
b') a'"' + (all b" -
'a'"' b" :) c'!'"' + (a""""B"
a'"' b' ) c ] d'!"
訓
​a'""" b'"") c ] d'
a!'""' b' ); c'"]d
— a" b'''') c""'] d
+[(a"" bill!
+[(a' b'"" - a'""' b' ) c " + (a"""" b! a!! (!!!!) c! + (all bl
+[(a'""' b' - a b'!!!) c'"' + (a""' b!!!! - a!" b'").c! + (a' b'll a"" b') c''''] d"
+ [(a!""" """" — a""' b'""") c'' + (a" a""" b'') c!!! + (a'""' b" -a" b'""') c!""'] d'
a!!!!
-+ [(a""' b! - a! B!!! ) c "
a'"' 6'' ).c' + (a' b" - a"! b') c'"' ]d""
b!!!!
+ (a" b'" -
B!!!
e
a b'!! ) c' ' + (a' bili
b!!!!) c'!!! + (a!!! b!!!!
a!!!
B!!!!) c'
+ (a' b'll! —
-
a!'""' b!! ) c
]d' )
a
a!"' b'"""") c ] d"
a" b"") c
] det
·
SPECIOUS ARITHMETICK.
бI
LXXXIX.
oe
niſh as many
mited and
When there are as many fimple Equations given as Quantities re- Problems
quired, generally ſpeaking, the Problem is limited, but in fome parti- which fur-
cular Cafes it may be unlimited or indeterminate, and in others impoffi- equations as
ble, this happens when the common Denominator is found equal to no- quantities
thing; that is, if there be two Equations given, when a b'a' bo, required are
not always
if there be three, when (a b'a' b) c' + (a" b — a b") c' + (a' b" limited, but
— a" b'). c =.0, &c. then if the Quantities A', A', A", A"", &c. are in fome caſ--
ſuch that the Numerators are alſo equal to nothing, the Problem is inde- es are unli-
terminate; becauſe the Fractions which ſhould exprefs the Values of the in others im
unknown Quantities are indeterminate; but if the Quantities A, A', A", poffible.
&c. are fuch that the common Denominator being equal to nothing,
the Numerators or any one of them is not equal to nothing, the Prob-
lem is impoffible, or at leaſt the Quantities required are all or fome of
them greater than any affignable Quantities. For Example, let there be
given the two Equations 2 = 3 x
and =6x-
5
4y, there
will refult x= and. y Now. as o is less than any affignable
Quantity, it follows that x and y are greater than any affignable Quan-
tities.
2
2
= 3
2
3
2. J
5
If the unknown Quantities be difengaged according to the ordinary
Method, there would refult this abfurd Equation, for the first
Equation gives xy+, and the fecond xy+. Wherefore
5
y y 3
3 ý + z = z + %, or = {, which is abfurd, if x and y are affign-
able Quantities, but if they are greater than any affignable Quantities, it
may be faid without abfurdity, that xy+, at the fame Time
that x=; becauſe the finite Magnitudes and vanishing in
Refpect of the infinite Magnitudes x and y, the two Equations
x=3y+ 3 and x yare reduced to x = y, which involves
no Contradiction.
2
3
5
3
XC.
2
2
3
2
2
3
I
problems to
The greateſt Difficulty which commonly occurs in the Solution of a
Problem confifts in finding the Equations arifing from the Conditions,
becauſe it often happens that the Relations neceffary for forming Equa- Rules for
tions are not expreffed in the Conditions of the Problem. In this Cafe bringing
the Analyſts examine, 1°. whether fome Quantity known or unknown, equations.
whoſe Relation with the other Quantities can be expreffed by an Equa-
tion, may not be introduced into the Problem; and if no fuch Quantity
can be found, or if this Quantity is not fufficient, they examine, 2°.
whether among the known and unknown Quantities there be any which
may be expreffed by new Letters, whereby a new Equation may reſult.
To underſtand which, let the following Problem be propoſed,
62
ELEMENTS OF
{
1
Problem.
There are three Meadows a, b, c, of the fame Quality, each of a given
Extent, in which the Grafs grows uniformly; a Number of Oxen d will
eat up the Pafture a in a Number of Days e, and a Number of Oxen f will
eat up the Pafture ↳ in a Number of Days g. It is required to find the
Number of Oxen x which will eat up the Pafture c in a Number of Days h.
It is plain that the Conditions of the Problem do not exprefs the Re-
lations requifite for forming Equations, but as it is Queftion of the
Quantity of Grafs in each Meadow when the Oxen entered & what grew
during their Stay, I divide the Oxen of each Meadow into two Herds,
I ſuppoſe the firſt to eat up the Grafs grew in each Meadow when
they entered, and the fecond to eat up the Grafs which grows, hence
I fuppofe d y + z, ƒ = t + u, x = s + r.
f
I first confider the Oxen which eat up the Grafs already grew, and
obferving that the Number of Oxen to eat up a Meadow fhould be
greater in Proportion as the Meadow is greater and the Time is lefs,
I make the two Proportions y : *=
a
Ђ
a
and y:s=
e
g
e by
есу
from whence I deduce t =
and s
S=
ag
C
a b
I next confider the Oxen which eat up the Grafs which grows whilſt
the others eat up the Graſs already grew, and obferving that their Num-
ber fhould be greater in Proportion as the Meadows are greater, without
any Regard being had to the Time. I make the two Proportions
zua: b and z:ra: c, from whence I deduce the Values of
z and r, Viz. z =
4 and r =
b
Now having ſeven Equations and ſeven unknown Quantities, dy+z,
f = t + u, x = s+r, t =
a u
CZ
a
e by
a g
S=
,
ecy
ab
a u
C2
Z =
+=
a
Subftituting in the third Equation for s and r their Values, there reſults
си
+, and deducing from the Equations dy+,
e cy
a b
eby
and ƒ=
+u, y =
a g
b dg — afg and u =
gb-be
afg-ebd
a g
and
>
- a e
ecy
+ cu
ъя
fubftituting thofe Values of y and u in the Equation x =
I find x =
3
--
น
a c f g h — a c e f g — b c d e b + b c d e g
abab
a beb
ab
To apply this general Solution to an Example, let the firſt Meadow
contain 3 Acres, the fecond 10 Acres, and the third 24 Acres; and let
12 Oxen eat up the Paſture of the firſt in 4 Weeks, 21 Oxen eat up the
Pafture of the fecond in 9 Weeks; and let it be required to find how
many Oxen will eat up the third in 18 Weeks. By fubftituting thoſe
Values in the general Solution, the Number will be found to be 36.
SPECIOUS ARITHMETICK.
63
CHA P. II.
Of the Refolution of Equations of the fecond Degree.
HAVING fully explained what concerns the Solution of Problems
of the first Degree, Order requires that we ſhould pafs to thoſe of the
fecond Degree, which we propofe to treat of in this Chapter. As to
the Manner of expreffing their Conditions, it is the fame as for Prob-
lems of the firſt Degree, it is only to folve the Equations to which the
Problems are brought, that different Methods are employed according to
the Degrees of thofe Equations.. Of this we have an Inftance in the
following Problem, which in its full Extent includes Problems of every
Degree, and is not more difficult to be expreffed analytically in the moft
complicated. Cafe as in the moſt ſimple..
It
includes
A Merchant having placed a Sum a in Trade, finding bimfelf to be a Problem
loofer, is willing to withdraw at the End of the firft Year, but having which in its
miffed the Opportunity and not being able to obtain it until the End of the full extent
Second, third, or in general until the End of the ntb Year, he finds that the problems of
Sum is diminished by the Quantity b, more than it was at the End of the every de--
firft Year. It is required to determine how much per Cent. bis Lofs amounted gree.
to yearly.
Let x be the Number fought, that is, what each £100 loft at the
End of the firft Year. Making the Proportion 100: 100
100
100
a X
,
the fourth Term a X
100
100
or a X
x(1-
X=a:
X
100
-)
will expreſs what the Sum a is reduced to at the End of the first Year..
If this Proportion be continued by faying 100: 100
a X (100
10000
x)
a X (100 x)
2
:
the fourth Term
100
:)
2
or a X
) will exprefs what the Sum
a X (100
10000
ax (
(1 -
X
100
5)²
a is reduced to at the End of the fecond Year, and what the fame Sum
a is reduced to at the End of the third Year will be expreffed by
x
100
)
3
و
and in general what it is reduced to at the End of
ах
ax (
I
*(1-
100
the nth Year will be expreffed by a × (1
multiplied by the Quantity 1-
100
II.
that is, by a
raiſed to the Power n.
If now it was propofed to find the Equation to be folved, fuppofing
the Merchant to have withdrawn at the End of the fecond Year,, it is
64
ELEMENTS OF
Equation of
the forego-
manifeſt that the Quantity a X
Quantity a × (1 -—--
ing problem the Quantity a X (
for the fe-
cond degree will give a X
For the
: give a × (1 -
x
100
а х
x(:
(1-
duced to x²
X
ΙΘΟ
-):
2
2
(1 — _—_~_~_~)² ſhould be put equal to
100
diminiſhed by the Quantity b, which
1 * )² = ax (1
100
100
-) — b, or multiplying
by itſelf, as indicated by the Exponent 2,
2 X
100
+
+2
10000
=ax (1 ———*) — b, which is re-
b
100
-)
100-10000 Equation of the fecond De-
a
gree, for to folve which the foregoing Methods are infufficient.
III.
If the Merchant is fuppofed to withdraw at the End of the third
third degree Year, the Equation to. be folved will be a X I
a
x
ΙΘΟ
)
—b, which by multiplying
felf, as the Exponent 3 indicates, becomes
180) 3
X
twice by it-
100
ах
Or x3
3 x
+
3x2
100
10000
ex(
*3
1000000
-)
= ax (1
(1-
b
300x² + 20000 x 1000000
cult to be folved than the foregoing.
n.
IV..
a
100
, an Equation more diffi-
As to the other Cafes it is eafy to perceive how the Equations they
furniſh may be formed, and that the Equation will be always of a De-
gree expreffed by the Number . If this Equation in general without
ſpecifying the Number n be required, it may be obtained by employing
the general Expreſſion a x ( " of the Quantity to which a
and the Equation will be
ax(1-
is reduced to after the nth Year,
ах
100
*
100
-)½
-)”
I
= ax
ах
(
I
X
100
V.
100
b
a
-)"
X
100
-)-
b, or
We ſhall for the prefent confine ourſelves to the Inveſtigation of the
Solution of 'that Cafe of the Problem in which its Equation is of the fe-
cond Degree, that is, when it is x²
100 X=
b
10000
or ra-
ይ
SPECIOUS ARITHMETICK.
65
ther we ſhall inveſtigate a Method for folving in general all Equations of
the fecond Degree. Thoſe who are defirous of folving the higher Cafes
of the fame Problem will eaſily effect it after they have feen the general
Methods to be explained hereafter correfponding to the different De-
grees of thofe Equations.
on of the me
What moſt naturally occurs in fearching for a Method for folving in Inveſtigati-
general Equations of the fecond Degree, is to examine the Relation thod for folr
which fubfifts between thofe Equations and the Equations of the firft ing equati
Degree. Now it is evident that every Equation of the firft Degree will ons of the fe
cond degree
become one of the fecond, if the two Members be fquared, for Exam-
ple, x+ab becomes, when ſquared, x² + 2 a x + q² = b², it re-
mains therefore to know whether by a contrary Operation every Equa-
tion of the fecond Degree may not be reduced to one of the firſt.
=
Let us take, for Example, the Equation 2-px-q, and let us
try if x²-px-q is not the Square of fome Quantity, the firft Part
of which is x and the ſecond a known Quantity, in order to find by this
Means the Equation of the first Degree, which being fquared, would
become x² - px
p x = q. Now it is eaſy to perceive that 2px is
not a Square, but at the fame Time it is manifeft that it may be made
one by an Addition of fome Quantity, and we are at Liberty to make
this Addition, provided the fame Quantity be added to the other Side of
the Equation.
x²
To find this Quantity which added to x²-p x will render it a com-
pleat Square, we have no more to do than to compare it with the
Square x² + 2 a x + a², the Term px correfponding with 2 a x, p
will correſpond to 2 a, and confequently a top. Now as a² is what
compleats x²+2a x into a Square, the Square of -p, that is, p²
will compleat x2px into a Square, that is, x2 px + 4 p² will
be a Square, it is one in effect, Viz. that of xp when xp,
and that of p-x when x<p. Having therefore added p² to
the first Member of the Equation, the fame must be added to the ſecond,
and there refults x² - px + p² -p2-q. Now either the Quantity
x-p or the Quantity p-x multiplied by itſelf, gives x²-px + 1 p²,
wherefore xp or p-x will be equal to the Number which
multiplied by itſelf will give p²-4, which is expreffed in general The fignƒ,
thus, (q), the Sign being employed by the Analyfts to in- indicates
dicate that the fquare Root of the Quantity before which it is placed is
to be extracted.
4
I
4
(†
Employing therefore this Denomination x- - 3 p = √ (4 p² — q) and
q), from whence is deduced.
1/2 px = √(p²
x = = p ± √ ( = p²
lyfts to include in one
the ſquare
root.
An equation
of thefecond
q). The Sign+being employed by the Ana- degree has
and the fame Expreffion the two Values of x.
two roots.
•
3 I
66
ELEMENTS OF
VI.
Let us now apply this general Solution to the Equation x² 100 x
b
to which we were led in the foregoing Problem.
10000
a
Comparing this Equation with x²-px-q, we will have p 100,
q = 10000
b
2
a
=
=
and fubftituting thofe Values in the Place of p and q
p² -g, we will have
in the general Formula x = ±√²
xp √
*=50+ √(2500
10000
음​).
VII.
We may reduce to a more fimple Form the radical Part
Reduction
of the value
√(2500
of x by re-
folving the
IOOCO
b
-) of this Value of x, by obferving, that
Q
root of the the fquare Root of the Product of two or more Quantities is the Pro-
product into duct of the fquare Roots of thoſe Quantities; for refolving 2500
thoſe of its
factors.
this prob-
lem.
b
10000 into its Factors 2500, I
a
4 b
and extracting the
a
a
-)
Roots of thoſe two Quantities we will have 50 and √(1 – 4b
whofe Produ& 50 (1
that is, that the Value of
x
4 b )
-)
a
will be √(2500
10000
will be 50 ± 50/(1-4).
b
a
As to the Demonftration of this Principle, that the Root. of any
Product is obtained by multiplying the Roots of its Factors, it eaſily
may be found, by obferving that the Square of a Product, as a b, is
the Product a a X bb, of the Squares a à & b b of its. Factors e and b.
VIIL
6
To make Ufe of this Value of x, nothing more is required than to
Example of know the Ratio that b bears to a. Let b, for Example, be the 5 of a,
that is, let us fuppofe the Merchant to have found the Sum at the End
of the fecond Year, diminiſhed by a Quantity equal to the of the
Whole more than it was at the End of the first Year, according to this
46
Suppofition 4
a
√(146)
= 60 or 40.
a
46
a
6
25
24 and I
I
hence the Root
,
25
2.5
I
and confequently will give x 50 ± 50 X 3/3
x=
5
In effect thoſe two Values of x refolve the Equation x² — 100 x
b
2400 into which the Equation x²
100 x=
10000
a
b
6
for whether x be
a
25
2400.
is transformed by the Suppofition. of
fuppofed
60 or 40, x 100 x becomes
x² —
SPECIOUS ARITHMETICK.
67
The Neceffity of the two Solutions 60 and 40 will appear after a
more fatisfactory Manner, as follows. Let us fuppofe first x 60,
that is, that the Sum of £100000, for Example, is diminiſhed 60 per
Cent. yearly, it is manifeft that at the End of the firft Year it would
be reduced to £40000, and at the End of the ſecond Year it will be re-
duced to £16000; but £16000 are lefs than £40000 by £24000, which
are the of 100000, as the Problem requires.
6
25
Let us now fuppofe the fame Sum of £100000 to be diminiſhed 40
per Cent. yearly, at the End of the first Year it will be reduced to
£60000, and at the End of the fecond to £36000, but £36000 are lefs
than the Sum £60000 by the Quantity £24000, or the of £100000.
If b was ſuppoſed to be the
50 ± 50 √ 2/5 = 50 ± 30,
be found to folve the Problem.
Ъ
a
IX.
25
of a, then
that is, 80 or
X.
100 X =
25
25
50 ± 50 √ (1
50.50
16) Another ex-
20, both of which will ample,
3
4), or Third exam
ple in which
the root of a
negative
quantity be-
ing required
is impoffible
If be put, there will refult x = 50 ± 50 √ (1
50 ± 50 √ 1/1, but as the Squares of all Quantities are affirmative, it
follows that the Quantity ✔cannot be affigned, or which comes
to the fame that the Problem is impoffible in this Cafe.
Hence we may conclude that there is no poffible Value which fub-
ſtituted for x in the Equation x²
10000 will render the
two Members equal, or what comes to the fame Thing, that the Sum a
cannot be leffened each Year according to any given Proportion, ſo that
from the ſecond to the third Year the Diminution may amount to a
Quantity equal to a Third of the whole Sum. The Analyfts however
regard as a Kind of a Solution or Root of the Equation x2. 100 X
10000, the Value 50 ± 50√ which reſults from thence,
but they call it an imaginary Root, and this imaginary Root on Account
of the Sign is always confidered as a double Solution.
From the general Value of x, p±√ ( p² — g), it appears that as of the fe-
often as the Quantity defigned by q is negative and greater than p², whoſe roots
the two Roots of the Equation x2 — px -q will be both imaginary. are imagina
=
3
-
XI.
I
(=
I
Thoſe roots
are called
imaginary.
Equations
cond degree
ry.
Though the Roots of an Equation or the Values of the unknown The real
Quantity are real and pofitive, however they do not always, as in the and pofitive
foregoing Example both folve the Problem. Let the following Question
be propoſed.
values of
the un-
known quan
always folve
The Sum of £190 was divided between three Perfons; whofe Shares were tity do not
in geometrical Proportion, and the greateft of them exceeded the leaft by £50. the problem
What were the ſeveral Shares ?
If x be put for the leaſt of them, then the greateſt will be x + 50, Firſt exam-
the Sum of which two fubtracted from (190) the Whole, leaves 140-2x ple.
158
ELEMENTS OF
OF
for the mean Share. Therefore x
Therefore 140.
140 — 2 x
2 x 140
=
2x:x + 50,
and confequently (1402x)2x (x + 50), that is, 19600-
+ 4x² = x² + 50x, whence
610x
3
2,
19600
3
* 2
610 x
3
+
93025
9
610x=- 19600, and
560 x
3262
hence by compleating the Square
19600
93025
+
3
9
2
34225 and by
.9
extracting the Root x= 305
3
+ 185 or x 40 and x = 163,
3
"
=
two real and pofitive Roots, however the firſt Root only folves the
Problem, the other two Shares being 60 and 90 Pounds.
1
As to the Root 163 it folves the following Queſtion.
'उ
2
A Sum of Money was divided between three Perfons, whofe Shares were
in geometrical Proportion, the greateft exceeding the leaft by £50, and their
Sum exceeding the mean Share by £190. What were the feveral Shares?
For the analytical Expreffion of this Queftion is (2 x 140) = x²
50 x, from whence refults the fame Equation as in the foregoing
Queſtion, and in this Cafe x = 163 Becauſe the Equation
19600 560x + 4x² = x²+ 50 x, includes not only (1402x)2
= x² + 50x, but alſo (2 x 140)² = x² + 50 x, which are the
analitick Tranſlations of two very different Queftions, the firft of which
gives x 40, the latter 163, wherefore the Roots of an Equation,
though real and pofitive, do not both folve the Queſtion in the Senfe it
is propofed, but confidered in two different Lights, which Difference
cannot be expreffed analitically. For a further Illuftration, here follows
another Example.
XII.
Two Travellers fet out at the fame Time, the one from Dublin, the other
Second ex- from Londonderry, the former for Londonderry, the latter for Dublin;
ample.
when they met and had computed their fourney, it was found, that the for-
mer bad travelled thirty Miles more than the latter, and that at their Rate
of Travelling, the former expected to reach Londonderry in four Days, and
the latter to reach Dublin in nine Days; the Distance between Dublin and
Londonderry is required.
I put x for the Number of Miles between Dublin and Londonderry; and
fince the Travellers both together had travelled x Miles when they met;
it follows that the former had travelled 30 Miles, and the latter
*+30
2
2
- 30 Miles, according to the firſt Condition of the Problem, there-
fore the remaining Part of the formers Journey to Londonderry is
30 Miles, which he expects to perform in four Days, and the
*
2
SPECIOUS ARITHMETIC K.
69
2
remaining Part of the latters Journey is * +30 Miles, which he ex-
pecs to perform in nine Days. I now enquire into the Number of Days
each hath travelled already, and I fay
4:
4 X (x + 30)
X
x +30
30 :
2
2+
= 4
Number of Days travelled by the Former, and
X 30
x +30
X 39
:
=9:
2
2
9X (x -30)
= 9 : 2 X (*
x + 30
Number of Days travel-
led by the Latter, but as they both fet out at the fame Time and are
now met, they muſt have travelled the fame Number of Days, therefore
4X (x+30)
which after the neceffary Re-
9X (x
-
x +30
30)
=
30
x
ductions, becomes x² 156 x 900, whence by compleating the
Square x²-156x+6084 = 5184, confequently x 150 or x = 6,
therefore the Diffance between Dublin and Londonderry muft either be
150 or 6 Miles; but 6 Miles it cannot be, becauſe when the firſt Tra-
veller came up to the Latter, he had travelled 30 Miles more than him,
and had not yet reached Londonderry; therefore the Diſtance between
Dublin and Londonderry is 150 Miles; for then the firft Traveller muft
have made 75 + 15 or 90 Miles, and the fecond 75-15 or 60 Miles,
from the Time of their fetting out; therefore the Former has 60 Miles
and the Latter go Miles to travel; but if the Former could travel 60
Miles in 4 Days, he muſt at the fame Rate have travelled 90 Miles in
6 Days, and if the Latter could travel 90 Miles in 9 Days, he muſt
have travelled 60 Miles in 6 Days, as the Problem requires.
As to the fecond Value, x = 6, it folves the following Problem.
Two travellers fet out at the fame Time, the one from Dublin, the other
from Swords, the Former with a Defign to pass through Swords, and the
Latter with a Defign to travel the fame Way. The Former bad overtaken
the Latter, and having computed their Journey, it was found that they
bad both together travelled 30 Miles, that the Former bad paffed through
Swords 4 Days before, and that the Latter at his Rate of Travelling, was
a nine Days Journey diftant from Dublin. The Diffance of Dublin from
Swords is required.
For if x be put for the Number of Miles from Dublin to Swords, then
it is plain that the firft Traveller muſt have travelled more Miles than the
Latter by ; but they both together travelled 30 Miles when they met,.
30+x
wherefore the firſt muſt have travelled
and the Latter muſt
3
2
have travelled 30 Miles, therefore the Diſtance of the first from
Swords, after he had overtaken the fecond, was 30-* Miles, which
2.
170
ELEMENTS OF
General
folving equa
gree.
he had travelled in 4 Days, and the Diſtance of the fecond from Dublin
30. + x
Miles, which he could travel in 9 Days; therefore to find
3
་
was
how many Days each had trrvelled already, I fay 30: 30+
= 4:
_4X (30 + x)
30
30+x 30
2
:
2
2
3
Number of Days travelled by the Former, and
x
= 9:
led by the Latter,
muſt have both
4× (30 + x)
30- x
9X (30)
30+x
Number of Days travel-
and as they both fet out at the fame Time, they
travelled the fame Number of Days, wherefore
9X(30
9X (30- x), which after the neceffary Reduc-
30+x
156 x =
tions, becomes x²
900, the fame Equation as found
above, whofe Roots are x = 150 and x = ·6. Wherefore the Diſtance
between Dublin and Swords muft either be 6 or 150 Miles, but 150
Miles it cannot be, becauſe after the firft Traveller had paffed from
Dublin beyond Swords, and at laſt had overtaken the fecond, they had
both travelled but 30 Miles, therefore the Diſtance from Dublin to
Swords muſt be 6 Miles, and this Number will anſwer the Conditions of
the Problem, for then the firft Traveller when he had overtaken the
Latter, had travelled 15+ 3 or 18 Miles, and the Latter 15-3 or 12
Miles. Therefore the firft Traveller had got 12 Miles beyond Swords
in 4 Days Time, and the Latter was 18 Miles diftant from Dublin,
which he could travel in 9 Days, but at the Rate of 12 Miles in 4
Days, the Former must have performed his 18 Miles Journey in 6 Days,
and at the Rate of 18 Miles in 9 Days, the Latter must have per-
Formed his 12 Miles Journey alfo in 6 Days, therefore from the Time
of their firſt ſetting out to the Time of their Meeting, they had both
travelled the fame Number of Days as the Problem requires.
XIII.
It is eaſy to perceive that any Equation of the fecond Degree may be
rule for re- refolved by repeating the Procefs purfued in refolving the foregoing
tions of the Equations, which may be expreffed thus. 1°.Tranfpofe all the Terms that
fecond de involve the unknown Quantity to one Side, and the known Terms to
the other Side of the Equation. 2°. If the Square of the unknown
Quantity is affected with a Coeficient, you are to divide all the Terms
by that Coeficient, and if the Sign of that Power be negative you are
to change all the Signs of the Terms. 3°. Add to both Sides the Square of
half the Coeficient prefixed to the unknown Quantity itſelf, and the
Side of the Equation that involves the unknown Quantity will then be
a compleat Square. 4°. Extract the Square Root from both Sides of the
Equation, which you will find on one Side always to be the unknown
SPECIOUS ARITHMETICK.
71
Quantity, with half the forefaid Coeficient fubjoined to it, fo that by
tranfpofing this half you may obtain the Value of the unknown Quan-
tity expreffed in known Terms, for Example, fuppofe x²-6x=27,
adding 9 the Square of the half of 6 to both Sides, there refults
x²-6x+9=9+27=36, and extracting the Square Root you
will find x 3 = 6 and 3
6, according as x is> or < 6,
wherefore x = +3+6 or x = 9 and x = 3, which both refolve
the Equation x²-6x=27, in like Manner if x2+8x=9, adding
16 Square of the half of 8 to both Sides, there will refult x² + 8 x
+ 16 = 1.6 + 9 = 25, and extracting the Square Root you will find
x + 4
+5,
5, that is, x = 4 ± 5, 05x 9 and 1.
XIV.
To accuftom Beginners to the Difficulties which occur in the Solution
of Problems of the fecond Degree, here follows another Problem.
Let it be propofed to find in the Line which joins two Lights the Point
where a Body would be equally illuminated by them, according to this Prin-
ciple of Phyficks, that the Effect of a Light is four Times greater when it
is twice nearer, nine Times greater when it is three Times nearer, or in-
creaſes as the Square of the Distance decreaſes:
Let a exprefs the Distance between the two given Lights, and let
the Ratio of m to n exprefs that of the Effect of the leaft Light at
a certain Diſtance, to the Effect of the greateſt Light at the fame Dif-
tance.
Let exprefs the Diſtance of the leaft of the two Lights from a
Point taken at will in the Line which joins the two Lights, it is mani-
feft that x will exprefs the Diſtance of the other Light from the
fame Point, that the Squares of thofe two Diſtances will be x² and
x² - 2 a x + a², and confequently the Quantities which decreaſe as
the Squares of thofe Quantities increaſe will be to one another as
and
32
I
2ax+a²
I
I
to.
2.
24
I
2
From whence it follows that if the Lights were of equal Force, the
Effects they would each produce in this fame Point would be to each
other as
but the abfolute Quantities of
thoſe Lights being to each other in the Ratio of m to n, their Effects
therefore will be to each other as
;
2.
22x+2²
77
to
2.2
A-2
n
zax + a²·
Now that the Point taken at will may become the Point required, I
put thofe two Quantities equal to each other, which gives the Equation
m a²
2 a mx + m x² = n x²
To folve this Equation, I tranfpofe the
the other Member, and there refults (n
Terms m x² and 2 a m x into
m) x x + 2 a m x = m a 85.
Another
problem of
the fecond
degree..
72
ELEMENTS OF
2 a m
a a m
· or x x +
X-
n
m
n
m
Term, and there refults x² +
a a m m72
+
bers of this Equation the Square of half the Coeficient of the ſecond
m)²
I afterwards add to the two Mem-
2 am x
a² m²
a a m
+
=
n
m
(n
-m
n
m
a² m n
ኪኪ
(n
17
112) 23
(n
112) 2
,
by reduc-
ing the two Terms
a a m
a a m m
+
n
m
(n 712
m)2
tor.
Of the two
lues, one is
whofe fecond Member becomes
to the fame Denomina-
Then extracting the Square Root of the two Members, there refults
a m
* +
n
m
a a m 12
or x=
(n
By extracting the Root of the Part
a m
a.
+
✔m ñ.
12 - -- 772
n
12
a a
(n m) 2
which is a perfect
Square, and leaving under the radical Sign its Multiplier m n, which is
no Square, at leaſt for all the Values of m and n. Wherefore the two
Values of x which folve the foregoing Equation, and confequently the
Problem which leads to this Equation are expreffed by the Formula
a
✈ m.n, or x =
x-m±√mn.
a m
.12
772
12
12
XV.
11
112
From this Expreffion it appears that one of the Values is neceffa-
foregoing va rily negative and the other pofitive, for 1°. if we take the radical
neceffarily Quantity mn with the Sign
Quantity mn with the Sign, there is no doubt but the whole
pofitive and Quantity will be negative, 2°. if we take √ m n with the Sign +,
the other ne m+mn which we will have then will be poſitive, becauſe n be-
ing greater m, mn must be greater than m.
gative.
Ufe of the
lue.
XVI.
If we now enquire into the Ufe of the negative Value of the Dif
negative va- tance x of the leaft Light to the affumed Point, we will find by recall-
ing what had been demonftrated (Chap. I. Art. LXIII.) with Refpect to
thofe Values in Equations of the first Degree, that its Direction is op-
pofite to that of the first, that is, that the Point which it gives for
folving this Problem, inftead of being placed between the two Lights,
will be placed in the Line on the other Side of the weakeft Light.
There can be no Difficulty in admitting this Pofition of the negative
Value of x, when it is obferved, that this fame Value was found nega- .
tive only, becauſe the Problem was folved in the Suppofition that the
Point fought was placed between the two Lights; for if it had been
fuppofed (as it might be) to be placed in the Line which joins them,
produced on the Side of the weakest Light, there would have reſulted
another Computation relative to this Pofition, and x which in this Cafe
1
SPECIOUS ARITHMETICK.
73
is taken in the Line produced on the Side of the weakest Light would
be poſitive.
XVII.
In order to make this appear, we will refume the Problem, the Point
fought being fuppofed in the Line produced on the Side of the weakest
Light. The Diſtance of this Point from the leaft Light being expreffed
by x, its Diſtance from the greateſt Light will be expreffed by a +x,
the Squares of thoſe Diſtances will be x² and a² + 2 a x + x², and the
two Quantities of Light
which being
m
#
2
and
a² + 2 a x + x
x2
equal by the Conditions of the Problem, will give
M
x2
n
a² + 2 a x + x²
2
or m a² + 2 a m x + m x² = n·x²,
m
n
m
the firſt Value
or (n — m) x² 2 am x = m a², or x²
which being refolved will give x =
of which a × (m + √ mn)
n
m
lem in the Senſe it is propofed.
Q
2 am x
12
m
a x (m ± √ m n)
#
m
>
will be poſitive, and folves the Prob-
As to the fecond Value X(m— J·m n)
n
m
of the Distance x, being
negative, it ſhould be taken in an oppofite Direction to the first, that is,
the Point which it gives is not fituated in the Line, which joins the
two Lights produced, but in the Line itſelf.
Hence in this Solution the Values of x differ from thoſe in the for-
mer with Reſpect to the Signs, and thoſe two Solutions confirm what
has been already proved in the foregoing Chapter, Art. LXIII. that the
unknown Quantities which become negative, fhould be always confi-
dered as being of an oppofite Kind from what they have been fuppofed
in expreffing the Conditions of the Problem.
XVIII.
In order to remove whatever Difficulties may occur to the Reader Example of
with Reſpect to this Problem, we ſhall apply it to an Example. Let the forego-
us fuppofe n = 4 m, that is, that the greateſt Light has four Times the ing problem
Force of the other. Subftituting this Value of n in the general For-
mula of Art. XIV. x=
a
172
× (m ± √ m n), it will be transform-
n
a
ed into x =
3
nish two Points, both of which ſolve the Problem, one fituated between
× ± (21), that is, +a, ora, which fur-
3 K
74
ELEMENTS OF
!
The pofitive
the two Lights twice nearer the weakest than the ftrongeft, and the
other in the line produced, at a Diſtance from the weakest equal to the
Distance between the two Lights.
XIX.
The pofitive and negative Values of the unknown Quantity do not
& negative always, as in the foregoing Example both folve the Problem. Let the
unknown following Queftion be propofed.
values ofthe
quantity do
A Draper buys fome Ells of Cloth for 70 Crowns, and finds that if he
not always bad 4 Ells more, he bud bought every Ell two Crowns cheaper. How many
the problem Ells did he buy ?
both folve
Let x denote the Number of Ells fought, then dividing the whole Price
by the Number of Ells, we have 70 for the Price of one Ell, and
ly
X
70
x + 4
70
X
for the Price of one Ell if he had got 4 Ells more, confequent-
2 =
70
x+4
2 + 12, or x
the Problem, as to x
it folves the following.
=
whence x²+ 4 = 140, confequently
*.
10 and x = — 14, the firſt Value of x folves
[
14, it does not folve the propofed Queſtion,
A Draper buys fome Ells of Cloth for 70 Crowns, and finds that if he
bad 4 Ells lefs, be would have bought every Ell 2 Crowns dearer. How
many Ells did he buy ?
70
When a
1
In Effect the analitick Expreffion of this Queſtion is
+2=
X
70
"
or x2
4
4x140, from whence is deduced x 14 and
10, which are precifely the Roots of the foregoing Equation
with contrary Signs. From whence it appears that the negative Roots.
folve the Queſtion, not as it is propofed, but with fome Alterations which
conſiſt in ſubtracting what ſhould be added, or in adding what ſhould be
fubtracted. The Sign which precedes the Root x == 14, for
Example, indicating that in the foregoing Queftion the 4 Ells are to
be fubtracted, and the 2 Crowns to be added to the Price.
<
XX.
If the Roots of an Equation or Values of the unknown Quantity are
queſtion is both negative, then the Queſtion has been wrong ftated, as will appear
wrong fta- by the following Example.
ted the va-
lues of the
A and B take in Trade £2112 per Annum each, but A whofe Profits are
unknown 2 per Cent. greater than thofe of B, clears £100 per Annum more than B.
quantity be- What are the Profits of each per Cent. and what do they clear per Annum.
Let x denote what A gains per Cent. Now fince A for every £100
negative. he layed out in purchafing Goods, recovered in the Sales 100 + x
come both
Pounds, and 2112 being the whole Sum recovered, 100 + x:x=2112:
SPECIOUS ARITHMETIC K.
75
2112 *
100 + x
= 2112:
2112 x
100 + x
the whole Profit of A, and in like Manner 98 + x:x 2
2112 X
1
98 + x
4224 the whole Profit of B. Therefore
2112 x -- 4224
98 + x
= 100 by the Queſtion, hence
x²+ 198x=-5576, whence by compleating the Square and ex-
tracting the Root x99 65, both which Roots are negative,
which denotes that the Queftion fhould be expreffed thus:
±
A and B take in Trade £2112 per Annum each, but A whofe Loffes are
2 per Cent, less than thofe of B, lofes 100 per Annum lefs than B. What
are the Loffes each fuftained per Cent. and what did their Loſſes amount to
yearly?
In effect, fince A for every £100 he layed out in purchafing Goods,
recovered but 100 x Pounds, and £2112 being the whole Sum re-
2112 X
100
X
2112X(x+2)
covered, 100
Manner 98
x: x = 21T2 :
the whole Lofs of A, in like
x:x + 2 = 2112:
the whole Lofs
98
of B, wherefore
2112 x + 4224
2112 X
100 by the Quef-
98
100
tion. Hence x²-198 x
5576, and compleating the Square and ex-
tracting the fquare Root = 9965, which are the fame Roots as
found before but affirmative.
x
From which it appears that A loft £34 per Cent. and his whole Lofs
amounted to £1088 per Annum, and that B loft £36 per Cent, and his
whole Lofs amounts to £1188 per Annum,
XXI.
The Principles which we have explained are fufficient for folving all
Equations of the fecond Degree, but to render the Application of them
eaſy to Beginners, we fhall proceed to exercife them in the Refolution
of feveral Equations, there will refult this Advantage, that befides their
becoming better acquainted with the Method, they will learn at the
fame Time the new Operations of fpecious Arithmetick, which, without
doubt, are owing to the Reſearches which the first Analyfts have made
on the Equations of the fecond Degree.
Let b x² = 2 c² x + 2 c² a, tranfpofing the Terms affected with x Other exam
on one Side, and dividing all the Terms by the Coeficient of xx, we will ples of the
have x²
2 c² x
b
refolution of
2 c² a
b
to the two Members of which, adding
equations of
the fecond
degrec.
64, and afterwards extracting the Square Root, we will have
c4
62
さん
​x = — ± √ √ (20² ab + + +
c4
)
= ( + c√ (zab + (²)
c2
b
Ъ
7.6
ELEMENTS OF
· Let f² + g² — 2 8 x + x² =
mm - n n
nn
2
2
'm² x²
ni
which I first reduce to
-) x² 28
·) ×. x² + 2 8 x = f² + g², and afterwards to
28n2
m2
X ==
f² m² + g² n²
m²
f2 m² + g² m²
n
,
or x² +
g2 n4
2.g
1 g n² x
m²
262 +
मेरे
+
g² n4
るる
​2
(m²
12272
* +
+gg
(mm
g n n
m2
122
+
-ff n4
(m² — n²)².
nn)²
m m - n n
+
x=
12
- ffnn)
m m
n n
ffnnmm g g n n m m —
23
from whence is deduced
n √. (ƒ ƒ m m + g g mm -
m m
n n
n),
or
× [— gn± √ (ƒ ƒ mm + g g mm — ƒ ƒn-n)]...
Let a b c — a f² + 2 a ƒ z = a z² -b z² be given, whofe Terms
being difpofed in order, will become z-
a
z a f
b
a b c
af²
Z z =
and compleating the Square z²
2 af
a
b
z +
a² f2
(a - b)2
aabc-abbe+abff
,
hence z=
(a - b)²
་
a
b
Pracefs of
the extracti-
Let the Equation 4 a²
af±√ (aabc-abbc abf2
2 x² + 2 ax = 18 a b -18 b4 be pro-
pofed, difpofing its Terms in order and reducing it, there refults:
x² — a x = 2 a² — 9 a b + 9 b², and compleating the Square, be-
comes x² — a x + 4 a² = 12 a² — 9, a·b + b², which gives
x = ÷ α + √ (2 a²
9 ab + 9 b²).
The young Analyſt would have eafily reduced this Quantity if he
had perceived, and he could not but have perceived by what precedes,
that the Square of a Quantity compofed of two Terms, is equal to the
Sum of the Squares of each of thoſe two Terms, and to double of the
Product of thofe two Terms.
4
For finding in the Quantity a² — 9 a b + 9 b, the Terms & a²
and 9 b², which are the Squares of a and 3 b, and the Term 9 a b
which is double the Product of a into 3 b. It is eafy to conclude that
this Quantity 2 a² — 9, a b + 9 b is the Square of a 3 b, there-
fore inſtead of the Expreffion (4² — 9 a b + 9 b²), we may write
fimply a 3b: Wherefore the Value of x is a ± 2a + 3b,
+
that is, either 2.4 3.b, or a 3 b. In effect, both thofe Values
folve the given Equation.
3
2
—
√✓
XXII.
2
Among the different Equations of the fecond Degree, which are re-
quired to be reſolved, Cafes fimilar to the foregoing may occur; it is
fquare root therefore neceffary to have a general Method for difcovering fuch Quan-
on of the
SPECIOUS ARITHMETICK.
77
tities as are Squares, and for finding their Roots; this Method is eafily explained in
deduced from the Principles employed in the foregoing Example. The an example.
Proceſs of this Method applyed to an Example, is as follows.
Let the fquare Root of the Quantity 30 a b + 962 + 25 a² be
required.
25a² + 30 ba + 9 b²
25 22
30ba + 9.6².
10 a
a + 3 b
50+36
· 30 ba
982-
a
I first range the Terms of this Quantity according to the Dimenſions
of the Letter a, for Example, as above.
A
I afterwards extract the Root of the firft Term 25 a² which is 5 a,-
which will be. the firft Term of the Root,, and I write it down befide
the propofed Quantity 25 a2 + 30 b a + 9 b², feparating them by a
Line in order to avoid Confufion. I then write under the propofed:
Quantity the Square 25 a² of 5 a, prefixing to it the Sign. I draw
a Line and reduce, and there remains 30 a b + 9 b², which I write un-
der the Line, which being done, I double 5.a which gives 10 a, and
divide the first Term 30 a b of the Quantity 30 a b + 9 b² by 10 a,
and I write the Quotient 3 b, which is the fecond Term of the Root
fought, befide 5 a, and I place it at the fame Time befide ro a, and
I multiply the fuperior. Quantity by this. new Term 3.b of the Root,
obferving as in Diviſion to change the Signs, in writing the Product un-
der the Quantity 30 a b + 9 b2, then reducing and finding that all the
Terms deſtroy each other, I conclude that 5a+3b is the Root required.
L
XXIII.
To render the Method of extracting the fquare Root familiar to Be-
ginners, here follow more Examples,
Another ex
ample of the..
extraction of
Let it be propoſed to extract the fquare Root of the Quantity the fquare.
4.a? ·4 ab +4αc + 62
root,
2 cb + c², whofe Terms are ranged ac-
cording to the Dimenfions of the Letter a.
4.882
4 a²
4.ab + 4.α6 + b² — abat.c²
4a-2b+c
2 a
6+6
40
b
4 b a + 4 c a + b²
2cb+c²
+4ba
62
4 ca
4ca
Q
20b+c²
+2cb
79
ELEMENTS OF
Example 3.
944
2
24 a² x² of 16 24
9x4
+ 12 6²x²
1262
+ 24 a²x²
+ 126²x² 16 a² 62
+464
242²x² + 16 a4
16 a² 62
+464
16 a4
62
4 a²
+262
322
4 a²
+262
I 2 62 x²+
16 a² 62
4
64
Example 4.
2 y² +41
2
34 +423
8y + 4
J² + 23
2
·4
23² + 2y
+423
- 83+ 4
433
432
4y²
87 +4
+ 4x² + 8-4
O
XXIV.
Method of The fquare Root of any Number may be found out after the fame
extracting Manner. If it be a Number under 100, its neareſt ſquare Root is
the fquare found by the following Table.
root of num
bers.
Squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
Roots I, 2, 3, 4, 5, 6, 7, 8, 9, IO.
Where it is eaſy to obferve, that the Square of a fimple Number
cannot confift of more than two Places of Figures, fince the Square of
10 the leaft Number confifting of two Places is 100, the leaſt Number
confifting of three Places. That the Square of a Number conſiſting of
two Places, cannot confift of more than four Places; for the Square of
100, the leaſt Number confifting of three Places, is 10000, the leaſt
Number confifting of five Places; and in general, that the Square of
any Number cannot confift of more than the Double of its Places.
From whence it follows that a Number confifting of leſs than three
Places, can have but one Figure in its Root; that a Number confifting
of more than three, but lefs than five Places, can have but two Figures
in its Root; that a Number confifting of more than five, but less than
feven Places, can have but three Figures in its Root, and fo on, taking
for Limits the odd Numbers 1, 3, 5, 7, 9, Ir, &c. whofe common
SPECIOUS ARITHMETICK.
79
Difference is 2. Confequently the Number of Places which the fquare
Root of any Number confifts of, is difcoverable by Infpection. How the
Numbers correfponding to thoſe Places are found, we ſhall now explain.
XXV.
First ex-
Let it be propofed to extract the fquare Root of 99856, as it con-
fifts of five Places, I conclude that the Root will confift of three. To ample.
determine the Numbers correfponding to thofe Places, I exprefs them
by x, y and z, confequently x² + 2 x y + y² + (2 x + 2 j) z + x²
99856. Hence the Difficulty is reduced to find the Terms of this
Quantity in the given Number 99856.
-
The first Number x being Hundreds, its Square x² will be
Tens of Thoufands, or Tens of Thouſands with Hundreds of
Thouſands; confequently this Term fhould be contained in 9,
the Number correfponding to the fifth Place of the Propofed,
and its fquare Root 3 (Hundreds) is the firft Figure of the Root,
which I place in the Quotient, and fubtract the Square of 3
(Hundreds) from the propoſed Number.
6
9.98.56(316
9
I
98
X161
626)
3 7 56
3756
The Number y being Tens and multiplied by 2 x, the Dou-
ble of the firſt, the Product ſhould be Thoufands or Thoufands 626
with Tens of Thoufands, confequently the Term 2 x y fhould x6
be contained in 9, the Number correfponding to the fourth
Place of the Prepofed. I divide therefore 9 by 6, the Quotient
is I (Ten) which being the fecond Part of the Root, I place it
after the firft Figure 3, and as p2 the Square of the fecond Part of the
Root is Hundreds, I conclude that the Terms 2xy + y² fhould be
contained in 98, the Numbers correfponding to the third and fourth
Places of the Propofed. I therefore multiply 6 Hundreds and 1 Ten or
61 by 1, and ſubtract the Product from 98, and the Remainder is 37
(Hundreds)..
It remains to find in thegiven Number the Terms 2 x x+2yz, the Pro-
duct of double the first Part of the Root (Hundreds) into the laſt (Units)
which will produce either Hundreds or Tens of Hundreds, and the Pro-
duct of the Double of the ſecond Part of the Root (1 Ten) into the laſt
(Units), from whence can arife only Tens or Tens with Hundreds, they
will therefore be contained in 375, omitting the 6 Units which does not
affect thoſe two Products. I divide therefore 375 by 62, the Double of
the first and fecond Figures of the Root, and the Quotient is 6, which
being the third Figure of the Root, I place it after the fecond (1), and
as 22 the Square of this third Figure is Units or Units with Tens, I con-
clude that 3756 fhould contain the Terms (2 x + 2 y) ≈ + z², the
Product of Double of the first and fecond Figures of the Root into the
third, together with the Square of the third; I multiply therefore 626
by 6, and fubtract the Product from 3756, and as there nothing re-
mains, I conclude that 316 is the Root required.
0
80
ELEMENTS OF
on of the
XXVI.
Procefs of It is eaſy to perceive that the fquare Root of any Number may be
the extracti- difcovered by the fame Method of Reafoning, the Procefs of which may
fquare root be expreffed thus. Put a Point in the Place of the Tens, and omitting
of numbers. one, point every other Figure towards the left Hand, and by these
Points the Number will be diftinguiſhed into as many Periods as there
are Figures in the Root. Then find the fquare Root of the first Period,
and it will give the first Figure of the Root, fubtract its Square from
that Period, and annex the fecond Period of the given Number to the
Remainder, then divide this new Number (neglecting its laft Figure) by
the Double of the first Figure of the Root, and quote the Number of
Times, then annexing the Quotient to that Double, multiply the Num-
ber thence arifing by the faid Quotient, and fubtract the Product from
the whole Dividend, as before. Annex the third Part of the given
Number to the Remainder, which is to be managed exactly as the laſt,
and proceed thus until all the Periods are brought down. If at laft there
be no Remainder, then will the Quotient exprefs the true Root.
Second ex
.ample.
*XXVII.
·27.39.47-5.6 (5 2 3 4
25
102239
X2/204
12) 23
Let the fquare Root of 27394756 be
required. I first point it into Periods of
two Figures each, and as there are four
Periods, I conclude there will be four Places
in the Root, then I find the nearest fquare
Root of 27 to be 5, which therefore is the
first Figure of the Root. I fubtract 25,
the Square of 5 from 27, and to the Re-
mainder 2, I annex the fecond Period 39, 1043 3547
and I divide (neglecting the laſt Figure 9) X3/3129
by the Double of 5 or by 10, and I place the
Quotient after 5, and then multiply 102
+
X 4/41856
by 2, and fubtract the Product 204 from 1046441856
239; then to the Remainder 35, I an-
nex the third Period 47, and dividing
3547 (neglecting the laft Figure 7) by the Double of 52, that is,
by 104, I place the Quotient after 2, and multiplying 1043 by the
Quotient 3, I find the Product to be 3129, which I fubtract, from
the Dividend 3547, and to the Remainder 418 I annex the laft Period
56, and dividing 41856 (neglecting the laſt Figure 6) by the Double of
523, that is, by 1046 I place the Quotient after 3, and multiplying
10464 by the Quotient 4, I find the Product to be 41856, which fub-
tracted from the Dividend and leaving no Remainder, the exa& Root
muſt be 5234.
SPECIOUS ARITHMETICK.
81
XXVIII.
122
X 2
38.94.89.(6 2 4.
36
294
244
5089
X4
4976
1 2 4 4
Let the fquare Root of 389489 be required.
After having firft Pointed it, I then find the
neareft fquare Root of 38 to be 6 which there-
fore is the firſt Figure of the Root; I fubtract
36,the Square of 6, from 38, and to the Remain-
der I annex the fecond Period 94, and I di-
vide (neglecting the laſt Figure 4) by the dou-
ble of 6 or by 12 and I place the Quotient after
´6, and then multiply 122 by 2, and ſubtract
the Product 244 from 294. then to the Re-
mainder (50) I annex the third Period 89,
and dividing 5089(neglecting the laft Figure
by the Double of 62, or by 124, I place the
Quotient after 2, and Multiplying 1244 by the Quotient 4, I find the
Product to be 5976, which fubftracted from the Dividend leaves the Re-
mainder 113, and as there are no more Periods to bring down there are
to be no more Figures in the Root; becauſe a Number confifting of fix
Places cannot have more than three Figures in its Root; but as 624 Mul-
tiplyed by Itſelf gives for Product 389376, which is lefs than the pro-
pofed Number 389489, and that 625 multiplyed by Itfelf gives for Pro-
duct 390625, which is greater than the propofed Number, I conclude
that 624 is the neareſt Root in Integers.
XXIX.
II 3
Third ex-
ample.
which is not
As this Reaſoning may be applied to any other Example whatſoever, The root of
it appears in general, that the fquare Root a of any Number A, found a number
by the foregoing Method, when there is a Remainder left, is lefs than
a+1 but greater than a; confequently if it be determinable it will be
expreffing any proper Fraction reduced to its leaft
at
Terms.
771
m
n
n
Now fince
an + m
n
n
a perfect
power of
the fame
degree as
the root re-
quired is
not determi-
is a Fraction reduced to its leaft Terms, then nable,
will be alfo a Fraction reduced to its leaft Terms; for if n
and a n + m have a common Divifor, then the Denominator of the
when reduced would be leſs than n, confequently
Fraction
a n + m
the Fraction
-n
172
n
has not been reduced to its leaft Terms, which
is againſt the Suppofition. if therefore the fquare Root of a Number
which is not an exact Square is determinable. A Fraction reduced to its
leaſt Terms when ſquared muſt give an Integer, or which comes to the
:
34
S
ELEMENTS OF
1.
fame Thing, two Numbers a and b that have no common Divifor and
two Others c and d that have no common Divifer with one another or
Method of
approximat-
ing to the
fquare root
of numbers.
with the two Firſt, in their Product
common Divifor.
{
a c
bd2
ac and b d would have
But it is Eafy to perceive 1º that a c and b can have no common Divifor
a c
er
b
cannot be reduced, for if poffible let
b
a
a c
Ъ
C
but
e
e b
gb
b
a
fince
is a
bgb, b is a Divifor of b, but b has no common Divifor with a or c
wherefore neither has b for if brs and, for Example, a = ts, then
b=grs, that is a, and b would have a common Divifor which is againſt
the Suppofition, Again fince'a ceb,
Fraction reduced to its leaft Terms, wherefore b is a Divifor of c, and
confequently b and c have a common Divifor which is likewiſe againſt the
Suppofition. In like manner it will appear 2° that a c and d can have no
common Divifor, and in general that any two Numbers which have no
common Divifor with a third Number, their Product will have no com-
mon Divifor with this third Number, wherefore fince b and d have no
common Divifor with a c their Product bd can have no common Divifor
1
a
with a c. hence we may conclude that if is a Fraction reduced to
its leaft Terms
a3.
63
b
and in general
an
611
will be a Fraction
a2
62
reduced to its leaſt Terms, wherefore a Fraction either pure or mixed
raiſed to any Power will give always a Fraction, wherefore if the fquare
Root, cube Root &c. of an integer Number is not an Integer, it cannot be
determined as alfo the Root of a Fraction or of an Integer joined to à
Fraction which when reduced to a Fraction each of its Terms is not a
perfect Power of the fame Degree as the Root required, for each Term
of the Fraction may be conſidered as an Integer a part, whofe Root can-
not be determined.
XXX.
Though the exact Root of an Integer which is not a perfect Square
cannot be found, however we may Approximate to it to any degree of
Exactnefs, for Example, to extract the fquare Root of 389489, inſtead
of 389489 we may write 3894890000 which is equal to it, obferving
10000
that the Denominator 10000 be a fquare Number, that is, fhould contain
an even number of Cyphers then extracting the fquare Root of the Nume-
rator which may be found true to an Unit, and dividing this Root by 100
which is the fquare Root of the Denominator, the fquare Root of
SPECIOUS ARITHMETICK
83
389489c000
10000
wrote
true to
2
I
or of 389489, will be obtained true to
if we had
2
100
389489000000
1000000
I
1000
>
it is eafy to perceive that the Root would be found
and ſo on. Hence in extracting the fquare Root after the
Number propoſed is gone through, if there is a Remainder the Operation
being continued by adding periods of Cyphers to that Remainder, the true
Root will be obtained in Decimals to any degree of Exa&neſs.
XXXI.
3 8.9 4.8 9 ( 6 2 4 › 0 9 0 5 Firſt Exam-
ple.
36
I 22
294
1 2 2 ) 2
X 2/244
1244) 598998
X4/497 6
12480)
хо
1 2 4 809)
X9
I 1.3-
· 3.00
113
I 1 3 0 0 0
1 1 2 3 2 8 1
To Approximate, for Example, to
the Root of 389489, having found
its neareſt Root in Integers, I annex
to the Remainder 113 a period of
Cyphers, I divide (neglecting the
laft Figure) by the double of
624 or by 1248, and place the
Quotient (0) after 4 and then
multiply 12480 by o and Sub-
tract (0) from 11300 then to the
Remainder (11300) I annex ano-
ther Period of Cyphers, and divid-
ing 1130000 (neglecting the laft
Figure) by the double of 6240 or by
12480 I place the Quotient after o,
and multiplying 124809 by the
Quotient 9 I fubtract the Product
1123281 from 1130000, and to the
Remainder 6719 I annex another Period which I manage as the laſt,
proceeding thus it is Manifeſt that the Root may be Approximated to any
degree of Exactnefs. If we neglect the Remainder 4780975, the Root
will be
we take
6240905
10000
6240906
10000
which is true to
1248180) 671900
хо
1 2 48 1805)
I
10000
X5
0
67190000
624090 2 5
478097.5
and
6240905
10000
for if inſtead of
,
great, fince this Quantity
or 389489 +
100000000
7700836
I Q0000000
which is greater
the Root would be too
ſquared gives 38948907700836
than 389489.
XXXII.
After the fame way the fquare Root of decimal Numbers, may be ex- Second
tracted, but before the Operation the Number of decimal Places are to be example.
made even, and after the Operation as many places are to be pointed off in
84
ELEMENTS OF
the Root as there are Periods in the Fractional part of the propofed Num-
ber. Thus if the fquare Root of 3297,6 be required True to four deci-
mal Places, inſtead of 3297,6 I write 329760000000 which is equal to
100000000
it, whofe Root is 57,4247 in like manner to extract the fquare Root of
0,99856 true to fix decimal places, inſtead of 0,99856 I write
998560000000
1000000000000
which is equal to it, whofe Root is 0,999279 as
will appear by the following Diagrams.
32:97;.6.0 (57, 4.2.47
25
107797
X 7749
1 8
}
0,99.8 5.6 0 (0, 9 9 9 2 7 9
8 I
X9
1885
170 1
1 1 4 8 2 28400
I 1 4 4 4 8 6 o
4)
X4 4576
X2/22964
1 1 4 8 4 4
4 4
5 4 3 6 0 o
0
X 4
4 5 9 3 7 6
459376
1 9 8 9 1 8 4 6 0
X 9/1790 I
1998 2 )
1.99.847.
X
55900
39964
1593000
1 3 9 8 9 29
I
1 1 4 8
4 8
7 8 4 2 2 4 0 0
1 9 9 8 5 4 9
1 9 4 6 7 100
X 3
039409
X9
X 91
1 7 9 8 6 9 4 1
382991.
1 4 8 0 1 59
XXXIII.
Befides the foregoing Method for Approximating to the fquare Root of
Another Numbers, the Analyfts have imagined others which we fhall proceed to
method of explain.
approximat-
ing to the
Let it be propoſed to Approximate to the fquare Root of 50, the near-
fquare root eft lefs Root of 50 in Integers is 7, let 7+ be the true Root, where-
of numbers. fore 49 +14% + x² = 50, or 1 + 142 +2²o: let this Quanti-
ty be multiplied by 1 + A2+ B2, the Product will be
2
Z
- 1 + (14 ~~ 4) 2 + (1 + 14 A− B ) ≈² + (A+ 14 B) z³ + B x4 = 0
in which the Terms (1+14A-B) z2 and (A + 14 B) 23 may
be made to vaniſh by determining A and B by means of the Equations
I + 14 A B=0 and A+ 14 B = 0, from whence reſults
A
14
197
and B =
A
14
the foregoing Equation, we will have-1 +
197
Z
I
>
ſubſtituting thoſe Values in
197
(14+) = +
z4.
=,0
197
Quantity 24
becoines
= x + (14+ - 113177 ) 2
-) x = 0;
which by rejecting the exceeding ſmall
14
197
z=0; hence 197 +2772% = 0, and
SPECIOUS ARITHMETICK.
85
197
=,0710678, which Value is true to the laſt decimal Place,
2772
and if more terms of the Series 1 + Az+ B z² + Cz³ &c. had been
taken, the Root would be Exacter in Proportion.
m
22
In general to Approximate to the fquare Root of any Number A let r
be the neareſt leſs Root in Integers and r + z the true Root, then will
8-² + 21 z+2² = A, 2 r z + z²= Ar², or —
(putting Ar² = m) then multiplying by 1+Az there will refult
m A
I
2 r
+x+
A
2 t
m
+(.I
21
I
here by making
2 y
Equation becomes
2 r
Z
2 r
+ A= 0, A is found
x + ( — — —
² + 1) x² +
z3 0,
2 r
I
and our
m
+1 +
21
4
=) ==
I
Z
442
by neglecting the laſt Term as being very fmall, will
2.
=
230, which
be reduced to
I
m
I
2 ↑
+ (1 + 1 )
帆
​zo, whence ≈ =
Z
z
2 p² + = m
4 r
To obtain a more approximate Value of z, we have only to multiply
22
====
2 r
= 0 by 1 + AzB z² whence there arifes
I
m
+ x +
2 y
m. B
+ (1 -
4) = + ( 2
A-
+ 4— ~ P ) x² + ( − 1 + B) =³ + ——— * = •
B
21
m
2 1
m
2
I
2.1
2 +
A
2 +
in which the third and fourth Terms will vaniſh by determining A and B
by means of the Equations,
from whence A is found =
L
2 t
2
4x² + 17
-I
=0&
A
21
.I
m B
+ A-
21
2 r
and B
+(
2 m² + m
z +
21 + 3 m
21 (4²+m)
M
21*
+(
242 + 12
2 g. 2 + 1/2 ??l
m
)
m
zo and z =
X
wherefore
2 r
which by rejecting the laft Term as exceeding fimall,
A
+B=0,
24
I
;
4x² + 212
m
x4 = 0;
24
will become
2 go² + // m
2 r 2 gize + mz
To obtain ſtill a more Approximate Value of z, it fuffices to multiply
21*
-2
771
2
+x+
2 ↑
122
+(1
2 **
B
m A
2 r
by 1 + Az + B z² + C z³, & there refults
-)=+(
+(-2, + C) 84 +
r
C
Ze pr
I
+ A
m B
2 +
2 y
25.
A
2
Z3
) x² + ( 141, + B = m1 C ) z³
27
27
86
ELEMENTS OF
in which Equation the third, fourth, and fifth Terms may be made to
vanith by determining A, B, and C by means of the Equations
A
B
I
27 + 1 -
m B
A
= 0,
2r
+ B.
m C
whence B
2 r
21A+I
2 r
2 yr
2 r
C=
+C=0
B
"
and C =
772
Firft exam-
ple.
exterminating C,there arifes 2r B+ A
773
A+
the value of B, becomes 42 4+21
fore A
771
2+
+(
4.² + m
8+3+4rm
"
and C=
16 +4 + 12 p² m + m²
16 +4 +8 gik 772
from which, rejecting the laft
83 m + ar m²
16 rt † 12 r² m + m²
27 B+ A
Z
+
172
B
2 +
+
27
o,which by fubftituting
2rA+I A
2 772
I
8r3+4rm
+ =0, where-
16 r4 + 8 p² m
11
and of courſe
z50,
Term as exceeding fmall, you will find
2
r m (2 r² + m)
giz (4 ri+ 3/71) + 4 m²
XXXIV.
To fhew the Uſe and great
Example, let it be propofed to extract the fquare Root of 441.
Suppofing r to be affumed = 20, then m = 41, and z =
Exactnefs of thefe Approximations by an
16.40
1641
by the first Approximation.
&
1
confequently r+z = 21
1641
67281
I
By the Second r+2=20+
= 21+
67280
2758481
more nearly.
275880
I
*
And by the Third r + z =
20 +
= 21
2758481
2758481
still more nearly.
Second Ex-
ample,
ni=8,
8, and z
80
204
Let it be propofed to extract the fquare Root of 108, here r = 10,
=,392. Hence the Root required is 10,392
nearly, which value is true to the last decimal Place.
By the Second r
+ z = 10 +
And by the Third
laft decimal Place.
1632
4160
+2=10+ 16640
42416
= 10,39230 more nearly.
= 10,3923048, true to the
It is to be obferved that a vulgar Fraction cannot always rigourouſly
and exactly be reduced to a Decimal. let, for Example, be pro-
poſed to be reduced to a Decimal
و
then r=
PX 10"
but
2
SPECIOUS ARITHMETICK.
87
10" = 2ª × 5"; now £X 2" × 5″
q
cannot give an Integer r, unleſs 9
A vulgar
be fraction can-
not always
be
fome Power of 2, or of 5, or of 2 X 5, lefs than n, foris fuppofed to be be reduced
9
a Fraction reduced to its leaft Terms, in every other Cafe X 10"
never give an Integer r, but it is manifeft the greater n
can
is, the nearer
да
11
will approach to , the Error being always lefs than
I
ΙΟ
IJ
Π
P
9
n
fince di-
viding p X 10" by q the Quotient which refults, and which is too little
will be too great if increaſed by an Unit, wherefore < Land·
+ + 1
p.
>
*
ION
9
to a
mal.
IOл
9
XXXV.
To fhew the uſe of the foregoing Rules, in the Solution of Problems,
here follow fome Examples.
A Gentleman left an fate of r Pounds per Annum to his Son, who. be- Problem,
ing a Minor, bis Guardian allowed him a Sum a the first Year, and puts out
at Intereft the Remainder ↑ -a. He allowed him the fecond Year a Sum b,
and the over-plus rb was alſo put out at Intereft, and the Guardianship be
ing ended a few Days after, the Revenue of the young Gentleman was
found to be Increafed during the two Year's Guardianship by a Sum d.
What rate of Interefi was allow'd.
100
100
X
Let expreſs the yearly Intereſt of 100%. I fay 100: =ra:
X
***
Sum added to the Revenue of the young Gentleman at the End
of the first Year, wherefore the Sum faved at the End of the first Year
will be ra+, the Intereft of which Sum at the End of the
Second Year will be
X
*
a
a
b
+
,which together with
X
the Inte-
reft of rb put out at Intereft at the beginning of the fecond Year,
will express the Increaſe, which the Revenue of the young Gentleman
has received during the two year's Guardianfhip. Confequently
27
+
X
d, putting 2 r- a — · b = m, r — a = c(to
C
abridge the Computation) there reſults the Equation + √2 = d, or
71 X
*+2
d
X
and compleating the Square and extracting the
88
ELEMENTS OF
fquare Root x =
m
2 d
m²
±√ ( ½ + + + + ) =
XXXVI.
4d2
m± √4cd+ m²
24
To apply this general Solution to an Example, let the Eftate be 4800 %
Application per Annum, and ſuppoſe the young Gentleman to have expended the firſt
of the fore- Year 2400/. and the fecond Year 3120/. and let the increaſe of his Re-
going Pro-
venue at the End of the ſecond Year be found to be 3567. 13s. 4d.
We will have r = 4800, a = 2400, b = 3120, d = 1070
2 r. — a
blem to an
example.
Another
Problem.
x =
X
=
m ± √
3
b = m=4080, Subftituting thofe Values in the Expreffion
(4 c d + m²
2 d
3 (4080 ± 4480)
2140
•
we will have x = 3 [4080 ±✓ (20070400) 1
2140
wherefore x = 12 and Confequently the rate
of Intereſt allowed was 81. 6s. 8d. per Cent. in Effect the Intereft of
2400/. for one Year at 81. 6s. 8d. per Cent. amounts to 2007, confe-
quently the Sum faved at the End of the first Year will be 2600l. the
intereſt of which for one Year at 81. 6s. 8d.per Cent. will be 2167. 1 35. 4d.
and the Intereſt of 1680/. at 87. 6s. 8d. for one Year is 140/. and con-
fequently the increaſe of the Revenue of the young Gentleman during the
two Years Guardianship will be 356/. 135. 4d. agreeable to the Conditi-
ons of the Problem.
ΙΟ
If the Revenue of the young Gentleman was fuppofed to be increaſed
4321. during the two Years Guardianship, then x will be found = 19
and confequently the rate of Intereſt allowed will be 10 per Cent. in ef-
fect the Intereſt of 2400/. for one Year at 10 per Cent. will be 2407, the
Sum faved therefore, at the End of the firſt Year will be 2640/. the In-
tereſt of which for one Year at 10 per Cent. will be 2641. and the Inte-
reft of 16801. for one Year at 10 per Cent. will be 1687, wherefore the
increaſe of the Revenue of the young Gentleman during the two Years
Guardianſhip will be 432. as the Conditions of the Problem require.
XXXVII..
Two Notes one of 1201. payable in 6 Months and the other of 1501.
payable in 9 Months, were diſcounted for 81. 10s. what rate of Interest
were they discounted at.
Let x denote the Intereſt of one Pound for 12 Months, then the a-
mount of 17, in 6 Months being 1+ and in 9 Months 1 +
2
x
3x
4
the preſent Value of the Bill due at the end of the 6 Months will there-
I 20¹
fore be
! + // * '
and that of the Bill due at the end of 9 Months
>
150
1+x.
SPECIOUS ARITHMETIC K.
89
120
I + 1/2 x
whence we have
+
=261,5 by the Question, which,
+ 150 + 75 x = 261,5 × (1
261,5 × (8+ 10 x + 3 x²)
8
= 3x²+10x+8, that is,
from which we have x² +
150
1 + 2 x
120 - 150
8, 5
by Reduction, becomes 120 + 90 x
+ $ x + 3 x²) or, 270 + 165 x
(270 + 165 x ) X 8
therefore
261, 5
x=3x²+10x+8,
2160
+
261, 5
1320
261, 5
2590
3 X 523
136
x =
3 X 523
compleating the Square,
&c.
x will be found
√
136
,
whence by
+ (1205)²] -
1569
1569
1295
1569
=
✓ ( 136 × 1569 + 1295 × 1295 ) — 1295 = 0, 05093, which
1569
multiplied by 100 gives £5, 093, or £5. 1s. 10d. ž. nearly, for the Rate
per Cent. at which the Notes were diſcounted.
XXXVIII.
In the different Examples which we have given of the Refolution of
Equations of the fecond Degree, the young Analyft could fcarce meet
with any Difficulty except when it was propofed to reduce radical
Quantities by taking from under the Sign, the fquare Quantities which Examples
were Factors of the radical Quantity: In Effect this Operation is the of the re-
niceft that occurs in the Refolution of Equations of the fecond Degree; duction of
to affift therefore the young Analyſt to perform it with Facility, here fol- radical quan
low fome Examples.
√ 48 a a b c = 4 a √ 3 bc, ✓ (a 3 b + 4 a abb +4 a b³)
p z z
30 b
-
a a b b m m
pp z z
+
4 a a m3
-)
a b b =
98
7
2
XXXIX.
CC
a m
p z
tities.
(a + 2b Jab),
C
v ( b b + 4 mp),
675
за
which have
No fooner had Equations of the fecond Degree made known the ir- Quantities
rational or incommenſurable Quantities (it is thus Quantities which have not an exact
no exact Root are called) but the Analyfts found it neceffary to perform root are cal-
on thoſe Quantities the fame Operations as on rational or commenfurable led incom-
Quantities, that is, they had to add, fubtract, multiply, and divide or irrational
Quantities either entirely incommenfurable, or partly incommenfurable quantities.
and partly commenfurable.
As to the Addition and Subtraction of radical Quantities, the only
Difficulty confifts in reducing them to their leaſt Expreffions.
menfurable
The additi-
on and fub-
traction of
3 M
90
ELEMENTS OF
irrational
quantities
prefuppofes
only their
reduction to
their leaſt
terms.
For Example, Let it be propofed to add 48 a b b and b✔ 75 a to-
gether, I reduce the firſt of thoſe Quantities to 46 √3 a, and the ſecond
to 5 b√ 3 a, whoſe Sum is 9 6 √3a.
In like Manner ✔ 48 c³
a b3
CC
a3 b.
+
I
16
27
9
4 a a b b + 4 a b³) =
√ 19 23 =
32 6 √ 3c,
a
✔ab..
2 C
a a
✓ 3
- bc) x
ab
a + c
✓ (a3b.
2 C
A
心
​√(Jaa
cc
-)
b c ↓ a b
a + c
XL.
Multiplica-
tion of
Surds.
3 aa + Bac +36c
With Reſpect to Multiplication, if the Quantities to be multiplied are
both incommenfurable, there is no more to be done than to multiply
the Quantities which are under the radical Sign, and ſet the common ra-
dical Sign over the Products; which afterwards is to be reduced to its
leaſt Expreffion.
Let it be propoſed for Example, to multiply ✔✅ a b by ✔ a c, I write
down a a b c or a ↓ bc. In like Manner
f
√ 3 c d X √ 4 ƒ gc = √ 1 2 f g c c d = 2 c √ 3 f g d.
When the radical Quantities to be multiplied are equal, the Multipli-
cation is performed by taking away the radical Sign. The Product, for
Example, of a³ c d into √ a³ c d is a³ c d.
If the Quantity by which the Surd is to be multiplied be rational, it
fuffices to write it before the Sign between Parentheſes when it conſiſts
of feveral Terms; and before it can be brought under the Sign it muſt
be firſt ſquared.
For Example, the Product of a + b into
ffg
و
or
√ [£ £ 8
£fe x (a + b)], or
(2+6) a2-bb
or √
[ -f B g
2 a
a
a a + b b
C
b
X
3 a
6 aa X (a a + bb)
✔c d
√
a a
2
ffg. is
b b
(a a ±z a b + b b )
a a
×
f√
b b
(ag + bx)
-
√(a
(a a + b b )
d
6a4 + 6a a b b
bb).
√ c d
-]
√ (a + b) X √ ( a − b ) = √ (a a -
If the Quantities to be multiplied confift of feveral Parts, either all
irrational or partly irrational and partly rational, the Operation is per-
formed by obferving the fame Rules as in the Multiplication of com-
pound Quantities, for Example, (3 a b c — 2 b √ a c) X 2 c √ a b =
6 a b c √ a c — 4 a b c √ b c. [ √ a b + √(a a − x x )] X
[√ab + √(aa-xx)] = aa+ab-xx + 2 √ (a³ b — a b xx).
[a + √(aa - b b ) ] x [a + √ (a a − b b ) ] = 2a a b b +
2a√(aa — bb), [ a + √ (a a — xx)] × [a - √ (a a — xx)]
SPECIOUS ARITHMETICK.
91
2 ax
=xx,
C.
a3
2ax
a
+
C
C
a3
x3a+
C
√
abb
baax
=
C
C
a a b
a a b
C
C
XLI.
-30-
Irrational Quantities are divided one by the other, by dividing the Divifion of
Quantities under the Sign and ſetting the Sign over the Quotient.
incommen-
furable quan
If an irrational Quantity is to be divided by a rational one, the irra- tities.
tional Quantity is to be placed above a ſmall Line and the rational
Quantity under it, and before it can be brought under the Sign it muſt
be firſt ſquared.
If the Surds have any rational Coeficients, their Quotient is to be
prefixed to the radical Quotient. All which will appear plain by the
following Examples.
√ a b
✔
√ a
=3a√3c,
= b
1 2 a c√ 6 b c
= C √ C₂
4 c √ 2 b
↓ a a b b — b b x x
a c √ b c
= √b,
a √ b
√(aa- xx)
√(a + x)
= √(ax),
a + x
a
X
a
If the Quantities to be divided confiſt of feveral Parts rational and
irrational, the Operation will be performed as the Divifion of compound
Quantities.
+ b b the
a b b c
For Example, a4 + 64 being divided by a² + a b √ 2
Quotient will be found to be a² ab √ 2+ b², a3 b
a a b √ b c + b bebe divided by abc the Quotient is aab-bbc.
In like Manner, a³ — a b c + a a √ b c — b c
abc the Quotient will be a a + b c + 2a√ bc.
XLII.
be divided by
The foregoing Principles are fufficient for refolving Equations of the
fecond Degree when they include only one unknown Quantity; but as
there frequently occur Problems where feveral unknown Quantities are
concerned, we shall proceed to explain how they are to be managed;
with this View we fhall propofe the following Problems.
lem of the
un-
known quan
tities are
The Joint-Stock of two Partners A and B was £165. (a), A's Money First Prob-
was in Trade 12 Months, and B's 8 Months, when they shared Stock and ſecond de-
Gain, A received £67. (b) and B £126. (c) what was each Man's Stock? gree where
Suppoſe A advanced x Pounds, then B advanced a — x Pounds, and two
their respective Shares of the Gain may be confidered as the Intereft of
their respective Capitals during the Time they were in Trade, the Rate concerned.
of Intereft being the fame for both: Let z therefore denote the Intereſt
of £1. for 12 Months, then the Amount of £1. for 12 Months being
If, and in 8 Months 1+ z, the Dividend of A will be x (1 + z),
I
ELEMENTS
OF
Second Pro-
blem,
Third Pro-
blem.
2
3
the Dividend of B (a — x) × (1 + z), hence x (1 + z) + (a
x (i + z) = b+c, from whence is deduced z = 3 m
x)
3 a
x + za
(putting bc = m to abridge the Computation) fubftituting this
Value of z in xxx, the Dividend of A, there will refult
= d, or x² + 3 mxax-dx=2ad,
adn compleating the Square and extracting
the Root there will refult x =— ± n ± √ (2 ad + 1/2 n²) = 5.5°
x +
3 ax
3 m x
x + 2 a
now putting 3 m
XLIII.
A Grocer fold 80lb. of Mace, and 100lb. of Cloves for £65. be fold 60lb.
more of Cloves for £20. than be did of Mace for £10. what Price did be
fell each at ?
Suppoſe he fold the Mace at x and the Cloves at y Shill. a b. then
the Number of Pounds of Mace
x.: 1 lb.
200
X
£10. or 200s. :
fold for £10. and j : 1 lb. = £20. or 400 s. :
y
Pounds of Cloves fold for £20.
and
200
*
+ 60 =
400
130
- 8 x
ΙΟ
= wherefore
tion I deduce
20 x
10 + 3x
1300 80x
1300, or x²
ولا
400
y
Number of
+ 100 y
now £65. or 1300 = 80
by the Queſtion. From the firft Equa-
=y, and from the fecond Equation
130 8x
ΙΟ
20✰
10 + 3x
+390x — 24 x² = 200 x, that is, 24 x² IIO X
or
1300, and compleating the Square
ΓΙΟ
x =
24
24
and extracting the Root, there reſults x
confequently y = 5.
XLIV.
55
185
=
or x = 10,
24
24
The Sum and the Sum of the Squares of threee Numbers in geometrical
Proportion being given to find the Numbers.
= x.z;
Let the Sum of the three Numbers fought be denoted by x, y, z,
from the Nature of Proportion x : y = y: z, that is, y y
moreover, fince their Sum is given, denoting this Sum by a we will have
x + 1 + z = a, and expreffing the Sum of their Squares by bb we
will have by the laſt Condition of the Problem x x + y y + z z = b b.
To make uſe of thofe three Equations, I first exterminate z by
Means of its Value a-x-y deduced from the Equation x+y+z = a;
fubftituting therefore this Value of z in the two other Equations they
will be transformed into 2 y y + 2 x x + 2xy +aa-2ax — 2 az
SPECIOUS ARITHMETICK.
93
x x
bb, and y y = ax
To exterminate out of thoſe
x y.
two Equations one of the two unknown Quantities they include, it fuf-
fices to find the Value of this unknown Quantity in each of thofe Equa-
tions, and to equate them; let x be the unknown Quantity to be ex-
terminated, the firft Equation will give
x = − § y + ÷ a ± √(·
플
​+
b b
2
1
a a
4
and the fecond xy + a ± √ ( a² —
Equating thoſe two Values of
½ y + // a
+ ÷ • ± √(
ས
x, there will
b b
2
a a
4
3),
å y y + z a y
y²).
ay —
refult the Equation
+ y)
2 y y + 1 a y
= −y + 1 a ± √ ( 1 a a—ay — ayy) to folve which
+들다
​I obſerve firft, that the Terms ・y + a being common to both
Members of the Equation, the Equation may be reduced to
#yy + ±ay) = ±√(‡ a a— ‡a y — ‡ yy)
b b
2
a a
4
the two Members of which being fquared, and reduced, there refults
bb, from whence is deduced y =
ay = — a a
a a
2 a
b b
this Value of y fubftituted in one of the foregoing Values of x, will give
b b
x = a +
at
4 a
ог х
a a
16
— 4 a a + 1 b b — 35 a a + 3 b b
bb + aa±√(10 bba a — 3a4—3.64)
ΙΟ
4 a
the Values of x and y in the Equation za
a a + b b ± √(10bba a
2 =
•
-- 36),
and fubftituting
4. R
3 a4 3 64)
x
y we will have
XLV.
16 aa
-):
ons folved
A's in this Solution we arrived at a very fimple Value of y, after hav- The forego
ing had very complex Radicals, it fhould incline us to think that this ing equati-
Value might be obtained after a more concife Manner: And it was not by another
difficult to imagine the following Method.
Let the two Equations x²- a x + yx=
and x² + xy
b b
a a
2
— y²+ay
2
axyy be refumed; fubtracting one from the
other there reſults o =
y =
other of
2 a
b b
2
a a
2
+ay, from whence is deduced
which being fubftituted in either one or the
Equations, gives
a a
b b
thoſe two
a a x − b b x
x² +
a x =
24.
— a4 + 2 a a b b
4 a a
64
,
or
Method,
1
94
When the
values of
two un-
known quan
tities in a
x2
ELEMENTS OF
(b b + à a) x
2 a
a4 + 2 a a b b — b4
=
from whence
4 a a
is deduced the fame Value of x as above.
XLVI.
To apply this general Solution to an Example, let the Sum of the
three Numbers in continual Proportion be 20, and the Sum of their
Squares 140; in this Cafe a
in this Cafe a 20, and b b =
y = 6 ½, and x = 6 ±±√3; Viz. 6
༨
16
the greateſt of the three Numbers fought, and 6 4
140, confequently
+ √3 is
5
16
16
V3 돟 ​the
leaft. For as the Values of x and z in the foregoing Solution are exac-
ly alike, x denotes ambiguouſly either of the extreme Numbers, and
thence there will come out two Values either of which may be x,
the other being z.
XLVII.
When there happens to be fuch an Affinity or Similitude of the Re-
lation of the two Terms to the other Terms of the Queftion, that in
making uſe of either of them, their Values would be brought out ex-
actly alike; (which will be eaſily feen) the Analyſts to avoid Ambigui-
problem are ty make ufe of neither of them, but in their Room chufe fome third
brought out which fhall bear a like Relation to both, as fuppofe the half Sum or
like, how half Difference, or any other Quantity related to both indifferently and
the ambigu- without alike.
ity thence
perfectly a-
arifing may
be avoided.
a a
b b
24
Thus in the precedent Problem, having found y =
to avoid the Ambiguity arifing from the Subftitution of this Value in
one of the foregoing Values, of x, I put
=u, and as
X
Z
2
x + x.
a
y
a a
+ b b
confequently
2
2
4 a
X
a a + b b
4 a
u and z =
a. a + b b
тъ
น ;
but
4 a
a 4 + z a a b b + b 4
a4
za a b b + b4
X Z =yy
นน
16 a a
10 b b a a
wherefore u u
√ (10 b b a a
3 a
น
4 a
11
4 a a
64
64)___, conſequently
b b + a a + ✓ (10 bbaa — 3 a4 —
b b + a a
√(10
2 =
Ѣ
4
Q
· 3 ·64)
and
a a
3a4
3 64)
a
h
3 a
16 a a
a 4
3
and
4
3
SPECIOUS ARITHMETICK.
95
XLVIII.
the fecond
In this Problem, there occurred Quantities which deftroyed each Example of
other, whereby the Computation was rendered extremely fimple, but equations of
as this feldom happens in Equations of the fecond Degree including fe- degree in-
veral unknown Quantities, we fhall proceed to explain the Methods cluding two
employed by the Analyfts in the more complicated Cafes. Let, for unknown
quantities
Example, the two Equations x2+4x-2xy=aa+2yy and qui
2ax + xy = 2 a a yy be propofed, the firſt of thoſe cated than
Equations gives x = a + y ± √( &a a — ay + 3yy) and the the forego--
· =
ing.
other gives x = a — y ± √(зaa ayyy), equating
(3
thoſe two Values, and reducing, there refults
за
-
3 a + 1 ± √ ( { a a − a y+3 y y) = ± √(3 a a—ay — ‡yy).
3
5
2
To make the Radicals diſappear in this Equation, I firft fquare the
two Members and there refults yy — ay + 2 a a ± (3 y − 3 a)
2 aa±
39)
√ ( & aa - ay + 3y y) + aa—ay + 3y y 3 aa-ay — yy,
which ftill contains a radical Quantity. To make it diſappear I write
down this Equation thus,
4
4
3
+
a4
2 a³y +
105
4-
more compli
- ½ aa + 2 ay — 6 y y = ± (3 y — 3 a) √ ( — a a — ay + 3y y)
reducing it and leaving the radical Quantity ± (3 + 3 a)
√ ( & aa — ay + 3yy) alone on one Side of the Equation; fquaring
afterwards the two Members
a ay y
4
54 a y³.
+36y4 = (911 y 18 ay + 9 aa) X (sa a — ay + 3yy)
which when reduced becomes 94 +9 aµ³-30аay + 27 a³· y = 11 a 4.
Equation refulting from the two foregoing ones, which is to be refolved The refult--
to obtain the Solution of the Problem which furnished thofe Equati- ing equati-
ons: From whence it appears that in Equations of the fecond De- of the un-
gree involving feveral unknown Quantities it is not as in Equations known quan
of the firft, in which the refulting Equation when one of the un- terminated.
known Quantities is exterminated never rifes to a Degree higher than
that of the Equations themſelves.
XLIX.
on when one
tities is ex-
which the
The Equation which refults when an unknown Quantity is exter- Another
minated out of Equations of the fecond Degree, may be obtained with- method by
out being at the Trouble of firſt folving them, and afterwards making unknown
their Radicals diſappear.
quantity is
ed in the
That this may appear let the two foregoing Equations be refumed, and exterminat--
from each be deduced the Value of xx, the first will give
foregoing
x²=aa + 2y y a x+2xy, the ſecond x² = 2 a a
Equating thoſe thofe two Values, we will have a a + 2 y y + 2 x y
y y x y + 2 a x, from whence is deduced
ax = 2 a a
3 y y
3 a
a a
32
,
-yy+2ax
xy. example..
which being fubftituted in one or other of the
f
96
ELEMENTS OF
The forego-
ing method
two given Equations, in x x
2 a x + xy = 2 a ay y, for Exam-
(v 2 a) X (31 y
ple, gives —994 — 6 aayy + a4 +
2
(3 a 3y)²
yy which when reduced, becomes
30 a ayy + 27 a3 y
= 2 a a
949 a y³
L.
за 33
11 a4 the fame above.
a a)
Let the two Equations 2+ a⋅ x y + b x = c y² + dy + e and
applied to x² + ƒxy + 8 x = by²+ iy + k be given, which are the most com-
another ex- plicated of the fecond Degree involving two unknown Quantities.
ample.
1
X
-
Deducing a Value of xx from each of thefe Equations and equating
them, we will have (af) X x y + (b − g) × x = (c — b) x y²+
(di)Xyek, which, by putting, to abridge the Computation,
a f = 1;
l; b g
c b =
= which; open; d-i-p; c = k = 1;
e e
will be transformed into 1x y + m x = n y² + py + q. From
whence is deduced x = ny² + py + q. Which fubftituted in
ly + m
one or other of the two given Equations, in the firft, for Example, there
(ny²+py +9) 2
will refult
(ay+b) x (ny² + py + q)
(1 y + m )²
ly + m
+
=cy² + dy + e, or (n y² + p y + q)² + (a y + b) × (n y² +by+q)
X (ly + m) = (c y² + dy + e) × (ly + m)².
