■.mm 
 
 LIBRARY OF CONGRESS. 
 
 PRESEN fED t 
 
 UlflTED STATES OP AMEJiiUA. 
 
 

 f^^^ms^ 
 
 mm 
 
 ^JX 
 
 ml 
 
 
 m)^^f^mM-m>z^. 
 
 
 «^^^M^»^ 
 
 
 ^mr^MTT 
 
REMARKS 
 
 ON 
 
 MATHEMATICAL OR DEMONSTRATIVE 
 REASONING: 
 
 ITS CONNEXION WITH LOGIC; 
 AND ITS APPLICATION TO SCIENCE, 
 
 PHYSICAL AND METAPHYSICAL, 
 WITH REFERENCE TO SOME RECENT PUBLICATIONS. 
 
 Vld^l EDWARD ^AGART, F.G.S. 
 
 .•.IINISTER OF THE CHAPEL llf LITTLE PORTLAND-STREET, REGENT-STREET. 
 
 ^ " The light of human minds is perspicuous words ; but by definition first 
 snuffed, and purged from ambiguity." — Hobbes. 
 
 " It is an old remark that geometry is the best logic." — Berkeley's Analyst. 
 
 LONDON: 
 JOHN GREEN, 121, NEWGATE STREET. 
 
 1837. 
 
^0 
 
 1^ 
 
 .Tlil) 1!Y lUCIIAnC KINDER, 
 NEW STREET, FETTER LAN 
 
THE REV. W. TURNER, JUN. 
 OF HALIFAX. 
 
 My Dear Sir, 
 To your valued instructions in Manchester College, 
 York, I am indebted for much of the pleasure which I 
 have occasionally derived from metaphysical inquiries 
 and discussions. In the metaphysical as well as theo- 
 logical department of that Institution, we were taught to 
 study with care and candour the best works, not to cavil 
 and confute, " nor to believe and take for granted, but 
 to weigh and consider." To you, therefore, as one well 
 skilled in weighing arguments and detecting fallacies, 
 I beg leave to inscribe this little volume, and remain 
 Your grateful pupil, 
 
 EDWARD TAGART. 
 
 Bayswater, 
 September 1S37. 
 
 a 2 
 
PREFACE. 
 
 The remarks here offered to the reader 
 were commenced some time ago, with the 
 design of adapting them to the pages of a 
 Heview or Magazine. Hence they have 
 perhaps too much of a critical and contro- 
 versial air for a distinct Essay on an abstract 
 subject. But the remarks grew under my 
 hands ; and becoming, as they grew, less and 
 less fit for any existing periodical, they are 
 now presented, but with great diffidence, in 
 
Vi PREFACE. 
 
 a separate form — with what advantage, the 
 reader of course will judge. 
 
 The well-instructed student of mathema- 
 tics, of logic, of the nature and theory of 
 language, and of what is called moral evi- 
 dence, will be apt to remark at the close of the 
 work, that it makes no specific addition to the 
 amount of his knowledge. But if the views 
 presented even to him be admitted to be 
 correct as far as they go ; if some thoughts 
 are here conveniently brought together which 
 must at least be sought for in widely-scat- 
 tered sources ; if they have the good eftect of 
 awakening the attention of the less profound 
 to important points connected with the sub- 
 ject before him, hitherto overlooked ; if they 
 suggest or stimulate inquiries worthy of con- 
 tinuance, — the publication will not be in vain. 
 Much use has been made of the sentiments 
 
PREFACE. Vll 
 
 of others, so as to form a sort of philo- 
 sopliical discussion in which many authors 
 are made to speak for themselves. But an 
 ample apology for this, if one be necessary, 
 will be found in the words of Dr. Law in his 
 preface to the translation of Dr. King's Essay 
 on the Origin of Evil : — 
 
 " A writer often does more good by show- 
 ing the use of some of those many volumes 
 which we have already, than by offering new 
 ones, though this be of much less advantage 
 to his own character. I determined there- 
 fore not to say anything myself where I 
 could bring another conveniently to say it 
 for me ; and transcribed only so much from 
 others as was judged absolutely necessary to 
 give the reader a short view of the subject, 
 and by that sketch to induce those who have 
 leisure, opportunity, and inclination to go 
 
Vlli PREFACE. 
 
 further and consult the originals, and to 
 afford some present satisfaction to those who 
 have not, 
 
 " But how judiciously this is performed, 
 the notes themselves must testify." 
 
CONTENTS. 
 
 Page. 
 Introductor))' remarks ..... 1 
 
 Mathematical reasoning sets out fj'om definitions . 5 
 Tliese definitions settle the meaning of terms . 1 
 
 These terms, signs of ideas of figure and quantity — 
 
 ideas originating in sensible impressions . . 20 
 
 Mathematical reasoning supported by diagrams, or 
 
 evidence of the senses . - ... 22 
 Mr. Whewell's language on experience as the source 
 
 of mathematical conceptions, criticised . . 27 
 
 Beddoes on Demonstrative Evidence, and Playfair 
 
 on Beddoes, considered . . . . .33 
 
 Influence of habit, or constant connexion of the 
 
 terms with the same ideas, in producing assent to 
 
 mathematical processes . . . . .36 
 
 Fewness of premises and of terms in n:iathematical 
 
 reasoning ....... 38 
 
 Distinctness and simplicity of ideas of number and 
 
 figure 40 
 
 Final and essential characteristic of deinonstrative 
 
 reasoning ....... 44 
 
CONTENTS. 
 
 SECTION II. 
 
 Page. 
 
 Mr. Dugald Stewart's, Dr. Wlaately's, and the Edin- 
 burgh reviewer's remarks on mathematical and 
 general reasoning contrasted . . . .50 
 Sir John Herschel's character of Sir Isaac New- 
 ton 57 
 
 Whately's opinion on the sameness of the reasoning 
 
 processes asserted and vindicated . . .60 
 
 Account of logic in the Encycloppcdia Britannica . 63 
 Logic another term for reasoning . . .65 
 
 Whately's logic considered . . . . .68 
 
 His analysis of arguments . _ . . . .73 
 
 Nature of syllogism . . . . . .74 
 
 Comparison of' logical and mathematical reasoning . 80 
 
 SECTION III. 
 
 The connexion between language and reasoning in 
 general ........ 92 
 
 Approach to mathematical exactness in metaphysical 
 sciences, how attainable . . . . .95 
 
 Distinction between mathematics which commence, 
 and inquiries which end, with definitions . .96 
 
 Demonstration not always necessary . . . 1 00 
 
 On demonstrative reasoning in physical science . 102 
 
CONTENTS. 
 
 Advantages of physical science. . 
 Works of reference .... 
 Cuvier on the stud}^ of natural history . 
 Metaphysical science .... 
 Metaphysical discussions concern the mean 
 
 words ...... 
 
 Stewart's and Mackintosh's dissertations 
 Stewart's estimate of Locke 
 Mill's Fragment on Mackintosh . 
 Mackintosh on Hartley 
 Laplace's Essay on Prohabilities . 
 Pla}-fair on Laplace .... 
 
 Laplace on association 
 
 Character of Mr. Austin's work, " The Prov: 
 
 Jurisprudence determined" 
 Austin's remarks on demonstration connecte- 
 
 ethics ...... 
 
 ng of 
 
 nee of 
 with 
 
 XI 
 
 Page. 
 
 104 
 107 
 
 108 
 109 
 
 111 
 112 
 113 
 118 
 121 
 124 
 126 
 127 
 
 130 
 
 