If we now perform the Operations which are indicated, re-
duce and order the Equation, there will reſult
4
+
+
2
(bln+amn + alp + 2 n p - 2 m lc - 1² d)
a l n + n n
bmn+qal+pbl+pam+ p² + 2 u q
a
I n + n n
12
·13
C
m² c — 2 m l d — e l²
12
32
C
m² d
2 mel
a.
n + n n 12 c
y.
92
a l n + n²
b m q
12 c
blq + a mq + b m p + 2p9 -
m² e
Whatever
y in two e-
, Equation of the fourth Degree which
refults from two of the moſt general Equations of the ſecond.
LI.
If we had two Equations fuch as xxy+axy = a b b and
be the di- xxyy + c cy x = a², they cannot be confidered as Equations of the
menfions of fecond Degree, becauſe the unknown Product of 2 intoy is of three Di-
quations if menfions, and that of x into y2 is of four Dimenſions; however x may be
thoſe of x do exterminated out of thofe two Equations by the foregoing Method.
not rife high That this may appear let A exprefs all the Quantities compofed of y
it be ex and known Quantities which affect 2 to whatever Degree they rife, (or
terminated in the Language of the Analyſts any Function of y) in one of the given
Equations; let B exprefs the Function of y which affects x in the fame
going me-
hod. Equation, and C the Quantity which is not affected by any Power of
er than two
may
by the fore-
1
SPECIOUS ARITHMETICK.
97
=
x, that is, let the firft Equation be Ax2 + B x C, and in like
Manner let the fecond be A x² + B' x = C.
From the firſt I deduce x²
C'
302
B' x
A'
CA
B Ax
A C'
CA'
A B'
Β Α'
C. Bx
A
and from the fecond
"
, equating thofe two Values there refults
A C' A B', from whence is deduced
which being fubftituted in the Equation
C'
Ax² + B x = C, gives AX (ACCA)²+ BX (ABBA') X
(ACCA) = CX (ABBA'), in which fubftituting for
A, B, C, A', B', C', the Functions of y their Values, the Equation fought
will be obtained.
LII.
this when the
If the unknown Quantity which is to be exterminated out of two How the
foregoing
given Equations, rifes to higher Dimenfions than two, it is eafy to per- method is to
ceive that by an Operation fimilar to the foregoing they may be tranf- be applied
formed into two others of lower Dimenfions, and by repeating
Operation the unknown Quantity will at length be taken away.
If we had, for Example, y³= x y y + 3 x and y y = x x
to take away y I multiply the latter Equation by y, and you have mentions
3 = xxy x y y
-3y, of as many Dimenfions as the former, now than two.
by making the Values of y3 equal to one another, I have x y y + 3 x =
3y, where y is depreffed to two Dimenfions, from
x x y - хуу
whence I deduce y y =
moft fimple one of the
I find y =
2 x3
3 x x
x y
3;
xxy — 31—3* by this therefore, and the
2 x
Equations first propofed yyxx — x y — 3,
which Value fubftituted in y y=xxxy-3
3
31
as being the moſt fimple, there refults
4x6 — 12 x4 + 9 x x
9x4 - 18 xx+9
18xx
X X
2x4 + 3xx 3, whence 4x6—12 x¹+ 9 x x = 9 x6— 18 x 4 + 9 x²
3 x x 3
= x4+9x
6 x6 + 9 x² + 6x4
2.7 x² + 54. x² - 27, which when
reduced becomes x618x445 x² +27=0.
9x2
LIII.
quantity to
be extermi-
nated rifes
to higher di-
were more
If there were three Equations given and as many unknown Quantities If there
of different Dimenfions, mixed promifcuoufly in each Equation, they than two un
may likewife by the foregoing Method be reduced into one, affected with known quan
only one unknown Quantity; for neglecting at firſt one of the three un- tities and e-
known Quantities, two of the three Equations will be fufficient to find ventheymay
quations gi-
an Equation involving the unknown Quantity which was not attended to, bereducedto
3 N
98
OF
ELEMENTS OF
one by the
foregoing
method.
Inconveni-
ency to
and either of the two others, and the fame Operation being performed
on one of the Equations employed in the first and on the third Equation,
there will refult another Equation involving the fame unknown Quanti-
ties, hence the Queſtion is reduced to that where two Equations involv-
ing each two unknown Quantities are concerned, whence a new Equa-
tion will be deduced, affected with only one unknown Quantity.
If there were four Equations given and as many unknown Quantities
mixed promifcuoufly in each Equation, the Question will be reduced in
like Manner to three Equations and three unknown Quantities, and after-
wards to two Equations and two unknown Quantities, and at length to
one Equation and one unknown Quantity; and the fame Method is to
be purfued if there were a greater Number of Equations and unknown
Quantities.
LIV.
When the Quantity to be exterminated is of feveral Dimenſions, there
would be required a very laborious Calculus to exterminate it out of the
foregoing Equations by the foregoing Method, we fhall therefore proceed to explain
how the Analyſts have remedied this Inconveniency.
which the
method is
liable.
How it has
been remov-
ed.
2
A x" + B x” ~ ¹+C x
+Cx²+Dx¬³+Ex~4....V=0(L)
Let {4
Ax""' + B' x " ~ I
B'x" 1/ → I 1/2
+ C'x" →² + D'x-3+ Ex4....V=0(L)
be two Equations in which A, B, C, D, &c. denote any Function of
J's
to exterminate x out of thofe two Equations, it is fufficient to find a
Function of x by which the firft Equation being multiplied, and another
Function of x by which the fecond Equation being multiplied in the Sum
of the Products, the Terms including the Powers of x will be deftroyed.
+N.x
SMx" + Nx?
+P: xn
Rx
2 M'x" + N'x" ~ +P¹ x² + 2x²
+pi
let then {MN"
12 - D
I
2
2
+Qx
T
3 + R· x
4...
T
After the Multiplication and Addition of the
111#
be thoſe two Functions.
Products there will refult,
AM x
仇​子​九
​+ B M x
111 + 11 I
+CMx
A' M' x" +"' + A N x
九千​九​〜 I
+BN
MN
+ B'M' x"
x'
?
3
+ "'+A Px"
+A'N' x”! +*~¹+ C'M' ½”
+ D M x
~² + C N x
2
"+B P
x
m + H
V T
V T=0
3
VITI
"+" 3
² + A 2x 112 113:
+ 1831 — 2:
+ B'N'x™ +
→
² + D' M' x
M1 + 1/ 3°
+ AP! ~"" + /1/~
171 |
x
2
+ C N! x
"
+ 171, ~ 3
?
+ B' P' "
+ A Q x
+ 111 — 3·
1181 | 12 |
3:
Where I obferve, 1°. That as the Condition that the Terms including
the Powers of x fhould be deftroyed, furnish the Equations by Means of
which the unknown Coeficients M, N, &c. are to be determined,
m+n must be equal to mn; whence refults the following Equa
tions for determining thofe Coeficients.
SPECIOUS ARITHMETICK.
99
AMA Mò
·AN+A'N' + B M + B' M'
AP+AP+BN + B'N' + CM + C M'
AQ+AQ! + BP + B'P'+CN+CN+D M÷D'M' =。
AR+AR+B Q + B' Q + CP+CP+DN+D'N' + EM÷EM=0
Confequently, VT+V'T'0
I.
I obferve, 2º. That as the Number of Terms including the Powers of
x is m+n, the Number of Equations they furniſh when fuppofed equal
to Nothing, will be alfo m + n, and as each Term of thefe Equations in-
clude an unknown Quantity, Viz. the Coeficient of a Term of the af-
fumed Multinomials, it follows (LXXXIX. Chap. 1.) that the Number
of undetermined Coeficients n + i + n + 1 will be m + n + 1,
+i+n+I
or m' + n' + 1; confequently n = m' I and n' m
Hence
when two Equations each involving two unknown quantities are given,
to exterminate one of them, it is fufficient to multiply the firſt Equa-
tion by a Multinomial as M x™”/ + N x™! +, &c. m'˜de-`
noting the Degree of the ſecond Equation; and the fecond by a Multi-
monial as M' x + N' x − 2 &c. m denoting the Degree of the firſt
Equation: To add thoſe two Products, and to put the Coeficients of
each Power of x in this Sum equal to Nothing, the laſt Term of the
Sum will be the Equation in y.
177
I
171
LV.
t
ac Newton's
For Example, let it be propoſed to exterminate x out of the Equations Inveſtigati
A x² + B x + C = 0 and A' x² + B' x + Co, multiplying on of Sir Ia-
A x² + B x + C by M x + N, and A' x² + B' x + C = o by rule for ex-
M' x + N', and adding the Products we will have (AM + A' M') x3 terminating
+(BM÷AN + B'M' + A' N') x² + (C M + B N + C'M' + B'N') x an unknown
+NG+CNo, the Condition that the Terms in this Equation in- two dimen-
quantity of
cluding the Powers of x ſhould be deſtroyed will give AM + A'M' 0, fions in each
B M + AN + B'M' + A' N' = 0, CM+BN+CM+ B'N' = 0, the forego-
equation by
confequently NC+CNo, and as there is one Coeficient undeter- ing Method.
mined. I determine M by fuppofing it A', which gives MA,
=
AN+ANA B' — BA, BN + B'N' AC-CA'
CA', from
A B' B' BAB' A A'C' + CA' A'
which I deduce N =
N =
AAC-ACA
A B'
A B'
ABB'+BBA
·BA'
and
BA
and thoſe Values of
N and N' being fubftituted in the Equation CN+C'No there
refults ACB' B' BCA BACA C + CCAA + AAC' C
-ACAC-ABB'C' + BBA Co, that is
RULE I.
(AC-BB-2 CA') AC'+ (BC'— CB') BA' + (AB'B' + CA'A') C=0.
If it was propofed to exterminate out of the Equations Example.
**+5x-3yyo and 3 - 21 + 4 = o, I respectively ſub-
** y 0,
$
1
1·00
Inveftigati-
on of Sir Ifa-
an unknown
and of two
ELEMENTS OF
fitute in the preceding Expreffion for A, B, C; A', B'C'; thefe Quanti
ties, 1, 5,311; 3,-21, 4; and duly obferving the Signs + and -,
there arifes (4+ 10 y + 18 y y) × 4 + (20 b y³) × 15 + (4 yy
-2713) X-3yyo, or 16 + 40 y + 72 jy + 300 90 y3
+- 69 14=0.
LVI.
Let it be propoſed to exterminate x out of the Equations A 3 + B x²
ac Newton's +C x + D =
o, and Ax² + B'x + C'= o, multiplying A׳ + B x²
Rule for ex-C x + D by Mx +N and A' x² + B' x + C' by M' x² + N'x + P'
terminating and adding their Products there will refult (AM + A'M' ) x4
quantity of (BM+AN + B'M'+A'N') x3 + (CM+ BN+CM+B'N'+AP)²
three dimen (D M + CN + C′ N' + B' P') x + DN + C' Po. The Con-
tions in one, dition that the Terms of this Equation including the Powers of x ſhould
dimenfions be deftroyed, will give AM+AM=0, BM+AN+ B'M'+ A'N'= 0,
in the other CM+ BN + CM + B'N'+ A'P' 0, DM+CN+C'N'+ B'P'=0,
equation by
the forego- confequently DNC Po, and as one of the Coeficients is unde-
ing Method, termined I determine M by fuppofing it A', which gives M'A',
ANANA B' - BA', BNB'N' + APACCA,
CN CN + B' P' — DA', from thofe three laft Equations I
deduce the Values of N and P', which may readily be obtained by
Means of the Formulas of Article LXXXI. Chap. 1. by putting = N,
y = P', z = N', mA B' - BA', n= AC-CA', p = — DA',
a = A, b = 0, c = A'; d= B, f= B', e = A'; g C, b = B', k C',
ek mfb m + c b n - cep
gives
N
then x ==
N=
and
*
y
—
a ƒ b + c d b
=
ceg
= =
a e k
AA'B'C'-BA'ACAB'3+ BA'B'² + AA'B'C' — CA'A'B' + DA'3
AACA B' BBA BCAA
akngen + dep - af p + gfm - d k m
ack a b f + d b c
B'B'
ce &
A²C2—2ACA'C'+C²A-BDA+ADA'B'÷ACB'—BCA'B'—ABB'C'
Example.
A A C' A B' B' B A B’
CAA'
becomes
+BBỰC
fubftituting thofe Values of N and N' in the Equation D N + C'P' 0,
there refults ADA'B'C' -BDA'A'C' ADB'3 +BDA'B'B' + ADA'B'C'
- CDA A'B' + D D A'3 + A A C3 ACACCACACC
+CCAAC-BDA ACADA'B'C' + ACB'B'C'— BCA B' C'
ABB'C' C + B BACC = o.
—
Or A AC3 — AB B' C 2 A CA' C'C' + BACC BCA B'C'
- 2 BDA A CACB'B' C+CCA AC AD B'3 — CD A2 B'
+ 3 AD A' B' C + B D A B¹² + D D A³ = 0, that is,
RULE II.
(AC-BB-2 CA) ACC (BCCB' 2 DA') BA' C'
+
+(CC-DB'') (AB'B'+CA'A') + (3 AB'C' + BB'B' + DA'A') DA'= 0.
If it was propofed to exterminate x out of the Equations p3-xy-3x=0
and yy + xy-x+3=0, I fubftitute in the precedent Expreffion
SPECIOUS ARITHMETICK.
IOI
for A, B, C, D ; A', B' C', and x, thofe Quantities 1, x, 0, -3*;
I, x, − x x + 3, and y refpectively; and there comes out (3-xx+xx)X
(9-6xx + x4) + (→ 3x + x³ + 6 x) (→ 3x+x³) +3 x x x x x
+(9x3x3 — x3 — 3 x) X-3x= 0, then blotting out the fu-
perfluous Quantities and multiplying, you have
وا
27 18 x x + 3 x4
9 x x + x6 + 3 x4
and ordering x + 18 x4 — 45 xx+27=0.
LVII.
18 x² + 12 x4 = 0,
Let it be propofed to exterminate x out of the Equations
Inveſtigati-
ac Newton's
four dimen-
dimenfions
A x² + B x³ + Ċ x² + D ≈ + E = 0, and A' x² + B' x + C' = 0, on of Sir Ifa-
C
I multiply the first by Mx+N and the fecond by M'x3+N'x²+0'x+Q' «Rule for ex-
and adding their Products there refults, (A M A M ) x terminating
+(BM+AN+B'M'+ A'N') x++ (CM+BN+CM+ B'N'+AO)x3 an unknown
+(DM+CN+C'N'+B'O' + A'Q') x²+ (EM + DN+B''+CO)x quantity of
+ EN + CQ' o, and the Condition that all the Terms in this fions in one,
Equation including the Powers of x fhould be deftroyed, gives the and of_two
Equations A M+ A'M' = 0, B M÷AN + B' MA'N' = 0, in the other
CM+BN+CM+B'N'+A′0'=0, DM+CN+CN+B'O+AQ'= 0; equation by
EM + DN + B'Q' + C'O' = o, confequently E N + C' Q = 0, the forego-
and as one of the Coeficients is undetermined I determine M by fuppofing
it equal to A'; whence
M'=-A, ANA'N' AB'- A'B, BN + B'N' + A'0' AC' — CA',
CN+ A'Q'+ B'O' + C'N'DA, DN+BQ + C'O' = — EA'.
I deduce from thoſe four laſt Equations the Values of N and 2 by
Help of the Formulas of Art. LXXXVII, by putting x = N, y=N',
Z O', t = Q'sa A, b = A',co, d = 0, a — A B'—BA', a' —B,
b' = B', c' = A', d'o, a' AC' — A' C, a" C, b" = C', c" = B',
d" = Â', a"
d'' = A',
a"!!
c'''
A' D, a""' = D, b'' = 0,
A'' = — A' D; c"" = D, b = 0,
A' E.
=
=
C', b" C, c"
=
c"" = C', d" B',
c'll
B',
=
In this Caſe the Numerator of the Value of x is reduced to
[(a b' — a' b) c'' + (a" b — a b') c'] d'""' + [(a' b — a b') c'"' — a""' b c'] d'
Confequently the Numerator of the Value of N will be A B'^ — B A'B'³¨
− Â Д' В¹² С' + C A2 B¹2 D A3 B' A A' B¹² C' + BA½ B' C!
+ A A² C² — CA3 CA A' B2 C + E A+ + B A¹² B' C'.
And the Numerator of the Value of t being reduced to
[(a b'_a'b) c'' + (a" b——a b'') c']´a""'+[(a' b—ab') c'"—a" b' c'] a″ +[a""bc"
+ (a b'" — a" b) c'""'] a' + [— a'""' b' c +- a"" b" c'+ (a" b'—a'b'') c"]a.
Confequently, the Numerator of the Value of 2 will be
AE A' B2 BEA² B'
+ADA' B' C + D² A3 +
- 2 ACAC² + C² A¹² C —
~ B D A² C + AC B2 C —
2
CEA3 + AE A2 C DB A½ C
ADA B C C D A¹² B' + A2 C13
A D B'3 + B DA B2 + ADA'B'C'
BCA BCA BB' C² + B² A C'²
ing Method..
102
ELEMENTS OF
wherefore E N + CQ, o will be transformed into
AE B4 B EA'B'3-AE A' B2C+CE A2B'2-DE A'3 B' - AEA'B'C
+BEA'B'C' + AE A2C2—CE A3C-AEA'B'C' + BEA'B'C' + E² A¹4
-AEA B C + BEA2 B' CCEA3 C+ AE A2 C2 BD A2 C2
+ AD A' B' C¹² + D² A'3 C + ADA' B' C2 CD A2 B' C¹
+ A² C14 — 2 A CA' C3 + C2 A2 C2 AD B3 C+ BDA B2 C'
+ AD A'B'C'² BD A2 C2AC B2C2BCA B' CAB B'C'3
Or into,
+ B² A' C¹³ = o.
A2 C42 A CA'
+ AC B¹² C2
CD A B C
+ BDA B2 C +
+ A¹4
E2 A4
C3-AB B' C3+ B2A'C3 BCA'B'C22BDA2 C2
B²A'C'³ —
AD B'3 C + AEB42 AEA B'2 C+ C² A¹² C¹2
+ CE A2 B2 2 CEA3 C + 3 ADA'B' Cz
D² A¹³C + 2 AE A2 C2 + 3 BEA2 B' C-DEA³B'
BEA B'32 AEA B2 Co.
RULE III.
–
That is (AC' BB'2 CA) A C3+ (BC-C B' 2 DA') BA'C'2
+ (A B¹² + CA2) (CC2 - D B C + EB' B' 2 EAC')
+(3AB'C' + B B¹² + D A¹²) DAC + (2 AC2 + 3B B'C' — DA'B'
B42) EA2+(BB-24G) E4 B2 = 0.
† B'
Example.
ac Newton's
rule for ex-
C')
If it was propoſed to exterminate x out of the Equations
x4 tu y y 3x²
y4
140 x x + y² = 0, and x x + y x 20 x + y y = 0;
Subſtitute for A, B, C, D, E; A', B', C', reſpectively, 1, 0, y jy
140, 0, yª; I, y 20, and yy; and there will come out
(−y + 280) × 35+ (2 y 3 — 40 +260) X (260 μ4 — 40 μsj
15 y y µ4
+3j4 × 14-2 y y X (36 40 15+40014)=0; and by Multiplica-
x
tion 1600 16 20800 5 67600 y40, and by Reduction
y
4 y y 52 y + 169 =
-
0.
LVIII.
3
+3
4015)
Inveſtigati- Let it be propoſed to exterminate x out of the Equations
on of Sir Ifa- A x3 + B x² + C x + D = 0, 'A' x³ + B' x² + C'. x + D' = 0, mul-
tiplying A x3+B x² + Cx+D by M x² + N x + 0, and A' x³ + B' x²
terminating +C' x + D' by M' x² + N' x + O', and adding their Products there
an unknown will refult (A M + A' M') x5 + (B M + A N + B' M' + A' N') x¹
quantity of +(CM+ BN + AO+C' M' + B' N' + A' O') x³ + (D M + C N
Lions in each +BO+ DM + C'N' + B' Q') x² + (DN + CO + D'N' + C'O') ¿
equation by O D + O' D' = 0.
three dimen-
the forego-
ing Method.
=
The Condition that the Powers of x fhould be deſtroyed in this Equa-
tion will give AM+AM' 0, BM+AN+BM+A'N'o
CM+ BN + AO+ CM+ B' N' + A' O' = 0, D M +CN+BO
+ DM' + C'N' + B' O' — 0, DN + D'N' + CO + C'O' = 0,
confequently D 0 + D'O' = 0, and as there is one of the Coeficients in-
determined, I determine M by fuppofing it = A'; whence
—
MA, AN+AN ABBA', BN + B'N'+AO + A' O'
=AC-CA, CN+CN+BO+ B'O' AD' DA,
D N + D'N' + CO + C' O' = 0.
·
4
SPECIOUS ARITHMETICK.
103.
I deduce from the four laft Equations the Values of O and O' by the
Help of the Formulas of Art. LXXXVII. by putting
X N, y = N', z = 0, t = 0', a = A, b = A', c = 0, d = 0, a =
ABBA'; a' = B, b' = B', c' = A, d' = A, a' AC
CA;
a" C, b" — C', c'" B, d" B',. a" AD' - DA'; a"" = D,
= — —
b!!! =D', ¿'"' — C, d'"' = C',
C", a"" =0.
=
=
In this Cafe the Numerator of the Value of z will be reduced to
[(a b' — a' b) a" + (a" b — a b") a'+ (a' b" — a" b') a] d'""' +[(a b'""'—a'"'b) a'
+ (a'"' b' — a b''') a] d" + [(a""' b — a b'") a" + (a" b""" — a'" b") a] d'
confequently the Numerator of the Value of O will be,
A B C D'
A2 C3
ABAC D' ADA'B'C' + B D A¹² C + ACA C²
C² A¹² C' + A C A C¹² + A B B' C'²
12
ACBC-B²ĄC¹²
2.
12
+ BCA BCA B' C D'AD A'B'C'ACAB'D' + CD A²B'
+ADB'3 A B B¹2 D' B D A B¹² + B² A' B' D' + AD A¨² D'
D² A3 + AD A'² D' -- A CA' B' D' — A D 'A' B' C
A² A' D'2
-B C A¹² D' + B D A² G'.
And the Numerator of the Value of t being reduced to
[ (a'ba b') c'!! + (a b!!! -
bill a"" b) c'] a" + [(a""_b— a b''') c''
+ (a b" — a" b) c'''] a' + [(a' b'"' — a'"' b').c' + (a'll b" -
+ a" b' — a' b") c'"] a.
-
Confequently the Numerator of the Value of O' will he
A² C B' D' B C D A2 + A C D A B'
A B C A' D'
A² D A' D'
A² DA'D' +A D² A¹² +
B D C A² + A B C A' D'
+ C3 A¹² + A B² B' D'
all bil ) cl
+ ₤3 D'=
A B C D¹
ABD A' C'
+ A2 C C12
ABD B2
A C² A' C
AC² AC
B3 A' D' +
+ A² D B' CA2 C B' D'ABDA C+ A B C A D
A B C B'C' B C² A'B' + B² CA' C'.
2
·
B² D. A'B'
+ A C² B'2
Wherefore DO †· D' O' = 0, will be transformed into
A² D B' C D' A B D A C D' A D² A' B' C ' + B D² A¹² Cl
+ACD ACA² DC3—C² DA¹² C+ ACD A C¹²+ABDB'C'2
- ACD B2 CB2 DA CBC DA B'C' + A² DB' C' D'
B² · A²´D
-ADA'B' CACDA'B' D'+CD2A B' + AD2 B3 ABD B'2D'
-B-D2A B'2 + B2 DA B' D' + AD2 A12 D' A2 DAD2 D3A3
D² B²
+·A·Ð²A D'ACDA'B'DA DA B' C-BCDAD! + BD² A¹²C'
B'C' DBD²
+ A B C A D'. - A C B D – B C D A D +ACD A B D
A² D'2
² A'
+- A³ D¹³ A² DA' D¹² — A² DA' D'² + AD2A2D+ABDA CD
D/3
ABCD2 B C D A ¹ D' + A B C A D'² + ƪ С C²D—AC²Ãˆ˜C'D'
A C² A CD + C³ A² D² + A B² B'D:2 · ABD B¹² D' — B³ A D'2
+ B² D A B D + A² D·B'ˆ'C' 'D' —¨´A² C B' D'2
7
',
12
+ AB CA' D'² + AG² B² D'
+ B² CA' C' D'o, or into
ABCB C D'
A B·D˜¯Â· CD'
- B C² A B' D'
A²D C¹³ + A B D B'C'² + 2 A C DA C² + ²C C²² D' — A B C B'C' D'
2 A C² A' CD'ABDA C'D'
B2 DA C2BCD A B C
B²
104
ELEMENTS OF
{
2 BD2 A2C + A³D¹³ — AB C'D'2 2 ACB' D'2
13
2
3
12
3 A¹DAD¹²
A CD B¹² C' + AD²B3 + AC2B2D2 ABD BD C²DA ¹C'
+CD2 A!² B' + C3 A¹²D'— 2 BĊDA¹²D' — 3 AD ABC' — BD² A'B'²
- D3A3+3ADAD +- 3 ABC A'D + B2CA'C'D'_BC A B D
— BCDA 2Ď — B3A'D'2+2 B2DA'B'D'+AB² B'D'² + 3 A2DB'C'D'
+ ACD A' B'D' o. That is,
RULE IV.
12
(AC' - BB' - 2 CA') X (AD C2— ACC'D') + (A D' + BC'
- CB' - 2 DA') BDA'C' + (~AD'+BC-2 CB'+3DA') X AAD¹²
×
+ (CDC' — D D B C C D' + 2 B D D') X (A B'² + CA²)
+ (3 A B'C' + B B'B' + D A2 — 3 A A' D') D DA' + (— 3A D'
- BC'+CBD A') BCA' D' + (B D' - 2 DB') X B B A' D'
+ ( − B B D — 3 A DC' — CDA') X A B D 0.
× ≤
LIX.
Let the Equations Ax4+ B x³ + C x² + D x + E = o, and
'A' x² + B' x³ + C' x² + D'x + E o, be propofed; the refulting
Equation after x is exterminated will be found to be,
RULE V.
+3(BA-AB') (E B'-BE') (EC-CE
2
(EA'—A E')^—3(BA'—A B') (E B'—BE)-3(BA-AB') (DB'—BD') (ED'—DI
-2 (CA' —A C') (E C'—CE!) ( +2(DA'—AD')²(E C'—CE')
(DA'—AD')(E B'—BE) + (CA—A C') (E B'—B E')²
(DA'—AD')(DC'—CD')
(DA' —AD') (DC-CD'))+2(CA'—A C') (DA'—AD') (ED'—DI
(C A'—A·C') (CB'—B C') (ED'—DI
+2(BA'—AB')
X
(A' EAE')2
X
(EA'—AE')
(DA'—AD') (E B'—BE') + (CA'—AC')²(EC'—CE')
- (DA-AD¹³ (BA-AB') (DB-BD')-2(BA-AB') (DA-AD') (EC-CL
(DA'—AD')³—(BA'—AB')(DB'—BD')²
+2(BA'—AB')
(DC'—CD')
+ (E B'—BE').
+ (CA-AC')
(BA'—AB')
+ (BA'—AB')
(DA'—AD') (DC-CD')
(CA' — AC₁) 2
-AC¹)
(CA'—AC¹)²
(DA-AD') (DB-BD')
(CB-BC) (E B'-BE')
(CB'—BC') (DC'—CD') j
X
(ED'—DE')
(BA-AB') (CB-BC') (EC'-CI
+2(BA'-AB') (EB-BE') (DB'—BI
(CA'—AC') (DA'—AD') (E B'—Bİ
X
(EC' — CE')
(BA-AB') (EB'—B E') ·
- (BA'—AB') (DC'—CD') }
X
(E B'—BE')²,
SPECIOUS ARITHMETICK.
105
How an un-
known quan
fions not ex-
ceeding four
This Rule will ferve to exterminate an unknown Quantity of a Number
of Dimenſions in each Equation not exceeding four; for if the unknown
Quantity is only of three Dimenfions in the Equation L', we have no
more to do than to put A' o, that is, to omit all the Terms in the tity of a num
refulting Equation among whofe Factors A' is included, and then the ber of dimen
whole Equation will be divifible by A. And if the unknown Quantity
only of two Dimenfions in the Equation L', we are to omit alſo all the in each equa
Terms among whofe Factors B' is included; and the refulting Equation tion may be
will be again divifible by A, &c. But if the unknown Quantity rifes only ed by the
to three Dimenfions in the Equation L, we are to omit all the Terms in- preceding
cluding E, and the whole Equation will be divifible by E'. And if the un- Rule.
known Quantity rifes only to two Dimenfions in the Equation L, we
are alfo to omit all the Terms which include D, and to divide a fecond
Time the Equation by E', and ſo on.
exterminat-
a y² = 0, by Example.
For Example, to exterminate x out of thoſe two Equations
x3. 2 a x² + 4 ay x y3o, and a x² + y² x
the foregoing Rule; I put E' =
O, A = 0; A = 1, B
2 a, C
a y², D' = y²,
4 a y, D = —
C'
12
a, B'
E-
y³, and E = o.
Whence the Terms in the Formula, among whoſe Factors A', B', or
E are included, will vaniſh, and the remaining Terms will be diviſible
by E' and A2; confequently the Formula will be reduced to
A² E/3 2 ACC E/2 A B D E + 3 A D D' C'E' +
A CD¹² E' + B² C'E¹2 2 B D C '2 E
A D D¹³ + D² C¹3 — C D
C¹² D' + B D C ' D¹² + C2 C¹2 E' B C C D E =0.
In which fubftituting for E', D', C'; A, B, C, D, their Values,
— a³ 36 —— 8 a4 35 + 2 a³ p6 + a² y? — 4 a² y² + 4 45 y + + 4 + 35
+ y⁹ + a³ y6 + 4 a³ y6 + 2 a² y7 16 a5 №4 — 8 a4 y5
which after the Terms are reduced and ordered becomes,
12 a4y5———-—- 12 a5 p4 = 0 = y5 ÷ a² µ³ + 6 a³ y²
7
39 + a² y² + 6 a³ y6
12 at y
12 45.
3
7
- y
LX.
4
The foregoing Method for exterminating an unknown Quantity of How the de
feveral Dimenſions in two Equations, each involving two unknown gree of the
Quantities, alfo ferves to determine the Degree of the refulting Equation. equation is
refulting
Let the higheft Dimenfions of y in A be p, of y in B, p+1, of y in determined.
C, p + 2, &c. In like Manner let the higheſt Dimenfion of y in A'
be p', of y in B' be p' + 1, of y in C', p' + 2, &c. Then the higheſt
Dimensions of y in the Coeficients of each of the unknown Quantities,
included in the Equations arifing from the Suppofition, that the Terms
affected by the Powers of x in the Sum of the two Products of the pro-
pofed Equations into the undetermined Multinomials fhould be deſtroyed,
form fo many arithmetical Progreffions, (it is thus, a Series of Quantities,
that increase or decreaſe by the fame conftant Difference, is called) having
the fame common Difference.
30
106
ELEMENTS OF
G
Limit which
**
*
****
*
* A
*
Р
р
D'
p'
pti. p'ti
p'.
Fix
pti p'ti
pti p'ti p+2 p²+2
pta p'ta p+3 p+3
pti pti p+2 p'+2 p+3 p'+3 p+4 +4
&c. &c. &c. &c. &c. &c. &c. &c. &c.
As will appear by the Inſpection of thofe Equations, (Art. LIV.)
P
&c.
p'
I
It will alfo appear by the Inſpection of the Formulas of Art. LXXXVII.
Chap. 1. that each of the Terms of the refulting Equation after the unknown
Quantities are exterminated, confifts of the fame Number m+n+1
of the Coeficients of thofe unknown Quantities, and that two Coeficients-
of the fame unknown Quantity do not enter the fame Term: Whence
it is eaſy to conclude, that the higheſt Dimenſion of y in the refulting Equa-
tion will be expreffed by the Sumof m+n+1 Terms of m+n+1 the
foregoing Progreffions, of which Terms two are not in the fame Column
or upon the fame Line. Now upon Examination it will be found that
m+n+1 Terms of m+n+1 of thofe Progreffions taken in this Man-
ner will always make the fame Sum. For Example, If we take five Terms
of five of thoſe Progreffions in fuch a Manner that two of thoſe Terms
are not contained in the fame Column, or are upon the fame Line, their
Sum will be found to be 3 p + 2p+6; and the general Expreſſion of
this Sum will be S + (m+ n + 1) × ( " + n), S denoting the
Sum of the Terms which compofe the firſt Line, if the Progreffions were
continued fo far; that is, of p, p 1, p 2, p 3, continued to
+ Terms, and of p', p-1, p' — 2, p' — 3, &c. continued to n'+
nti
Terms; wherefore S (2 p—n)
confequently, if the higheſt Dimenfion
be expreffed by G, then
= (m+n+1) ( m² + 1 ) + (2 p + n)
2
n
2
m
2
-)
+ (2 p' — 1') x ( 1 + 1))
2
;
of y in the refulting Equation.
.ས
n + i
+ (2 p' + n)
n'
n² + 1;
2
2.
I
in which fubftituting for n and n' their Values 'I and m — 1
there refults G = mm² + pm + p' m.
LXT.
It is eafy to perceive, that the Degree of the refulting Equation when
an unknown Quantity of feveral Dimenſions in each Equation is exter-
the degree minated, can never exceed m m' + pm + p' m, p and p' expreffing the
of the refult Exceſs of the greateſt Sum of the Exponents of the unknown Quanti-
ing equation ties in any one Term above the greatest Exponent of the Quantity to
ceed. be exterminated in each Equation; for Example, if there were given
the two Equations a3x5 y 2 24 32 x3 + 38 x — q² = 0, and
can not ex-
-
SPECIOUS ARITHMETICK.
107
=
3,
a³x³-3 a³x y²+ y5x—6— a² = 0; the greateft Sum of the Exponents
of x and y in any one Term of the firft Equation is 9, which fubtracted
from 5 the greateſt Exponent of x gives p 4; and the greateſt Sum
of the Exponents of and in any one Term of the fecond Equation
which is 6. fubtracted from 3 the greateſt Exponent of x gives p
now x being exterminated out of the Equations a 4x5 2·at p6 x³
+ y° x
a⁹= 0, a³ j³ x3 3 a³ 15 x + 15 x - 36 a4 = 0, the
Degree of the refulting Equation by the foregoing Art. will not exceed
5 X 3 + 4×3 +5 X 3, that is, the 42d. Degree, much lefs then
will the Equation in y deduced from the two propofed Equations afcend
to the 42d. Degree.
Let
8
A x
A x
172/
LXII.
171
171.
+B x
+ B' x
+ B" x
m
I
+Cx
2
7721
I
+ a² x
772/
24
+o
+ D x
+ D' x
##
3
3 +... V
+
132// I
3 +
=0
Ax 771 ||
772/1-2
+C" x +D"
+...
of three e-
be three Equations; in which A, B, C, &c. A', B', C', &c. A", B", C", How an un-
&c. denote Functions of two unknown Quantities y and z, and known knownquan
tity of leve-
Quantities, and let the Dimenſions of A, B, C, &c. be expreffed by ral dimenfi-
p, p + 1, p + 2, &c. thoſe of A', B', C', &c. by 'p', p' + 1, p' + 2, ons is exter-
&c. and thofe of A", B", "C", "&c. by p", p" + I, p" + 2, &c. to ob- minated out
tain the two Equations which refult when x is exterminated, it fuffices quations by
to find three Functions of x as fimple as poffible, by which the three the forego-
Equations being multiplied refpectively, in the Sum of the three Pro- ing method.
ducts the Terms affected by the Powers of x will be deftroyed, and
three other Functions of x different from the firft, as fimple alfo as pof-
fible endued with the fame Property.
+"!
Let
SM x"
+ N *"
#
+ P x +2
"
ช
I
I
+P" +"!!!
3 +
T
T'
M' "' + N' P 1 - 2 + 2, +"1 - 3 +
x"
·M" x + N
+ 2"
11-711-3+ ... T"
24
be the three firſt Functions, performing the Multiplications, and adding
the three Products there will refult,
' M x
M'x'
m) + " + A N≈
ma
712 + 11
+ D⋅ Mx
172 - #
+ B M x
m In - I
+ B N
+ C M x
x
m In
2
111 - #
2
+ C N x
I
+ C'M' x
#11/111 - 2
+ B P≈
+ A Q≈
{
1"M" x”"+""+ B' M' x''+ "/ — I
+ A' N' x™+ #1
+ B"M"""+ ||||~—-
+A"N"
¤//— I
+ AP x
777
/ + #11 - 2
112 + # 3
#+#
3 ... V T
VT =
3
3
1/2 + "
"It 3
-
+ B' M' x •111 + 1 - 2 + D'M' x”+”! ****
+ AP' x
+C"M" x™"† 1 ||~~. 2
+ B" N'' +”¹¹¡| +# //- 2
if All P!! x|| + #1||~ 2
x
Xx
3
3
3
3
+ C'N' x™+ #1] →
+ ·B' P'x"; † nj –
+ A' Q' x™! + "'¡ ~
+D"M"x
+C"N"x”/+#/- 3
+ B" P" ~"+#}}~ 3
+ A! Q !! x # ||| + # // mm
|| | || | || ||- 3
**
VIT
..TH
108
ELEMENTS OF
n' = m
n"
2 and n
=
LXIII.
x
+n"
Let us ſuppoſe first mn m' + n' and m+n>m"+n", or at
furtheft equal. It is obvious that in Order to deſtroy all the Terms af-
fected with x, the Number of undetermined Coeficients must be m+n+1,
wherefore m + n + 1 n + I + n + 1 + n" +1, wherefore
m' — n' 2. This being fuppofed, put-
ting the Coeficients of each Power of equal to Nothing, there will
refult a Series of Equations; in which it is eaſy to perceive, that the
higheſt Dimenſions of the fame unknown Quantity, M or N &c. form an
arithmetical Progreffion, and that all thofe Progreffions have the fame
common Difference; whence by a fimilar Reafoning to that employed
in Article LX. it will appear, that the Number G expreffing the
higheſt Dimenſion of the Equation in y and z refulting from the Combi-
m+ n
十几
​nation of m+n+1 Equations, will be GS+ (m + n + 1)
S
expreffing the Sum of the Terms which compofe the firft Line if the
Progreffions were continued fo far. Now it is eaſy to perceive
1º. That the Numbers which denote the Dimenſions which the Co-
eficients M, N, P, &c. fhould have in the firſt Equation if they were
found there, are p, p − 1, p — 2, p 3, &c. the Number of Terms
being n + 1, whoſe Sum is expreffed by (2 p—n) × (" ±¹).
X
2
2
2º. That the Numbers which in the fame Suppofition denote the Di-
menfions of M', N', P', &c. are p', p' — I, p'-2, &c. continued to
a Number of Terms
n' + I, whofe Sum is expreffed by
"'+1). 3°. In Order to find the Numbers which
(~2 p' — π') x
("
2
denote in the firſt Equation the Dimenſions of M", N", P", &c. if
thoſe Quantities were found there, I obferve, that as the Place of the
Term A" M" x "+" in the Order of the Powers of x is denoted by,
m + n m"
n" + 1; ſo will its Place in the Order of the Equati-
ons be denoted by the fame Number; confequently, to difcover what
fhould be the Dimenſion of the Coeficient M" in the firſt Equation,
we muſt retrograde m+n-m"n" Equations; confequently, the
Number which denotes this Dimenſion will be pl - 771 n + m" + n";
and confequently, the Numbers which denote in the first Equation the
Dimenſions of M", N", P", &c. if thoſe Quantities were found there,
are p" n+m" + n", pl 171 n+m" + n" — I, p".
+m" + 12" — 2, &c. continued to a Number of Terms
n"
whoſe Sum will be (2 p″ — 2 m — 2 n + 2 m" + 2 7") x (·
771
Conſequently, G= (m+n+ 1 )
(
m n
2
n2")
7) (
" + " ) + (2 p − n)
-)+(2p=
n+ 2 m"+n')(
+ (2 p' — 18') ( "' + 1 ) + (2 p" — 2 m — 2
— ») (- -) + -—= 113
77
12-
72"
"+ 1,
n" + i
-).
2
n + 1
2
n'" + i
2
-),
SPECIOUS ARITHMETICK.
rog
Or by fubftituting for n and n' their Values found above,
G = m m' + p m' + p'm
in m' + m" р - p' + p" + I
-(p + p' - p' + m + m' — m" — n"— 2) n". A Quantity which
(+かー​が​十​m+m
is undetermined fince n" has not been determined.
LXIV.
Hence the Equation in y and z may be obtained different Ways, but
and confe-
it is manifeft that that which gives for G the leaft Quantity is only to be How n" is
chofen; and it is from this Condition that n" is to be determined. To determined
determine therefore this extreme Value of G we are to find the limitting quently G.
Ratio of the Cotemporary Increments of the two undetermined Quan-
tities G and n" in the Equation G = m m' + pm' + p' m 771 m'+m"
-p-p' + p" + 1 = (p + p' — p" + m + m'm" — n" — 2) n",
and to put this Ratio equal to Nothing.
p17"
p'n"
12
p — p' + p + i
"
"
771 12
Let d G and d n" denote the cotemporary Increments of G and n",
then G+dG= m m'+- p m' + p' m 171 m' + m"
m dn
p d n"
p'd n" + p" π" + p" d n"
12"
'm'n'
m' d n" + m" n'"+m" dn" + n' n"+ 2 n" d n'" + d n'd n" + 2 13
+ 2 d n"; and fince Gm m' + p m' + p' m m m' + m"
+p" + 1-pn" — p' n" + p" n'mn'
A n"
·
p' d n" + p" d n'
d G =
+ 2nd n" + d n" dn" + 2 d·n"..
d G
And
d n"
wherefore,
p- p' + p"
p' + pil
m
m
n"
m' n" + m" n"+n
m d n'' —
-
11 2 n".
2n
m' d n" + m' d n
m' + m" + 2 n" + d n'" + 2 şi
m' + m" + 2 n" + d n" + 2, ex-
preffes the Ratio of any cotemporary Increments of G and n", and this
Ratio is always greater than -p-p' + p" -m -
but d n' decreaſing, this Quantity will alfo decreaſe,
d n" as fmall as we pleafe, we may make — þ -
+m" + 2 n" + d n' + 2 approach as near as we
+p" 171 m' + m² + 2 n" + 2; wherefore,
m² + in" + 2 "+2 is the Limit of -P.
+ m" + 2 12" + d n" + 2; that is, of the Ratio
+m" n"
ni+m" + 2 n'"+2;
and as we may take
ľ + p"
pleafe to
p- p' + p
p' +
771
771'
が
​771
d G
d n
;
which gives
m"
p"
m+m' + m"+p+p+p"
2
when made equal
to Nothing. But it will not always be poffible to employ this Value,
1º. Becauſe " fhould be a pofitive Quantity. 2º. Becauſe
m+m' + m" + pt ptp"
cannot always be reduced to an Integer;
3º. Becauſe m+n fhould be greater than "+n", or at least equal to
it. 4. Becauſe m' being ſuppoſed lefs than m orm, it follows, that
2
2 thould be greater than n", or at leaſt = n". We fhall therefore
}
110
OF
ELEMENTS
affume "=
!
Obfervati-
ons which
n, a', n".
w + m² + m" te+ p + p" ta
2
m'i · p" — I
fuppofing, a to be the leaſt Quantity that can anfwer thofe Conditions.
This Value of n" being fubftituted in G there will refult,
`'G = m m' + m m'! + m p' + m p' + m' m" + m' p + ni' p" + m" þ
2
m"
2
+m" p² + pp" + p p + = - (^+x+x+~+~+~)"
p'
(으
​LXV.
2
Though there cannot be found a Value for a which will anſwer thoſe
four Conditions, we are not to conclude, that from the Combination of
the three Equations a more fimple Equation cannot be derived, than if
they were combined two by two; for we are to obſerve, that thoſe
ferve to find Conditions arife from the Suppofition, that m+n= m' + n', and
the values of m+ n > m" + n" &c. now there was no Reafon why we fhould fuppofe
m + n = m'n', rather than m+n=m"+n" or m' + n' = m'"+n"".
It will therefore be proper to make three different Tryals. And if they
all three give pofitive Values for n, n', n", (as ſmall a Quantity as pof-
fible being put for a) we are to employ the two Reſults which give the
leaſt Value for G. and if there refult pofitive Values for n, n', n', only
two different Ways, they are to be employed, provided they give a lefs
Value for G than would refult from the Combination of the Equations
two by two, and if there can be found poſitive Values for n, n', n", but
one Way, we are to conclude, that the unknown Quantity is once to
be exterminated by combining only two of the three Equations toge-
ther. Finally, if there be no Potlibility of obtaining for n, n', n', pofi-
tive Values without rendering the Value of G greater than it would be
if the Equations were combined two by two, we are to have Recourſe
to this laft Method, but this laft Cafe will ſcarcely ever happen, if it
happens at all: For Example, it can never occur when pp'p"
one Equation at leaſt will be found more fimple than by combining the
Equations two by two, except when fome of the Quantities m, m', m",
are equal to 1.
+
LXVI.
= 0,
The foregoing Principles are fufficient for finding in all Cafes the two
Equations in y and z. We fhall now proceed to illuſtrate them by fome
Examples.
1°. Let p = p'= po, and m=m=m", then " 3 m — a
771 ---- I
2
a
2
mta-
2
九十
​ta
α
2
π =
12 =
2
2
2
If m is even, then a may be put o, and the Value of G will be found
2
m²; confequently, lefs than would refult from the Combination of
the Equations two by two, we may alfo put a 2, except when
m2, and then G=
3x² + 1
4
SPECIOUS ARITHMETICK.
III
3 m² + 1
4
which
If m is odd, put a 1, and there will refult G =
Value is leſs than would refult from the Combination of the Equations
two by two; now as there is no Reaſon why the equal Values of n and
n' ſhould be employed with Refpect to two of three Equations, rather
than with Reſpect of the two others, we are to make Ufe of them both
Ways, and there will refult two Equations in y and z, each of a Degree
expreffed by 3 m² + 1
4
77
2
2
>
Rules deriv-
ed from the
principles
nating an
Hence, if there are three Equations given each of a Degree in
which the higheſt Dimenfion of x is expreffed by m, if m be an even
Number we are to multiply each Equation by an undetermined Multino- precedent
mial of a Degree expreffed by
and by Means of the Equa- for ext
tions refulting from the Suppofition that the Powers of x in the Sum of unknown
the three Products fhould be deſtroyed, an Equation in y and z will be quantity of
obtained of a Degree expreffed by m². To obtain another Equation, menfions in
we are to multiply two of the three propofed Equations by a Multino- three equa--
mial of a Degree expreffed by and the third by a Multinomial of
171
P
772
2
4
>
a Degree expreſſed by
three Products the Terms including the Powers of
ſult an Equation in y and z of a Degree expreffed by
But if m be an odd Number, we are to multiply the
Equation by a Multinomial of a Degree expreffed by
and by putting in the Sum of the
171
177
o, there will re-
m² + 1.
first and fecond
I
,
and the
2
3
3 we are to
2
third by a Multinomial of a Degree expreffed by
multiply alfo the first and third Equations, each by a Multinomial of a
Degree expreffed by
Degree expreffed by
tinomial of a Degree
Degree
2
༣
and the fecond by a Multinomial of a
m
I
2
771
3
>
or elſe the ſecond and third by a Mul-
2
#IL
1
and the first by a Multinomial of a
a.
2
and there will refult two Equations in y.and s, each
of a Degree expreffed by
3 m² + 1
4
LIVIL
From whence we may perceive, that by combining the Equations wo
by two, there will refult an Equation in y and z of higher Dimenfions
feveral di-
tions.
1
112
ELEMENTS OF
than when the three Equations are combined together; for by the first
Inconveni- Method the refulting Equation will rife in general to a Number of Di-
menfions expreſſed by m², and by the fecond to 32 m² ( 3 m² + 1)² when
ency to
which the
method of
exterminat-
ing an un-
known quan
tity by com-
paring the
equations
-)
2
I
m is an even Number, and to 3 m² + 1)² Dimenſions, when m is
odd.
171
2
(3
3.
4
7
This Difference is rendered more fenfible by the following Table.
4... 5... 6...
16
= 16...81 ... 256... 625... 1296... 2401
G 12 • 49... 156...361... 756... 1369
By ft. Method, G
two by two By 2d. Method, (G
is liable.
in
•
n'
LXVIII.
2º. Let m = 6, m' = 5, mn" = 3, p = 4, p' = 5, p
:refult n" =
4
2
a
12 + a
n' =
· 12 =
2
19, there will
10 t a
2
The leaſt Value that can be given to a is a = — 4, and then G=83. But if
we invert the Order of the Equations, and write down m = 6, m'
=
.m" = 5, p = 4,. p'
'11' =
Q
2
given to a is a
18, and then G
3,
the leaſt Value that can be
=
=
19, p" 5, there will refult n"=
20
2
12
a
18.
n =
2
104.
4, there will refult n"
n"=
20
Q
2
a
n' =
14
a
18
2 n
2
the leaſt Value that can be gi-
2
2
Inverting again the Order of the Equations, and writing down m=5, m'=3,
m" = 6, p = 5, p' = 19, p"
ven to a is ſtill a = 18, and then G will be found = 85; but this
Combination cannot take Place becauſe it gives m +n<m"+n"&c. and
if the Equations be combined two by two, there will be found G = 80,
G 144, and G125. Wherefore, in this Cafe, one of the Equa-
tions in x and y is found by combining the Equations together, and the
other by combining two of them together; we are to multiply the
Equations indicated by m = 6, m' = 5, m" = 3; I fay, multiply thofe
Equations by the Multinomials indicated by n = 3, n = 4, n"=0, and
the two Equations to be combined to obtain the fecond Equation in
y and z, are thoſe indicated by m = 6, m' =
LXIX.
3º. Let m = 7,
7, m' — ·6, m" = 5, p
n
2
a
€
5•
***
3, p'
4つ​が
​4, p"
a
I
n' =
3
72
,
2
refult n"=
Value therefore that can be given to a is a 3;
cauſe it gives m + n <m" n'; let then a = 5,
52; changing the Order of the Equations,
a
2
2, there will
the leaft
but it is uſeleſs, be-
and G will be found
and putting m7,
m' = 5, m'' = 6, p = 3, p' = 2, p!' = 4, there will refult"
·m'
་ང
2
a
SPECIOUS ARITHMETICK.
113
# = 5 +
2
a
>
72 =
a
+ª, the leaſt Value that can be given
2
to a is a = — 1, but it is uſeleſs, becauſe it gives m+n<m" + n"..
Let a = 1 there will refult G = 52, as it ſhould be, fince the Multi-
plicators of the Equations will be the fame in this Cafe as in the foregoing.
Let then a 3 there will refult G54, changing again the Order of
the Equations, and writing down m = 6, m' = 5, m" = 7, p = 4,
3 + a
p' = 2;, p'l 3; there will refult n" —
n
2
; the Suppofitions of a
5
a
n' =
2
2
I, a = I cannot take
Place. Let then a 3, then G = 52, which give the fame Multi-
plicators as were found above, and for the fame Equations. If a be put
equal to 5, then G = 56, combining the Equations two by two, there
refults G 88,
88, G = 64, G
64, G62. Wherefore, the two Equations in
y and z will be found by combining the three Equations together, the
one being of 52 Dimenſions, and the other of 54.
LXX.
If feur Equations are given involving four unknown Quantities, to ex-
terminate one of them as x ; let m, m', m", m'"', denote the Exponents of x
in the four Equations, let the Dimenſions of the Coeficients of the fe- How an un--
knownquan
veral Powers of x in the firft Equation be expreffed by p, p +1, +2, tity of feve-
p+ 3, &c. in the fecond by p', p' + 1, p² + 2; p + 3, &c. in the
third by p", p" + 1, p" + 2, &c. and in the fourth by p'", p'"
p' + 2, p'"' + 3, &c.
+ I,
ral dimenfi-
onsisted out
ons is exter-
of four equa
tions by the
Let n, n', n'', n'"', denote the Exponents of the undetermined Multi-
nomials by which the Equations being multiplied, in the Sum of the four foregoing
Products the Powers of x will be deſtroyed.
By a fimilar Reaſoning to that employed above, it will appear, that
m+n fhould be put equal to m' + n', and m + n+1=n+i+n+I
+n" + i + n'" + 1; from whence is deduced n m'
and n' = m — n'
nil
n"
72111
3
3; alſo m + n ſhould be >m" + n" or at
"", or at furtheſt equal.
furtheft equal, and m + n > m'" +
It will appear likewife that S =
+ (2 px) (+) + (2 p" - 2 m
(2 p
1 + 1
2
-)
n) (-"
−2
+ 2
" + ") (~" + i)
method.
+(2 p'"'— 2 m − 2 n + 2 m'"' + n'"')
(
"'" + 1-');
n
I
Conſequently, G = (m + n + 1 ) ( " + " ) + (2 p − ") ( " + 1 )
2
2
+ (2 p' — N') ( ~ + 1) + (2 p"-2 m2n+2"+") (++)
3 P..
114
ELEMENTS OF
n""
+ (2.`p!" —— 2 m — 2 n + 2 m'" + "'"') ('2" + 1),
or fubftituting for n and n' their Values,
G =
m m' + p m² + p' in
+ "'" + p"" + 3 — 12" 12"
2
2 771 2ß 2 m'
2 p' + m" + p"
(m+m'—m"+p+p'—p".—n"n""—3)n"
n"
12'11
3) n'".
(111 + 111' m"" +p+p' - p'll
To find the Values of n" and n" which fubftituted in the preceding
Expreffion of G will render it the leaft poffible, we are to find, 1°. The
limiting Ratio of the cotemporary Increments of G and n", confidered
as the only Quantities undetermined in the preceding Equation, and to
put this Limit equal to Nothing; then to find the limiting Ratio of the
cotemporary Increments of G and n"", fuppofed to be the only Quan-
tities undetermined in the fame Equation, and to put this Limit alfo
equal to Nothing. Theſe Operations being performed you will find,
m+m²+m"+m"+p+p+p" + p""
-12"
3
in"
m + m² + m" + m"" te té' + p" + p'"'
3
p"
m""
I, and
· p'. I,
Mill =
but becauſe n, n', n", n'"', ſhould be poſitive integer Numbers, and an-
fwer the Conditions mentioned above.
We muſt put n"—_"+m'+m"+m'"'+p+p'+p"+p—a
General ex- and n'''
preffion of
3
m+ m'+ m"+m"+p+ p'+ p" + p"--b
3
— m''—p"'—I,
I,
In pill
the degree of a and b expreffing the leaft Numbers that can anſwer thofe Conditions.
Subftituting in G thofe Values of n" and "", there will refult
thereſulting
equations af
known quan
+ m p" + m p'"' + m' m"
+ m' p'""' + m" "m""""
Þ + m"" p' + m'"p" tp pil
ter an un- Gm m' + m m" + m_m" + m p'
+ m'm" + m' m'" + m' p + m² p"
+ m" p + m" p' + m" pl" + m'"' p + m'
tp p''' + p' p' tp' p'" + pll p'll
tity of feve-
ral dimenſi-
ons is exter-
minated out
of four equa
tions.
n'"l
+
a² +ab+b²
Let p
12=
n
772
9
Ъ
3
m+m'+m"+m" +p+p+p" + p.) 2
3
(
LXXI.
3
p" =p"" =0, m = m' = m'
m", then
3
m
a
n"
3, n' =
m + a + b = 3,
3
3
m + a + b - 3.
3
It is eaſy to perceive firft, that whatever Values poſitive or negative
are given to a and b, if m be leſs than 3, it will not be poffible to ren-
der n, n', n'', n'", at the fame Time pofitive. In effect if m = 2, as in
this Cafe, x² and x are the only Quantities to be exterminated, the
SPECIOUS ARITHMETICK.
115
Equations are to be combined three by three two different Ways, and the
third Equation will be obtained by combining two of them together..
If m
m3, then by putting a = 0, and b = 0, pofitive Values will
be obtained for n, n', n', n'"', confequently the four Equations may be
combined together; the two other Equations without x will be obtained
by combining the Equations three by three, two different Ways.
But when m is greater than 3, there are three Cafes to be confidered,
either m is exactly divifible by 3, or after the Diviſion there remains 1,
or there remains 2.
Rules deriv-
FIRST CASE. To obtain in this Cafe the firft Equation without x, ed from the
put a = 0, b = 0; to obtain the fecond, put a 3, b = 0; and to preceding
obtain the third, put ao, and 63. The correfponding Values of principles
G will be G =
m², G 3 m² + 1, G = 3 m² 1. From whence for extermi-
+
it is eafy to conclude that the Degree of the refulting Equation after all known quan
the unknown Quantities are exterminated, will be expreffed by
( m² + 3 m²) (3 m² + 3 m² + 1).
2.
3.
(}
2
3.
nating an un
tity of leve-
ral dimenfi-
ons in four
1, equations..
I
SECOND CASE. To obtain the firft Equation without x, put a
b=1, and becauſe by changing the Order of the Equations, there will
refult a different Factor for the fame Equation, the Suppofition of a 1
and balone will furnish the three Factors to be applied to each-
Equation to obtain the three Equations without x. For Example, if
m4 then n"" =0, n'' = 0, r
0, n = n
= 1, n = 1, and m, m', &c. being
all equal, it is indifferent to which of the four Equations the Values of
nor of n' are referred, confequently there may be formed three Combi-
nations, each of which will furniſh a different Equation. The Value of G
will be the fame in each Cafe and 2m², from whence it is eaſy
to conclude that the Degree of the refulting Equation after all the un-
3:
known Quantities are exterminated will be expreffed by (m + m² + 1)².
3
THIRD CASE. The three Equations without will be obtained by put--
ting a=2, b = 2, and by applying the Factors three different Ways to
the propofed Equations, each of which will rife to a Degree expreffed by
2 m² +4, from whence it is eafy to conclude that the Degree of the
m4 + 4 m² + 4 m² + 4 m² + z
3
3
3
refulting Equation will be
m!!!
m'
(
17).
Let now p2, p′ = 0, p" — 1, p"" = 1, m = 7, m² = 6, m" = 5,-
12 =
m!!.
4; we will have n!"
a+b-4. Let b =
3
8
b
n!!
3
10, n² = 1, n = 0, and G
12'
3
a+b=1
3.
1 and a = 5, we will have n'"
=
3,
=
26. If we put a
— 1, b — 5,
.1.16
ELEMENTS OF
known quan
n"
there will refult n'" = 1, n" = 2, n' = 1, no,
= 0, and G = 26. If
we put b = 2, a 2, then n"" = 2, n'' = 1, n' = 1, n = o, and
G25. Now whatever Alteration is made in the Order of the Equa-
tions, there will refult the fame Values for G, if not greater; it remains
therefore to examine whether more fimple Values may not be obtained
for G, by combining the Equations three by three, or two by two. The
Calculation being performed, it will appear that the Equations indicated
·by m' = 6, m" = 5, m"" = 4, p' = 0,
o, p" = 1, p''! =1, will give
G24, and thoſe indicated by m7, m" = 5, m'" = 4, p = 2,
ガ
​p'll =1, will give G 25; wherefore in order to obtain the
three fimpleft Equations that can refult after x is exterminated, we are
to compare the four propofed Equations three by three, the two different
Ways, juft mentioned; and the four together after the Manner indica-
ted by the Combination which gave G 25. The Combinations of the
Equations two by two, all give Values for G greater than thoſe already
found.
= 1,
LXXII.
If five Equations are given involving five unknown Quantities, to ex-
terminate one of them as x, let m, m', m", m'", m'"", denote the Expo-
nents of the Degree of x in the propoſed Equations, and let p,p′,p",p"""p"""
How an un- exprefs the Dimenfions of the firft Coeficient of thoſe Equations, the
tity of feve- Dimenfions of the other Coeficients being ſuppoſed to obſerve the fame
ral dimenfi- Law as heretofore; by a ſimilar Reaſoning to that employed for 2, 3, 4 un-
ons is exter- known Quantities, it will appear that m + n = m' + n', m + n+1=
of five equa- n + 1 + n + 1 + n" + i + n'" + i + n'"""' + 1, from whence is
tions by the deduced n = m
m' — n''
4, and n' m
n"
It will be demonftrated in like Manner that
minated out
foregoing
method.
n'"
721111
- n'"
4.
G=(m+n+ 1) (”+”) + (2 p − n ) ( " + ¹ ) + (2 p' — n') ( "' + ¹)
+(2p"—2m—2n+2m"+n") ( "'"'+'")+(2p!"—2m—2n+2m'"+n") (-
+(2p!'"'—2m—2n+2m'""'+-'"'"'") ("'""' + 1));
-'-')=mm'+pm'+p'm—
2
2
リ
​-n"
=mm'+pm'+p'm—3m—3p—3 m'
-3p'+m"+p"+m'"+p""'+m"""'+p"""'—n"'n""'—n"'n'""'—n'"'n'""' +6—(p+p'
·p"+m+m'-m"-n"-n'"-n!!!_ ·4)n"'—(p+p'—p"" + m + m' — mill.
'n'"'"'—4)n'""'—(p+p'—p'""' +m+m'-m'""' "—n'""'—n'"""'—4) n'""".
·4)
The Condition that G fhould be the leaſt Number poffible, requires
that we ſhould put equal to nothing, 1°. The limiting Ratio of the co-
temporary Increments of G and n', confidered as the only Quantities
undetermined in the preceding Expreffion. 2°. The limiting Ratio of
the cotemporary Increments of G and n", confidered as the only un-
determined Quantities in the foregoing Expreffion. 3°. The limiting
SPECIOUS ARITHMETICK.
117
Ratio of the cotemporary Increments of G and """, confidered as the
only undetermined Quantities in the aforefaid Expreffion. Thefe Ope-
rations being performed, we will have
""
n'"
n"
1111
m+m+m"+m""+m'""'+p+ § +p"+p'"' +p''!!
4
m+m' +m"+m" + m" +p+ p'+ p" + p" + pull
4
m+m'+m"+m" + m'""" +p+ p'+ p" + p" + p!!!
4
— m" — p" — 1,
m'"'— p'"'— I,
m pl - I,
but becauſe n, n'"', n'""' fhould be integer Numbers and pofitive, and fuch
that m+n>m"+n", m + n > m"" + n"", m + n > m'""'+n'""', or at the
furthest equal.
I put n'= m+m'+m" +m"+m"+p+p'+p" +p"" +p"-a
ก
72'111
4
m+m'+m"+m'"'+m'""'+p+p' +p" +p'"' +p"""—b
4
m+m'+m"+m'"'+m'""'+p+p'tp" +p""' +p"""".
4
C
m
-p"—I,
—m''''—p''''—I,
a, b, c being the leaft Numbers pofitive or negative that can anſwer
thoſe Conditions.
If we fubftitute in G thofe Values of n", n'", n'""', we will have
G=m m' + m
m' + m m" + m
General ex-
preffion of
m" + mm"" + m m'""' + m
p'
+m p'''' +- m' m''
+ m' p''"' + m'' m'"'
m"
+m""m""" +m""
+mp + m'"' p'
+m"""p" + m"""p"
+ m' m'"' + m'
m'""' + m' p
+m p" + m
+ m'
"'p"
p''
the degree
of the refult
+ m'
p'"'
+ m" m'""'+m"
p
ing equati-
+ m'"'p"
tp p"
tp p'"'
+m" p'
+ m'" p'""' + m'""'p
tp p''" + p p" to p
+m" p'" +m" p'""
+m"""p"
ons after an
+p' p''"' + p" p'"' +p" p''"'
+ p!!! p!!!!
+
16
6
Let p =p'=p" =p"""
a² + b² + c² + ab + ac + b c
m+m'+m" + m'"+ m'""'+p+p't p" + p'"' + p
pl =0,
=0,m=m' — m'
unknown
quantity of
feveral di-
menfions is
exterminat-
ed out of
five equati
ons.
4
LXXIII.
-]
m'll
n'
122-c-4
4
m-b-4
ガニ
​M-a-4
,
,n':
m'''', then
m+a+b+c−4,
4
4
m+a+b+c=4.
n =
4
Whence if m is leſs than 4, there are no Values which ſubſtituted for
a, b, c will render n, n', n'', n'"', n pofitive, that is, the unknown
Quantity is to be exterminated by the Rules laid down for a leffer Num-
ber of unknown Quantities.
If m = 4, the unknown Quantity may be exterminated by combining
the five Equations together but only one Way, by putting a=bco.
Rules deriv-
ed from the
118
ELEMENTS OF
!
preceding If m be greater than 4, there are four Cafes to be confidered, either
principles
Por experim is exactly divifible by 4, or after the Divifion there remains,
natinganua or in fine 3.
knownquan
tity of feve
+
冲
​*
:
or 2,
FIRST CASE. The first Equation without x will be obtained by put-
ral dimenfi- ting a = 0, b=0, c'—0, the fecond by putting ao, b
the third by putting a
ting a = 4, b = 0,
ons in five
equations.
Og b
c = 0.
I
8'
گپ
0,= 4,
4, co, in fine, the fourth by put-
The correfponding Values of G will be
G = { m² + 1, G = { m² + 1.
•
1.:
G = { m², G = { m² + 1,
SECOND CASE. Put a T, br, c = 1, and the Values which
refult for n, n', &c. being.not all of the fame Degree, they may be ap-
plied four different Ways to the five Equations, whereby
tions without x will be obtained each of a Degree, G =
2
the four Equa-
5. m² + 3.
5.m 8
THIRD CASE. Put a 2, b = 2, c = 2; proceeding as in the fe-
cond Cafe, the four Equations without x will be obtained; which will
rife each to a Degree expreffed by
-5 m²
8.
-3
+ 3/4
2
•
FOURTH CASE. Put a 3, 63, c=3, and proceeding as in
the two foregoing Cafes, the four Equations without x will be obtained,
each of a Degree expreffed by & m² + 27.
LXXIV.
In general if there are N unknown Quantities and N Equations given;
from all that precedes it is eaſy to conclude,
1º. That if A denotes the Sum of the Quantities m, m', m", &c.
and p, p', p", &c. Then
General ex-
preffion of
A
a
A
b
the degree
n"
m"
N
p"
I, nill
m!!!
I-
N — I
Bill
I,
of the refult
A
M!!!!
A
d
ons after an
unknown.
NI
p!!!!
-N. --I-
-p!!!!!!
I,
m ·
ing equati- n =
quantity of and fo on; likewiſe n
feveral di-
menfions is
exterminat n'=
ed out of N
equations.
A + a + b + c + d +, &c.
NI
- m'
A+a+b+c+d+, &c.
N I
p-p'
I.
2º. That if B denotes the Sum of the Squares together with the Sum
of the Products of the Quantities a, b, c, &c. taken two by two, and
Cthe Sum of the Products of the Quantities m, m',m",&c.p‚p'‚p"‚p""',&c.
multiplied two by two, omitting the Products mp, mp, m" p', &c. and
the Product pp' of the two Quantities, which belong to the Equations
in which mn was fuppofed equal to m'n', there will refult in general
XA².
B
N-2
G=C+(N—1)² —(N—1)—~
2
ง
=a+
B
N-2
(N—1)²´´´´ 2(N—1)
2
A
NI
SPECIOUS ARITHMETICK.
1.19-
CHA P. III.
Of the Refolution of Equations of all Degrees which confift of two Terms
only, of thofe confifting of three Terms that can be reduced by the Method
of Equations of the fecond Degree, to Equations confifting of two Terms:
with different Operations relative to thofe Equations, fuch as the raifing
of Quantities to any Power, the Extraction of Roots, the Reduction of
radical Quantities, &c. and of the Reduction of Equations by furd Divifors.
HAVING fully treated of Problems of the fecond Degree, we
fhall now proceed to explain how to folve Problems of higher Degrees,
beginning with thofe which produce Equations confifting of two Terms;
of which Kind is the following, which in its full Extent produces Equa-
tions of every Degree confifting of two Terms.
I.
full extent
produces e-
A young Gentleman borrowed a Sum of Money a, for the Loan of which Problem
be was to allow Intereft payable at ftated Terms, but it not fuiting bis Con- which in its
venience to pay it at the End of the firft Term, the Lender agreed that it
fhould be continually added to the Principle after it came due, fo that at the quations of
End of each Term, the Sum or Amount should become a new Principle for every degree.
the fucceeding Term, until the Whole fhould be difcharged at one Payment. two terms.
confifting of
Now after a certain Time t, the Debt amounted to a Sum г.
The Rate of
Intereft allowed is required?
Let the arbitrary Number (but commonly 100) that the Rate of In-
tereſt is computed from, be expreffed by d, and the Intereſt of this
Number for one Term by i. Now the Principal of the firſt Term be-
ing a, its Intereſt will be
Sum due at the End of the first Term will be ad.ai
a i
; to which a or
· d
d
ad being added, the
d+ i
a X.
d
d
=ax
the Principal of the ſecond Term being adai, its Intereſt will be
aid+ai to which the Principal reduced to the fame Denomi
22
,
nator, d being added, the Sum due at the End of the fecond Term will
a d² + zaid + a i²
be
d?.
=ax.
(+).
d i
d
In like Manner it will
appear that the Sum due at the End of the third Term will be
a d³ + 3 a i d² + 3 a i² d + a i3 (4++)³. Without pro-
d3
= ax
(土​)
ceeding any further it is eaſy to perceive that thofe different Refults form
a Series of Quantities which increaſe by one common Multiplicator, or in
120
ELEMENTS OF
Equations
the Language of the Analyſts, a geometrical Progreffion; the firſt Term
being a, and the common Multiplicator, which for brevity Sake I
fhall denote by x. The Term of the Progreffion where x is raiſed to the
Power whofe Exponent is 1, expreffes the Sum due at the End of the
firſt Term, that where x is raiſed to the Power whofe Exponent is 2,
the Sum due at the End of the fecond Term, and in general the Term
of the Progreffion where x is raiſed to the Power whofe Exponent is t,
expreffes the Sum r, due at the Expiration of the Time t; whence we
have ra xt, which Equation being folved, the Value of x, and con-
ſequently the Rate of Intereft i = x d-d will be obtained.
II.
Let us firſt fuppofe the Equation to be of the third Degree, that is,
ax³r, to folve this Equation, it is eaſy to perceive that it fuffices to
of the third
degree con-
take away from x3 its Coeficient, and extract the cube Root of the two
fifting of two Members. The Character employed by the Analyſts to exprefs the cube
Root, is the fame as for the fquare Root, with 3 put over it.
Thus to express the Value of deduced from the Equation a x3 r,
3, they write down x = /. If, for Example, the
terms.
The figure 3
is put over
the charac- or
ter to de-
note the
cube root.
+
a
3
a
=
Sum lent a be £1000 and remained unpaid 3 Years amounted to £1331,
then the Amount of £1 in one Year, or x=3/1, 331.
III.
It is to be obſerved that we are not at Liberty to prefix the Sign + or
to the radical Sign of the cube Root, as to that of the fquare
Root, but that the Sign of the cube Root is the fame as the Sign of the
Quantity whofe Root is required, becauſe the Cube of a poſitive Quan-
tity is pofitive, and that of a negative Quantity negative. However,
have but one though a Cube has only one Root, and this Root has only one Sign, we
fign prefixed are not to conclude that an Equation as a x³r, will give only one Va-
lue for x. That this may appear, let the Equation a x3r be reduced
A radical
cube can
to it.
3
que
to this Form x³
=0, and its Root x =
, alfo to this
a
a
3
Form x
=0, then dividing x³ —
x3
by-the
a
Quotient will be an Equation of the fecond Degree, which will contain
the other two Roots.
a
To perform this Operation with lefs Trouble, I put =c3, where-
by the preceding Equations are transformed into x3c30, and
x-co, dividing the first by the fecond, the Quotient will be found
to be x² + cx + c²=o, whoſe two Roots are expreſſed by
C
x = -1/C ± √-2, and will become the fecond and third Values
SPECIOUS ARITHMETICK.
I 21
b
3
of x in the Equation x3
when for c is fubftituted its Value
a
a
IV.
I 3
Ъ
be
2
a
± √(-
The Subftitution being made, the two Values of x will be found to
ЪЪ
b) for it is eaſy to per-
a a
3
3
4
3
b
3
ceive that the Square of
that is, the product of
a
b
into
a
3
b
3
b b
fhould be
a
a a
and that in general the Multiplication of How cubi-
cal radicals
cube Roots, as that of ſquare Roots, is performed by multiplying first the are multi-
Quantities which are under the radical Sign, and afterwards prefixing this flied into
Sign to their Product.
The three Roots of the propofed Equation a x3 = b, are therefore
b b
+√(-),
V.
3
b
I
==
3
x =
a
2
i
b
a
3 3
4
b b
a a
Ι 3
2
b
a
3 3
4
a
one another
Roots of an
x equation of
the third de
gree conûft-
the firſt real, and the two ing of two
the firſt real, and the
others imaginary, which however may be faid to refolve the propoſed
Equation.
VI.
terms.
of equations
In like Manner an Equation confifting of two Terms of any Degree may
be folved, by employing a radical having for Exponent the Exponent of Refolution
the unknown Quantity in the propofed Equation. For Example, the confifting of
Equation ax = or xt = being propofed, there would refult
== 'N = ·
W
a
a
If t is an odd Number, this Quantity is neceffarily negative or poſitive,
according as is negative or pofitive: ift is an even Number the Root will
have as the ſquare Root the Sign, and will be reel whenis pofi-
a
a
a
tive, but when — is negative ( being an even Number) the two Roots
expreffed by will be both imaginary, whence all the Equa-
±√
t
a
two terms of
every de-
gree.
Thofe equa-
=승
​can have no more than two reel
a
tions can on
ly have two
tions expreffed in general by xt =
Roots, the other Roots being neceffarily imaginary, confequently the Pro- real roots.
blems which produce thofe Equations can have but two Solutions.
зе
122
ELEMENTS OF
Example. 4
Another ex-
ample.
Obfervati-
ons on the
1,2, the
For Example, if the Debt was £10000. and remained unpaid for four
Years amounted to £20736. the Equation to be folved would be
*4 = 2,0736, the two real Roots of which are 1,2 and
two imaginary ones +✔
folves the Queflion as it is propoſed, and gives 20 per Cent. for the Rate
of Intereft required.
144 and ✓ 144; however only one
If the Debt was £100000. and remained unpaid for five Years a-
mounted to £161051. the Equation to be folved in this Cafe would be
x5 = 1,61051, which has only one real Root + 1,1, the other Roots are
found by refolving the Equation x+1,1x3+1,21x²+1,331x+1,4641=0
which arifes by dividing the Equation x5-1,610510 by x-1,I=0;
it is eaſy to perceive without refolving this Equation that its Roots are all
imaginary, for if any of them was real they would refolve alfo the Equa-
tion x5 1,61051, confequently fome other Number befides 1,1 raifed
to the fifth Power would give 1, 61051.
VII.
To compleat the Refolution of Equations confifting of two Terms, it
involution remains to fhew how the radical Quantities they produce are reduced to
of quantities their loweft Terms, or how their Roots are extracted when they are per-
fect Powers.
Rule for in-
volving fin-
gle quanti-
ties.
Example.
ล
To enable the young Analyfts to diftinguifh thofe Cafes it will be ne-
ceffary to make ſome Obfervations on the Converſe of this Operation, that
is, on the raifing Quantities to any Power.
If a Quantity as a bed was propoſed to be raiſed to a Power m, it is
eafy to perceive that this Power would be am fm cm dm
If it was propofed to raiſe to any Power a Quantity as
a b c
de
having
a Divifor, it is manifeft that the Divifor and Dividend muſt be raiſed to the
propofed Power, and if the Quantities had Coeficients they muſt likewife
be raiſed to the fame Power alfo, if the Factors of the given Quantity are
Taifed to any Powers, they will then be raiſed to a new Power, whofe Expo-
nent will be the Product of the Exponent they had already into that of the
Power to which they are propoſed to be raiſed, thus
raiſed to the
Power
amq brq
спа
2 a
a² 63
3 c4
will give
3
8-a6 69
27 012
ambr
1
;
13
railed to the Power q will give
C
; all which is very eaſy to conceive, and is a neceffary Confe-
quence of this Principle that a Quantity raiſed to any Power is the Refult
of the Multiplication of this Quantity into itſelf as many Times lefs one,
as there are Units in the Exponent of the 'Power.
If the Sum of £1000. was lent at 6 per Cent. per Annum, and remained
unpaid 3 Years, and it were required to find the Sum due at the Ex-
SPECIOUS ARITHMETICK.
123
piration of that Time, in this Cafe a 1000, d= 100, i = 6, t = 3,
d + i
106
53
=*=
; confequently the Sum due raxt
d
100
50
=
1000×(-
3
53
50
148877
148877
= 1000 X
125000
125
£1191. 125.
2
VIII.
It is now eafy to perceive that the Converfe of this Operation, that is
the Extraction of Roots is performed by dividing the Exponents of the
Parts or Factors of the Quantity by the Number that denominates the
Root required, whether thole Factors are in the Numerator or in the De-
nominator. Let it be propoſed for Example to extract the cube Root of
0366
لام
>
Evolution
of fingle
I divide the Exponents 3, 6, 9 by 3, and affume the Quotients quantities.
1, 2, 3 for the new Exponents of the fame Letters, whence
be the cube Root fought.
a bz
c3
will
If the Quantity had a Coeficient its cube Root is alfo to be extracted;
thus the cube Root of
8a3b6
·69
will be 2ab2.
In like Manner the Root of the fourth Power extracted out of
16 a4 68
d4 c¹²
12
will be
2ab2
dc3
In general, the Root n of the Quantity
àm fr
cq.
will be expreffed by
In
an
9
C.ft
let the particular Values of m, r, q, n be what they will,
tional
whence in the Place of any Quantity that has a fractional Exponent we
may ſubſtitute the Root of a Quantity whofe Exponent is an Integer, by What frac-
placing above the radical Sign the Denominator of this Fraction, and ers denote.
placing within the radical Sign the Quantity having for Exponent the Nu-
merator of this Fraction, that is "
am br
112
9
Ca
pow.
If for Example the Sum of £1000. was lent at 7 per Cent. per Annum Example.
and remained unpaid one Year, two Months and thirteen Days, and the
Sum due at the Expiration of that Time be required.
In this Cafe a = 1000, d = 100, i = 71,
d+i
d
107
100
215
200
43
49.
t
438
· 6:
365
75
Years, confequently the Sum
$
ELEMENTS OF
124
duera x² = 1000 X
43
40
6
5
=1000 // 43].
IX.
40.
If it was required to find what principal Sum a, let out at a certain
What nega-Rate of Intereft, would amount to a Sum r at the Expiration of a certain
tive powers.
denote.. Time, then a = =rx, thofe two Expreffions being equal,
Example
what the
power o de-
notes.
Examples
of the fore-
going Re-
ductions and
Transforma
Lions.
A
χε
が
​for if they be multiplied by the fame Power of x there will refult the
fame Produa, for Example, x2 Xrxt=rxt =
x2t X. r and in ge-
neral any Quantity placed in the Denominator of a Fraction may be tran-
fpoſed to the Numerator if the Sign of its Exponent be changed.
xt
To apply this general Solution to an Example, let it be required to find
what principal Sum of Money lent out at 6 per Cent per Annum would
amount to ₤385 135. 7d. at the Expiration of 7 Years, here r=385,681,1
x = 1,06 and t =7, confequently a = £256 10s. When to
then 4+i
xt
or x is equal Unity, confequently x° = 1.
X.
As the foregoing Reductions and Transformations are neceffary in an
infinite Number of Operations, to render the Application of them eafy
to Beginners, here follow fome Examples.
ca-1
ला
2
Í
3>
a
23.
=a3b-3 C)
a3:
2 62
1 =
b3 c
ab
= √ a-3, 63,
3
3
2.
3
63
b
a
a3
a
3
24.
3
3
a
b
3.
2305
2
ole
r
I
b
b
I
3
3.
लात.
b√b¾√a
2.
a
1
3
Xa c3
a bz
C
mla
of
ક્ષત
a 3
a
12
b
4
4
, 3
+lerg
C.
1.
1
35
12
~I~ WIT
b
C 3
int m
ml +
C 4
3'
a
12 t a5 635
3√ 64
4
in m
C· 3
+
rifl
a
ml +
fi
in I ct
3
4
c² 12
2
b
5
a 12 b
= a
ac²
6 3√ b
4
मल
C
b.
35
124.
SPECIOUS ARITHMETICK.
125
J
XI.
When there is only a Part or Factor of the propofed Quantity whofe
Root can be extracted, the Root of this Part is placed before the Sign and
the other Factors under the Sign, let it be propofed, for Example, to ex-
tract the Root of the fifth Power out of 32 210 68
Power of
2a2b
of the Product of _32 a10 b5
243 c5
into
down
2 22 Ž
3 c
b
5
3 6
63
2629
which is compofed Extraction
of the roots
486 c7
the firſt Factor being the fifth of imperfect
and the other having no Root of the fifth Power I ſet
63
262
for the Root required.
8 a3 b + 16 a³ c
will be
54 d
a3 b
a3
с
is the
powers,
In like Manner the cube Root of
+, becauſe the Quantity 84% +16 43 c
.2
3
6
2
a
3
d
843
Product of
6+2c
into
27
2 d
>
a,
54 d
the first of which Quantities is a per-
fect Cube, that of 2 and the fecond has no cube Root. In like Man-
3.
5 32a9+128 a6 b3—160 a5 b4
ner A
366
XII.
2.a 5 5/ a++ 4 a 63—564
b
36
When the Quantity whofe Root is propofed to be extracted is compoſed
as the foregoing ones of feveral Terms, and after having ſeparated from
thoſe Terms fuch Quantities as they all include that are perfect Powers
of the fame Name with the Root, it is fufpected that the Remainder
is a perfect Power of fome commenfurable Quantity compofed of ſeveral
Terms; the Operation to be performed to diſcover when this happens is
fomewhat more difficult, to find the Method to be purfued in this Opera-
tion, we fhall make fome Obfervations on the Converſe, that is, on the
raiſing of compound Quantities to any Power, from whence we will de-
rive Rules for extracting the Roots of thofe kind of Quantities beginning
by the cube Root.
XUL
nomial con--
Let the ſimpleft compound Quantity us be affumed, and let it be In what the
raiſed first to the Cube, or third Power. Firft by multiplying uz into cube of a bi
itſelf we ſhall have u u + 2 u z + z z, which is the Square or fecond fits.
Power of + z, and this again multiplied by uz gives
23 + 3, 26: 26 2 + 3 uz z + z³ or the Cube of u+z, whence it ap-
pears, that a Quantity confifting of two Parts when raifed to the Cube
gives, 1º. The Cube of the firſt Part, 20. The Triple of the Square of
this first Part multiplied into the fecond Part, 3. The Triple of the first
126
ELEMENTS OF
tities.
Part multiplied into the Square of the fecond, 4°. The Cube of the
fecond.
XIV.
The cube Root therefore of any compound Quantity will be difcovered
after this Manner, first look out for a Term which is a Cube and this
Method of Cube correfponding to u³ place its Root in the Quotient, which will cor-
extracting
the cube refpond to u, then fubtract its Cube from the propofed Quantity, and di-
root of com vide the Remainder by the Triple of the Square of this Root, the Quo-
pound quan tient will be the fecond Member of the Roof, and will correfpond to z;
then multiply this Term by the Quantity correfponding to 3 uu+3uz+zz,
that is, by the Triple of the Square of the firft Member, together with
the Triple of the Product of the first Quantity into the fecond, and the
Square of the fecond, fubtract the Product from the aforefaid Remainder,
and if nothing remains then the cube Root of the Binomial correfponding
to uz is obtained: If there ſtill remain feveral Terms, to diſcover whe-
ther the propofed Quantity is not the Cube of a Trinomial, thoſe two
Terms must be employed in the fame Manner as the firft was, to find the
fecond.
XV.
Some Examples will illuftrate this Method. Let the cube Root of the
First exam- Quantity 86+60 b2 4 + 150 64y² + 125 66 be required.
ple.
2
2
(124 + 30 b² y²+2564
·8-76 + 60 y 4 b²+15064y²+12566 2 y²+ 562
-836
60b2y4 +150 643²+125 66
-60 b² y4—1:50 b4 y² — 125 b6
O
O
7
{
I extract the cube Root of the firft Term 86, and fet down this Root
2 y² befide the propoſed Quantity, and then write down the Cube of 2 p
under this first Term, obferving to change its Sign, that is, prefixing to it
the Sign, the Subtraction or Reduction being performed, I fet down
the Remainder 60 b² y4 + 150 b4 j² + 125 66, and place above 2 y² the
Triple of its Square, that is, 124, I then divide the firft Term '60 b² y4
by 12y4, and place the Quotient 5 b befide 2 y2, I then add to 124 the
Product 30 622 of the Triple of 2 y into 5 b, and I add to thoſe two
first Terms 25 64 the Square of 5 b², I then multiply 5 b2 by thoſe three
Terms, and I fet down their Products with their Signs. changed under the
Quantity 60 62 4 + 150 64 y² + 125.66, and after the Reduction there
being no Remainder left, I conclude that the propofed Quantity is a per-
fect Cube, and that its Root is 2 j² + 5 b².
SPECIOUS ARITHMETICK.
127
XVI.
4
x6+66x5+216²x4+4463x3+6364x²+5465x+2766 ¿ 3x4+126x3+216²x² + 1863 x + 96
66x5 +216²x4+4463x3
-6bx5 -126² x 4 — 863x3
96²x4+3663x3+63b4x²+54b5x +2766
-9b² x 4 —— 36 b 3 x3—63b4x²-5465x
O
0
O
-2766
0
3x4+ 66x3+ 4b2x²
1 x²+ 26x + 362
If the cube Root of x6+66 x5+21 6²x²+44 6³x³+63 84x²+5465x+2726
was required, which it is eafy to perceive fhould confift of more than two
Terms, by an Operation fimilar to that employed in the foregoing Exam-
ple, the two firft Terms of the Root will be found to be x² + 2 bx ; but
as there is a Remainder 9 b²x4 + 36 b³ x³ + 63 bª x² + 54 65 x + 2765,
I divide the firſt Term 9 b² x4 of this Remainder by 3 x4 triple of the
Square of ², becauſe this Term is the first of thoſe which arifes by trip-
ling the Square of the Quantity xx + 2 bx which actually correfponds to
the first Part (called u Art. xIII.) of the cube Root fought, having divided
9 b² x4 by 3 x4, I fet down the Quotient 3 b2 befide xx+2bx, I then
add together the Triple of the Square of x x + 2 bx, the Triple of the
Product of xx + 2 bx into 3 b2 and the Square of 3.62 and there reſults
3x² + 12 b x3 + 21 b²x² + 18 63 x¹ +964, which I multiply by 3 b²,
and I fet down the Terms with their Signs changed under the Quantity
962x4 + 36.63.x3 + 63 64 x² + 54 65 x + 27 ¿, and as after the Re-
duction there is no Remainder left I conclude that x² + 2 bx + b² is the
cube Root of the propofed Quantity.
2
XVII.
Second ex-
ample.
The cube Root of any Number may be found after the fame Manner, Method of
if it be a Number under 1000 its neareſt cube Root is found by the fol- extracting
lowing Table.
דר
ΙΟ
Cubes 1 8 27 64 125 216 343 512 729 1000
Roots I .2 3 .4 5 6. 7 .8 9
where it is eafy to obferve that the Cube of a fimple Number cannot con-
fift of more than three Places of Figures, fince the Cube of 10 the leaſt
Number confifling of two Places is 1000, the leaft Number confifting
of four Places of Figures, that the Cube of a Number confifting of
two Places of Figures cannot confift of more than fix Places, fince the
Cube of 100 the leaft Number confifting of three Places of Figures is
1000000, the leaft Number confifting of feven Places: Proceeding in
this Manner it will appear that the Cube of any Number cannot conſiſt
}
the cube
root of num
bers.
128
ELEMENTS OF
by an exam
ple.
of more than the Triple of its Places, and in general, that the Power p of
a Number confifting of n Places cannot confift of more than pn Places; from
whence it follows, that a Number confifting of lefs than four Places of Fi-
gures can have but one Figure in its Root, that a Number confifting of
more than four but less than feven Places can have but two Figures in its
Root, that a Number confifting of more than feven but lefs than ten Pla-
ces of Figures can have but three Places of Figures in its Root and fo on,
taking for Limits the Numbers 1, 4, 7, 10, 13, &c. whoſe common
Difference is 3, confequently the Number of Places which the cube Root
of any Number confifts of is difcoverable by Infpection: How the Num-
bers correfponding to thofe Places are found we ſhall now proceed to explain.
XVIII.
5,305,472(174.41,&c.
I
4305
300
2100
1470
343
3913
392472
86700
346800
8160
64
355024
Let it be propoſed to extract the cube Root of
The forego- 5,395,472 as it confifts of feven Places of Fi-
ing method gures, I conclude that the Root will confiſt of
explained three, to determine the Numbers correſponding
to thofe Places, I exprefs them by x, y & z, con-
fequently x³-3x²y+3xy²+y³+3[x²+2xy+y²]z
+3[a+y] x²+z35,305,472: Hence the
Difficulty is reduced to find the Terms of this
Quantity in the given Number. It has been pro-
ved that the Product of two Numbers multiplied
into one another has as many Places of Figures to
the right Hand of it as there are to the right
Hand both of the Multiplcand and Multiplicator,
confequently the Product of three Numbers mul-
tiplied into one another will always have as many
Places of Figures to the right Hand of it, as there
are to the right Hand of the three Numbers mul-
tiplied together. Now the first Number x being
hundreds its Cube x3 will have fix Places of Fi-
gures to the right Hand of it, whence feparating
by a Line the fix Figures 305472 to the right
Hand of the propofed Number, The Cube of
the unknown Number x of Hundreds of the cube
Root will be found in the Part 5 to the left Hand
of the feparating Line, and will be the greateſt
Cube contained in this Part 5, but I is the
greateſt Cube it contains, confequently I the
cube Root of 1 is the Number of Hundreds of
the Root which I therefore fet down in the Quotient and deducting 1
from 5 the propoſed Number is reduced to 4305472.
37448000
9082800
36331200
83520
64
364'4784
1033216000
912460800
912460800
52320
912513121
I
120702879, &c.
SPECIOUS ARITHMETICK.
129
The Square of a Number of Hundreds which has four Places of Figures
to the right Hand of it, being multiplied by a Number of Tens which
has one Place of Figures to the right Hand of it, the Produ&t will have
five Places of Figures to the right Hand of it: wherefore the Term 3x²y,
will be contained in the Part 43 which has five Figures to the right
Hand of it. But the Product arifing from the Multiplication of three
times the Square of the Number of Hundreds multiplied into a Number
of Tens, being divided by three Times the Square of the Number of
Hundreds, will manifeftly give the Number of Tens, whence the Number
of Tens of the Root will be found by dividing 43 by 3, triple the
Square of the Number 1 Hundred of the Root; to determine the Part
of this Quotient to be employed, I obferve that 3x²y + 3 x y² + y³, or
three Times the Square of the Number of Hundreds of the Root multi-
plied into the Number of Tens, together with three Times the Square of
the Number of Tens into the Number of Hundreds, and the Cube of the
Number of Tens ſhould not exceed 4305, fince the Cube of the Num-
ber of Tens ſhould have three Places of Figures to the right Hand of it:
dividing therefore 43 by 3 or 4305 by 300, inſtead of the whole
Quotient 14 reſulting from this Divifion I find only 7 to anſwer the above
Condition; I therefore fet down 7 in the Root to the right Hand of the
I Hundred, and dedu&t 300 X 7 + 7 X 7 X 10 X 3 + 7X 7 X 7 or
2100 + 1470 + 343 or 3913 from 4305, and to the Remainder 392 I
annex the remaining Figures of the propofed Number, and there reſults
392472, which fhould contain the Terms 3 (x² + 2 x y + y² ) %
+ 3 (x + y) z² + z³, or three Times the Square of the 17 Tens mul-
tiplied by the unknown Number of Units, together with three Times
the Square of the Number of Units, multiplied into the Number of Tens,
and the Cube of the Number of Units: Now becauſe three Times the
Square of the 17 Tens multiplied into the Number of Units, fhould have
two Figures to the right Hand, it will be contained in the Part 3924,
which has two Figures to the right Hand of it, whence dividing 3924
by 867 three Times the Square of 17, or dividing 392472 by 86700 the
Quotient 4 will be the Number of Units of the Root, which I fet down to
the right Hand of the 17 Tens already found, and deduc
3 X 170.X 170 X 4 + 3 x 170 x 16+ 64, or 346800 +8160 + 64,
that is, 355024 from 392472 and there remains 37448, whence 174 is
the cube Root required of the greateſt Cube contained in the propofed
Number 5305472, which exceeds this Cube by 37448.
XIX.
Method of
When a Number is not an exact Cube, as the foregoing one, we may approxima-
however approximate to its cube Root to any Degree of Exactnefs; for ting to the
Example, to extract the cube Root of 5305472, inſtead of 5305472 numbers.
cube root of
3 R
}
130
Procefs of
cube root of
numbers..
ELEMENTS OF
it is eaſy to perceive that we may take 5305472000000
1000000
which is equal
to it; obferving that the Denominator fhould be a cube Number, then ex-
tracting the cube Root of the Numerator which may be found true to an
Unit, and dividing this Root by 100 the cube Root of the Denominator;
the cube Root of -5305472000000
or of 5305472 will be obtained true to
3000000
k
: If we had wrote
100'
5305472000000000
the Root would be found true to
1000000000
I
1000
,
it eaſy to perceive that
and fo on, whence in extracting
the cube Root after the propoſed Number is gone through, if there is a
Remainder, the Operation being continued by adding Periods of Cyphers
to that Remainder,, the true Root will be be obtained in Decimals to any
Degree of Exactnefs, as in the Example.
XX.
It is eafy to perceive, that the cube Root of any Number may be dif
the extracti covered by the fame Method of reaſoning employed in the foregoing Ex-
on of the ample, the Procefs of which may be expreffed thus; point every third Fi-
gure beginning with the Units Place, and by theſe Points the Number will
be diftinguiſhed into as many Periods as there are Figures in the Root;
then find by the Table of Cubes the greateſt Cube in the firſt pointed Pe--
riod of the propofed Number, the cube Root of which place in the Quo-
tient to the right Hand, and fubtract the Cube thereof from the firſt point-
ed Period, bringing down to the Remainder the fecond Period; then divide
this Refolvend by thrice the Square of the firft Quotient Figure, confider-
ed as expreffing a Number of Tens, whereby the fecond Quotient Figure
of the required Root will be found; continue this Method till all the Pe-
riods are brought down, and then, if the propoſed Number be a perfe&
Cube, there will be no Remainder but if fomething fhould remain, annex
Cyphers three at a Time, and carry on the Extraction decimally to any
propoſed Degree of Exactnefs.
Faveftigati-
on of Sir Ifa..
ac Newton's
:
XXI.
As by raifing the Binomial uz to the third Power Rules are obtained
for extracting the cube Root of Quantities; in like Manner, by raiſing it
to the fourth, fifth, &c. Powers, Rules may be derived for extracting the
Roots of the fourth, fifth, &c. Powers of Quantities: But to compre-
hend under one general Rule, the Extraction of Roots, the Analyſts have
fought a general Expreffion for the Power m of u+z.
m
To find this general Value of ( + z) or of u + z multiplied into
itſelf as often as there are Units lefs one in m; let us firſt examine the
theorem for Product arifing from the Multiplication of feveral binomial Factors
raifing a bi-
x + a, x + b, x + c, x + d, &c. and endeavour to diſcover the Law
nomial to
any power. obferved in the Formation of all the Terms of this Product.
SPECIOUS ARITHMETICK.
131
Multiplying all thofe Factors into one another there refults,
5+(a+b+c+d+e)x²+(ab+ac+ad+ae+bc+bd+be+cd+ce+de) x³
+(abc + a b d +abe+acd+ace+ade + b c d + b c e + b de +-cde) x²
+ (a b c d + a bce + abde + acde+bcde)x + abcde.
From the Inspection of this Product it is plain, 19. That the firſt
Term is raifed to a Power equal to the Number of Factors.
✔
2º. That the fecond Term contains x raiſed to a Power leſs by an Unit,
having for Coeficient the Sum of all the Letters (a, b, c, d, e,).
3º. That the third Term is compoſed of x raiſed to a Power lefs by
two Units, with a Coeficient equal to the Sum of all the Products that can
be made by multiplying any two of the Letters (a, b, c, d, &c.) by one
another.
2
4°.That the 4th Term includes x raifed to a Power lefs by three Units,
with a Coeficient equal to the Sum of all the Products that can be made
by multiplying into one another any three of the Letters (a, b, c, d, e, &c.);
and after the fame Manner all the other Terms are formed.
Applying therefore thofe Obfervations to the preſent Cafe, in which all
the Factors are equal, and their Number is expreffed in general by m, it
will appear that the firft Term will be
that the first Term will be "that the ſecond Term will be
multiplied by ma, fince b, c, &c. are equal to a, and their Number
is m; that the third Term will be "-2 with a Coeficient equal to a²,
repeated as often as there are Rectangles a b, a c, bc, &c. in the Coefici-
ent of the third Term of the Product of the Factors x+a, *+b, x+c; &c.
the Number of which is fuppofed to be m : fince all the Products
ab, ác, bc, &c. fhould be all equal each to a², when b, c, &c. are
equal to a that the fourth Term will be 3, with a Coeficient equal
to a³, repeated as often as there are Products a b c, a b d, a cd, bed, &c.
in the Coeficient of the fourth Term of the Product of the Factors
x+a, x+b, « + c, z + d, &c. whofe Number is my and after the
fame Manner all the other Terms are formed.
+
}
m
The Queſtion is therefore reduced to know how many different Products
a Number m of Letters will admit of, when taken two by two, three by
three, &c. for fuppofing that thoſe Numbers were found, and were ex-
preffed by A, B, C, D, &c. there would refult "" + max" ¹+A a² x
+ Ba³ x² - 3 + Ca4x4 + D a5 x-5, &c. for the Value fought of
(x+a)m
M--- 2
To find first how many different Products a Number m of Letters
a, b, c, d, &c. taken two by two will admit of, I obferve that when all
thofe Products are formed, there will be twice as many Letters fet down as
there are Terms.
I obſerve next, that each of the Letters a, b, c, &c. fhould be repeated
the fame Number of Times, and that each Letter, being multiplied by all
the reſt and not by itſelf, can not be repeated more than Times ;
I
1
132
ELEMENTS OF
wherefore the Number of Letters to be fet down in forming all thofe Pro-
duas fhould be m X (m 1); wherefore the Number of all thofe Pro-
ducts fhould be m × (m−1) and this is the Value of A, or of the
2
>
Coeficient of the third Term of the Formula fought.
To find the Coeficient of the fourth Term, that is, the Number of
different Products of three Letters a b c, ab d, a c d, b c d, &c. that a
Number m of Letters a, b, c, d, &c. will admit of, taken three by three,
I obſerve firſt that this Number ſhould be the one third of the Letters fet
down in forming thofe Products.
I obſerve next, that each of thofe Letters fhould be repeated the fame
Number of Times, and that this Number is the fame as that which ex-
preffes how many different Products all the other Letters, taken two by two,
will admit of; for it is manifeft, that each Letter a for Example, fhould
be joined to all the Products bc, b d, c d, &c. of the other Letters, ta-
ken two by two.
The number of Times therefore that each of the Letters a, b, c, d, &c.
ſhould be repeated, is the fame as that which expreffes how many dif
ferent Products a Number m I of Letters b, c, d, &c. taken two by
two, will admit of. Now we have ſeen that when the Number of Letters
was m, the Number of their Products, when taken two by two, was ex-
preffed by the one Half of the Number m, multiplied by the Number
Wherefore, when the Number of Letters is m
m
I.
take the one Half
that is,
(m
m
2
I
I, we fhould
of this Number, and multiply it by m-2,
1) X (m — 2). _ expreffes the Number of Times that each
2
of the Letters a, b, c, &c. fhould be repeated in all the Products in
Queftion; and as the Number of thofe Letters is m, confequently,
mx (m-1) × (m − 2)
I X
will be the Number of all the Letters fet
down; wherefore the Number fought of the Products of three Letters
mx(m-1)X (m — 2)
abc, a b d, &c. will be
and this is the Va-
2 X 3
2
lue of B, or of the Coeficient of the fourth Term.
To find the Coeficient C of the fifth Term, that is, the Number of
Products of four Letters, that a Number m of Letters will admit of; I
obſerve, that this Number ſhould be the one fourth of all the Letters fet
down in the Products; that each of thofe Letters fhould be repeated the
fame Number of Times, and ſhould be combined with all the Products of
three Letters, that a Number m 1 of Letters will admit of: and in
fine, that thofe Products of three Letters, that a Number m 1 of Let-
SPECIOUS ARITHMETICK.
133
ters will admit of, fhould be expreſſed by
I
(m-1)x(m2)X (m3)
2 X 3
m X (m − 1) X (m − 2)
2 X 3
for the fame Reaſon that
expreffes the Num-
ber of Products of three Letters, that a Number m Letters will admit of;
hence the Number fought of the Products of four Letters a b c d, abce, &c.
m × (m − 1) × (m − 2) X(m — 3)__
will be expreffed by
X
X
2 X 3 X 4
Forming after the fame Manner the other Coeficients, and fubftituting
in the foregoing Formula in the Room of A, B, C, D, E, &c. their Va-
lues thus found, there will refult at length > + max
m
171
Σ
m
I
+ m x
a² xm
2
+
m × (m − 1) X (m
2)
a3 x
3
2
2 X 3
m. X. (m
1 ) x ( m
+
2.)
2 X 3 X 4
x ( m
3)
a 4 x
4
+
-
mx (m − 1). X (m— 2) X (m − 3) × (m—4)
2 X 3 X 4-X 5
for the Power m of x + a.
In like Manner, the Value of
m X (m − 1)
+ z² um -
+
+
2
a5-5, &c.
General for
mula for rai
24
u+z" will be um zu
mx (m − 1) X (m -2)
协
​23.
M
น
3
fing + z ta
4 &c..
m X (m 1) X (m 2.) x (m − 3).
2 X 3 X 4
As to the Value
2 X 3
24·2-4,
of (uz)", it is manifeft, in order to obtain it, it
fuffices to make z negative in this Formula,
which will transform it into
-1)X(m—2)
m
น-พรน +
171. I
mx(m-1)
2
2
z² 2m - 2
ซีน +
mx(m-1
2 X 3
974
+
น
A
2 X 3 X 4
XXH.
*
น
the power #
m × (m − 1) × (m −2) × (m −3) z4 µm -4, &c.
TIL
To raiſe any Binomial to a given Power by means of the foregoing For-
mula, we have no more to do but to fubftitute in the foregoing Value of Application
(uz) in the Place of u the firft Term of the given Binomial, in the of the fore-
Place of z the fecond, and in the Place of m the Exponent of the Power to la to an ex-
which the propofed Binomial is to be raiſed.
за
Let it be propofed, for Example, to raife 3 ac2bd to the fifth
Power, I put 3 acu,
น, 2 b d = 2,5
u" = (3 a c) 5 = 243 a5 c5, m zu"
m X (m I)
— — 810 a4 b cªd,
2
-1080 a³ b b c 3 d d,
2,5 = m, and there reſults
= 5 x − (2b d) X (3 a c A
x² 2”—2= 10 X. 4b b d d x ( 3. ac)3
2.2 น
I
m X (m—1) X (m—2)
).
= 10 X = (2 b d ) 3 × ( 3.a 6)² =
z 3
น
171 --
720 a² 63: c.2 da
·
going formu
ample.
2 X 3
}
134
}
ELEMENTS OF
How the
foregoing
formula
་
2).
(E_244.). X (7 216) X ( 244.) 24
2 X 3 X 4
(m—3)
4
Z น
x¹ u” — 4 = 5 x − (2b d)ª × 3⋅ a c
3.
น
25 μm-5
m × (m −1) × (m − 2) × (m—3) X(m—4)
2 X 3 X 4 X 5
— —
= 240 a b4 c d4,
=IX (2 b d )5 × (3 a c)° 32 65 d5.
As to the other Terms, their Coeficients including among their Factors
m5, which is equal to nothing when 5, they ſhould all vaniſh;
whence the Value of (3 a c 2 bc)5 will be 243 a5 c5 810 a4 b c4 d
+ 1080 a3 b2 c3 d2 720 a² 63 c² 63 + 240 a 64 cid² - 32 65 d5.
XXIII.
If a Quantity confifting of more than two Terms is to be involved, it
may be eaſily effected by the fame Method. If it be a Trinomial, for
may be ap- Example, calling u the firft Term of this Trinomial, z the Sum of the
two other Terms, the Difficulty of raifing a Trinomial to any Power will
quantities
confifting of be reduced to that of raifing a Binomial to the fame Power, fince each of
m × (m − 1)
X
plied to
11: I
24
more than
two terms. the Terms mu
น. 22, &c. does not include
any Quantity to be involved more compofed than the Binomial.
Example.
24
XXIV.
To fhew by an Example how the foregoing Formula is to be employed
for raifing to any Power a Quantity confifting of more than two Terms,
let it be propoſed to raiſe a + 2 bc to the fourth Power. I put m=4,
u = a, z=2bc, and fubftituting thofe Values in the Formula, there
a², m z u = 4 × (28 - c) X a³8 a³ b— 4 à³ c.
X
a3b- a
112
refults u
m X (m
1)
2
+ 6a²c²,
111
I
z² u" — 2 = 6 × (2 b — c)2X a² = 24 a² b²
=
mx (m-1) x (m2)
3
24 a² b'c
น
23 212 3
j
4 X (26
4×
c)3 X a
2 X 3
48 a
b b c
+
24 a b
c c
4 a c
3
1
2)
172
4
= (26
4X3 X 2
2X:3
32 a b 3
m × (m − 1) × (m − × m→ 31 24 26"
X
X
2 X 3 X 4
=6b4 — 4 × (26)³× c+
+
萨
​4 X 3 X 2 X 1
4 -
X
น
·4 × 3 × (2; b) & Xc² —
2 X 3 X 4
and confequently, (a + 2b —
2
>
c) 4
X26x2
16 64 — 32,63 c + 24 b b c c — 8 b c³ + c²,
*
c) 4 = a4 + 8. a³ b — 4·a³ c + 24.a² b²
24a² bic + 6 a² c² + 32 a·b3 48-a b² c + 24 ab
+1664 32 b³ c + 24 b² c²
3263
c+
8·6c3+64.
XXV.
4 ac3
To find the Roots of Numbers by means of the foregoing Theorem,
every fourth, fifth, and in general every mth Figure, beginning from the
Units Place, is to be pointed, according as it is the Root of the fourth,
fifth, or in general of the mth Power that is required; and if there be
SPECIOUS ARITHMETICK.
135
3
trafting the
from the
any Decimals annexed to the Number, they are to be pointed after the General
fame Manner, proceeding from the Units face towards the right Hand. rule for ex-
whereby it is eafy to perceive, that the Number will be divided into fo roots of num
many Periods as there are Figures in the Root required. Then inquire bers derived
which is the greateft m Power in the firft Period to the Left of the given foregoing
Number, and the Root of that-Power will be the first Figure of the Root theorem,.
required, which call u, and fubtract from this firft Period. To find
the fecond Figure. z. of the. Root to the Remainder annex the first Figure
of the ſecond Period, and this Number will contain the Term m u
of the Binomial uz raiſed to the Power. m, whence if mu
be divided by mu
and after the Divifion is performed, if what re-
171
172
mains of the two firft Periods does not, exceed
171 mm I
Z:
MI
m X m - I 772 2
2
Ka
น
≈2
ut -3x3 +
m. m
·
I m
2
m
3
4
22
+
772
•
172.
I
►
m
22.
2 X 3.
12
น
2 X 3 X 4
24 +, &c..
the Quotient will be the fecond Figure, of the Root. The third Figure
will be found by Means of the two firft, as the fecond was found by the
firſt, and afterwards the fourth Figure (if there be a fourth Period), after.
the fame Männer from the three first.
XXVI
11
Let the fifth Root of 6436343 be required,. I. firſt point it 64,36343,
the Formula of the fifth Power is 25+ 5 u4* z + 10 u³* z² +10 u² z³ of this me--
Application
+ 5 u z¹ + 25; wherefore 64 is either equal to, or contains 15; it is not thod to an
equal to it, fince no Root raiſed to the fifth Power will give 64; confe- example..
quently it contains it, but the greateſt fifth Power in 64 is 32, whofe fifth
Root is 2; wherefore 2 u. I fubtract 32 from 64, and to the Remain--
der 32 I annex 3, the firft Figure of the fecond Period, and there refults-
323, now u=16, and 5. u480; I therefore divide 323. by 80, and the
Quotient is 4, leaving a Remainder 3, which annexed to the remaining
Figures in the propofed Number, there refults 36343, which fhould not
exceed 10 u³ z² + 10 u² z³ + 5. u zª + 25, but 10 3 is equal to
280 Thouſands, 10 u2z3 to 2560 Hundreds, which added to the 1280
Thouſands already found, makes 15360 Hundreds, and as this Number. ex--
ceeds 363 Hundreds that remain in the propofed Power, the fecond Part of
the Root 4 has not been rightly determined: Let the fecond Figure of the
Root be 3, then 3 X 80 240, and 323. 240 83, wherefore
there will remain 836343 which fhould not-exceed 10 u3ig2 + 10 u² 23
+5u2+ + 25; now 10 u3z2720 Thouſands, and 10 u² z³ = 1080*
Hundreds, which added to 720 Thoufands, makes 8280. Hundreds, and
5x4810 Tens, which added to 8280 Hundreds, makes 83610 Tens;
laftly, 25-243, which added to the 83610 Tens, makes 836343.; where
fore. 3 is the fecond Figure of the Root required..
4
23
136
ELEMENTS OF
2
n
I
n
2
2
3
N
I
I
+
Z
24
To approximate to the
XXVII.
Root of any Number N, let r denote the
neareſt leſs Root in Integers, and r the true Root: Then will
i
N=r"+n" I z+n.
+ n
3__ ‚”—44,&c. or
n
n-I
n-2
N
gets 222 +12.
2x²+n.
p" — 3 23
2
3
n-
m
+ %
4
nr
2
n
I
n
2.
12
+
2
3
4
General me
z3
thod of ap- (putting N-r" m). Multiplying this Equation by I + A z, there
proximating will refult,
to the roots
of numbers.
=
n
2
2
3
༣
Ι
24, &c. — 0.
r3
m
1-
+(
I
m d
Ngor
-): =+(
พ. I
I
2
y
+ 1)
ZL
+
(
n
I
n
2
I
n
'I
A
+
2
3
2
)
23
n-I
of
2
1-2
3
n-3
4
I
73 +
n-2
A
z4, &c.=
0
L.
2
3
go2
n
I
2
ተ
12
Ι
I
2
jr
+(₁+
n-I m
•
in which the third Term will be deſtroyed by determining A by Means of
the Equation
found =
m
and by fubftituting this Value above, we have
+ A= 0; from which Equation A is
2 Ng
1) %
n+1
4
n+1 · n−1 · n—2
4 X 3 X 2
24
r3
n pl
nti
n+1⋅n—1 11-2 2~3
5 X 4 X 3 X 2
the Terms of which Series decreaſes very fwiftly, neglecting, therefore, all
the Terms after the fecond as exceeding fmall, it will be reduced to
m
+(x+
ท -I. m.
Q N zz
-)
2 = o, wheńce z =
r m
n-I
nr" +
⚫ m.
2
•
2- I
L.
I
23.
3
jin 2
z5
goof
+49
&c.=0
To obtain a more approximate Value of x, it is fufficient to multiply.
In
n—I
I
N
+8+
22 +
nr
2
*
NI
2
n
N2
I
4
+3
m A
+
whence
+(
+ +
n
2
n-I
•
m
3
12 - I
12.ካ
n-2
2 X 3 X géte
2
-1)+
n gin
go!
2 X r
n
I.
A
+
n
2
A
3
дода
+
N2
3
I
42
z3
24, &c. = o, by 1 + Az + B z²,
-) =+ (-
NI
2 r
+ B) x² + (·
n
2+
=
(-
n
+A-
m B
I n
2. n
3
2 X 3 X 4 X 3
)x4, &c. = 0.
SPECIOUS ARITHMETICK.
137
In which Equation the third and fourth Terms will be deftroyed, by
determining A and B by Means of the Equations
+A-
2Xr
m B
NI
2-2
2 3X+2
+
2
Let the latter of thofe Equations be multiplied by
add the two Products together, ſo ſhall
nr
In
4+B=0
que
and then
J
n-
I
n-I
+A+
22-2
m
n-I
m A
2
2
" + I
+
3
nr
2
1 "
1- I
I
+
whence is deduced
2
12 I
2
1-2
772
1 I
3
nr
-A
712
I +
nr
ʼn — I
2
To render this Value of A more fimple, I obferve, that if two
Quantities be decreafed by two other fmall Quantities nearly in the fame
Ratio with the two firft, that their Difference will ftill be in the fame Ra--
tio with the two first Quantities very near; wherefore, feeing the Nu--
merator of this Fraction is in Proportion to the Denominator nearly as
11-I · 11-2
771
11
I
: I, or as
2
**
2 X 3
n
I
n
2
272
let
2
3
n
2
172
3
rator, and
1 yell I
N-2
m
nj
ከ "I
3
Nr"
be therefore taken from the Nume-
from the Denominator; by this Means the
Fraction itself will be reduced to
I
———. (n − 1)
2
I
or to
r + (n +
1)
. — — — ·
n r
I
I
M
nr
I
I)
2
I + ( m. + I)
6
n jn
,
m
whence putting
No gutt
A
P
ጾ ተ
we will have z =
third Degree.
6
n
£+ (n + 1) X / p
p + (n + 1) × & 2 for an Approximation of the
1+(2-n − 1) . P
I
33
+C
If more Terms of the Series 1 + A z + B x² + C z³, &c. are taken
the Root will be found exacter in Proportion; for Example, if four Terms.
of this Series were employed, there would refult for an Approximation of
× (n − 1) + (n-1 X (2n-1) x ½
1 + 1/2 n p
the fourth Degree
1/2
A =
p + / n p ²
I
p
and z =
2 -
I
I +
2 n
pt
1) X (12 - I 1)
2-
p²
12.
3 S
138
{
ELEMENTS OF
But both thofe Expreffions, in Cafes where p is a proper Fraction, will be
better adapted to Practice by making
N. gon
I
ข
and
N.
12
p
Ι
fubftituting
Application
of the fore-
going ap-
proximati-
ons to an ex
ample.
x=rt === x
for p its Equal; whence after proper Reductions
6v+n+1
60+47-2
nearly; or xr+
XXVIII.
rx (2v + n)
X(2v+2n-1)+{X(2n−1)X(n−1)
To fhew the Ufe and great Exactness of thofe Approximations by an
Example, let the Equation x3 = 500 be given, then aſſuming r = 8 we
ི་
and confequently r + x = 832
have
by the first Approximation.
= 7,937
32
508
508
By the fecond we have v=
12 -1
Ngus
3 X 512
hence by the fe
12
I
cond Approximation = 8
x 8-
X
16
768 +4
·768 + 10
= 8
191
3032
8 X 253
128 × 251 +
फाल
1
Ufe of the
= 7,93700527 nearly. And by the third x 8
= 8
-6072
96389
= 7,9370052599.
When a near Value of z is determined, then adding it to r, ſubſtitute
the Aggregate in the Place of N in the Formulas, and you will by a new
Operation obtain a more correct Value of the Root required; and by
thus proceeding you may arrive at any Degree of Exactnefs.
XXIX
To fhew the Uſe of the foregoing Rules in the Solution of Problems,
foregoing, here follow fome Examples.
rules in the
folution of
First exam-
Suppoſe that out of a Cafk holding c Gallons of Wine when full, a cer-
problems. tain Quantity x was drawn and the Cafk filled up with Water, and that
the fame Quantity of the Mixture was afterwards drawn, and ſupplied
by Water ſeveral Times, and then it appeared that befides Water there
were but r Gallons of Wine left in the Cafk: How much Wine was
drawn out each Time ?
ple.
C - x expreffes the Quantity of Wine left after the firft Drawing;
the Quantity of Wine drawn out at
but c:c-
*
*:
(c-x)Xx
fecond Drawing, and c
Quantity of Wine left
(cm 3 ) 2
Alfo c:
2
C
(c = x) ·
*
C
after the third Drawing:
(c =
x) 2
x)². x
c2
(c) 2
C
out at the fourth Drawing, and
(c) 2
the
C
the Quantity of Wine drawn
(c) 20
66
(c)3
C C
SPECIOUS ARITHMETICK. -
139
equal the Quantity of Wine left after the fourth Drawing; then
equal the Quantity of Wine drawn
(c-x)t-I
C:
(x)-1.x
=X:
C#
ct
I
out at the Drawing, and
ct
2
(c~~x)-1.x
ct
the Quantity of Wine left after the t Drawing. Now
I
(c-x)t
ct-
(c = x)+
ct
by the Queſtion, or (c-x)=rct-1 and c-x=rct-1] and c-rct-1]
C
"
=X.
To apply this general Solution to an Example, let the Contents of the
Cafk be 81 Gallons, the Wine remaining 16 Gallons, the Number of
Times Liquor was drawn out 4, then x 88-4/8503056-27 Gallons.
XXX,
If every Century the Number of Men were doubled, what would be the Second ex--
annual. Increaſe?
ample.
Let the primitive Number of Men be expreffed by n, and let their
Number be increaſed annually by their Part.
I
X
Their Number at the End of the first Year will be expreffed by
n
n +
at the End of the fecond Year by
or by (1+*)*
by ( 1 + * )
100
tion (1+x) 100
n + n x
n; at the End of the third Year by (
n. at the End of the 100th Year.
+
n + n x
22
+x
X
3
n; and
Now by the Quef
I
-
100
n2n, wherefore
=
I + x
2
X
I + *
Whence
10070240
10000000
wherefore x =
10000000
70240
= 144
}
nearly, which ſhews that the Number of Men ſhould be annually increaſed
by their
I
144
Part, in order that every Century their Number ſhould be
doubled: It is therefore no way furprizing that the World ſhould have been
peopled by one Pair.
XXXI.
After the Annalyfts had found the general Theorem for the Involution
of Binomials, they foon perceived that it might be extended to other
Powers befides thofe whofe Exponents are whole and pofitive Numbers;
having found, for Example, that a
concluded, without Doubt, that to
#1
might be fubftituted for a; they
extract the Root n of any compound.
1
140
ELEMENTS OF
n
Quantity reprefented by u + z; it fufficed to fuppoſe m = I in the
foregoing Formula, or what amounted to the fame thing that
น +
I
n
น
I
or the Root n of u + z.
2
โล
2
I
1
I
X
I
I
n
n
2
72
น
x² + &c. = (4+ x)*¸
I
(u + z)"
, or (u + z)—" = u¬"
น
ทน
n X
n
1
2
น
In like Manner, that
२
2
&c. In one Word the Order and
Generality that the Analyſts always found in the analitick Operations, eaſily
fuggeſted to them, that though the foregoing Formula had been inveſti-
gated only in the Suppofition that m was a whole and pofitive Number,
however that it might be applied to all the other Values of m.
Let it first be propofed to fhew that the Formula in Queftion ferves to
raife a Quantity to a Power whofe Exponent is a Fraction, or what
amounts to the fame thing, let it be propofed to prove that
y
X
I
#
n
n
*
n
n
·(u + x) = u +
น
น z +
น
n
2
I
2
x² +, &c.
#
which by dividing the two Members by u
go
and putting
2
=p becomes
น
y
X
I
71
X
24
(1+p)" =1+ =—=—=p+
n
n
n
n
カート
​p²+
n
n
2
p³ +, &c.
2 X 3
To prove this Equation it fuffices to fhew that its two Members being
raiſed to the fame Power n, there will refult equal Quantities, that is, putting
go
**
go
*
ga
X
I
X
IX
2
N
n
12
n
n
s= p+
p² +
n
2
2 X 3
p³ +, &c.
(I s)”.
(I + p)r = (1 + 5)".
Now as r and n are two whole Numbers, the Members of the forego-
ing Equation may be raiſed to the Powers indicated by their Exponents, by
Means of the general Formula (u+z)".
Whence
go (1)
Itrpt
p²+
2
n(n-i)
=x+nst
2
r(x-1) (1-2)
2X3
n(n-1) (n-2)
2 X 3
·83+
53+
2 X 3 X 4
n(n-1) (n-2) (n-3)
r(r—1) (r—2) (r—3)_p4+,&c.
54&c
2 X 3 X 4
Equation, we must find by
Means of the Value of s thoſe of s², s³, s4, &c, and then multiply the Va
The Problem being reduced to prove this
SPECIOUS ARITHMETICK.
141
lue of by n, that of 2 by
that of 4 by-
n (n − 1)
that of s³ by
2
n (n−1) (n−2)
2X3
n (n−1) (n−2) (n—3), &c. to obtain the Values of all the
2 X 3 X 4
Terms of the ſecond Member of the foregoing Equation, thoſe Values be-
ing found, and fet down under one another, there refults
-p4+, &c.
r
you
j
r
да
g
I
2
*3
1+ns=
itn. — f.fn.
n n
N 12
n
71 11
72
71
H
n.n-I
+.
NNI 2
2
nn— I p² go
-p²+n.·
2 X 3
·p³+n.
2
+
n.n-1.1-2
2 X 3
← 53 =
n.ni.n—2 +3
2 X 3
nnn-2.1-3
+
4
2 X 3 X 4
+
N.N-I
2
なる
​• I
2
n² n
2
な
​n
I 2
p4+,&c.
f
n3p3 4
N
n nr. 2
2 X 3
313 r
2123 12
·sp4+, &c.
+4
+
2 X 3 X 4
78
n
II
+, &c.
n n—1.n—2 1—3
p++,&c.
2 X 3 X 4 nፋ
+, &c.
Reducing the Terms of the fecond Member of this Equation, there
nn-I
n.nn-2
refults, 1+ns+
-52-f
s³+
2
I
2 X 3.
1.4-[1-2
$4+, &c.
to
The gene-
ral formula
p4+, &c. found for
n.n~I.n-2.12-3
2 X 3 X 4
4.7—1.j—— 2 go. ·3
p3+
2 X 3 X 4
Wherefore the general
rem found for the Involution of Binomials will ferve alfo for their
=1+rp + "== p²+
2
2 X 3
which is preciſely the foregoing Equation.
lution.
XXXI.
the involu-
Theo- tion of bino-
mials, ferves
Evo- alfo for their
Let it now be propoſed to prove that the fame Formula ferves to raiſe
Quantities to negative Powers, either Integral or Fractional, or that
go
#
(2+2) =26
3*
+
*"
ተ
-2
+++
I
72
11
n
n
4
N
x+
น
2²+
2
2 X 3
12
and putting
which Equation by multiplying the two Members by u
evolution.
**
--
น z³+,&c.
צוא
Z
is reduced to
2.
да
*
+1
+z
18
n
17
n
n
n
(17p)
pt
n
2
2 X 3
It is manifeft that inftead of the Quantity (1 + p)
*
g
-+1 -+2+-+3
·p4+,&c.
n n
B
n
2 X 3 X 4
we may ſet
n
142
ELEMENTS OF
down
I
(1+1)
I
*
confequently the foregoing Equation is transformed into
11
3
j
да
11
n
= I
72
2
+
que
**
+ I
+2
n
12
n
2 X 3
p³ +, &c.
(itp)
12
But as we can ſubſtitute for (1 + p) its Value deduced from the For-
mula of the foregoing Article, it is manifeft that it fuffices to prove that
I
y
j
да
↑
11
I
2
2
n
n
It -pt
82+
n
17
ニー ​3
3
12
n
2
2 X 3
p3+
n
n
n
n
2 X 3 X 4
p4+,&c.
+
*
*
да
1
+ I
+ I
•
-+2
n
it
n
n
n
=I
$2
·p³+
12
n
n
n
72
2
-p4—,&c.
If
*
n
2 X 3
+1 · 2+2 · — +3
2 X 3 X 4
Now to prove this Equation, it fuffices to multiply the fecond Member
by the Denominator of the first, and to make it appear that the Produc
that refults is Unity, the Numerator of the first Member: which happens
in effect; for performing the Multiplication, there refults.
1'
•
n
2
11
+1
$2
2 X 3
n
n
+2
2 X 3 X 4
*
ጥ
jo
↑ ↑
+ I.
+2
+1. = +2. — +3
83+
n
12
n
n
p4, &c.
j
1"
go
12
p²+
+ / 5
j
j
↑
+1.
n
$3
11 12
11
n
n
12
12 n
2
2 X 3 X 4
p4+, &c.
y
4
go
jo
jo
j
j
jo
I
1
I
+1
η
n
n
n
n
n
+
p³ +
72
n
n
2
2
2 X 3
p4~, &c.
y
"
j
y
j
24
2
12
12
12
n
n
n
+
2 X 3
2 X 3
p4 +, &c.
+
1*
I.
2
-3
n n
12
n
+
p4 —, &c.
15
2 X 3 X 4
+, &c.
whofe first Term, which is Unity, is the only one that remains after the
the Reduction.
XXXIII.
Since to extract any Root of a given Quantity, is the fame Thing as to
raiſe that Quantity to a Power, whofe Exponent is a Fraction that has its
SPECIOUS ARITHMETICK.
143
of the fore-
Denominator equal to the Number that expreffes what Kind of Root is Application
to be extracted, "it is plain that the Root of any Quantity may be found by going formu
the foregoing Formula, by putting moror, &c. according as it la for find-
is the fquare Root, cube Root, or the Root of the fourth Power, &c. ing the rocts
of perfect
that is required; and if the propofed Quantity is an exa& Square, or an powers of
exact Cube, &c. the Series will ſtop of itſelf, the Numerator of one of compound
quantities.
the Terms, in this Cafe vaniſhing; if not, it may be carried on to any
Number of Terms, as will appear by the following Examples.
Let it be propofed to extract the fquare Root of 1 + 26 + b², put-
ting u=1, z=2b+ b², and m in the foregoing Formula, I find
the first Term u = 1, the ſecond
Z
TAL
mn X (m- 1) 111
Z
W
m X (m-
the 5th.
น
B2
=
*” — 2 22 — — — b² —
=
2)
(m
1) X (22
2 X 3
m X (m
17 m Ι
mu z = 6 + 1/6², the third
22
I
I
Z
63-64; the 4th. Term.
8
I
um = 3 x3 = =
= 63 + 3 64 + 3 - 65 + 1/65 66
น
2
2) X (m ༣)
1) X (m
2 X 3 X 4
5
128
5 65 - 15- 66 - 15867
4
4
171
21
4 24 =
8
동 ​64
68. The 6th., &c. and the
Sum of all thofe Terms will be the Root required.
I
Adding all thofe Terms, I obſerve that the firſt 1 remains entirely, that the
I
I
2d. b + — b² is reduced to b, becauſe the Part b² of this ſecond Term
2
2
is deſtroyed by the fame Quantity with a negative Sign in the third Term
I
64; that what remains of the third Term
I
I
62
63
2
2
I
I
63
64 after
I
62 is deftroyed, ſhould be likewiſe en-
8
2
2
I
63+
3
64 + 3 / 65+ 66. of
2
369
8
tirely deſtroyed by the fourth
4
I
16
I
which after the Reduction there only remains 64 + 365 + 1/16 6.
8
this Remainder of the fourth Term is deftroyed in like Manner by the fifth
Term, and continuing the Reduction further on, it will appear that the
Whole vanishes except + b, which is the Root required.
XXXIV.
Let the Root of the 5th. Power of the Quantity a+b be required, Application
fubftituting the greateſt of the two Parts of this Quantity, which I fuppofe of the fore-
to be a in the Room of 1, the leaft Quantity b in the Room of z, and going formu
la for find-
I
in the Room of m, there will refult for the Root required
5
144
ELEMENTS OF
ing the roots
of imperfect
powers of
compound
quantities.
The forego-
I X4 X9X 14 X 19
{
14
१
5
+
a
b
1 X 4
5
a
62+
1 X 4 X 9
นา
a
63
5
2 X 25
2 X 3 X 125
19
24
1 X 4 X 9 X 14
5
5
64+
2 X3 X4 X 625
I
5
b
or ax(1+.
5 a
2 bb.
+
25 a a
a
a
b5~,&c.
2164
625 a4
+, &c.)
2X 3X 4X5X3125
663
125 03
Though it be requifite to carry on this Series to an infinite Number of
Terms, in order that it may exprefs exactly the Root required, however,
as this Series will converge fooner, the greater a is in Refpect of b, by
taking a Number of its Terms we fhall approximate to the Root to any
affigned Degree of Exactneſs.
I
I
Let, for Example, b = a the fix firſt Terms of the Series will be
5
I
a × (1+
50
10
100000),
2
3
+
1250
62500
21
6210000
+
399
1561500000
XXXV.
which converges very fwiftly.
Let it be propoſed to approximate to the Quotient of
a a
b
I bring
x + 6 •
ing formula the Quantity xb from the Denominator to the Numerator, by chang-
applied to ing the Sign of the Exponent I into its oppofite 1, fo that the Expreffi-
raifing quan
tities to ne- on ftands thus, a a X (x + b)~ ': I then reduce (x+6)¹ into a Se-
gative pow- ries, by fubftituting the greateſt of the two Parts x in the Room of u, and
the leaft in the Room of z, and I in the Room of m, and I find
ers.
Difficulty
that occurs
when the
a a
aax(x+b)−1 = +
X
If in the propofed Quantity
a a
1º.
b- b
a a
I
I
a a b
+
a a b 2
3
a263
F
*4
+
a² 64
*5
+&c.
XXXVI.
a a
"
b + x
b was, then we will have
22 × (1+1+1+1+1+1+1+, &c.), whoſe
b
two terms Value is infinite, which is no more than what we knew already, fince
of the deno-
minator are
a a
equal.
bb
{
a a
O
exprefs an infinite Number.
a a
2º.
b+b'
a a
whence 6+6
a a
The Sign being employed by the Analyſts to
a a
27 × (1 − 1 + 1 —, &c.) =
2
x (0+o+, &c.
b
o, which is not true, confequently a Fraction whofe
}
SPECIOUS ARITHMETICK.
145
Denominator confifts of two equal Terms cannot in that State be reduced
into a Series: To remedy this Inconveniency, the Analyfts divide the Sum
of the two Parts of the Denominator into other unequal Parts: For Ex-
ample, in the propoſed Fraction let c = 2b+x, and the Frac-
tion will be transformed into
C
a a
a a
b + b
the Series arifing from it will be equal
whoſe Denominator being 26,
a a
6+6'
XXXVII.
may be re-
The Utility of the foregoing Formula is not confined to the finding by All kinds of
Approximation all Sorts of fractional or negative Powers; it is of infinite quantities
Ufe for reducing complicated Radicals: thus, if it was propofed to reduce duced into
[CA - (cctrr) x²]
to an infinite Series, I bring the Quantity foregoing
√(CA
4
√(C4.
2
CCX
x²)
ccx2) from the Denominator to the Numerator, by changing
the Sign of the Exponent into its oppofite, fo that the Expreffon ftands
플
​thus, [c4(cc-rr) x²]
1444
x (c4 - cc x²) which by fubftituting
cc + jr will become
for Brevity fake, a for cc and b for
14
2
(a² + bx²) x (a² — a x²)
플
​>
reducing thoſe two Factors into infinite
Series, and afterwards multiplying them into one another, there refults,
3a²+2abb2
8a4
a4 + 20 a³ b — 6 m² b² + 4a b3
I +
e + b
292
x² +
a
+
_35
128 48
of
√ (a² + bx²)
√ (a²
a 102)
x²)
5a³+3a²b—a b²+b3
*4 +
*б
16 26
-564
x8, &c. for the Value
feries by tho
formula.
XXXVIII.
of all forts of
radical quen
We have ſeen (Chap. II. Art. XXXIX, XL, XLI.) how the Analyfts Addition &
perform upon Radicals of the fecond Degree the Operations of Addition, fubtraction
Subtraction, Multiplication, and Diviſion; thofe Operations being equally
neceflary for radical Quantities of higher Degrees, we fhall proceed to ex- tities.
plain what thoſe new Radicals require.
As to Addition and Subtraction, they require no more than what has been
faid already of thofe Operations upon Radicals of the fecond Degree: it
fuffices to reduce each Radical to its moſt fimple Expreffion, and to add and
fubtract them as commenfurable Quantities.
Let it be propofed, for Example, to fubtract
3
(b4+2 a b³) from
3/ (8 a³ b + 16 a4); I transform the firft Quantity into b 3/(b + 2a),
and the fecond into 2 a 3/(b + 2 a), and then ſubtracting one from the
1
3 T
146
ELEMENTS OF
•
the other, the Difference is (2 a + b) 3/(b + 2 a): In like Manner,
3 a 4/(.16 68 + 32 b4a4) being added to 4 b 4ƒ(a4 b4 + 2 a8) their Sum
will be 10 a b 4 (64 +2a4).
XXXIX.
As to Multiplication and Divifion, if the radical Quantities have
Multiplica-
tion and Di- the fame Exponent, the Method is the fame as for Radicals of the fecond
Degree; it ſuffices to operate upon the Quantities to which the radical
dical quan- Sign is prefixed, and fet the fame radical Sign over their Product or
vifion of ra-
tities that
have the
fame expo-
pent.
Quotient.
Thus,
9 a² 63
4
5 ayy × 3/7 ay z =
3/35 a a y³ z,
or y 3/35 a² z
X √27 a³ 66 =
243 a5 b9
3 a b
5 64
4
3/(a²b²+64) divided by 3
a
86
4
gives for Quotient 3/
8 a² b3+865
a²
-62
९
=263 / 22=62
2
a²+b²
3/(a2x3
ફે
x5)
9
3√ (a b4 +b4x)
XL.
x
a
X
b
b
To perform
But to perform thofe Operations on radical Quantities of different Ex-
thofe opera- ponents, they muſt firſt be reduced to others of equivalent Value that ſhall
tions upon have the fame radical Sign.
radical quan
tities of dif- Let, for Example, 3 a b and 5 a bb be propoſed to be reduced to
ferent expo- the fame radical Sign, I raiſe a b to the fifth Power, and fet down 15 in
are firit to be the Place of 3 and there refults. 15 a5 b5 = 3√ a b, I likewife raiſe
reduced to a b to the third Power, and fet down 15 inftead of 3y, whence 5y a bb
I
15 a3 b6.
the fame ex-
ponent.
Method of
Now if it was propoſed to multiply 3 a b by 5y a b b the Product would
be 15y a8 b', if the firft was to be divided by the fecond the Quotient would
be 15
a5 65
a3 b6
I 5
b.
In like Manner the Product of 3y a2 64 into y a4 b5 would be 6√ a¹6b23
=a2 63 6✓ a4 b6.
In general to reduce two radical Quantities" at be and / ar bs to the
performing fame Exponent, the firſt muſt be transformed into "*/ ap" bq", and the ſe-
this reduc- cond into my arm bms. If they are to be multiplied, their Product would
be map + rm bgnsm, and if the first is to be divided by the fecond,
the Quotient will be "ap" - rm fqu— sm.
tion.
mn/
bqn
It is eaſy to perceive that when the Exponents of the radical Quantities
have a common Divifor, there is no Neceffity of reducing each Radi-
cal into another, whofe Exponent is the Product of the two firft Expo-
nents; for Example, if a b and a b³ be propoſed, the firſt will be tranf-
formed into 4/42 62; in like Manner if 4/a3 b, ab are transformed in-
to ¹²/a9 b³ and ¹²✅a¹² b².
12
12
12
4
SPECIOUS ARITHMETIC K.
$47
XLI.
The foregoing Operations may be performed after another Manner, by
confidering the radical Quantities as Powers with fractional Exponents.
p q
Thus to multiply "√ a b by "✔ab is the fame Thing as to multiply
{
P
q
b" by a" b", whofe Product is a
771
£ +
1 + 1/
patri
sm + qr
mn
or a
b
Another
Method of
performing
If it was propoſed to divide the firft Quantity "ap ba by the fecond the forego-
"arbs the Quotient would be a
1/
------
or a
pr
s m
MA
b
VILN
ing operatio
QDS,
If it was propofed to divide 2 at 63% c² by Pv a²m bc, transforming
31
I
the firſt Quantity into a 2PbPc P
2m
n
2.m
#
and the ſecond into a p b p c p
and ſubtracting the Exponents 2,7
the ſecond Quantity from the Exponents
M
I
, of the Letters a, b, c in
3n
2p 20 p
of the fame Let-
ters in the firſt Quantity, the Quotient will be found to be
X
3122 *
2p b2p x1=
b
2P
3m
a 2p
XLI.
Compound Surds are fuch as confift of two or more joined together;
fimple Surds by being multiplied into themſelves give at length rational
Quantities, yet compound Surds multiplied into themfelves, commonly
give ſtill irational Products, but when any compound Surd is propofed,
there is another compound Surd, which multiplied into it, gives a rational
Product; thus a+b multiplied by ab gives a-b. whence
Fractions affected with furd Quantities, may be reduced to a more fimple
Expreffion, by multiplying the Denominator of the Fraction by that
Surd which will give a rational Product, and multiplying the Numerator
by the fame Surd.
XLIII
Let amb exprefs any binomial Surd, and let it be propofed to trans-
form it into another Binomial, in which the Quantity a fhall have a given
Exponent n, and the Quantity ỏ ſhall not be multiplied by a, that is,
}
148
ELEMENTS OF
let it be propofed to find a multinomial, by which the propoſed Bi-
nomial being multiplied, the firft Term of the Product, fhall be a”; and
all the other Terms except the laſt, ſhall deſtroy each other.
Let a* by + a≈ bu be any two Terms of this Multinomial immediately
fucceeding each other, and let them be multiplied by the propofed Binomial,
Inveftiga- the Product thence arifing will be am+x by+ax ¿l + y +ax+x tx + aż bit, and
Surd which the firft Term of the Product fhould be an, m + x = n, x = n—m
multiplied and yo, wherefore the firft Term of the Multiplicator is a
tion of the
into a pro-
pofed Surd
gives a ra-
tional Pro-
duct,
going The
72
To find the fecond Term, I fubftitute this firft Term in the affumed Pro-
duct, whence there refults a +an-m bl + am+xlu + az ¿itu but + ar—mbl
+ am+z = 0. Conſequently the fecond Term of the Multiplicator
tu
fhould be negative, and becaufe nm = m + z & Iu, Confequently
2 m and 2 lu+1, this fecond Term will be - at 2112 bl.
To find the third Term of the Multiplicator, I fubftitute this fecond Term
in the affumed Product, and there reſults an — an—m bl. + an-mŀl + an=212.621.
Let now the laſt Term of this Product, be put equal to the first Term of
the affumed Product, becaufe thofe Terms fhould deftroy each other.
Z
a”
022
· 2 ·621 + am+zby = 0,
o, whence n 2; m = m + x, x =
1-3 m and y = 21 and 1+ y = 31, Confequently the third Term of the
Multiplicator will be + a3b2l. To find the fourth Term, I fubftitute
the foregoing Values in the affumed Product, and there refults a"-an- mbl
+an—m bl—an 212 621 +-an. 2112 626 +ax 3 m b31 + ant bu+abitu
whence n.
3 m = m + z & z = n 4 m and u 37
3 and += 4!
Confequently the fourth Term of the Multiplicator will be
and in General, the p Term will be a"-p" b (p-1)%
2m
a
นะ
4m b
b31
Now let the p Term be the laft Ferm of the Multiplicator, fince the
laft Term of the Product. refults from the Multiplication of the laft. Term
of the Multiplicator into the laſt Term of the propofed Binomial, and that
it contains none of the Powers of a, it will be expreffed by
nl
b (p − 1)1 + 1 — f m
as n
n
m
-
#
pmo and of Courſe, p= and its
112.
Sign is pofitive only, when is an odd Number, fince all the Terms of
the Multiplicator expreffed by even Numbers, are Negative...
nl
XLIV.
!
Application Whence in General, when any Quantity is divided by a Binomial Surd
of the fore-ambl where I and m reprefent any Fractions whatſoever, by taking n, the
orem for re- leaft integer Number, fuch that will be an Integer, and multiplying
ducing
Fractions both Numerator and-Denominator by a”.
m. I a
#- 2.11 bit an
37.621
involving
a 4b 31, &c. the Denominator of the Product will become rational
+
fard Quan
m
SPECIOUS ARITHMETICK.
OUS
149
and equal to a + b then dividing all the Members of the Numerator
tities to a
more fimple
by this rational Quantiy, the Quote arifing, will be that of the propoſed Expreflion.
Quantity divided by the Binomial Surd expreffed in it's laft Terms.
3
√5-√2
3√5+ 3√² = √5+ √23
√6
Thus
3
3/20
3/20
3√√16+2+3√4·
3/20
3/4 3/2 3/4-3/2
—
X
3√16 +2+3√ 4 = 3√/4-3/2x
√42+√18
4
2·3/2+2+3√ √ 4
23√√2 + 2 + 3√ Ä
2/10
2√2-3√3
23 √ 40 + 23√ 20 + ³√ 80
2
= 2 3√5 X 3/20 X³ío, alſo
4./20 +4./10X³./ 3 + 2 √√20 ׳√9+6√io+ ³/20⋅× ³/3 + 3 √10 × ³√92
I
−8√5 — 4√ 10 ×· 3.√✓. 3 — 8√5׳√9 −6/10 −6 × √5 × 3√3—3√1º × √9ª
XLV.
To compleat the Refolution of Equations, confifting of two Terms,
it remains to explain how the unknown Quantity is difengaged when it enters
the Exponent.
да
Thus if it was required to find in what Time x a Sum a lent out at In-
tereſt at a given Rate per Cent. would amount Principle and Intereft to a
Süm r, the Equation to be folved would be rapor
Equation the unknown Quantity enters the Exponent.
XLVI.
2=t, in which
a
&c.
tion of the
Since the Exponent x includes all determined Numbers if in the Room Inveſtiga- -
of x be ſubſtituted fucceffively all the integer Numbers there will reſult. Method of
1228 57. 16: 15. p4.p³. p². p². po.. p²
po.. p². p²..p3.p4
folving E-
which forms a geometrical Progreffion. Now, if between the Terms of quations
this Progreffion, there be inferted a Number of Terms n, it is eafy to confifting of
perceive that they will be expreffed by the Powers of p: with fractional Ex-Te
ponents, and that there will refult a new Progreffion, the Multiplicator being unknown
two Terms
Quantity
Exponent,
I and that ni may be conceived fo great and confequently the Pro-enters the
greffion may be conceived to increaſe fo flowly, that all the Numbers from
Ito co, and from 1 to 0, will be found there, p being fuppofed to be an
Integer, and greater than Unity.
XLVII.
Let for Example p 10, then the Progreffion will be transformed into
10000000
I
1000000
"
I
100000
I
I
I
ร
10000
1000
100
IQ
1, 10, 100, 1000, г0000, 100000, &c..
1
750
ELEMENTS OF
Now, between 1 & 10, there are eight whole Numbers, 2, 3, 4, 5, 6,
7, 8, 9, and between 10 and 100, there are 89 whole Numbers, viz.
11, 12 &c. but if between each of the Terms of this Progreffion, there
be inferted a Number of Terms n, there will refult a new Progreffion, the
I
I
common Multiplicator being 10 "+' among whofe Terms thofe interme-
diate Numbers will be found.
To find the Term of this Progreffion for Example, which is equal to 3,
I proceed thus, the Number 3 required being one of the Terms of a geo-
metrical Progreffion, of which I & 10 are the Extreames; I inveſtigate
the middle Term of this Progreffion; now, this Term being equally
diftant from the Extreams I and 10, thofe three Terms the firft I the
middle Term and the laft Term 10 will be in continued Proportion, ſo that
the middle Term will be equal to IX 10 or 10,5, which I find to be
3,16227766. The Number 3, 16227766 being greater than 3, I invefti-
gate a new Term that will be the Mean of a continued geometrical Propor-
tion, having for Extreams 1 & 3,16227766, which will be 1X3,16227766
or 10,25 = 1,77827941. As the Number 3 is greater than 1, 77827941, or
10,25 and less than 3,16227766 or 10,5, I inveſtigate a new Term, a mean
proportional beween 1,77827941,and 3,16227766, which will be the fquare
Root of 10,25 X 10.5 that is of 3, 16227766 X 1, 77827941, or of
5,6234132514809806, which is 10-375 or 2, 37137370. As the Number
3 is greater than the new Term, 2,37137370 and lefs than 3,16227766, I
inveſtigate a mean Proportional between those two Terms by extracting the
fquare Root of 10,375 X 105 that is of 2,87137370 X 3,16227766 which
is 2,73841962. As the Number 3 is greater than the new Term 2,73841962
and less than 3,16227766, I find a mean Proportional between thofe two
Terms by exracting the fquare Root of 10,4375.X 10,5 that is of 2,73841962
X 3,16227766 or of 8,6596431880316892 which will be 10,46875 or
2,94272717. The Number 3 being ftill greater than the laſt Term
2,94272717 and lefs than the firft 3, 16227766, I inveſtigate a mean Pro-
portional between theſe two Terms, by extracting the fquare Root of
10,46875 X 10,5, or of 9,305.7203891660222, which will be 10,484375, or
3,05052789.
Proceeding in this Manner to inveſtigate new Mean proportionals between
two Terms, one greater and the other leſs than 3, after having performed
26 of thoſe Operations, I find 10,47712125 3,00000000, which does not
differ from 3 by a decimal Unit of the eighth Order, theſe 26 mean Pro-
portionals are fet down in the following Table, each of them being placed
between the Extreams, and to indicate the Order of the Operations, the
two firſt Extreams are marked by the Letters A and B, and the mean Term
placed between them by C, the other Means are denoted by the following
Letters of the Alphabet.
SPECIOUS ARITHMETICK.
151-
XLVII.
A
C
100,00000000 = 1,00000000 P
100,50000000 = 3,16227766
2
B
IQ1,00000000
N
= 10,00000000
100,47717285 = 3,00035655
10°,47711182 = 2,99993491
10°,47705078-
= 2,99951334
A 100,00000000
1,00000000
2
D
I00,25000000
C
I0°,50000000.
D 100,25000000
1,77827941 R
3,16227766 P
1,77827941 R
10º,47711182 = 2,99993491
10,47714233 = 3,00014572
E 10º,37500000
2,37137370
S
C I0°,50000000
3,16227766
Q
இ.
10,47712708 3,00004031
100,47611182
100,47717285 = 3,00035655
10º,47714233 = 3,00014572
= 2,99993491
E
F
100,37500000 =
F 10°,43750000
C 100,50000000 =
10º,34750000
Ꮐ 100,46875000 =
C 100,50000000 =
G 100,46875000 =
H 10°,48437500
2,37137370
S
100,47712708 .3,00004031
2.73841962 T
100,47711945
2,99998751
3,16227766
இ
2
10,47711182
2,99993491
C
10°,50000000
2,73841962
2,94272717 Ꮴ
3,16227766
2,94272717 V
3,05052789 X
3,16227766 T
T
10º,47711945 = 2,99998751-
AS
10,47712326
100,47712708
3,00001390
3,0000403I
10º,47712326 = 3,00001390
100,47712135 = 3,00000070
10º,47711945 = 2.99998751
H 100,48437500 =
I
10,47656250
3,05052789 X
2,99614286
10047712135 = 3,00000070
G
10°,46875000
2,94272717
T
I
100,47656250 = 2,99614286
r
K 100,48046875 =
3,02321308
H 100,48437500
3,05052789
K | 100,48046875
3,02321308
L 10,47851563
3,00964753
I 100,47656250 =
2,99614286
L
100,47851563 =
M 10,47753906
3,00964753
3,00288762
I 10°,47656250 =
2,99612286
M100,47753906
3,00288762 BB.
N
10,47705078
2,99951334 CC
X
I 100,47656250 = 2,99614286 X
N 100,47705078 = 2,99951334
10°,47729492 =
M100,47753906
T
CC
3,00120000
DD
P
100,47729492
10047717285
N 10047705078
3,00288762
3,00120000
3,00035655 EE
2,99951334
BB
B B
DD
= 2,99997739
10,47712135 =
3,00000070
10º,47712088 = 2,99999739
10,47712112 = 2,99999908
100,47712135 = 3.00000070
100,47712112 = 2,99999908
10°,47712123 = 2,99999989
100,47712135
3,00000070
100,47712123 = 2,99999989
10º,47712129 = 3,00000029
100,47712135 = 3,00000070
100,47712129 = 3,00000029
10,47712126
3,00000009
100,47712123 = 2,99999989
100,47712126 = 3,00000009
100,47712125 = 3,00000000
10,47712123 = 2,99999989
NK
100,47712040 2,99999410
100,47711945 = 2,99998751
100,47712040 = 2,99999410
10,47712088
X
Z
A A
X
A A
B B
1152
What is
meant by
the Loga
rithm of a
Number.
*
譬如
​SPECIOUS ARITHMETIC K.
By the fame Method of Proceeding, the Terms correfponding to the
-other intermediate Numbers will be found, by inveſtigating mean Propor-
tionals between andro, for the Numbers less than 10 between 10 &
100, for the Numbers lefs than 100, but greater than 10; between 100
& 1000, for the Numbers less than 1000 and greater than 100, &c.
XLVIII.
Whence if p be given, and if px.
be given, and if pb, we can find the Value of x fuch
that pb: this Value of x is called the Logarithm of b, and p the Baſe
of the Logarithm.
There are therefore as many different Syſtems of Logarithms, as there
are different Numbers p which may be affumed for the Bafe, but in two
Syſtems the Logarithms of the fame Number are always in a conſtant
Ratio. Let the Bafe of one Syſtem = p, and of the other q, and the
Logarithms Logarithm of the Number b in the firft Syftem= x, & in the ſecond=y,
'How the
of Numbers
of one Syf-
tem are de- then = b and qy=b, wherefore px = qy confequently p
p= pr
Aduced from
Xx
q y whence
thofe of ano- the Fraction is always the fame whatever Number is affumed for b.
ther.
In every
卜
​}
Syftem the
Logarithm
of Unity
iso.
The Lo-
*is
3
Whence if one Syftem of Logarithms of all Numbers were computed, the
Logarithms of any other Syſtem may be found by the Rule of Three.
Thus if the Logarithms correfponding to the Bafe 10, were calculated, the
Logarithms correfponding to any other Bafe for Example 3 may be found.
Let the Logarithm y of the Number b correfponding to the Bale 3 be re-
quired, the Logarithm of the fame Number b correfponding to the Bafe
io being given. Since Log. 3, correfponding to the Bafe 100,47712125
and Log. 3, correfponding to Baſe 31. 0,47712125: 1x y, where-
X
0,47712125
*
1=
:
fore y
= 2,095933 X.x, if therefore all the Logarithms
correſponding to the Baſe 10, be multiplied by 2,095933, there will reſult
the Logarithms correfponding to the Bafe 3.
XLIX.
In every Syftem of Logarithms, Log. 10, for if in the Equation
xb, we put b=1, then = 0.
x
The Logarithms of Numbers greater than Unity, are pofitive, thus,
Log. pi; Log. p22; Log. p33, &c. But the Logarithms of
Numbers lefs than Unity, but pofitive, are negative, for Log. —=—1;
I
garithms of Log. 22
negative
Numbers
are ima-
ginary.
I
2, Log.
p3
Numbers are imaginary.
·I
3, &c. and the Logarithms of negative
In like Manner if Log. px, then Log. p²
neral Log. p" nx or Log. pn Log. p,
2x Log. p33x and in ge-
on Account of x= Log. p.
1
SPECIOUS ARITHMETICK.
153
x
The loga-
product of
rithm of the
whence the Logarithm of any Power of p is equal to the Logarithm of
multiplied by the Exponent of the Power; thus Log. √p== Log. P,
But if Log. ay. is equal to
Log, p˜—— Log. p; and fo on.
I
p
=
two numbers
the Sum of
Log. JP.
and Log. bx: becauſe a=py, and b=p, then Log. a b = y +x, whence the loga-
the Logarithm of the Product of two Numbers, is equal to the Sum of rithms of the
the Logarithms of the Factors. In like Manner it will appear that factors, and
Log. 2=y-x=Log. a― Log. b, hence the Logarithm of a Fraction
is equal to the Logarithm of the Numerator lefs the Logarithm of the
Dénominator.
a
Log. r- Log. a
Log. p
Whence from the Equation pris deduced x =
it being indifferent what Syftem of Logarithms is employed (Art. XLVIII).
L.
The Refolution therefore of Exponential Equations, or of thofe in which
the unknown Quantity enters the Exponent, is reduced to the Inveſtigation
of a general Formula for finding the Logarithm of any given Number.
y'
the loga-
rithm of
their quo-
tient, to
their diffe--
rence.
general for--
Let y, y', y" exprefs any three Terms of the Progreffion of Art. XLVI. Inveſtiga-
immediately fucceeding each other; we will have y: y=y: y", or tion of a
y' : y =y" :y', Conſequently yyyyyy', or (putting mula for
dy dy' finding the
y! — y=dy and y"—y' = dy! ) dy : y = dy' : 'y', wherefore
ابو
dy
whence is a permanent Quantity.
لا
y
=
Let the Exponent of the Power of p which is equal y be expreffed by x,
the Exponent of the Power of p, which is equal y' be expreffed by x + d x,
the Exponent of the Power of p which is equal y will be expreffed by
x + 2.dx, becauſe the Exponents of the Terms of this Progreffion increaſe
by the ſame conftant Difference dx, whence affuming M for a determined
Quantity, dx =
› confequently is the Value
of the Ratio of the cotemporary Increments of any Number y and of its
Logarithm x.
M dy
y
or
dx
dy
M
J
M
J
dx
To find the Ratio of the Quantities y and x from that of their Incre-
ments, I put y=1+, and there refults dy dz, and =
M =MX (1 + z)
r+z
I multiply this Value of
M⋅ (z — x² + 23 —
·
dx
dz
dy
dx
dz
—M(1−x+x² — z³+z4—25+ &c).
by z, and confequently there will arife
24 + 25 — z6+ &c.) which Terms I di--
logarithm of`
any given
number. -
vide feverally by their Number of Dimenſions, and the Refult:
3. U
154
ELEMENTS OF
'I
I
I
-1
I
I
MX ( 2 -
x²+
≈3 24 +
25
2
3
4
5
6
26 +, &c).
I put = x, to which Equation no conftant Quantity is to be added, be-
cauſe when zo it is reduced to O, as it really fhould, for when zo
J =1+x=1,
= 1, whofe Logarithm o. And by this Equation is expreſſed
the Ratio between x &y, fince from this Equation we can return back to
the propofed Equation.
LI.
By this Formula, the Logarithm of any Number greater than Unity,
but less than 2 may be found; I fay lefs than 2, for if + is equal 2,
then the two Terms of the Denominator will be equal, and the Series will
be erroneous, and much more fo when z exceeds 2, however, the foregoing
Formula may ſerve for finding the Logarithm of any Number greater than
Unity, by calculating the Logarithms of fuch Numbers that are leſs than 2,
which multiplied into one another, or divided by each other, produce the
How it may propofed Number.
Inconve-
niency to
which the
*foregoing
.method is
Miable.
be removed.
Anotherme-
rhod for
finding the
logarithm of
any number
For Example, finding that
12
10
8
ΙΟ
12
X
IO
X
9
ΙΟ
I.
I, 2 X 1, 2
0, 8 X 0, 9
=2,
I calculate by the foregoing Formula the Logarithm of 1, 2, putting
zo, 2; the Logarithm of o, 8 fuppofing z=0, 2, and the Lo-
garithm of 0, 9 by putting z=— o, 1. I add together the Logarithms
of o, 8 & 0, 9, and deduct their Sum from the Double of the Logarithm
of 1, 2, and there reſults M X 0,693147180559, &c. for the Logarithm
of 2 required.
To determine the Value of M correfponding, for Example, to the
Syſtem of Logarithms whofe Bafe is 10, I firſt inveſtigate the Logarithm of
10, finding that 2X 2 X 210, I triple the Logarithm already found,
o, 8
and deduct from it the Logarithm of o, 8, the Remainder will be
MX 2, 302585092994, &c. is the Logarithm of 10; now the Lo-
garithm of 10 correfponding to the Bafe 10 is 1, and of Courſe we will have
1 = MX 2,302585092994, wherefore the Value of M correfponding to
the Syftem of Logarithms, whofe Bafe is 10, will be o, 43429448, &c.
LII.
The foregoing Method of finding the Logarithms of Numbers being not
very ready, we fhall proceed to explain how the Analyfts have remedied
this Inconveniency.
Let the Sum of any two Numbers be expreffed by z,
rence by v, confequently the greateſt will be z +
v. Now let y=
บ
and their Diffe-
v, and the leaft
七十
​z + v
and fuppofing &
62
SPECIOUS ARITHMETICK.
F55
to be a permanent Quantity, we will have dy =
Wherefore the Equation.
the Equation.
d x x ( z v) (z v - do)
2 Z
do
(≈—v) (2—~—dv)
will be transformed into
d x
M
dy
y
MX (2)
d x
or into
22. do
2 z. M
z+v
do
2 z. M
Wherefore
,
(z
≈ + v) x (≈
do)
V
dv)
(~+~) X (≈-0
V
expreffes the Ratio of any Cotemporary Increments of x and v, and this
Ratio is always greater than
22. M
(2+·~) (~~~)
(≈
(≈
but do decreaſing, this
Quantity will alſo decreaſe, and as we may take do as fmall as we pleaſe,
we may make
to
2 z. M
(≈+v)X(x —v-d) approach as near as we pleaſe-
(≈ + v) (≈ — ~)
d x
2 z. M
wherefore
2 z. M
(≈+v) (~—0)
is the Limit of
the Ratio
do
To find the Ratio of the Quantities and v, from the limiting Ratio of
their cotemporary Increments
2 z. M
(≈+v) (2−v)
2 M z
-
2 M. z. (2²— v² ) —¹ — 2
M(
—
22
24
+
06
+
23
+
≈5
+ &c
27
dx
c);
I multiply this Value of
23
~5
2 M ( 2/2 + + +
Z
z3
z5
verally by the Number of
by v, and confequently there will arife.
27.
their Dimenſions, and I put the Reſult
7
+ &c), which Terms I divide ſe-
2 M ( /2/ +
(-
Z
3 23
323
+
25
5 25
+ + &c) equal x,
3
07
727
to which Equation no conftant Quantity is to be added, becauſe when vo
it is reduced to o, as it really fhould, for when vo, the Fraction
= 0.
z+v
&
is reduced to =1, whofe Logarithm
2
≈
LI II.
Whence if the Sum
Difference by v, and if
of two Numbers be expreffed by z, and their
2 M v
be put A,
=B,
A q²
Da²
Z
= Ag
Cv²
24
=D, &c. the Logarithm of the Quotient of the greateſt divided
by the leaſt will be expreffed by A+B+C+D+ &
3
156
ELEMENTS OF
For Example, putting M=1, the Logarithm p of the Quotient of*·
126 divided by 125 may be found, the Logarithm q of the Quotient
of 225 divided by 224, the Logarithmr of the Quotient of 2401 di-
vided by 2400, and the Logarithm of the Quotient of 4375 divided
by 4374.
But p Log. 126 Log. 125 Log. 2. 32. 7-Log. 53 Log. 2
pLog. — = =
+2 Log. 33 Log. 5+ Log. 7
q= Log. 225-Log. 224
+ 2 Log. 3 + 2 Log. 5
Log. 7.
r=Log. 2401 - Log. 2400
Log. 32.52
- Log. 25.7=-5 Log. 2
Log. 74
2
Log. 25. 3. 5²=-5 Log..2
Log. 3-2 Log. 5+ 4 Log. 7.
S=Log. 4375— Log. 4374 = Log. 54.7
7 Log. 3+ 4 Log. 5 + Log. 7.
Log. 2. 37=- Log. 2.
Whence a p a Log. 2 + 2 a Log. 3
a Log. 2 + 2 a Log. 3 — 3 a Log. 5+a Log. 7.
bq=5b Log. 2+2b Log. 3. + 2b Log. 5. —b Log. 7.
cr= 5 c Log. 2 -c c Log. 3 2 c Log. 5+ 4c Log. 7.
df d Log. 2-7 d Log. 3 + 4d Log. 5+d Log. 7.
Let a, b, c, d be fuch, that ap + by + c r + df = Log. 2 + Log. 5,
whence arifes the following Equations for determining thofe Quantities,
a - 5 6 — 5 c — d = 1
I; 2a + 2 b c—7d=0; 3a+26-2 6
+4
+ 4 d = 1; a—b+4c+d=0; which give a 239, b= 90, c:
=
90, c——63,
d=103; wherefore 239 + 90 q-63r+ 103, or 2, 302585092924,
Log. 10, the fame as before, and the Value of M correfponding to
the Syftem whofe Bafe is 10, will be found to be 0,434294481903, and
fubftituting for M this Value, there will reſult
=
M (202p+76q — 53 r + 87 ) Log. 7.
M (167 p +639-44 r +72)= Log. 5.
M(114 p +43 730 r+49) Log. 3.
=
1 — M (167 p +63944r + 72S)= Log. 2.
I
44r+72S)
Having M, and the Logarithms of the prime Numbers 2, 3, 5, 7 below
10, to obtain the Logarithms of the prime Numbers 11, 13, 17, 19 &c.
Above 10, let A be any prime Number, I fuppofe that the Logarithm of all
the prime Numbers that precede A have been found, then
Log. A=
2
[Log.
A2
AL
+ Log. ( A− 1 ) + Log. ( A + 1). ].
If the Logarithms of all the prime Numbers that precede A, are not
known, let the Logarithm of any other Number B be given. Find the
Logarithm of the Fraction if A be greater than B, or of if it is
A
B
B
A
leſs: This Logarithm being found, add it, in the firſt Caſe, to the given Lo-
SPECIOUS ARITHMETICK.
157
A
garithm of B, and there will refult the Logarithm of B, that is, of A.
X
B
B
In the ſecond Cafe, deduct the Logarithm of the Fraction from the
A
B
B
Ā
given Logarithm of B, and there will refult the Logarithm of
B X
A
B
=
= A.
미녀
​LIV.
Application
of the fore-
To fhew the Uſe of theſe Formulas by an Example, let it be required to going for-
find in what Time 5757. will raiſe a Stock of 756 1. 13 s. 2 d. 4 at
per Cent.
}
4
In this Cafe we have p=1,04, a=575, and r=756, 66. Whence
Log. 756, 66 — Log. 5757 the Number of Years required.
I
Log. 1, 04
LV.
mulas to an
example.
rithms of
bale to the
The Logarithms correfponding to the Baſe 10, befides their Utility which The loga
they have in common with other Syſtems, are the beſt adapted to numeral numbers
Arithmetick. For fince the Logarithms of all Numbers except of the correspond-
Powers of 10 are expreffed in decimal Fractions, the Logarithms of Num- ing to the
bers between 1 and 10, will be contained within the Limits o and I, the belt adapted
•Logarithms of the Numbers between 10 & 100 will be contained within to numeral
the Limits 1 and 2, and fo on. Every Logarithm therefore is made up of arithmetick
and why.
an Integer and a decimal Fraction, the Integer is called the Characteriſtick,
and the decimal Fraction the Mantiſſa. The Characteriſtick confifts of as
many Units lefs one as there are. Places of Figures in the Number to which
it correfponds, the Characteriſtick, for Example, of the Logarithm of the
Number 78509 will be 4, becauſe it confifts of five Figures. Whence,
from the Logarithm of any Number it is perceivable, how many Places of
Figures it confifts of: Thus the Number correfponding to the Logarithm
7,5804631 will confift of 8 Places of Figures.
LVI.
If the Logarithms differ only by their Characteriſticks, the Numbers to
which they correfpond, will be to each other as a Power of 10 to 1, and
confequently will be expreffed by the fame fignificative Figures, thus the
Numbers correfponding to the Logarithms 4,9130187, and 6,9130187
will be 81850 & 8185000, the Number correlponding to the Logarithm
3,9130187 will be 8185, and the Number correfponding to the Logarithm
0,9130187 will be. 8,185: The Mantifla therefore indicates the Figures by
which the Number is expreffed, and the Characteriſtick fhews how many
Figures to the right Hand are to be feparated by a Comma. Thus, if the Lo-
K
358
ELEMENTS OF
Ufe of lo-
perations.
garithm 2, 7603429 was found, the Mantiffa indicates thofe Figures 5758945,
and the Characteriſtick 2 determines the Number correfponding to this Lo-
garithm to be 575,8945. if the Characteriſtick was o, the Number corre-
fponding would be 5, 758945, if the Characteriſtick was-1, the Num-
ber correfponding would be ten Times lefs, viz. 0,57589450, and if the
Characteriſtick was 2, the Number correfponding would be ten Times
lefs, viz. 0,05758945, &c. in the Room of thofe negative Characteriſticks,
-1,2,3, &c. the Analyfts fet down 9, 8, 7, &c. it being under-
ftood that thofe Logarithms are to be diminifhed by 10.
LVII.
The Tables of Logarithms are of great Ufe for performing with eaſe,
garithms for and Expedition the Operations in numeral Arithmetick, becauſe, by the
performing help of thofe Tables, not only the Logarithm of any given Number, but
numeral o alfo the Number correfponding to any given Logarithm may be found.
A Debt for Example of a 1000 l. 3 Years due, was acquitted Principal
and Intereft, by paying 11917. 2 let the Rate of Intereft allowed be
required.
First ex-
ample.
Second ex-
ample.
Here a 1000l. r=1191/
Confequently we will have p√
3,0759179 — 3,0000000
3.
Log.r-Log. a
X
125
2
125
> x=3, are given to find p
or Log. p
a
0, 0759179
= 0,0253059: Since
3
d + i
2
ď
0, 0253059 is the Logarithm of p or of
adding to it the Log.
of d or of 100, the Sum 2, 0253059 is the Log. of di, but to this Lo-
garithm correfponds the Number 106, wherefore di=106, where-
forei6, the rate of Intereft required.
The Sum of 1000l. was borrowed at 7 per Cent. per Annum, the Sum
due at the End of three Years feven Months and 15 Days are required.
Here a 1000, i7, d= 100, p=
107 플
​100.
215
200
43
40
264 Years, confequently rp, or Log. r =
264 × 0,0314085
73
1320
365
73
Log. a +x Log. p.
3, 0000000 +
= 3, 0000000 + 0,
1135869 = 3,
correfponds the Number 1298 29
required.
1135869 to which Logarithm
which expreffes the Sum due
30
SPECIOUS ARITHMETICK.
159
LVIII.
If it be agreed between the Borrower and the Lender, that the Intereft tho'
not paid when it becomes due, is not to be added to the Principle, ſo that the
Amount ſhall be converted into a new Principle for the fucceeding Term ;
the Sumr due at the Expiration of any limited Time will be expreffed by
ti
d
ti
ȧ + a + i = a(1 + ii)
= a(1 + i) and in this Cafe the Lender is faid to
ceive fimple Intereſt for the Loan of his Money.
Now comparing this Expreffion a ( 1 +
ti
d
.)
In what cafes
compund in-
re- cereft is fa-
vourable to
the borrow
er, and in
of the Sum due, at what cafes
the Expiration of a Number of Years t, computed (in the Language
+
the Analyſts) at ſimple Intereſt, with the former Expreffion a (di)
diſadvan-
of tageous
of the Sum due at the Expiration of the fame Number of Years, com-
puted at compound Intereft, it will appear 1° that if t is an Integer greate,
ti
t
than Unity, ( d++ )' > (x + -—-——) for ( 4++)' =
dt
d'
+
tid-s
dt
+
t. t- 1. i2 di~2
2 dr.
+
ti
Now this Quantity is manifeftly equal to
t. t — 1. t − 2:23 £t—3
2 X 3 dr
++ a real and po-
ſitive Quantity, conſequently it is greater than (1 + i).
2º if t=1 it will eaſily appear that thoſe two Quantities will be equal.
3° if t=
Þ
I
P
ti
we will have (d+i < 1 + 1/
or +
dp
+á
for raiſing the Quantities on both Sides to the Power p there will refult
on one Side dți, and on the other I++ a real and poſitive
Quantity.
+
✔
i
d
4º from whence we may conclude in general, that if t is any fractional
Number greater than Unity, a
-)' > a+ tia, and if t is a
(d+i); >
Fraction leſs than Unity, a (d+i)' > a+
tia
Whence it appears, that when compound Intereft is allowed for the Loan
of Money, the Sum due at the Expiration of any Time exceeding a Year,
is greater than it would be if fimple Intereft was allowed, that the Sum due
at the Expiration of any Time lefs, is lefs than it would be, if fimple Intereſt
was allowed, wherefore, if compound Intereft is advantageous to the Lender,
+ &c.
160
ELEMENTS OF
Problem.
in certain Cafes it is alfo to the Borrower in others, the Compenfation
it is true is not equal, fince the Advantage of the Borrower ends with the
first Year, and that of the Lender goes on continually increafing as the Num-
ber of Years increaſes.
LIX.
A Sum a was lent, on Condition that the Principal and Intereft at a certain
Rate i per Cent per Annum, ſhould be difcharged in a Number n of equal Pay-
ments, one at the end of every Year; and that at each Payment, the Interest
then due should be cleared, and the Remainder by which fuch Payment ex-
ceeds the Intereft applied to reduce the Principal, the Value of the Payments is
required.
To folve this Queftion, I obferve that the firſt Payment is compofed of
two Parts, one of which is the Intereft of the Principal at the fuppofed
Rate, due at the End of the first Year; the other is a Portion of the Prin
cipal taken to compleat the Payment. That the Principal being diminiſhed by
the first Payment, the Intereft due at the End of the fecond Year will be
leſs than the former, and confequently the Portion of the Principal to be
taken to compleat the fecond Payment, will be greater than the Portion taken
to compleat the firſt, and ſo on for the fucceeding Years. Whence there
reſults two Series, the one decreafing, whofe Terms exprefs the Intereft due
at the End of every Year, the other increaſing whofe Terms repreſent the
different Portions of the Principal taken to compleat the Payments. It is
this laft Series we fhall confider, and to difcover the Law of it, let
z, y, x &c. exprefs the Portions of the Principal taken to compleat the firſt,
fecond, third &c. Payments, ſo that z + y + x + &c. = a.
T
a i
The firſt Payment will be 41+%, the 2d ai-si
-
d
+y, the Third
ai — zi—yi + x &c. As thoſe Payments are ſuppoſed to be equal, by
d
comparing the firſt with the ſecond, the fecond with the third,
reſult ſo many different Equations. Whence is deduced y=%X
2
&c. there will
d + i
d
× (1+i), &c. from whence it appears, that the Series in
d + i
Queſtion is a geometrical Progreffion, which by putting
d=p, will
be expreffed as follows, z, zp, zp², z p³ &c. z p²-3, z pr−2, zp
To find the first Term of this Progreffion, I obferve that the Sum of all the
*Terms except the firſt, is equal to the Sum of all the Terms except the laſt,
multiplied by the common Ratio, wherefore a — z = a p − z p” whence
„ap - a = z p” or z = a+
to which adding the Intereft of
$18m I
I
£
1
SPECIOUS ARITHMETICK.
the Capital due at the End of the first Year,
there refults ra× = + p − 1 ) ·
to the ſame Denominator) ▾ = a × (
I
r
I
12
p" I
p"
which is aX(p − 1 ),
or (reducing the whole
-), or in
, or in Logarithms,
▪
rói
of the fore
Log. r = Log. (p" + 1 —p") — Log. (p" — 1).
Let the Sum lent, for Example be 1201. 5 s. Intereft 6 per Cent. and to Application
be reimburſed in 7 equal Payments. Here a = 120, 25, p=1,06, n=7 going pro
whence Log.rLog. 10, 84852-Log. 0, 50361 =1,035369-9, 702113 blem to an
1, 333256, whence r 217. 10s. 9 d.
LX.
When a Debt is diſcharged by many equal Payments, and the Intereſt
due at the Time of each Payment is cleared, before any Part of the Princi-
pal, compound Intereſt is allowed to the Lender. To make this appear, I
argue thus, a Perlon who borrows a Sum a will owe at the End of the firſt
d+i
Year, a X, but by Hypothefis he reimburſes at the End of this
d
Year a Sumr, wherefore he will owe at the Beginning of the ſecond Year
d+i
r, and at the End of the fecond Year he will owe
d
* [ + + + ]
a.X
ax [ d + ] — r (confidered as Principal) multiplied by, that
d
example.
When a
is,
debt is dit-
d
ax [ d + i ]² — r
-rx.
d. i
X.
d
charged by.
feveral e-
4++; and fince at the End of the Year
he
qual pay-
ments com-
a
rx
x
d+i
terelt is al-
d
reimburſes a Sum r, he will owe at the Beginning of the the third
• [ 4 + + ]² - r
[ 4 + 1 ]³ -
a X
d
2
Year pound in-
r, and at the beginning of the 4th Year lowed to the
d+i
[4 + + ]² – r x ¹ti
at the End of the nth Year.
+
a X
[ d ÷ ÷ ] ~ ~ ~
d + i
[ < ÷ ÷ ] "˜¯
d
i
r, and in general
- · [ 4 ]”
Wherefore, if the Payment fhould be made in a Number'n Years, the
foregoing Quantity must be put equal to nothing. Now I obferve, that in
this laft Quantity all the Terms that are multiplied by r form a geometrical
Progreſſion, of which the firſt Term is[4], and the laſt Term
1. Now, the Sum of this Progreffion, putting for Brevity's fake,
will be equal to pr divided by p1, confequently
dți
=
ap" — r X
[-
X. ===0, conſequently r = 4 ×
o,
p"
-- I
I
Value as found, before.
}
3. X
p»+1
・pr
the fame
Ꮧ
lender!
+62
<
ELEMENTS OF
What is
-meant by
.annuities.
Refolution
of the va-
rious quef-
LX.I
When a Sum of Money is lent on Condition that the Borrower ſhall dif-
charge both Principal and Intereft in a Number of equal Payments, to be
made yearly, fuch Payments is called an Annuity. In Computations relating to
'Annuities therefore, there are four Things to be confidered, the Sum lent a,
the Rent received every Year r, the Rate per Cent i, the Number of Years
n that the Annuity is paid; and any three of thofe four Things being given,
the other may be found, as will appear by the following Examples.
Let it be required to find what a yearly Rent r is worth in ready Money
for its Continuancen Years, the Rate i per Cent per Annum, compound
Intereft being allowed to the Purchaſer.
Here r, n, p are given to find a, which is equal to r
Log. (r.
117) — Log. (p − 1).
p n
I
.pn
, or
I
To apply this general Solution to an Example, let the Rent be 30%
tions relat- per Annum, to be continued 7 Years, and let the prefent Worth be required,
allowing 6 per cent compound Intereft to the Purchaſer.
n = 7, and p = 1,06 and of Courfe
—
ing to an-
nuities.
Here we have r = 30,
Log. a = Log. 10,048
Log. 10,048 - Log. 0,06 1,002079—8,778151 = 2,223928″
and confequently a 167. 9 s. 5 d.
LXII.
Let it be required to find for what Time n a Sum a will purchaſe an
Annuity of r Pounds per Annum, at the Rate i per Cent compound Intereſt.
p"
je
Here a, r, & p are given to find n. Becauſe p” = r+a-ap
n =
Log.r Log. (r+a-ap)
Log. p
ар
Let it be required to find, for Example, for what Time will 1671. 9s. 5 d.
purchaſe an Annuity of 30l. per Annum, at 6 per Cent
this Cafe.a167,416,r=30,p=1,106, hence n =
7 Years, the Time required.
LXIII.
compound Intereft. In
Log.30-Log. 19,9517
Log, 1,06
Annuities are faid to be in Arrears when they are payable or due either
yearly or half yearly, &c. and are unpaid for any Number of Payments,
to compute the Amount m of all thofe Payments, allowing any Rate of In-
tereft for their Forbearance.
Suppoſer the first Years Rent, then rpr will be the Amount of the
first Years Rent more the ſecond Years Rent, and rp²+rp+r the Amount
of the firſt and ſecond Years Rent, more the third Years Rent, and in gene-
ral the Amount of the n-1 Years Rent more then Years Rent will be ex-
preffed by rp" + rp² =² + rp" - 3
++
2
SPECIOUS ARITHMETICK.
163
Now the Sum of this Progreffion is
r pr
p- I
m the Amount re-
quired, or Log. m — Log. (r p” — r) — Log. ( p — 1 ).
LXIV.
To apply this Formula to an Example, let a yearly Rent of 30%. be un-
Refolution
paid 9 Years, and let its Amount at the Rate of 6 per Cent per Annum, com-
of the va-
pound Intereſt be required.
rious quef-
Here r = 30, n=9&p=1,06 & Log. m=Log. 20, 68353 — Log. 0,06 tions relat--
=1,315626-8,778151=2,537475, and confequently m = 3441. 14s. 6d.
LXV.
Let it be required to find what Annuity r forborn n Years, will raife a
Stock m, at a Rate i per Cent compound Intereft.
Here m, p, n are given to find r, which is equal to
Log. r = Log. (mp
m) – Log. (p-1)
mx (p-1)
11
P
OF
Let a = 344.5 s. n = 8, and i 31. 10 s. Confequently p= 1,035,
wherefore Log.r=Log. 12,04875—Log: 0,316803=1,080908-9,500785,-
1,580123, and confequently r = 38,0297 1. = 381. os. 7 d. per
Annum.
LXVI.
Let it be required to find in what Time n, an Annuity of r pounds per
Annum, will raiſe a Stock m, at a Rate i per Cent. Compound Intereft.
Log.(mp+r-m)—Log.r
Here m, p, r are given to find n, which is equal to
Log.p.
Let m 344,25, r 38,0297, and i=37. 10 s. hence p = 1,035
Log. 50.07845-Log. 38,0297-8 Years, the Time required.
and n. =
Log. 1,035
LXVII.
A's all freehold or real Eſtates may be confidered as Annuities to com-
tinue forever, to obtain their true Value a, it fuffices to put the Quan-
tity a p". r p"
a
P
p²
2
#1 - I
rp"
2
go p” ~
3
r p"
4
да
pr
", or
continued to an infinite Number of
Terms equal to nothing, which on Account of all the Terms that are
multiply'd by r being a geometrical Progreffion infinitely decreafing,
and confequently equal to
gives a =
p- I
I
is reduced to a
=o which
p- I
Let it be required, for Example, to find the Value of a freehold Eftate of
25 1. per Annum, allowing 51. 1os. per Cent, &c. Compound Intereft to
the Buyer:
ing to an
nuities in
arrears.
1
164
Problems
ELEMENTS OF
Here are given r = 25! i = 54 10s. and confequently 1,055
wherefore, a 454. 10s. 103d. the Value required.
LXVIII.
After having folved all the Difficulties that could occur in the Solution of
Problems of every Degree, producing Equations confifting of two Terms,
Order requires that we fhould proceed to explain how to folve Problems
producing Equations of every Degree confifting of three Terms, but the
Analyſts have not as yet found a general Method for every Problem of this
Kind, the following in its full Extent, produces Equations of every Degree,
confifting of three Terms that have been compleatly folved.
LXIX.
Let it be required to find two Numbers, whofe Product is a, and the Dif-
producing ference of their nth Powers is b.
equations
confifting of To folve this Queſtion let the leffer Number be x, then the greater is
three terms
folved by
a"
a
the method Now, by the Queſtion b = →→ x", or b x" = a"-x, confequently
for equa-
tions of the
fecond de-
gree.
Examlpe of
n
b
2
2+ bx"a", to refolve this Equation, I add the Square of to
both Sides, and there refults x2" + b x² + & 6² = a" + 4 b² whoſe Root
is xˆ±&b=+√ ( a” + & b²), & x" = − & b + √ ( a + // b² ),
wherefore x = "v [ — { b± v ( a" + b² )].
=
By means of this Formula, all Equations may be refolved confifting of
three Terms, in the first of which, the Index of the unknown Quantity x,
is double of the Index of x in the ſecond Term, and the third Term is a
given Quantity, and it is eaſy to perceive, that all Equations in this general
Formula x² + bx” = a" cannot have more than four real Roots, and
will only have two when n is an odd Number.
LXX.
To apply this general Solution to an Example, let it be required to find
the forego two Numbers whole Product is 15, and the Difference of their Squares is 16.
Here a = 15,6 16, and n = 2, confequently-`
ing method.
Another
example.
x = ±√ [−8 ±√289] =√±8 ± 17 =+ √9=3.
LXXI,
2
Let x4-bbxbbcc be propofed to be folved, adding on both
Sides 464 Square of one half of the Coeficient of xx, there will refult
x4 - bb x x + 1 64 = = 64 + bbcc, whofe fquare Root is x² - 1/2 bb
= ½ 2
= ± b√ ( bb +cc), whence is deduced xx bb + b√ ( b b + cc),
and con'equently x = ±v [÷bb ± b√ ( 4 b b + cc)] fufceptible, of
two real Values, and of two imaginary ones.
The two firſt are
±√ [ 1/6 6+ 6 √ ( / b b + cc)], the two others are
x =
SPECIOUS ARITHMETICK.
165
:
x = ± √ [ ÷ bb — b √ ( & b b + cc)] which are neceffarily imaginary,
becauſe b√(bbc) is greater than bb.
x3.
LXXI.
=
Third ex-
Let the Equation x6 — 2 a² b x³ a6 be propofed, adding to both
Sides a4 b b, there refults 6-2 a ab x3 + a + b⋅ b = a + b b + aº, whose amples
fquare Root is x³-a ab=a² √ (a² + b²), or x3 = a²b±a² √ (a² + bb),
and confequently x = 3√ [a ab ± a² √(a² + b²)], fufceptible of two
real Values, one pofitive and the other negative; the other four Roots are
imaginary.
4
LXXIII.
bb)
1
ample.
Let the Equation (a a + b b) x x = a a b b be propofed, ad-
ding to both Sides & a++ aabb + 64, there will refult x4-(a a+bb)xx
1/2
+4a4+ a ab² + 1 b4 = a4 — a a b b + 64, both Members be- Fourth ex-
· ½ 1
ing perfect Squares, extracting therefore the fquare Root, there will refult
x x − 1 a a − 1 b b = ± ÷ a a +bb, which gives xxa a and
xx bb, that is, xa and xb, which are the four Roots of
the Equation 4 — (a a + b.b) xxa a b⋅b.
-
LXXIV.
Let the Equation 4 (2gb+ 4 ff) x x = - g g b b be propofed
to be folved, adding to both Sides the Square of half the Coeficient of the
fecond Term, and extracting the fquare Root of both Members, there re- Fifth ex-
fülts x²-gb 2 ƒ ƒ = ± 2ƒ √ (gb + ƒƒ) which gives x
2f ff)
± √ [gb + 2 ƒ ƒ ± 2 ƒ V(gb + ƒ ƒ )]•
Now on Examination I find that this Quantity is a perfect Square, that
of ƒ ±√ (ff +gb), for the Term 2 ƒ√(g b +ƒƒ) is double of the
Product of ƒ into (gb+ƒƒ), and the Quantity gb+2 f f contains
the Square off, and the Squareff + gb of the radical Part.
Whence the foregoing Value of x is reduced to f ± √ (ƒƒ + g·b)
and -ƒ±√ (ƒƒ + g b ), and thoſe are the four Roots of the Equation
x4 — 2 gb x x − 4 f f x x = -g bb.
**
LXXV.
In Order to diſcover when the Roots of Surds may be expreffed exactly
by other Surds, as in the foregoing Example, and how thofe Roots may
be found, I obferve that the Root of a Quantity compofed of two Parts,
one of which is rational and the other a radical Quantity of the fecond De-
gree, fhould itſelf confift of two Parts, and that one of them at leaſt ſhould
be a radical Quantity.
Let then A+B exprefs in general the propofed Quantity, A denoting
the rational Quantity, and B a radical of the fecond Degree, and let p+q
exprefs the Root fought.
I now obſerve, that whether p denotes the radical Quantity, or whether
it be expreffed by g, or that p and q are both Radicals, in the Square
ample.
*
166
ELEMENTS OF
Method of
1
2
p² + 2 p q + q², the Term 2 p q can only be a Radical; comparing there-
fore this Square with the given Quantity, 2 p q will reprefent B and p²+q²,
A; that is, we have the two Equations p² + q² = A and 2 p q
for finding pand q. From the fecond I deduce p = which fubftituted
B2
492
2
B
29
B2
= B
finding the in the firſt gives 12, + q² = A, or qª — A q² = — or q² A
fquare root
of quantities
4
partynatics = √(AA- BB), confequently g✓ [ § A ± ± √ ( A ² —B²)]..
+ ½ q =
nal and part- Subitituting afterwards this Value of q in the Equation p² + q² = A, or
ly irrational. p = ± √(A-g2) we have p = ± √ [ { A = ± √ (A² — B²)]; from
2
p:
whence it follows, that the Root required of the Quantity A+B is
±√[ ÷ A ± § √ (A ² — B2)], or fimply.
± √ [ ÷ ^ ± ÷ √ (A² → B²)]
A
± √ [ { ^ — ÷ √ ( A² —— B²)].
± √ [ & A + ÷ √ (A² — B²)]
As to the Signs which ſhould affect the two Parts [A+ ¾ √ (A²—B²)] --
and √ [ ✯ A — √(AB)] of the Root required of AB, they are
the fame when the radical B is pofitive, but different when B is negative;
for it is eaſy to perceive that in general p+q, or pq being the
Root of A+B = p p + q q + 2 p q, p q, or pq is the Root
of AB = p p + q q − 2 p q.
В
LXXVI.
That the Quantity ±√[‡A+ ž√(A²—B²)] ± √ [÷A—¥√ ·(A²—B²)]`
found for the Root of the Quantity A+B may be expreffed by a Bino
mial Surd, the Quantity A B² muſt be a perfect Square; which will
never fail to happen as often as the Root of A+B can be extracted:
To make this appear we have only to obferve that p q is expreffed by
(AB) at the fame Time that p + q is expreffed by (A + B);
from whence it is eafy to conclude, that (AB) × √(A+B), or
L.(.A.2. B²) = (p − q) x (p + q), or p² — q a commenfurable
Quantity.
LXXVII.
4
To fhew the Application of the foregoing Method, let the Quantity
a. a + 2c√(a acc) be propofed, comparing it with A+B we have
Application
Pre- a² - A and B = 2c √(a²-2); and confequently ✓ (42—B²)= a a—2 cc,
of the
going me- whence √ A+ √(A² — B²)] = √(a2c2); in like Manner,
thod to an √ √ ( 2 -B2)]= c; that is, the Root fought is c+(aa-cc,
[ 3 A — — (A² —
example.
or -C
√(aa
1
1/2
cc).
2
LXXVIII.
If it was propofed to extract the fquare Root of 16+6√7, putting
A 16 and B6/7, there refults (AAB B) 2, and confe-
A=
=
quently [+√(AA-BB)]=3, and √√(AA—BB)]=√7,
whence 37, or 3√7 will be the Root required..
#
SPECIOUS ARITHMETICK.
167
LXXIX.
F
Let the Root of a p—2a√ (ap — a a) be required, putting Aap Another ex-
and B = 2a√(ap a a), there refults √(A A—B B)= a p
ample.
2 aa
and √ [A+ √(AA — BB)] = √(a p—aa): In like Manner,
√ [ { ^ ~ { √(A² — B²)] = a; that is, the Root fought will be
A
a
½
12
a a) a, or a-
√(a.p..
a a).
- LXXX.
If the Quantity 62
propoſed, then A =
-----
a b + 3
b²— a b +
a² + 2 √ (a b3 —
a a, B = 2√ a b³
and √(A² — B²) = √√ b4 — 6 a b³ +
4
19
2
a262
—
2 a² b² + a3 b) was Third ex-
— 2 a² b² + 4 a³ b, ample.
3a3b+
2
I
04
16
a a, which will give √[A + ÷ √ ( A ² — B²) ]
4
= b b −3 ab +
= √(bb — 2 a b + a a),
whence the Root fought is a
-√(66-2 a b + a a).
and VA√ (A² ~ B²)] = √ab;
V[
b+ √(6 6 2 a b + & a a), or — √ab
LXXXI.
After the Analyſts had found out a Method for difcovering among the
Quantities partly rational and partly irrational fuch as were perfect Squares,
they, without Doubt, fought alfo how to diftinguish thoſe that were perfect
Cubes or higher Powers, as being abfolutely neceffary to compleat the Refo- Method of
finding the
lution of Equations contained in the Formula X +ax b, or cube root of
" ~ [ — & a ± √ (b+4a a)]. Let us first examine what must be quantities,
done when m3, that is, to find the cube Root of any Quantity A + B
A+B partly ratio-
in which A is rational, and B a Radical of the fecond Degree.
2 m
#2
I obſerve firſt, that the cube Root of a Quantity of this Nature can-
not include more than one Radical of the fecond Degree, for it is eafy to
perceive, that the Cube of a Quanty fuch as mn which contains
two of thofe Radicals, will neceffarily be affected by thoſe fame Radicals.
I obſerve next, that the fame cube Root fought cannot contain any other
kind of Radicals except a cubical Radical, and which is common to the
two Parts of the Root. Such would be, for Example, the Quantity of
3√ m + √ § × ³√✅ m, whoſe Cube mƒ³+3m ƒ g+ (3 m ff + mg) √5,
f 3,
as the propofed Quantity A+B confifts of a commenfurable Quantity,
and a radical Part of the fecond Degree.
3
Now let p+q exprefs the Root fought being the Part affected by the
Radical of the fecond Degree, whether it be affected as well as p by a cu-
bical Radical or net.
It is manifeft that the Cube p3 + 3 p² q + s p q 2+3 will not contain
any other Radical, but the Radical of the fecond Degree, which is in q, and
that only the Terms 3 p p q and q3 will be affected by this Radical; I
therefore compare thote two Terms with the given Quantity B and the
two others with A, which gives the Equations A = p³ + 3 p q² and
3 34
B = 3 p² q + q
naland part
ly irrational.
168
ELEMENTS OF
of the fore-
going me-
thod to an
example.
=
9
3
To folve thofe two Equations we muſt firſt exterminate one of the un-
known Quantities p or q, which may be effected by any of the Methods
explained in the fecond Chapter, but much eaſier by obferving, that if p+q
is the cube Root of p³ + 3 p²q + 3p q² + q³, the firſt Part of which
p³ + 3 p q² = A, and the fecond 3 p q + q3 = B, p—q will neceffarily
be the cube Root of A-B; which is in this Cafe = p³ + 3 p q² 3 p²q-q³,
whence A + B = (p + q)3 and AB (pq3; confequently,
(A + B) (A — B) = (p + q)³ ( p − q)³, or AA — B B = (pp — qq)³
or ³√(A A
3
BB = p² q², or n= p² q², in which n is given, and
muſt be either a commenfurable Quantity, or a fimple cubical Radical.
From the Equation n = p² q², or q² = p² —n, and from the Equa-
—
tion A p³ + 3 p q² is deduced 4 p3
3 p n Ao, by Means of
which Equation the Part p of the cubical Root fought may be obtained,
and when found, the fecond Part of the Root fought will be obtained by
Means of the Equation q√(p² — n).
99) 3
As to the radical Sign it will be pofitive, if the Radical of the propofed
Quantity has the Sign+, and negative, if the Radical of the propoſed
Quantity has the Sign; for it is eaſy to perceive, that the radical Part
of the Cube of p + q which is (3 p p + qq) × q will always have the
fame Sign as q.
If the Root of the propofed Quantity A+B does not contain a cubical
Radical affecting all its Terms, n or 3√(A ABB) will be a commen-
furable Quantity, and confequently, the Equation 4 p³
3 p n - A = 0
will not be affected by any radical Quantity, and as in this Cafe will be
Fational, we cannot fail of finding it by inveſtigating all the Divifors of this
Equation.
If the Root fought fhould have its two Parts affected by a cubical Ra-
dical, which will happen as often as A ABB is not a perfect Cube,
then AABB muſt be mulitiplied by a Quantity 2, fuch as that the
Product A2 2 B² 2 may be a perfect Cube; and now instead of
A + B we muſt extract the cube Root of (A + B) X √ Q, which be-
ing found and divided by the cube Root of ✔✅ 2 will give the Root re-
quired.
LXXXII.
[s aa - ab 8 a az a363, and nor
(2
3
12 Bab
To fhew the Application of this Method, let it be required to find the
Application the cube Root of the Quantity 7 a3-3 a² b+(5 a a—a b) × √(2 a a—a b).
Comparing this Quantity with A+B I find 7 a3 3 a b = A,
(2 a a — a b)] = B; and confequently, A2 B² =
ao 3a5 3
³/ (A¹ - B2): =-aatab,
fubftituting this Value of n, as alfo that of A in the Equation
4p³ — 3pn — A=0, there refults 4p3 + 3paa-3 pab — 7 a³ + 3a² b;
which is divifible by pa, that is, the Value of p-is a. Subftituting this
Value of p in the Equation q(ppm), there refults q=√(zaa—ab),
whence the Root fought is a + (za a
ab).
SPECIOUS ARITHMETIC K.
1.69
·
LXXXIII.
Let it be required to find the cube Root of 2 a ac-abc
•[ 2 a
b v (a acc
bbcc)], putting 2a a c
.
b b c
a b c
b b c
B2
= A and B = [2 a b v (a a c c — bbcc)], there reſults A²
= 2 b+ c c — 2 a b3 cc, which is not a perfect Cube. To diſcover what will
make it a perfect Cube, I refolve it into its Factors, and it becomes Another ex-
2 xccx (b − a) × 63; whence it appears that by multiplying by
X
4x (b− a)²
CC
it will become a perfect Cube, that of 2 × ( b
-a) xb,
or of 2bb — 2 a b; and confequently if the propofed Quantity be multi-
fquare Root of 4X (b − a)²
26 — 2.22
C
plied by
Quantity [(2 a a —
a b
CC
there will refult a new
b b) x ( 2 b 2a) (2 a b)
X(26-2a)Xv(aa-bb)], the firft Part of which reprefenting A, and
the fecond B, will give for
Whence the Equation 4 p3
3
(A ABB), that is, for n, 2 b b 2 a b.
A o will be transformed into
3pn
p=
4 p³ — 3 p x ( 2 b b — 2 a b) + (b b + a b −2 a a) X. (2 b — 2 a) = 0,.
which is divifible by pab, that is, ba, which Value being
ſubſtituted in q = v(pp-n) there refults q√(a a- bb); wherefore
b — a √(a a bb) will be the cube Root of the Product of the pro-
26-2; wherefore
a
[ba√(a²-bb)] × 3/c
3√(2.6—24)
pofed Quantity into
C
is the cube Root required of the propoſed Quantity.
LXXXIV.
If the Terms of the Quantity whofe cube Root is to be extracted have
Diviſors, they are first to be reduced to the fame Denominator, and then the
cube Root of the Numerator is to be divided by that of the Denominator.
LXXXV.
If it be propoſed to extract the Root of a numerical Quantity partly ra-
tional and partly irrational, it may be performed much eaſier than by the
foregoing Method.
=
2
ample.
Method of
quantities,
nal and part-
For fuppofing first, that the Root fought ſhould not be affected by a cubical finding the
Radical, but fhould confift of an Integer, and a radical Part, alfo an Integer cube roots of
Number, becauſe pq (AB) when p+q=3/ A+B we will numeral
¾ √ ( A − B ) + ³ √✓ ( A+B )
have p =
; whence the rational Part of partly ratio-
the Root will be obtained by computing in the neareft integer Numbers, ly irrational.
(A-B) and 3/(A+B), and taking the one half of theſe two Numbers.
For affuming for ³√(A — B) and for 3/(A+B) the neareſt integer Num-
bers, neglecting the fractional Parts, the Error that can arife in each of theſe
Quantities will be less than, and confequently the Integer Number that
reſults for the Value of ¸³√( 4 − B ) + VA+B
that is, for p, will not
2
3 Y
*7O
ELEMENTS OF
differ from its true Value by Unity, and as this Value of p fhould be an In-
teger, it will be exactly determined by this Means.
Having thus obtained the Value of p, and that of n or of 3/(AA—B B)
being given, fubftituting the Values of p and n in q√(pp-n), the fecond
Part of the cube Root required will be obtained.
-
If the Root fought fhould have its two Terms affected by a cubical Ra-
dical, which happens as often as A2 B2 is not a perfect Cube, then as in
B²
fpecious Quantities we muft extract the cube Root of A2+B√Q, Q
being a Number fuch as that A2 2- B2 Q may be a perfect Cube, and
having found this Root, divide it by the cube Root of √2
3
LXXXVI.
3
To fhew the Application of this Method, let the Cube Root of 7 +5√2
be required, putting A=7, B= 5√ 2, I find n or ³/ ( A² — B²) — — 1,
I afterwards obferve that the Value of 3/(A+B), or of 3/(7 +5√2) is
nearer 2 than 3. I therefore affume 2 for its Value, in like Manner I obſerve
Application
³/
of the fore- that 3 (A —B), or ³√(7 — 5 √2) is nearer o than 1, I affume o for this
Quantity, whence p, or
3/ (A+B) + 3√(A→B)
= 1. I fubſtitute
this Value of pinq = √(pp—n), and I find q = 2, whence I con-
clude, that if the cube Root of the propofed Quantity 7+52 can be ex-
tracted, it will be 1+2, and in Effect, I +2 railed to the Cube,
giveş 7 +5√2.
going me-
thod to an
example.
I
LXXXVII.
2
Let the cube Root of 5 + 33 be required. Here A A-BB-2,
but 2 not being a perfect Cube, I examine what Number 2, by which
5 +33 being multiplied, will render A ABB a perfect Cube,
and finding 2 for this Number, I inveſtigate the cube Root of 10 + 6 √ 3,
3 A
Another ex- and I find n=2, and the neareſt Value of (A+B)+³/ (4—B)'
ample.
The fore-
or of
2
2
3/ (10 +6√/ 3) +37 (10-6√3) in Integers to be 1, fubfti-
tuting thoſe Values of p and n in q = √ ( p p −n) I find q =√3; and
finding upon Examination that p + q or i + √3 is the cube Root of
+√3 is the cube Root fought of
10 + 6 √ 3, I conclude that
3/2
5 + 3√3.
LXXXVII.
The Computation for determining p may be rendered more fimple by
going me- obferving that inſtead
thod ren-
dered more
fimple.
¾ / ( A + B) +
24
B)
we may put
V
of ( A + B ) + ³ √ ( A − B )
2
√√ (4 +B)
•
Since n or 3✓ (Az
B2)===
3/ (A
n
SPECIOUS ARITHMETICK.
171
$4
2
¾√(A+B) ׳√ (A—B). Which is fimpler than 3√(A+B) +³/(A—B)
becauſe it is eaſier to divide the Number n already found by 3 (A + B)
than to compute feparately 3/A- B).
LXXXIX.
of the new
method.
To fhew the Application of this new Formula, let it be applied to the
Example of Art. LXXXV. where A was 7, and B52. After having Application
found, as in the fame Article, that n =— 1, and that 3 ( A + B-)
in the neareſt integer Number, to be 2, inſtead of ſearching, as in the fame
Article, the cube Root of 752 in the neareft integer Number, I di-
vide n or
1 by the Value 2 of ¾³/ (A + B) which gives
3
being fubſtituted in the foregoing Formula, which expreffes the Value of p,
n
3√ A+ B+ 3/ (A+B)
2
2
which
, or I (affuming the
found in the above-men-
2
there reſults
neareſt integer Number) for the Value of p, as
3. (A+B) + 3
(A — B)
the Remainder
>
2
tioned Article by the Formula
of the Operation is performed as in the ſame Article.
XC.
It is to be obſerved however, that the new Formula may lead into Error,
if A and B have not the fame Sign, for when theſe Quantities have diffe-
rent Signs, the real Value of 3(A+B) may be fo fmall with refpect to
n, that the neareſt integer Number affumed for this Value, will give for This new
n
3 ( +B) + 3√(A+B)
-
2
method de-
fective wher
A and B
a Number that will differ from its true
have diffe-
rent figns.
Value by feveral Units. For Example, if it was required to extra& the cube
Root of 45
29√2, putting A = 45 and B=
29 K 2, the Value of
3(A+B) or of 3 (45292) in the neareſt integer Number
will be 1, and as 3 (A2 B2) is here equal to 7, the Value of p, or of
n
¾ √ ( A − B ) + −¾ √ (A+B)
2
will be found in this Cafe to be 4,
which in Reality is only equal to 3, as may be proved by the Expreffion
3√(A+B) + 3√ (A~ B.)
2
of the foregoing Method.
But provided that this new Method of finding the Value of p is exact,
when A and B have the fame Sign, it is of little Confequence whether it be
applicable or not, when thofe Quantities have different Signs. For it is
eaſy to perceive that in this Cafe, we have only to ſuppoſe A and B to be
1
172
ELEMENTS
OF
What is to
both pofitive, and extract the Root p + q, then givep the fame Sign that
A has, and q the fame Sign that B has.
1
It remains then to examine, whether when A and B have the fame Sign,
or what amounts to the fame Thing, if A and B be both pofitive, we
3.√ ( A + B) + 3√( A + B
be done in may ſubſtitute in
this cafe.
2
n
in the Room of
3√(A+B) its Value in the neareſt integer Number, fo that the Value
which will reſult will not differ from its true Value by Unity, to be affured
of it, let the Value of 3√(A+B) in the neareſt integer Number be fup-
pofed to differ from its true Value by, which is more than can ever hap-
3√ (A + B) ± 3 + 3 √ (A+B) ±
2
n
pen. In this Cafe, the Value of p will be
2/2/
to fhew that this Expreffion of p can not differ from it's true Value by
Unity, let p+q be ſubſtituted in the Room of 3 (A
- B) and pp + q q
p + q ± 3 +
p p + q q
in the Room of n, and there will refult
from
2
I
Cafe in
which the
two fore-
going me.
thods fail.
which deducting p, the Remainder
±9.+ 1/2
2p+29±1
will express the diffe-
rence between the Value of p in the neareſt Integer, and its true Value,
now, it is manifeft, that this Quantity can never equal, for in the firft
Expreffion
9 +
I
2 p + 29+ I
that it includes, the Numerator q + being
lefs than the one half of 2 9 + 1, will be much lets than the one half of
I
9 + 1/1/2
2p+29 - I
which it like-
2p+2 9+1: and in the fecond Expreffion
wife includes, the Numerator - q, being less than the one Half of 24,
will be alfo less than the one half of 29 + 2 p — I.
Whence the Value of p determined by the foregoing Method, cannot
differ from its true Value by Unity, and confequently as often as a Quantity
A+ B, partly rational and partly irrational, (in which only Integers enter
either under the radical Sign, or before this Sign)-will have a cube Root
p+q, expreffed in Integers, this Root may be found by the foregoing
Method.
}
XCI.
But if the Quantity A+B, though clear of Fractions, fhall have a
fractional Root,fuch as the Quantity 2+5, whofe cube Root is +ž√ 5,
it cannot be found by either of the two foregoing Methods.
To remedy this Inconveniency, the Analyſts have fought in a direct Man-
ner all the fractional Roots, which when cubed, become integer Numbers.
SPECIOUS ARITHMETICK.
173
£
Let all thofe Roots be expreffed by +; p and m expreffing Inte-
m
n
gers that have no common Divifor, and q the Root of an integer Num-
ber, which will admit of no Reduction with the Number n.
p3
Quantity to the Cube, there will reſult +
2
3 p2
n m²
92
m3
3 p q 2
m n²
Raifing this
for the rational
How to re-
.medy this
Part, and [
[+]× g for the irrational Part. Now, by the deficiency.
n3
9
Conditions of the Problem, the firſt Part
ficient
3p p
n m²
352
Let
m² n
99
n3
$3
m3
39
2
+311, and the Coe-
m 1
2
+ of the fecond Part ſhould be integer Numbers.
+2be put equal to b, which will expreſs an in-
23
m4
7722
teger Number. Then q²=bn33 p² n² ; but this Quantity by Hy-
potheſis ſhould be an integer Number; confequently 32 n2 fhould be
alſo an Integer; wherefore 3 2
confequently n fhould be a Multiple of m.
Now let nm 1, confequently q²
p.3
thoſe Values of n and q² in
1722
fhould be likewife an Integer, and
= b m³ 13 — 3 p2 12, fubftituting
m3
+
3p q2
m n²
8 p3
m3
,
this Quantity will become
after the Reductions
+3p bl, which ſhould be an Integer, but
and m having no common Divifor, this Quantity cannot be an integer
Number, unleſs m be 1 or 2. As to n or m, it is eafy to perceive that it
fhould be equal to m, becauſe the Equation q² = b m³ 13 — 3 p² 12
will give ÷ = √ (b m³ 1 — 3 p²) and confequently
9
I
n
771
2
9
ml
that is,
-√ (b-m³ ! — 3 p²) whence it appears that the fecond Part of the
Root can have no other Denominator than what the first has, and confe-
quently muſt be either 2 or 1.
When therefore the Root of a Quantity A+ B, the rational Part of
which A, and the irrational Part B is expreffed in Integers, cannot be found
by the foregoing Method, we have no more to do than to muliply this
Quantity by 8, and inveftigate by the foregoing Method the Cube Root of
the Quantity refulting from this Multiplication, and if this does not fucceed,
no farther Trial is to be made: if it fucceeds, the one Half of the cube Root
found, will be the Root required.
$
}
174
ELEMENTS OF
What is to
be done
when the
cube root
XCII.
When the Number A and the Radical B are Fractions, they are to be
reduced to the fame Denominator, and the Root of the Numerator and
Denominator are to be extracted feparately. Thus, to extract the cube
Root of√242 12/
this reduced to a common Denominator is
✔968 25 and the Roots of the Numerator and Denominator found
2
".
I
ſeparately give the Root 2√21. And if you are to extract the cube
3√2
Root out of 3 3993 + √ 17578125, divide, its Parts by the common
Divifor ³/3, and the Quotient being 11 + √125, the cube Root of the
propoſed Quantity will be found by taking the cube Roots of 3√3, and of
II + √125, and multiplying them into each other.
XCIJI.
If it be propoſed to extract the cube Root of a Quantity compoſed of
two Radicals of the fecond Degree, whether it be a fpecious or numeral
Quantity, we have no more to do than to multiply it by the Cube of one
of the Radicals that it contains, the Product that reſults being a Quantity
the Sum of partly rational and partly irrational, Its Root may be extracted by the
foregoing Method, which when found is to be divided by the Radical, by the
Cube of which the propoſed Quantity was multiplied.
fhould be
two radicals,
How the
root denomi
XCIV.
If the fourth Root of a Quantity as A+B was required; first, the
nated by an fquare Root is to be extracted, which, if it cannot be found, much leſs
even num- can the Root of the fourth Power be found. In like Manner, all even
ber is ex-
tracted. Roots are to be fought, continually depreffing them by extracting the
fquare Root.
In general, if the m
XCV.
Root of the Quantity A + B was required, then
(p + q)" = A + B, and raifing p+q to the Power m, it will appear
172
that A is the Sum of all the odd Terms p",
m. (m
1)
ከዚ
q² p
2
2
m- 4 &c. and that B is the Sum of
Inveſtigat.
m (m
N- 1) (m 2) (m — 3)
of the rule
24
2 X 4
q4 p²
172
m(
m (m − 1) (m — 2)
2 X 3
93 pm - 3.
for extract-
ing the root all the even Terms m q p
of any pow-
er of
tities partly m (m − 1) (m − 2) (m — 3)
–
2 X 3 X 4 X 5
(m—4)
rational and
partly irra-
tional. A+B is the mth Power of p+q,
confequently
or pp
772
(A-B) X
"/
95 pm-5, &c. whence, when
A-B will be the mth Power of p-q;
(A + B) = ( p + q ) × ( p − 9 ),
X
m²/(A+B) + "/(A—B)
2
alfo
q q = n, or q = V p² -
99
n₂
SPECIOUS ARITHMETICK.
175 +
11
=
p+q+p -q,
, or p =
2
m
#
m√ (A + B) + m√ (A+B)
2
becauſe
•
"(AA-BB) = "/ (A + B) X / (A — B), and thus from the
Compofition of the Binomial A + B we are lead to its Refolution, when
A is rational, and A2 B2 is a perfect m Power.
B²
XCVI.
If the mth Root of A2 B2 cannot be taken, multiply 42 B2 by a
Number 2, fuch as that the Product may be the leaft perfe& m Power
n" — A² 2
42 2 - B² 2; and now inftead of 4+ B extract the m Root of
(A + B) × VQ, as above, and when found, divide it by 2/2, and the
Quotient will be the Root required.
171
To find this leaft perfect Power m (n") that fhall be a Multiple of
A2 B2 by a whole Number 2, let this given Number A²
B² be re-
prefented by the Product a b d f, whoſe fingle Divifors let be a a a...
b b b .... d, f, and the Product of theſe Divifors raifed to the Power m,
which is a bid f" divided by a" bPdf, will give the Quotient a"- *
fm -p d f"
a whole Number, provided fome Index as n
or be not greater than m; if it is, take, inſtead of the ſingle Divifor, a or b,
a² or b², a³ or b³, &c. till there be no negative Index in the Quotient, that
is, till 2 be a whole Number.
172
I
XCVII.
Having fhewn how Equations are folved by compleating them into per-
fect Squares, the Square on the right Hand Side being a Quantity entirely Refolution
known, we fhall now proceed to explain how Equations are compleated in- of equations
to perfect Squares, the Squares on both Sides being affected by the un- of even di-
known Quantity. Let the Equation 4 9 x³ + 15 x² — 27x+9= 0; mentions by
be propoſed to be folved according to the Rules of the Extraction ing them in
of the fquare Root, I find the two firft Terms of the Root of this Quan-
tity to be x2
x. Let the whole Root be expreffed by plained by
*Z
№4
2
2
x4
x + A; now obferving that the laft Term 9 of the
propoſed Equation is a perfect Square, as likewiſe the laft Term of the af-
fumed Equation (x²— ———x+4)² = x^—9׳+(24+ 31 ) x²—94x+4².
= 3 which will transform this Equation into
I put
A
105
4
2
2
4
*4 9x3 + *2 27 +9, whofe Terms agree with all the
Terms of the propofed Equation, except the third; the Difference of thoſe
Terms being 45 x²; to compleat therefore the propofed Equation into a
2
compleat-
to perfect
fquares ex-
an example.
176
ELEMENTS OF
1
perfect Square, add to both Sides of it 45x2, and extracting the Square
2
Root of it, I find x²
-2x+3=
x
3√5, and conſequently,
2
2
9 ± 3
39
27
+
4
4
8
√5].
XCVIII.
ax:
2
Let the Equation x42 a x3 + (2 a² — c c ) x² — 2 a³ x + a4=0
Another ex- be propofed to be folved, according to the Rules of the Extraction of Roots
ample, I find the two firft Terms of the Root of this Quantity to be x²
Let the whole Root be expreffed by x2 ax + A; now obſerving that
the laſt. Term at of the propoſed Equation to be a perfect Square, as like-
wife the laſt Term of the affumed Equation (x2 − a x + A )² =
*4 2 à x³ + ( 2 A + a²) x² 2 a A x + A 2
་ ༢
: I put A = a?
which will transform this Equation into x4-2 a x³+ 3 a²x²-2 a³ x + ·a4,
whoſe Terms agree with the Terms of the propofed Equation, except the
third, the Difference of thofe Terms being (3 a² 2 a² + c²) x² =
( a² + c² ) x²; to compleat therefore the propofed Equation into a per-
-fect Square, I add to both Sides of it (a² + c²) x², and extracting the
fquare Root of it, I find x²- a x + a² = x √ (a² + c²), and confe-
quently a ÷ v(a a+cc) ± √ [( cc aa+a)v (aa+cc)].
XCXIX.
2
=
9
2
4.
Let in general the Equation 4 + px³ + q x² + rx + s = o, be
propoſed to be folved, in which p, q, r, s, are the given Coeficients, and
Inveſtigati- the Equation is clear of Fractions and Surds;
on of the
method of
compleat-
ing equati-
ons of four
dimenfions
into perfect
fquares.
1
px + Q
x² + ½ p x + &
x² + ½ px
x4 px³
+p x³ +
q x² + xx+s
2 x2 +
2
2 +2
+ px³ +
Px3
+62
q x² + rx + s
= p²x²
p² x2
= p²) x² + rx + s
+ A²x² + 2ABx + B²
2 2x²
Qpx— Q²
0
According to the Rules of the Extraction of Roots I find the two firſt
Terms of the Root of this Quantity to be x2px. Let the Square
A A x² + 2 A B x + B2 compleat it into a perfect Square; fo that the
whole Root fhall be expreffed by x² + px + Q
14
I obſerve that p2+A2, if they are Fractions they ſhould have the
fame Denominator, fince they deftroy each other, or their Refult is an In-
teger; confequently 2 is an Integer, or half of an Integer.
SPECIOUS
177
ARITHMETICK.
=
=
Now from the Equations q- p²+A²—2 Q=0,r+2AB—pQ=0,
and s + B2 22=0, putting q ppa we have 2 2A+ α,
Þ Q = 2 AB +r, Q²= B²+s; fubftituting therefore the Value of 2, The reduc-
as given by the firft Equation, in the other two,
4
a 2
we fhall get tion of equa
:
them into
wiſe be ef-
≥ p A² — 2 A B = 6, and .A4+ ½ α A ² — B² = Suppofing tions by
$
B = r = ½ α p, and sa a. In which Equations the unknown compleating
Quantities appertaining to the latter of the two affumed Squares are only perfect
concerned, and from which their Values might be found; but as the refult- fquares can-
ing Equation, when one of the Quantities is exterminated, riſes to the fixth not other-
Dimenſion, and would require more Trouble to reduce it than even the fected than
original one propounded, little Advantage would be reaped there from: by trial.
The Analyſts therefore, inſtead of proceeding in a direct Manner, have en-
deavoured to diſcover fuch Properties or Relations of thofe Quantities as
might enable them to guefs at their Values, which may be afterwards
tried by Means of the Equations here exhibited.
C.
kvn, and
To explain what they have imagined in this Reſpect, it is to be ob
ſerved, that A and B may be either rational Quantities or Radicals, and in
this laſt Cafe, as A²x² + 2 A B x + B2 fhould be a rational Quantity,
A and B muſt be affected by the fame Radical. Let then A
B=1√n, by this Means the two Equations derived above will be chan-
ged to p
2 k l n B, and kn² + ½ a k² n − 1² n = },
or top k²
and ½ k4 n + a k² — 212
212 =
ſpectively.
p k² n
B
2 k l =
n
25
re-
2
Now fince k and I are Integers, or Halves of Integers, confequently n an
B
Integer, it is plain, that and 2 muſt be Integers likewife, or
n
n
at
are
leaſt the Halves of Integers, and confequently that n (whofe Value we
ſeeking) ought to be fome common integral Divifor of @ and 2 §.
Moreover, with Regard to k and 1, it is evident from the first of thoſe
B
Equations & p k — 2 1 = — that the former k ought to be ſome Diviſor
k.
β
11
k
of, and that if the Quotient be taken from ✯ p k the Remain-
der will be the Double of I.
It further appears, from the Equations 2=
AZ
A 2+ cz
2
and Q²=B²+s,
by fubftituting for A and B their Equals n k² and n 12, that 2 will be
How equa
tions of four
dimenfions
may be re-
duced in
this manner
3 Z
178
ELEMENTS OF
œ + n k²
and 12 =
22-5
n
2
, from the former of which Q
will be known, when n and k are known; by Means whereof and the other
Equation, may be a fecond Time found, and the Agreement or Coin-
1
cidence of this Value, with that before determined for 1, will be a Proof
that n and k have been rightly affumed, and that adding to the given
Equation the Quantity n× (k ≈ + 1)² it will be compleated into the
Square (x²+ p x + 2)².
CI.
Since the foregoing Method of Solution depends upon the affuming pro-
The num- per Diviſors of ß, 2 and for the Values of n and k, it will be ex-
ber of trials
B
12
diminished pedient, in Order to diminiſh the Number of Trials, to diſcover fuch of
from the the Divifors as are not to the Purpoſe, which may be effected from the
confiderati- Confideration of the Properties of even and odd Numbers.
on of even
and odd
numbers.
q
a+nk²
2
being previouſly transformed to
2
//
2
f
In Order to which 2 =
n k² = 2 2 α = 2 2
9 + 3/2 p ] ²
= p]² - ƒ (by putting
22=ƒ). It is evident from thence, that if p be an odd Num-
ber p² 4 f, and confequently its Equal 4 n k², will likewiſe be an odd
Number, becauſe an even Number 4 f, fubtracted from the Square of an
odd one, always leaves odd; therefore feeing 4 k² X n is here an odd Num-
ber, both n and 4 k² must be odd, (for the Product of two even Numbers,
or of an odd one and an even one is even and not odd); whence it follows,
becauſe 2 k ] is odd, that 2 k muſt be odd too; and confequently k the
Half of an odd Number.
Now ſeeing p, n, and 2 k are all of them odd Numbers (when p is fuch)
they may therefore be expreffed by 2 a + 1, 2 b + 1 and 2 c + 1, re-
fpectively; a, b, and c being Integers In Confequence of which Affump-
tion, the Equation 4 n k = p²-4f will by Subftitution be changed to
8 b c² + 8 b c + 2 b + 4 c² +40 +1=4a² +49 +1
40+ I 4 f, or
2bc² + 2 b c + ½ b + c² + c =
a² + a f; from whence it is ma-
nifeft, as all the Terms but b are known to be Integers, that must
be an Integer likewife: And fo b being an even Number, it follows that
n, or 2 b + 1, muſt be double of an even Number, (or a Multiple of 4)
increaſed by Unity; therefore all the Divifors of 6 and 2 that have not
this Property, may be fafely rejected, as not for the Purpote.
In like Manner, if p be even the fame Limitations will take Place, pro-
vided that r is odd, which will be the Cafe when 2 is the Half of an odd
Number; for when 2, is an Integer, A² = p² -f, and B2 22- 5,
B² =
being Integers, their Product A B² will be an Integer, and confequently,
the Iquare Root thereof A B being rational, will likewiſe be an Integer,
and fop 2 and 2 A B being both even Numbers, their Difference r, as
SPECIOUS ARITHMETICK.
179
given by the Equation p 2 = 2A B+r, would be even, and not odd.
Let then 2
2x + 1
then 2
2
4x²+4x+1-45
4
but B², or its Equal n 12, being here equal to the Square of Half of an odð
Number (2) joined to an Integer (-s), I will be alfo Half of an odd
Number.
Let then /= 2+1, then n = 22-s= 4ny²+4y+n
2
12
confequently 4*²+4* + 1 = 4 £ = nx
4*²+4x+1-
4
I + 4x+4x² - 45'
I + 4 + 4 32
2
4
(+9+4+1), and
2
=1+4 × (
に
​x+x²
1 + 4 + 43²
2
-);
whence n is poſitive, and the Double of an even Number, or a Multriple
x + x²
y 12
1 +43 + 4 312
is
of 4 increaſed by Unity, when the Fraction
pofitive, but negative and a Multiple of 4 lefs Unity, when this Fraction is
negative.
From thofe feveral Conclufions is derived the following Rule for the
Reduction of Equations of four Dimenſions, contained in the Formula
x4 + p x³ + q x² + r x + 5 = 0.
CII.
a p, and
В
ก
for reducing
Make a = 9 = p²; B = r
4 a α, then put
for n fome common integral Divifor of B and 2, that is neither a Square, General
nor divisible by a Square, and which being divided by 4 fhall leave Unity, if rule derived
n be pofitive, but a Multiple of 4 lefs Unity if n is negative, if either porr from the
foregoing
be odd. Put alfo for k fome Divifor of if p be even, or Half of the odd conclufions
Divifor if p be odd: Take the Quotient from p k, and call Half the Re- an equation
2
a + 1 k²
mainder 1. Make 2=
and try if n divides 22,-s, and
the Root of the Quotient be equal to I; if ſo it happen, then the propoſed
Equation, by Means of the Values thus determined, will be reduced to
x x + ½ px + 2 = + √ nx (k x + 1).
2
"
That the Divifor n ought not here to be a Square, is evident from what has
been already remarked, fince both A and B would then be rational Quan-
tities; and that it ought not to be divifible by a Square, will alfo aqpear, if it
be confidered that and in the Equation, k√n A and 1√n = B
are to be taken the greateft, and n the leaft, that the Cafe will admit of.
I
Let the Equation x 12 x
+
CIII.
17o be propofe to be reduced.
of four di-
menfions.
180
ELEMENTS OF
Application
of the fore-
· going rule
to an ex-
ample.
Here p = 0, q = 0, r = 12, s=-17; confequently, ∞ =
17, and B and 2 ; that is, 12 and
8 = 12, Š
only 2 for a common Divifor, it must be n = 2. Again
В
34, having
= 6,
n
3,
whofe Divifors 1, 2, 3, 6, are to be fucceffively put for k, and
3
2
— I, -, for I refpectively, but a + n k²
2
S
n
equal to 2, and
; that is, k² is
and when the even Divifors 2 and
6 are ſubſtituted for k, 2 becomes 4 and 36, and 22
:
=1
s being an
3,
2 = I, and
+
n l, that is,
odd Number, is not diviſible by n = 2; wherefore 2 and 6 are to be fet
afide but when I and 3 are written for k, 2 is 1 or 9, and 22-s is 18 or
98 refpectively; which Numbers can be divided by 2, and the Roots of
the Quotients extracted being 3 and ±7; but only one of them, Viz.
3 coincides with 7. I put therefore k = 1,
I, I
adding to both Sides of the Equation n k² x² + 2 n k l x
2x2 12x18, there refults x4 + 2x² + 1 = 2x²
and extracting the Root of each, x²+1=± √2 × (x —
again, extracting the Root of this laft, the four Values of x according to
the Varieties in the Signs are ½ √ 2 + √ (· 3 √ 2 − 1 ),
플​),
- 3 √2 × √. ( 3 √ 2 − ) ; ± √ 2 + √ (− 3 √ 2 − 1)
/ √2 - √(- 3 √ 2 — 2).
Let the Equation x4
poſed to be reduced.
1
I
12 x + 18;
3), and
CIV.
6x3
- 58x2 114x
ΙΙ
II = o be pro-
58, r = 114, S -II, confequently
B, and
1135. The Numbers 8 and 2,
4533 have but one common Divifor 3, that is
The forego-
Here p=.
6, q
ing rule ap-
67=α,
315
plied to an-
other exam- that is,
315 and
ple.
2
В
11
3, and the Divifors of
105 =
are 3, 5, 7, 15, 21, 35, and io5.
n
В
Wherefore I firſt make Trial with 3k and dividing or 105 by it,
get the Quotient 35, and this fubtracted from
26, whoſe half, 13, ought to be equal to 7, but
n
pk-3 X 3, leaves
a+nk² -67+27
2
or
2
that is,
20 is equal to 2, and 22 —'s 411, which is indeed diviſible
by n = 3; but the Root of the Quotient 137 cannot be extracted, there-
fore I reject the Divifor 3, and try with 5k, by which, dividing
SPECIOUS ARITHMETICK.
181.
02
n
=-105 the Quotient is 21, and this taken from pk-3X5,
- 67+75 = 4
a + n k²
leaves 6=21. At the fame Time 2 = 2
2
and 23
s, or 16+ 11 is divifible by n, and the Root of the Quotient 9,
that is 3 coincides with . Whence I conclude, that putting 3, k = 5,
2 = 4, n = 3, adding to both Sides of the Equation, the Quantity
n k²x² + 2 n k lx+n, that is, 75x2+ 90 x + 27, and extracting
the Roots, it will be
x² + ÷ px+2= √n × (kx+1), or x²-3x+4=±√3× (5x+3)
3±5√3 ± √/17 ± 21 X
and again extracting the Root, x=
CV.
2
21 X 3.
2
No Regard in the foregoing Rule is had to that Circumftance in which ß
happens to be nothing. In this Cafe, the Equation p k²
2 kl
B becomes pk² - 2 kl=0; where one Root, or Value of k foregoing
11
α
n
Cafe in
which the
rule cannot
be applied.
must neceffarily be nothing. Therefore 2 being, we have l✔
√ (2² — s) = √( = ∞²
s) = √( // α ² —s); fo that by a direct Proceſs our given
Equation is reduced to x²+px + ÷ α = √( — α a-s) wherein a is
given 9 pp.
q #
Thus the Equation x4 + 2 x3 37 x² — 38 x + 1 = 0, where obferved in
α=38, B = o, is reduced to x²+19 + 6 √ 10.
60,
I
=
Befides the Value of ko when = 0, there are two other Values
which equally fulfil the feveral Conditions required, and bring out the very
fame Conclufion.
=
2
For fubftracting the Square of half the Second of the original Equations,
22— a = AA, p 2-r=2 AB, Q-s B2: from the Product of the
other two, there will be obtained the Equation 23+ q Q²+(‡ pr—s) ×2
+ 1/2 × (as rr) o, wherein the unknown Quantity 2, is alone
Xias 4
concerned, which Equation being of three Dimenfions, the Root 2, and con-
fequently k =
22-
N
will admit of three different Values.
Rule to be
this cafe.
to an exam-
Thus the Value of k in the propofed Equation x+2x3—37x²—38x+1=0 Application
may be 0, 3, or 4, or which comes to the fame, the Equation itſelf may be of this rule
reduced to x²+x—19=±6√10, x²+x+7=±√5×(3x+2), ple.
or to x² + x— 3 = ±√2 × (4 ×+ 2), all which are, in Effect, but
one and the fame Equation, as will appear by fquaring both Sides of each,
182
ELEMENTS OF
Third ex-
ample.
and properly tranfpofing; from whence, in every Cafe, the given Equation
x4 + 2 x³
37 x²
38x+1=0 will emerge.
CVI.
Let the Equation x43x3 + 44x²-123 + 97o be pro-
poſed to be reduced.
*
Here p = 3, 9 = 44, r = — 123, s = 97, q— p² = 44
=
—
9
4
167
βα
>
4
N
_501
8
"
*
"
// α p = −
123 +
501
8
984 + 501
8
483
8
a a
11
Во
+
27889
1552
4
27889
27889
16
·
;
26337
4
8779; whence B
it muſt be n = 3; wherefore = +
— a a
a a = 97
=
the Divifors of 8 are
3, 7, 23, and thoſe of È are
and 2, having only 3 for a common Divifor,
16
16
3, and
В
483
161
= +
: I
ท
8X3
8
therefore try whether k = +
(becauſe p is an odd Number) will
2
fucceed.
p k
В
21
23
44
_
11
II;
n k
and 1 = -
2
I I
2
167-147
4
4
«+nk²
147
3
Whence is deduced
49
X
2
8
2
4
+
2, and 2² =
·25
;
2
5
8
Q? 2
25 — 388
8
363. This Number divided by
4
3 gives
121
4
I I
whoſe Root is +
= 1; wherefore the
2
4.2
x +
2
2
=(
7
X
2
propofed Equation can be reduced, and there
I I
refults
2
-) √ - 3
Application
of the fore-
going rule
CVII.
If the higheſt Term of the Equation to be reduced has a Coeficient dif-
ferent from Unity, the Equation may be transformed into one that fhall
for reducing have the Coeficient of the higheſt Term Unity, and the foregoing Rules
equations may be applied to the new Equation.
having frace
ted terms.
Thus, if the Equation m x + px³ + q x² + rx + s = o be mul-
tiplied by m³, it will be transformed into
4
3
m² x³ + m³ p x3 + m³ q x² + m³ rx + m³ so.
SPECIOUS ARITHMETICK.
183
Let my, and 4+ pp³ + mq y y + mm r y + m³ s = o will
refult; to which the foregoing Rules may be applied.
2
For Example, if the Equation 2 x4-6x3+137x²-400x+315=0
was propoſed, here 2 x =y, and confequently m = 2, m² = 4; m³ — 8.
Subftituting y for x, and multiplying the Coeficient of the third Term by
2, the fourth by 4, and the laſt by 8, there will reſult,
146 №3 + 274 2 1600 y +25200.
1600, s 2520, 92274-9
—
Here p-6, q=274, r
=265 = ∞,
βα
2
3 X 265
=
795, and r
// pa
Example,
1600 + 795 — — 805 = 6, s
s-
— a α = 2520
70225
4
60145
= 3.
4
The Divifors of B are 5,
23, 523; wherefore n =
whence
В
n
Let k = 7,
2
805
5
B
n k
7,
—
23, and thoſe of
are
5,
5, or n=- 23. I put n=-5,
+161, and k=7, or k= 23.
21
23 =
1 —— 22; whence is deduced _*+nk²
2
44 = 2, and
265 — 5 × 49 —10=2
2
and 22=100, 22—s=100—2520—— 2420, and
22—5
n
2420 = 484 = ( — 22)² = 12; the Equation therefore reduced is
5.
33 + 10 = (7 3 — 22) √ — 5, and reſtoring 4 x² for y² and
2 x for y, and dividing all the Terms by 4, x²
3
x +
ང
2
2
=(=/=
ΣΤ
X
) ✓
2
24
CVIII.
Application
going rule
Hitherto we have applied the Rule to the Extraction of furd Roots, to of the fore
apply the fame to the Extraction of rational Roots, it fuffices to make ufe
of Unity for the Quantity n, and by this Means we may find whether an
Equation that has no fracted or furd Terms can admit of any Divifor, either
rational or furd, of two Dimenſions.
Thus, if the Equation x4 x³ − 5 x² + 12 x 60 was propo-
fed, by fubftituting I, 5, 12, and 6, før p, 4, 7 and s re-
for extract-
ing the ra-
tional roots
of equations
of four di-
menfions.
a
184
ELEMENTS OF
fpectively, I find a 5
Divifors of the Quantity
-
B
n
I
Halves whereof (if p be odd)
I
5
I find —— pk —
2
2
a + n k k
2
β
n k
,
or
B = 9 3 /==
, and putting n = 1, the
75
8
are 1, 3, 5, 15, 25, 75, the
are to be tried for k, and for k trying
-5, and its Half
5
1. Alfo
8
=6, the Root whereof
4
== 2, and
= 2, and 22-
12
1/2
agrees with 1. I therefore conclude, that the Quantities n, k, 1, 2, are
rightly found, and having added to each Part of the Equation the Terms
n k² x² + 2 n klx+nll, that is, 6 x x 12 x + 6, the Root
may be extracted on both Sides, and by that Extraction there will come
out xx+x+2=±√nx (kx+1), that is, xxx+1/2
= ± 1 × (2 × — 2 ), or x x -3x+3=0, and xx+2x-2=0,
+ 1/2 x
and fo by theſe two quadratick Equations the biquadratick One propofed
may be divided.
CIX.
B
R
If at any Time there are many Divifors of the Quantity fo that it
may be too difficult to try all of them for k, their Number may be dimi-
niſhed by ſeeking all the Divifors of the Quantity a srr. For the
Obfervation Quantity Qought to be equal to fome of thofe, or to the Half of fome
which ferve odd one; for a s
Q — al n², and taking from both r² =
+ n² k² 12, ſeeing the Remainder
to diminish
2
- a
the number 22
= p² Q2 p Q n k l
of trials.
α
Q² — (a + n k² )
Q
2
× n 1 ² — 1 p² 2² + p Q n k l =
2 2 X 11 12 p² Q² + p Qn k l has 2 in every Term,
the Thing is manifeft.
P
I
Thus in the laſt Example « s
rris
- r is –
2, fome one of whoſe
3
4.
2
Divifors 1, 3, 9, or of them halved
سا
, ought to be 2;
2
2
2
Wherefore by trying fingly the halved Divifors of the Quantity
and for k. I reject all that do not
글​, 롤​, 돌​, 돌​,
25
2
2
I
21
2
make a + — — 11 k²,
or
8
2
I
+ →→k k; that is, I to be one of
쯍​+
B
Viz.
n
75
SPECIOUS ARITHMETICK.
185
the Numbers 1, 3, 9,
3
2, 2; but by writing,,,
2
2
3
2
2
are found
15, &c. for k, there come out reſpectively - - 2 + 1/
I
2
51
+
&c. for 2, out of which only
2
3/4 and 2/3/4/1
I
I
3
9
2
2
3, and 2
༢
2=
or
2
2
2, and confequently
among the aforefaid Numbers 1, 3, 9,
the reft being rejected, either k will be =
5
k =
2
I
and 2; which two Cafes let be examined.
2
CX.
If the Equation to be reduced is of fix Dimenfions, let it be
s
x6 + p x5 + q x² + r x³ + 5 x² + 1 x + vo, and let there be af
fumed (x3 + p x² + 2 ≈ + R)² − ( A x² + B x + C) ²
1 x
= x6 + p x5 + q x¹ + r x³ + s x² + 1 x + v=o, which, by Invo-
lution and Tranſpofition, will give
Inveſtigati-
on of the
method of
reducing e-
quations of
(2 Q+ & p²—q) x4+(2R+pQ−r)x³+(pR+Q²− s) x²+(2 QR—†)x
+ R² — v = A² x4 + 2A B x³ + ( 2 4 C + B² ) x² + 2 B C x + C², fix dimenfi-
From whence by equating the Coeficients of the homologous Powers, ons by trial.
and writing aq − & pp,
We have 1°. 2 Q— a = A ²;
Z-
2
3°. p R +23 — s = 2 AC + B²;
5º. R² v = G².
2
If now the Value of 2= ½ A²
Equations, be fubftitued for 2,
2 R + ½ p A ² + // p a
2
I
Q—
2°. 2 R + p Q− r = 2 AB;
4°. 2 QR — t = 2 BC;
-
+ ½ a, as given by the firſt of theſe
in the fecond, we ſhall get
= 2 AB, and confequently,
R = AB — Þ 4² + — £ ( by putting ß = r — § p a), which
4
2
Value, together with that of 2,, being fubftituted in the three remaining
Equations, we fhall have,
1º. p A B — — p² A² + ½ p B + &
2º. A³ B — § p Aª + § ß Ą² + ®
a
A¹ + — α A² + — «²—s=2AC+B²
A В ~ ‡ p a Ą² + ÷ « ß — t = 2BC
4 A
B
186
ELEMENTS OF
2
3°. A² B² — þ A³ B+ A B+
which by putting y
Ө
0 = v
=
I
16
p²A4 — pß A²+ § ß² — y = C²
—
~ — — α α, »t« ß, and
} r a a, n =
spß, =
BB, will be reduced to
p A B — — p² A² + — A4 + ½ a A² —
A3 B — Þ A² + ½ ß A² + a A B
— 2 AC + B²
=
१
4 p a A ²
2
I
and A² B² — §
reſpectively.
§ A³ B+ B AB +
p²
n = 2 B C
=
10
1
P
A+ — ‡ p ß A ² — 0 — C²
Now as the Values of A, B, and C, are either rational Quantities, or if
irrational, affected by the fame radical Quantity; let them be expreffed by
k√n, l√n, and mn, refpectively; then Subſtitution being made, and
every Equation divided by n, we ſhall have
FH k2
FHk
p n k¹ + ——— ß k² + a kl
B
1º. pk l-
I
I
I
p² k² +
n k++
a k2
3
4
4
2
= 2 km + 12
12
2º. n k³l-
I
I
α
2
24
pa k²
n
=2/m
4
12
p² n k+ —
I
4
Þ B k² ——— =m²
Ө
3°. n k² 1² — — p n k³ 1 + B k l + ——
2
>
n
From whence it appears (fince k, l, and m, are here confidered as In-
tegers, or as the Halves of fuch) that, and 0, ought to be all of
them divifible by n, or which is the fame, that n ought to be ſome com-
mon integral Divifor of the Quantities, n, and 0.
With Regard to k, let the feveral Terms in the former Part of the firſt
of our three Equations, in which k is found, (in order to abreviate the
Work) be denoted by Fk, then will the Equation itſelf be changed to
F k
= 2 km + 12, and in the very fame Manner our other
72
two Equations will be changed to G k
H k
= m², reſpectively.
22
"
= 21 m, and
n
Let now the Square of Half the Second of thefe Equations be fubtracted
from the Product of the first and third, then will
FkO
n
Hk C
+
१०
* *
I
G k n
2
-
G² k² +
༢༠ =2 km³,
n
n n
2 n
n n
which, by dividing the whole by 2 k, and putting a 50
at Length become,
4 n², will
2nn
m³ :
I
I
I
G² k
HX
GX
++
1FX
+
=
n
2
n
k
SPECIOUS ARITHMETICK.
187
1
λ
2 n n
where ¿,", and 9, being all diviſible by their common Divifor n, (as is
fhewn above); it is manifeft, that (in order that m may be an In-
teger, or the Half of an Integer), ought alfo to be divifible by its Divifor
k; that is, k ought to be fome Diviſor of the Quantity
λ
2 n n
Again, with Regard to I, let the Value of RA B± p Q + ± r,
as given by the fecond of the five original Equations, be fubftituted in the
fourth, by which Means we have 2 QA B = p 2² + r 2− t = 2 B C,
r2-
or 2 2 n k l-p 2+r 2− t = 2 n lm (becauſe A=k/n, B=l/n,
C = m√n) and confequently
до
2
pQ2
n l
t
= 2 m―2k 2;
where 2 m — 2 k 2, being an Integer, it is evident that
"2-22 — t
r Q p Q
n l
muſt be an Integer alfo ; and confequently, I fome Divifor of the Quantity
да
& Q — p Q²
t from whence
11
=
being found in Numbers, the Value
of RAB ÷ p Q + ÷ r = n k l − p 2+r will be had like-
wife; and then by Means of the three laft of the five original Equations,
the Value of m may be alſo found three different Ways, and the Truth of
the Solution thereby confirmed: For thofe Equations, by fubftituting for
A, B, and C, their Equals k√n, √n, and mn, do become
R²— v = n m², 2 QR− t = 2 nlm, and p R+22-s2nkm+nll;
from the firft of which m
✓
R2
12
m =
QR — 3 t
n l
>
and from the third m =
; from the fecond
22+PR-all — s
2 n k
which Values therefore, when Q, R, &c. are rightly affumed, will be
all found equal among themſelves.
CXI.
from the
confiderati-
As the foregoing Method of Solution depends upon the affuming proper The num-
Divifors of,,, &c. for n, &c. To bring the Work into lefs Compaſs, ber of trials
it will be expedient to reject fuch Divifors of thoſe Quantities as are not to diminiſhed
the Purpoſe, and this may be effected as in the precedent Cafe, from the
Confideration of the Properties of even and odd Numbers, the Reaſoning on of the
thereon being the very fame in this Cafe as in the former, which therefore properties
it will be unneceffary to repeat. And from the Conclufions above derived
is deduced the following Rule for reducing an Equation of fix Dimenfions bers.
x6 + Ŕ x5 + q x² + rx³ + s x² + + x + 0 = 3.
جب
of even and
odd num-
•
188
General
rule derived
from the
method for
ELEMENTS OF
CXII.
// s
Make qppa, r — pa = 6, 5 — = p² = 7, y — — a α = 8,
// α B = n, v 1 B B = 0, 80
ท
ท
Then for n take of the Terms 2, n, 20, fome common Integer Divifor,
foregoing that is not a Square, and that likewife is not divifible by a Square, and
reducing an which alfo divided by the Number 4, ſhall leave Unity, provided any one of the
equation of Terms p, r, t, be odd. For k take fome Integer Divifor of the Quantity
fix dimen-
Lions.
λ
2 n n
if p be even, or the Half of an odd Divifor, if p be odd, for a
take the Quantity, a + n kk, take for 1 fome One of the Divifors
2r-Q2p-t
of the Quantity
if Qbe an Integer, or the Half of
an odd Diviſor, if Qbe a Fraction that has for its Denominator the Num_
QrQQp—t be nothing. And for
÷
n
>
21
- ~
ber 2; or 0, if that Dividual
R the Quantity r — Qp + nkl. Then try if RR can be di-
vided by n, and the Root of the Quotient extracted; and befides, if that
as to the Quantity
Root be equal as well to the Quantity
22+PR-nll-s
2 n k
QR t
n l
; if all theſe bappen, call that Root m, and i
n
Room of the Equation propofed, write this x3 + 3 px x + Qx + R
± √ n x ( k x² + 1 x + m ) ; for this Equation, by Squaring its
Parts, and taking from both Sides the Terms on the right Hand, will pro-
duce the Equation propoſed.
24
CXIII.
Let x-2 ax5 +2bb x² + aabb x3 — (2 aa bb — 2a³6 + 4ab³) x²
of the fore. + 3a a bª — aª b b = o be propoſed to be reduced.
Application
going rule
to an exam-
ple.
4
Here writing
0, and 3 a a b4 —
come out 2 b b
4
—
2a, +2bb, +2 a bb, — 2 a ab b + 2 a³ b 4 ab³,
a^ b b for p, q, r, s, t, and v, refpectively, there will
a a = ∞, 4 a b b
6, 2 a³ b + 2 a ab b
4 a 63 — a² = 4, —
40
a3
bª + 2 a³ b + 3a a b b −4 ab3----
64
4 a 64 = n,
n, and
5
a4 = 5.
=
4
122 45
Ө
2 b b, or
// a5
3 a3 b b
—a a b4 +a+bb
and the common Divifor of the Terms 2, 7 and 2 0, is a a
266 a a, according as a a, or 2 bb, is the greater. But let a a be great-
er than 2 b b, and a a 2 b b will be n; for n muſt always be affirma-
—
tive. Moreover we have,
"
1
2
11
१
n
3 +2 a b b, and
2
B
a a + 2 a b + → bb,
་
4
I
1
I
2
— a² + — a abb, and
a4
4
2
SPECIOUS ARITHMETICK.
189.
""
I
I
2 n
I
X
or
26
a5b
I a 4 bb
N
8 nn
2nn
4
8
+ — — 43 63
3
a² 64, the Divifors whereof are 1, a, a a; but be-
8
is
2
cauſe ✓nk cannot be of more than one Dimenfion, and that
of one, therefore k will be of none; and, confequently, can only be a
Number. Wherefore, rejecting a and a a, there remains only 1 for k, be-
fides + n k k gives o for E, and
I
2
2
Q+
2.20 -
n
is allo
Nothing; and confequently 1, which ought to be its Divifor, will be No--
thing. Laftly, r — ž p Q + n kl gives a bb for R,
2 a a b4 + at bb, which may be divided by n, or ea
Root of the Quotient a a bb be extracted; and that Root
4
-
ly, Viz.ab is not unequal to the indefinite Quantity
and R R v is
2b b, and the
taken negative-
Q R - ½ pt
nl
22+pRnll-s
z n k
9
or, but equal to the definite Quantity
wherefore that Rootab will be m, and in the Room of the Equation
propoſed, there may be writ x3pxx+Qx+R=vnX (kxx+lx+m),
that is, x3 ax x + a b b = √ (aa-2b b) x ( x x — ab);
the Truth of which Conclufion you may prove by fquaring the Parts of the
Equation found, and taking away the Terms on the right Hand from both
Sides; for from that Operation will be produced the Equation-
x6 — 2 ax5 + 2 b b x² + 2 a b b x³ ( 2 a a b b — 2 a³ b² + 4 a 63) x²
+ 3a a b¹ — at bbo, which was propofed to be reduced.
— .
CXIV.
When ao; in which Cafe k (or one Value of k at leaſt) will alſo
beo, the Reduction will be performed by a direct Proceſs. For k
being Nothing, the three Equations wherein k, l, and m are first introduced which the
will become
ع
R
0.
21 m, and
17
n
171
2-
Cafe in
foreging
rieved in
obſerved in
this cafe.
= 1²,
=m²; whence rule cannot
be applied.
l·√√ n = √~ }, m√n=0; and confequently x=0 ² = 0,
3,
Rule to be
— —
as it ought to be; therefore, by fubftituting thofe Values, and writing al-
fo, inſtead of 2 and R, their Equals, and 8, the Equation given is
here reduced to x³ + 1 × x² + =
1 ∞ x + B = + x√− ¿± √ — 0.
1
3
CXV.
If the Equation propofed is of eight Dimenfions, let it be
sx4
z
品
​x8 + p x7 + q x6 + r x 5 + s x² + + x³ + v x² + w x + x = 0;
then by affuming (x++ ½ p x³+Qx²+Rx+S)²— (A x³+B x²+Cx+D)ª
=x8 +p x7 + qx² + r x² + s x² + ¢ x² + v x²+wx+x=0;
190
ELEMENTS OF
Method for
diſcovering
and proceeding as in the former Cafes we ſhall have here, 1º. 22-=A²,
2º. 2R+pQ−r= 2 A B, 3°. 2 S + pR+Q2—s=2 A Ċ + B B,
4°. p S + 2 QR — t = 2 AD + 2 B C, 5º. 2 QS + R R — u
= 2 B D + C C, 60. 2 RS w=2 CD. 70. SS z = DD.
Putting now (as before) A = k √ n, B = 1 √ n, C = m ↓ n,
I
whether an Dbn; putting alfo, to fhorten the Work, Q
equation of
fions can be
reduced.
Q'n' +
I
2
น
eight dimen RR' n + & B, S = S'n+y; that is, let the Quotients of Q, R,
and S, when divided by the common Divifor n, be Q', R', S', and the Re-
mainders, B, andy, reſpectively: Then to determine theſe laſt,
which muſt be firft known, before n can be known, let Subſtitution be
made in the ſecond and third Equations, every where difregarding fuch
Terms wherein n and its Powers are involved. Thus Subftitution being
made in the fecond Equation, we have 2 R'n + B + p Qn + ½ par
2 kln; where the homologous Terms in which n enters not are B,
pa, and -r: The others therefore being here difregarded, we have
B+플 ​pa r = 0, or B = r pa; in the very fame Manner from
a
the third Equation y ++ — α a — s = 0; and confequently,
푸다
​y = s
Σ
ax.
Let Subftitution be now made in the fourth, fifth, fixth, and feventh,
a
V,
Equations (ftill diſregarding fuch Terms as would involve the Powers of n)
and there will come out, 10. // p r + 1/2 α ß t, 20. ay + B B
30. By w, 40. yy z. Now, as all the other Terms that would
arife in theſe Equations (beſides thoſe put down) are affected with n, and
I
are therefore divifible thereby; it is manifeft, that the four Quantities
21/22 ay
β β, ω w — By, and z
I
t
즐 ​py
½ a ß, v
444
γγ;
here brought out, which, for Brevity lake, I reprefent feverally by. d, e, sin,
muſt likewiſe be all of them divifible, by the fame common Diviſor n,
when the Equation given is capable of being reduced. If therefore no fuch
common Divifor, (under the Reſtrictions ſpecified in the preceding Caſes,
depending on any Coeficient p, r, t, or w, in a Place denominated from an
even Number, being an odd Number,) can be diſcovered, the Work will
then be at an End.
CXVI.
From the fame Method of Operation, which may be looked upon as a
fort of Examination, whether the Equation be reducible or not, we may
General me find all the Quantities, to which n ought to be a common Divifor, when
thod for dif- the Equation given is of 10, 12, or a greater Number of Dimenſions.
covering
whether an Let x + p x² - 1 + q x² m²+r x² m −3+s x²-4+t x²-5+ &c. =0
x² +px +qx2m-2
2+rx²m
equation of be given, and let there be aſſumed,
172
4.
J
SPECIOUS ARITHMETICK.
igr
X
I
- I
½
of even di-
can be re-
[x”++ px”~¹ +-(Q_n+ž a) x™¬²+(R'n+ B)x−3+(S! n+)-4&c.]2 any numb-
— n (k` x¹² - ¹ +1 x²+mx-3&c.)²= x² +p x² - 1+q x² - 2 &c. menfions
then by fquaring " + &px" - ' + ( Q' n + ½ a ) x2, &c. and duced.
tranſpoſing x² + px²-1+q x², &c. it will appear that
the Terms of this Equation, in which n enters not, will be,
2 m
x
I
2 m
β
Y
Xx2112
2
+pa
2 m
Xx
3+
P
Xx
x x 2 172
#12 - 4+
力
​Xx2=5+
α
-9
В
5+ 1/2 a
X +2 11
gu
BB
From the former Half of which Terms all the Quantities a, B, y, &c. will
be determined by affuming the Coeficients equal to Nothing: Thus we
have = 9 ipp, B = r ½ pa, y = s 2. P. B
d = t — // pra B, &c. and thoſe Quantities being known, the Coe-
ficients of the remaining Terms will likewife be known, which ought all
of them to be diviſible by n, in Order that the Reduction may fucceed;
that is, they ought to be fuch, as to admit of a common Diviſor (n) un-
der the Reſtrictions before ſpecified.
12
I I
ΤΟ
CXVII.
For Example, if the Equation given is of twelve Dimenfions, as
x¹² + px¹¹ + 9 x¹0 + r x9 + s x8 + t x7 + v x6 + a x5 +
+ cx³ + dx² + ex + f = o, we ſhould have a q
b
4 Application
of the fore-
going me-
PP thod to an
B = r
Ε
E = V
½ pd
½ pa, y = s ž p B — — a a, d.
d = t
py-a ß, and
// ay BB, and the Coeficients of the other fix
example.
1
Terms (whereof n ought to be a common Divifor) would be
E
½ pe + ½ ad + ½ By —
½re + dd d, ½ da
a, & a s
I
+ / B d + 1 2 x —b, žße + — y d—c,
ß yy žyd
and
e,
εε
- f.
Thefe Operations for finding of n will moft commonly end the Work.
If fuch à Value however ſhould be found for n as to anfwer all the Condi-
tions above ſpecified, and confequently there are Hopes of a future Reduc-
tion. It may be tried by the following Steps, which though applied only
for reducing an Equation of eight Dimenfions, yet it is eafy to perceive,
are general for all Equations.
CXVIII.
z
Becauſe by the 7th Equation SS - ≈ = n b²,
fquare Number b², to which, after it is multiplied
of the Equation z, being added with its proper Sign
if n be even, feek a
into n, the laft Term
+ n b², ſhall make
192
ELEMENTS OF
(
q,r,
a fquare Number, which may be expeditiously performed, by adding to z
when ʼn is an even Number, or to 4 z when it is odd, thofe Quantities
fucceffively n, 3 n, 5 n, 7 n, 9 n, 11 n, and fo on till the Sum becomes
equal to fome Number in the Table of fquare Numbers; and if no fuch
Number occurs, before the fquare Root of that Sum, augmented by the
fquare Root of the Excefs of that Sum, above the laft Term of the Equa-
tion, is four Times greater than the greateſt of the Terms of the propofed
Equation p, q, r, s, t, v, &c. There will be no Occafion to try any
farther; for then the Equation cannot be reduced: But if fuch a fquare
Number does occur, let its Root be S, if n be even, or 2 S, if n be odd;
and call
but if n is odd, then Sand b may be Fractions
that have 2 for their Denominator: And if one is a Fraction, the other ought
to be ſo too, which is alſo to be obſerved of the Numbers R and m, Qand 1,
p and k, hereafter to be found. And all the Numbers S and b within this
rule for re- Limit must be collected in a Catalogue.
'General
ducing equa
tions that
12
= b;
Having thus found n, b, and S; k is next to be found by fucceffively try-
have been ing all Numbers which do not make n k+p four Times greater than
found by the the greateſt Term of the Equation; when k is had, Qis to be found by
foregoing
method to the Equation Q=
-be reducible
nk² + a
2
tried for / which do not make n
eft Term of the Equation, and
n p k² + 2 B
4
R =
Qbeing found, all Numbers are to be
IQ greater than quadruple the great-
being found we have
+nkl. Laftly to find m, all Numbers are
fucceffively to be tried, which do not make n m + R greater than quadruple
the greateſt Term of the Equation. Of the Values of the Letters regiſtered.
in the Catalogue, thoſe only are to be affumed which agree when found fe-
veral Ways, thus, Sz n b by the 7th Equation; and allo
PR 22 + nll
2
dent b√SS —
b=
n
=
+
+nk m by the third; and its Correfpon-
Z
by the 7th Equation, and =
PS+2QR-1-2 nl m
2 n k
228 + R²
n m²
z n l
by the 5th, and
by the fourth; alfo =
2 RS w
2 n m
by the 6th Equation; which Values therefore, when
Q, R, &c. are rightly affumed, will be all found equal among themſelves,
and the Equation propofed will be then reduced to
4
3
x² + & p x³ + Q x² + R * + S = √ n × ( k x³ + 1 x² + mx + b)
px³
SPECIOUS ARITHMETICK.
193
Let x8 + 4x7
be propofed to be reduced.
Here q
I
4
= pß
0 = B, s — 3 p B — — a a
5 +
5
CXIX.
xó̟ — 10 x5 +5 x4
5 x3.
I
4:
v4
25
= 5
= − 2 = x, t −
4
IO X X —IO X-
5 = a, r— pa=
5
4
-10 x-5=0,
10 +0
1/2
p r — ±aß
5
1
- d v
ay — BB — 10 —
=
25.
2
8
10 = 8, Z
25
-5 64
11
5,
105
8
345
64
, w
1/4 B y
", therefore,
2 ♪, 21, 2, 8, refpectively, are
105
,
20, and
4
345, and their common Divifor 5,
8.
which Application
of the fore-
divided by 4, leaves I, as it ought, becauſe the Terms is odd. Since going rule
therefore, the common Divifor n, or 5, is found, which gives Hope to a fu- to an exam-
ture Reduction, and becauſe it is odd, to 42, or 20, I fucceffively add ple.
n, 3n, 5n, 7⋅n, 9n, &c. or 5, 15, 25, 35, 45, &c. and there arifes
-15, 0, 25, 60, 105, 160, 225, 300, 385, 480, 585, 700, 825,
960, 1105, 1260, 1425, 1600. of which only 0, 25, 225, and 1600,
are Squares.
Wherefore, the Halves of thofe Roots 0,
5, 15
5,
1 ,20 are to be collected in
2 2
SS-
Z.
that is, I, 3 3 7
12
2 2
ود
a Table for the Values of S, & the Values of
reſpectively for b. But becaufe S+n b, if 20 be taken for S, and 9 for b,
becomes 65, a Number greater than four Times the greateſt Term of the
Equation. Therefore I reject 20 and 9, and write only the reft in the
5, 15.
Table as follows, b {1. 3, 1, } s {0. 5,
2
2
2 2
Then I try for k, all the Numbers which do not make 25k greater
than 40, (four Times the greateſt Term of the Equation), that is, the Num-
8, 7, — 6, — 5, — 4, — 3, 2, — 1, 0, 1, 2, 3, 4, 5, 6, 7,
bers.
putting
or
n k k + a
2
60, 15, 20, 15, 0,
2 ,
175
2
5 kk-5
>
that is, the Numbers: 315
, 120,
2
2
120 reſpec-
5, 0, 15, 20, 75, 60, 175,
༡;
24
2
2
2
tively for 2: But fince 2n 1, and much more Qought not to be greater
than 40, I perceive I am to reject
Correfpondents 8, 7, 6,
315, 120, 125 and 60, and their
2
5, 5, 6, 7 and confequently that only
— 4, — 3, ——-2, -1, 0, 1, 2, 3, 4 must refpectively be tried for k,
4 B
194
ELEMENTS OF
and 75, 20, 1, 0, -1,
2
2
Let us therefore try
W
will be fucceffively to be
greater than 40, that is,
I
2
0,
75
15, 20, 25, refpectively for Q.
2
for k, and o for 2, and in this Caſe for 1 there
tried all the Numbers which do not make 2+nl
all the Numbers between 10 and
IO; and for
+ n kl,
2 ß - n p k k
4
R you are relpectively to try the Numbers
or-5-57, that is,-55,50,45,-40, -35, -30,25,-20,
—15,—10,—5,0,5,10,15,20, 25, 35, 40, 45, the three former of which,.
and the laft, becaule they are greater than 40, may be neglected. Let us
try therefore, 2 for 1, and 5 for R, and in this Cafe, for m there will be
befides to be tried all the Numbers which do not make R ± nm, or 5 ± 5m
greater than 40, that is, all the Numbers between 7 and-y, and fee whether
Q2−pR+nll
or not, by putting
that is, 520+ 20 or 52 H
2
5 m=S, that is, if any of thefe Numbers
¡H + n k m, or
ད
2
45.
༢༨
2
55
2
2
65
2
25
- 15
5
5
IS
2
2
2
2
1
25
དང
2
35
25, 45
2
Numbers o, £
+
vla
2,
15
65, 75, 85, is equal to any of the
2
2
2
+, which were brought into the Catalogue
2
for S, and we meet with four of theſe
15
5
5
15
to which
2
2 2
2
Anſwer +
7
+
+ +
7
2
2
2
2
3
±2, ±2,± written for b in the fame Table, &
2, 1, 0, — 1 ſubſtituted for m, but let us try - or S, 1 for m, and
± 3 for b, and you will have
+
5
2
2 RS- น
- 25+ 10
3
2 n m
10
2
nm m
25 +10
- 5
Τ
· and
20
-10+5 + 20
3
=
10
2
2
and
2 QS + R R -
2 n.!
2 nk
pS + 2 QRt 2 nl m
Wherefore, fince there comes out in all Cafes - 3, orb, I conclude all the
Numbers to be rightly found, and confequently that in the Room of the
Equation propofed, we may write x42x3 + 5 x − 2 1
=√5 × (-x³-2x+x-1). For by fquaring the Parts of
SPECIOUS ARITHMETICK.
195
this, there will be produced that Equation of eight Dimenfions which was
at firſt propoſed.
m
CXX.
and fourth.
Whatever Efforts the Analyfts have made, to find a general Method for Refolution
folving Problems producing Equations of every Degree, befides thofe ex- of Equa-
preffed by ax"= b, and ax2 + bxm =c, they have not hitherto fuc- tions of third
ceeded, except for fuch as produce Equations of the third and fourth De- degree.
gree, which we ſhall proceed to explain, beginning by the Refolution of
Problems, producing Equations of the third Degree.
CXXI.
The Compound Intereft of a certain Sum of Money, put out for 4 Years,
amounted to a Sum a, but the fimple Intereft thereof for the fame Time, and
at the fame Rate, would have been a Sum b. What was the Sum put out ?
And what the Rate of Intereſt ?
I
3
46
I
Let y denote the Intereft of 17. for one Year, therefore fince the fimple
Intereſt of 1 1. for 4 Years, is 4y, and the compound Intereft (1+1)+
or 4 y +6y ² + 4 p³+p4, we have, as 4y+6y²+4y³ +-y4:4y = a: b
by the Nature of the Queſtion, and confequently 6y+4y²+y3 =
from the Refolution of which Equation, y will be found, and confequently
the Rate of Intereſt required.
CXXII.
4a
b
of the third.
Let in General the Equation to be folved, be y3+ dy²+ey+f=0; Equation
if we could reduce thisEquation to another, including only the first and laft the moit.
Term, it is manifeft that the Equation would be folved. Now the moſt na- complicated
tural Method of transforming an Equation into another, in which we are degree..
left at Liberty to make fome Variation, is to fubftitute in this Equation in
the Room of the unknown Quantity fome other Quantity, in which a Letter
is left undetermined in order to make uſe of it at pleaſure.
3
CXXIIE
Let then be fubftituted in this Equation, in the Room of y, and
there will reſult x³ + (3 r + d) x x + ( 3 r r + 2 dr + e) x Transfor
+3+dx²+er+fo. Asr in this Equation, is at our Difpofal, it is mation by
eafy to perceive, that by its Means, any Term may be taken away out of wh
the Equation.
means of
which any
term is
taken away,
I
Let, for Example, 3. r+dbe put =0, or r
d, and the from an
3
Equations.
I
Equation is reduced to x3+ (e
ا احد
x + 227 ď³ —
2
de
d3
+ƒ=0,
3
3
the ſecond Term vaniſhing. If 3 r r + 2 dr + e be pat = 0, 05.
196
ELEMENTS OF.
土
​e
I
· d± √ [− ÷ + dd], the third Term will vanifh,
3
3
but the two others will remain.
9
It is eaſy to conceive, that by fubftituting xr in the Room of y, no
more than one Term at a Time can be made to vaniſh, but if by this Tran
formation, we have not entirely effected the Reduction of the propofed.
Equation to two Terms, at leaſt the Queſtion has been rendered more
fimple, fince an Equation confifting only of three Terms remains to be
folved.
x3
d2
3
— 2 ) x + 12/27
de
3
Of the two transformed Equations that refult by taking away either the
2d or 3d Term, the firſt, ׳ + (e —
23 ====² + f = 0
is the moſt fimple, which we fhall therefore endeavour to reſolve,
by ftill trying to diminish the Number of its Terms: but before we
proceed to this Reſearch, we fhall explain how the Method employed for
transforming an Equation of the third Degree, and exterminating its
intermediate Terms, may be applied to Equations of every Degree.
CXXIV.
Let, for Example, the Equation propofed be y4+a y3+by² + cy+d=0,
The forego- the moſt general of the fourth Degree, fubftituting xr in the Room of
there refults.
ing trans-
mation ap-
plied to an
Equation of
the fourth
degree.
ولا
g3
x4+(4 r+ a) x³ +(6 r²+3 a r+b) x²+(4 r³ +3 a r² + 2 b r + c) x
r4 + a r³ + b x²+cr+d=0.
3
In which putting 4r+ao, or ra, there will refult an Equa]
tion of the fourth Degree wanting the fecond Term.
In like Manner, if r be determined by Means of the Equation
6 r² + 3ar+b= o, there will refult an Equation of the fourth Degree,
The fecond wanting the third Term, the fourth Term in like Manner may be made to
vaniſh by ſolving and Equation of the 3d Degree, &c. But the Analyſts ſeldom
or never take away any Term but the Second, becauſe the other Terms
ed and why. cannot be taken away without employing Calculations complicated with
Radicals.
term moſt
commonly
exterminat-
The fecond
term exter-
minated in
an Equation
of the fifth
degree.
CXXV
4
By fubftituting in the Equation 35 + a p4 + 6 µ³ +cy² + dy + e = 0,
xr for y, there will refult x 5 + (5 r +: a) x + &c. in which the fecond
Term will be taken away by folving the Equation 5+a=0, or putting
.I
ra and in General, in an Equation of the mih Degree repre-
5 1/2
1}{- I
7II
be put
putxr it will
+ &c. whereby it appears
In an Equa- fented by yay +bym―² + &c. = 0, if y
tion of the be transformed into x" + (mr + a) x
tnth degree.
that the fecond Term may be taken away from the propofed Equation by
putting y=x-
a
773
2
SPECIOUS ARITHMETICK.
197
CXXVI.
After this little Digreffion, let us proceed to refolve the Equation
of the third Degree, x3 + (e-
d
3
2
27
de
3
+1)=0,
to which the general Equation 3 + d y² + ey+fo, was reduced
I
by putting y =≈ ——, d, and to ſhorten the Calculation, let it be ex-
3
preffed thus, x3 + px+9=0.
Purſuing the fame Method already employed, let the foregoing Equation Refolution
be again transformed, by fubftituting, for Example, uz in the Room of of the gene-
x, not with a View to take away a Term of this Equation, as before, for it ral Equation
x3 + px +
would foon appear that the Term taken away would return, but to reſolve q=0.
this Equation into others more fimple. Without perceiving that fuch a Me-
thod would fucceed; yet it is manifeft, that the Transformation of an E-
quation into another, in which a Letter is left undetermined, cannot but be
of Ule.
P
u+z being fubftituted for x, there refults 23+ 3 u uz + 3 uz ž
+ z³ + p u + p z + q = 0. Let one of the unknown Quantities z
or z be fuppofed fuch, that u3 +23+q=0. In this Cafe we will have
3 u² z + 3 uz ²+pu+pz=o, which being divided by u + 2, gives
3u2+p=0, or u — — By which Equation one of the unknown
Quantities introduced may be exterminated; to find the Equation that will
determine the other unknown Quantity, let this Value of a be ſubſtituted
in the firft Equation u3+z3 + q = 0, and there will refult
p3
223
32
27233 +23+2=0, or z6 + q z³ =
27
I
p3
whence is deduced
27
x = √ [ − = ? ± √ √ ÷ 2² +³)]. Wherefore u or—
2
I
11
3
2
9
3
27
I
³V[− ÷ 1± √(¦ ¦ &² + — 1³)]
2
27
p
35
which is reduced to
³ // [ 39 ± √ ( & q² + p³)]; for it is eaſy to perceive that the Product of
27
27
7
±√ ( 19² + 1/1/17³) into +9 ± √ ($q² + = p³) is- p³ con-
9
fequently is the Product of the cube Roots of thoſe Quantities.
Ι
3
Now,
adding together thofe Values of u and %, we will have u + z or
2
7
2
(q² + p³)], or
x = 3√ [− = 9±√ ({ q² + — p³)] — ³/ [ + ÷ 9 ±
x = V [− ÷ 9 + √ (§ q² + § p ³)] — 3 √ [ + § 9 + √
for the Value of x will be the fame, whether the Radical ✓
be taken pofitively or negatively.
(
2
÷ q² + z p³)]
2
I
(49² + 237 p³)
198
ELEMENTS OF
,
ing formula
exprefles
3
1
}
CXXVII.
The foregoing Formula expreffes only one of the Values of the three
The forego Roots of the Equation, to find the two others we muft divide the Equation
x³ +px + q = o by the Root found, and the Equation of the fecond
only one of Degree that refults for Quotient, will when refolved, give the two other
Roots of the Roots required.
the three
equation.
I
27
To find the general Expreffion of theſe two Roots, I put, in order
to abbreviate the Work [9 + √ (399 + p³)]=m and
3/ 3 =
13 // [~ 9 + √ ( qq + p³)]=n, then obferving that m np and
3 — 4
723 = q, I divide the Equation x3 + + q = 0 by x + m
inveſtigat- and there retults the Equation x² + nx
ing the two
Method of
m3
other roots., whofe two Roots, are x
Cafe in
which the
value of
cannot be
diſcovered
3
3
m
2
x
n,
nix + mm+ nn + mno
± 3 × ( m + n) √ — 3, or
3
I
I
¿ ³
// [ ÷ 9 + ³ √ ( 4 qq + 27 p³)] — § ³¾/ [—±q+³/ ± 99 + 1/7 p³)]
±[÷3/[iš 9+√ (¥ qq+ 27 p³)]+š√[~\ q+√(✯ qq+27 p³)]]X√—3
which are neceffarily imaginary when✓ ( q² + p³) is a real Quantity.
CXXVIIL
I
27
I.
27
The foregoing Solution of Equations of the third Degree is liable to this
Inconveniency, that it gives no Value for x when 3 is negative, and
greater than 99, in this Cafe, the Value of the Quantity (39²)
by means of is imaginary, whence the two Quantities 3 [−9+ √ (— q² + 2 p³)]
the forego- and 3 [ / 9 + √ ( q² + p³)] which compofe the Value of x are
ing formula
imaginary, we are not however to conclude, that the Value of x in
of the ima- this Cafe is imaginary, it being real, as may be proved after the following.
ginary quan- Mannner.
on account
tities it in-
cludes.
The value
of x how-
ever in this
cafe is real.
Let the Value
Σ
I
q and b
I
Z
I
I in the
[—³q+√ (‡.q²+ z p³)]—³/ [ \ 9+√íš 99+ 27 p³)].
of x be refumed, and let a be put in the Room of
Room of √(99 + 27 p³) fuppofed imaginary, and there will refult:
x = 3√(− a + b √ − 1 ) − 3 √ (a + b −1.) Now 3√ (— a+b √ — 1)
(~
and √(a+b√
into Series by Means of the
binomial Theorem.
1) being reduced
a³ +
1
413
1
3 b v
bv ---- I
a 62
3
9.
1 I
The fir will become
8
-
a
363. ✓ - 1+
✓-it
10
δι
243
a 3 64 + &c.
+
And the Second will become:
5
21
a
8
3:63
+1-
83/x+
10
a
243.
}
MIMI
a
3°
64
a
12
9
&c. and x will be =
SPECIOUS ARITHMETICK.
199
2 4
I
I
3
2
3
a 3 62 +
10
a
243
17
308a-366 + &c. or
0501
I
x (1 + 1 ==
10 64
15466
922
243a4
+
tirely real.
0561 a6
&c.) Expreffion en-
2a3 X
Wherefore when p³ is negative and greater than
I
27
1 3
9
99, the Value
3√ [ − / 9 + √ ( 27 p³ + 4 q² ) ] — ³/ [ { g + √ ( z p³ + — q²)]
of x, though exhibited under an imaginary Form, is however real. But
the Analysts have not been hitherto able to give it a real Form, without ad-
mitting an infinite Number of Terms in its Expreffion.
CXXIX.
approximat-
Though the Value of x cannot be determined exactly, we may however Method of
approximate to it by Means of the foregoing Series; for fuppofing a greater ing to the
than b, the Terms of this Series will decreaſe very ſwiftly, and a few of the value of x
fuft Terms will give a near Value of the Root required. But if a is lefs in this cale.
than b, in finding the Value of 3 (− a + by — 1) & 3√ (a+b√ — 1)
by Means of the binomial Theorem, b I muſt be put for the first
Term of the Binomial, and a for the Second, and there will refult for the
Value of x, or of ¾√ (—a+b √ — 1) — ³√ (a+b√ −1), the Quantity
(-a+br
√
—
3
14
20
a5 — 748 b 3 a7 + &c.
8
+
3
b 3
10
a
b
81
3 a³ +
44-6-34 45
729
19693
ta 2
or
22 a4
I
36 3
2
2764
243 64
+
374 съб
6561 66
&c.
24
Expreffion entirely Real, and whofe Terms decreaſe very fwiftly, when a
is lefs than b.
CXXX.
x = 1 2 3³ √// [ / 9 + √ (499 + 7 p³)] — ÷ ³ √✓ [ ÷ 9 + v (
I
27
I
other values
real in the
Of the three Roots of the Equation x3+p x + q = 0, thofe The two
99+ 27 p³)] of x, are
플​3
+27
±[{√[{q+√ († q²+ šp³)]+š√[−‡q+v (499+77 p³)]]×√ −3 fame cafe.
are both imaginary, when (99 +27 p³) is a real Quantity. We
fhall now fhew that thefe two Roots are real when (997 p³.)
+ 27
is imaginary, that is, when p is negative, and p3 is greater than 99.
ž
For changing by Means of the Denominations already employed,
the Expreffion of thofe two Values of y into 3 (a+br—1)
—±³ √(a+b v−1) ±[ ½³√ (a+b √✓ − 1 ) + ž ³✓↓ (−a+b√−1)]X√−3
and reducing it into Series, there will refult for the the two Values of x.
2
3
200
ELEMENTS OF
1
How from
the roots of
a 3 x 1 +
b b
+
9 aa
22 64
1064
243 a4
+
154 b6
6561 a5
&c. +
Ѣ
-XI-
37466
2353
a V 3
5bb
27 aa
243 a4
6561 26+ &c. Expreffion entirely real.
CXXXI.
As the most general Equation of the third Degree reprefented by
ej
the Equa- 73+ dy² + e + ƒ = o is reduced to x³ + (
tion x3+px
+qo are
deduced the
de
3
d2
32² ) x + 3 d³
d3
3
I
+f=0, or to x3 + x + q = o, by fuppofing y=x-d
p
3
roots of the it follows that the Values of y in the general Equation y3+dy² + e +ƒ=0
equation
will be found by refolving this laft Equation x3+px+q=0, and de-
I
j³ + dy²+ ey
+f=o. ducting d from its three Roots.
0.
3
CXXXII.
Whence it appears, that every Equation of third Degree has at leaſt one
An Equa- real Root, and that the two others are either both real or imaginary.
tion of the
third degree
has three
one real root
I
To diſcover which of thoſe two Cafes happens in any propofed Equation
of the third Degree, the fecond Term must be exterminated, in order to
real roots, or compare it with the Equation x3 + x + q = 0; which being done, if p
px
and two im- be pofitive, or if negative, 3 is not greater than 72, the Equation
poffible will have only one real Root expreffed by the Formula of Art. cxxvi. But if
p is negative, and p3 is greater than 99, the three Roots will be real,
How to di- but cannot otherwife be determined than by Approximation.
roots.
ftinguish
thoſe cafes.
27
27
I
27
qq the firſt Value of x expreffed by
3
I
If p3 is negative and equal to
³ √ [ — /3/ 9 + √ ( = q² + // p³) ] — ³ √ [ 3 q + √ ( = q² + 27 p³ ) ] is
reduced to 239 and the two other Values expreffed in general by
1 2 3 √ [ 32 9 + √ (1 9 9 + ½ p³ ) ] — ±³ √ [ − ÷ 9 + √ ( — 9 9 + 3µ³)]
when 273 + 13 √ §7+√(§ 99+27p³)]+ž³√[−‡q+√(† 99 + 37 p³)]]X√ — 3
is reduced to +3/q.. Confequently in this Caſe the three Roots of the
Equation are real.
Nature of
the roots
is negative
and equal
//99.
I
ΣΤ
CXXXIII.
To folve the Equation y3 + 4.y² + 6y=
I
да
4a.
·46
b
I
ΣΤ
produced by the
Application foregoing Problem: Let a= 344,81 1; b=3201; whereby this Equation
of the fore- is transformed into y+4y²+6y=0,310125r. I firft exterminate the
going Me-
thods to the fecond Term by putting y=x
reſolution of
the propoſed the Equation x3 + 0,666 x
problem.
J
4
and by this Subſtitution there reſults
3
3, 5693840; which compared with
x² + px+9=0, gives p=0,666 &c and q = 3,569384: thoſe
Values being fubſtituted in ¾ (29 +27 p³), this Quantity becomes
I
SPECIOUS ARITHMETICK.
201
I
7
I
27
33,19538 which is a real Quantity, whence the Value required of x will
be obtained by Means of the Formula of Art. cxxvI. Subſtituting therefore
in 3 // [ - ÷ 9 + √ (499 + 37 p³) ] — ¾ [ ÷ q + √ (#99 + 7p³],
- 1,784692 for q, and 3,19538 for✓ (99+p3); this Value of
x becomes 3/1,784692 +√ 3,19538—3/ — 1,784692 † √ 3,19538
+
and fubftituting this Value of x in y=x-1,333, &c. there refults
y=3✓ [1,784692+√3,19538]—³√[−1,784692+√3,19538]— 1,333
,05, which is the only real Root of the Equation, therefore the Rate of
Intereſt was 5 per Cent. and the Sum put out 1600/
CXXXIV.
duced to
Every Equation of the fixth Degree, in which the Dimenfions of the Equation of
unknown Quantity are all even, may be refolved by the foregoing Method. the fixth de-
Let for Example, the Equation 26 +9 z¹+39 zz +55=0, be propofed, gree re-
putting zz equal to a new unknown Quantity x lefs the third Part of the one of the
Coeficient of the fecond Term, that is, putting zz=x- 3, the Equation third.
will be transformed into 3. † 12 x − 8
80, which is only of the third
Degree, and wants the fecond Term.
-
7
ΣΤ
Comparing now this Equation with x3 + x + qo, we have p = 12,
q= 8; and confequently (#p³ + = 99)=80; whence x
or 3√/ - / 9 + √ ( p³ + q² ) . ] − 3 √✓/ [ & 9 + √ ( 2, p³ + 499)}
= p3
= 3√ (4+ √ 80) — 3√(√/ 80-4), and fince by Suppofition
ZZ = x 3, or 2 =
3), we will have the unknown Quantity
fought √ [3√ (4 + √ 80) — 3√√ √ (804)
x. +
3√ √ (80 — 4)-3] and the
two Values given by this Exprefion are the only real Ones of the Six that
the propofed Equation admits of
z
(
•
Equation of
a higher de-
gree reduced
in like man-
the third,.
In general, all Equations as z3" + az²" + bz" +co, is reduced to her to one of
one of the third Degree wanting the fecond Term, by putting 2=x
=X - a.
CXXXV.
Let it be propofed to divide the Number 24 into two fuch Parts, that the
Difference of their Cubes may be 3584.
3
L
an Equation
Let 12+ exprefs the greater Part, and 12 the Leffer, then will
(12 + x)3 — (12-x)3 = 3584. That is, x3 + 432 x 1792, which Another
=
being compared with x3+px+9=0, I find p=432, 9-1792, and as problem
pis pofitive, I conclude Art. cxxxII. that the Formula of Art. cxxvI will fuc- producing
ceed in this Cafe, fubftituting in Effect the Values of p and of 7 in the general of the third
Formula x=3/ [— g + √ ( z p³ + — qq ] −3₁/ [3 g + √ (37 p³ + 99 ) I loved by
degree
we will have x3/896 + √37888003896 + √3788800, the forego-
which by the Rule of Art. LXXXVI. will be found to be reducible, fince ing method.
896+√3788800 is the Cube of 2+✓ 148 and 896+✓ 3788800 -is
the Cube of 2 + √148, whence the Value of x is reduced to 4, whence
8 and 16 are the two required Numbers.
7
+
4. C
202
ELEMENTS OF
Problem
producing
an Equation
of the third
degree that
cannot be
folved by
the formula
of Art.
1.
cxxvi.
Application
of the me-
thod of Art.
cxxix. for
approximat-
roots of the
ing problem.
CXXXVI.
What two Numbers are thofe, whofe Product is 98, and if from the Square
of the greater, the leffer be taken, the Remainder will be 90.
Suppoſe x the Greateſt, and y the leaft of thoſe Numbers, then
xy=98, or y
y = 90, or y=xx
I
27
98
X
& x x
I
=06
98
or x39098. Comparing this Equation with the general Equation
x³+px+9=0, we have p—90, q—— 98. Now p being negative,
and p3 being greater than 99, the Equation cannot be refolved by the
foregoing Formula. I therefore employ the Method of Art. cxxix. to find
an opproximate Value of x.
I
3
To this End, I fubftitute for p and q their Values — 90, and — 98,
in x = 3√//[−9+ √ ( 27 p³ + q q ) ] — ³ √ [ 1 g + √ (27 p³ +299)]
and there Reſults = 3√ [49+√−24599]—³√[— 49+√(−24599)]
which compared with 3 (a + 6 √ − 1 ) — 3 √ ( a + b √ − 1 or
x
2 a
X- I +
36 3
522
2762
b
22.44
3
274 аб
+
243.64 6561 66
I
&c. gives a 49
bb = +24599, which are to be ſubſtituted in this infinite Series.
To perform this Subſtitution, I firſt extract the cube Root of 24599 to
2
obtain b³ 3 which I find to be nearly 29,08, and confequently
>
2 a
2
363
ვა
or
ing to the
Equation
produced by
98
the forego-3
is nearly
24599
10000
8902.
a a
Squaring afterwards a, and dividing it by bb, I find
to
0,0976, whofe Square 0,00952 is the Value of as to the Value of
аб
66
be nearly
b b
a4
64
-and of the higher Powers, they are to be disregarded in this Series, which
222
converges very ſwiftly. Subſtituting thoſe Values of 22, in the infinite
a4
64
Series, I find-1,104 for the Value of x given by the Formula of Art. cxXVI.
to obtain the other two Values of x, which are alſo real, Art. cxxx11. I divide
the Equation x390x 980, by x+1, 104 and there refults for
Quotient 2 I, 104x-88, 781 with a Remainder 0,0142, which may
be rejected as exceeding fmall. So that xx1, 104 x88, 7810,
may be confidered as the exact Quotient arifing from the Divifion of
90 x 98 o by x + 1, 104, and as the Product of
the two Roots fought; refolving therefore this Equation, I find
x=0,552 ± 89, 0857, that is, +9,990 and - 8,886, confequently,
the three Values of x in the propofed Equation x390x 98 0,
x3
-
— =
SPECIOUS ARITHMETICK.
203
are1, 104; 9,990; 8, 886, whereof the fecond only is for our
Purpoſe.
Inconve-
niency to
which the
foregoing
approximat-
The foregoing Method of refolving by Approximation, Equations of the
third Degree, whofe three Roots are real, is liable to this Inconveniency, Method of
that when a differs but little from b, the Terms of the Series expreffing the ing to the
Value of x, decreaſe fo flowly, that a great Number will be required to roots of E-
give a near Value of the Root, whereby the Computation is rendered ex- quations of
treamly laborious. To remedy which, the Analyfts have fought a Me- degree is
thod in general more commodious in Practice, which we fhall proceed liable..
to explain.
CXXXVII.
With this View, let the Equation x3 px + q = 0, or rather
x3 — px + q = 0, (the Cafe in which the Method of Art. CXXVI. fails, being
comprehended among thofe in which p is negative) be refumed, which
for greater Simplicity I propofe to reduce to this Form, z-z=r,
to effect which, I put x = mz, which transforms the Equation.
x³-px+qo into z3-
whence it appears;
૨૩.
Þ
2
m.m.
q
77
that by putting m = √ p, if q is pofitive, and m=p, if qis nega-
✔p,
tive, the Equation x p x + q = 0 will be reduced to the Form
Z3 31
Z=1.
3
the third
ge+-
Another
neral me-
thod of ap-
proximations
modious in.
4
more com-
3
I now obſerve, that if this Equation be of the Number of thofe that
cannot be folved by the Formula of Art. cxxvI. r muſt be less than
or, and at the fame Time one of the Roots of the Equation must be practice..
pofitive, and greater than Unity, but lefs than, for if exceeded §,
there would reſult for r, that is, for z
za Number greater than √›
and confequently the Equation 2-zr, would be of the Number of
thoſe that can be folved by Means of the Formula of Art. cxxvI.
3.
3
4
This being premiſed, I put z= 1+d, and fubftituting this Value in
the Equation 3-zr, there refults + 2 d + 3 ♪ ♪ +♪³ = 0,
which multiplied by four Terms of the Series 1+AS+ BS² + Cd? + &c..
the Product will be
·⋅r + (2· — r Ạ) d +(3 + 2 A − r B) ♪² + (1+3 A+ 2 B − r C) d³
+ ( A +3 B+ 2 C ) d4 + ( B + 3 C) 25+ Cd5,.
in which the third, fourth and fifth Terms may be made to vanish, by de--
termining A, B, and C by Means of the Equations 3 + 2 A―r B = 0;
1 + 3 4 + 2 B r C = 0, A + 3 B + 2 C = 0, whence is deduced
B =
2 A
+
ダ
​3
a=
whence exterminating C,
or
2 B
3 A
0 = 2 C + 3 B + Aj
+
+
2 B
+
3 A
+ =/ +
you
j
[+] × B B + [ 32 + 1 ] × A +
2
*
11
3 B
24.
09.
A
+ 1 = 0,
2
204
ELEMENTS OF
which by fubftituting the Value of B becomes
I
( 8 + 12 r + p² ) ×A + 12 + 9r+ 2 r = 0, whence A is found
4
2
↑
[ ÷ + ² ] ×
[ +/- + - +
[
~² ² ²
2 A
+ ? ]
+ [ ÷ +
A
2
¦ ] × ^ + ¦ = 0, 0%,
—
1
] × ^ +
6
+
9
21
+ =0, multiplying by 2 r²,
p2
12 + 11: j
B =
14 + 3*
8+127+29
8 + 12r + ga²
15+
8+127+82
rt
+[ 16+36x+13+2
31-6r
8+127+7 ] 85
fubftituting thoſe Values in the foregoing Equation there reſults
8+12+2
which by rejecting the two laſt Terms as exceeding
+ 8r + 12 r² + g3 = ( 16 + 36 r +
= 8
8x+12+2+3
16+36r+1329
Conſequently 1 + d or z =
I
CXXXVIII.
15
8+12r+ph
do=
ſmall, is reduced to,
13 r²); hence
d;
16+140+25+2 af gr 3
16+36r+1382
To diſcover the Error produced in the Value of z by rejecting the
Terms
31-6 r
2
8+12+
25
15 -
8+12r+p²
ተ
d, it fuffices to try
that Cafe in which thofe Terms are greateft, let then z or I + be fup-
The forego- pofed to be equal to, which is greater than can ever happen, when the
ing method
of approxi- propofed Equation is of the Number of thofe that cannot be folved by the
mation give Method of Art. cxxvI. in this Cafe r = √, and the foregoing Method
1596+1192/ inftead of the true Value
1452+972√
the value of
x true to the will give for the Value of z,
fifth decimal
place.
39
√, now it is eafy to perceive
one ten thouſanth Part of an
I
3
that theſe two Quantities differ only by the
Unit, and this is confequently the greateſt
16+44-+-25²+x3
for the pofi-
16+36r+132
Error that can be committed by taking
tive Root of the Equation z zr, if this Equation be of the Num-
ber of thoſe to which the Formula of Art. cxxvI. cannot be applied,
After the Value of z has been computed by the Expreffion
26 +44 +252 +
16 +36 r + 13 ge²
approximate Value of x.
3
!
it muſt be multiplied by m, to obtain an
SPECIOUS ARITHMETICK.
205
CXXXIX.
If ſtill a greater Degree of Exa&nefs is required, more Terms of the
Series + A + B s² + C ♪ a &c. must be taken in.. Now, it is evident.
that in all Caſes, B =
2 A
=
+ 3, c=
2.B
*
+
3 A
+
;
2 C
D= -+
go
3 B
A
+
2 D
3 C
B
; E
+
+
j
+
j
.t
the laſt Value is always equal to nothing.
A
gr
&c. where
Subſtituting in the firſt of thoſe Equations B = 24 + 3, 2 and
2
for and reſpectively, it becomes B= 24 + 4, this Value
2 B
+
3 <
A
+ //
до
gr.
↑
*
wrote in the fecond Equation C =
of
9
Method of
approximat-
Bing to the
value of x
to any af
gives figned degree
of exactness.
2 Q A
C =
2.9
+ +
3 A
I
j
ተ
+, which by fubftituting R
for
22+3
and
29+1
reſpectively, becomes R A+r'.
*
ተ
and r',
Thoſe Values of B and C fubftituted in the third Equation, gives
↑
2 RA
D=
+ 2x +
324
S and for
2R+32+1
s
iga
SA+s.
34
A
+32 +4, which by fubftituting
and 2 +39 reſpectively, becomes
ige
After the fame Manner the fourth Equation will be reduced to
E=TA+, by ſubſtituting Tand t for 25 + 3R + 2,
reſpectively, &c.
*
25+3r+9
?
Whence is derived the following Rule for obtaining an Approximation
to take in as many Terms of the propoſed Series as you pleaſe. Put
2 = 2; R = 22+ 3
j
**
V = 2 T + 3 S + R
T
;
&c.
S=
2 R 2+
q = 3, r' = 2r+ 1,5 = 20²+39,
*
ga.
? +32 + 1; 7 = 28+ 3R+Q;
je
25+3r'+y,
=
=
27+35+41
Divide the laſt of theſe Quantities q, r, s, t, &c. by the correſponding
206
ELEMENTS OF
Quantity of the upper Series 2, R, S, &c. then
የ
+
2 + rx ² ² ² + + x 2
I
I
I
I
&c. or
gat
2
t 2
2 + rx
=
+
+
+
+
*
R *
t
ซี
I
+
&c.
will be fo'many different Values of 8, whereof each is more exact than the
preceding One.
CXL.
To fhew the Application of the foregoing Approximations by an Ex-
Application ample, let the Equation ≈3 — z= be propoſed.
of this Me-
thcd to an
Example.
3
Here 2 = 6, 2 = 9,
ge
2 R
3
= 3, Hence 2=2=6, R= 22+ 3 = 45,
S = ²² +32 + 4 = 327, T =
j
V=2T+38 + R
*
+3
8+3R+ 2 = 2385,
j
= 9, r' = 29+ 1
2 5 + 3 r + q
=17388, Alfo q=
3
8 = 211 +39
= 423, t=
=
9
gal
57
=
6
R
45,
S
ગ્
gr
= 57,
= 3078, v = 21+3s+&!
gu
Problem
producing
of the thirda
=22446, Therfore
22446
7
17388.
2= 1, 137158.
=
423
t
3078
327, T
2385.
From whence &=0, 137158, and confequently
CXLI
There are three Numbers in geometrical Progreffion whofe Sum is a, and
the Sum of their Cubes b; what are thofe Numbers.
If x, y and z denote the Numbers required, then by the Queſtion
an Equation x + y + z = a, x² + y³ + z² = 6, or by Tranſpoſition x+z= ay
degree and x³+z3=b — y³ & ( x + x)3 = (ay)3. Now the Difference
this method. of thoſe two laſt Equations is 3 x² + 3xx²=aª— 3 a² y + 3 ay²-b₁
folved by
z
but x+x=ay and x 2 = y², whence y3 - a²y +
Let a and b be fuch that a2
Equation be transformed into y3
a3
b
13, &
3
a3- b
3
5. Then will the
13y+5=0, and fubftituting—≈ √ 13.
SPECIOUS ARITHMETICK.
207
for y, it will become z² Z
13√13, whence z is deduced, and
conſequently y=— 3,784341, &c.
CXLII.
Having fhewn how to folve Problems producing Equations of the third
Degree, we ſhall proceed to explain the Methods employed by the Analyſts
for folving Problems producing Equations of the fourth Degree, of which
the following is one of the most general of this Order.
The Sum a, the Sum of the Squares b, the Sum of the Cubes c, and the
Sum of the Biquadrates d, of any four Numbers being given; to determine
the Numbers.
Problema
producing
u, an Equation
the moſt
Let the four Numbers be denoted by x, y, z and u, and put A—a—
B = b — u², C=cu³, and Ddut. From whence by the complicated
Conditions of the Problem.
2
x+y+z= A, x² + y² + x²=B, x³+y³+z³=C,x¹+y4+z¹=D
Now, if the ſecond of theſe Equations be fubftracted from the Square of
the first, we ſhall have 2 x y +2 x z + 2y z = A² — B.
D.
And if, in the like Manner, the fourth Equation be fubtracted from the
Square of the 2d, we ſhall have, 2 x² y² + 2 x² x²+ 2 y² x² = B²
Moreover, if from the Square of the former of theſe laſt Equations, the
Double of the Latter be deducted, there will come out
—
B — B² + 2 D ;
A¹ — 2 A² B — B² + 2 D ;
8 x² y z + 8 y² x z + 8 z² x y = A¹ — 2 A²
or, 8 x y z x ( x + y + z) =
A4 2 A2 B B² + 2 D
whence, x y z =
8 A
(becauſe x++z=A}
Again, by multiplying the first and fifth Equations into each other, we get
2 x²y+2x² z+2 y² x + 2 y² z+2x²x+2x² y +6 x y z— A³—A B,
and by multiplying the first and third, Equations together, there arifes
׳ + µ³ + ׳ + x² y + x² x + y² x + y² z + z² x + z² y = A B.
x3 x3
The Double of which laſt taken from the Precedent, leaves
6 x y z − 2 x³ — 2 µ³ — 2 x3 = A³ — 3 A B, and this added to
2 x3 + 2 µ³ + 2 z³ = 2 C, gives 6 x y z—₤3 — 3A B+ 2 C.
x3
23
Hence
A3 — 3AB2C
= xyz
6
44
-
2 Ą² B — B2
8 A
- 2 D
and confequently by Reduction A4-6 4² B+8 AC+3 B² — 6 D = 0.
In which Equation let the feveral Values of A, B, C and D be
of the fourth
degree.
***
208
ELEMENTS OF
B
now fubftituted, and dividing the Whole by 24, we at Length, have
น
24 — a u³ +
+
२२
2
b
Q3
X u²
3
ab + 2c
6
X u
3
62
6 d
=0.
Whoſe Roots Anſwer
24
6a² b + 8 ac+
all the Conditions of the Problem.
CXLIII.
Let in general u4 + a u³ + b u² + cu+do reprefent all Equations
Refolution of the fourth Degree, the Difficulty is foon reduced to ſolve an Equation
ral Equation reprefented by z4 + p z² + q z +r=0 by putting u = z
of the gene-
of the fourth
degree.
TheRefolu-
tion of an
4.a.
4:
Now, the Manner the moft natural of attempting the Refolution of
this Equation, is to confider it as the Product of two Equations of the
fecond Degree, aud fo contrive that the Determination of the Coeficients
affecting the Terms of thofe Equations of the fecond Degree, depend on
Equations eafier to be folved than the propofed one.
و
Let firſt zz +x+o be taken for one of thofe Equations; it
is manifeft that the other fhould have for fecond Term xz, fince the
Product of thofe two Equations fhould give an Equation wanting the fecond
Term. Let then zz - xz + s = 0, be taken for this fecond Equation,
multiplying thoſe two Equations into one another there will refult.
2.
z
24 + (s = x² + 1 ) x x + ( 5 x − t x) x + tso, which being
compared with the propofed Equation, gives for determining s, t, x the
three Equations s x² + 1 = p; sx 1 x = q;ts = r.
To make Ufe of thofe three Equations, I multiply the first by x, and
tben add it to the fecond Equation, and there refults 2 sxx3px + 1,
whence is deduced s = £+px + x³
2 x
21x
ts=r gives t=
x3 + px +8.
which fubftituted in the Equation
and fubftituting thofe two Values of
Equation of s and in the Equations x tx== q, there refults at Length
the third de-
gree depends
on one of
the third de-
3+ px + 9
2
2442
x³ + px + q = 9, oг, x 6+2px4+(pp—4r)x²—q²—0,
x3
gree which Equation of the fixth Degree, which by the Method of Art. cxxxiv. is tranf
is called by formed into one of the Third, whence the Difficulty of Equations of the
the Analyfts
the reduced fourth Degree is reduced to that of the Third, for this laft Equation (which
Equation. is called the reduced) being folved, we have no more to do but to fubftitute
the Value it gives for x, in the Equations -+s = 0, ≈≈+x2+1=0,
≈≈
SPECIOUS ARITHMETICK.
209
or rather in z z - ≈ + 1/2 × × + 3/2 p +
zz + xx+
27
xx +p+
9
x
2 x
= 0 and in
o, and afterwards folve thofe Equations,
or what amounts to the fame, fubftitute the Value of x in the Roots
2], &z=− ± √ √ [-
21
**]
+ /1/ xx
x² + p + 1/2
9
of thoſe two Equations, and the four Roots required of the Equation
4
x² + p z² + q z + r = 0, and confequently of the propofed Equation,
u4 + au³ + bu² + cu + do will be obtained.
CXLIV.
X
Roots of an
It appears at firſt View, from the Expreffion of thoſe Values, that in the
fourth Degree, as in the Third, one Expreffion cannot be found for all the The four
Roots of the Equation. However, when it is obferved that the Quantity Equation of
included in the two foregoing Expreffions is neceffarily a fquare Radical, the fourth
fince it reſulted from the Refolution of an Equation in which x is always of degree may
be expreſſed
an even Number of Dimenſions, it will eafily appear that each of thofe by o
Expreffions may denote four Roots, the firft being then expreffed thus, formula.
≈ = ± ± × ± √ ( − 1 xx
and the Second.
thus, ≈ = = ÷ + √ ( 4 x x −
x
27
9
2 X
:)
xx.+p+
Thofe two Equations in Appearance differing; may render the foregoing
Reaſoning ſuſpe&, fince it ſeems to lead to this Abfurdity, that an Equation
of the fourth Degree may have eight Roots, but this Difference is only ap--
parent, for the Identity of thofe two Expreffions is reduced to that of
"
27
x x
xx +p+
x
} and of √ √✅/ (− 4 xx−1 p = ½), that
is, of — xxp, and of
2
2 r
xx + p = 1/2
tity of thoſe two laft Quantities cannot fail to take
determined by Means of the Equation +2 p
But the Iden-
Place; when x has been
+ (p² - 4+) x²-g²=0;.
4 D
410
ELEMENTS OF
fince this Equation has been manifeftly deduced from the Equation
- 1 x x − 1 p =
劲​干
​9
2 r
2 x
x
CXLV.
1
x
z=±÷×
The firſt, z=±± √√ − × × − p
૧.
2 x
of the two fore-
going Expreffions is to be chofen, as being the moft Commodious.
CXLVI.
Since the Equation from whence is deduced the Value of x, gives necef-
farily three Values of x preceded by the Sign, and there is no Reaſon of
preferring one of thoſe Values to another, befides, it is known, that an
Equation of the fourth Degree cannot have more than four Roots, it feems
natural to conclude, that any one of thoſe three Values of x preceded by
Roots of the the Sign, may be employed indifferently, and ſtill the fame Expreffion
reduced for the four Values of z be deduced.
Let which
ever of the
Equation be
employed
the fame
Roots will
fult for the
propofed
Equation.
.
x
9
To render this manifeft, the most natural Method would be to
find the three Values of x preceded by given by the Equation
always re- x6+2 p *4 + (pp - 4r) x² — q² = o, and fubftitute them one
after the other in the Expreffion = ±±√√− 1 × × 30 F
to be affured of the Identity of the three different Expreffions arifing
from thoſe Subſtitutions. But the Calculation that this Method would
require is fo long, that one could ſcarce have Patience to continue it to the
End, we ſhall therefore proceed to explain how the Analyſts have remedied
this Inconveniency.
}
2 *
I obſerve firſt, that whatever Value of x is fubftituted, in the
general Expreffion ≈= ± 1 ×±√/−1 ××−3 µF
2x
**, the four Va-
lues of z expreffed at once by this Quantity, may be repreſented by i + k,
i-k; -i+1, i i-l; or what amounts to the fame Thing, that
the four Roots of the Equation z4 + p zz + qz +ro, may be re-
preſented by z -
i k, z i + k, z + i −1, z + i + l ; i
گیا
denoting the Part of the Value of z, k and I the two Quantities
9
√ — 1xx-p—2—, and
一
​9
— — + 2², multiplying
***
XX ------ 1 p
x
therefore thoſe four Roots into one another, there refults the Equation
SPECIOUS ARITHMETICK.
211
2
z¹ — ( 2 ii + k k + Il ) z ² — (2ikk — 2ill) z+i4— į² k² — j² 12+k² 12 = 0
(2
which compared with 24+ z² + qz +ro, gives the Equations
2
p. — — 212 — k² — 1², q = — 2 i k² + 2 ill, r = i^ — j² k² — j² 1²² + k² 12,
12
by Means of which, the Equation x6 + 2 p x² + ( p² — 4r) x² — q = 0,
x²—
is transformed into.
*4
-
2
if
x6—(4i²+2k²+21²)x4+(8 i²k² +8i²]²+k4—2k² 12 +14) x ² —4j²k4+8;²k²]²—4;214—0
whofe Roots are, x=2i; x=±k± 1;x= + k + 1; now,
±
fubſtitute any one of thote Values of x in the four Expreffions contained in
*
= ± ÷ × ± √ ( −
X X
/ x
さ
​z=t÷xt/(mžxx+ii+音​« +÷尸 ​¥
k²
9
2 x
i 12
に
​-)
we
or rather in
i k²
-), & there
will ſtill refult the four Values of z, i + k, i −k, −i+1, −i− 1.
CXLVII.
It follows from hence, that the Roots of an Equation of the fourth
Degree are all four real, or all four imaginary; or of the four Roots two
are real and the other two imaginary, for it is not poffible to make any
other Suppofitions with Refpect to the four Roots of the given Equation, tion of the
fince it has been proved that thofe four Roots are all expreffed at once by the fourth de-
**
.)..
Formula ≈ = ±±±√(— ± ×× — ± p + −2%).
x
An Equa-
gree is ex-
actly refolv
able when
of its four
roots two
two imagi-
nary.
It follows alſo from hence, that when of the four Roots of an Equation
of the fourth Degree two are imaginary, and two are real, the reduced are real and
xo + 2px + + (p²
4 r) x²
990, will be of the Number of
Equations folvable by the Formula of Art. cxxvI, and confequently in this
Cafe the Equation z++p+ qz+ro, can be compleatly folved.
but the contrary happens when the four Roots are all real or all imaginary.
That this may appear, we are to confider that the general reduced Equation
x6—(4;²+212 +2 k²)x4+(8;² k²+8;² 1²+k+—2k² 1²+14) x²—4i²k++8;²k 2/2 — 4; 2/4 = 0
is the Product of three Roots xx-4ii, xx-k-2kl—11, x²-k²+2 kl — II :
Now, if one of the two Quantities kor only are imaginary, the
two Roots, x² — kk 2 k 1 — 11, xx
l
kk + 2 kl I are imaginary,
and confequently the reduced Equation is in this Cafe folvable by the For-
mula of Art. CXXVI.
-----
But if k and I are both real, or both imaginary, the three Roots The con-
x x 4ii, xx. k k
2 k l — 11, xx
trary hap-
- k k + 2 k l 1 of the re- pens when
duced Equation will be all three real, and confequently cannot be obtained the four
by the Formula of Art, cxxvI. whence in the fourth Degree as in the Third real or all
the Formulas of the Refolution can be applied only to Equations that have imaginary.
two impoffible Roots.
roots are all
212
ELEMENTS OF
cafe of the
four real
roots from
CXLVIII.
*
How to di- When an Equation of the fourth Degree is propoſed to be folved, and
itinguish the by its Means the reduced Equation x6-2px + (pp — 4 r) x² — qq=0,
has been formed, if it be found to be of the Number of thofe that are not
that of the refolvable by the Formula of Art. cxxvI. and it is propoſed to find whe-
four imagi- ther in this Cafe the four Roots are real, or whether they are all four ima-
ginary, it may be effected from the two following Obfervations.
nary ones.
Conditions
real roots.
2
10. When the four Roots are real, the general reduced Equation
of the four 6-2×(2 ¿²+k²+1²) x4+ [81² (k²+12)+(k²—12)2] x² — 4 ¿² ( k² ² — 1 ² ) ² = 0,
has neceffarily the fecond Term negative, and the third Term poſitive,
fince 2X (2 i² + k² + 12 ) can neither vaniſh, nor be pofitive, when
, k, l, are real Qantities, nor can 8 iiX (k² + 1² ) + ( k² — 12 ) 2
likewife vanish, or be negative, under the fame Reſtrictions.
Conditions
of the four
roots.
}
2
20. But if the four Roots are imaginary, or what amounts to the fame
Thing, if k k and I are negative, the reduced which is then expreſſed thus,
imaginary x+2(k²+1²—2ii)x4+[(k²+1²)(k²+1²—8ii) — 4k² 12]x² ·4 ¿² (k² — 1²)2—0,
cannot have at the fame Time the fecond Term negative, and the third
pofitive, for if k²+12 is lefs than 2 which would render the ſecond Term
negative, the third Term whofe Coeficient is (k+12)X(k² +1²—8;²)—4k² 12.
will be neceffarily negative.
Every
-CLXIX.
The Inſpection of the Equation 24-( ? ¿² +k²+1²) z²——(2 ik²—2i ll )x—&c.
given in the Art.
in the Art. CXLVI. furniſhes an Obfervation whereby fome-
times may be diſcovered, if an Equation that ſhould have its four Roots
either all real or all imaginary, is in the firft of thoſe two Cafes or in the
Equation of fecond. The Obfervation is, that every Equation of the fourth Degree,
the fourth wanting the fecond Term, having the third Term pofitive, neceffarily con-
degree tains impoffible Roots. Since the third Term of all thofe Equations repre-
fecond term fented by (→ 2 ii + kk +11) zz can never be pofitive, when ii, kk, II,
are pofitive; that is, when the Roots are real. Knowing therefore that an
Equation of the fourth Degree has impoffible Roots, and being affured on
the other Hand, that it should have its four Roots all real, or all im-
poffible, it is eafy to perceive which of thoſe two Cafes fhould take Place.
wanting the
and whofe
third term
is pofitive
has imagi-
nary roots.
How to ob-
CL.
When it has been found, that the four Roots of an Equation of the fourth
tain approxi- Degree are all real, one of the Roots of its reduced Equation is to be fought
by the Method of Art. cxxix. and being fubftituted in the general Formula
imate values
SPECIOUS ARITHMETICK.
213
9
* = ± ± √ [ - + x x − ],
р
T
-2 x
of the four
there will refult the roots whea
they are
real.
Values of the four Roots required.
CLI.
To apply the foregoing Rules to the Solution of the Equation,
a²-b
3 ab +20
ab + 2e x16+ a4—6a²b+8ac+3b²—6d
264 423 +
a3
-X22
2
produced by the foregoing Problem, let a
+
=
—
24
16, b = 58, c—— 28,
z—
น
3
tion pro-
propoſed
folved by
d= 2222, whereby the Equation will be reduced to 24+16 u³ +99 u²
+228 u +1440, and fubftituting in this Equation z 4 for u, to
make the 2d Term vaniſh, it will be transformed into 24+3x-52%+48=0,
which being compared with the general Equation 2 4 + px²+qx+r=o, The Equi-
will give 3; 9=-52; 48; and confequently the reduced duced by the
p =
r =
Equation will be x6 6x4-183x²-27040, in which exter- Problem
minating the ſecond Term, by ſubſtituting y 2 in the Room of x², the forego-
there will reſult y³ 195 y 2322 = 0, which is refolvable by the ing method.
Formula of Art. CXXVI. and fhews confequently, that the propofed Equa-
tion is of the Number of thoſe that can be exactly folved, that is, of thoſe
that have two real and two impoffible Roots. In order to find them, I
employ the Formula of Art. cxxvI. to folve y3 195 y — 2322 = 0,
2322=0,
and the Value it gives is y=3/(1161+1073296)—3/(— 1161+V1073296)
which is reduced to 3/(1161 + 1036 ) — ³√/ (—1161 † 1036), or
to 3/2197 + 3/135, or at Length to 18; fubftituting this Value of
y in x = √(x-2), there refults x164, which fubftituted in
3
the general Value of z=±1×±√ (− i **
×
9
· F
24.
2 *
there will refult for the two real Roots of the
--),
Equation
13
x² + 3x² - 52 z+48=0; x=2 ± √ ( − 4 − 3 + 1½³),
2
2
or z=2±1, that is, either 3, or 1, and for the two impoffible Roots
x=
2
±√(-4
3
2
13
2
·),orz
= ·2 ± √ — 12,
4, there will refult
Then fubftituting theſe four Values in uz
for the four Roots of the propofed Equation
1
214
ELEMENTS OF
u4+1·6u³+ 99u²+228 +144=0, u=—1, u =—3,
CLII.
Another
Problem
producing
3. น
4-6 ±√12.
The Sum (a), and the Sum of the Cubes (b), of five Numbers in continued
geometrical Proportion being given; to find the Numbers.
Let x and y denote the three middle Numbers taken in Order. Then
x; z
x 24
Z
Z
will be the first Number, and the laft; and we ſhall have
+x+x+y+
2.J
Z
Jaz
хоб
= a,
Z.
z3
+ x3 + x3 + 3 + 15 = 6
уб
≈3
by the Queſtion.. Put u =x+y, then from the firft Equation
an Equation x²
2
and cubing the two Members
ofthe fourth
+
32
= a
น
Z
degree
Z
folved by
яб
the forego-
ing method.
z3
+[-
2
-]× x² 1² + 15 = (a — 4 — z.)³, whence
x6
23
+ [3x² + 3y²]
26
23
x
2
43
23
น
2
+ = (au-x)3. —
2)3 (3α
(3a3u - 3 x) X x², and
x3+y³ =(x+y. )3 — (x+y) ×3 x y = u³ — 3 u z², & fubftituting thoſe
Values above, we have (au-x)3 (3 a—3 u—3≈) Xx²+u³-3x²+z³ —b
and a z — u? — uz + z² = 0, the firſt of which Equations by Reduction,
becomes a³-3 a2X(u+z)+3.ax(x²+2 u x)—3 u² 3 u x²+3x³—b,
from whence the other Equation multiplied by 3 z being fubtracted, there.
remains a³-3 a² × ( u + ≈) + 3 a × (u² + 2 u ≈ — x²)=b, therefore.
is² + 2 uz
护
​น
·a × (u +·x.): =
Equation, u² + 2 u ≈
b
за
z:
z
a²
but, by the fecond
3.
z ≈2 — a ≈ + u z, whence, by Subſtitution
b
a²
a z + u z—a X:( u+x) =
3 a
3
that is, uz-az =
{
a²
b
or a z
= d
3.
3.a
ux=d(by putting
b a²
3 a 3
d) from which the fecond
Equation being fubtracted, there refults u²x²=d, wherein let
az-d
Z..
the Value of u (found from the former Equation) be now ſubſtituted, and we
fhall have
z²=d, and confequently, there will refult.
24
(ax-d)2
Z
24
(a² — d) Xx² + 2 addo,, whence z.will be found.
SPECIOUS ARITHMETICK.
215
CLIII.
Now let a and b-be fuch that
whereby the foregoing Equation will be transformed into z4+3x²+2x-3=0,
Comparing this Equation with the general Equation z4+p x²+qz+r=0,
we will have p=3, q = 2, r = 3. And theſe Values being ſubſtituted
in the reduced Equation x6 + 2 p x 4 + ( p p − 4 r) x x — 9 q=d,
will transform it into x6+ 6 x² + 21 x2
x+6x+ x²
a + d = 3, 2 a d=2, and d23,
-
4 = 0.
To folve this Equation, I put xu-2, to make the ſecond Term
vaniſh, whereby it will be reduced to 23+ 9 u300, which accord-
ing to Art. CXXXIV. is of the Number of thoſe which have only one real
Root, and confequently may be refolved by Means of the general Formula
of Art. cxxvI. Whence it appears that the propofed Equation has twe
real Roots and two imaginary ones.
That the propoſed Equation has imaginary Roots, might be perceived by
obſerving that the Coeficient of its third Term 2 ≈ is poſitive, (Art. CLIII.
Refolving now the Equation 23+ 9 u 30 = o, by the Formula of
Art. cxxvI. I find u (15+ 6 √ 7 ) + 3/√ (15 - 6 √7)
wherefore ±√(2), or x = ±√[V15+6√/7)+¾/(15—6√/7) — 2].
This Value of x being fubftituted in the general Value, or Formula
√
x= ±±±√ ( − ÷ **-), which in the preſent
Cafe is
3
2
I
± ÷ ×± √ (− = ~ ~ − 2) there will refalt for the
x = /
+
two real Roots, z=− 3 √ [³ (15+ 6 √7) + ³/ (15 — 6/7 — 2)]
I
—
±√/ [−± √(15+6/7)—43/(15—6√/7)—1+, √[V (15+0√7)+√(15—6√/7)—2]
and the two imaginary Ones. =√ [¾(15+6√7)+3/ (15— 6√/7)—2}
I
`±√/[−$V(15+6√/7)—#V(15—6/7)—1- √[³¾/(15+6√7)+¾³/ (¢5—6v7)—2.
a]
CLIV.
(15+6√7)+{√(15—6V7)
What two Numbers are thofe, if from the Sum of the Square of the
Greater, and fix Times the Leffer, be deducted the Cabe of the Leffer, the
Remainder will be 8; and their Product multipțied by the Greater gives 1.
If be the Greater & the Leffer of thofe Numbers, then xx+6— x³-8
:
}
2.16
ELEMENTS OF
1
Third
Problem
producing
or xx — 23 6x+8, and x² 2=1, 05 xx= then 23-6x+8=-,
2
Z
or 24 — 6 x² + 8-10, comparing this Equation with the general
an Equation One, I find p=6, q=8, r = 1, and confequently the reduced
of the fourth Equation will be x6 12x² + 40 x
degree
folved by
640, in which putting
x²u+4 to make the fecond Term vanith, there refults 3-8-320,
2
The forego which has only one real Root, expreffed by
ing method.
3√(16+-3803)+3√(16—80) = 2²/(6/3+10)+23/(6√/3—10)
/
V
3.
which by employing the Method of
u = 2 × ( 1 +√ 3 ) − 2 X ( 1 − √ 3 )
—
√ 3
√ 3
Art. LXXXV. is reduced to
=4, fubftituting this Value
of u in x = √(u+4), there reſults =8, whereby the general Equation
x
必​社 ​+플 ​± √(− 1 **
아
​-
干
​is transformed into
2x
== F
±
2 (1 + √2)
1
== √²±√(IV 2), which gives
for the two real Roots of the propoſed Equation, and + √2 ± √ (1 ~ √√/2.)
for the two imaginary Ones.
CLV.
Let it be required to divide the Number 5 into two fuch Parts, that if
the Square of the Leffer be deducted from it, and the Remainder multiplied by
the Square of the Leffer, and four Times the Leffer deducted from the Product,
the laft Remainder will be 29.
Fourth
problem
producing
an Equation
of the fourth
degreefolved 24
by the fore-
going me-
thod.
Let z
= 0.
≈ the leffer Part, then (5
5 ±² + 4 + 29 =
4+p x² + q = + r=0; gives p
quently the reduced
which by putting
Equation will be x6
23
373
น
10622
27
3
=
x²)`X.x² - 4% 29, or
This Equation compared with
5,9=4, r = 29, and confe-
10 4 — 91 x² — 16 =0,.
IO
+, will be transformed into
u +
3
o, Now, as this Equation cannot be folved
by the Formula of Art. cxxvI. and confequently is of the Number of
thoſe whofe three Roots are real, it follows that the Roots required of the
Equation 24-5 x²+4x+29≈o, are either all four real or imaginary,
but obferving that when the Roots are real, the fecond Term of the reduced-
Equation is negative and the third pofitive, I conclude that the propoſed
Equation has all its Roots imaginary fince the third Term91 x2 of its
reduced Equation is negative.
SPECIOUS ARITHMETICK.
ic
217
CLVI.
A Number of Yards of two Sorts of Cloth were bought for 111. each
Yard of the first Sort coft as many Pounds as there were Yards, and each Yard
of the fecond Sort coft il. Now, if the Number of Yards of the two Sorts
of Cloth were interchanged, the Coft (regulated as before) would have been
but 9 1. How many Yards of each were there.
4 = 22,
blem pro-
whence Equation of
Let the Number of Yards of each Sort of Cloth be expreffed by z & y Fifth Pros
reſpectively, then z² + y = 11, or zz = 11~
11-y,
y, and y y + z = 0, ducing an
or z=9yy, confequently 81-18 ² +
11 — y = y4 — 18281, or 418 y²+; +
To folve this Equation,
I first find the
206 — 36x4 + 44 རྒྱུས · 1 =0, or u³ — 388 u
-
I
it is reduced by putting x² + 12.
w
70 = 0.
the fourth
degree
folved by
reduced Equation the forego-
29290, to which ing method;
Now, as this Equation has its three Roots real, and that the fecond
Term 36x4 is negative, whilft the third 44 is pofitive, it follows
Art. CXLVIII. that the propofed Equation has all its four Roots real;
To find thofe Roots I employ the Method of Art. cxxxvII. to folve the
Equation u3-388 u — 29290, and ſubſtitute 22, 74, the Value of
in the Equation =√ (ù+ 12), which gives 5, 894 for x, and ſubſti-
u, ≈
tuting this Value of ♬ in y = ± 1 x − √
(− \ x² − ÷ 1 = 2 )
干 ​2x
there will refult 3,426; 2,467 ; — 2, 315, 3, 579 for the four Roots
of the propofed Equation, of which only 2,467 is to the Purpoſe.
CLVI.
—
propofed
of Radicals.
Having fhewn in the Refolution of Equations of the fecond, third, and Method of
fourth Degree, how the Analyfts by Means of the radical Signs have been clearing any
able to express the Value of the unknown Quantity in thofe Equations, it Equation
remains to explain how the Equation from whence any propofed radical Ex-
preffion is derived may be found, for Example, how to find the Equation
whofe Root is x = 3/ ab3 +3✓ a² b + 3√ a² c, that, in which
x= √(a² +63) — 3 ( a³ — 63), &c.
-
To folve all Problems of this Kind, or which comes to the fame, to take
away any Number of Radicals out of an Equation, the Analyfts operate as
follows, they fubftitute in the Room of each Radical an unknown Quantity,
whence there refults 1°. A new Equation clear of Radicals, 2°. As many
Equations confifting of two Terms as there were Radicals in the propofed
Equation. Now each of thofe Equations confifting of two Terms, will be
4 E
218
ELEMENTS OF
cleared of Radicals by raifing its two Members to the Power indicated by
the Exponent of the radical Sign that one of its two Terms contains.
Whence we have no more to do than to exterminate out of all thofe
Equations cleared of Radicals, the unknown Quantities that have
introduced.
CLVIII.
been
Let it be propofed to take away the Radicals out of the Equation
x = 3√ a b² + 3a d2, by putting 3 aby, and adds, there
3
refults the three Equations xyz, y3 = a b², x3 = a d², deducing
from the first yz, and fubftituting it in the Second, there refults
The forego-3-3x² + 3x x²-z³ ab2, out of which, it remains to exter-
ing method
illustrated minate z by Means of the Equation 23 — a d².
by an ex-
ample.
Z
2
Z3
To effect which, I fubftitute in the first of thoſe two Equations
·
2
x 3 3 x² x + 3 x x² — z³ — a b² in the Room of z³, a d² given by
by the Second, and it becomes x3 -3x²+3x² — a d² = a b², whence
a d² + a b² x3 + 3x²≈
is deduced ² =
; multiplying afterwards the
3 *
two Members of this Equation by z, and fubftituting in the Room of z³ its
ad²z+ab²z-x3x+3x²xx
Value a d², there refults a new Equation a d²=
which gives zz =
11
3x ·
3 a d² x — a d² z — a b² z + x3 z
3x2
Z
Equating thofe two Values of zz, I deduce an Equation in which is
x4 + 2 a d² x - a b²
only of one Dimenſion, and folving it, I find z =
a b² + a d² + 2x3
which being fubſtituted in one of the foregoing Equations, for Example, in
x³-3x² +3 x z² — a d²—a b², gives at Length x9—3 a d² x6—3 ab² x6
+3a² 64 x³+3a² d4x3—21 a² d²b² x3——a³ b6+3 a3b4d²+3a3d4 b²+a3d6.
CLIX.
Sometimes Equations may be cleared of Radicals without having Recourfe
to the foregoing Method, by only tranfpofing the Terms, and raifing the
two Members to the Power indicated by the Radical which then will be alone
in one of the Members, for Example, if it was propofed to clear the Equa-
tion xy+ (a³ + √ a5 x) of Radicals, tranfpofingy into the firft
Member, xy=3/ ( a³ + √ a5 x ), and raifing both Members to the
Cube, x3 — 3 x² y + 3x y² — y3 = a + √45x, and tranfpofing 43 we
a5
€
have x³-3x² + 3 x y² — 13 — a3 a4x, and fquaring both Mem-
x3—3 y y3 = √✓
bers there will refult an Equation clear of Radicals..
SPECIOUS ARITHMETICK.
210
CHA P. iv.
Of the Nature and the Number of the Roots of Equations of all Degrees; of
the Method of finding the Equations of a lower Degree that are their
Divifors, with the Methods of approximating to the Roots of both numeral
and literal Equations of every Degree.
THE
HE Refolution of an Equation of the fifth Degree might perhaps be
made to depend on the Refolution of one of the Fourth, that of an Equation
of the fixth Degree on the Reſolution of one of the Fifth, &c. But this
Method the Analyfts have not purſued.
ུམཱ{
Having found that Equations of the fecond Degree have two Roots, thofe
of the Third three, thofe of the Fourth four; they were inclined to think
that in general an Equation has as many Roots as there are Units in the
Number of its Dimenſions, and to be affured of this Truth, inſtead of in-
veſtigating the Roots of an Equation, they fought the Equation that would
have given Quantities for its Roots, after the Manner we are about to ex-
plain in the Reſolution of the following Problem.
I.
The Sum (a), and the Sum of the m Powers (c) of four Numbers in
continued geometrical Proportion being given, to determine the Numbers.
If u and y be affumed to reprefent the two middle Numbers, then from
the Nature of continued Proportionals, the two Extreams will be expreffed by
นน
y
22m
17
J
and, and ſo we ſhall
น
+ u™+y™+
have
fo
u༢
J
น
+ i + y + 2²²
we will have "+ j™
mx = 4 ×
2, 171
2
Z
17
= b,
u
= a, and
producing
an Equation
the moſt
complicated
of the th
now putting y=x, and uy=%, degree.
P
mx”−²z+ m ×
m3
2
1-4 2
m —
5z³ +mx・
7-5
6
m
X
X
X 824
&c.
2
3
2
3
4
น
น
and becauſe
+
y
น
s)
x, the Sum of the mtb Powers of the
two Extreams — (~—-—-≈)”¨¨¨¨× (-x) −x+m׳ (a) — 4x² &c.
we therefore have
m × × [x − 2 + ( a − x ) ” −2] +
m-2x+mX
3
2
220
ELEMENTS OF
which Equation by writing
x3
2x+a
inſtead of its equal z,
becomes
x™+ (a—x)™
--
× [x” — ²+(4−x) "~²] +™m ===
171.
m(m—3) ×
*б
2
(2x+α)²
X [2017-4
+(a
An Equa-
mx3
2x+a
x)
"-4 &c.] which being folved, the Value of x will
be found, from which the feveral Values of z, u, y will alſo become
known.
in x
171
qx²
II.
+ px
MI
+ qx²
2 + &c.
-
tion may be I lay that
refolved into
as many
fin de
e fac
as there
the highett
dimeon of
are units in
the un-
known
quantity.
+1
܂
r = 0, ex-
Let x" + px" + 9x-2 + &c.
prefs in general an Equation of the mth Degree clear of Radicals and
Fractions. To find the Value of x in this Equation, is the fame as to
find a Quantity, pofitive, or negative, real or imaginary, which fubftituted
ro, in the Place of x,
will make all the Terms vanish. Let us fuppofe this Quantity a to be found,
+q
712-2 + &c.
ro, will be divisible by
x-a for it is manifeſt 1°. that fince x is only of one Dimenſion in the Divifor,
by the Rules of Divifion the Operation may be carried on until a Remainder
is obtained that does not contain . Let then be the Quotient, it is mani-
feft, that if to the Product of the Quotient into the Divifor x-a, the Re-
mainder R be added, there will refult a Quantity equal to the Dividend,
now by fubftituting in the Dividend, a for x, all the Terms by Hypothefis
vanifh, therefore the Quantity (≈ − a ) × 2 + R fhould alfo vanish,
if x be puta, but by putting xa, this Quantity becomes (a-a)2+R,
and fince (aa) 2+R=0, Ro, wherefore the Divifion is performed
without leaving a Remainder, wherefore x” +p x +qx™-²+ &c. --- r
is exactly divifible by x-a.
*11 I
Let a Quantity b when fubftituted in the Room of x in the Quotient 2.
refulting from the Divifion, make all the Terms of this Quotient vanifh, it
will be divifible by x b; and it is manifeſt that if b when ſubſtituted in the
Room of x makes all the Terms of the Quotient vanifh, it will allo
make the Dividend vaniſh, for the Dividend, (xa) 2 therefore every
Suppofition that reduces to nothing, will alfo reduce the Dividend to no-
thing, wherefore - b will alfo divide the Dividend exactly.
>
C
In like Manner, if c be a Quantity which fubftituted in the Room of x,
will make the Quotient divided by x-b vanish, the new Quotient and
confequently the Dividend will be divifible by x-c, we will therefore have
as many fimple Quantities xe, x— b, x —c, &c. as there are Units in
m, which multiplied by one another, will produce the propoſed Equation,
in the Room, therefore of the propofed Equation we may ſubſtitute
( 26 - a) ( x − b ) (* − c ) &c, = 0.
SPECIOUS ARITHMETICK.
221
III.
=
When a Quantity is fuppofed equal to nothing, one of its Factors must
be equal to nothing, whence the propofed Equation is the Product of
x— a = o into ( x
o into (b) (x —c) &c. or of xbo into
(x — a) (x —c) &c. in each of thofe Cafes x is the fame Quantity in
the fame Cafe, and different in the different Caſes, thus x² — (a+b) x+ab
is the Product of x - a = o into ( x
o into (x —b) or of x-bo into x
bo into xa
this Equation x² ( a + b) x + ab
bab=0 that reſults by fubftituting a for x, and
a a
ва
a
An Equa
of as many
folutions as
there are
fimple fac-
tors multi-
tion admits
reprefents thefe two
plied by one
another that.
produce it..
bb — a b b babo that refults by fubftituting b for x: In the firft
of theſe Cafes, and its Powers,repreſent a and its Powers. in the ſecond, the
Letter and its Powers reprefent b and its Powers, whence the Equation
x" + px + qx"
ro really reprefents as many fimple
Equations as there are Units in the higheſt Dimenfion m of the unknown
Quantity.
1- I
773-2
IV.
The Product of x
-
a into x-b cannot be equal to another Product
e
x
f
=
*
b
and confequently
No Equa--
tion can
have more
≈ —e into x — ƒ for if it was then
2
a
roots than it
contains di-
menfions of
known
x — a muſt be diviſible by x —ƒ as alfo xe by x
e by x-b, which cannot
be, or x —ƒ and x b muſt have a common Divifor, as alfo x- the un-
and x—e which likewife is impoffible, wherefore every Quantity x²+x+q quantity.
in which x rifes to two Dimenſions cannot be produced but by the Multipli-
cation of two fimple Factors x
a, x b, and by no other than thofe
two, wherefore in an Equation of the fecond Degree x cannot poffibly have
more than two different Values a and b.
a into x — b
In like Manner it will appear, that the Product of x-a into
into xc cannot be equal to x e into x-ƒ into x
a
( x − } ) ( x − 8 )
−
tors x-
−
X
(x—b)(x-6)
g, for then
wherefore the Denominators
of thoſe Factors ſhould have a common Divifor as likewiſe their Numera:
a, x
e, which cannot be, wherefore in an Equation of the
third Degree, and in general in any Equation, the unknown Quantity cannot
have more Values either real or imaginary than there are Units in the De
gree of the Equation.
222
ELEMENTS OF
Objection
againſt the
*
v.
Againft the foregoing Demonftration the following Objection may be
ftarted. Let a 4,617,c=7, e8 and x=2, then (a) (b)
=—2×—13=-5×—6—(~ — 7) (✯ = 8) = (x − c ) (x — e) • ´
tion folved. it is true, that in fome Cafes by giving a certain Value (a) (x — b)
x (x à)
foregoing
demonftra
General
: = ( × — ¿ ³) : ( x − e), but as x is a general and undetermined Quantity,
=(x
the foregoing Equation ſhould take Place, whatever Value is given to x
which is impoffible, in Effect let x=a theǹ (á—a) (a—b) = (a—c) (ame)
that is, (ac) (a —e) =0, which cannot be, fince cand e differ
from a and b.
vi.
Some of thoſe Quantities a, b, c, &c. or all of them, may denote real
Quantities, equal, or unequal, or fimple imaginary Quantities, as B√√✓ — 1,
or mixed ones, as ▲ † B √ — 1, to thoſe two Forms all imaginary
Quantities whatſoever being reducible, A and B expreffing real Quantities.
10. a+b√1±g+b√−1, may be reduced to the Form of
A+B √ — 1, for a+b √ 1±g±b√-1=a±8+ (b ±6) √ — 1
b) - I
whatſoever. A+ B-1, by putting a g=A, and b + b = B.
form that
may be gi-
ven to every
imaginary
quantity
=
I
2º. (a + b √ − 1) × (g+b √
for (a + b√ — 1) ×
wherefore a gbb
a+bil
30.
8+b√-1
− 1) may be reduced to A+B√ — 1.
(g+b √√ − 1) = (ag — bb + b g + ab) √ — I;
A, and bg + a b = B.
= A + B √
j
I, for
a + bi/E...
g + b √ I
(a+b)−1 ) × (8—b√ — 1 )
(g+b)−1 ) × ( ġ—bij — 1 )
(åg +bb + bg — ab ) ;√ — 1
gg+bb
wherefore
ag+bb
88+ b b
=A, and
bgab
-B.
88+ bb
8 + b √
-- I
=A+B √1, for fuppofing it
8 + b √
4°. ( a + bij — 1 )
to be fo, Log. ( a + b √ — 1 )
= Log. (A+B √1), or
(8 + b√ - 1) Log. (a + b √ —I= Log. (A + B √ − 1),
wherefore fince the limiting Ratio of the cotemporary Increments of
A + B √1, and of its Logarithm, is equal to the limiting Ratio of the
SPECIOUS ARITHMETICK.
22$
cotemporary Increments of (a+b)
8 + b√ - 1 I
and of its Logarithm,
g+1 being fuppofed to be a permanent Quantity, we will have
b
db I
(N) (8 + b √ − 1 ) x ( da + d b√
a+b √
I
-):
=
dAdB -1
√
A+B√ I
and
(da+db,/—1)(a—b√—1)
(a+b√-1(a-b√−1}
gada+gbdb-abdb+bbda.
(bada+hbdb+gâdb—gb da)↓ I
aa + bb
+
·aa + bb
(8+b√-1)x(da + db √ =+) = (8+b√—1) x²
=
a+by
I
-)=
X-
and performing the fame Operation upon the fecond Member of the Equation
dA+ dB √ I Ι (dA+dB √−1)X(4—Bv—1)
I
(N), we will have
A + B √
(A+B√—1) × (A—B√—1
Ad A+ Bd B
(4 dB-B dĄ)√1
I
X
A A+B B-
and confequently
AABB
gada+gbdb-abdb + bbda + (bada + bbdb +gadb — gbda) ✔ I
a a + b b ..
√
AdA+BdB + (AdB-BdA) I
A A + B B-
2
Now becauſe the real
Part of the firſt Member is equal to the real Part of the Second, and the
imaginary Part to the imaginary Part, we will have the two following
Equations, (0) gada +g b d b—a b d b+bb ďa
a a + b b
(bada + b b d b + g a d b − g b d a) ↓
& (P)
a a + b b
wherefore (2)
AdA+BdB-
= gx
AA BB
ada+bdb
Á а A + B à B
AABB
(AdB-BdA)/—I
AA+BB
adbbda
aa+bb -}-bx{ }
aa+bb
I now obferve, that if the Logarithm of √ ( A A+B B) be expreffed
by x the limiting Ratio of the cotemporary Increments of √ (AA+BB):
& of its Logarithm x will be expreffed by the Equation
&
+
g
√(aa+bb)
AdA+BdB
AA+BB
dx:
In like Manner, if the Logarithm of √ (aa +66) be expreffed by y,.
the Equation g X
=dy will exprefs the limiting Ratio of
a da + b d b
a a + bb
a+bb)
y,
the cotemporary Increments of ✓ (a a +66) and of its Logarithm y
*
7
224
ELEMENTS OF
and if we put
bx{
a db-bd a
a a + b b
-}
d 2, and confider
this Equation as expreffing the limiting Ratio of the cotemporary Incre-
adb-bda
ments of two variable Quantities, and ſ-bx{ -}
that from which it has been deduced, or Log.c
aa+bb
J-b { adb = bda}
a a +
Z
2,
expreffing the Bafe of the Logarithms Art. XLVII. Chap. 111. we will have
Log. √ ( A A+B B) = Log. (aa+bb) + Log. c
g
or (AA+B B) = √(a a + bb) x c
- bf {
adb-bda
aa+bb
-bf{ adb=bda }
aa+bb
-}
Performing upon the Equation (P) the fame Operation that was per-
formed upon the Equation (O), this Equation (P) after dividing both
Members 'by-1, and reducing becomes
ad a + b d b
شما کم
adb-bda
} +8x { aa + bb
= b Log. √ (a a + b b ), as to
Ad B- Bd A
в
(( R )
=b
bX
AA + B B
x{
a da + b d b
a a + b b
-}·
Now, S. bx
}
S
{-
A d B - B d A
AA BB
aa+bb
it Expreffes an Arc whoſe Tangent is
B
B
for let *
A
exprefs an Arc whofe Tangent is and let this Arc a receive the In-
crement da, then d Tang. a
Tang. (ada) =
2
=
A
Tang. (a+da) - Tang a, but
Tang a + tang da
tang, a X tang da
tang. datang. a X tang. d a
1- tang. a. tang da
which Ratio is less than
I
and
>
1 + tang. 2a
—
confequently d tang. u
tang. da
1-tang a tang da
d tang. a
1+tang.¹ a
but continually approaches to it
when de decreaſes, and coincides with it when da vaniſhes, and becauſe in
this Cafe da and tang. da continually approach to a Ratio of Equality,
SPECIOUS ARITHMETICK.
225
1 + tang.² a
Increments of the Arc and its Tangent, in which Expreffion fubftituting for
da =
d. tang. “
will exprefs the limiting Ratio of the cotemporary
tang, its Value
B
A
and for d. tang. a its Value
AdB Bd A
A A
fults da =
Ad B Bd A
AABB
adbbda
there re-
In like Manner it will appear that
S.8x aa+bb expreffes the Product arifing from the Multiplication
of gʻinto an Arc whoſe Tangent is 2
Whence to reduce (a+b)
a
5+ b√√
by
1
to the Form A+B√—I',
it fuffices to defcribe a Circle with a Radius equal to the Value found for
-
- bf { adb = b da}
g
✓ (A A+B B), that is, equal to (a a+bb) x C
aa+bb S.
and take on the Circumference of this Circle, an Arc equal to the Value
alfo found of ƒ {AB=Bd4} that, is, equal to h Log.√(aa+66)+8x
f{
a d b - b d a
aa+bb
+
-};
B
the Sinus and A the Cofinus.
x
will be the Tangent of this Arc; B will be
**
VII..
in the Eqna--
If one of the Terms of a Multinomial is an imaginary Radical, as for
Example, ab 1, the Radical may be made to vanish, by
multiplying it by another Multinomial that differs from it only by the Sign The imagi
that precedes the Radical. Whence it is only the Product of x-a-b — I
bI fons cannot
nary Expref-
into xa+b1, that can make the Radical in the propofed Quantity difappear
diſappear, by giving the Product x² - 2 a x + a² + b. Becauſe it is only tion pro
in this Cafe that the particular Products of each real Term into b√duced but
deſtroy each other by contrary Signs, and in the fame Cafe it is manifeft Number is
that the Term b that contains the Product of the two Radicals ba
and b ✓
I is neceffarily pofitive. Whence there are never in any E-
quation whofe Coeficients are real Quantities, fingle imaginary Roots, or an
odd Number of imaginary Roots; but the Roots become imaginary.
in Pairs, and an Equation of an odd Number of Dimenfions, bas al-
ways one real Root.
4. F
when their
even..
I
226
ELEMENTS OF
Method of
finding the
Syfterns of
factors cor-
reſponding
mulas of the
fecond,
third, &c.
degree.
VIII.
Let a, b, c, d, e, f, g, &c. exprefs real and pofitive Quantities,
a being greater than b, b greater than c, greater than d, d greater
than e, &c. The Syftems of Factors of the Formula's of the fecond Degree.
I. x² + m x + n = o, II. x²· mx + n=0, III. x²+mx
IV. x2
m x ng V. 2 n=0, VI x
.11 0.
In which m and ʼn denote real and pofitive Numbers, will be as follows,
to the For- ( a ) ( x + a); (x+a)(x−a); (x — a) (x—a); (x +a) (x+b);
(x + a) (x
(x+a)(x−b; (x—a) (x+b); (x—a) (x——b); (x+a√ — 1 ) ( x—a₁/— 1);
(x+a+a√—1) (x+a—a√√—1); (x—ata₁√√—1) (x—a—a
a — a₁ — 1);
(x+a+b√−1) (x+a−b√√−1);' (x−a+b√− 1 ) (x—a—b√ — 1) ;
(x+b+a₁~1.) (x+b—a√−1); (x−b+a√—1) (x—b—a√√✅—1);
In like Manner the Syftems of Factors of the Formula's of the third,
fourth, fifth, &c. Degree will be found.
IX.
To diſcover the Nature, that is, the Sign of each Coeficient of each
Syftem of Factors correfponding to Equations of the fecond, third, fourth, &c.
Degree, the Factors of thofe Syftems are to be multiplied into one another.
When feveral Syftems of Factors produce the fame Formula, to deter-
When feve- mine the nature of the Roots of any propofed Equation contained in this For-
ral fyftems
of Factors mula, it will be requifite to find the Syftem of Factors that will produce
correſpond it, how this may be effected by Means of the Coeficients of its Terms, we
to the fame fhall proceed to explain.
Formula
how to find
the ſyſtem
of factors
corre pond-
ing to any
propoſed
Equation
I obferve that there are as many Quantities a, b, c, d, in each Syſtem of
Factors, as there are Coeficients m, n, p, &c. Whence there will refult
as many Equations between a, b, c, d, &c. as there are Coeficients.
Confequently if fome of the Quantities a, b, c, &c. are equal, there will
be more Equations than unknown Quantities, by which Means an Equation.
contained in will be obtained, out of which the unknown Quantities a, b, c, &c. have
been exterminated, which will ferve to determine whether the propofed
Equation has equal Roots or not, and in Caſe it has, the Form and Signs
of its Roots.
this For-
mula.
But if all the Quantities a, b, c, d, &c. are unequal, there will refult
only as many Equations as there are unknown Quantities, and confequently
they cannot all be exterminated. But by putting fome of them equal,a Quantity
greater than o in fome Cafes, and lefs than o in other Cafes will be obtained,
which will ſerve to determine whether the Roots of any propofed Equation are
all unequal, and in Cafe they are, the form and Signs of thofe Roots.
SPECIOUS ARITHMETICK.
227
X.
Thus, 10. let x²+mx+n=(x+a) (x+a) = x² + 2 ax+a²=0;
then m = 2a, n=a², conſequently m²
2
4 = 0.
n
I)
20. Let x²+m x + n = (x+a+a√ - 1) ( x + a—a—1)
= x²+2ax + 2 a² = 0, then m 2 a, n = 2 a².
1712
1712
2n = 0,
=
Conſequently
3°. Let x²+mx + n = ( x + a ) ( x + b ), if b be put equal to a
the Quantity m² 4 no, wherefore if b be lefs then a, the Quantity
4 n will be greater or leſs than o, to find which of the two fhould
take Place, let a = 2 and 6 = 1, then m =3, n=2, which Values being
ſubſtituted in the Quantity m² 4n there will refult 32 22. 2 >0.
—— 2².
Whence I conclude that m²-4 n >o is the Condition for this Syſtem.
4°. Let x²+mx + n = (x + a + b √ − 1 ) ( x + a − b √ -1)
if b be put equal o, this Syftem will coincide with the firft Syftem, and
confequently m²-4=0, but if b be greater or lefs than o, this Quantity
will be greater or lefs than o, and on Examination it will be found to be
lefs than o, whence the Condition for this Syftem will be m² - 4 n < 0.
+#
n
2—2n>0.
/—1),
Hb be put equal a, this Syftem will coincide with the Second, and con-
fequently m² 2 no, but when b is lefs than a this Quantity will be
greater or lefs than o, and on Examination it will be found to be leſs than o,
whence the Conditions of this Syſtem are m² 41 <0. m² 21>0.
5°. Let x²+mx+n= (x + b + a √ − 1 ) (x+b-a/-1), 6 being hereby are
put equal to a this Syftem will coincide with the Second, and confequently difcovered
m² — 2n=0,.wherefore when bis leſs than a, this Quantity is greater or leſs
than o, it is eafy to perceive that in order to diftinguith this Syftem from the
foregoing one, that the Condition for this Syftem fhould be m² -2 n < 0.
It is thus the following Tables were conftructed, which when com-
pleated, will ferve to determine in any propofed Equation, the Nature and
Number of its Roots, that is, whether they are real or imaginary, equal or
unequal, pofitive or negative.
I. {x²+mx+n=0
(x+a)(x+a)
I
the nature
and number
of the roots
of any pro--
pofed Equa-
tion.
II. {(x²
mx + n = 0}
772 -47=0
m²—4n=0(x—a)(x—a
2
(x+a+a√/−1)(x+a~a√√/—1)--m²—2n=0(x−a+a√—1)(x—a—a√/—1) m²—211—0
(x+a)(x+b)
m² 4n>0(x-a) (x-b⋅ )
(x+a+b√−1)(x+a−b√−1) m²-4n<0
m²-4n>>
m²-4n<0
2_2120
m
m² — 2 n ) ( x − a+b√ √ ~1) (x—a—b√√—1)
(x+b+a√−1)(x+b—a√−1) m²−2n<0(x−b+a√—1) (x—b—a√√~1) ~~m
-
− −
III. x²+mx n = (x + a ) ( x − b) IV. x² — m x n = (x — a) (x + b)
Vị đổ trà ( tay I) (a VI) VI.
x² +
2
2
− 11 = (x + a) (x — α)
211 <0.
228
ELEMENTS OF
(x + a) (x + a) (x + a)
=
I. { x³ + m x²+ux+p
•}.
112
(x + a) (x + a √ − 1 ) (x — a NI)
(x + a) (x+a+a√−1)(x+a—a₁/—1) -
( x + a ) ( x + b) (x+b)
278=0
— 3 n = 0, m³ — 27 po
722
n=0, m?
m?3
4 m² — 9 n =
-9n=0, 2m3
270=0
{2m³-9mn+27p<0
(x + a) ( x + a) (~ + 6 ) } 4(m²—3n) (n²—3mp)—(mn—9p)²—0 S2m³-9mn+27p>0
(x+a) (x+b√ − 1 ) ( x − b √ — 1)
·(x+b)(x+a√ — I) (x — a√1)
(x+a)(x+b+b₁√√→1)(x+b—b√ √—1) ?
}
M 11
m3
p=0{
n> o
{ m²
7722
12 < 0
4m² 9n>0
(x+b)(x+a+a√~1)(x+a-a√√−1)} 2 (m²—n) (n²—mp)—(mn—p)²= 0 14m² —9n<0
(x+b)(x+a+b√√−1)(x+a—b√/−1))
√ 4m²-9n>o
(x+a)(x+b+a√−1)(x+b—a√√−1 ) 100n(m²+1)²-(2m³+17mn+p)'=04m
4m²-9n<0
(x+a)(x−b+a√—1) (≈—6—a√/—1)
(x+b)(x+b+a√~1)(x+b-a₁—1)
(x+a (x+a+b√✅~1)(x+a—b₁√√—1)
(x + a) (x + b) (x + c)
(x+a)(x+b+c√−1)(x+b~c√√—1)
}
2 m3—9 mn+27p=0{
4 (172² - 3 12) (122
m²
1722
72
>
9 mn+27 p = 0 { 4 m² —9n<0
>o
4m² ~gn >0
(n² —3 m p) — (mn — 9 p)² > o
98)
4(m² -3n) (n²—3mp)—(mn~9p)² <0
2(m²☛n) (n²—mp)—(mn—p)²>o
2 m³ 9mn+27 p >0,4 m²-9″> 0
2
mn—p>0,2(m² n) (n² —mp)—(mn—p)²<o
2
(x+a)(x+c+6√~x)(x+c—b√/−1) {100m(m² (n)—(2m³+17mn+p)²<0,m²—n>0
(x+a)(x—c+b√√—1)(x-c-b√√-1) mn-p<0, 100n(m²+n)²—(2m³+17mn+p)²<0
(x+b)(x+a+c√√−1)(x+a—c√√—1)
4(m² —3n) (n²—3mp)—(mn—9p)² <o
100 n(m²+n) 2-(2m³ +17mn+p)² <o
2m³ 9mn-27p<0,4m²-9n>0
(x+6)(x+c+a√−1)(x+c—a√—1) { mn—p>0,2m3—9mn+27p>0
100n(m²+n)² — (2m³+17mn+p)²>0,4m²—9n<0
-p<0,100n(m²+n)²—(2m³+17mn+p)²>0
m²-n<0
2
(x+c)(x+a+b√/−1)(x+a−b√√−x) { 100n(m²+n)²—(2m³ +17mn+p)²>0
(x+c)(x+b+a√−1)(x+b—a√ —1)
(x+a)(x+a)(x—b)
2(m²—n) (n²—mp)-(mn-p)2>0
2m³—9mn+27p<0
2(m²—n) (n²—mp)—(mn-p)²<o
4m²-9n<0
II. {x³ +mx² + nx− p = 0
=0}
4(m²—3n) (n²+3mp)—(mn+9p)²=0
(x − a)(x+b+ a√=1(x+b—B√ — 1 § 100n(m² + n)²—(2m³+17mn—p)²=0 { 2m 3-p>0
(x=a) (x+b+av=3(x+b=av = }}
I
2m3—p<o
SPECIOUS ARITHMETICK.
229
(x—b)(x+b+a√−1)(x+b—av—1)
(x—b)(x+a+aƒ−1)(x+a—a√—1)
(x+a)(a+b)(x—c
(x~a)(x+c+bv−1)(x+c—b√−1}
(x—b)(x+a+a/—1)(x +a—c√—1)
(x—b)(x+c+a√−1)(x+c−av—1)
(x−c)(x+a+b√−1)(x+a−b√ —1)
(x−c)(x+b+a√−1)(x+b—a√−1)
(x+a)(x—b)(x−b)
2m³+mn-p=0
2(m²—n)(n²+mp)—(mn+p)¹—o
4(m²—3n)(n²+3mp)~(mn+9p)²>o
{ 100n(m²+n)²➡(2m³+17mn—p)²<o
2m³ -p<o
4(m²—3n)(n²+3mp)~(mn+9p)*<o
100n(m²+n)²(2m³ +17mn-p)² <0
2m³ —p>o
S100n(m²+n)²—(2m³+17mn-p)²>o
{2m³+mn p<0
100n(m²+n)²(2m³ +17mn—p)²>0
2(m²—n)(n²+mp)—(mn+p)^>o
3.
2m³ po
S 2m³+mn—p>0
( 2(m²—n)(n²+mp)—(mn+p)² <o
III. {x³ + mx² — n x
nx +
(x+a)(x−b+b✓—1)(x—b—b√—1)
(x+a)(x—b)(x−c)
(x+a)(x−8+c√/−1)(x—b—c√—1)
(x+a)(x-6+b√—1) (*~———-b√/~1)
(x−a)(x—a)(x+b)
-
+x=0}
4 (m²+3n)(~3mp+n²)—(mn+9p)²=ɔ
2(m²+n)(—mp+n²)—(mn+p)²=0)
4(m²+3n)(−3mp+n²)—(mn+98)²>o
{ 4(m²+-3n)(—3mp+n²)—(mn+9p)² <o
{2(m²+n)(—mp+n²)-(mn+p)²>o
2 (m²+n)(~mp+n²)—(mn+p)²<a
IV. {x³-x²+x+p=0}
n
4 (m²—3n}(n²+3mp)—(mn+9p)²=0
(≈+b)(x−a+b√—1) (*~a~b √/—1)
− b √ — 1}) } 100n(m²+n)² —— (2m³+17mn—p)²=0} 2m³
(x+a)(x—b+a√-1)(x-b—a√.
(x+6)(x−b+a✓—1)(x—b—8√-
(x+b)(x—a+a₁/—1)(x—a—a√-1)
(x−a)(x—b)(x+c)
(x+a)(x~c+b√—1)(x——b₁/—I
(x+b)(x—a+v√√/—1)(x—————
2 m³+mn-p=0
2m³ —p>0
2m³ po
2 (m²—n)(n²+mp)—(mn+p)²=0
4(m²—3n){n²+3mp)—(mu+9p)²>o
{ 100n(m²+n)²=— (2m²+17mn—p)² <0
2m³ p>0
4(m²—3n)(n²+3mp)—(mn+9p)"<
100n(m²+u)² — (2m * +17mn—p)² <0
2 7713-p <0
230
ELEMENTS OF
(x+6)(x+c+a√/—1)(x——a√/—1).
(x+c)(x—a+b√—1)(x——a——b√—1)
(x+c)(x—b+a/—1)(x—b—a√—1)
(x+a)(x+a)(x—a)
(x+a)(x+b)(x—b)
(x+a) (x—a)(x+b)
(x-a)(x+b)(x+b)
(x+a) (x+a)(x—b)
V. { x²+mx².
{
100 n (m²+n)2-(2m³ — 17mn-p)">0
2m³+mn-p<0
3
100n (m²+n)²(2m³ +17mn-p)²>o
2(m² 11)(n²+mp)—(mn+p)²>o
2m³ po
3
2m3+•mn—p>0
2(m² —n)(n²+mp)—(mn+p)² <0
V. { x²+mx²-nx — p=0
(x—a)(x+·b+b√−1)(x+b—b√✓✓—1)
(x—a)(x+a+b√−1)(x+a—b√—1)
(x+a)(x+b)(x−c)
(x+a)(x−b)(x+c)
(x~~a)(x+b)(x+ic)
(x—a)(x+b+ci/—1) (x+b—c₁/—1)
(x−a))x+c+b√✓−1)(x+·c—b√√/—I
(x−b)(x+a+c√/−1)(x+a−c√—I
M²
}
17112
p=0}
·n=0,m³—p=0
5 m²-n>o
{ m²--n<o
2
•n<o
{
m²—n<0
} 4(m²+3n) (n²+3mp)—(—mn+9p)²=0 { m²-n>o
2(m²+n)(x²+mp)-(—mn+p)²=0
2 m³ —mn p=0
m3
{ 4 (m² + 3n ) ( n² + 3m²
{ 4 (m²+3n) (n²+3mp)~(~nin+9p)²>0.
mn—p<0,m² —n>0
{
mn-po
4(m²+3n)(n²+3mp)—-(—mn+9p)²>
mn—p<0,m²-1 <0
4(m²+3n) (n²+3mp)—(—mn+9p.)² <o
2(m²+n)(n²+mp)~(~mn+p)² >o
2m3-mn-p<0,m²-n<0
2(m²+n)(n²+mp)~(~mn+p)² <0
4(m²+3n)(n²+3mp)~(~mn+9p)²<0
{4(m² + 3n) (m² + 3 mp) (
2m³—mn—p>0,m²———n>o
mx² + nx- -p=0}
(x—a)(x—a)(x—a)
VI. {*³
m³—3n=0,m³—27p=0.
m²—n=0,m³—p=0
4m²-9n=0,2m³—27p=0
2m³—9mn27p>0
(x−a) (x+a√—1)(x—a√—I
(x-a)(x-a+a√~1)(x—a—a—1)
(x-a)(x—b)(x-b - } 4 (m^—3n) (n²—3mp)—(mn—9p)² = 0 { 2m³ —9mn+21 p <0
(x—a)(x—a)(x—b)
m
-n>o
} mn—p=0 m²-n<0
(x—b) (x—aja√/—1)(x—a—a√—x)} 2(m²—n) (n²—mp)—(mn—p)²—o { 4m²—9n>o
(x—a)(x+b√—1)(x-b/~1)
I
(x—b)(x+a√−1)(x—a√—1)
(x~a)(x—b+b√ƒ—1) (x—b—b√√/——1)
(x—b)(x—a+b√—1)(x—a—b₁—1)
(x−a) (x+b+av—1) (x+b—a√·
(x—a)(x+b+ax—1)(x+b—a√—1)
(x—a)(x—b+a√—1) (x−b—a√√—1) 100n(m²+n)2-(2m³+17mn+p)²=0
2
4m²-9n<0
4m²-9n>0
4m²-9n<
m²-n>o
m²-n<0
SPECIOUS ARITHMETICK.
231
(x-b)(x~bta₁/ -1 ) ( x—ba√1)
(x—a) (x—a+b✓—1)(x—a—b√ √✓ — 1
(x—e)(x—b)(x—c)
4m²—9n>o
4(m² — 3n) (n² ——3mp)—(mn—9p)²>0
4(m² ———3n) (n² —3mp)—(mn—9p)² <o
(x—a)
(x—b·†¤₁/—1) (x—b—c.—1 ) 2 ( m²—n) (n²—mp)—(mn—p)² >0
(x—a)
(1-^*)(1/9+みき​)(ローン​)
2m3—9mn+27p>0,4m²—9n>0
mn—p>0,2(m²—n) (n²—mp)—(mn—p)²<o
(x+c—b₁/—1) { 100n(m²+r)²(2m³ +17mn+p)² <o
(x+c+b√/—1)
(x—a)(x—c+b√/−1)(x—c—b√√—1)
2
I
mn-p<0,100n(m²+n)2-(2m3+17mn+p)² <0
S4(m² - 3n) (n² — 3mp) —(mn—9p)² <0
(x—b)(x—a+c√√—1) (x—a—c.—1) 100n(m²+n)2-(2m3+17mn+p)² <0
2m³—9mn+27p<0,4m²—9n>o
-12
(x—b) (x+c†a√/—1) (xt—a√/—1) { mn—p<0,100n(m²+n)²—(2m³+17mn+p)²>0
(x—b)(x—c+a√—1)(x—c—a √✓−1 ) { mn—p>0,2m³ —9mn+27 p>0
نه
2
I
100n(m²+1)2-(2m3+17mn+p)²>0,4m²-9n<o
(x—c)(x—a+b√√—1(x—a—b√-1) { 100n(m²+n) 2— (2m3+17mn+p)² >0
}(1ヮーター​x)(1+qx)
I
3
2(m²—n)(n²—mp)—(mn—p)²>o
-9mn+27p<0.
n'
(x—c) (x—b+a√/—1) (x—b—a—1) { 2 (12)(2m)_(mn—p)² <0,4m²—9n<0
(x+a)(x—a)(x—a)
(x+a)(x—b)(x —b)
(x+b)(x-a) (x-a)
(x—a)(x—b)(x+6)
(x+a)(x—a)(x—b)
VII. { x³ — mx² — nx+p=0 }
m²—no,m³—p=0
Sm²-n>o
} 4(m²+311) (n²+3mp)—(—mn+9p)²=0
{{.
{
m²-n<Ⓒ
(m²-n>0
}
mn—p=0 { m² = no
p=0{;
(x+a)(x—b+b√—1)(x—b—b.j—1) 2{m²+n)(n²+mp)—(—mn+p)²=0
(x+a)(x—a+b₁j——1) (.x—a—b√—-1) 2m³—mn—p=0
{
4(m²+3n)(n²+3mp)—(—mn+9p)²>0
mn-po,m²-1 <0
(x+a) (x—b) (x−c) ·
(x—a)(x+b)'x—c)
(x-a)(x—b)(x+c)
B
mn-p>o
{ 4(m²+3n) (n²+3mp)—(—mn+9p)²>0
\ mn—p<0,m²-n>o
S4(m² — 3n) (n² +3 mp)—(—mn+9p)² <0
(x+a)(x—b+c√/—1) (x—b—c√—1) { 2 (m²+n) (n²+mp)—-(—-mn+p)²>o
12
2m³-mn-p<0,m²-n<0
(x+a)(x—c+b√—1) (x—c—b √—1) 2 (m²+n) (n²+mp)—(—mn+p)² <0
(x+b) (x—atc√—1)·(x—a—ex/—1) { 4(m²+3n) (n²+3mp)—(—mn+9p)² <0
(2m 3-mn-p>0,² 110
232
ELEMENTS OF
(x+a)(x—b)(x—b)
VIII. x³+mx²-1138+p=0.
(x+a)(x—b+b√—1)(x—b—b√/—1)
(x+a)(x—b)(x—c)
(x+a)(x−b+c√√—1)(x−b—c√↓— I
(x+a)(x—c+b√—1)(x—c—b√√/—1)
+3
IX. x³- mx
(x—a)(x+b)(x+b)
(x−a)(x+b+b√−1)(x+b—b√✓—1).
I
(x-a)(x+b)(x+c)
(x−a)(x+b)+¤√/—1)(x+b—cq/—1)
(x−α)(x+c+b√−1)(x+c—b√√✓−1)
2
2
4(m²+3n)(x²-3mp)—(mn+9p)²=0
2(m²+n)(n²—mp)~(mn+p)²=0
4 (m²+3n)(n²—3mp)—(mn+gp ) ²>o»
4(m²+3n)(n²—3mp)—(mn+9p)² <0
2(m²+n)(n² —mp)—(mn+p)²>0
2(m²+1)(m²
2(m² + n) (n² —mp)—(mn+p)² <0
nx
p=0.
4(m²+3n)(n² —3;mp)—(mn+9p)?
2(m²+n)(n²—mp)—(mn+p)²—0
4(m²+3n) (n² —3mp)— (mn+9p)²>0.
S4(m²+3n) (n² 3mp)-(mn-9p)2 <0
=
2(m²+n)(n²—mp)—(mn+p)²>0
2(m²+n) (n²-mp)-(mn+p)²<0:
2
x³+mx²+p=(x+·a)(x—c+b√√—1) (x—c—b√√
—ctib√/~1)(x—c—b√—1)=0
I
(x—a)(x+a+a√√)—1)(x+a—a₁/—1).
(x+a)(x+a)(x—b)
x3 + mx²-p=√(x+a)(x+6)(x—c)
(x~a)(x+c+b√—1) (x+c—b√✅—1)
·(x—b)(x+a+c√~1)(x+a—c√−1)
(x+a)(x—a+a√−1)(x—a—a√/~1)
(x—a)(x—a)(x+6)
2
x3 - =·
mix² + p = {(x−a)(x—b)(x+c)
(x+?)(x~c+b√—1)(x—c—b√√—1)
·(x+6)(x —a+c₁/—1)(x—a—c
x³-mx²-p(x—a)(x+c+b√. -1}(x+c—b₁√√—1)
(x+a)(x—b+a√√—1)(x—b—a√—1.)
x² + nx + p = {(x+a)(x=c+b₁√√—1)(x—
x3 — nx+p=
C-
√~1)
(x+b)(x—6+a√√—1)(x—c—a√√~1)
(x+a)(x—b)(x—b) -
(x+a)(x−b+b✓—1)(x—b—b√—1).
(x+a)(x—b (x—c)
(x+a}(x—b+c√√/−1)(x—b—c /—1)
(x+a)(x—c+b√/—1)(x-c—ba√—1
(x-a)(x+b+a√ - 1)(x+b-a√√/—1)
I
x³ + nx-p = {(x—a) (x+c+b√√−1)(x+c-b₁—1).
· (x—b)(x+c+a√—1) (xt—a√~1)
(x−a)(x+b)(x+6)
x3. NX
=
(x—a)(x+b+b₁/—1)(x+b—b√√—1)
(x−a)(x+b)(x+c)
(x~—a)(x+b+c√—1). (x+b—c√—1)
(x—a)(x+c+b✓—1)(x+c—b₁/—1)
-
2 m³ p=0.
m3-
4m3-27p=0
4m³ 27p>0
2m³ p<0
4m3-27p<0,2m³ —po
2m³ po
4m3—27p=0
4m3—27p>0
2m³ p<0
4m3—27p<0,2m³ —p>0
100n3 — p² = 0
100n3—p² <☺
10073-20
3
4r³ 27p² o
2 123-p²:
4n3-27p² <o
{ 27
S 4n3—27 p² <0
2.273 — p². >a
$23.
100% —p².
10un³—p² <0
100n3—p²
4π³. — 27 p² -0,
2n3p2-0
423-27 p² 0
S4n³-27p² <0·
2n3—p²>o
2n3—p² <o
x³+p=(ñ+a)(x−c+b√—1)(x—6—b√✓−1); x׳—p=(x—a)(x+c+b√—1)(x—6—b√/—1)
SPECIOUS ARITHMETICK.
233
1
XX...
M --- I
H
2
+r=0, Method of
or negative, it will be a Divi- finding the
If one of the Roots of x" + px +qx"
is an integer Number a, either pofitive
for of the laft Term r; for a" + pa
conféquently a +pa +qa
I
m
2
17
am - 1 + pam - ² + 9am-
3
m-it
172
十九
​I +qa
2
111 ---
2
commenfu-
+na+r=0,`rable roots
of an equa-
+na= r, wherefore tion.
The first Member of
*
a
which Equation is an Integer, as being compofed of Integers, wherefore
Q
is an Integer, wherefore a is one of the Diviſors of r.
Whence is
derived a Method for inveſtigating all the commenfurable Roots of an
Equation; for they ſhould all be found by trying the Divifion of the Equa-
tion, by x more or lefs each of the Divifors of the laft Term.
For Example, fuppofe the Equation x35x+7x30 is to
be refolved, the Divifors of the laft Term
3, being either I, or
3, or + 1, or +3; I try the Division by x I, X 3, ≈ † 1,
3; the Divifion by x and x-3 fucceeding, I perceive that
the Equation may be wrote thus, (x I) x (≈ 1) × (x − 3)
whereby it appears that the propofed Equation has three commenfurable
Roots, one of which is +3, and the other two are equal, and expreſſed
each by +1.
≈
I
= 0,
When an Equation is not divifible by a fimple Equation compoſed of
x + or
fome one of the Diviſors of the laſt Term, it will be a Sign
that this Equation has no commenfurable Root.
XXI.
tion whoſe
unknown
It may be objected against the foregoing Method of finding the com- In an equa-
menfurable Roots of an Equation, that if any of the Roots of this Equa- coeficients
tion is a Fraction, it will not be poffible to find it among the Divifors are Inte-
of the laft Term: becauſe if fractional Divifors be admitted in a Num- gers the
ber, fuch Divifors may be found without End. But this Objection is
quantity
eafily removed, by obferving that in an Equation clear of Fractions, and cannot be a
having the Coeficient of its higheſt Term Unit, the Value of the un- fraction.
known Quantity cannot be a Fraction (), whofe Numerator and De-
nomninator are rational and integer Numbers.
A
Let for Example x3 + px² + qx + r = 0, and let - be ſuppoſed
b
to be a Root of this Equation, then² + 2 + r = 0, and
3
2
bz
да
b
a³ + pba² + qb² a² +63 r = o, confequently by the foregoing Article
the Integer Number a fhould be a Divifor of the laft Term 63 r. Now,
་
4 G
234
ELEMENTS OF
Transfor-
means of
which an
equation
may be
a
Ъ
3
fince, a and b have no common Divifor, the Fraction being by Hy-
pothefis, reduced to its lowest Terms, it follows that a and b³ have no
common Divifor, Art. xxIx. Chap. 11. Wherefore a muft be a Divifor
of r, wherefore rna, ´n being an integer Number; whence we have
a³ +pba² +qb² a+nb³ a=0, wherefore a² +pb a+q bª +n b³ =0,
therefore, by the foregoing Article, a fhould be a Divifor of the laft Term
qb²+nb³, and confequently of g+bn, wherefore q+bn=ma, whence
a² + p b a + b² m a = 0, and a + b + b² mo, and of courſe
m b, that is, = an integer Number which is contrary to
P
the Hypotheſis.
b
2
XXII.
When a Problem produces an Equation whofe Coeficients are Fractions,
the commenfurable Roots it may contain, cannot be found by the foregoing
Method; however the Problem by a fimple Transformation may be
changed into another, in which the Equation to be refolved is clear of
cleared of Fractions, and having the Coeficient of its higheft Term, Unit.
Suppoſe the Equation propoſed is x3 + x²+==+=
fractions.
a
Б
C
t
F
04
by putting the unknown Quantity x equal to another y divided by fome
undetermined Number m, I transform the Equation into a new one,
3
ڈیو
7123
ay 2
су
+63 + + + = 0,
d
2
b m
772
in which I perceive, that if
a m
Cm
ь
d
2
and
em 3
F
of y³ +
or
a m
b
C m²
}² +
J
d
· +.
is diviſible by b,
e m³
<= 0,
by d, and by fig
will be Integers, we have therefore no more to do
than to affume for the Product of the Numbers b, d, f, and if thoſe
Numbers are not prime to each other, it will be eaſy to find a Number
leſs than their Product that will be divifible by all the three.
The Equation being transformed into another clear of Fractions, the
commenfurable Roots of this.laft Equation are to be inveftigated by the
foregoing Method; and if none are found, neither will the firft have any,
fince x ſuppoſed commenfurable, cannot give an incommenfurable Quan-
tity when divided by the Number m, which is alfo commenfurable.
XXIII.
The foregoing Method is liable to this Inconveniency, that when the
Inconveni- laft Term has very many Divifors, the Labour will be exceffive to try all
the Diviſions that this Method preſcribes; how the Analyſts have abridged
foregoing
it, by limiting to a ſmall Number the Divifors that are to be tried, we ſhall
method is now proceed to explain.
ency to
which the
liable.
SPECIOUS ARITHMETIC K.
235
XXIV.
ons that
have fery-
They obſerved first, that if in the Root+a of any Equation, or if in Obfervati-
the Divifor x+a of any Quantity, & be put equal to a given Number,
the Number into which the Divifor will then be transformed, will be a ed to ren
Divifor of the propofed Quantity, in which x has been put equal to the der the
fame Number, that is, for Example, if in the Quantity 3-2x-21 of foregoing
which x-3 is a Divifor, x be put = 5, the Number 94 into which
x32x21 is transformed by this Suppofition is neceffarily divifible by feat.
the Number 2 into which x-3 is transformed by the fame Suppofition.
I;
They then ſuppoſed in the Quantity whoſe Divifor they fought, fuc-
ceffively equal to feveral Numbers, fuch, for Example, as + 1, 0,
Suppofitions that ſhould be firft made, becauſe they are the eaſieſt
fubftituted, they afterwards fought the Divifors of the Numbers into which
the propofed Quantity is transformed by thoſe Subſtitutions, and obſerved:
1. That among all the Divifors of the Number arifing from the Sup-
pofition of x+1 in the propofed Quantity, the Number I + a fhould be
found, fince x + a is the Divifor ſought.
=
2º. That among all the Diviſors of the Number arifing from the Sup-
pofition of x=0, which are no other than the Divifors of the laft Term
of the propofed Quantity, the Number a fhould be found.
3º. That among all the Divifors of the Number arifing from the
Suppofition of x = I ſhould be the Number
XXV.
I + a.
method
more per-
Fundamen-
commenfu-
Now as the Numbers + a, a, Ita are neceffarily fuch, that the
firft exceeds the fecond by Unity, and that the fecond exceeds the third tal princi-
alfo by Unity; it follows that of all the Divifors of the Number arifing ple for find-
from the Suppofition of x=0, thoſe only can exprefs a, that are exceeded in the
by Unity by fome one of the Divifors of the Number arifing from the rable roots
Suppofition of x = 1, and exceed at the fame Time by Unity fome one of an equa
of the Divifors of the Number arifing from the Suppoſition of x==1.
If ſeveral Numbers are found among the Divifors arifing from the
Suppofition of xo, that have the above mentioned Conditions, to reduce
the Divifors of the laſt Term to ftill more narrow Limits, x is to be put
I.
2, and then obferve if among the Divifors of the Number arifing from
this Suppofition, there be found any that exceed by Unity thofe arifing
from the Suppofition of x = 1, and fo on.
It is eaſy to perceive that the Trial that is to be made of all thoſe Di-
vifors fhould be double, that is, that each of them fhould be taken as well
negatively as poſitively.
To illuftrate this Method and to render the Application of it more eafy,
we fhall proceed to give fome Examples fhewing the Order to be obſerved
in the Calculation to avoid mistakes, and to abridge, as much as poflible, the
Labour of the Calculator.
tion.
1
236
ELEMENTS OF
XXVI.
1
Let it be propofed to find the commenfurable Roots of the Equation
x3—2x² - 13x+6=0; or which is the fame, let it be propoſed to
find the Divifors of one Dimenfion of the Quantity
2
~ 34
1 1 1
x3
2
2x
13x+6.
I 8
I
O
6'
I
-I
Ι
16
I
2
2 2 2
4
8
'3
6
+4
+3
4. 8
16
+2
2 16
Application
of the fore-
I firſt ſet down one under the other the Suppofitions 1, 0, -1, to be made
for x
going me- x; I fet down afterwards befide each of thofe Numbers, the Num-
thod to an bers 8, + 6, + 16, or fimply 8, 6, 16, (becauſe the Signs do
example.
not affect the Divifors) into which the Quantity x3 2x2 13x+6
is transformed by thofe Suppofitions, and I feparate them from the firſt
Numbers by a vertical Line. I fet down in a third Column the Numbers
1, 2, 4, 8; 1, 2, 3, 6; 1, 2, 4, 8, 16, which are the Diviſors of the
foregoing Numbers; the four firft befide 8, of which they are the
Divifors; the four that follow befide 6, and the five laſt befide 16.
✰=
·
This being done, to find among the Divifors, 1, 2, 3, 6, of the
Number 6 arifing from the Suppofition of x=0, that which is to be added
or deducted from x to obtain the Divifor fought, or rather to exclude
from all thoſe Divifors, thofe that have not the required Conditions; I
obferve 1. that 1, which is the firſt of thoſe Divifors, cannot be admit--
ted, whether it be taken with the Sign + or with the Sign; for if it
be taken with the Sign+, that is, if x + 1 be confidered as the Divifor
fought, this Divifor becoming 2 by the Suppofition of x+1, ando by
the Suppofition of I, we fhould find 2 among the Numbers of the
firſt Line, and o among thoſe of the third: Now the ſecond of thoſe Con--
ditions is not anſwered. As to I, it cannot be admitted either; that is,
X I is not the Divifor fought, for this Divifor becoming o by the
Suppoſition of x=+1, and 2 by the Suppofition of x=
fhould find o among the Numbers of the firſt Line, and the Number 2
among thofe of the fecond. Now only the fecond of thoſe two Conditions
is anſwered. I next perceive that the. Divifor 2 is alfo to be rejected,
becauſe if it be taken with the Sign+, that is, if x +2 be confidered as
the Divifor fought, we would have + 3 by the Suppofition of x = + 1,
and I, by the Suppofition of x=-1, which would require that the
Number 3 fhould be found in the firſt Line, and I in the third: Now the
firft of thoſe two Conditions is not fulfilled. And if 2 be taken with the
Sign, that is, if x 2 be confidered as the Divifor, we would have
I, we
1 and 3 for the Suppofitions of x=+1 and of x =—1, which
SPECIOUS ARITHMETICK.
237
would require that I ſhould be found in the firſt Line and in the third:
the first of which Conditions is only fulfilled.
I
Having excluded 1 and 2, I affume the Divifor 3, and I perceive that
taking it with the Sign+, that is, confidering x + 3 as the Divifor fought,
we should find +4 by the Suppofition of x=+1, and +2 by the
Suppofition of X
I and in effect I find 4 in the firft Line and 2 in
the third, wherefore +3 has the Conditions required, I therefore fet it
down in the fecond Line, that is, oppofite the Number of which it is a
Divifor. I fet down at the fame Time the Numbers + 4 and † 2 in the
Line above it, and below it, not that thofe Numbers are to be joined to x
to ferve as Diviſors to the propofed Quantity, but becauſe the Trial of
the Diviſors not being compleated, other Numbers befides + 3 might be
found having the required Conditions, and in this Cafe new Suppofitions
are to be made, to diſcover among thoſe Numbers thoſe that are to be
excluded. I now try if 3 taken with the Sign will fucceed, that is, if
-3 has the Conditions required to be a Divifor of the propofed Quan-.
tity; in this Cafe we ſhould have 2 and 4 by the Suppfitions of
x = +1, and of x
x= I, which happens in effect, whence x 3 has the
Conditions required to be a Divifor as well as x+3, I therefore fet down.
in a fifth vertical Column 2g. — 35 4..
•
Laftly, I try 6, and I perceive that if it be taken with the Sign + we
fhould find + 7 and +5 in the Lines above it and below it, which does
not happen; and if it be taken with the Sign we ſhould find — 5 and.
-7 in the fame Lines, which likewife does not happen. I conclude:
therefore that the propofed Quantity can have but x-3 and x+3, for.
Divifors of one Dimenſion..
*
To diſcover if there is the fame Foundation for trying the Divifion by
3 as by x + 3; I obferve that if a fourth Line was made by fuppofing
x= 2, we fhould find -5 for the fourth Term of the Column
-2, 3, 4 and 1 for the fourth Term of the Column +4,
+3, +2; for it is manifeft that the Divifor x 3, would become — 5,
by the Suppofition of x=2, and that the Divifor x +3 would be-
come by the fame Supppofition. But by putting x-2 in the
propofed Quantity x32x13x+6, it becomes 16 which is not di-
viſible. by 5.and is by 1. Wherefore 3 cannot be a Divifor of this
Quantity, whence if it has one, it muſt be x +3, I therefore divide
x3 — 2x²-13x+6. byx+3 which fucceeds, and gives for exact Quo-
tient xx
5242.
In Numbers fo fimple as 8, 6, 16, it was an eaſy Matter not to omit
any of their Divifors, becauſe thoſe Numbers have but few Divifors, but
in Numbers that have many Divifors, to obtain them all, they must be
fought with a certain Order, we ſhall therefore proceed to explain by an
Example how the Analyſts perform this Operation.
238
ELEMENTS. OF
{
the divi-
fors of any
number.
XXVII.
Let it be propoſed to find all the Diviſors of the number 120. I firſt
I
120
2
бо
4. 2
30
8. 4. 2
15
24. 12. 6. 3
5
I
Method of draw a vertical Line to the left
finding all Hand of this Number, and to
the left Hand of this Line op-
propofed pofite to 120 I fet down 1, as
being its first Divifor. I af-
terwards try whether 2 will di-
vide 120, and as the Divifion
fucceeds, I fet down 2 to the left Hand of the vertical Line, oppoſite to
60 the Quotient of the Divifion, which I fet down to the right Hand
of the fame Line.
120. 60. 40. 30. 20. 15. 10. 5
I afterwards try the Divifion by 2, which fucceeds, and gives 30 for
Quotient, I then fet down the new Divifor 2 under the firft, and 30
under 60. I multiply at the fame Time the new Divifor 2 by the one
above it 2, and fet down the Product 4 to the left Hand of the fecond
2, as being a new Divifor of the propoſed Number 120.
As 30 is divifible by 2, I fet down 2 again to the left Hand of the
vertical Line in the fourth Row, and the Quotient 15 to the right
Hand oppofite to it in the fame Row. I multiply at the fame Time the
new Divifor 2 by 4, which gives 8 for a new Divifor of the propoſed
Number. But omit multiplying it by the firft Diviſors, becauſe there
would refult 4, which has been already fet down.
15 not being divifible by 2, I try the Divifion by 3, which fucceeds,
and gives 5 for the Quotient, which I fet down to the right Hand in
the fifth Row, and the Divifor 3 to the left; I afterwards multiply 3
by 2, by 4 and by 8, which I find in the fuperior Lines, and fet
down to the left Hand of 3, the Products 6, 12, 24, which are, as is
manifeft, new Divifors of the propoſed Number.
5 having no Divifor I fet it down to the left of the vertical Line
in the fifth Row, and I fet down at the fame Time the Product of this
Number into all the foregoing Diviſors 2, 3, 4, 6, 8, 12, 24, and there
reſults 10, 15, 20, 30, 40, 60, 120, which I fet down in the ſame
Row to the left Hand of 5.
the left Hand of the
After the fame Man-
Which being done, all the Numbers which are to
vertical Line from 1 to 120 are the Diviſors of 120.
ner all the Diviſors of any other Number may be found by dividing it by
its leaſt Divifor that exceeds Unit, and the Quotient again by its leaft
Divifor, and fo on, till you have a Quotient that is not divifible by any
Number greater than Unit. This Quotient, with theſe Divifors, will
be the firft or fimple Diviſors of the Number, and the Products of the
Multiplication of any 2, 3, 4, &c. of them will be the compound Divifors.
SPECIOUS ARITHMETICK.
239
XXVIII.
Let it now be propoſed to find the commenſurable Roots of the Equa-
x5
12.x + 5x3 ·61 x² + 22 X · 120 = 0.
tion.
1. | 165 | I
3. 5. 11. 15 · 33 · 55 · 165
O 120 I
•
•
1+ 3-11
12
2.3.4.5.6.8.10.12.15.20. 24. 30.40. 60. 120 +2
I 221 I 13.27.221
+13
example of
finding the
of an equa-
Having fet down in a firft vertical Column 1, 0, 1 for the Values
to be fucceffively given to x; and in another vertical Column the Num- Another
bers 165, 120, 221 refulting from the Subſtitution of thoſe Values in the
Quantity x5 12 x4 + 5 ×
61 x² + 22 x 120. I place in a third commenfu-
Column the Divifors of thofe three Numbers, each oppofite the Num-
ber that produced them. This being done, I examine firft if among tion by the
the Divifors of 120 the Number 2 has the required Conditions, and I foregoing
perceive that when taken with the Signit agrees with the Numbers method,
and I above and below it. I therefore fet down in the fourth vertical
Column +3, +2, + L. But the fame Number taken with theSign —
does not fucceed, as 3 is not found below it.
3
Examining in like Manner all the other Divifors of 120, I obſerve,
that the Number 12 taken with the Sign has the required Conditions,
provided the Numbers II and 13 that are above and below it are alfo
taken with the Sign —, I therefore fet down in a fifth vertical Column
the three Numbers
1.2.2 13.
[I,
To diſcover now by which of the two Quantities
2
+ 2 or x 12
the Divifion is to be tried, I obferve, that if it was by the firft,
ſubſtituted for x in the Equation fhould make it vaniſh, and as this does
not happen, I conclude that the Diviſion by x 12 is the only one to
be tried, which fucceeds and gives x45x2-x+ 10 for Quotient,
whence 12 is one of the Values of x in the given Equation, and the
only commenfurable one.
XXIX.
Let it be propoſed to find the commenfurable Roots of the Equation.
x6
4x5 +34
4x5+3
2
74
I 30
2
О
•12x3.. 5 x² + 11 x + 36· =0
1. 2. 3. 5. 10. 15 . 30
036 I 2.3.4.6.9. 12. 18.
I
•
+ 3
36
+2
3
+ I
· 4·
20.40
2.4.5.8.10
•
2
ара
Third
plication of
the method
of finding
the com-
3+10 menfurable
+9
51+
I to
I 40
Having fet down in a firſt vertical Column the Values 1, 0,
be given to x; in a fecond, the Numbers 30, 36, 40, into which the
propofed Quantity is transformed by thefe Suppofitions, and in a third,
all the Divifors 1, 2, 3, 5, 6, 10, 15, 30 of the Number 30; the Di-
roots in an
equation.
240
OF
ELEMENTS
reduced ra-
viſors 1, 2, 3, 4, 6, 9, 12, 18, 36 of the Number 36, the Divifors 1, 2,
4, 5, 8, 10, 20, 40 of the Number 40; I find among all thofe Divifors
four Columns that have the required Conditions, the firft + 3, + 2,
+ 1, the fecond — 2,
3, -4,
4,5, the fourth
+ 10, +9, + 8.
3,5, the third
I
To difcover which of thofe Progreflions are to be rejected, I first put
X 2, and ſet down 2 above 1 in the first Column, I afterwards fet
down in the fecond Column 74, the Number into which the propoſed
Quantity is transformed by fuppofing x 2. I now perceive without
taking the Trouble of inveftigating the Divifors of this Number, that
the two Columns + 3, +2, + 1, and + 10, + 9, + 8, are to be
rejected, for if the firft could be admitted, we ſhould find + 4 among
the Divifors of 74, and if the fecond could be admitted, we ſhould find
+11 among the fame Divifors.
But I perceive, that the Numbers
I and 2, that the Columns
29 3,
4, and
3, 4, 5 require, are Divifors of 74. I
therefore fet down the Numbers
I and 2 above
2 and
3
in the ſecond and third Colums, I afterwards try to exclude one of theſe
219 3,-4, 5; which
is eafily effected, fince if the first was to be admitted, the Equation
thould be reduced to o by putting x 3, which not happening, there
remains only the Column I, 2, 3, 4, 5 to be tried, that
is, the only Divifor to be tried is x - 4, which fucceeds, and gives for
Quotient x5 + 3 x3 - 5x 9. whence +4 is one of the Values of
x in the propoſed Equation, and the only commenſurable one.
two Columns I, 2, 3, -4; and
XXX.
Advantage When an Equation of the fourth Degree has been found to have its
of invefti- four Roots real, before its reduced Equation is attempted to be folved by
gating the
commenfu- Approximation, in order to obtain the Value of x to ſubſtitute it in that of z
rable divi- (Art. CL. Chap. III.); it will be proper to examine by the foregoing Method
fors in the whether any of its Roots are commenfurable if it has but one commen-
ther than in furable Root, the other thre cannot be found otherwife than by Approxi-
the propof- mation, viz. by folving an Equation of the third Degree, whoſe three
ed equation Roots are real; but if an Equation of the fourth Degree has two com-
fourth de- menfurable Roots, its Refolution may be compleatly effected, fince when
theſe two Roots are found, there will only remain to be folved an Equa-
tion of the fecond Degree.
of the
gree.
XXXI.
When an Equation of the fourth Degree has two commenfurable Roots.
it will be eaſier to find them by Means of the reduced Equation than
by itself; for it is manifeft that in this Cafe, in the four Values of z
repreſented in general by i+k; i k ; — i + 1 ; —il; i can only
be a commenfurable Quantity, and confequently in the Root xx
4ii
SPECIOUS ARITHMETICK.
241
of the reduced Equation, 4 ii will reprefent a commenfurable and Square
Quantity, whence we are to felect out of the Divifors of the laſt Term,
only ſquare Quantities, affected with the Sign-
Likewife if the Equation zª +p z x + q x + r = o can be refolved
into two Equations of the fecond Degree the Coeficients of whofe Terms
are commenfurable Quantities, we are to inveſtigate the Divifors of the
reduced Equation, and felect out of the Divifors of the laſt Term only
fquare Quantities, and affected with the Sign.
XXXII.
Tho an Equation of the fourth Degree has not two commenfurable How to dif-
Roots, or is not refolvable into two Equations of the fecond Degree, equations
tinguish the
the Coeficients of whoſe Terms are commenfurable Quantities, it may of the
be refolvable into two Equations of the fecond Degree, the Coeficients of fourth de-
whoſe Terms are ſquare Radicals, in this Caſe the Part+x of the roots are
I
I
9
Expreſſion ± ÷ +√ (− − × × — — Þ + 2) of the Value
+ *+
2
XX
2x
4
of z, or which is the fame, the Parti common to the four Roots
i + k, i k, · i + 1, i 1, ſhould be a fimple Radical
of the fecond Degree, becauſe in the reduced Equation x is every
where of an even Number of Dimenfions, conſequently in all Equations
of this Nature, their reduced Equation is diviſible by xxa commen-
furable Quantity, and of courſe Equations of the fourth Degree are
refolvable by the foregoing Method, when their reduced Equation has
Divifors of this Form.
gree whofe
only affect-
ed by
fquare ra-
dicals.
XXXIII.
illuftrated
Let, for Example, the Equation of Art. CLI. Chap. III. + 16 u³ The fore-
214
+ 99 u² + 228 u + 144 = o be propofed to be folved, to diſcover going ob-
whether it has two rational Roots, I fele&t out of the Diviſors of the laſt fervations
Term 2704 of its reduced Equation, x6 +6 x 4 183x² - 27040, by an ex-
thoſe that are Squares, and that are affected with the Sign and I find ample.
that this reduced Equation has for Divifor x x 160, whence the
Roots of the propofed Equation are
the above mentioned
Article.
found as in
2
62
6 ~2
In like Manner, if the Equation of Art. CLIV. Chap. III. 24
+8% −1 =o was propofed to be folved, it would appear by this
Method that its reduced Equation x6 - 12x4 + 40 x 2 640, has
for Divifor xx 8, whence are obtained the Roots of the propoſed
Equation.
To fhow the Uſe of the foregoing Rules in the Solution of Problems,
here follow fome Examples:
4 H
242
ELEMENTS OF
XXXIV.
Two Maſons A and B jointly perform a Piece of Work in 12 Days; now, if
the Sum of the Days in which they could each have ſeparately performed the
fame, be multiplied by the Days in which A alone (he working quicker than B)
could have done it, the Product will be 1000, in what Time could each do
it ?
Suppoſe A could do it alone in x Days, and B in y Days, then
the Work done by A, and y: 1 = 12: = the
x: I = 12 :
12
Another
problem
folved by
the forego-
ing me-
thod.
12
X
12
Work done by B, in 12 Days: then +
or 12 x = xy·
12y, then y
, confequently
x -12
y
12x
x
12
y
= 1, or 12 x + 12y=xy,
12XX
= xy,
Third pro-
blem folved
by the fore-
going me-
thod.
but (x + y) × x = xx + xy
or x 3
=
1000, that is, xx +
x—12000, then x 3
X - 12
12xx
X - 12
= 1000,
1000 x+12000=0.
−12xx +12 xx=1000
Now, by the foregoing Method, there will be found the Divifors 21, 20,
19, differing by Unity, but x 20 will divide the Equation, wherefore
x20, and y = 30.
XXXV.
A Mafon had two cubical Pieces of Marble, containing between them 1072
folid Inches, and being feverally placed in his Yard, they stand upon 130 Square
Inches of the Ground, what are the Sides of thofe Cubes?
then x 3
and x
2
2
xxyy
Suppoſe x the Side of the greater, and y the Side of the leſſer Cube,
+ y³
3 1072; or x3 = 1072 y³ and x x + y y = 130,
130
yy; but x6 =
hence y6 195 14 1072 y3
foregoing Method, there will be
differing by Unity, but y
y = 7 and x = 9.
=
24
= (1072 3) (130-yy)3;
+25350 y 523908 = 0; and by the
found among the Divifors 9, 8, 7, 6, 5,
7 will divide the Equation, wherefore
XXXVI.
If the propofed Equation has no fimple Divifor, then we are to enquire
if it has not ſome quadratick Divifor, (if itſelf is an Equation of more
than three Dimenſions) the Solution of Equations of the fecond Degree
being as compleat as that of the firft. How the Analyſts have proceed-
Inveftigati- ed in this Reſearch we fhall therefore continue to explain.
on of the
method of
quadratick
commenfu-
Let x x + bx + co exprefs the Equation of the fecond Degree
finding the that may be one of the Factors of a given Equation, or what amounts
to the fame, let x x + bx + c be the Divifor ſought of a given Quan-
rable fac- tity, by putting xo in this Divifor, it is manifeft, that it will be re-
tors of any duced to the Number c, and that this Number will be one of the Di-
propofed
viſors of the laſt Term of the given Quantity.
equation.
SPECIOUS
243
ARITHMETICK.
2
2
If we afterwards put x = +I in the Divifor x² + b + xc, it will
be transformed into I + b + c, which will be one of the Divifors of the
Number that refults, by putting likewife in the propofed Quantity
x = 1. Wherefore, if all the Divifors of this Number be inveſtigated,
and Unity be deducted from each of them affe&ted as well with the Sign
as with the Sign, among the Numbers poſitive and negative, re-
fulting from this Operation, the Number expreffing b + c will be
found.
If laftly we put x1, as well in the propofed Quantity as in the
Divifor xx + bx + c, which then becomes I b+c, it is eaſy to
perceive that if all the Diviſors of the Number that reſults be inveſti-
gated, and Unity be deducted from each of them affected as well with
the Sign+as with the Sign, among the Numbers pofitive and ne-
gative reſulting from this Operation, the Number expreffing — b + c
will be found.
Now, as c is an arithmetical Mean between b + c and —b + c, it fol-
lows that out of the three Series of Numbers expreffing b+c, c, − b + c
thoſe only that are in arithmetical Progreffion are to be felected. When
three Numbers in arithmetical Progreffion are found, it is manifeſt the
one correſponding to the Suppofition of x = o is to be affumed for c,
and as the Number that correfponds to the Suppofition of x = 1,
will reprefent b+c, it is manifeft, that by deducting the first from the
fecond, the Quantity b will be obtained; fubftituting afterwards thofe
two Numbers in the Room of b and c in x² + bx + c, there will re-
fult a Divifor for the propofed Quantity to be tried, and the only one
to be tried, if but one arithmetical Progreffion has been found among all
the Series of Numbers furniſhed by the Divifors of the Quantity in which
x has been made fucceffively = 1, 0, - 1. If feveral arithmetical Pro-
greffions have been found, thofe that are to be rejected will be found
much in the fame Manner as in the Caſe of fimple Diviſors, by making
new Suppofitions for x, as - 2, 3, or + 2, + 3, &c.
2, or
For if in the propofed Quantity we put, for Example, x=-2, x=-3,
&c. it is manifeft that all the Divifors, as well pofitive as negative of
the Number that refults, will be expreffed by 4-2b+c, or 9-36+
c, &c. into which x² + bx + c is transformed, when x =
x=3, &c. and deducting from all thofe fame Divifors 4, 9, &c.
the remainders will be expreffed by 2 b+c, — 3 b + c, &c. Now,
26+c,
3b+c, &c. being Terms of the arithmetical Pro-
greffion b+c, c, bc, no more remains to be done than to ſearch
among the Progreffions already found, for one that is not altered by
thefe new Values of x, which will ferve to determine the Numbers
b and c.
+
244
ELEMENTS OF
| 1
→ O
XXXVII.
Let it be propoſed, for Example, to find a Diviſor of two Dimenſions
of the Quantity
r
I
21
I. 3.7.21
65
1. 5. 13.65
2
125
1. 5. 25. 125
51. 3. 7.21.2
3
୨
1472 49. 147
Application
of the fore-
going me-
thod to an
example.
!
2
x5 + 3x4 + 2 x3 + 8 x² - 36 x + 21.
2, O
-21, -7, -3, −1, 1, 3, 7, 21
-66,-14, -6, -2, 0, 4, 12, 64
4-129,-29,−9,−5,
2
-24
1
3
4 14
3, 1, 21, 121
I
—156, —58, —30, —16,—12, —10,2
—8, -6, -2, 12, 40, 138
12
I that are
I first fet down in a vertical Column the Values 1, 0,
propoſed to be given to x, I ſet down afterwards in a ſecond vertical Co-
lumn the Numbers 1, 21, 65, into which the propofed Quantity is tranf-
formed by thofe Suppofitions.
I fet down in like Manner in a third Column, befide the firft Num-
ber, its only Divifor I; befide 21 its Divifors 1, 3, 7, 21; and beſide
65 its Divifors 1, 5, 13, 65.
This being done, I let down in a fourth Column the Numbers I, 0, 1,
Squares of 1, 0, I wrote in the firſt Column, and confequently the
Values of xx in the fame Suppofitions of 1, 0, 1. Laſtly, I form a
fifth Column from thofe Conditions:
1º. That the firft Line may be
taken with the Sign and with the
—
2, 0, found by dedu&ting I from I
Sign +.
2º. That the fecond Line may be compofed of the Numbers 21,
7, 39
I, + I, +.7, +21, the fame as the Diviſors which are
fet down befide them, but taken with the Sign and with the Sign +.
3°. That the Numbers of the third Line may be 66, 14, 6,
2
2, 0, + 4, + 12, +64, the firſt of which 66, 14, — 6,
found by deducting I from the Numbers 65, 13, 5, I affected with the
Sign, and the others o, +4, +12, +64 found by deducting I from
the fame Numbers 1, 5, 13, 65 affected with the Sign +.
To determine afterwards the arithmetical Progreffions contained in
thoſe three feries of Numbers. I begin by taking in the firft Line the
Number 2 for the first Term of a Progreffion, I take fucceffively for
fecond Terms all thofe of the fecond Line; and feek at the fame Time
the third Terms which theſe two firſt would require, and I examine
which of thoſe third Terms are found in the third Line: Now,
2 and
21 fhould give for third Term 40, which is not found in the third
Line, I therefore reject 21, and next affume 7 for fecond Term, and
12 for third Term, and that 12 is not found in
as, it ſhould give
the third Line, I reject alfo — 7, as likewife-3, becauſe this laſt ſhould
give 4, which is not found in the third Line.
SPECIOUS ARITHMETICK.
245
As to 1, fince it gives o for third Term, and o is found in the third
Line, I fet down in a fixth Column the Progreffion 2, I, O, like-
wife taken for fecond Term, giving +4 which is alfo found in the
third Line, I fet down the ſecond Progreffion-2, +1, +4, and as
+7 and + 21 fhould give each a third Term, which is not found in the
third Line, I reject them. Having determined the Progreffions that can
begin by 2, I paſs to thoſe whoſe firſt Term will be o, and to find them
I take, as I did for 2, all the Numbers of the fecond Line one after the
other.
I perceive firſt, that
21 fhould give for third Term
6 and
7,
14 which
14;
42, which is
not found in the third Line; I next perceive that 7 gives
is found there, I therefore fet down the Progreffion 0,
likewife
third Line; I fet down the Progreffions 0,
As to the Numbers + 1, +7, +21, becauſe the third Terms they give
are not found in the third Line, I reje&t them.
3 and - 1, giving
2, which are alfo found in the
3, 6, and o, I, 2.
To diſcover now which of thofe Progreffions are to be rejected, I put
x = — 2, and ſet down 2 in the firft Column; obferving at the fame
Time to place, 1º. In the fecond Column, 125 Number into which the
propoſed Quantity is transformed, by fubftituting in it this Value of x.
2º. In the third, the Numbers 1, 5, 25, 125, Divifors of 125. 3º. In
the fourth Column, the Number 4 ſquare of 2, and Value of xx.in
this Suppofition. 4°. In the fifth Column, the Numbers 129, 29,
9, 5, found by deducting 4 from the Numbers 125, 25, 5, 1, taken
with the Sign, and the Numbers 3, +1, +21, +121, found by
deducting 4 from the fame Numbers 1, 5, 25, 125, taken with the Sign +.
By this Means, I perceive that the two Progreffions - 2, + 1, + 4,
and 0,7,14, are to be rejected, becauſe they ſhould give for fourth
Term +7 and 21 which are not found in the fourth Line.
r
2,
But the Progreffions 2, I, 0; 0, — 3, 6; 0, I,
giving for fourth Terms +1, -9, -3, which are found in this fourth
Line, to exclude at leaſt one of thoſe Progreffions I have need of a new
Suppofition. I therefore put x=-3, whereby the propofed Quantity
is transformed into 147. I therefore fet down 147 in the fecond Column,
and befide it, in the third, its Diviſors 1, 3, 7, 21, 49, 147; I likewiſe
fet down in the fourth Column 9 Square of 3, or Value of x x in the
new Suppofition; laftly, I fet down in the fifth Column the Numbers
156, - 58, 30, 16, 12, 10, 8,
6,
+ 12, + 40, +138, found by deducting 9 from the Numbers
1, 3, 7, 21, 49, 147, taken with the Sign and with the Sign +.
By this Means, I find that the Progreffions
0,
I,
21
2, I, 0, + 1;
219 3, fhould be rejected, but that the Progreffion
·☺, — 3, — 6, —9, fhould be retained; becauſe the two fifth Terms of the
246
OF
ELEMENTS
firſt Progreffions ſhould be +2 and
4, which are not found in the
fifth Line, whilſt the fifth Term of the Progreffion 0, — 3, -6, — 9
12 which is found there.
is
After having reduced all the Progreffions to one, viz. o, — 3, — 6, &c.
I take in this Progreffion the Number 3, which, in the fixth Co-
lumn correſponds to the Suppofition of x = 0, to exprefs the Term cof
the Divifor fought x x + bx + c. I afterwards take in the fixth Column
likewife, o which correfponds to the Suppofition of x 1, and which
according to the foregoing Principles fhould exprefs b+c; whence de-
ducting c or 3 from o, or from b+c, I find + 3 for b, and confe-
quently the Divifor fought x² + bx + c is xx+3x-3, if the pro-
pofed Quantity has one of two Dimenſions; I therefore try the Diviſion of
this Quantity x5 + 3x² + 2x² +8x² - 36 x + 21 by xx+3x-32
which fucceeds, and gives x3 + 5x7 for Quotient.
XXXVIII.
Let it be propoſed to find a Diviſor of two Dimenſions of the Quantity
x4 + 6 x³ + 11 x² + 10 x + 5.
21331.7.19.1334-137,-23,-11,-5,-3₂+3,
Ι
Ι
1 33 1.3.II. 33
о
I
511.5
in m
II
+35,+1291+151+:
I-34,12,-4,-2, 0, +2, +10, +32 +10 +2
05, 1, +1, +5
-2, 0-
+51+1
O
311.3
4
·7,523, — I
Application
going me-
nother ex-
ample.
-
-
I fet down in a firft vertical Column the Values 2, 1, 0, I, 21
of the fore- which are propoſed to be given to x; I afterwards fet down in the ſecond
thod to a vertical Column the Numbers 133, 33, 5, 1, 3, into which the propoſed
Quantity is transformed by thofe Suppofitions; I fet down in the third
Column oppofite to, thofe Numbers all their Divifors. In the fourth the
Values 4, 1, 0, 1, 4 of xx, reſulting from the Suppofitions made for x
in the firſt Column; laſtly, in the fifth Column I fet down, for the firſt
Line, the Numbers 137,-23, II, 5, 3, +3, +15, +129
found by dedu&ting 4 from the Numbers 133, 19, 7, 1, taken firft with
the Sign and afterwards with the Sign +. Likewife in the fecond Line
the Numbers 34, 12,
-2, 0, +2, + 10, + 32 produced by
deducting I from the Numbers 33, II, 3, 1, taken firft with the Sign
and afterwards with the Sign +. I proceed in like Manner to find the
other Lines. All thoſe Numbers being fet down, I take in the fifth Line
of the fifth Column, the firft Number-7 for firft Term of a Progref-
ſion, and taking at the fame Time 2 in the fourth Line for fecond
Term, I perceive that the third fhonld be + 3, and is not found in the
third Line; I therefore paſs to o, which by affuming- 7 for firſt Term,
will give + 7 for third Term, and as + 7 is not found in the third Line,
4,
SPECIOUS ARITHMETICK.
247
I conclude that among the Numbers in the fifth Column, there is no Pro-
greffion that can begin by 7.
2
I therefore take — 5 for firft Term, and I perceive that affuming
for fecond Term, the third will be + 1 which is found in the third
Line, but gives for fourth Term +4, which is not found in the ſecond
Line; I therefore reject 2, and affume o for fecond Term, and there
reſults +5, 10, 15, for third, fourth, and fifth Terms; and as
all theſe Numbers are found in the third, fecond, and firft Lines, I fet
down in the fixth Column the Numbers 15, 10, 5, 0, ·5.
3,
3
for firft Term, I perceive that neither
2,
2
Afterwards taking
nor o can be affumed for fecond Terms, becauſe the firft would produce
the Progreffions
I, 0, +1, the laſt Term of which is
not found in the firft Line; and the fecond would produce the Pro-
greffion-3, 0, +3, &c. which fails counting from the third Term
+3 which is not found in the third Line.
It remains to take 1 for firft Term, I firſt give it 2 for fecond
Term which does not fucceed, but affuming o for fecond Term, there
refult for third, fourth, and fifth Terms, the Numbers + 1, +2, +3
which are in the third, fecond, and firft Lines; I therefore fet down in
fixth Column the Numbers + 3, + 2, + 1, 0, -
I.
This being done, I perceive that there is no further Occafion of giving
x new Values to exclude one of the two Progreffions, becauſe the given
Quantity being of four Dimenſions, muſt neceffarily have two Divifors
of two Dimenſions, or none at all; for when a Quantity of four Dimen-
ſions has been found to have a Diviſor of two Dimenſions, the Quotient
which is likewiſe a Divifor, will be alfo of two Dimenſions.
According to this Obfervation, I take indifferently either of the two
foregoing Progreffions, the firſt, for example, in which 5 correſponding
to the Suppofition of x = 0, and 10 to the Suppofition of x = 1, there
reſults c = = 6
5 and b + c = 10, that is, b = 5. Whence the Divifor fur-
niſhed by this Progreffion is xx+5x+5. I divide therefore the pro-
pofed Quantity by xx + 5 x + 5, and the Divifion fucceeding, gives
for Quotient x x + x + 1, which is the Divifor that the other Prc-
greffion would have furnished.
2
XXXIX.
finding the
If it be propoſed to find the Divifors of an Equation, fuch as
б p+ — µ.3 21 µ² +39 + 200, whofe firft Term has a Co-
eficient different from Unity, they may be obtained by employing the Method of
foregoing Principles without being at the Trouble of transforming it into divifors of
one that ſhall have the Coeficient of the firſt term Unit, as may be effec-
ted by the Method of Art. XXII.
To make this appear, we ſhall firſt examine what concerns Diviſors of
one Dimenſion. Let mx + a reprefent the fimple Divifor of any pro- equation
one dimen-
fion when
the higheſt
term of the
248
ELEMENTS OF
cient diffe-
rent from
unity.
I,
has a coefi- pofed Equation. It is manifeft, that if x be put fucceffively equal to
2, I, 0,
2, &c. This Divifor will beocme in all thofe Cafes
2 m + a, m + a, a,
2 m
m + a,
--a, &c. Quantities that are
in arithmetical Progreffion in which it is eaſy to obſerve,
1º. That the Difference m of all the Terms is the Coeficient of x in.
the Divifor.
2º. That this fame Difference m is a Divifor of the Coeficient of the
firſt Term of the propofed Quantity.
3º. That the Term a correfponding to the Suppofition of x = 0, is
the Part of the Divifor clear of x.
4°. That the fame Quantities in arithmetical Progreffion will be Di-
viſors of the given Quantity, in which x has been fucceffively put equal
to 2, 1, 0, Į, 2, &c.
From which it follows, that in Order to diſcover ſuch a fimple Di-
vifor when there is any, the fame Proceſs is to be purſued as above for
the three firſt Columns, Then among all thoſe Numbers a Progreffion
is to be looked out for, whofe common Difference is a Divifor of the
Coeficient of the first Term of the propoſed Quantity. Laſtly, to em-
ploy this Progreffion, in the Room of a in m xa the Term of the
Progreffion correfponding to the Suppofition of xo is to be ſubſti-
tuted, and in the Room of m the Number found by deducting any Term
of the Progreffion from that immediately above it.
XL.
Let it be required to find, for Example, the Divifors of the Quan-
tity
*3
21 x² + 3 x + 20
2 . 3. 5. 6. 10 . 15 · 30
I. 7
6 x4
2
30
I
I
O 20
I
3
I
•
3.
2 34
I
.
2.17.34
I 2º 4. 5. 10. 20
•
ΙΟ
7
4
I
2
I
2
range as ufual in the firft Column the Suppofitions 2, 1, 0, —1,
to be made for x, in the fecond, the Numbers 30, 7, 20, 3, 34 into
which the propofed Quantity is transformed by thofe Suppofitions.
Laftly, in the third all the Divifors of thofe Numbers.
This being done, in order to difcover among all the Numbers of the
third Column fome Progreffion that will ferve to find the Divifor fought;
I try the firſt Number 1 of the firſt Line, and I perceive that affuming
for fecond Term the firft Number I of the fecond Line, the Progreffion
I, I, I, &c. that refults cannot be admitted, becauſe it cannot repreſent
the Divifor fought m x + a which fhould neceffarily vary according to
the different Values given to x. I perceive likewife, that 7 cannot be
SPECIOUS ARITHMETICK.
249
affumed for fecond Term, becaufe the third Term produced by this
Suppofition would be 13, which is not found in the third Line.
Taking afterwards I with the Sign, I perceive that the I of the
fecond Line cannot be affumed for fecond Term, becauſe the third would
be 3, which is not found in the third Line. As to 7, it is ufeleſs to try
whether the third Term it gives is found in the third Line, fince the
Difference of I, to 7 is 8, which is not a Divifor of the Coeficient of
the firſt Term of the propoſed Quantity. Wherefore I whether taken with
the Sign, or with the Sign, is to be rejected.
Taking in like Manner all the other Numbers of the firſt Line, I find
only 10 that can have the required Conditions. Affuming 7 for fecond
Term, it gives the Progreffion 10, 7, 4, 1, 2 whofe common Diffe-
rence is 3, Diviſor of the Coeficient of 6 x³.
Setting down therefore this Progreffion in the fourth Column, I take
the Term + 4 correfponding to the Suppofition of xo to expreſs the
Part a of the Divifor fought, and deducting the fame Term +- 4 from the
Term above it +7 which repreſents m + a, I find 3 for the Expreffion
of m, that is, the Divifor fought if there is one, muſt be 3 ≈ + 4. I
therefore try the Divifion by this Quantity, which fucceeds, and gives
3x + 5 for Quotient.
2 x3
3 x
2
In like Manner, if it was required to find the Divifors of the Quantity
2 4 x5 - 50 x 49x3 140x² + 64≈ + 30
2
I 23
2
42
•
I. 2 3. 6.
7. 14 · 21 · 42
I
3
•
30
297
•
I
1 . 2 . 3 . 5 . 6 . 10 . 15. 30
•
I 3. 9. II. 27 · 33 · 99 297
Here the Divifors to be tried are 2 x
I, 4 x
+3+3
+ I
I
+7
+ I
I
5
5
ΙΙ
3
9
5 and 6 x — 5; of
which 6x-5 fucceeds, there coming out 4 x4-5 x3 + 4x²
XLI.
20x—6.
To fhew the Uſe of the foregoing Rules in the Solution of Problems, Problem
here follow fome Examples:
24
folved by
going rules.
A Mafon had 2 cubical Pieces of Marble, the Side of the one exceeded the the fore-
Side of the other by 7 Inches, and the folid Inches in both made 873; what
were their Dimenſions?
If the Side of the leffer Cube be expreffed by x, then the Side of the
greater will be x + 7; and x ³ + (x + 7) ³ = 873 by Queft.
3 3
Hence 2 x3 + 21 x² + 147 x 5300: Now, by the foregoing
Method I find the Progreffion 7, 5, 3, whofe common Difference is 2 a
Divifor of the Coeficient of 2 x³, and taking the Term 5 correfpond-
ing to the Suppofition of x = 0, to exprefs a and 2 for m, I find that
2 x3 + 21 x +147x 530 is divifible by 2x-5, giving for Quo-
tient x² + 13x + 106, whence 2 x = 5, and x
24
2
5
2
250
ELEMENTS OF
Another
problem
folved by
the fore-
going rules.
XLII.
There are two Numbers, the Difference of whofe Cubes is 604, and if
their Difference be multiplied by the greater, the Product will be 36: What
are thofe Numbers ?
If x the greater and y the leffer of theſe Numbers; confequently
(x — y') × x = 36, or x — y = 36
y3604, by Queſt. and
Alfo x3
x
dividing the fecond Equation by the firſt, x² + xy + y²=
604x151 x
36
9
alfo by ſecond x
Difference of the two laft is 3
2
2 x y +.y
2
36 × 36-
1296x
The
XX
XX
X
151* -
y =
9
1296
therefore
x x
Method of
two dimen-
11664
151-271
y3604, whence 27 y4-151
3 +16308y79540=0.
Now by the foregoing Method there will be found the Progreffion 7, 6,
5, 4, 3, whofe common Difference is 1 a Divifor of 27 the Coeficient of
4
y the higheſt Power of y, and taking the Term 5 correfponding to
the Suppofition of yo to exprefs a, and 1 for m, I find that
27 y 151y3 +16308y79540 o is divifible by y — 5, giving
for Quotient 27 y316y280y+15908; Wherefore y 5.
=
4
—
XLIII.
We ſhall now procceed to explain how Divifors of two Dimenſions,
finding the when the higheſt Power of x has a Coeficient, are inveſtigated. Let
divifors of m x² + bx + c reprefent the Divifor fought of a given Quantity, it
Aons when is manifeft, that the laſt Term c will be a Divifor of the laft Term of
the given Quantity, and that m will be a Divifor of the Coeficient of the
higheſt Power of x in the fame Quantity.
* has a
coeficient.
Affuming therefore one of the Divifors of the Coeficient of the firſt
Term of the propofed Quantity to exprefsm, and performing the fame
Operation as in Art. XXXVI. with this difference, that deducting instead of
the Squares 16, 9, 4, &c. the Product of thefe Squares into the Num-
ber taken for m, in like Manner b and c will be found.
2
If the Divifor that refults, by fubftituting in m x² + bx + c, for m
the affumed Number, and for b and c thofe that have been determined in
confequence of this Affumption, does not fucceed, another of the Di-
vifors of the Cocficient of the firft Term of the given Quantity is to
be affumed to exprefs m, and the Calculation compleated as before. "If
after having tried all the Divifors of the Coeficient of the firſt Term,
no Divifor is found by this Method, you may conclude that the propoſed
Quantity has no commenfurable Divifor.
1
SPECIOUS ARITHMETICK.
251
XLIV.
Suppofe, for Example, you are to find a Divifor of two Dimenfions.
of the Quantity
4 x5 + 16 x4 — 22 x³ — 14 x² — 56 x + 77
2 117 1.3.9.13.39.1178-125-47,-21,-17,-11,-9,-7,-5,1,5,31,109 |+ 5
I
51.5
0 77 17.11.77
2-7,3,1,3
0-77,-11,—7,—1,I,7,11,77
-155—53,—19,—11,—5,—3,—1,1,7,15,49,151
-1 153 1.3.9.17,51.1532
1
-2 437,1.19.23.437 8-445-31—27,−9,−7,11,15,429
I,
3
II
-19
-27
method to
Having placed, as ufual, in the first Column the Numbers 2, 1, 0, Application.
-1, -2, &c. to which x is fuppofed fucceffively equal; in the fe- of this
cond, the Numbers 117, 5, 77, 153, 437 into which the propofed Quan- an ex-
tity is transformed by fubftituting in it thofe Values of x; in the third, all ample.
the Diviſors of the Numbers in the fecond: I firſt try if it has a Diviſor
of two Dimenſions, the Coeficient of whofe firft Term is Unit, and not
finding any, I affume for m, that is, for the Coeficient of the firſt Term
of the Divifor, the Number 2 which is one of the Divifors of the Coe-
ficient 4 of the firft Term of the given Quantity.
I then fet down in the fourth Column, inſtead of the Squares of the
Numbers of the first, the Product of thoſe fame Squares into 2 fuppoſed.
Value of m, that is, I fet down in the fourth Column the Numbers 8, 2,
o, 2, 8. I afterwards deduct theſe Numbers from all thofe of the third
Column taken with the Sign and with the Sign +, which gives for
the first Line of the fifth Column 125, - 47, 21, 17, II,
- 9, - 7, − 5, + 1, + 5, + 31, +- 109; for the fecond Line
3, &c.
2
-
7
II coг-
All thoſe Numbers being fet down, I feek for arithmetical Progreffions
among thoſe Numbers, and I find but the Progreffion +5, — 3, — II,
- 19, 27 which I fet down in the fixth Column; now taking
refponding to the Suppofition of x = 0, to exprefs. the Number c, and
deducting this Number from 3 which is above it, I find the Remain-
der 8 for to exprefs b. The Divifor therefore that refults from the Sup-
poſition of m = 2 is 2 x² + 8 x II; and trying the Divifion it fuc-
ceeds, giving for Quotient 2 x3-7, and without taking the Trouble
of going thro' the Calculation that the Suppofition of m = 4 would re- Every
quire, I perceive that it would not fucceed, becauſe then the Quotient quantity of
7 could be refolved into Factors of two Dimenſions, which is dimenfions
impoffible. Whence the propofed Quantity has no other Divifor of that has di-
two Dimenfions but 2x² + 8 x
When the Quantity whofe Divifors are to be inveftigated does not ex- of a lefs
ceed the fifth Degree, they may be found by the foregoing Methods; number of
for if this Quantity is found by thofe Methods to have no fimple or qua- than three.
dratick Divifor, we may alfo conclude that it has no cubick Divifor.
2x3
· II.
lefs than fix
vifors muft,
have fome
dimenfions
252
ELEMENTS OF
If the quan-
XLV.
But if the Quantity rifes to fix or more Dimenſions, it may poffibly
tity has fix not be refolvable but into Factors of more Dimenfions than two, the
menfions it Method to be purfued for finding thofe Divifors is founded upon the fame
Principles as the preceding ones.
or more di-
may have
only divi..
fors of three
1
Thus if it be a cubick Divifor that is required to be found, let it be
, by fuppofing x equal to the Terms of the
or more di- m x 3
menfions.
2
arithmetical Progreffion, it will be as follows:
x=
3
27 m
x= 2
8 m
9 n + 3r
41+2r
5
27 m
9 n
8 m
4 n
3rts
arts
5 n
3 n
X=
I
m
n+ j
m
n
x=
О
о
r+s
+s
れれれ
​r 2n
r 2n
T
2n
n
r
m
12
S
m
n + r + s
Method for
vifois of
three or
more di-
menfions.
Where the firſt Differences are not themſelves in arithmetical Progref-
finding di- fion, as in the laſt Cafe, but the Differences of its Terms, or the ſecond
Differences, are in arithmetical Progreffion, the common Difference
being 2 n, whence n is known. The Quantity r is found in the Column
of the fecond Differences, and s is always to be affumed fome Diviſor of
the laſt Term of the propofed Equation, as m is of the Coeficient of the
firft Term; whence all the Coeficients of a Diviform x3nx²+rx-s,
with which Trial is to be made, may be determined.
If it is a Divifor of four Dimenſions that is required, by proceeding
in like Manner, you will obtain a Series of Differences whofe fecond
Differences are in arithmetical Progreffion. If it is a Divifor of five
Dimenſions that is required, you will obtain in the fame Manner a Pro-
greffion whofe third Differences will be in arithmetical Progreffion, and
by obferving theſe Progreffions, you may diſcover Rules for determining
the Coeficients of the Divifor required.
XLVI.
Hitherto we have only fhewn how to find the Divifors of Numeral
Equations, but as literal Equations may alſo have commenfurable Di-
vifors, we fhall proceed to explain how they are to be inveſtigated.
Suppofe, 1º. That the Equation includes only one known Letter with
x, and that this Equation, to ufe the Language of the Analyfts, is homo-
geneous, that is, has all its Terms raifed to the fame Dimenfion, fuch
for Example, as the Equation x3 ax²-10a2x+6 a³. It fuffices
to fubftitute Unity in the Room of the given Letter a of this Equation,
and find its Divifors by the preceding Rules. Theſe Divifors being
found, if they are of one Dimenfion, the Letter a is to be fubftituted
again befide the Number which ferves for fecond Term. If the Divifor
SPECIOUS ARITHMETICK.
253
2
12 a3 be propofed, by
is of two Dimenfions, a is to be placed after the Coeficient of the fecond
Term, and a a after the Number that ferves for third Term.
Let the Quantity x3 + 4 ax²- 17 a2 x
1, that Quantity becomes x3 + 4x²
3, whence x-3 a is a Divifor of the propofed Quan-
putting a
Divifor is x
tity.
3
2
17 ።
12 whofe
In like Manner if 2 x5 +5 a x4—3 a² x³ — 8 a³ x² - 20 a¹ x + 1245
was propofed, by putting a I, it will become 2 x5 + 5x4 - 3x3
82 20 x + 12, whofe Diviſor of two Dimenfions is found to
be 2x² + 5x-3. Subftituting again in this Divifor a befide 5, and aa
befide 3, there refults 2 x² + 5 a x 3 aa for the Divifor of two
Dimenfions of 2 x5 +5 axt
24
4
2.
3 a²² x 3
XLVI.
2
8 a3 x²
20 a¹ x + 12a5.
a
the divifors
involving
two letters.
The fame Rules may ferve for diſcovering the Divifors in an homo- Method of
geneous Equation involving three Letters, but by Help of fome Obfer- finding all
vations of Calculation which naturally occur, they may be found after
much eaſier Manner.
Suppofe, 1°. That the Quantity given which involves three Letters in a quan-
a, b, x, fhould have for Divifor a Quantity that includes only two ing three.
tity involv
Letters x and a, for Example. Since this Diviſor, be it what it will, tho'
it does not contain b, divides the propoſed Quantity in which benters, the
Value of the Letter b muſt not affect the Divifion, and confequently this
Diviſion may be performed even when bo.
Wherefore if ¿ be put
o in the given Quantity, the Quantity arifing from this Suppofition
muſt have for common Divifor with the whole Quantity, the Divifor
fought, whence by the Rules of Art. LXXIII. Chap. I. the Divifor fought
of whatever Number of Dimenfions it, be provided it contains but two
Letters, may be found.
2
XLVII.
x4
3
a3
To fhew the Application of this Method, let, 1°. The Quantity
x² + a x² + 2 a² ²x² + 3 a³ x + a b b x + a² + a abb be propofed,
of which it is required to find a Divifor involving only the Letters a, x.
By putting bo, there refults + ax³ + 2 a²²x² + 3 a³ x + a4
whofe common Divifor with the entire Quantity, or which is the fame,
with the Remainder a bb x + a a b b is x + a, which is therefore ne-
ceffarily a Diviſor of the given Quantity, and the greateſt it can admit
of that does not contain b.
3
x3
ab
XLVIII.
3
2
Example.
Let x54a x4 +бa ax³-ab x3 + abb x² + 2 a ab x²-4a³ x² Another
— 2 aabb x — 2 a³ b x + 2 a 3 b b, be propofed; by putting bo in Example.
this Quantity, there refults x5 — 4 a x² + 6 a ax³-4a³ x2 whofe
greateſt common Divifor with the propofed Quantity, or which comes to
the fame, with the Remainder ab x3 + a b b x² + 2 a ab x²
2
}
254
Method of
one dimen-
fion involv-
ing three
letters.
a³
ELEMENTS
OF
~2 a abb x — 2 a 3 b x + 2 a3 bb, that is, the greateſt common Di-
viſor of the Quantities x3
24
+ bx² + 2a x 2 a b x
4 a x² + 6 a a x
26 a x + 2 a a b.
4 a³, and x3
Now, if thoſe two
Quantities have a common Divifor that does not contain b, it will a'fo
be common to the two Parts x³ + 2 ax 2 a² x and b x zabx
+2 a2 b of the laft of thofe two Quantities; but the common Divifor
of thoſe two Parts can only be x x 2 a x + 2 a², I try therefore if it
divides alfo x3
2
3
4 a x² + 6 a ax
3
2
2
4 a³, and as it divides it in effect,
I conclude that it is the Divifor fought of the propofed Quantity.
XLIX.
Suppofe now that the propofed Quantity, involving three Letters whofe
finding the Divifors are required has no Divifor, compofed only of two Letters, or
divifors of if it has, that they have been found and fet apart. To find afterwards the
Divifors involving three Letters and of one Dimenfion, that it may con-
tain, I exprefs this Divifor by mx+na+pb, m, n, p denoting Num-
bers, I now obſerve that if a, x, b be put fucceffively o in this Di-
vifor, there will refult the three Quantities m x + pb, na + p b,
mx+na, fuch, that the two Terms that each of them contain is found
repeated in the two other Quantities. mx +pb, for Example, given by
the Suppofition of a = o, is compofed of m. x which is found in mx + n a
arifing from the Suppoſition of b = 0, and. of pb which is found in
na +pb arifing from the Suppofition of x = 0; I perceive at the ſame
Time, that the Sum of theſe three Quantities m x + n a, m z + pb,
na + pb is the double of the Divifor mx + na +p b.
Now, as thofe three Quantities are neceffarliyDivifors of thoſe to which
the propoſed Quantity is reduced by the fame Suppofitions of a, x, b
cqual o, it follows that to find the Diviſors of this Quantity which are
of one Dimenfion and involve three Letters, we must first fet down
ſeparately the three Quantities into which the propoſed one is transform-
ed, by the Suppofition of a, x, bo, then fet down befide each of
thoſe new Quantities all its Divifors involving two Letters of one Di-
menfion. This being done, we muſt fele&t three Diviſors out of thoſe
three Claffes of Divifors involving two Letters, having the Conditions
above-mentioned, viz. that the two Terms of which each of thoſe Di-
viſors confifts, is found in the other two Divifors. Thofe three Divifors
being thus found, the Half of their Sum will be the Divifor of the pro-
pofed Quantity, if it has one.
If it be found neceffary in order to have in one of thoſe three Divifors
of two Letters, the two Terms which ſhould be the Repetition of thoſe
that are in the two others, to change their Signs, it is eafy to perceive
that this Change is admiffable, fince in general a Quantity that divides
another, will fill divide it, when the Signs of all its Terms are changed.
SPECIOUS ARITHMETICK.
255
L.
To fhew the Application of this Method, let it be propofed to find the
Diviſors involving three Letters of the Quantity
2
2 x³ + 7 ax² - 36 x² + 5a² x → 3 ab x + 4 b² x + 10 ab² — 6b3
10 ab²
3
2
·6b3
5a — 3x, 10a 6b
2 x³ — 36 x² + 4 b bx-6b3| 2x-3%
5 a
3 b
2 x
2 x³ + 7 α x² + 5a²x
36
2x + 5a
2x + 5 a
3
-
5 a
5a+2x-36
Application
thod to an
Having fet down in a vertical Column the three Quantities 10 a 62-663; of the fore-
2 x³-3 b x² + 4 b b x - 6b3; 2 x3 + 7 α x2 +5 a2 x, into which going me-
this Quantity is transformed by the Suppofition of x, b
a, equal o ; example,
I fet down in a ſecond vertical Column oppofite each of theſe three
Quantities, their Divifors of one Dimenfion involving two Letters.
The first gives 5 a 3 6 and 10 a 6b; the ſecond only 2 x
·3b,
and the third 2x + 5 a. I now perceive that if of the two Divifors
3 b, 10 a 6 b, the firſt 5a — 3 b be taken; it will have with
the two Divifors 2 x 3b, 2 x + 5a, the Conditions required. For
this first Divifor 5 a-3b contains 5a, which is repeated in the Divifor
2 x + 5 a, and 36 which is repeated in 2x-3b; likewife
b contains 2 x, which is repeated in 2 x + 5 a, and
3 b which is repeated in 5 a 3 b; laftly, 2 x + 5 a is
compoſed of 2 x and of +5a, which are repeated in the two others
5 a — 3b, 2 x — 3 b. I therefore, according to the foregoing Rule,
add theſe three Divifors together, and there refults 4 x 6b+ 10 a,
the one half of which 2 x 365 a is the Divifor fought. In effect
trying the Divifion, I find for Quotient x²+ax + 2 bb.
2 x
3
3b.
LI.
Let it be propoſed to find the Diviſors of one Dimenſion involving
three Letters, of the Quantity
8x42a
2 2
2a x3 —10bx3 — 3a²x
3a²x² — 5 a b x² - 12ab²x +9 a² b²+15ab3
9 a² b² + 15 ab 3
8x4 10 b x3
3 a + 5b, 9 a + 15 b
5b, 8 x 10
3 a, 2 x + a
4 x
8x4 — 2 ax³-3a²x² 4x3
2
—
3 a
56
4x-5 b
4x-5 b
4 x
30 - 56
a 5 b Applica
2
tion of the
method to
> another
Putting in this Quantity x, a, b fucceffivelyo, there refults the three foregoing
Quantities 9 a² b² + 15 ab³, 8x4-10 bx³, 8x4 — 2 ax³ — 3 a²x²,
which I fet down one under the other in a vertical Column. I fet down example.
3,
256
ELEMENTS OF
The fore-
going me-
thod ap-
plied for
finding the
divifors
involving
three let-
ters at the
fame time
as thofe in-
volving
two letters
5 6, 8
in another vertical Column befide each of thefe Quantities, their Divi-
fors of one Dimenfion involving two Letters; thofe of the first are
3 a +5 6 and 9 a 15b; thofe of the fecond 4 ≈
and thofe of the third 4 x
x 10b;
3a and 2x + a.
It is now eafy to perceive that the three Divifors 3 a + 5 b, 4x-56,
and 4x3 a have the Conditions required, provided the Signs of the
firſt be changed, that is, by fetting it down thus -34-5b; I there-
fore fet apart thoſe three Divifors in the fourth Column, I add them to-
gether, and I take the one half of their Sum, which gives 4x-34 - 5 b
for the Divifor fought, and trying the Divifion, it fucce:ds, giving for
exact Quotient 2 x3 + x²-3 ab².
tax
LII.
In thoſe two Examples the Divifors of one Letter of each of the three
Quantities of the firft Column were not fet down, becauſe thoſe Di-
viſors could never be the Quantities into which the Divifor involving
three Letters is transformed by the Suppofition of x, a, b equal o, and
befides it has been ſuppoſed that it has been found by the Method of
Art. XLVI. that the propofed Quantity has no Divifor involving two Let-
ters, but if the propofed Quantity has Divifors of this Sort, they may
be found at the fame Time as thoſe involving three Letters, by the fore-
going Method, provided they be but of one Dimenſion.
Suppoſe, for Example, you are to find the Divifors of the quantity
3
2
16 x³ + 166 x² - 48 ax² + 35 a²x-16 a b x-6 a³ + 3 a² b.
- 6 a³ +3aab
3
16 x² + 166xx
+16b
| a,za,—2a+b,—6a+3b
Sx,2x,4x,8x, 16x,x+b,2x+2b
24x+46,8x+86,16+166
16x³-48axx+35aax-6a³ | a—4x,3a—4x,x— 2α
a² x
a
20
4
za+b
x+b
X-20
4x
a-4x34x
a-4x | 30-4x | x—2a+b
Having fet down in a firſt Column the three Quantities-6a3+3 a² b,
16x3 + 16b x x, 16x3 48a x² + 35 a
6 a³ into which this
ба 3
Quantity is transformed by the Suppofition of x, a, b equal o; I fet down
in the fecond Column, and in the firft Line a, 3 a, 2a + b, 6a+3b
Divifors of one Dimenfion involving one and two Letters of the Quantity
— 6 a³ + 3a² b, likewife I fet down in the fecond Line the Divifors
x, 2 x, 4 x, 8 x, 16 x, x + b, 2 x + 2 b, 4 x + 46, 8x + 8 b₂
b,
16 x + 166, of the fecond Quantity 16 x3 + 16 bx x: and in the third
Line, a
2 4 Diviſors of the third Quantity,
16x3
4x, 3 a
48 a a⋅ x + 35a²
4 x, x
X 603.
Now on account of the great Number of thefe Divifors, in order to
omit none of them that may have the Conditions required, I obferve
:
SPECIOUS ARITHMETICK.
257
much the fame Order as in trying the numerical Divifors. Comparing the
firſt Diviſor of the firft Line with all thoſe of the other Lines, and per-
forming afterwards the fame Operation for each of the other Divifors of
the first Line, I perceive firft, That if a is a Part of a Divifor of the
propofed Quantity, it muſt be a Divifor containing only a and x, be-
cauſe if it contained another Term b, this Divifor could not be reduced.
to a by the Suppofition of x = o. In this Cafe therefore we have only
to chuſe among the five firft Diviſors x, 2x, 4 x, 8 x, 16 x. Now,
as among all thofe Divifors, only 4 x is repeated in the third (provided that
this Diviſor is affected with the Sign), and at the fame Time of all
the Diviſors of the third Line, there is only the first a- 4x which in-
cludes the fame Term a of the firft Line. I conclude that if a be a Part
of a Divifor, this Divifor muſt be a 4 x; I therefore fet it down a
Part. I afterwards pafs to 3 4, and as I find it repeated in the Divifor
34-4x of the third Line, and that the other Term 4 x of the fame
Divifor is found among the Diviſors of the ſecond Line, after changing
the Sign of this Divifor; I conclude that 3 a 4 x may be alſo a Divi-
for of the propofed Quantity, and I fet it apart likewiſe in order to try it.
As to2 ab it cannot alone be a Divifor of the propoſed Quantity,
becauſe then among the Divifors of 16 x³ + 16 b xx the Term b fhould
be found, to which 2 a + b is reduced by the Suppofition of a = 0.
It remains therefore to try if it be not a Part of a Divifor in which x en-
ters; I obferve that of all the Divifors of the ſecond Line, only x + b
can be compared with it, becauſe it is the only Divifor that has the Term
bin common with it. I perceive likewife, that of all the Divifors of the
third Line only x 2 a can be compared with the fame Divifor 2a+b,
becauſe it is the only one that contains the Term 2 a. I obferve,
laftly, that as the two Terms of the Divifor 2a + b are repeated in
the two other Divifors x + b, x 2 a, fo likewife the two Terms of
the Divifor x + b are repeated in the two others 20+ b, x
and reciprocally that the two Terms of the Divifor x-2 a are repeated
in the two others x + b, 2a + b. Whence I conclude that the three
Divifors 2 a + b, x + b, ≈ 2 a, have the required Conditions to
form a Divifor, I therefore add them together, and fet apart the one Half
* 2 a+b of their Sum for a Divifor to be tried. As to the Divi-
for 6 a +36, I perceive at once that neither of its two Terms are repeat-
ed among the Divifors of the other Lines, and confequently it is to be
rejected.
24,
Having thus found the three Divifors a 4x, 3 a 4 x, x = 2a+b
with which Trial is to be made, I try the Divifion by the laft, which fuc-
ceeds, and gives for Quotient 3 a a 16 a x + 16 xx, which I divide
afterwards by 3 - 4 x. The Divifion likewife fucceeds, and gives for
A
4 K
258
ELEMENTS OF
3
Method of
divifors of
Quotient the first Divifor a-4x, whence the propofed Quantity is the
Product of thofe three Divifors.
LIII.
If the propoſed Quantity has no Divifor of one Dimenfion, we are
finding the to enquire if it has not fome quadratic Divifor. Let m x²+nax+pbx
two dimen- +9 a² +rab + sbb express this Divifor; putting fucceffively x = 0,
qa² +rab+sbb
fons invol- a = 0, b = 0, in this Divifor, there refults
ving three
letters.
Application
of this
method to
20 ex-
ample.
2
2
2
ga² +rab+ s b b ; m x² + p b x + s b b ; m x² + na x + qa²;
which are all three Divifors of the Quantities into which the propofed
one is transformed by the fame Suppofitions of x, a, b equal o. More-
over, each of thofe Divifors is fuch, that the Terms affected with fquare
Letters are repeated in the two other Divifors, whilft the Terms, that
contain a Product of two Letters are alone of their Species. When,
therefore, among the Divifors of two Dimenfions and involving two
Letters of the Quantities to which the propofed Quantity is reduced by
the foregoing Suppofitions, three are found having the above-mentioned
Conditions, by adding them together, and taking the one half of all the
Terms affected with Squares, and leaving thofe that are rectangles en-
tire, a Diviſor of two Dimenfions to be tried will be obtained.
LIV.
To ſhow the Application of this Method, let it be propofed to difcover
whether the following Quantity, which has no fimple Divifor, has any
quadratick Divifors:
2
2
3
x5—4a x4 — 5a² x³+46²x³ — a³x² — 14a b²x²+364x — a¹x. — 3 a³b²
2
~30+b²
a²,ab,b²,3a²,3ab,3b²
x²+36²x²+b²
2
*5 +46² × 3+3b4x
x5—4ax4—5a²x³-a3x²-ax x²+ax
2
2
362
2
62
x² + b²
+62
2
+ax
2
x² + 3.6²
x +4x
સ
x²+b²+axx²+3b²+ax
ſet down in a firſt Column the three Quantities-3a3b²; x5 +46² 3
$4
*;
+5
2
3
23 x²
2
+3
4 a x4 — 5 a² *
a4x; into which the
propofed Quantity is transformed by the Suppofitions of x, a, b equal o.
I afterwards fet down befide thefe Quantities their Divifors of two Di-
menſions; the firſt gives a², a b, b², 312, 3. a b, 3 b. b; the ſecond
x² +36²; x² + b²; the third only x²+ax. In this Method, the
Divifors confifting only of one Term are not to be rejected, even tho' it
has been found that the propofed Quantity has no Divifor involving two
Letters, becauſe a Divifor confifting of three Terms and of two Dimen-
fions, may be reduced to one Term, by fuppofing one of the Letters
equal o.
SPECIOUS ARITHMETICK.
259
2
a
2
2
It remains now to try all the Divifors of the firft Line; I perceive,
1º. That the firft 2 is to be rejected, becauſe this Square is not repeated
in the other Lines. I pafs afterwards to a b, and as this Diviſor does not
contain any Term affected with a a or with bb, I conclude that the Di-
vifor, of which it might be a Part, cannot have, beſides this Term, any
other but xx, or a x, or bx, whereby the Compariſon of a b with the
Divifors x²+3 b2, x² + b² being excluded, it follows that if a b be a
Part of a Divifor of the propofed Quantity, this Divifor muſt be x x
+ ax + ba; but at the fame Time, I perceive that x x + ax + b a
cannot be a Divifor of the propofed Quantity, fince it would be reduced
to xx by the Suppofition of a=o, and that xx is not one of the Divifors
of the fecond Line. Wherefore the Divifor a b is likewife to be rejected.
As to the Divifor bb, I find it repeated in the Divifor x2 +b of the
ſecond Line, and finding that the fame Divifor x² + b², has x² in com-
mon with the Divifor of the third Line, I conclude that x² + b² +ax
has the Conditions required. I afterwards pafs to the other Divifors of
the firſt Line, and I perceive, 1°. That 3 a a and 3 ab are to be re-
jected, in like Manner as a² and a b; I obſerve afterwards that 3
is repeated in the Divifor x2 + 3 62 and x2 in xx+ax, whence I
conclude that x x + 3 b b + ax has alfo the Conditions required.
Trying thoſe two Divifors, I find that the fecond only fucceeds, giv-
ing for Quotient x35 ax² + b² x — a³.
2
2
2
2
2
2
2
2
2
2
2
b b
Inftead of trying all the Divifors of the firft Line, we might, by try-
ing that one alone which is contained in the laſt Line, find much more
readier that x² + ax + b² and x² + a x + 3b² are the only Divifors to
be tried. For obferving that the Divifor x²+ax contains x² which
is repeated in x² + 362 and in x² + b², and that thoſe two laft contain,
the one 3 b², and the other b², which are each found among the Divifors
of the firſt Line, it is eaſy to conclude that x² + a x + b² and
x² + a x + 3 b¹ have the required Conditions, and that they are the
only ones, fince if there were others, they would give different Quantities
from xx+ax by fuppofing b = 0, or different Quantities from x²+362
and x²+b² by fuppofing a = 0.
2
2
2
2
LV.
diviſors are
If fome Letter in the propoſed Quantity is only of one Dimenfion, it is Cafe in
eaſy to perceive that only one of the Divifors of this Quantity can contain which the
it, whence there will be at least one Diviſor that does not contain it, and found much
confequently to find this Divifor (Art. XLVI.) you may ſeek for the great- easier than
eft common Divifor of the Terms in which that Letter is found, and at the by the fore;
going me.
remaining Terms in which it is not found. Thus, in the Quantity hod.
Зах
3
8aaxx + 18 a³ x + c x³ acxx 8aacx+6a³c-8a4,
the common Divifor of the Terms cx³-acxx—8aacx+6a³c and of
x4—3ax3—8aaxx+18a3x-8a, viz. xx+2ax-2aa, will divide the
whole quantity.
3
¦
260
Another
example.
3
ELEMENTS OF
LVI.
x4
Let it now be propoſed to find the Divifors of 2 x5 +3α x² + b² x3
— a² x³ + 4 a b² x² + 6 a² b² x + 2 a b4 — 2 a3 62 involving two
or three Letters, either of one or two Dimenfions,
I |2ab4—2a3b4
2
2 ab4a3bb
2a,a
2ab,ab,2b,b
2ab²,ab²,2b²,626
b³ —ab², 2ab²-2a²b, ab² —a²b, 2b²-2ab, b²—ab, 2ba-2à², ba—a²,2b-2a, b—a
2b³—2ab², ab³—a²b², 2ab3—2a2b²)
¿³+ab²,2ab²+2a²b, ab²+a²b‚2b²+2ab,b²+ab,2ba+2a², ba+a², 2b+2a, b+a,
263+2ab²,ab³+a²b³‚2ab³+2a²b²)
2ab³—2a³b, ab³—a³b, 2b³—2a³b, b³—a²b, 2b²a—2à³, b²a—a³, 2b²—2a², b—a
¿ª—a²b², 234—2a²b², abª—a³b², 2ab42a3b2)
2
2
3
2x5 +3 ax4+b² x3 — a² x3 +4 ab² x² + 6 a² b² x+2ab4 — 2 a³ b²
|a,2a,b,2b,b—a,2b—2a,b+a,2b+2a
2ab4-2a3b2
2x5+b²x3
bb,2bb,ab, 2ab,bb-aa,2bb-2aa, ba-aa,
bataa, bb-ab, bb+ab, 2ba-2 aa,
2ba+2aa, 2bb2ab, 2bb+2ab
x,xx,2xx+bb
x5+3ax4—a²x³]x,x²,2x²+3ax— na
64
a262
b3 — a²b
b²- a²
62.
b + a
a b
2 ab
b ba a
2xx+bb
x x
x x
* X
x x
2
3
४ ४
2x² + 3 ax— a a
²+al x²+2ab2x²+b²+3ax—a
2
2
—ab x² —2ab
2 a b4 — 2 a³ b², 2 x5 + b² x³, and 2 x5 + 3 a x4 - a2 x3 being the
Quantities to which the propofed one is reduced by the Suppofition of
x, a, b equal o, oppofite each of theſe Quantities muſt be ranged all
the Diviſors they may admit of, as well of one Dimenſion as of two.
As the first of thofe three Quantities admits of a great Number, in order
to omit none of them, I follow the fame Order as was obſerved in the in-
Application veftigation of the numerical Divifors.
of the
method of
Art. XXIV.
Having fet down this firft Quantity 2 a b4 - 2 a3 b b, and drawn a Line
to the left Hand of it, I write down Unity, as being its first Divifor; I
for finding afterwards fet down 2 under 1, becauſe it is after r the fimpleft Divifor
all the divi- that this Quantity will admit of, and I fet down to the right Hand of the
fame Line a b4a3 62. I afterwards divide this Quantity by a, and ſet
down a to the left Hand of the vertical Line, writing down at the fame Time to
fors of lite-
ral quan-
tities.
ex
2
SPECIOUS ARITHMETICK.
261
{
the right Hand the Quotient 4a2 b2. I next multiply a by 2, and
fet down 2 a to the left Hand of a, as being a new Divifor of the pro-
pofed Quantity; I afterwards divide b4 — a2 b2 by b, and fet down the
Divifor b to the left Hand, and the Quotient 63a2b to the right.
Laftly, I multiply b by 2, by a, by 2 a; and ſet down to the left Hand
of b, the Products 2b, a b, 2 ab, as being new Divifors of the propofed
Quantity.
zbb-
2
a by 2,
22,
The propofed Quantity being reduced to 63 - a² b, I divide it again
by b, which I fet down to the left Hand, and the Quotient b² a to
the right; I omit multiplying b by 2, or by a, or by 2 a, becauſe there
would refult the Divifors already found; but I multiply it by 7 and by 2 b,
which produces the new Divifors of two Dimenſions b b and 2 b b ; if
Divifors of three Dimenfions were to be admitted, as may be neceffary
on other Occaſions, then b ſhould be likewiſe multiplied by a b and 2 a b.
After having reduced the Quantity to bba a, I perceive that it is
diviſible by b a, and the Quotient is b + a. I therefore fet down one
to the left and the other to the right Hand, and I multiply &
by a, by 2 a, by b, and by 2 b, which give for new Divifors, 26
b a — a a, b b — ab,b²,2b², 2ba —2aa, 2bb. 2 a b. And if Diviſors of
three or four Dimenſions were required, I ſhould have alſo multiplied
b — a by a b, 2 a b, a bb, z a b b. The Quantity remaining bo
admitting of no Diviſor, I ſet it down to the left Hand, and I multiply
it by 2, by a, by b, by 2 b, 2 a, b
2 a, which give for
new Divifors of one and two Dimenfions, 2 b+ 2 a, ba
2 ba + 2 a a, b b + a b, 2 b b + z a b, b b — a a, 2 b b 2 a a. If
Diviſors of three, four, five Dimenſions, that is, all the Diviſors that
the propoſed Quantity can admit of were required, I would have alſo
multiplied ba by a b, 2 a b, bb, 2 b b, a bb, za bb, ba — a a,.
2 ba
2 a a, b b a b, ż b b
zab, abb -aab, zabb za² b,.
2 a b², a b³ a² b², zab3 2a2 62.
b3 a b², 2b3
This being done, I fet down all
ſions, a, 2 a, b, 2 b, b. 4,26
a b
a, 26
+aa,
the Diviſors of one and two Dimen-
2 a, b + a, 2.6 + 2 a, a b, 2 ab,
a a, 2
z ba 2 aa, b² a b, z b²
zab, ba+aa, 2 ba+zaa,
b² + ba, 2 bb + 2 a b, beſide the Quantity 2 a b4 — 2 a³ b² which
a3
produced them. I afterwards fet down beſide the Quantity 2 x5 + b² x ³
its Diviſors of one and two Dimenſions x, 2 x x, x x + bb, and beſide
the Quantity 2 x 5 + 3 ax4 - a2 x3 its Divifors of one and two Di-
menſions x, x 2x + 34 x a².
2
>
2
3
Now, trying all thofe Divifors to diſcover thoſe that are to be rejected,
I foon perceive that all thofe of one Dimenfion are of this Sort; for x
being the only Quantity of one Dimenſion in the fecond and third Lines,
I conclude that it muſt be the only Divifor of one Dimenſion that the
propofed Quantity can admit of; fince if the Divifor of one Dimenfion
262
ELEMENTS OF
How the
included either a Term affected with a, or one affected with b, the Terra
affected with a, would remain in the Divifors given by the Suppoſition of
b = o, and the one affected with b would remain in the Divifors given
by the Suppofition of a = 0; but x is not a Divifor of the propofed
Quantity, confequently it has no Divifor of one Dimenſion.
24
I pafs next to the Divifors of two Dimenfions, beginning by x which
I take in the third Line; finding it alfo in the fecond Line, I concludé
that if it makes a Part of a Divifor, it can only want a Term affected with
the Rectangle ab; in effect if there were Terms in this Divifor affected
with a a, with bb, with ax, or with b x, thoſe affected with bb or with b*
would not have difappeared by the Suppofition of a=0; and thoſe
affected with a a, or with ax, would not have vanished by the Suppo-
fition of b=0. But I find in the firft Line ab, 2ab; wherefore
xx+ab, xx+2ab, xx
ab,
2 ab, are the Divifors to be
tried.
I afterwards paſs to the Divifor 2x² + 3ax-da, and I find the Term
2x² repeated in the Divifor 2 x x + b b above it, as likewiſe the Term
- aa repeated in feveral Diviſors that are in the upper Lines; but of
all the Diviſors in which it is repeated there is only bb-aa, which has
the Term bb common with the Divifor 2xx + bb. Whence it is only
with 2xx + bb, and bb
that 2x²+3ax
aa can concur to form
a Divifor that will have the Conditions required, and this Divifor which
is 2x²+3ax — aa+bb is confequently to be tried.
aa,
I perceive that it is uſeleſs to ſeek for another Divifor, becauſe if there
could be found one that could not be diſcovered by trying the Diviſors of
the third Line, it would confiſt of one Term affected with ab: Now it is
eaſy to perceive that the propofed Quantity admits of no Divifor of this
kind. I therefore attempt the Divifion by 2x² + 3ax — a² +bb, which
fucceds, and gives for Quotient x3 +2abb, whereby it appears that none
of the Quantities x x + ab, xx—ab, xx + 2ab, xx
X X -------- 2 ab can divide
the propoſed one.
LVII.
1
If the propofed Quantity is of more than five Dimenſions, and having
found that it has no Divifor of one or of two Dimenſions, then we are
to enquire by a fimilar Proceſs if it has not fome Divifor of three or four
Dimenſions.
LVIII.
When the propofed Quantity is not homogeneous, that is, when all its
divifors of Terms are not of the fame Dimenfions, it is to be rendered homogeneous
quantities
that are not by means of a new Letter, and the Divifors of this new Quantity being
homogene found by the preceding Rules, the new Letter introduced is to be exter-
minated by putting it equal to Unit, and by this Means the Divifors re-
found. quired of the propofed Quantity will be obtained. Suppoſe, for Example,
ous are
SPECIOUS ARITHMETICK.
263
2
the Quantity to be y6 + by5-by-y²+by-b; I multiply the Term
4 by a, to make it of fix Dimenſions. For the fame Reafon I multi-
ply y² by at, by by at, and b by a5, whence there refults the Quantity
yo + bys — a by 4 + atyy + as by-a5b, which by the foregoing Method
is found to be the Product of yyabby into y4a4.
abby into y4a4. I fuppoſe a=1
in thoſe Factors, and there refults yy-bby, and y+1 for the two
Factors of the propofed Quantity p6 + by5 — by4 + y²+ by —b.
a4
LVIX.
To find the incommenfurable Roots of Equations of every Degree, Method of
1º. Subſtitute fucceffively in the propofed Equation 1, 2, 3, 4, &c. in place ting to the
approxima-
of the unknown Quantity, until the Quantity that reſults after each Sub- roots of
ſtitution changes its Sign, then the Root y will be a pofitive Number numeral
greater than that which immediately precedes the Change of the Sign of
the Reſult, and leſs than that where this Change happens.
equations
involving
one un-
2º. Inſtead of putting the firſt Member of the Equation equal o, put known
it; then find the limiting Ratio of the cotemporary Increments of
quantity.
y and z, and let it be expreffed by A. Subftitute the Value of y, eſtimated
pretty near the Truth by the foregoing Method, in the Equation, as alfo
in the Value of A, and let the Error or refulting Number in the former,
be divided by this numerical Value of A, and the Quotient be fubtracted
from the faid former Value of y, and from thence will ariſe a new Value
of that Quantity much nearer to the truth than the former, wherewith
proceeding as before, another new value may be had, and fo another, &c.
till we arrive to any Degree of Accuracy defired.
LX.
Let it be propoſed to find a Value of y in the Equation y4 +2y3-36vy
+5y
116=0.
If
y
I
If y
2
I + 2
16+ 16
36 + 5
144 + 10
116
-144
116—218
If
= 3
81 54
then
If J
256 + 128
324 +15 —
576 +20
116
116=-
-290
116= -288
If y 5
625 +250-
900 + 25 —
116
-116
If
6-
1296 +432 —
1296 + 30—116=+346
Whence I conclude that one of the Values of y will be greater than 5,
but lefs than 6. Let it be fuppoſed to be 5, 5, to find a more approximate
Value of it, I inveſtigate the limiting Ratio of the cotemporary Increments
dy and dz of y and z in the Equation 4+2y336yy + 5y—116=z,
which I find to be 433 +6yy 723 +5, then becaufe
Y-
14+2133691 +5y116 = y.
413+6yy723 +5
буу
I fubftitute in the firft Mem-
ber of this Equation, in Place of y, its Value. 5,5 already found, whence
we have 5,5-
70,3125
456
= 5,340 for a new Value of
ولا
with which pro-
264
ELEMENTS OF
of the fore-
ceeding as before the next Value of y will be found to be 5,3354, and
from thence the third Value of y = 5,335438, which is true to the laft
Decimal Place.
LXI.
Let y (1 − y) + √ (1 − 2 33) + √ (1 — 33 ³) — 2
-2=0, be propoſed.
Application one of the Values of y in this Equation will be nearly 0,5. To find a more
going me- approximate Value of it, I inveftigate the limiting Ratio of the
thod to an- cotemporary Increments d y and d z of } and Z in the Equation
other ex-(1-3) + ✓ (1 − 2 y3) +√ (1
− — 31³) - 2 = 2, which I find
9yy
ample.
When
there are
and as
many
to be
I
-
2 y
2√(1-3) √(1—2y}) z√√(1—3y³
; whence we have
0.204
0,5 + =0,557 for the next Value of y, from whence by proceed-
3-545
ing as before, the next following will be found 0.5516, &c.
LXII.
When there are two Equations given, and as many Quantities (x and y)
to be determined, find the Equation expreffing the limiting Ratio of the
cotemporary Increments dx and dy of x and y in both Equations, and in
two equa- the former collect all the Terms, affected with dx, under their proper Signs,
tions given, and having divided by dx put the Quotient = 4, and let the remaining
Terms, divided by dy be repreſented by B: in like Manner, having divided
Quantities the Terms in the latter affected with dx, by dx, let the Quotient be put
and the Reft divided by dy, b. Affume the Values of x and y
determine pretty near the Truth, and fubftitute in both the Equations, marking the
Error in each, and let theſe Errors, whether pofitive or negative, be figni-
Quantities. fied by A and r refpactively, fubftitute likewife in the Values of A, B, a, b,
BrabR aR—Ar
be converted into Numbers, and reſpectively
x and y
how to
thoſe
Applica-
tion of the
foregoing
method to
an ex-
ample.
= A,
and let
and
Ab-aB Ab—aB
added to the former Values of x and y, and thereby new Values of thoſe
Quantities will be obtained; from whence, by repeating the Operation, the
true Values may be approximated ad Libitum.
LXIII.
Let there be given the Equations y + √ (yy
x + √ ( y y + x) — 12 = 0; to find x and y.
The Equations expreffing the Relation of the cotemporary Increments*
xx)
10 = 0, and
dx, and d y, of x, and y, being here dy +
¿ x +
and dx+
+
لو
y dy
√(3y-xx)
*) √(yy-xx)
= 0, we have A equal √(yy —≈≈)'
y d v + // d x
=0, or dy +
√(xy + x)
2 d x3
ydy
√ (33 + x) √ (y + x)
(3y+x)
y
**
y dy xdx
= 0, and
√ (yy − xx)
x dx
B equal 1 +
√(3y+xx)
>
a=1+
44
√ (33 + x)'
and b =
SPECIOUS ARITHMETICK.
265
Let be fuppofed equal 5, and y = 6; then will R equal
x
— .68,
1.5, B equal 2. 8, a equal 1. 1, b equal 9; there-
Br
-.6, A equal
b R
fore
A b
23, and
a B
a R Ar
A b — a B
=.37, and the new Values
of x and y equal to 5.23, and 6.37 reſpectively; the next Values will
come out 5, 23263, and 6, 36898.
LXIV..
* *
Let 49 X (x
(x+y)²
2
— 25 X ( 1
(1 -
(1+ y)
-) = 0, and 81 X Application
x x
x
x y
of the fore-
going me--
(I
(I + y)
2
49 X (
y
(I + x )²
y
50 x
98 x
50 x
+
(x+y) *
B
2
I
(1 + y)²
( x +- y ) ³
j
I
@=
+* J
2 x y
(1 + x)
3], and b ==
( 1 + y) ³
162 **
=o. Here we have thod to 100-
2 ple.
39
ther Exam-
(1+)
I
A equal 49 × (1+
162 * f
(1+ y)²
x[
+49xX
2
I
+49 × [-
y y
Suppoſe x =
= .8,
A= 68, B
+
68, B = 20, 7,
(1 + x)
2
and 6, then will be found R. 45, r = 2, 66,
x and y equal to ,799,
y=
a
0=- -131, b = 146, and the next Values of
and ,582, with which repeating the Operation,
the next following will come out,79912 and,58138.
LXV.
-
of the me-
When an Equation contains two unknown Quantities x and y, and
it is propoſed to exprefs the Root y in a Series of Terms including the
Quantity x with the other known Quantities; to find the Value of y
in fuch a converging Series, it is neceſſary to confider x either 1º. as very Fundamen
fmall, or 20 as very great, or 3°. as differing very little from fome given tal principle
Quantity; that fo by fuppofing x to be very ſmall the Series may con- thed of S-
verge wherein the Rooty is expreffed by a Progreffion of Terms, in ries.
which the Dimenfions of increaſe in the Numerators, 2° by fuppofing
* to be very great that Series may converge, in which the Dimenfions of
x continually increaſe in the Denominators of the Terms, or that 3º by
fuppofing x to differ but very little from fome given Quantity, fome
Letter as z being fubftituted for that Difference, it may come in the
Place of x conſidered as very fmall.
LXVI.
Thus if the Equation to be refolved was x ²y + a y
ax=0,
if x be conſidered as very little, the Value of y may exhibited
in a Series having its Terms compoſed of the Powers of di-`
vided by thofe of a with their refpective Coeficients.
when is very little
x²
in refpect of a, the Terms x,
>
Since
*
4 L
266
ELEMENTS OF
1
Divifion of
x
4
942
&c. decreaſe very quickly. If x vaniſhes in respect of
a
23
ſecond Term will vaniſh in reſpect of the firſt, ſince *:
3
2
a, the
:x=x: a, and
after the fame Manner vaniſhes in reſpect of the Term immmedi-
ately preceding it.
a
But if x be confidered as very great, the Value of y may be exhibited
in a Series whofe Terms are compofed of the Powers of a divided by
thoſe of x with their refpective Coeficients, fince when x is very great
in reſpect of a, then a is very great in refpect of, and
in reſpect
a
of
fo that the Terms a,
x
3
a
a4
5
a
>
X
2
a
&c. in this Cafe decreaſe
very ſwiftly; in either Cafe, the Series converge fwiftly that confiſt of
fuch Terms; and a few of the firft Terms will give a near Value of the
Root required.
Whence it appears that the Roots of Equations involving two unknown
Series into Quantities may be exhibited by two Sorts of Series, whereof one con-
2fcending verges ſo much ſooner as the unknown Quantity it contains is lefs, and the
and defcend-other converges fo much fooner as the unknown Quantity is greater. In
ing ones.
the Terms of the former; which are called afcending Series, the Expo-
nents of the unknown Quantity continually increaſe, but continually de-
creaſe in the Terms of the latter..
An Equati-
on lofes
terms
LXVII.
To determine the first Term of the Series, exhibiting the Value of
an unknown Quantity, in an Equation that involves two unknown Quan-
fome of its tities, as of y in the Equation x 2y+ay 2a2x0, the Analyfts af-
when one of fume together fuch Terms of the Equation as will be found to be-
its unknown come vaſtly greater than the other Terms, that is, which give a Value
quantities of y which fubftituted for it in all the Terms of the Equation, ſhall raiſe the
indefinitely Dimenſions of the other Terms all above or all below the Dimenfions of
great or in the affumed Terms, according as x is fuppofed to be vaftly little or vaſtly
definitely
little. great in refpect of a.
is fuppofed
LXVIII.
To determine the Terms in any propofed Equation that the Suppofition
Method of of x or of y being indefinitely great or indefinitely little, will render
fading the
greateſt vaftly greater than the other Terms: I obferve firft, that one Term of
terms of as the Equation cannot be ſuppoſed vaſtly greater than all the reft, for all
equation by the other Terms would vanifh in refpect of it, and this Term then alone
dufion, would be equal to nothing, which is abfurd. The greateſt Terms there-
fore of an Equation are at leaſt two in Number, but there is no Reafon
way of ex-
SPECIOUS ARITHMETICK.
267
why any two Terms may not be fuppofed the greateſt of the Equation,
unlefs, the Confequences arifing from this Suppofition deftroy the Suppo-
fition itſelf.
2
2
Thus in the propofed Equation x y + a y² — a² x 0, x being con-
fidered as indifinitely great, if xy and a y be fuppofed to be the great-
eft Terms of the Equation, then ²y + ay² = 0, or x²y-ay², and Application
2y+ay² =
. dividing by-ay, y—— which ſubſtituted for x, the Equation be- thod to an
@omes
x+ +4
a
a
+ a²x=0, 2
o, where the affumed Terms ² ya y² are
Q
of more Dimenſions than the other ax, and confequently may be fup-
poſed to be the greateſt Terms of the Equation.
2
૨૨
X
If y anda² be fuppofed to be the greateſt Terms of the Equa
tion, then x²y-a2x=0, which by tranfpofing and dividing by ✰² gives
y =
a²
2
x
which fubftituted for y, then a² x +
where the aflumed Terms x²y,
والا
a5
*
2
a² x = 0,
a 2 X are of more Dimenſions
than ay", and confequently may be fuppofed to be the greateſt Terms
of the Equation.
2
a²x be ſuppoſed to be the greateſt Terms of
Finally, if ay and
the Equation, then a y² a² x = 0,
wherefore y = a
1 : 2 1: 2
X
I : 2
5:2
x +a²x a² x
2
which fubftituted for y, the Equation becomes a
where the affumed Terms are of lefs Dimenfions than the other x² y,
and confequently cannot be fuppofed to be the greateſt Terms of the
Equation.
→
2
x being fuppofed indifinitely little, if ay and ax be fuppofed to
be the greateſt Terms of the Equation, then ay2-a²xo, which gives
y = a¹² x1:2 which fubftituted for y, the Equation becomes
X 5 : ²+a²x—a²x—o, where the affumed Terms ay2-a²x are of leſs
Dimenſions than x²y, and confequently may be fuppofed to be the
greateſt Terms of the Equation.
a
I : 2
2
If x²y+ay² be ſuppoſed to be the greateſt Terms of the Eqation,
then x²y+ay²=0, which gives y————, which fubftituted for
24
+2
2
ونا
the
Equation becomes + -a²x=0, where the affumed Terms x²y, ay²
a
a
are of more Dimenfions than a²x, and confequently cannot be fup-
poſed to be the greateſt Terms of the Equation.
If x²y and a²x be fuppofed to be the greateſt Terms of the Equa-
tion, then
*=
2
which fubftituted for y, the Equation becomes
of this me
example.
268
ELEMENTS OF
Ufe of the
analytick tri
angle in this
refearch.
a²x+
x=0, where the affumed Terms 2y and a²x are of
more Dimenſions than ay², confequently cannot be fuppos'd to be the great-
eft Terms of the Equation; whence it appears that when x is fuppofed in-
definitely great, the propofed Equation x2y+ay2-a2x=0, is reduced to
the two following ones xy+ay20, or x²+ayo, and x2y-a2x=o, or
xy—aa=0; and when x is fuppofed indifinitely little, it is reduced to
ay² — a²x = 0, or y²-ax=0.
2
LXIX..
2
What was eaſy in an Equation confifting only of three Terms, would
be very laborious if a more complex Equation was propofed, the Num-
ber of Terms occafioning a vaft Number of Trials for the moſt Part
unſucceſsful. How the Analyfts have remedied this Inconveniency, we
fhall therefore proceed to explain.
C
03
Kes
درو
x¹y
*
.8
४
2
6
X
४
10
་
x 5 x 3
x5y?
x5
25
४
345
4
x + y +
375x
3,4
x²y 5 | x² y ≤
+
43
x+y³ | x+y² | x+y
x343
*
3y
3.2
3
x
2 2
2
2.6
X
x
2x3
7
xy
xy5
4
xy
xx 3
ху
6
5
4
3
N
N
xy
4
x
४
*
1
m
C
Ι
B
A
Let B A and AC be drawn at Right Angles to each other, and let
them be divided into equal Parts, and from the Points of Divifion let
Lines be drawn meeting each other at Right Angles, which will form fo
many equal Squares as in the Figure; in theſe Squares place the Powers
of x from A towards C, and the Powers of y from A towards B, and in
any other Square place that Power of y that is dire@ly below it in the
Line A B, ſo that the Index of y in any Square may exprefs its Diſtance
from the Line A C, and the Index of x in any Square may expreſs its
Diſtance from the Line A B.
1
SPECIOUS ARITHMETICK.
269
:.
Of this Figure or Triangle we are to obferve, that being placed upon
the Side A B, when x is fuppofed indefinitely great, or indefinitely little,
(and on the Side A C wheny is fuppofed indefinitely great or indefinitely
little) of all the Terms in the fame vertical Column, that only can be
confidered as one of the greateſt Terms of the Equation, which is placed
in the higheſt Square of this Column, when x is fuppofed indefinitely great, Property of
the analytick
or that which is placed in the loweſt Square of this Column, when x is
triangle.
fuppofed indefinitely little; for y having the fame Exponent in all the
Terms in the fame Column, their Subordination entirely depends on
the Exponent of x; wherefore x being fuppofed indefinitely great, the
greatest Term in the Column is that in which has the greateſt Expo-
nent, which is placed in the higheft Square of this Column, and x be-
ing fuppofed indefinitely little, the greateft Term in the Column is that
in which has the leaft Exponent, or which is placed in the loweſt
Square of this Column.
LXX.
The foregoing Obfervation, as is eafy to perceive, ferves to diminiſh
the Number of Comparifons requifite to be made, for difcovering the
greateſt Terms of an Equation, but in order to avoid every ufelefs Com-
pariſon, we are further to obſerve, that if every where for y, be fubfti-
tuted the Value expreffed in the Powers of x, that arifes for it, by fup-
pofing any two Terms of the Fquation equal, the Dimenfions of x in all
the Terms that are found in the ſame ſtraight Line will be equal,
but the Dimenſions of x in the Terms above that Line will be greater The expo-
than in thoſe in that Line, and the Dimenſions of x in the Terms below nents of the
the faid Line will be lefs than its Dimenſions in that Line.
2
Thus if the two Terms 2 and 3 y² be fuppofed equal, we
I
*
I
Vx
*
플
​=x X
, and fubftituting this Value for y
x4
terms which
fame
find are in the
in all ftraight line
are in arith-
&c. metical pro-
greffion.
y²===, and y ===
the Squares, the Dimenſions of x in the Terms, x+ y², x³ y, x6 8,
which are found in the fame ſtraight Line with x2 and x3y², will be 2,
but the Dimenfions of x in all the Terms above that Line will be more than
2, and in all the Terms below that Line will be leſs than 2. For Exam-
ple, fubftituting the foregoing Value of y in the Term + y³, which is
above the Line, it will be transformed into x52, whofe exponent exceeds
2, and if it be fubftituted in the Term 3 y³ below the Line, this Term
will be transformed into 32 whofe Exponent is lefs than 2.
x
LXXI.
3
3
The foregoing Property of the Analytick Triangle is founded upon this
Principle, that the Squares whofe Centers are in a ftraight Line contain
Terms in which the Exponents of x and y are in Arithmetical Progref-
fion: which is manifeft with respect to the Terms in the vertical Co-
lumn AC, or horofontal Line AB, and their Parallels, in the Terms of
the fame Column the Exponents of y are the fame, and thofe of a form
370
ELEMENTS OF
the Arithmetical Progreffion 1, 2, 3, 4, &c. in the Terms upon the
fame Line, the Exponents of x are the fame, and thofe of y form the
Arithmetical Progreffion 1, 2, 3, 4, &c. As to the Terms taken in
any oblique ſtraight Line, let us fuppofe that this ſtraight Line parting
from the Center of a Square, traverfes k Lines and Columns before it
Demonftra- paffes through the Center of another Square, it is manifeft, that fince
foregoing
the Squares are ranged uniformly, it muſt traverſe k Lines. and 7 Co-
property of lumns before it paffes thro' the Center of a third Square, and as many more
the analytic before it can attain the fourth, and ſo on. Wherefore, fince the Expo-
triangle nent of x is increaſed by Unity in afcending a Line, and the Exponent of
y is increaſed by Unity in traverfing a Column, from right to left, if the
tion of the
ences of
171
72
Term placed in the firſt Square be xy, that placed in the fecond will
m+k r+l
bex
m+2k +21
y, that in the third x y th
"
and fo on, in which
the Exponents of x form an Arithmetical Progreffion, m, m+k,m+2k, &c.
and thoſe of y the Progreffion n, n+1, n + 2 1, &c.
M
LXXII.
"
m + k n+l
y
*
x
m + 2k 7 +21
>
&c. in
Converſely, Terms as x y
which the Exponents both of x and of y are in Arithmetical Progreffion
whofe Differences are k and 1, thofe Terms are placed in Squares whofe
Centers are under a Ruler traverfing at the fame Time k Lines and /
Columns, which determines the Inclination of this Ruler to the Sides of
the Triangle.
For the Inclination of the Ruler to the Sides of the 'Triangle, and
The ratio of the Ratio of k to depend entirely of each other, fince k and I are the
the differ- Number of Lines and the Number of Columns which the Ruler at the
thoſe progres fame Time traverſes. If k furpaffes 1, the Ruler is more inclined to the
fions depend Columns than to the Lines, and cuts off a greater Portion of the Side
AC of the Triangle, than of the Side AB. In general, fince the Ruler
the ruler to traverſes k Lines in traverfing / Columns, it will traverſe, from Column to
on the incli.
nation of
the fides of
k
T
the triangle. Column a Number of Lines expreffed by, whether denotes an Inte-
The terms
Atraight
ger or a Fraction, if the Ruler traverfes two Columns in traverfing
one Line, it will traverſe but Half a Line in traverfing one Column.
LXXIII.
Therefore if on the Surface of the Analitic Triangle there be drawn
which are in two parallel ftraight Lines, and confequently inclined as much one as the
two parallel other to the Lines and Columns, the Squares whofe Centers thofe
lines have ſtraight Lines traverſe, contain Terms in which the Exponents of x and
exponents y are in Arithmetical Progreffion, which in both one and the other
Thoſe under the firſt ſtraight
arithmetical ſtraight Line have the fame Difference.
172 # mtk ntl
7 +21
m + z k
progreffion s
&c. thofe
Line being, for Example, xy, x
*
y.
y.. >
which form
SPECIOUS ARITHMETICK.
271
under the ſecond may be expreffed by y, x
P 9
x
m² + ak n + 21
x
y
>
&c. and xy,
1
#+k q+1 ptak q+21 having the
-y *
n
"
&c. fame com-
n.. mtk nt! mon diffèr-
Converfely, if two Series of Terms be affumed as x y, ** y
p q
ptk qt p+2k q+21
y
> x
y
&c. in each
of which the Exponents of x and y are in Arithmetical Progreffion, hav-
ing the fame common Difference, the ftraight Lines which pafs through
the Centers of the Cells of thofe two Series of Terins will be parallel.
LXXIV.
ence,
Such being the Difpofition of the Terms in the analytick Triangle, if
any two Terms be fuppofed equal, and a Ruler be applied to the Centers of terms that
the Cells of thofe two Terms. I fay, that if in all the Terms placed in are in the
ſame ſtraight:
the Cells through whofe Centers the Ruler paffes, be fuftituted for y, line are of
the Value that ariſes for it from the foregoing Suppoſition, the Di- the fame or-
menfions of In thofe Terms will be all equal.
*
For as their Exponents are in Arithmetical Progreffion, they may be re-
prefented by the Series xyx
m+4k +41
n
y
der, if two of
them be fup-
pofed to be
m+3k n +31 of the fame
y >
Now let any two of them as
order.
17 m+k x+!
n
*
m + 2k +21
3
*
"
X
y
x
mtsk nts
y
&c.
m + 2k n +2Į
mtak ntal
x
y
and *
m+5k nts!
y
*
y
be fuppofed equal, then
m + 2 k
n+21
x
y
m+sk +51
#
X
y
3 k 31
k l
2 k
21
, or IX
و و
, confequently x y = 1, x
y = 1, &c.
m + 2k n +2
*
whence the Dimenſions of x, in all the Terms under the Ruler
m
n m†k n+1
# + z k
n+21
x
,
y
1
a k zl
* y, x 3
m
#
&c. refulting from the Multiplication
3 k 31
of x
y, &c. is m m-
p+k q + l
2+1
menfions of x in the Terms xy, x y
y into I, xy, X y, *
P 9
nk: l, and the Di-
&c. which are all
found in the fame ftraight Line parallel to the Ruler will be pq kl.
LXXV.
two undeter
The Dimenſion of x in all the Terms that are under the Ruler may Ratio of the
be alfo found by examining where the Ruler cuts the first Column. If orders of the
it paffes through the Center of fome Cell of this Column, it is manifeft, mined quan
that the Exponent of x in the Term placed in this Cell will reprefent tities in this
the Dimenſions of x in all the Terms under the Ruler, and if the Ru- fuppofition.
272
ELEMENTS OF
ler paffes between the Centers of two Cells, the Point through which
it paffes will still ferve to determine the Dimenfions of x in thofe
Terms, whofe Exponent in this Cafe is a Fraction. We are to conceive
this Column, and in general each Column, as a ſtraight Line divided into
equal Parts by the Centers of the Cells; the Terms x, x², x³, &c.
whofe Exponents are Integers, placed in the Points of Diviſion, and the
1:2 1+1:3 +3:4
Terms, as x
&c. whofe Exponents are Frac-
tions or mixed Numbers, placed in the Points which divides the Inter-
vals between the Centers in the fame Ratio, as Unity is divided by the
Exponent of Fraction, which forms or concurs to form the Exponent of x Hence
the order of
the Terms
that are in
the fame
*
1:2
,
x
2
we are to conceive x placed precicely in the Middle betweeen
I
Araight line. Or I, and x¹,
The terms
that are a-
bove this
ftraight line
are of a fu-
perior or-
der: thofe
و
and x
1+1:3
placed in the firſt Point of the Subdivifion
of the Interval between x and x2 into three equal Parts, confequently
I
2
if the Ruler paffes, for Example, between the Centers of x and of
x' but three Times nearer the first than the laft, that is, through the
2+3:4
X
Point where x is conceived to be placed, we are to conclude
that the Dimenſions of x in all the Terms under the Ruler is 2.
The Exponent of the Dimenfions of x in the 'Terms under the Ruler is
negative when it cuts the first Column produced below the firſt Line, which
k
m P
т > 12
or
9
In this Cafe,
happens when < n k : l, or p < q k : l, that is, when
then the Exponent m-nk: lor pqk: I is negative.
the first Column is to be conceived as a ſtraight Line produced below the
Point, and divided into Parts equal to thoſe that are above the Point,
and the Terms whofe Exponents are negative, as x
that are be placed in the Points of Diviſion.
low it of an
inferior or-
der.
LXXVI.
I
و
ناک و
3
x, &c.
Since therefore of two parallel ftraight Lines drawn on the Surface of
the analytick Triangle, that which is above the other cuts the firft Co-
lumn in a higher Point, the Terms in this Line are all of higher Di-
menfions of x than thoſe in the Line below it. Thus the Ruler paffing
m + k n+1 mtak n+al
through the Terms x y, x J
&c. which are
y
all of m nk: / Dimenfions of x. Any other Term is of higher of
lower Dimenfions of x, according as it is placed in a Cell whofe Center
is above or below the Ruler.
&
m
X
>
LXXVII.
>
This is the Principle which determines the Comparifons to be made
for finding the greateſt Terms of an Equation. When x is fuppofed
SPECIOUS ARITHMETICK.
273
indefinitely great, it is manifeſt that it is uſeleſs to confider two Terms as
being the greateſt of the Equation, if the Ruler, applied to the Cen-
ters of their Cells, leaves above them any other Term, for this Term
being of more Dimenfions of x than thofe through which the Ruler
paffes, it would therefore be indefinitely greater than thoſe which were
fuppofed to be the greateft, which is abfurd.
When x is fuppofed indefinitely little, it is ufelefs to fuppofe any two
Terms to be the greateſt of the Equation, if the Ruler, applied to the
Centers of their Cells leaves below them any Term of the Equation,
for this Term being of lower Dimenſions of x than thofe through which
the Ruler paffes, it would be indefinitely greater than thoſe which were
ſuppoſed to be the greateft, which is abfurd.
LXXVIII.
Whence is deduced the following Rule, for difcovering what Terms
ought to be affumed from an Equation, in order to give a Value for x,
or y, which shall make the other Terms, all of higher, or all of lower
Dimenſions of x or of y, than the affumed Terms.
After having traced the analytic Triangle, range each Term of the Rule deriv
Equation in its proper Cell, or what in Practice is more commodious, ed from the
form & Triangle with Points difpofed in Quincunx, then change into an foregoing
Aſteriſk each Point that repreſents a Square, which contains the fame principles,
for diſcover-
Dimenſions of x and y, as the Terms in the Equation. Or a Triangle ing the great
may be conſtructed of Wood or of Ivory, pierced with fmall Holes and eft terms of
an equation.
ranged at equal Diſtances, and parallel to the Sides of the Triangle,
and the Holes which repreſent the Cells in which the Terms of the
Equation are placed, may be ftopped with Pegs.
Then the Triangle being couched on the Line without x, if x be
fuppofed indefinitely great or indefinitely ſmall, or on the Line without
y, if y be fuppofed indefinitely great or indefinitely ſmall, affume when
the variable Quantity is fuppofed indefinitely great, fuch Terms as lie
in a ſtraight Line, fo that the other Terms fall all below the ftraight
Line. This ftraight Line which thus determines the greateſt Terms of
the Equation, is called by the Analyfts, a Superior Determinator. The
fame Operation may furniſh feveral of them.
But if x or y be fuppofed indefinitely little, affume fuch Terms as lie
in a ſtraight Line, fo that the other Terms lie all above the ſtraight
Line, this ſtraight Line or thoſe ftraight Lines, for there may be ſeve-
ral of them, are called by the Analifts, Inferior Determinators, becauſe
they determine the greateſt Terms of the Equation, being thoſe which
are placed in the Cells through whofe Centers they pafs.
4 M
274
ELEMENTS OF
Application
of the fore-
going rule
to an exam-
ple
ed the Triangle with Points,
LXXIX.
y+ay².
Let the Equation propoſed be x² y + ay² — a² x=0, having deſcrib-
and converted into Afterifks the Points
which repreſent the Cells 2y, p2 and x, it will appear
B
B
A
B
A
that there are only three Determinators, AB, BC, CA, of which, when
the Triangle is couched on the Line without x, two AB, AC, are
fuperior, and one inferior B C; but when the Triangle is couched on
the Line without y, ihe Determinator A B is a ſuperior one, and AC and
B C are inferior ones.
The Determinator A B gives the Equation x²y+ay²= 0, or
xx+ayo, the Determinator AG gives xy-a²xo, or xy-aa=0,
and the Determinator BC gives a p²-a2x=0, or yy-ax=0.
—
SPECIOUS ARITHMETICK.
275
Whence being ſuppoſed indefinitely great, the propofed Equation
is reduced to thoſe two xx+ay=0, and xy—a a=0, given by the
Determinators AB, AC, which are fuperior ones, when the Triangle
is couched on the Line without x.
But if x is fuppofed indefinitely little, the Equation is reduced to
yy
—ax=0, given by the Determinator B C, which is an inferior one,
in the fame Poſition of the Triangle.
If y be fuppofed indefinitely great, the Equation is reduced to
xx+ay=o, given by the Determinator AB, which is a fuperior one,
when the Triangle is couched on the Line without y.
But if y be fuppofed indefinitely little, the Equation will be reduced
aao, and y y-ax=o, given by the Determinators AC,
B C, which are inferior ones in this fame Pofition of the Triangle.
to xy
LXXX.
3
ટ
Let xxyy+a x y² + b x² y + c x³ - d d xy+eexx+ƒ³g=0μ
be propoſed.
After having formed the Triangle
with Points, and converted into Afte-
riſks the Points which correfpond to
the Terms of the Equation, it will
appear that all the Afteriſks may be
included in the Pentagon ABCDE,
there are therefore five Determinators,
which give the five following Equa-
tions.
B**
*D
A
a
= 0.
AB gives ƒ³ y+ax y² = 0, or, dividing by ay, xy+£2
BC gives a x y²+x2 y2 =0, or, dividing by x y², x+a=0..
CD gives x² y²+cx3o, or, dividing by x2, yy+cx=0.
DE gives c x³ +ee x² = o, or, dividing by c x², x+ =—= 0.
e e
C
a%
And £ £ gives-e ex² +ƒ³y= 0, or, dividing by ee, xx+ £y=0.
E A
If x be fuppofed indefinitely great, the Triangle being couched on the
Line without x, then C D is the only fuperior Determinator. Wherefore
this Suppofition reduces the propofed Equation to yy+cx=0.
If x be fuppofed indefinitely little, the Triangle remaining in the
fame Poſition, the inferior Determinators are AB and AF, which give the
Application
of the fore-
going rule
to another
example
276
ELEMENTS OF
Equations xy + £
=0, and xx+
a
3
f3
e e
y= 0, to which two Equa-
tions the propofed one is reduced by fuppofing x indefinitely fmall.
If y be fuppofed indefinitely great, the Triangle being couched on the
Line without y, there are three fuperior Determinators AB, BC, CD,
which give the Eqations xy +
11=0, x+a=0, yy+cx=0, to
which the propofed Equation is reduced by fuppofing y indefinitely
great.
a
Roots of the
3
But if y be fuppofed indefinitely little, there are two inferior De-
terminators, AE, E.D, which give the Equations, x x + £³
O,
ев
and x +
e e
C
y =
= 0, to which the propofed Equation is reduced by
the Suppofition of y indefinitely little.
LXXXI.
A Determinator may pass through more than two Cells, and then the
Equation it gives confifts of more than two Terms, but this Equation
may be refolved into feveral fimple Equations. If the Determinator
m n m+k n+l m tak mal
paſſes thro' the Cells x y, X y
y &c. it will give
mtk x+l
equation gi- a x y + b x } +0x
x
"
m +2k +21
y +dx
m+3k n +37
"
&c. =0,
ven by a de- in which Equation, the Terms correfponding to the empty Cells
terminator. will have their Coeficents a, b, c, d, &c. = 0. All the Terms of
n
k Z
this Equation being divifible by xy, it may be reduced to a+bxy
2 k 27
3 k 31
+ cx y + d x y
&c. = 0,
o, or fuppofing x y = 2, to
a + b z + czz+dz³, &c. = 0.
Let, R, r, p, &c. exprefs the Roots of this Equation, it may there-
be refolved into thofe Equations z
R=0, 2 -=0, 2-p=0, &c.
k l
kl
That is, x y—R=0, x y
r = 0, x y
= 0, &c. which are
k
?
k
k
y = rx
, y =
PX
2
&c. or infine
to
1:/
k: l
I :
k: 2
y = r
X
y = P x
2
&c.
reduced to y=Rx
1: -kil
J=R *
SPECIOUS ARITHMETIC K.
277
LXXXII.
1:1 -1:1
1:1-171
Thoſe Coeficients R, r, p, &c. of the Equations y == R
y =
P
I l -kil
x
>
X y=r x
&c. may be imaginary, they are all fo, when
the Roots of the Equation a+b+cz z+dz³, &c. = o, are imagi- They may
nary, which may happen as often as the Equation is of an even Num- be imagina
ber of Dimenfions, when the compleat Number of its Terms is odd, 7.
when the Determinator paffes through an odd Number of Cells, reck-
oning from the first full Cell to the laft. But when this Number of
Cells is even, the Equation a+bz+czz, &c. o, being of an odd
Number of Dimenfions, has at least one real Root. And in parti-
cular its Roots cannot be imaginary, when the Determinator traverſes
only two full Cells, which are in two contiguous horizontal or vertical
m + k n + 1
y
172 #
Columns. For in this Cafe, the Equation being a x y + bx
(kor being equal to Unity, on
account of the Contiguity
k
of the Columns) we will have a + b x y = 0, or a + bx y
= 0, or a + bx y = 0, that
1
4
a
is, x=-
, or y
b
k
до
1:/
-kil
1:1
k : l
X
P
x
LXXXIIF.
f:/ -kil
The Coeficents R, r, p, &c. may be real, and the Quantities R x
&c. nevertheless imaginary, or half
imaginary. A Root is faid to be half Imaginary, ſuch as y ax, which is real
when x is pofitive, and imaginary when x is negative, and is faid to be
entirely imaginary, as -xx, which is imaginary, whatever Value
pofitive or negative is given to x.
v
Since the even Powers of a pofitive or negative Root, are neceffarily
pofitive, but the odd Powers are pofitive, if the Root is pofitive, and
negative if the Root is negative, it is manifeſt that an odd Root is al-
ways real, whatever is the Power of the Quaatity whofe Root is extract-
ed, but that an even Root can only be real when the Power is pofitive.
Therefore it this Power is an even Power of a variable Quantity, the
even Root will be real or imaginary, according as the Power is taken
pofitively or negatively, that is, according as it is affected with a pofi-
tive or negative Coeficent, but if the Power, whofe even Root is extract-
ed is an odd Power of a variable Quantity, the Root is half imaginary.
1 :/ -k; /
-kil ľ
Thus in the Equation y = R
k
x =vRx if is an odd Num-
Or half ima
ginary.
278
ELEMFNTS OF
Obfervati-
ber, y is always a real Quantity, but if I is an even Number, y is half
imaginary, k being odd; and kleing even, y is real when R is pofitive,
and imaginary when R is negative.
LXXXIV.
k
We are further to obferve with respect to the Exponent-7 of x,
::/ -kil
in the Equation y = R 20 given by the Determinator.
10. That it is negative, whenk and I have the fame Sign, which happens
when the Arithmetical Progreſſions m, mk, m + 2k, &c. n, n + 1,
n+21. &c. of the Exponents of x and of y in the Terms which are
under the Ruler, (Art.LXXI.) are both increafing or both decreaſing. Then
the Ruler recedes at the fame Time from the firſt horizontal Line, and
from the first vertical Line; it only cuts one of thofe two Lines, or
fets cut from the Point of the analytick Triangle. In this Cafe the Sup-
pofition of x indefinitely great, renders y = R
1 : l - k: l
x
or
R
k
x
indefinitely little, and the Suppofition of x indefinitely little, renders
indefinitely great.
2º. That this Exponent
k
J
is pofitive, when I and k have contrary
ons refpect- Signs, which happens as often as one of the two Arithmetical Progref-
ing the ex-
ponent of fions m, m+k, m + 2k, &c. n, n + 1, n +2, &c. is increafing and
thofe Roots. the other decreafing, when the Ruler approaches one of the Sides of
the Triangle whilft it recedes from the other, when it cuts them
both any where elfe, but in the Point of the Triangle.
In this Cafe, the Suppofition of x indefinitely great or indefi-
I :/ k: l
or
alfo y = R X
:
(Rx²) indefinitely
! ! !
nitely little, will render
great or indefinitely little. They are of the fame Number of Dimenfi
k
ons if 14/17
= 1, if k = 1, if the Determinator is equally inclined to the
k
I
two Lines. But if >1, if k> I, if the Determinator is more in-
clined to the vertical Columns than to the horizontal ones, retrenches
a greater Portion of the Line without y, than of the Line without x,
then y = R X is of a higher Order than x, and of a lower Or-
I :l kil
k
der than x, if
< 1, if k < !, if the Determinator retrenches a
1
SPECIOUS ARITHMETICK.
279
lefs Portion of the Side of the Triangle without y, than of the Side
without x.
3º. If ko, which happens when the Determinator is paral-
1:1 -kil
k:
lel to the Line without X, then y = R x is reduced to
I : /
y = R x
O
I
=R.
wherefore
gives only finit Values for y.
being fuppofed indefinitely great,
4°. And for a fimilar Reafon, when the Determinator is parallel to
the Side of the Triangle without y, reduces to o, it is obvious that
y being fuppofed indefinitely great, gives for x only finit Values deter-
mined by the Roots of the Equation a x
x₂+
m+ak n+al
y
&c. = o, which fince
k
viding by x y to a + bx + cx
2 k
LXXXV.
>
11
m+k n + l
y + bx }
o, is reduced, by di-
&c. o.
Thoſe Particulars refpecting the Method of finding the greateſt terms of
an Equation being premiſed, let the Va'ue of y deduced from any propoſed
b
k
Equation involving x and y be expreffel by the Series Ax +Bx +cx +-Dx, &c.
in which the Exponents b, i, k, l, &c. increaſe or decreaſe accord-
ing as the Series is an afcending or defcending one. A, B, C, D, &c.
are the Coeficents of the fucceffive 'Terms, and as it is poffible that
one or other of them may vanish with all thofe that follow, it may
happen that the Number of Terms of the Series is finite, and then it
expreffes the the exact Value of y.
LXXXVI.
To determine as many Terms of the Series as you pleafe, proceed
thus. To find the firft Term. Suppofe x indefinitely great, if a de-
ſcending Series is required, but indefinitely ſmall, if an afcending Series
is required. This Suppofition reduces the Series to its firft Term Ax
for if the Series is a defcending one, the Exponent b is greater than i, k
b
or 1, &c. and x being fuppofed indefinitely great, the Power x is inde-
k
1
b
finitely greater than the others x, x, x,&c. (Art.LXXVI.) which may
be neglected without Error, and if the Series is an aſcending one, the Ex-
b
ponent b is the leaſt, and ≈ being indefinitely finall, the Power x will
Investigati-
on of the
fucceffive
terms of #
feries.
}
280
ELEMENTS OF
make all the others vanifh, therefore the Suppofition of x indefinitely
great, in a deſcending Series, and that of x indefinitely little, in an af-
b
cending Seires, reduces it to yAx. But thoſe fame Suppofitions re-
duce the propofed Equation to one or feveral Equations, fuch as y =
I
1:1
R
- k:/
given by the fuper, or infer. Determinators, (Art. LXXVIII.)
I:/
wherefore AR and b
k
the Determinators therefore ferve
1
to determine the first 'Term of the Series, or Series, when the propof-
ed Equation furniſh ſeveral.
The following Terms are found after the fame Manner. Let u ex-
i
k
prefs the Sum of the Terms Bx + Cx +Dx, &c. which fucceed the
b
first, confequently yAxu, this Vaiue of y, fubftituted in the pro-
poſed Equation, transforms it into another, including the variable Quan-
tities u and x; let x be fuppofed in this Equation indefinitely great, when
the Series required is a defcending one, and indefinitely little, when
the Series required is an afcending one and the fuperior or in-
ferior Determinators will give one or feveral Equations, fuch as
}:/ --kil
21 = R
(Art. LXXIX.) but the fame Suppofitions of x inde-
b
finitely great or indefinitely ſmall, reduces the Series u = B x + Cx &c.
i
to u = B x therefore B = R
>
1:2
k
and i
ī
hence the Deter-
minators of this first transformed Equation will give the fecond Term
i
Bx of the Series.
Transforming a new the Equation, fuppofing u=Bxt, where t
k
expreffes all the Terms Cx+Dx, &c. which follow the fecond Term
of the Series; and the Determinators of this fecond transformed Equa-
k
tion will give the third Term Cx of the Series, and proceeding after
the fame Manner, the fourth Term will be obtained, and the following
Terms to the laft, if the Number of Terms is finite, or at leaſt as
many as the End propofed may require, if the Number of Terms of
of the Series be infinite.
SPECIOUS ARITHMETICK.
281
But we are to obferve in the Courfe of thofe Operations, that the
Nature of afcending Series require that the Exponents of x fhould con-
tinually increaſe, and in defcending Series thote Exponents ſhould con-
tinually decreafe. Wherefore, tho at the first Operation performed on
the propoied Equation, all the fuperior Determinators are to be taken into
Confideration, to obtain all the defcending Series, or all the inferior Deter-
minators to obtain all the aſcending Series: In the fubfequent. Operations,
no Attention is to be paid to the fuperior Determinators which will give
the fame or a greater Exponent than was obtained by the precedent Õpe-
ration, nor to the inferior Determinators which will give the fame or a
lefs Exponent than what was obtained by the foregoing. Operation. If
there are no other Determinators, the Courſe of the Operations is finiſh-
ed and the Series is ended.
LXXXVII.
a x 3
0,
Let the Equation a y³--
3 x3 y
be propofed, and let it be placed on the
analytic Triangle, and this Triangle being
couched on the Line without x, there is but
one inferior Determinator, which, paffing thro'
the Cells y3 and x³, gives the Equation
3,
a 33
= 0, or y = x, which is the
first Term of an afcending Series. To find
a x3
3
x3
How to
or forked.
the fecond, I fubſtitute x + u for y, and the Equation will be transformed know when
into a x³ + 3 au x x + 3 au u x + a u³ x4 — x3 U a x³ = 0, a ſeries be-
or 3 aux x + 34 uux + au x4 u = 0, which being placed comes ima-
in its Turn on the analytic Triangle, gives two inferior Denominators, imaginary,
ginary, half
one which paſſes thro' the Cells u³, uux, uzx,
is to be rejected, becauſe it would give u=Rx,
this fecond Exponent of x being the fame
as was found by the firft Operation; but the
other Determinator, which paffes thro' the Cells
uxx and x4 gives the Equation 3 auxx-x¹=0,
which is the fecond Term of the
OT #=
xx
за
Series. The third Term will be found by fubfti-
xx
tuting +t for u in the foregoing Equation,
3 a
which transformsit into 4 +3 at xx+
tx4
+
3 a
+ ttxx+at3
+3attx+
$5
+2tx³ + 3 at tx +
3 a
3
3 a
† x³ = 0, or 3 a t x x + 1 x
26
27 aa
³
x6
fx4
+
27a a
3 a
+ + + x x + a 13 = 0₂
t t
4 N
282
ELEMENTS OF
which being placed on the analytic Triangle,
one Determinator gives — Rx, and which
confequently fhould be rejected, but the
other inferior Determinator which paffes
thro' t x`x and x6, gives the Equation
яб
27aa
= 0, or t =
x4
81 23,
3 at x x +
which is the third Term of the Series, for
*
за
x x
J
} = x + u, and u =
3a +t, and
x4
xx
24
#
&c.
за
81a3
81as,&c.hencey=x+
the Law of the Progreffion of the Terms being evident.
LXXXVIII.
Let the Equation y y 2 x y + x x
2 ay + ax + aao be
Application propofed, from whence is to be deduced the Value of y in x expreffed by
of the fore- an afcending Series.
going me-
thod to an
The Equation being placed on the analytic Triangle,
example. gives but one inferior Determinator couched on the Line
which gives yy 2 a j + a a = 0, which
without
ولا
=
a = o, or y = a;
has only one Root, but double, y
au being fubftituted for y, gives the transformed
Equation a a + 2 au + u u 2 ax 2ux + xx
2a a 2 au + a x + a a = 0, or u u
2 u x + x x = 0,
placing it on the analytic Triangle, I find but one inferior
Determinator, which gives the Equation u u — a x = 0,
±√ax=±ª
1:2 1:2
or u = ±√ ax =
for u in the Equation u u
X
—
a x
1:2 1:2
. Subftitutinga x +t
ax - 2 u x + x = 0,
which will transform it into a x ±2 a
I : 2 3:2
-ax F 2a
F2 X
1:2
2
1:2
1:2
t
x
+tt
2 + x + x
X
<= 0,
o, or
1:2 1+1:2
*
7
1:2
+ 2 a t x + tt 2 a X
2 tx + x x = 0, which
xx
put on the analytic Triangle, and as two of its Terms ± 2 a
1:2 1 + 1:2
and F 2 a X
have no Cells to lodge in,
let them be placed (as explained Art. Lxxv.) be-
tween two Cells, viz. the firft in the fecond Column
between the Cells t x2 or t, and t x¹, and the ſecond in.
the firſt Column between the Cell x' and the Cell x 2 ;
there will then be two inferior Determinators, one paf-
fing thro' tt and tx
1:2
1:2
1:2
1:2
t x
givingt-Rx is to be rejected, as having the fame
SPECIOUS ARITHMETICK.
283
Exponent as the precedent Term; the other Determinator paffing thro' the
Cells t x
I: 2
1 + 1; 2
and x
I: 2
1:2
I: 2 1+1:2
, gives +2 a tx +2a X
ort=x, which is the third Term of the Series.
To obtain the fourth, x+s is to be ſubſtituted for in the laſt Equation
1:2 I : 2
+2a tx +tt 2a x
transform it into 2 a
I: 2
+ss+2a X
I: 2
+24 SX
1.: 2
1+1:2
1:2 1+1:2
-2 tx +
x x = 0, which will
1:2 1+1 2
X
1:2
± 2 a
tt2
2 x x
5 X6
I: 2
+rs=0; thofe Terms being placed on
the analytic Triangle gives only one Determinator, which
I : 2
gives s Rx ; this Exponent being less than the
preceding one, is to be rejected. Hence the Series is ended
for y = a + u = a + √ a x + t = a± √ ax+x.
= 0, or into
On examining what this-laft Determinator gives, or its Equation
± 2 a
1:2
142
sx +ss = 0, we will find it has two Roots, 1°. s = 0,:
1:2 1:2-
which terminates the Series; 2º. s = 2 à X + 2 √ ax,:
which ſubſtituted in ya√ a x + x +'s, gives y a ±√ a x
+ x ± 2 √ ax, which is the fame as y a ± √ a x + x, which is
therefore the exact Value of y, which may be eaſily verified, fince by
taking away the Irrationality, the Equation y = a + √ ax + x will be
transformed into the propofed Equation py2 xxx−2 ay+ax.
+aa=0..
yy
LXXXIX.
+
of the fore-·
It is to be obſerved, 1°. that if among the Equations which the fuc- Application
ceffive Determinators furnifh, there be found any one whoſe Roots are going me-
all imaginary, the whole Series which the firft Terms feemed to an- thod to
nounce will become imaginary, for one imaginary Term renders the another
example.
whole Sum, of which it is a Part, imaginary, unleſs what is imaginary
in one Term be deſtroyed by what is imaginary in another Term, which
cannot happen in this Cafe,, where x in each Term has a different Ex--
ponent.
2º. That if among the Terms of a Series there be any which are half
imaginary, the Series is half imaginary, that is, imaginary when is fup-
pofed negative, and real when x is fuppofed pofitive or reciprocally.
3°. That if among the Equations which ferve to determine the fuccef--
five Terms of a Series, there be found any which have feveral real Roots;
284
ELEMENTS OF
then the Series becomes forked as it were, and is multiplied into as many
Series as there are real Roots, as often that happens.
XC.
2
Let the Value of yinx, expreffed by an afcending Series
deduced from x + x² + ayy 2 a² ya 30
Example of be required.
an imagin-
ary feries.
Example of
a feries half
imaginary.
This Equation placed on the analytic Triangle
couched on the Line without x, furniſhes but one in-
ferior Determinator, which gives the Equation
ayy-2 a² y + a30, which has only one Root,
but double, viz. y a = 0, or y = a; let then
2
=a+u, and fubftituting this Value of y in the propofed Equation,
it will transform it into x3+ax² +ux² + au u = 0,
which being placed on the analytic Triangle, furnishes
but one inferior Determinator, which gives
a uu+axx =0, whofe Roots u =+V/ -xx are
imaginary; confequently the Value of y in a cannot
be expreffed by an afcending Series, becaufe the only
one which could exprefs this Value would be
y = a + u a + √ x x, &c. which is ima-
ginary.
XCI
Let the Equation x2y+ayy-2axy+axx=0
be propofed, from whence is to be deduced the Value
of y in x, expreffed by an afcending Series. This
Equation being placed on the analytic Triangle, fur-
nishes but one inferior Determinator, paffing thro?
the Cells yy, xy, xx, which gives the Equation
zy — 2 á x y + axxo, which has only one
Root, but double, y=x. Subftituting therefore
xu for y in the propofed Equation, it will be
a y y
༢
transformed into x3 + x² u+au uo, which being placed on the
analytic Triangle, furniſhes but one inferior Deter-
minator, which gives the Equation au u + x³ — 0,
which has two Roots u+and+3
fubftituting
X
3
t
- ===+1 = a
a
Q
1: 2
a
3:2
X
+t
for u in the first transformed Equation, the fecond
1:2
will be a
3+1:2
1:2
3:2
+txx+2a tx-x +att = 0,
SPECIOUS ARITHMETICK.
85
which being placed on the analytic Triangle, fur-
nifhes two inferior Determinators, one ufelefs, be-
3:2
cauſe paffing thro' the Cells t´t, tx , it gives
3:2
>
t = R x where x has the fame Exponent as
in the precedent Term; the other paffing thro'
t x
3 : 2
and x
I : 2
±2a
3 + 1 2
>
gives the Equation
3:2
1:2
3 +1:2
± a
o, or t
or t =
wherefore the three firſt Terms of the Serics are +/-
x3
a
※3
a
12
2 a
2 a
From the Inspection of which it appears, 1°. that the Series is ima-
ginary, when ≈ is fuppofed pofitive, becaufe then-is
-is an imagi-
nary Quantity; but if x be fuppofed negative, the Series is real, and then
2º. The Series is double, becaufe the Term - has equally the
Sign and the Sign, the Equation a u ux30, which furniſh-
ed this Term, having two real Roots #= + √
wherefore in reality there are two Series, the three firſt Terms of one
?
+3
,andu
‚andų——
a
x3
a
are x + √
*3
x x
and of the other x
2 a
XCII.
x3
XX
«
24
The general Method being attended with tedious Calculations, we ſhall
now proceed to explain how the Analyſts have abridged them.
transform-
We are to obferve, that the Subftitution of Ax+u for y (Art. LXXXVI.) Manner the
in any Term of the propofed Equation, transforms it into as many Terms terms of a
as there are Columns, from the firft to that in which it is placed inclu- ed equation
fively; each Term being placed in a different Column, and all thofe are placed
Terms being fituated on the fame ftraight Line, parallel to the Determi-
nator which gave the Equation y = A x
ג'
b
h
b
For the Power n of u+ Ax being u+n Ax u +
I
n.n
I .2
nk
A x น +, &c. to  x
2
2 b
11
2
n
12
I . 11
2
3 3 h 1-
•
A x
21
+
1.2.3
#
m
or the laſt Term, which being fubftituted for y in a Term as x y
in the an-
nalytic
triangle.
286
ELEMENTS OF
of the Order m + nh, y being of the Order h (Art. LXXIV.) and which is
m *
placed in a Column preceded by n others, will, transform it into xu
24 m + 2b n 2
+ n Ax
m+b
na
n. n
I
น
+
A x
I. 2
น +,&c. to A. x
m + a
which are all of the Order m + n h. But all the Terms which are of the
fame Order, are placed in a right Line, parallel to the Determinator
hence all the Terms into which "" istransformed are in a right Line parallel
b
to the Determinator which gave yAx; and it is evident, that the Term
m "
#
m + h
xocupies the Cell in which the transformed Termx y was lodged
in a Column preceded by others; that the ſecond Term x น
is in a Column preceded by - 1 Columns; that the third x
1
* m + xb
is in the next Column, and fo on to the laft Д x
Example. firſt Column, or in the Line of the Powers of x.
ولا
m+2b
น
which is in the
Thus when the Equation x2 y2 + a y3
+ bxy ² + c x² y + d d x y +ƒ³ xo is
placed on the analytic Triangle, couched on
the Line without x, it will give four Determi-
nators; firft an horizontal one, which paffes
thro' the Cells x2 y2 and x²y, which gives
for y a conftant Value (Art. LXXXIV.) & which
may be expreffed by A; fubftituting A+ u for y
in the propoſed Equation, it will be transform-
ed into A²x² + 2 A u x² + u² x² + A³ a
+ 3 4² a u + 3 A a u² + a u³ + A² b x + 2 A b u x + b u²
+ Acx²+cu x² + A d d x + d du x +ƒ³ x = 0, from whence it
appears, that the Terms
2
2
x*y´
2
73
x²y
2
202 21.2
213
3
น
of the propofed
x y² duce
Equation, pro-
duce in the
transformed
Equation
* * *
x y
, น
2
24.
21, 20
2,
>
u',
uQ
24
x u
,
2
x
169202
X
xu, x
Xu, X
*
*
-*
米​ナナーナ
​x
Thus each Term has given one to each Column which precedes it,
and thoſe Terms are on an horizontal Line, that is, parallel to the De-
Terminator which gave y = 4.
SPECIOUS ARITHMETICK.
287
24
The fecond Determinator of the propofed Equation paffed thro' the
Cells y3 and x2 y2, and gave y = A x: the Subftitution of A x²+u for
y, transforms the Equation into A² x6 + 2 A u x² + u² x² + ³ a x˜
+3 Aau x² + 3 Ä a u²x² + a u³ + A² b x5 + 2 A bu x³ + bu² x
+ Acx² + cu x² + A d d x ³ + ddux+ƒ³ x = 0, therefore the
Terms
x 2 y 2
13
x y²
X J
X
u, xo
43
2
>
2
have
2
**
3
X u, x5
given) x²u, x4
xu, x³
X
X
1
and placing the transformed Equation on the analytic Triangle, it will
appear that all the Terms into which a Term of the propofed Equation
had been transformed, are placed in the fame ftraight Line parallel to the
Determinator, which gave the Equation y = Ax2, the fame Thing may
be verified with Reſpect to the two other Determinators of the propofed
Equation; the third paffes thro' the Cells y3 and x, and gives an Equa-
tion of this Form y = Ax ; fubftituting therefore A x +u for y,
and the transformed Equation will be A² x
2 2
3
1:3
24
1+1x3
2:3
2.
2+2:3
1:3
น
ولا
2+1:3
+2 Aux
3
+au+
2+1:3
2 1:3
+ u x + A a x + 3 A a u x +3Аau x
A
2 1 +2:3
A b x
A d d x
2...2
y
1+1:3
+2 Abux
+ bu x + Acx
+ d d û x +ƒ³ x=0, hence the Terms
2 2 21:3
x u x
2+2:3
2, X
3
113 2 2:3
>
X
u, x
u, x
I+2:3
U₂ X
73
2
น
2 1+1:3
x y² give xu,x
x²y
x y
મ
2
x u, x
Xu, X
21:3
1+1:3
2
+cux +
ངས
288
ELEMENTS OF
This transformed Equation being placed on the analytic Triangle, it
will appear that each Term of the propofed Equation has given a Term
to each Column that precedes it; and that thofe Terms are on a firaight
1:2
>
Line, parallel to the Determinator which gave y = Ax but as the
Exponent of the Order of y is a Fraction, it was neceffary, in order
to place thoſe Terms, to divide into three equal Parts the Intervals of the
con.ciguous Cells in the fame Column (Art, Lxxv.)
2
Finally, the fourth Determinator of the propofed Equation paffing thro'
x y and x, gives yAx and the Subftitution of Ax
>
2
+u for y,
charges the propofed Equation into A +2 A u x + u u x x +
3
2
3
2
A a t
+zA aux
2
2 A bu + b u ~ + A c x
therefore the Terms
3
2
+ 3 A a 20 21 2 +an + A b x
+ cux + Add + d du x +ƒ³ x =
24
+
O,
x²y
2
13
x y
2
xy
x y
K
2
produces {
r
2, 2
X 11
3
>
ว
xu,
I 24
24
-3
22 **
21
>
Xx
U, X
I
X U
x2
X 1,
X
21, X
, >
X
X
Here the fame Rule is obferved, but as the negative Exponent (— 1)
of the Order of y, introduces Terms in which the Exponent of x is nega-
tive, it was neceffary, in order to place thofe Terms, to produce the Tri-
angle below the Line without x (Art. LXXV.)
XCIIL.
Thofe Examples fhew, that when feveral Terms of the propofed E-
quation are under the Determinator, or any of its Paralells, or are of the
fame Order, they are transformed into Terms, which ranged in the fame
ftraight Line, feveral of them fometimes lodge together in the fame Cell.
Thus in the foregoing Example, if the horizontal Determinator be
employed, the Terms x2 y2 and c x²y, which were under this Deter-
SPECIOUS ARITHMETICK.
289
1
2
minator, were transformed into x2 u² + (2 A + c) x² u + (A² + Ac) x²,
and the Terms b x y², d d xy, f³ x, which are under a ftraight Line,
parallel to this Determinator, have given b x u² + (2 A b + d d) x u
+ (A² b + A d d + f³) x, which are ſtill under the fame ſtraight Line.
Wherefore, when there are feveral Terms of the fame Order, as
m + b # − 1
B
172 n
axy + bx y
mt2b n - 2
+c x
و و
&c. continued to x
b
m + nb
>
which are all of the Order m + n h, fince y (= A x ) is of the Order h,
b
the Subſtitution of Ax+u for y transforms it into a Series of Terms,
171
7
fuch as Px u + Q¤
m + n.b
nued to Z X
m+hn - Į
18+2b # 24
H + Rx
น +, &c. conti-
which alſo are all of the Order m†nh, in which
the Exponents of x form an arithmetical Progreffion, the common Diffe-
rence being h, and the Exponents of another arithmetical Progreffion, the
common Difference being 1.
น
The Coeficients P, Q, R, &c. of thofe Terms, may be calculated by
a contracted Rule, derived from the following Confideration: The Term
171
axy,
by the ſubſtitution of u+Ax for y, is transformed into a x
+ na Ax
u
m + b
n
n
I
2 m + 2 k *
•
น
+
a A x
น
I . 2
11. n
I . 71
2
3
#2 + 3b
*
3
+
1 A x
26
,
&c. the Term b x
Y
1.2.3
is transformed into b x
12 + b n
I
11
T
1 + 2 b
2
น
+
b A x
น
+
I
71
I. 11
2
2 m + 3 b n
3
m + 2 4
b A x
14
y
I. 2
m+27
M
2
wr + 3b n
3
2
into cx
21
+
C Ax
16
I
3
y
into d x
И
&c.
m + b
I
71 ·3
&c.
d x
m + 3b n
1/1 #1
a x y + b x
is transformed into a x
I
n
+ "= 1 b
I
b A + c)
+")
11
I . 11
2
I. 2
2
X
&c. the Term cx
m + 3b n − 3
111 11
+ cx
m + 2 b
за
u + (na A + b) x
m+ 2 b
༡
น
b¹² + " — ² c A
A
I
is deduced the following Rule:
2
+("
+ d)
•
&c. and the Term
Wherefore, the Sum
+dx
m + 3 b
71 + h π-1
m
m + 3 b
I
26
•
น
I
I . 12
2.3
11
3
J
I . 2
I
* A
2
3
2
a. A +
,&c. from whence
40
I
1
290
Abridged
method of
requifite to
feries.
ELEMENTS OF
b.
1
Write in the firft Line, all the Terms of the fame Order, or even the
performing whole Equation, diſtinguiſhing, for Conveniency's fake, the Orders of its.
the trans- Terms, changing, if you pleafe, y into u; I fay if you pleafe, for it will
formations be found upon Trial, that it will be more commodious not to do it; but
be made for then it muſt be obſerved, that y, which denotes before the Operation the
finding the whole Series, and during the Operation its firſt Term only, denotes,
terms of a during the fecond Operation, the fecond Term, and, during the third-
Operation, the third Term, &c. Having wrote the propofed Equation
in the firft Line, and ranged the Terms according to their different Or-
ders, multiply each Term by the Exponent of y and by Ax, and
divide thofe Products by y, and this will give a fecond Line; multiply,
each Term of this fecond Line by half the Exponent of y and by A x.
and divide the Products by y, and a third Line will be obtained. Each
Term of this Line, multiplied by one third of the Exponent of
and by
Ax, and divided by y, will give a fourth Line. And this Operation
is to be continued until there remains but Terms without y; then the
Sum of all thofe Lines will be the transformed Equation.
b
XCIV.
2 2
y
2
b
2
Thus in the Equation of the foregoing Article, if we employ the hori-
zontal Determinator, which gave the Equation x² y²+cx² y = 0,
Application
of this me- or y
c, there will reſult A c and h = 0; and the Operation
thod to an is performed thus:
example.
I, Order.
II. Qrder,
བལ་ལི་དམ་
2
x² y² + c x² y
+ bxy² + d d xy + ƒ³ x
2
I
2
I
0
X-c)
24
-2 c x² y = c c x
133
2 b c x y
14
c d d x
0.
+ccxx
+bcc x
O
O
III.Order.
+ ay³
ау
3
3
1
засуу
2
2
3accy
118
a c3
2
The transformed Equation, therefore is x2 y2 + (0 — 2 c) x² 3
+ (cc — c c) x² + b x y ² + (d d − 2 b c) x y + (ƒ³ — c d d + b c c ) x
'+ays
3 a c y y + z acc y a c³ = 0, in which the fecond
Term is reduced to cx²y, and the third vanishes.
—
3
SPECIOUS ARITHMETICK.
291
+
If in the fame Equation we would employ the Determinator that paffes
2
thro' the Cells c x² y and f³ x, which gives y
will be performed thus:
I.Order.
II. Order.
24 2,
f3 the Operation Application
CX
of this me-
thod to
III.Order. IV.Order.
another
example.
3
cx²y +ƒ³ x + x² y² + ddxy + bxyy + a y³
I
О
2
I
2
3
f3
X
CX
·ƒ³ x
2 f3 x y
fad
ƒ³d d
26f37
3afyy
о
C
14
2
f
C
CX
I
2
О
NÍN
+
f6
CC
О
+
b fo
CCX
O
3 a ƒ6 y
C C X X
I
3
a f9
+
3x3
2
Whence the transformed Equation is cx² + (ƒ³ — ƒ³)….x + x² y
2
+ ( d d — ²ƒ³) x y — (f3 d d — to ) + b x y y —
C
-)
C
CC
+ay
3
3 a f³ y y
+
3 a foy
a f9
2 24
C X
x
c3 x3
bf6
2bf3y
+
C
CCX
=0, in which the fecond
Term diſappears.
If the Sum ax y
XCV.
# + π-1 I
y
Ternis of the fame Order whatever, is
لو
+ bx
万
​A x
which is the Value of u,
+cx
212 +2 13 14 In
#
y
&c. of
In what
cafes fome
terms of
diviſible, once or ſeveral Times, by of the
the Sum of the Terms into which thoſe
b
transform-
tions are.
it is transformed by fubftituting + Ax for y, is alfo divifible, the fame ed equa-
Number of Times, by u, becauſe thoſe two Sums
only differ by their wanting
Expreffion. But the Terms of the transformed conftitute a Series
M
(Art.cIII.) Px u + Qx
m+b
m + b =
- I
m + 2
B
#1 -- 2
11
+ Rx
LL
+, &c.
122 + (1 − 1 ) }
terminated by the Terms + X ≈
+Zx
1 x b
m + (1 − 2) 15 24
کا
This Series cannot be divifible by u, unleſs the laſt Term
o and Zo; it cannot be divifible by u, or twice by u, unleſs the
two laſt Terms Y and Z be o; it cannot be diviſible by u³,
3 or three
Times by ", unleſs X, Y, and Z be = o. In general, as many Times
>
393
ELEMENTS OF.
&
asu, or rather ” A X
b
,
which is its Value, divides the Sum of the
Terms of any Order, fo many Terms of this Order are wanting in the
transformed in the first Columns, for the Terms Z, Y, X, &c. are thofe
which are placed in the firft, fecond, third, &c. Columns.
B
Wherefore, fince yAx o is one of the Roots furnished by the
b
Determinator, yAx divides at least once, the Sum of the Terms
which are under this Determinator; confequently, in the transformed
Equation is neceffarily wanting the Term correfponding to the Point
where the Determinator cuts the firft vertical Line, and in the transform-
ed Equation, will be wanting the two Terms correfponding to the Points
where the Determinator cuts the first and fecond Column, if yAx,
divides the Sum of the Terms which are under the Determinator, twice,
A x
万
​}
55
if y
=o be a double Root of the Equation which this Deter-
minator furniſhes; but if yAxo be a triple Root of this Equa-
tion, in the transformed Equation will be wanting the Terms correfpond-
ing to the Points where the Determinator croffes the three firſt Columns,
and so on.
P
P
1
{
R*
*-*-*-*
Pp p
Р Р
R q 9
XCVI.
If therefore P Q repreſents a Determinator, and yAxo be
fimple Root of the Equation which it furniſhes, in the transformed Equa
SPECIOUS ARITHMETICK.
293
tion under this Determinator will be wanting only the Term which
ſhould occupy the Cell Q in the first Column QÓ, at leaft the Term
q in the ſecond Column will not be wanting. Pq is alſo a Determinator
of the transformed, for all the Terms of the transformed are below (or
above) thofe under P q, as were all the Terms of the propofed Equation
(Art. LXXXVI.); but it is a ufelefs Determinator, becaufe P q being Part
of PQ has the fame Inclination to the Lines and Columns as PQ
b
&
Wherefore PQ, having given y = Ax, Pq will give u Bx (Art.
LXXII.) confequently having employed PQ, the Determinator Pq is
wfelefs; but from the Point q there parts another Determinator which
paffes thro' the higheft or loweft full Cell of the firft Column Q_O, and
which gives an Equation by which the ſecond Term of the Series is de-
termined.
*-*
*V
оо
S
Что о
9%
о
о
T
R$
S
p
P
P
V
Ӧ
P p p R o q a
b
But if yAxo be a multiple Root of the Equation furniſhed by How the
the Determinator PQ, for Example, a triple Root; then in the trans- irregular
formed the Cells Q, q, q, remain empty, and the Determinator P q is terms of a
terminated at the Cell R on the fourth Column, it is uſeleſs, for the feries are
inveftigat-
Reafons already given, to employ this Determinator. But there parts ed.
another (RS) from the Cell R, which may terminate in S at the firſt
Column, and this is the fecond Determinator which gives an Equation
by which the fecond Term of the Series will be found.
It may happen that the Term S is wanting, and that the Determina-
tor RS is terminated on fome other Column than the firſt, as in T, and
then this Point T gives rife to another Determinator TV; and if this
does not reach the firſt Column, but is terminated fooner, there parts
from its Extremity another Determinator, and perhaps from that ano-
ther, &c. to that one which is terminated on the firft Column. Each
Z
294
How to
a feries
begins to
become
regular.
ELEMENTS OF
of thofe Determinators being differently inclined to the Lines and Co-
lumns, give a different Exponent to the Power of x in the fecond (or
third, fourth, &c.) Term; whereby the Series is forked into as many
Series as there are Roots in all the Equations given by all thofe Deter-
minators.
To invefligate which, it fuffices to confider the Determinator R T,
which parts frora the Point R, Extremity of the first Determinator P R
which has been neglected, and to make Ufe of all the Roots of the Equa-
tion which it furniſhes; one of thofe Roots is u — o, the Sum of the
Terms which are under this Determinator RT being diviſible by u, fince
RT does not reach the firſt Column, which is the Line without u; and
this Root being employed to obtain the following transformed Equation by
fubftituting o+t for u, which is effected by only writing t for u in all
the Terms of the Equation, which transformed confequently furniſhes
the fame Determinators as the foregoing one. And as PR and R Thave
been already employed, we arrive at the Determinator TV in the fame
Manner, as if at first we had paſſed from P R to TV, the Root u = 0 of
the Equation furniſhed by R T producing the fame Effect as if RT had
been neglected. It would be the fame, if the Determinator TV did not
reach to the firſt Column, but was accompanied with other Determinatorɛ.
XCVII.
ง
•
From whence it appears how the irregular Terms of the Series are to
be calculated. I call by this Name thofe which are given by Equations
which have ſeveral Roots, real or imaginary. But it is ſcarce poffible
that this Kind of Diſorder ſhould laft long; for as foon as we come to a
Determinator whofe Equation has no multiple Roots (or if a fimple Root
be employed, if this Equation has fimple and multiple Roots) in the
transformed Equation, there will be wanting of the Terms which are
placed under this Determinator, for Example, RS, but the Term S
know when which fhould be on the firſt Column (Art. xcv.). The Determi-
nator therefore of this transformed will part from the Cell T the higheſt
(or loweſt) of the fecond Column, and will pafs thro' the Cell V, alfo
the highest (or the loweft) of the full Cells in the firft Column; where-
fore, in the Equation furniſhed by this Determinator TV, the variable
Quantity u forExample does not furpass the firſt Degree, becauſe Tis on the
fecond Column, which is the Line ; and this Equation will only have
one Root, which will certainly be real; and then the Series becomes re-
gular, becauſe all the enfuing Determinators parting from the Point T
no Equations any more occurs which have feveral Roots; and all the
fucceeding Terms of the Series may be calculated with greater Facility
by a Method we fall now proceed to explain.
u
SPECIOUS ARITHMETICK.
295
XCVIII.
the form of
Suppoſe a Determinator to be obtained whofe Equation has a fimple Determi-
Root, and that this Root is employed; for greater Simplicity I fhall call nation of
it the first Determinator, abftracting from all the foregoing ones, if there a regular
were any; I fhall alfo call the Equation whieh it furnishes the propofed feries or the
one, altho' it may be one of the transformed ones. Let m exprefs the Or- feries of the
der of the Terms thro' which this firft Determinator paffes, and let of the
regular
る
​exponents
3- 1 x = 0·be a fimple Root of the Equation it furniſhes. Subfti- terms.
tuting Axu for y in the Sum of the Terms of the Order m, the
b
M
one which ſhould be on the firft Column would be x, we negle& its
Coeficient as being of no Confequence; but this Term is wanting, fince
yAx is fuppofed to divide the Sum of the. Terms of the Order
¿
-b
(Art. XCIII.). The Term x u which follows falls on the fecond
Column, and the Cell which it fills is the higheſt (or the loweſt) of the
full. Cells of this Column. If mn is the Exponent of the Terms of
the fecond Order (the Sign is for the defcending Series, the Sign+
for the afcending ones) the higheſt (or the loweſt) Term in the firft Cor
** 干
​hence the Determinator of this transformed paf-
lumn will be x
fing thro' the Cells x
b⋅ Ix
= Ex
>
17
b
# = n
u and x will give u=Bx.
for the. fecond Term of the Series.
m Fr−#+ &
In
Wherefore the Diffe-
and of
rence of the Exponents h and hn of the first Term Aw
b = "
the fecond B x of the Series, is the fame as that of the Exponents\
m and m nof the firſt and fecond Order.
u will remain
In all the following Transformations, ther Cell x
full, u being changed fucceffively into t, s, r, &c. and all the following
Determinators will part from this Cell to pass thro' the higheſt (or the
loweſt) of the full Cells of the firft Column; they pafs fucceffively thro'
the different Cells of this Column, becaufe at each transformation, the
Cell thro' which the Determinator paffed is emptied; but on the other
Hand, each transformation fills fome new Cells of this firft Column.
h I s
Fin
*
Hence when B-x + t is ſubſtituted for u in the first transformed,
# +38
F'4"
the Terms of the firſt Order m fill the Cells x
, *
&c. and the. Terme of the fecond. Order n alfo fill the Cells
>
X
296
ELEMENTS OF
772 + 2 x
FAL
I 3th
ཚེ་
X
,
# + j *
&c. in general x (j expreffing any Integer)
but the fame Subftitution in the Terms of the Order m +p will fill in the
firft Column, the Cells x
13 IP.
774
Ip I "
X
m + P = 2 2
&c. in
m F p = j n
Fp F n
general x
m + 2n
be the higheſt (or the loweft) full Cell in the firft Column,
M
b
m + 27
✰ and x
m + 3 r
and will
is the highest (or the
b = 37.
If x.
the third Determinator will pass thro' x
b I 2 r
give t=Cx
And if afterwards x
loweſt) of the full Cells of this Column, s will be = D x
and
thus the fucceffive Exponents of x in the Terms y, u, t, s, &c. of the
Series, will be h, hn, h2 n, h + 3 n, &c. in arithmetical Pro-
greffion, whofe Difference is n. The Series would have no other Terms,
if there were in the propofed Equation no other Terms but thofe of the
Orders m and m +n, all the Transformations, ad infinitum, would give
no other Terms but fuch as are contained under this general Expreffion
m I j n
X
(jis an Integer, or fometimes o). But if there be in the pro-
pofed Equation, Terms of another Order, whofe Exponent is m†p,
theCell x m will be once the higheſt (or the loweft) of the firft Co-
lumn; then the Determinator, which parts always from the Cell
72 + p
- b
121
b
་་
m
I pm I b
X
u (or x t, or x
h = P
(or t=Cx
s, &c.) will give u = B x
b = p
or s = D x
b + p
= B x
the Term in which
x has for Exponent hp is therefore one of the Terms of the Series.
b Ip
hIP
The ſubſtitution of B x +t for u (or Cx + for t, &c.) in
the Terms of the Orders mjn will fill, in the first Column, the Cells
- m = j n = p m + j n + 2 p m = j n = 3 p
X
2
&c. this firft Column
therefore will acquire Terms included under the general Expreffion
m F j n = j p
and the Determinator paffing fucceffively thro' thoſe
Terms, will give to the Series, Terms contained under this general Ex-
h = j n = j p
preffion Hx
,
If the propofed Equation included a fourth Order of Terms whofe Ex-
ponent is mq, the general Expreffion of the Terms of the Series would
b = j n = j p = j q
吁​!
be H x
Orders of Terms.
, and fo on, if there be a greater Number of
SPECIOUS ARITHMETICK.
297
XCIX.
Herce, when in the Calculation of a Series we come to the Regular
Terms, that is, when we come to a Determinator whofe Equation has no
multiple Roots, or when a fimple Root of the Equation furnished by a
Determinator is employed; the Succeffion of the Exponents of x in the
following Terms of the Series are easily found thus: take the Exponents
m, m + n, m + p, m + q, &c. of all the Orders of the Terms of the
Equation, and fubtract them all from the greater m (or fubtract from
them all, the leaft m) in order to obtain the Differences n, p, q, &c.
then add or fubtract from h, Exponent of the firſt Term, which is given
by the firſt Determinator, the multiples n, 2 n, 3 n, &c. of the first Rule de-
Difference; then add to, or fubtract from, all thofe Terms fucceffively, the forego-
the Multiples p, 2 †, 3 p, &c. of the fecond Difference; and add to, ing confide-
or fubtract from all thofe Terms, the Multiples q, 29, 39, &c. of the rations for
finding the
third Difference fucceffively, and continue thus until all the Differences fucceffion
are exhauſted. Finally, range all thofe Exponents according to their of expo-
Magnitude.
rived from
nents of the
regular
The arithmetical Progreffion which begins by h, and whofe Differ- terms of 1
ence is the greateſt common Diviſor of n, p, q, &c. includes thoſe feries.
Exponents, but contains ufelefs Terms, unleſs the leaft Difference # be
a common Divifor of all the other Differences.
C.
By this Rule, the Form of the Series is obtained, that is, the Succef-
fion of the Powers of x which form thoſe Terms; but the Coeficients of
thoſe Terms are alſo to be found, which is eafily effected, by giving to
each of the Powers of x, which enters the Series, an undetermined Co-
eficient A, B, C, D, &c. and ſubſtituting, in the Equation, for y this unde-
termined Series which expreffes its Value, and determining, one after the
other, each Coeficient A, B, C, D, &c. by means of the complex
Coeficient of each Term of the transformed Équation, which are equal
to Nothing, fince otherwiſe the Whole could not be equal to Nothing.
3 3
CI.
2 x5 y 2
a
Application
of this rule
example₫
Let us apply this Method to the Equation 6 x7
+ 4 a³ x³ y + 2 a5 x x — 3 a5 x y + a5 y yo, fuppofe the Value
of y in x, expreffed by an afcending Series, is required.
The Equation being placed on the analytic Triangle, there is only one to an
inferior Determinator, which gives the Equation a5 y y 3 a5 x y
+ 2 a5 x x = 0, or y y 3 x y + 2 x x = 0, which has two fimple
Roots y = x and y = 2x, in genèral y = Ax, wherefore 1.
Drawing ſtraight Lines parallel to the Determinator thro' all the Terms
of the Equation, it will appear that it is compoſed of three Orders of
Terms; the firft, which contains the Terms a5 y y, 3 a5 x y, 2 a5 x x,
thro' which paffes the Determinator, has 2 for its Exponent, becauſe
this ftraight Line cuts the firft Column in the Centre of the Cell x²;
4 P
298
ELEMENTS OF
6 x7
-2x512
-2 x5y²-
3 2 2
the fecond includes the Terms-a3 x2 y2
4
and +4 a³ x³ y, and its Exponent is 4,
a3 x3y,
becauſe the ſtraight Line which paffes
thro' the Cells x2 y2 and x3 y cuts the
firft Column in the Centre of the Cell x+;
and the third Order of Terms, which is
compofed of the Terms 2 x5 y² and
6
has, for a fimilar Reaſon, 7 for its
Exponent: thoſe three Orders of Terms
and their Exponents may be alſo diſcover-
ed, by fubftituting in the Equation for y
x7
Ъ
2
=Ax, which transforms
4
2 A² x7 A² a³ x² + 4 A a³ x4 + 2 a5 x x
S
its Value Ax
it into 6 x7
3 A a5 x x
+ A² a5 x x = 0, where it evidently appears, that the two firft Terms
are of the Order 7, the two following of the Order 4, and the three laſt
of the Order 2. Since an aſcending Series is required, the leaſt Expo-
nent 2 is to be fubtracted from the two others 4 and 7, by which the
Differences 2 and 5 are obtained; then the Numbers contained in the
general Expreffion h +jn +jp or 1 + 2 j + 5j is to be fought, by
adding to 1 (h), the Multiples of 2 (n), and adding to all thofe Numbers
the Multiples of 5 (p), which will give
I, 3, 5, 7, 9, II, &c.
6, 8, 10, 12, 14, 16, &c..
II, &c.
16, &c.
21, &c,
which being ranged according to their Magnitude, conftitute the Pro-
greffion 1, 3, 5, 6, 7, 8, 9, 10, 11, &c. the Form of the Series will be
thereforey A x + B x³ + C x5 + D x6 + E x7 + F x8 +, &c.
which Value of y ſubſtituted in the Equation gives
+6x7
-2 ² x¹
1
&c.
a3 A² x4_2a³ ABx6
-2a3 ACx8
&C.
a3 B2 x8
&c.
B
33
+4a³x³y=
+2a5xx+2a5 xx
+ 4 a³ A x² +4 a³ B x6+
+4a³ Сx8 + &c.
—3a5xy=—3a5 A xx-3 a5 В x²-3 a5 C x6—3 a5 D x7-3 a5 E x8
&c.
+ a5yy=+ a5 A² x x +2 a³ A B x¹+2a5 A Cx6+2a5 A Dx7+2a5 A Ex8 + &c.
+ a5 B²x6+
2
+2a5B Cx8+ &c
SPECIOUS ARITHMETICK.
299
2
Now fince 6 x7-25 y² — a³ x² y²+4a3 x3 y + 2 a5 xx-3 a5 xy
a5yyo, it follows, that the Sum of theſe Series involving muſt
vanish; but that cannot be, if the Coeficient of every particular Term
does not vaniſh. For every Term where x is infinitely little, is infinitely
greater than the following Terms: So that if every Term does not vaniſh
of itſelf, the addition or fubtraction of the following Terms, which are
infinitely less than it, or of the preceding Terms, which are infinitely
greater, cannot deſtroy it, and therefore the Whole cannot vaniſh. It
appears therefore, that the complex Coeficient of each Term being put
equal to nothing, will furniſh the following Equations :
2a53a5 A 45 AA 0, or AA-3A+2=0,
+ à5 =
— a³ A² + 4 a3 A-3 a5 B + 2 a5 AB = 0,
or A A — 4 A =
— 2 a³ A B + 4
4 B
or 2 A B
a
(2 a à A — 3 a a) B,
B-
AB=0,
a³ B - 3 a5 C + 2 a5 A C + a5 B² = 0,
a a BB:
B = (2a a A — 3 a a) C,
6- 2 A2 3 a5 D + 2 a5 A D = 0,
or 2 A A — 6 = (2 a a A.
3
3
3 a a) a³ D,
4a³ C
3
— 2 a³ AC — A3. B² + 4 a³ Č— 3 a5 E+ 2 a5 AE+2a5 BC= 0,
or 2 AC + B² - 4 C — 2 a² BC = (2 a² A — 3 a²) E,
&c.
for determining A, B, C, D, E,
&C.
&c. and give
A = I
or A 2
A A
B =
4 A
3
2 A B
C =
=
гаа А
4 B — a² B2
a a
B =
4
3 a a
a a
2 a a A
3 a a
\ 15
Q4
16
C =
1
a4
D =
2 AA — 6
4
(2 a a A
2 AC
BB.
E =
za a) az
4 C
a5
D=+
2
45
2
2 a² B C
&c.
z a a A — z a a
There are two afcending Series, y = x+
ΙΙΙ
II 2
=
E =
аб
a6
&c.
3x3
+ 15×5
+
III x7.
аб
a a
a4
+
4 x6
a5
&c. and y=2x
4x3
a a
16 x5
a4
236
+
112 x7
&c.
Q5
26
>
CII.
2 a5 x y
Application
of this rule
Let the Equation x7a3 x3 y + a³ x² y² + a5 y y —
+ a5 x x = o be propoſed, from which the Value of y in x, expreffed to another
by an afcending Series, is required to be deduced.
This Equation being placed on the analytic Triangle, there will be
found but one inferior Determinator, which gives the Equation
a5 y y — 2 a5 xy a5 xxoor y²-2xy + xxo, which has
+
example.
300.
ELEMENTS OF
1
3
3 3
2
two equal Roots, or one double Root, y=x, whence x + u being ſubſti-
tuted for J in the propofed Equation, it will be transformed into
x7 + a³ x³ u + a³ x² u² + a5 u uo, which being in its turn placed
on the analytic Triangle, there are two inferior Determinators, one of
which gives the Equation a5 u u+a³ x³ u = 0, or u =
3
ther gives the Equation a3 x3 + xq = 0, or u=—
u
X4
as
*3
aa
and the o-
both one and
the other Exponent of x exceeds the preceding one, confequently thoſe
two Determinators are uſeful; but it fuffices to employ the firſt, which
3
x3
a a
b
= Ax that is, h3 and A =
I
Qa
3
3 2
gives u = —
and fubftitut-
ing Ax³, or fimply x3, in the Equation x7 + a³ x·³ u + a³ x² u²
+as uuo, it will be changed into x7 + a³ x6 + a³ x8 + a5 x6 = 0,
which, it is manifeft, includes three Orders of Terms, whofe Exponents
are 6, 7, 8, and the Differences I, 2.
2. Since the leaft divides the greateſt,
the Succeffion of the Exponents of x will be the arithmetical Progreffion
3, 4, 5, 6, &c. whofe firft Term is 3 (4), and the Difference I (n), the
Form of the Series will be then u — A x3 + B x4 + C x5 +- D x6, &c.
and this Value of x ſubſtituted in the Equation gives
x7 = +
x7
+a³x³u = + a³ Axб +
+α3x²x²
2 2
a3 B x7 + a³ Cx8 + a³ D x9 + a³ Ex¹º +&c.
+ a³A²x8+2a3 ABx9+2a3A Cx¹? +&c.
3
+ a³ BB x¹º+&c.
+asu u = +85 A² x6+ 2 a5 A B x7 +2 a5 AСx8+2a5AD x9 +2 a5 A Ex¹º+&c.
+ a5 B B. x8+2 a5 B C x9+2a5BDx¹9+&c.
+ a5 CCx¹²+&c.
SPECIOUS ARITHMETICK.
301
From whence is deduced, by putting each complex Coeficient equal to
Nothing,
1 + a3 B + 2 a5 A Bo
a³ C+ a³ A A+ 2 a5 AC + a5 B B = 0,
a³ A a5 A A = 0
that is, A=
Τ
or A o
a a
I
B=
B=-
I
43
a3
3
C=
2
I
C=-
49
N
24
2
D=
a5
25
7 E= 4
аб
&C.
a³ D + 2 a³ A B + 2 a5 A D + 2 a5 B C = 0 D=
a³E+2a³ AC+a³ BB+2a5AE+2a5.BD+a5.CC≈0.E=
&c.
&c.
Hence u is expreffed by two Series, to each of which adding, we will
have two Values of y = x - x² + x²+ + 2 x 5 + 2x6 +
x4
and y = x
23
x5
24
+3
a a
2x6
a5
x4
a
3
4x7
a4
a5
7x7
25
&c.
, &c. this laſt Series is precifely
the fame as that which would be obtained by the Means of the Determi-
nator which gave the Equation —
3
3
X
3
x4
a3
for fubftituting x4 for u in
a5
the Equation x7+a³ x³ u + a³ x² ² + as u u= 0, it will be changed
into x7a3 x7 + a³ x² + a5 x80, in which, it is manifeft, there
are three Orders of Terms, whofe Exponents are 7, 8, 10, the Differences
are therefore 1, 3; and as the leaſt divides the greateſt, the Succeffion
of the Exponents of x is 4, 5, 6, 7, &c. and the Form of the Series
y = A x^ + B x5 + Cx6, &c. whofe Coeficients being determined, is
converted into
x4
a3
x5.
a4
2x6
as
4x7
26.
&c. hence this Determi-
nator gives only the fecond of the two Series which the other had fur--
niſhed.
2
CIII.
A deſcending Series is required, expreffing the Value of y in x, deduced
from this Equation x3 y3+3 a x² y² + 3 a²x² y + a³ xx - a³ x y
a² x + a5 = 0.
3 2
..
This Equation being placed on the analytic Triangle, furniſhes but one
fuperior Determinator, which gives the Equation ²y²+3ax²y³+3a²x²y
+ a³ x² = 0, or dividing by x², y³ + 3 a y² + 3 a² y + a³ = 0,
which has only one Root, but triple, ya; fubftituting therefore a + u
for y in the propofed Equation, it will be transformed into x23a3 x u
450, which being placed on the analytic Triangle, furnishes two
fuperior Determinators, one of which gives 2 43 a³ x u = 0, or
x น
a3
u = + a
±a
3:2
K
- 1; 2
24
and the other a³ x 1 + a5=0, oru = a²x
u
302
ELEMENTS OF
j
+
7
S
But it fuffices (Art. xcv.) to employ only the firft, which gives
I
2
2
and ſubſtituting x
will change it into x
I : 2
I: 2
I 2
a3 x
for u in the transformed Equation, it
น
+asx°
0, in which, it is mani-
I
feſt, there are only two Orders of Terms whofe Exponents are and o;
2
there is therefore only one Difference, the Succeffion of Exponents are
therefore ———,
2
2, &c. and the Form of the Series
I
3
I,
2
2
I 2
I
3:2
Ax
+ Cx
+ B x
x²u³ = A³×³+3AABx+3AACx +3AADx +3AAE x²+ &c.
&c. this Series fubftituted for u in
+3ABB
+6 ABC
+
B3
+6ABD
+3 ACC
+ 3 B B C
#4
Hld
-a3 D x
-a3 Ex
&c.
-a3xua³ Ax²-a3Bx-a3 Cx
5
5
+ a³ = + a³
furniſhes the following Equations:
A3 — a³ A=0
3 A AB — a³ B + a5 = 0
о
3 AAC+3 ABB-a3 Co
3:2
32
that is a or Ao
A+
+
I
B =—=—=aa
2
3 5:2
B = a a
Fac=0
C
. C =
C = +÷
3 A A D + 6 A B C + B³ — a³ D=0
· D-
8
La3
D23
D=—=— a³ D=a3
2
3AAE+6ABD+3ACC+3BBC—a³E=0 € = ±1057:2
E
128
-a
E=0
SPECIOUS ARITHMETICK.
303
Hence we have three defcending Series expreffing the Value of u in x,
3:2 - 1:2
I
I 3 5:2
3:2
I
24
viz. u = a
X
a a x
a
X
a3 x
2
8
2
105 7:2
-5:2
3:2
1:2
I
I'
+-
mloo
3
8
128
5:2
a
a
2
+ a³ x
аз
X
&c.
น
a
X
a a x
+
2
3:2
I
24
3
105 7:2
5:2
I.
*
a X
a
&c. u = a a x
2
128
, &c. this laft is precifely the fame as would be obtained by the
Means of the ſecond Determinator, which gave u a 12 x X
fubftituted for u in the Equation x² u³
I
K
2
23
a3
Ι
I
;
for x
xua5 transforms it into
a³ + a5 = 0, which fhews that there are two Orders of Terms,
whofe Exponents are I and o; there is therefore only one Difference
I
2
1, and the Form of the Series is Ax + Bx + Cx
which Value being fubftituted for u in the Equation
น
3
-4
+Dx
+ &c.
+ &c.
+3AADx
+6 A B C
B3
| +++
3
2
23
x² u³ = A³ x
A3
+ 3 A A B x
+3 AAC x
3 AB B
-2.
a3 D x
+ as
that is A
A =
a
a³Gx
a3 E x
4
-4
-a³xua³ A-a3 Bx -a3 Ax
=a5+ &c.
gives the following Equations:
a3 A+ a50
3.A
A3 a3 B о
3 AAB — a³ Co
3AAC+3 ABB-a3 D о
3 AAD + 6 ABC-B3a3 E
Wherefore ua²x
&c.
2
+a³x +3α+ x + 10 a5 x
at
4
B =
D
E
2
23
3 44
IO a5
49 a
&c.
4
ет
+49a6x
thus the Valuc fought of y in x is expreffed by three defcending
a a
Заа
a
a3
5
&c.
N = &c.
&c.
105 a3
+
2xx 128xx
Serics, y = a+a
a
a a 3 a a
a
a3
X
2 x
8 x
a
y=a-an
a a
y = a+
+ ४
a3
x x
2 x
+
+
3 a4
As
8x
+
2 xx
10 45
+ 4925
x4
105 23 24
128xx
&c.
*5.
&C.
304
ELEMENTS OF
Reverfion
of feries.
CIV.
When in an indeterminate Equation, the Value of one of the variable
Quantities (y) is found expreffed by an infinite Series, including in its
Terms the Powers of the other variable Quantity (x), and it is required
to find the Value of x in y, expreffed by an infinite Series, whofe Terms
contain the Powers of y, this Method is called the Reverfion of Series,
to which the preceding Doctrine may be conveniently applied.
3
3
Suppoſe we had found y = ax + b x² + c x³ + dx+ + ex5, &c.
whofe Coeficients a, b, c, d, &c. are known, and the Value of x in r is
required, I put oy+ax + b²x² + c x² + dx4, &c. in which x
being fuppofed very ſmall, it will appear (Art. CXIX) that the Form of
the Series muſt be xly+my² + ny³ + py4, &c. fubftituting this
Value of x in the Equation, we will have
y
y
+ ax = +aly+amy²+ any³+apy4+ any5+
+bx²= ·· +blly²+2bmly 3+bm²y4+2b1py5+
2
+2blny4+2bmny5+
ary6 &c.
b b n y6 &c.
2 b l q y6 &c.
+
2 bmp 26 &c.
+cx³ =
3
+ c 1³ y³ +3cllmy4+3clm²y5+
+dx4 =
+ex5=
+ƒx6=
3
cm³ y6 &c.
+3cllny5+ 6 clmny6 &c.
+ 3 c llp y6 &c.
+ d 14y++4dl³ my5+6 d l l mm y6 &c.
+
e 15 75+
+
4d13 nyб &c.
6
11
5e 14 m p6 &c.
fio y6 &c.
which gives the following Equations:
al x x = 0
am + bll=0
or / -
I
a
b
112
an+2bml+c13=0
12
ap+bm² + 2 bln + 3 cl lm + d 14 = 0, p =
&c.
a3
z b b — ac
a5
8
5 a b c — 5b ³ — a² d
al
&c.
and fubftituting thoſe Values in x=ly+my²+ny3+py4, &c. there will
by 2
23 +
2b bac
25
33 +
5 a
b
C
- 5 63
al
a3
refult x=
a
1464
Value required ofy.
21 a c b b + 6 a² b d + 3 a² c c = c A3
a9
a² d
34 +
y5,&c. which is the
SPECIOUS ARITHMETICK.
305
1
CV.
When a Series is regular, that is, when a fimple Root of the Equation
furnished by the Determinator is employed, the Difference (1) of the
b
b F#
Obfervati-
Exponents of x in the firft and the fecond (Ax, B x ) Terms of
the Series, is equal to the Difference of the Exponents (m, m + n) of
the firft and fecond Orders of Terms of the Equation; but it is net by may be
b
on where-
difcovered
ſo if the Root yAxo be Multiple. If the Degree of its Mul- in many
b
b
caſes if a
feries is
tiplicity be expreffed by j, that is, if yAx divides j Times, the half ima-
Sum of the Terms of the firft Order m, provided it does not divide the ginary,
Sum of the Terms of the fecond Order mn, the Difference A
j of
the Exponents of the first and fecond Terms of the Series A x + Bx, &c.
will be equal to, which is the Difference 1 of the Exponents of the
Orders m and mn divided by j, fo that in the fecond Term, the Ex-
ponent of ≈ will be (i = ) ½ = 7
For fince y
b
12
1
b
h
n
Axo is a Root whofe Degree of Multiplicity is j,
when Axu is fubftituted for y, there will be wanting in the trans-
formed Equation the Terms correfponding to the Points where the De-
terminator cuts the j firft Columns, (Art. xcv.) that is, the Terms
m - j h j
&c. continued to the Term x
26
مام شده
X
>
.b
ll, X
172 2 b 2
น which
>
will be the Term correfponding to the Extremity of this firft Determi-
nator; it is therefore from this Term that the ſecond Determinator parts,
which will pass thro' the first Term of the fecond Order x
will not be empty, fince y
Terms of the fecond Order.
h
112 F
#
whoſe Cell
Ax is not a Divifor of the Sum of the
Hence the Equation which this fecond.
Determinator furniſhes, will be of this Form
or u B x
j
or z
j m
= B x
I"
— m + j b
fore of the ſecond Term is =7-
h
12
吐
​,
jb j j m
u = B x
I"
the Exponent there-
4 Q
306
J
ELEMENTS OF, &c.
b
Hence we may difcover, without Calculation, whether a Series whoſe
firſt Term is real, be half imaginary; when the Root yAxo, fur-
niſhed by the first Term, dividing feveral Times the Terms of the
firſt Order m, does not divide thoſe of the ſecond Order m✈n, which is a
very ordinary Cafe. For if j, Degree of the multiplicity of this Root, is an
even Number, and n, Difference of the Exponents of the Orders, an odd
b
+;
Number; the fecond Term (B x ) of the Series is half imaginary
(Art. LXXXIII.) and it is real if j be odd, but if n and j be both even,
1
be real or imaginary, according as the Coeficients of the Terms x
and x
m+ n
it will
m j h j
772 ------ - b
น
have different Signs or the fame Sign, but it is on this ſecond
Term it depends that the Series be real, or entirely or half imaginary,
for the Equation which in the preſent Cafe furnishes this Term, confift-
ing only of two Terms, will have no multiple Roots; from hence forth,
the Series will become regular, and all its Terms, counting from the
third, will be real.
END of SPECIOUS ARITHMETICK.
ERR AT A.
Page 2. Art. 11. Line 25. for, its triple more 410 makes 490, read,
its triple more 410 makes 890.
P. 40. Art. LXIV, 1. 9. for c e = 18, b f
bƒ = 18.
P. 104. Art. LIX. 1. 5. for
— 3 (В A — A B') (Ē D' —
(E
P. 124. Art. x. 1. 6. for, 3,
12, read, ce= 12,
3 (B A' — A B') (E R' — B E') read,
a3
DE')
3 c5
b
-X, read,
3
a²c5
b
- X. Line 7.
3
导
​2
for, = a
read, a
For, ¿¹², read, c
+ 3/10.
Page 149. for, 3√
x
b
35
2
35
I 2
5 × 3✓ 20 X 3 10, read, 3√5 + 3√ 20
P. 160. laſt Line, for, a p—a = z p
#
read, a p— a
a = z p
for a +, read a x.
P. 223. 1. 7. for,
Ad AB d B
A A + B B
Ad A + B dB
X, read,
A A + B B +.
TABLE
O F
CONTENT S.
PREFACE.
A preliminary Difcourfe on the Nature, Divifions, and Utility of the Ma-
thematicks.
§1.Whence the Name of the Mathematicks
is derived,
2. Their Nature and Object,
Page i
3. Their Divifion, and their different Ex-
tent among the Ancients and Moderns, ib.
4. Metaphyfical Generation of thefe Sciences,
and of their different Branches,
5.
Vi
Ufeful Remark on what is called Abſtract
Mathematicks,
ix
6. Efteem they have been held in by the
Judicious in all Ages,
Opinion of Socrates examined,
X
xi
7. Anſwer to the Objections ſtarted by the
Scepticks and Epicureans againſt Mathe-
maticks,
xiv
8. Their Defence against thoſe who have
endeavoured to undervalue them,
xix
9. The Advantages accruing to mankind from
the Cultivation of the Mathematical Sci-
ences,
Curious Note refpecting fome Uſes to
which they have been applied by injudi-
ous Perfons,
ib.
10. Apology of Abſtract, and purely intel-
lectual Mathematical Enquiries, xxvii
11. View of the Inftructions given in the
Drawing-School-eſtabliſhed by the DUB-
LIN SOCIETY, purfuant to their Refolu-
tion of the fourth of February, 1768; to
enable Youth to become, Proficients in the
different Branches of that Art, and to pur-
fue with Succefs, geographical, nautical,
mechanical, commercial, and military
Studies,
PART the Firſt.
CHA P. I.
xxviii
Of Numbers, of the general Principles of numeral Arithmetick, and of the
Operations performed upon fimple Numbers.
§ I.
"WH
HAT is underſtood by number, I
The different kinds of units agreed
upon for the convenience of trade, ib.
Numbers are either concrete or abſtract, ib.
2. The formation of numbers,
2
Inconveniency of expreffing each number
ber by a particular word or character, ib.
3. Characters employed to reprefent num-
bers,
ib.
4. How the arithmetical characters are dif-
pofed to reprefent all poffible numbers,. ib.
All numbers from nothing to ninety nine
are reprefented by help of two characters, 3
Each character retains the name of the
number it reprefents,
ib.
5. All numbers from nothing to nine hundred
and ninety-nine are reprefented by help of
three characters,
ib.
TABLE OF
CONTENTS.
r
How a number reprefented by three cha-
racters is read,
ib.
4
6. All numbers poffible are reprefented by
help of ten characters,
ib.
Every three places of Figures are called
Periods, and have the fame names,
Names of the different periods of figures, 5
Rule for reading any propofed number, ib.
7. The number of grains of fand that would
form a globe as big as the earth propofed as
an example,
ib.
8. Of decimal parts,
6
9. Numbers fet down after the units place
are decimal parts,
ib.
10. Firſt manner of expreffing decimal num-
bers containing decimal figures,
7
Second manner of expreffing numbers con-
taining decimal figures,
jb.
In what a number containing decimal fi-
gures differs from one that has none, .8
Third manner of expreffing numbers con-
taining decimal figures,
ib.
11. How numbers confifting only of decimal
parts are reprefented,
ib.
12. Cyphers annexed to the right hand of the
comina or feparating point of a number do
not alter its value,
ib.
9
The different operations performed on
numbers, are reduced to addition, fub-
traction, multiplication, and divifion,
13. The numbers to be added fhould confift of
units of the fame fpecies,
ib.
ib.
14. Procefs of addition,
13. Firſt example of the addition of integers,
10
16. Second example of the addition of inte-
gers,
17. Eample of the addition of numbers
taining decimal figures,
I I
con-
ib.
18. Addition of decimal numbers performed
by reducing them to units of the fame de-
nomination,
1-2
ib.
19. Quantities to be fubtracted. one from the
other should be of the fame fpecies,
Procefs of fubtraction,
gery,
7
1*3
20. Firſt example of the 'fubtraction of inte-
ib.
21. Second example of the ſubtraction of in-
tegers,
14
22. Third example of the fubtraction of in-
tegers,
tion may be contracted,
16
28. An example of fubtraction performed by
the contracted method,
ib.
29. Subtraction employed to prove addition, ib.
Process of this operation explained by an
example,
17
30. Addition employed to prove fubtraction, ib.
Procefs of this operation explained by an
example,
18
31. The particular denominations given to the
numbers.concerned in multiplication, ib.
The lign X is employed to denote that two
quantities are to be multiplied one by the
other,
32.
ib.
Inconveniency of reducing multiplication
to addition,
ib.
33. The product of two figures multiplied into
one another ſhould have as many places
to the right hand as there are places to
the right hand in the multiplicand and
multiplicator,
19
34. The multiplicator is always an abftract
number,
ib.
35. In whatever order the factors of a multi-
plication are difpofed the product will be
always the fame,
20
36. Firft method of finding the product of two
numbers leſs than 10 by the table of Pytha-
goras,
21
37. Second method for finding the product of
two numbers lefs than 10,
22
38. Procefs of the multiplication of whole
numbers when the multiplicator is repre-
fented by one fignificative figure, ib.
39. First example of the multiplication of in-
tegers,
23
40. Second example of the multiplication of
integers,
ib.
41. Third example of the multiplication of
integers,
24
42. Procefs of the multiplication of integers
when the multiplicator is repreſented by
feveral fignificative figures,
ib.
43. First example of the multiplication of in-
tegers when they are both reprefented by
feveral fignificative figures,
ib.
44. Second example of the multiplication of
integers repreprefented by feveral fignifi-
cative figures,
25
45. Third example of the multiplication of
integers reprefented by feveral fignificative
figures,
ib.
24. Fourth example of the fubtraction of in-
tegers,
ib.
15
25. How the fubtraction of decimal numbers
is performed,
46. Procefs of the multiplication of numbers
containing decimal parts,
26
ib.
26. Example of the fubtraction of decimal
numbers,
27. How the foregoing operation of fuðtrac-
47. First example of the multiplication of
numbers containing decimal parts, ib.
ib.48. Second example of the multiplication of
numbers containing decimal parts, is,
TABLE OF CONTENT S.
parts by others that have none,
71. Example,
40
ib.
72. Method of approximating to the quotient
of a divifion when the divifor contains no
decimal parts,
41
73. The foregoing method explained by an
example,
ib.
74. Method of approximating to the quotient
of a divifion when the divifor is greater
than the dividend,
75. Firſt example,
76. Second example,
49. Third example of the multiplication of | 70. Diviſion of numbers containing decimal
numbers containing decimal parts,
27
50. Fourth example of the multiplication of
numbers containing decimal parts, ib.
51. How the operations in multiplication are
proved explained by an example, ib.
52. The particular denominations given to the
numbers concerned in divifion,
28
53. Different notions that may be formed of
divifion according as the divifar and divi-
dend are of the fame or of different fpe-
cies,
29
54. In the operation of divifion the number of
units of the dividend and divifor is only
confidered and not their ſpecies,
30
57. The multiplication or divifion of the di-
vidend and divifor by the fame number
does not alter the quotient,
31
58. The quotient of a dividend divided by a
compofit divifor is equal to the quotient
arifing from the continual divifion by the
component parts of the diviſor,
32
59. The divifion of any number lefs than 90
by another less than 10 performed by help
of the table of Pythagoras,
33
60. Proceſs of the method of dividing a num-
ber by another reprefented by one figure,
ib.
•
J
61. First example of the divifion of a number
by another repreſented by one figure, 34
62. Second example of the divifion of a num-
ber by another reprefented by one figure,
35
63. General method for dividing one number
by another,
ib.
64. The foregoing method explained by an
example,
36
65. General rule for finding the number of
places that a quotient of a divifion will
confift of,
37
Why in each operation of divifion a num-
ber greater than 9 is not fet down in the
quotient,
ib.
66. Second example of the divifion of integers
reprefented by feveral fignificative fi-
gures,
38
67. Third example of the division of inte-
gers reprefented by feveral fignificative
figures,
39
68. Proceſs of the divifion of numbers cou-
taining decimal parts,
ib.
ib.
42
ib.
77. A quotient of a divifion that cannot be
obtained without leaving a remainder is
expreffed by a decimal feries confifting of
equal periods of figures fucceeding each
other ad infinitum,
43
78. Property of the digit 9 which divides num-
bers lefs than itſelf,
44
79. Method of multiplying by numbers con-
fifting of equal periods of figures fucceed-
ing each other,
ib.
80. Property peculiar to all divifors lefs by an
unit than the terms of the progreffion 10,
100, 1000, 10,000, which divide numbers
leſs than themſelves,
45.
81. Method of finding from whence fimple and
compound repetends are derived, 45
S2. Method of finding from whence fmple and
compound repetends preceded by cyphers
are derived,
ib.
83. Method of finding from whence fimple and
compound repetends preceded by a cer-
tain number of decimal figures are de-
rived,
47
84. Method of contracting the operations of
divifion,
45
85. Spanish method of performing divifion ex-
plained by an example,
49
86. French method of performing divifion ex-
plained by an ex înрle,
50
87. Proof of the operations in diviſion made by
cafting out the nines, as in multiplica-
tion,
Example,
SI
52
88. Method of verifying each figure of the
quotient according as they are found, ib.
TABLE OF
CONTENT S.
1
1.
CHA P. II.
Of Fractions and their various Reductions, c. Of the Operations perform-
ed upon compound or applicate Numbers, and the Menfuration of Surfaces and
Solids.
Ο
RIGIN o´ fractions,
Notation fractions,
2. The different fpecies of fractions,
53
îb.
54
3. How a whole number is changed into a
fraction,
ib.
A fraction whofe numerator and denomi-
nator are equal is an unit,
55
4. The value of a fraction is not changed by
multiplying its numerator and denomina-
tor by the fame quantity,
ib.
5. Method of reducing fractions to their low-
eft terms,
ib.
6. The foregoing method of reducing fracti-
ons to their loweſt terms applied to an ex-
ample,
56
ib.
7. Grounds of the foregoing method,
8. Another method of reducing fractions to
their lowest terms,
57
9. The foregoing method applied to an ex-
àmple,
ib.
10. The foregoing method applied to another
example,
ib.
11. Method of reducing any two fractions to
the fame denomination,
58
12. Application of this method to an example,
ib.
13. General method of reducing any number
of fractions to the fame denominator, ib.
14: Application of this method to an example,
59
15. Inconveniency to which the foregoing me-
thod of reducing fractions to the fame de-
nomination is liable,
ib.
Method of remedying this inconveniency.
according to the two different cafes that
may occur,
Example of the firſt cafe,
16. Example of the fecond cafe,
19. Addition of fractions of the fame denomi-
60
ib.
ib.
17. Reduction of improper fractions,
Example,
61
ib.
nation,
62
Example,
ib.
Addition of fractions of different denomi-
nations,
Example,
ib.
ib.
21. Subtraction of fractions of the fame de-
nomination,
Subtraction of fractions of different deno-
minations,
63
ib.
23. Multiplication of fractions by whole num-
bers,
Firft method,
Example of this fiift method,
Second method,
Example of this fecond method,
ib.
64
ib.
ib.
ib.
24. Divifion of fractions by whole numbers, 65
First method explained by an example, ib.
Second method explained by an example,
ib.
The ſecond method not always practicable,
66
ib.
25. Multiplication by fractions,
26. Four different methods of multiplying a
fraction by a fraction,
ib.
27. Firſt method explained by an example, 67
28. Second method explained by an example,
ib.
29. Third method explained by an example, 68
30. Fourth method explained by an example,
ib.
31. The multiplication of a whole number by
a fraction is the fame as the multiplication
of a fraction by a whole number, 69
32. In what confifts the divifion by a fracti-
on,
ib.
The divifion by a fraction reduced to a
multiplication,
70
33. First method of dividing a fraction by a
fraction,
ib.
Second method of dividing a fraction by a
fraction,
it.
34. The different. fubdivifions of fractions and
the manner of expreffing them,
71
35. A fraction of a fraction is equal to the
product arising from the multiplication of
the two fractions by which it is expref-
fed,
72
A fraction of a fraction is always the fame
whatever way the fractions by which it is
expreffed are arranged,
it.
36. A fraction of a fraction of a fraction, &c.
is equal to a fraction having for numera-
tor the product of the numerators of all
the fractions by which it is expreffed, ard
for denominator the product of their der o-
minators,
ib.
37. Neceffity of employing units of differei t
denominations for meaſuring the fame
kind of magnitudes,
73
TABLE OF CONTENT S.
38. The different kinds of units employed in
73
money weights and meaſures,
39. Value of the different units employed in
money weights and meafures, and the
characters by which they are diſtinguiſhed
from each other,
ib.
41. Procefs of addition of mixt numbers,
43. Addition of pounds, fhillings, and pence,
explained by an example,
74
75
44. Addition of pounds, ounces, penny-weights
and grains, explained by an example, 76
45. Addition of yards, feet, inches, and lines,
ib.
explained by an example,
47. The proceſs of fubtraction of mixt num-
bers,
77
78
ib.
48..Subtraction of pounds, fhillings, and
pence,
Firft example,
49. Second example,
67. Multiplication of weights explained by an
example,
68. Multiplication of lines and furfaces,
69. Figure 1,
88
ib.
ib.
The area of a rectangle is found by mul-
tiplying the lineal meaſures in the bafe by
thole in the altitude,
Example,
70. Another example,
Figure 2,
89
ib.
ib.
90
71. The contiguous fides of the fuperficial
meaſures contained in any figure are the
lineal meaſures of its dimenfions,
Example,
ib.
ib.
72. The fuperficial meaſures are divided and
fubdivided into parts analogous to thole
of the lineal meaſures,
Example,
79 73. Another example,
ib.
50. Subtraction of pounds,
weights, and grains,
51. Subtraction of yards, feet, and inches, ex-
plained by an example,
80
52. A queſtion for practice in addition and fub-
traction of mixt numbers,
ib.
76.
ib.:
ib.
91
ounces, penny-74. The geometrical multiplication of a line
by a line is reduced to arithmetical mul-
tiplication, by afcribing to the multipli-
cand the two dimenfions affecting the fac-
i^.
tors,
Method of finding the area of a rectang e
whofe dimenſions are expreſſed in yards
and parts of a yard explained by an ex-
ample,
92
53. Procefs of the multiplication of mixt num-
bers,
ib.
54. The multiplication of pounds, fhillings,
and pence, explained by on example,
81
55. Table for finding what parts of the mul-
tiplicator fhould be taken for the different
numbers of fhillings, pence, and farthings,
in the multiplicand,
82
56. Method of finding at one operation the
pounds and fhillings arifing from the mul-
tiplication of a number of fhillings into a
whole number,
57. Firſt example,
58. Second example,
59. Third example,
83
ib.
ib.
84
60. Method of finding at one operation the
pounds, fhillings, pence, and farthings, a-
rifing from the multiplication of a number
of pence or farthings into a whole num-
bor,
61. First example,
62. Second example,
63. Third example,
77. Application of the foregoing method to
another example,
93
78. Method of reducing the parts of fuperficial
meaſures analogous to thofe of the lineal
meaſures into fquare meaſures,
94
79. The foregoing method applied to an ex-
ample,
80. Figure 4,
95
ib.
The content of any parallelogram is found
by multiplying the bafe into the altitude,
81. Figure 5,
ib.
96
The content of any triangle found by mul-
tiplying the bafe into half the altitude, ib.
82. Figure 6,
ib.
ib.
Another method of finding the area of a
triangle,
it.
ib.
85
83. Grounds of the foregoing method,
84. Figure 7,
ib.
97
ib.
86
Rule for finding the content of a quadri-
lateral figure whofe oppofite fides are pa-
rallel,
ib.
85. Figure 8,
it.
Method of finding the area of any right
lined figure by dividing it into triangles, 98
ib.
86. Figure 9,
ib.
Rule for finding the area of any regular
polygon,
ib.
64. Fourth example,
65. The foregoing method applied to an ex-
ample containing pounds, fhillings, pence,
and farthings,
ib.
66. Grounds of the foregoing method for mul-
tiplying fhillings by a whole number,
Grounds of the foregoing method for mul-
tiplying pence or farthings by a whole
Aumber,
87
Grounds of the foregoing rule for finding
the area of any regular polygon, it.
TABLE OF CONTENT S.
87. The area of a circle found by multiply-
ing its periphery by half its radius, 98
88. Figure 10,
99
Method of finding the ratio of the diame-
ter to the circumference,
Figure 11,
ib.
ib.
100
89. Method of finding the area of any circle
by means of the foregoing proportion of
the diameter to the circumference,
90. The area of a circle found more expedi-
tiouſly but lefs accurately by the propor-
tion of Archimedes,
91. Figure 12,
IOI
ib.
Rule for finding the area of a ſegment of
a circle,
ib.
92. Method of finding the area of a ſector of
a circle,
93. Fignres 13 and 14,
ib.
102
Method of finding the convex fuperficies
of a prifin,
Grounds of this method,
94. Figures 15 and 16,
Rule for determining the fuperficies of a
cylinder deduced from the foregoing me-
thod,
95. Figures 17 and 18,
folid meaſures contained in any folid are
the lineal meafares of its dimenfions, 108
107. Method of finding the content of a pa-
rallelepiped whoſe baſe is expreſſed in
fquare yards and parts of a fquare yard,
and altitude in yards and parts of a yard,
109
108. Method of finding the content of a pa-
rallelepiped whofe dimenfious are ex-
preſſed in yards and parts of a yard, ib.
109. Method of reducing the parts of folid
meaſures analagous to thofe of the lineal
meaſures into cubical meaſures,
110. The foregoing method applied to an ex-
ample,
111. Figure 23,
110
III
112
The content of any prifm is found by
multiplying the area of its bafe by its
height,
P
ib.
ib.
The content of a cylinder found by mul-
tiplying the area of its bafe by its alti-
tude,
ib.
112. Figure 24,
ib.
ib.
ib.
113. Figure 25,
ib.
The content of a pyramid or cone found
by multiplying the meaſure of its baſe
into 1-third of its altitude,
ib.
ib.
114. Figure 26,
113
ib.
ib
103
Method of finding the fuperficies of a re-
gular pyramid,
Grounds of this method,
.96. Rule for finding the fuperficies of a fruf-
tum of a pyramid,
Grounds of this rule,
ib.
ib.
How the fuperficies of an irregular pyra-
mid is determined,
Rule for finding the folid content of a
fruftum of a pyramid or cone,
ib.
115. Grounds of the foregoing rule for defer-
mining the folidity of a fruftum of a py-
ramid or cone,
ib.
116. Rule for finding the folid content of a
Sphere,
104
97. Figure 19,
ib.
114
Rule for determining the fuperficies
of a
117. Grounds of the foregoing rule,
ib.
ib.
cone,
Figure 27,
ib.
98. Method of finding the fuperficies
of a
fphere,
ib.
118. Rule for determining the folid content of
a fegment of a fphere,
115
Figure 20,
ib.
Figure 28,
ib.
Grounds of this method,
ib.
Grounds of the foregoing rule,
ib.
99. Figure 21,
105119.
Rule for determining the fuperficies of a
fegment of a fphere,
116
ib.
Grounds of this rule,
ib.
ib.
100. Figure 22,
The content of a parallepiped is found by
multiplying the area of its bafe by its
altitude,
101. Example,
102. Another example,
ib.
ib.
jb.
103. The folid meafures are divided and fub-
divided into parts analagous to thoſe of
the lineal meaſures,
Example,
104. Another example,
107
ib.
ib.
Method of dividing a mixed number by
a fimple one,
Method of dividing a mixed number by
another mixed number,
ib.
120. How to diſtinguiſh when the quotient
ſhould be a concrete or abſtract number,ib.
Dimenſions of the units of the quotient,
when the units of the dividend and divifor
confift of a certain number of dimenſions,
ib.
121. The divifion of pounds, fhillings, and
pence, by an abftract number explained.
by an example,
117
122. The divifion of weights by an abſtract
number explained by an example, 118
105. The length, breadth, and altitude of the [123, The divifion of pounds, fhillings, and
TABLE OF CONTENTS.
pence, by pounds, fhillings, and pence,
explained by an example,
119
124. Another method of dividing mixed num-
bers of the ſame ſpecies,
125. Divifion of fquare yards and parts of a
120:
fquare yard by lineal yards and parts of a
lineal yard explained by an example, ib.
126. Diviſion of fquare feet and parts of a
fquare foot by lineal feet and parts of a
lineal foot explained by an example, 121
127. Divifion of cubical yards and parts of a
§1. WHA
cubical yard by lineal yards and parts of a
lineal yard explained by an example, 122
128. Divifion of a number of yard-inch-inch,
feet-inch-inch, inch-inch-inch, by a nuia-
ber of inch-inch, explained by an ex-
ample,
123.
129. Extraction of the fquare root of com-
pound numbers explained by an example,
124
130. Extraction of the cube root of compound
numbers explained by an example, ib.
CHA P.
Of Proportion, and the principal Rules
HAT is meant by a ratio, 125
Difference between an arithme-
tical ratio and a geometrical ratio, ib.
Quantities of the fame kind can only have
a ratio to one another,
ib.
2. The arithmetical ratio is of the fame fpe-
cies with the quantities compared, ib.
The geometrical ratio is always an abſtract
number,
ib.
3. The geometrical ratio is expreffed by the
quotient of the divifion of the terms of the
ratio one by the other,
126
4. Two equal ratios form a proportion, ib.
5. Method of finding the fourth term of a pro-
portion,
>
ib.
127
|
III.
of Arithmetick which depend therein.
14. The fourth term required of a proportion
is of the fame fpecies with that which is
not confidered as an abſtract number, 131
16. Firſt example of the fimple rule of three
direct,
132
17. Second example of the fimple rule of
three direct,
ib.
18. Third example of the fimple rule of three
direct,
ib.
19. In what cafes the firft and third terms of
a rule of three thould be reduced to the
loweſt denomination they contain, 133
20. If the firft term of a role of three be
concrete unit it is not to be fuppreffed as
ufelefs,
Éxample,
7
ib.
ib.
134
ib.
21. How a rule of three inverfe is perform-
ed,
22. Example,
General rule for performing a fingle rule
of three either direct or inverfe,
135
ib.
23. Example,
Grounds of this method,
6. The fourth term of a proportion found by
multiplying the third and fecond together
and dividing the product by the firft term,
128
7. The tranſpoſition of the mean terms of a
proportion do not alter the fourth term, ib.
8. In every geometrical proportion the pro-24. How a compound rule of three is reduced
duct of the means is equal to the product to a fimple rule of three,
13
of the extremes,
ib. 25. The foregoing method explained by an
9. Any three terms of a proportion, being example,
given how to find the term that is want-26. Second example of the compound rule of
ing,
129
10. What is meant by the rule of three or
golden rule,
遗
​ib.
11. The different fpecies of rules of three, ib.
The fimple rule of three direct, ib.
The fimple rule of three inverfe, ib.
The compound rule of three direct and
inverſe,
'ib.
three direct,
ib.
137
27. Third example of the compound rule of
three direct,
138
38. Fourth example of the compound rule of
three direct,
139
29. The compound rule of three inverſe ex-
plained by an example,
ib.
30. Direct method of folving queſtions in the
compound rule of three inverſe,
31. Rule of conjunction,
140.
ib.
12. The fimple rule of three direct explained
by an example,
130
13. Of the three terms given in a rule of three 32. Application of the rule of conjunction to
the two which are of the fame fpecies an example,
ſhould be confidered as abſtract numbers, ib. 1:33. Another example,
141.
142
4 R
TABLE OF CONTENTS.
to find the intereſt,
143
34. Varieties in proportion,
143;
ib.
35. Things to which the questions in intereft
relate,
36. From the principal, rate, and time given,
!
The buying and felling prices given to find
the rate,
First example,
Second example,
2222
ib.
ib.
ib.
Third example,
Example,
ib.
Fourth example,
ib.
37. From the amount, rate, and time given,
62. Caſe II.
ib.
to find the principal,
ib.
The buying price and the rate per cent.
ib.
Example,
38. From the amount, principal, and time
given, to find the rate of intereft,
profit given, to find the advanced price, ib.
First example,
ib.
ib.
Second example,
ib.
ib.'
Example,
Third example,
ib.
39. From the principal, amount, and rate
given, to find the time,
Fourth example,
154
ib. |
63. Cafe III.
ib.
ib.
Example,
40. Things to which the queſtions in diſcount
The rate given and another proportional
to it required,
ib.
relate,
144
Firft example,
ib.
41. From the fum due, rate, and time given,
Second example,
ib.
to find the preſent worth,
ib.
64. Cafe IV.
ib.
Example,
ib.
The advanced price given to find the
43. From the fum due, prefent worth, and
prime coſt,
ib.
rate given, to find the time,
ib.
Firft example,
ib.
ib.
Example,
Second example,
ib.
44. From the fum due, prefent worth, and 65. What is meant by barter,
time given, to find the rate,
ib.
145
Method of folving queſtions in barter at
45. What is meant by tare and tret,
ib.
one operation,
ib.
46. Rule for deducting the tare,
ib.
Firſt example,
ib.
Rule for deducting tare and tret at one
operation explained by an example,
Second example,
155
ib.
Third example,
ib.
47. Rule of mixtures,
146
Fourth example,
ib.
48. First cafe,
ib.
66. From whence arifes the courfe of ex-
The quantities and prices of the ingredi-
ents given to find the rate,
change,
ib.
ib.
49. Firft example,
ib.
67. Manner of exchanging between England
and Ireland,
ib.
50. Second example,
ib.
Reduction of English money to its equiva-
51. Third example,
ib.
lent Irish,
ib.
147
Reduction of Irish money to its equivalent
English,
ib.
1
52. Second cafe,
The rates of two ingredients and that of
the mixture given to find the quantity of
each ingredient,
53. First example,
54. Second example,
55. Ufe of the foregoing example in gunnery,
56. Third cafe.
:
149
A compound of two fimples its total va-
Manner of exchanging with the Britiſh
plantations,
156
68. Exchange, applied to drawing and remit-
ing,
*a
ib.
ib.
148
ib.:
ib..
69. Manner of exchanging between England
and France,
ib.
4
Eftimation of the French coins,
:ib..
ib.
157
ib.
ib.
151
Reduction of money of Genoa to its equi-
valent fterling,
ib
lue, and that of an unit of each ingredi-
ent given to find the quantity of each, 150
57. Example,
70.
,ib.
58. A compound of any number of fimples its
total value, and that of an unit of each
ingredient given to find the quantity of
each,
59. Example,
ib.
60. How the profit or lofs made upon any ar-
ticle in trade is eftimated,
61. Cafe I.
-152
นั
153.
Reduction of English money to its equiva
lent French,
Manner of exchanging between England
and Genoa,
Eftimation of the coins of Genoa,
Reduction of sterling to its equivalent
money of Genoa,
71. Manner of exchanging between London
and Leghorn,
=
ib.
Eftimation of the coins of Leghorn,, ib.
TABLE OF CONTENTS.
Reduction of fterling to its equivalent
money of Leghorn,
$57
Reduction of Leghorn money to¹its equi-
valent ſterling,
72. Manner of exchanging between England
and Venice,
Eftimation of the coins of Hambourg, 161
Reduction of fterling to its equivalent
Hambourg money,
ib.
158
Reduction of Hambourg money to its equi-
valent ſterling,
ib.
ib.
ih.
77, Method of arbitrating exchanges explain-
ed by an example,
ib.
79. What is meant by ſtock,
162
Eſtimation of the Venetian coins,
Reduction of ſterling to its equivalent
Venetian money,
ib.
Reduction of Venetian money to its equi-
valent ſterling,
ib.
73. Manner of exchanging between England
and Portugal,
ib.
Eftimation of the coins of Portugal, } ib.
Rednction of fterling to its equivalent
Portugueſe money,
159
Reduction of Portugueſe money to its
equivalent fterling,
ib.
74. Manner of exchanging between England
and Spain,
Eſtimation of the Spaniſh coins,
Reduction of fterling to its equivalent
Spanish money,.
1
Reduction of Spanish money to its equi-
valent fterling,"
75. Manner of exchanging between England
and Holland,
ib.
་
1
The different government funds and ſtocks
of companies,
80. Computations in ftock-jobbing,
81. Table exhibiting the intrinfick
the different publick funds,
ib.
ib.
value of
163
ib.
ib.
rule of
164
The ufe of the foregoing table explained
by an example,
82. Rule of company,
General rule for performing a
company,
83. Application of the foregoing rule to an
example,
ib.
ib.
ih.
160
'86. The compound rule of company explained
by an example,
165
87. The different fpecies of rules of falſe
pofition,
ib.
84. A more commodious method of finding
the proportional parts,
ib.
166
88. In what the rule of fingle falfe pofition
confiſts,
ib.
ib.
89. The rule of fingle falſe pofition explained
by an example,
ib.
90. Application of the rule of fingle falfe pofi-
tion to another example,
167
91. The rule of double falfe pofition explain
ed by an example,
*
' ib.
Eftimation of the Dutch coins,
ib.
Reduction of Sterling to its equivalent
Dutch money,
ib.
Reduction of Dutch money to its equiva-
lent ſterling,
tib.
76. Manner of exchanging between England
and Hambourg,
J Y
1
7
1
2
11
PART the Second.
CHAP
I.
Óf the analytick, Method of expreſſing Problems by Equations, and of the Re-
E
folution of Equations of the first Degree.
XAMPLE of a problem, fimilar to
thofe which the firſt analyſts might
have propoſed to themſelves,
I
Solution of this problem fuch as might be
found without fpecious arithmetick, ib.
2. Algebraic method of expreffing the fore-
going problem,
{
The fign+denotes addition,
To folve an equation is to find the value of
the unknown quantity it includes, ib.
Refolution of the equation which expreffes
the foregoing problem,
ib.
The fymbol denotes fubtraction, ib.
4. Another folution of the foregoing pro-
3
2
blem,
ib.
ཀ ་
The fign
defigns equality,
. ib.
F
३
-5. Another example of the foregoing pro-
blem,
ib.
}
An equation is the equality of two quan-
6. Third example of the foregoing problem,
tities,
ib.
ib.
2
TABLE OF CONTENT S.
4
The fign X denotes multiplication,
7. A new problem of the fame nature with
the foregoing,
ib.
8. The analytick falution of a problem con-
fifts of two parts,
5
In the first the problem is expreffed by an
equation,
ib.
In the fecond the equation is folved, ib.
9. Equation's of the first degree are thofe in
which the unknown quantity is only mul-
tiplied or divided by known quantities, ib.
io. The terms of an equation are its parts fe-
parated by the figns or
ib.
11. Any term may be carried over from one
fide of the equation to the other by chang-
ing its fign,
6
12. The members of an equation are its two
parts feparated by the fign=
ib.
14. Method of making the multiplicator which
affects the unknown quantity diſappear, 7
15. Method of making the divifor which af
fects the unknown quantity difappear, ib.
16. Examples of equations of the first degree
folved by the foregoing principles, ib.
17. Method of making the fractions of an
equation diſappear,
ib.
18. Another method whereby they are made'
to diſappear all at once,
19. Third problem,
8
9
In fpecious as in numeral arithmetick a
bar is employed to indicate divifion, ib.
21. Another folution of the fame problem, 10
22. Fourth problem,
I I
Manner of expreffing proportions in ſpe-
cious arithmetick,
ib.
24. General folution of the foregoing pro-
blem,
13
The first letters of the alphabet ferve to
express the known quantities and the laſt
letters the unknown ones,
ib.
Letters which follow one another without
Affirmative terms are thofe which are
preceded by +,
ib.
Negative thofe which are preceded by,ib.
32. The algebraick addition is the fame ope-
ration as the foregoing,
19
33. In what fenfe a negative quantity can be
faid to be added,
ib.
34. From the foregoing operation the alge-
braick fubtraction is deduced,
Proceſs of fubtraction,
20
ib.
35. A quantity is increafed when a negative
quantity is fubtracted from it,
21
36. Third example of the refolution of literal
equations,
ib.
37. A number placed at the top of a letter to
the right denotes how often it is repeated
by multiplication, and in this cafe the let-
ter is faid to be raiſed to the power ex-
preffed by this number which is called
the exponent,
22
The numbers which are on the left and
on the fame line are called coeficients, ib.
38. Fourth example of the reſolution of lite-
ral equations,
ib.
39. Incomplex quantities are-thoſe which con-
fiſt only of one term,
23
{
Their multiplication deduced from the
foregoing examples,
ib.
40. Fifth example of the refolution of literal
equations,
ib.
41. Divifion of incomplex quantities deduced
from this example,
24
42. Sixth example of the refolution of literal
equations,
ib.
Uſe of parentheſes in fpecious arithmetick,
ib.
43. Multiplication of compound quantities or
multinomials deduced from the forego-
ing article,'
25
Example of the multiplication of multi-
nomials,
ib.
44. Fundamental principle of multiplication, 26
45. Method to be purſued in multiplication, ib.
Application of the foregoing method to an
example,
having any fign between them are to be
confidered as multiplied by each other, ib.
25. Example of the foregoing folution in num-46.
bers,
26. Fifth problem,
27. Example in numbers,
28. Another example,
29. The application of thofe rules has pro-
duced feveral operations of fpecious arith-48:
metick,.
ib.
27
29
General method of dividing compound
quantities,
F5
16
ib.
17
47. Sixth example of the refolution of literal
equations,
Manner of performing the divifion indi-
cated in this example,
ib.
ib.
Firft example of the refolution of literal
equations,
18
/1
30
30. Second example of the refolution of lite-
ral equations,
ib.
What is underſtood by ordering a quantity
according to a letter,
ib.
31. Reduction of quantities to their moſt fimi-
ple expreffion,
ib.
49. Application of the foregoing method to
an example,
31
Manner of avoiding working by conjec-
ture in divifion,
TABLE OF CONTENTS.
go. Another example,
2. Problem in which two unknown quanti-
ties are employed,
32
74. Firſt example,
75. Second example,
34
54. Application of the foregoing ſolution to
an example,
36
56. Problem in which two unknown quanti-
ties are employed,
ib.
57. Example of the foregoing problem in num-
bers,
58. Another example,
38
ib.
Singularity of the expreffions arrived at in
in this example,
ib.
Method of diſcovering their meaning, ib.
59. General theorems concerning the figns of
quotients or products,
39
60. Demonftration that -bintod is bd
+
when thoſe quantities are not preceded by
others,
ib.
61. The other cafes are demonſtrated in like
manner,
ib.
62. How the negative value found folves the
problem,
ib.
63. Unknown quantities becoming negative
are of an oppofite kind from what they
were fuppofed in the expreffion of the
problem, as alfo known quantities;
64. Example of the ufe of known quantities
made negative,
40
ib.
65. Another example in which the known
quantities are made negative,
41
66. Two equations whatſoever of the first de-
gree, including two unknown quantities,
may be reduced to the foregoing ones, ib.
Example,
67. Another example,
ib.
68. Another manner of folving the foregoing
example,
42
44
69. Compariſon of the two foregoing folu-
tions,
ib.
46
76. Third example,
ib.
49
ib.
77. Another manner of finding the greateſt
common divifor of the quantities in the
foregoing example,
50
78. Other quantities whofe greatest common
divifor is found independent of the fore-
going method,
ib.
79. When there are three quantities required
in a problem there must be three equa-
tions given to find them,
SI
How the values of the unknown quanti-
ties involved in thoſe equations are found,
ib.
80. Problem in which three unknown quan-
tities are required,
ib.
8r. Manner of abridging the calculations by
particular denominations,
52
83. All problems of the first degree in which
three quantities are required when re-
duced to equations are contained in the
foregoing problem,
53
84 In what cafes problems are indetermi-
nate,
Firſt example,
85. Second example,
54
ib.
56
86. Inconveniency to which the foregoing
method for exterminating unknown quan-
tities is liable,
57
Obfervations which have ferved to im-
prove this method,
ib.
87. General rule for finding the values of any
number of quantities required when as
many fimple equations are given, 58
88. Obfervations which have ferved to ren-
der this rule more fimple,
59
89. Problems which furnith as many equations
as quantities required are not always li-
mited, but in fome cafes are unlimited and
in others impoffible,
64
71. General method for finding the greateſt
common divifor of two numbers,
73. General method for finding the greatest 90. Rules for bringing problems to equat
common divifor of fpecious or algebraick
quantities,
48
tions,
Problem,
CHA P.
H.
ib.
62
St.
PROBLEM
Of the Refolution of Equations of the fecond Degree.
ROBLEM which in its full extent in-
cludes problems of every degree, 63
2: Equation of the foregoing problem for the
fecond degree,
3. For the third degree,
equations of the ſecond degree,
$
The fign✓ indicates the fquare root, ib,
An equation of the fecond degree has two
roots,.
ib.
64
ib.
7. Reduction of the value of x by refolving
the root of the product into thofe of its
factors,
66
$5
8. Example of this problem,
ib.
5. Inveſtigation of the method for folving
TABLE OF
OF CONTENT S.
9. Another example,
67
10. Third example in which the root of a
negative quantity being required is im-
poffible,
39. Quantities which have not an exa& root
are called incommenfurable or irration-
al quantities,
ib.
ib.
ib.
whofe
ib.
90
Thoſe roots are called imaginary,
Equations of the fecond degree
roots are imaginary,
11. The real and pofitive values of the un-
known quantity do not always folve the
problem,
Firſt example,
12. Second example,
ib.
ib.
68
13. General rule for refolving equations of
the ſecond degree,
༡༠
14. Another problem of the fecond degree, 71
15. Of the two foregoing values one is necef-
farily pofitive and the other negative, 72
16. Ufe of the negative value,
ib.
73
18. Example of the foregoing problem,
19. The pofitive and negative values of the
unknown quantity do not always both
folve the problem,
74
20. When a queſtion is wrong ftated the va-
lues of the unknown quantity become
both negative,
ib.
21. Other examples of the reſolution of equa-
tions of the ſecond degree,
75
22. Proceſs of the extraction of the fquare
root explained in an example,
76
23. Another example of the extraction of the
fquare root,
77
24. Method of extracting the fquare root of
numbers,
78
25. First example,
79
26. Procefs of the extraction of the ſquare
root of numbers,
80
ib.
81
27. Second example,
28. Third example,
29. The root of a number which is not a per-
fect power of the fame degree as the root
required is not determinable,
ib.
30. Method of approximating to the ſquare
root of numbers,
82
31. Firft example,
83
32. Second example,
ib.
33. Another method of approximating to the
fquare root of numbers,
84
34. First example,
86
Second example,
ib.
A vulgar fraction cannot always be re-
duced to a decimal,
87
35. Problem,
ib.
36. Application of the foregoing problem to
an example,
88
37. Another problem,
ib.
38. Examples of the reduction of radical
quantities,
89
The addition and fubtraction of irrational
quantities preſuppoſes only their reduction
to their leaſt terms,
40. Multiplication of furds,
ib.
41. Divifion of incommenfurable quantities, 91
42. Firſt problem of the ſecond degree where
two unknown-quantities are concerned, ib.
43. Second problem,
44. Third problem.
92
ib.
45. The foregoing equations folved by another
method,
93
47. When the values of two unknown quan-
tities in a problem are brought out per-
fectly alike, how the ambiguity thence
arifing may be avoided,
94-
48. Example of equations of the ſecond de-
gree including two unknown quantities
more complicated than the foregoing, 95
The refulting equation when one of the
unknown quantities is exterminated,
ib.
49. Another method by which the unknowa
quantity is exterminated-in the foregoing
example,
ib.
50. The foregoing method applied to another
example,
96
51. Whatever be the dimensions of y in two
equations if thofe of x do not rife higher
than two it may be exterminated by the
foregoing method,
ib.
52. How the foregoing method is to be ap-
plied when the quantity to be extermi-
nated rifes to higher dimenfions than
two,
97
53. If there were more than two unknown
quantities and equations given they may
be reduced to one by the foregoing me-
thod,
ib.
54. Inconveniency to which the foregoing me-
thod is liable,
How it has been removed,
98
ib.
55. Inveſtigation of Sir Ifaac Newton's rule
for exterminating an unknown quantity of
two dimenſions in each equation by the
foregoing method,
Example,
99
ib.
56. Inveſtigation of Sir Ifaac Newton's rule
for exterminating an unknown quantity of
three dimenfions in one, and of two di-
menfions in the other equation by the fore-
going method,
Example,
100
ib.
7. Inveſtigation of Sir Ifaac Newton's rule
for exterminating an unknown quantity
of four dimensions in one, and of twe
¢
TABLE OF CONTENTS.
dimenfions in the other equation by the
foregoing method,
Example,
IOI
102
58. Inveſtigation of Sir Ifaac Newton's rule
for exterminating an unknown quantity
of three dimenfions in each equation by
the foregoing method,
ib.
59. How an unknown quantity of a number
of dimenſions not exceeding four in each
equation may be exterminated by the pre-
ceding rule,
105
Example,
ib.
60. How the degree of the refulting equation
is determined,
ib.
61. Limit which the degree of the refulting
equation can not exceed,
106
62. How an unknown quantity of feveral di-
menfions is exterminated out of three
equations by the foregoing method, 107
64. How n" is determined and confequently
G,
65. Obfervations which ferve to find the va-
lues of п, n', x",
109
110
66. Rules derived from the precedent princi-
ples for exterminating an unknown quan-
tity of feveral dimenfions in three equa-
tions,
III
67. Inconveniency to which the method of
exterminating an unknown quantity by
comparing the equations two by two is
fiable,
I 12
113
70. How an unknown quantity of feveral di-
menſions is exterminated out of four equa-
tions by the foregoing method,
General expreffion of the degree of the
refulting equation after an unknown quan-
tity of feveral dimenfions is exterminated
out of four equations,
114
71. Rules derived from the preceding princi-
ples for exterminating an unknown quan-
tity of feveral dimenfions in four equa-
tions,
115
72. How an unknown quantity of feveral di-
menfions is exterminated out of five equa-
tions by the foregoing method,
General expreffion of the degree of the
refulting equation after an unknown
quantity of feveral dimenfions is exter-
minated out of five equations,
116
117
73. Rules derived from the preceding princi-
ples for exterminating an unknown quan-
tity of feveral dimenfions in five equa-
tions,
ib.
74. General expreffion of the degree of the
refulting equation after 20 unknown
quantity of feveral dimenfions is extermi-
nated out of Nequations,
118
CHA P.
III.
Of the Refolution of Equations of all Degrees which confift of two Terms only,
of thofe confifting of three Terms that can be reduced by the Method of Equa-
tions of the Second Degree, to Equations confifting of two Terms; with dif
ferent Operations relative to thofe Equations, fuch as the raising of Quan-
tities to any Power, the Extraction of Roots, the Reduction of radical Quan-
tities, &c. and of the Reduction of Equations by furd Divifers.
$1. ROBLEM which in its full extent pro-
PROBLEM
duces equations of every degree con-
fifting of two terms,
119
2. Equations of the third degree confifting
of two terms,
The figure 3 is put oyer the character/
to denote the cube root,
Thofe equations can only have two real
roots,
Example,
Another example,
ib.
122
ib.
7. Obfervations on the involution of quan-
tities,
1 20
ib.
ib.
3. A radical cube can have but one fign pre-
fixed to it,
Rule for involving Gingle quantities,
Example,
ib.
ib.
ib.
8. Evolution of fingle quantities,
123
4. How cubical radicals are multiplied into
one another,
121
What fracional powers denote,
Example,
ib.
ib.
5. Roots of an equation of the third degree
confifting of two terms,
9. What negative powers denote,
124
ib.
Example,
ib.
6. Refolution of equations confifting of two
terms of every degree,
What the power o denotes,
ib.
ib.
TABLE OF
CONTENTS.
10. Examples of the foregoing reductions and
transformations,
124
11. Extraction of the roots of imperfect pow-
ers,
125
13. In what the cube of a binomial confifts, ib.
14. Method of extracting the cube root of
compound quantities,
15. Firft example,
16. Second example,
126
ib.
127
17. Method of extracting the cube root of
numbers,
ib.
18. The foregoing method explained by an
example,
128
19. Method of approximating to the cube root
of numbers,
129
20. Procefs of the extraction of the cube root
of numbers,
130
21. Inveſtigation of Sir Ifaac Newton's theo-
rem for raiſing a binomial to any power, ib.
General formula for raifing + to the
power m,
133
22. Application of the foregoing formula to
an example,
ib.
23. How the foregoing formula may be ap-
plied to quantities confifting of more than
two terms,
24. Example,
134
ib.
25. General rule for extracting the roots of
numbers derived from the foregoing theo-
135
rem,
26. Application of this method to an example,
ib.
39. Multiplication and divifion of radical quan-
tities that have the fame exponent,
146
40. To perform thofe operations upon radical
quantities of different exponents they are
fift to be reduced to the fame exponent,
ib.
Method of performing this reduction, ib.
41. Another method of performing the fore-
going operations,
147
43. Investigation of the furd which multiplied
into a propofed furd gives a rational pro-
duct,
148
44. Application of the foregoing theorem for
reducing fractions involving furd quanti-
ties to a more fimple expreffion, ib.
46. Inveſtigation of the method of folving
equations confifting of two terms when
the unknown quantity enters the expo-
nent,
149
48. What is meant by the logarithm of a
a number,
152
How the logarithms of numbers of one
fyftem are deduced from thofe of another,
ib.
49. In every fyftem the logarithm of unity
iso,
ib.
The logarithms of negative numbers are
imaginary,
ib.
The logarithm of the product of two
numbers is equal to the fum of the lo-
garithms of the factors and the logarithm
of their quotient to their difference, 153
50. Inveſtigation of a general formula for
finding the logarithm of any given num-
ber,
ib.
27. General method of approximating to the
roots of numbers,
136
28. Application of the foregoing approximat-
ions to an example,
138
29. Ufe of the foregoing rules in the folution
51, Inconveniency to which the foregoing
method is liable,
154
of problems,
ib.
How it may be removed,
ib.
First example,
ib.
139
52. Another method for finding the logarithm
of any number,
ib.
54. Application of the foregoing formulas to
an example,
157
30. Second example,
The general formula found for the invo-
lution of binomials ferves alfo for their
evolution,
141
33. Application of the foregoing formula for
finding the roots of perfect powers of
compound quantities,
143
34. Application of the foregoing formula for
finding the roots of imperfect powers of
compound quantities,
ib.
35. The foregoing formula applied to raiſing
quantities to negative powers,
144
36. Difficulty that occurs when two terms of
the denominator are equal,
ib.
37. All kinds of quantities may be reduced in-
to feries by the foregoing formula, 145
38. Addition and fubtraction of all forts of 1a-
dical quantities,
ib.
55. The logarithms of numbers correfponding
to the bafe 10 the beſt adapted to numeral
arithmetick and why,
57. Uſe of logarithms for performing numeral
operations,
First example,
Second example,
ib.
158
ib.
ib.
58. In what cafes compound intereſt is favour-
able to the borrower and in what cafes
diſadvantageous,
59. Problem,
159
160
Application of the foregoing problem to
an example,
161
60. When a debt is diſcharged by ſeveral equal
TABLE or CONTENTS.
payments compound intereft is allowed to
the lender,
61. What is meant by annuities,
161
162
62. Refolution of the various questions relat-
ing to annuities,
ib.
64. Refolution of the various questions relat-
ing to annuities in arrears,
163
69. Problems producing equations confifting
of three terms folved by the method for
equations of the fecond degree,
70. Example of the foregoing method,
71. Another example,
72. Third example,
73. Fourth example,
74. Fifth example,
ib.
ing them into perfect fquare cannot o-
therwife be effected than by trial, 177
100. How equations of four dimenfions may
be reduced in this manner,
ib.
101. The number of trials diminiſhed from the
confideration of even and odd numbers, 178
102. General rule derived from the foregoing
conclufions for reducing an equation of
four dimenfions,
179
103. Application of the foregoing rule to an
example,
164
180
ib.
165
104. The foregoing rule applied to another
example,
ib.
ib.
75. Method of finding the ſquare root of quan-
tities partly rational and partly irrational,
166
77. Application of the foregoing method to an
cxample,
79. Another example,
80. Third example,
ib.
167
ib.
81. Method of finding the cube root of quanti-
ties partly rational and partly irrational, ib.
82. Application of the foregoing method to an
example,
83. Another example,
168
169
ib.
105. Cafe in which the foregoing rule cannot
be applied,
181
Rule to be obferved in this cafe, ib.
Application of this rule to an example, ib.
106. Third example,
182
107. Application of the foregoing rule for re-
ducing equations having fracted terms, ib.
Example,
183.
108. Application of the foregoing rule for ex-
tracting the rational roots of equations of
four dimenfions,
ib.
85. Method of finding the cube roots of nu-
meral quantities, partly rational and part-111.
ly irrational,
86. Application of the foregoing method
example,
ib.
to an
170
87. Another example,
ib.
88. The foregoing method rendered
fimple,
more
89. Application of the new method,
9o. This new method defective when
B have different figns,
171
A and
ib.
172
188
109. Obfervation which ferve to diminish the
number of trials,
184
110. Inveſtigation of the method of reducing
equations of fix dimenfions by trial, 185
The number of trials diminished from
the confideration of the properties of
even and odd numbers,
187
112. General rule derived from the foregoing
method for reducing an equation of fix
dimenfions,
ib.113. Application of the foregoing rule to an
example,
ib.
114. Cafe in which the foregoing rule cannot
be applied. Rule to be obſerved in this
cafe,
189
115. Method for difcovering whether an equa-
tion of eight dimenfions can be reduced,
116. General method for difcovering whether
an equation of any number of dimenfions
can be reduced,
ib.
117. Application of the foregoing method to
an example,
What is to be done in this cafe,
91. Cafe in which the two foregoing methods
fail,
ib.
173
How to remedy this deficiency,
93. What is to be done when the cube root
fhould be the fum of two radicals, 174
94. How the root denominated by an even
number is extracted,
ib.
95. Investigation of the rule for extracting the
root of any power of quantities partly ra-
tional and partly irrational,
ib.
97. Refolution of equations of even dimenſions
by compleating them into perfect squares
explained by an example,
175
176
98. Another example,
99. Inveſtigation of the method of compleat-
ing equations of four dimenfions into per-
fect squares,
ib.
The reduction of equations by compleat-
190
191
118. General rule for reducing equations that
have been found by the foregoing method
to be reducible,
192
119. Application of the foregoing rule to an
example,
193
120. Refolution of equations of the third and
fourth degree,
195
122. Equation the most complicated of the
third degree,
ib.
123. Transformation by means of which any
4 S
1
TABLE OF
term taken away from an equation, ib.
124. The foregoing transformation applied to
an equation of the fourth degree, 196
The fecond term moſt commonly exter-
minated and why,
1
CONTENTS.
ib.
125. The ſecond term exterminated in an
equation of the filth degree,
ib.
In an equation of the mth degree, ib.
126. Refolution of the general equation
x³ + px + 9 = 0,
197
127. The foregoing formula expreffes only one
of the three roots of the cquation, 198
Method of inveſtigating the two other
roots,
ib.
128. Cafe in which the value of x cannot be
diſcovered by means of the foregoing for-
mula on account of the imaginary quan-
tities it includes,
ib.
The value of x however in this cafe is
real,
ib.
129. Method of approximating to the value of
x in this cafe,
199
130. The two other values of x, are real in
the fame cafe,
ib.
131. How from the roots of the equation
x3 9
x³ + px + qo are deduced the
roots of the equation y³ + dy² + ey
+f=0,
3
200
132. An equation of the third degree has three
real roots, or one real root and two im-
poffible roots,
27
ib.
ib.
How to diftinguish thofe cafes,
Nature of the roots when p³ is nega-
tive and equal 99.
ib.
133. Application of the foregoing Methods to
the refolution of the propofed problem,ib.
134. Equation of the fixth degree reduced to
one of the third,
201
Equation of a higher degree reduced in
like manner to one of the third, ib,
135. Another problem producing an equation
of the third degree folved by the forego-
ing method,
ib.
136. Problem producing an equation of the
third degree that cannot be folved by the
formula of Art. cxxvi.
202
Application of the method of Art. cxxix.
for aproximating to the roots of the
equation produced by tho foregoing pro-
blem,
ib.
Inconveniency to which the foregoing me-
thod of approximating to the roots of e-
quations of the the third degree is liable,
203
137. Another general method of approxima-
tion more commodious in practice, ib.
138. The foregoing method of approximation
give the value of true to the fifth de-
cimal place,
204
139. Method of approximating to the value of
to any affigned degree of exactneſs, 205
140. Application of this method to an ex-
ample,
200
141. Problem producing an equation of the
third degree folved by this method, ib.
142. Problem producing an equation the moſt
complicated of the fourth degree. 207
143. Refolution of the general equation of the
fourth degree,
•
208
210
211
The refolution of an equation of the
third degree depends on one of the third
degree which is called by the Analyſts
the reduced equation,
ib.
144. The four roots of an equation of the
fourth degree may be expreffed by one
formula,
209
146. Let which ever of the roots of the redu-
ced equation be employed the fame roots
will always refult for the propoſed equa-
tion,
147. An equation of the fourth degree is ex-
actly refolvable when of its four roots
two are real and two imaginary,
The contrary happens when the four roots
are all real or imaginary,
ib.
148. How to diftinguish the cafe of the four
real roots from that of the four imaginary
ones,
Conditions of the four real roots, ib.
Conditions of the four imaginary roots, ib.
149. Every equation of the fourth degree
wanting the ſecond term and whofe third
term is pofitive has imaginary roots, ib.
150. How to obtain approximate values of the
four roots when they are real,
ib.
151. The equation produced by the propofed
problem folved by the foregoing method,
212
213
152. Another problem producing an equation
of the fourth degree folved by the fore-
going method,
214
154. Third problem producing an equation of
the fourth degree folved by the forgoing
method,
216
155. Fourth problem producing an equation of
the fourth degree folved by the foregoing
method,
ib.
156. Fifth problem producing an equation of
the fourth degree folved by the foregoing
method,
217
157. Method of clearing any propoſed equation
of radicals,
158. The foregoing method illuſtrated by an
example,
218
TABLE OF CONTENTS.
CHA P. IV.
Of the Nature and the Number of the Roots of Equations of all Degrees; of
the Method of finding the Equations of a lower degree that are their Divi-
fors, with the Methods of approximating to the Roots of both numeral and
literal Equations of every Degree.
$1. PROB
ROBLEM producing an equation the, 28. Another example of finding the commen-
moſt complicated of the nth degree, furable roots of an equation by the fore-
going method,
219
2. An equation may be refolved into as many
fimple factors as there are units in the
higheſt dimenfion of the unknown quan-
tity,
220
3. An equation admits of as many folutions as
there are fimple factors multiplied by one
another that produce it,
221
4. No equation can have more roots than it
contains dimenſions of the unknown quan-
tity,
ib.
5. Objection againſt the foregoing Demonftra-
tion folved,
222
6. General form that may be given to every
imaginary quantity whatfoever,
ib.
9.
7. The imaginary expreffions cannot difap-
pear in the eqnation produced but when
their number is even,
225
8. Method of finding the ſyſtems of factors
correfponding to the formulas of the fe-
cond, third, &c. degree,
226
When ſeveral ſyſtems of factors correſpond
to the fame formula how to find the fyftem
of factors correfponding to any propofed
equation contained in this formula, ib.
10. Tabl s whereby are difcovered the nature
and number of the roots of any propofed
equation,
20. Method of finding the commenfurable
roots of an equation,
227
233
21. In an equation whofe coeficients are inte-
gers the unknown quantity cannof bea
fraction,
ib.
22. Transformation by means of which an e-
quation may be cleared of fractions, 234
23. Inconveniency to which the foregoing me-
thod is liable,
ib.
24. Obfervations that have ferved to render
the foregoing method more perfect, 235
25. Fundamental principle for finding the
commenfurable roots of an equation, ib.
26. Application of the foregoing method to
an example,
236
239
29. Third application of the method of finding
the commenfurable roots in an equation,
ib.
30. Advantage of inveſtigating the commen-
farable divifors in the reduced rather than
in the propoſed equation of the fourth de-
gree,
240.
32. How to diftinguish the equations of the
fourth degree whofe roots are only affect-
ed by fquare radicals,
241
33. The foregoing obfervations illuftrated by
an example,
ib.
34. Another problem folved by the foregoing
method,
242
35. Third problem folved by the foregoing
method,
ib.
36. Investigation of the method of finding the
quadratick commenfurable factors of any
propofed equation,
ib.
37. Application of the foregoing method to an
example,
244
38. Application of the foregoing method to
an example,
246
39. Method of finding the divifors of one
dimenfion when the higheſt term of the
equation has a coeficient different from
unity,
247
41. Problem folved by the foregoing rules, 249
42. Another problem folved by the foregoing
rules,
250
43. Method of finding the divifors of two di-
menfions when x has a coeficient, 250
44. Application of this method to an example,
251
Every quantity of lefs than fix dimengons
that has divifors muft have fome of a lefs
number of dimenfions than three.
ib
45. If the quantity has fix or more dimenſions
it may have only divifors of three or more
dimenfions,
252
Method of finding divifors of three or
more dimenfions,
27. Method of finding all the divifors of any | 46.
propoſed number,
238
ib.
Method of finding all the divifors involv-
L
TABLE OF CONTENT S.
ing two letters in a quantity involving
three,
253
49. Method of finding the diviſors of one di-
menfion involving three letters. 254
50. Application of the foregoing method to
examples,
255
f
52. The foregoing method applied for finding
the divifors involving three letters at the
fame time as thofe involving 2 letters, 256
53. Method of finding the divifors of two di-
menſions involving three letters, 258
54. Application of this method to an example,
ib.
55. Cafe in which the divifors are found much
easier than by the foregoing method, 259
56. Another example,
260
Application of the method of Art xxiv.
for finding all the divifors of literal quan-
tities,
ib
58. How the divifors of quantities that are not
homogeneous are found,
262
59. Method of approximating to the roots of
numeral equations involving one unknown
quantity,
263
60. Application of the foregoing method to
examples,
ib.
62. When there are two equations given, and
as many quantities x and y how to deter-
mine thoſe quantities,
ib.
63. Application of the foregoing method to
examples,
ib.
method of
265
65. Fundamental principle of the
feries,
66. Diviſion of ſeries into aſcending and def-
cending ones,
266
67. An equation lofes fome of its terms when
one of its unknown quantities is fuppofed
indefinitely great or indefinitely little, ib.
68. Method of finding the greateſt terms of an
equation by way of exclufion,
ib.
Application of the foregoing method to an
example,
267
69. Ufe of the analytic triangle in this re-
fearch,
268
Property of the analytic triangle, 269
70. The exponents of the terms which are in
the fame ſtraight line are in arithmetical
progreffion,
ib.
270
71. Demonſtration of the foregoing property
of the analytic triangle,
73. The terms which are in two parallel
ftraight lines have exponents which form
arithmetical progreffions having the fame
common difference,
ib.
74. All the terms that are in the fame ſtraight
line are of the fame order, if two of them
be fuppofed to be of the fame order, 271
75. Ratio of the orders of the two undeter-
mined quantities in this fuppofition, ib.
Exponent of the order of the terms that
are in the fame ftraight line,
272
The terms that are above this ſtraight
line are of a fuperior order: thofe that
are below it of an inferior order, ib.
78. Rule derived from the foregoing princi-
ples, for diſcovering the greateſt terms
of an equation,
273
79. Application of the foregoing rule to ex-
amples,
274
81. Roots of the equation given by a deter-
minator,
82. They may be imaginary,
83. Or half imaginary,
276
277
ib.
86. Investigation of the fucceffive terms of a
feries,
279
87. How to know when a ſeries becomes
imaginary, half imaginary, or forked, 281
88. Application of the foregoing method to
examples,
282
90. Example of an imaginary feries, 284
91. Example of a feries half imaginary, ib.
92. Manner the terms of a transformed e-
quation are placed in the analytic tri-
nagle,
Example,
285
286
93. Abridged method of performing the tranf-
formations requisite to be made for find-
ing the terms of a ſeries,
290
94. Application of this method to an ex-
ample,
ib.
95. In what cafes fome of the terms of thoſe
transformed equations are wanting, ib.
96. How the irregular terms of a feries are
inveſtigated,
293
97. How to know when a feries begins to be-
come regular,
294
98. Determination of the form of a regular
ſeries or the exponents of the regular
terms,
295
99. Rule derived from the foregoing confi-
derations for finding the fucceffion of ex-
ponents of the regular terms of a feries,
297
101. Application of this rule to examples ib.
104. Reverfion of feries,
304
105. Obſervation whereby may be diſcovered
in many cafes if a feries is half imagi-
305
nary,
END OF THE TABLE OF CONTENTS.
pilate
plate. I.
Q
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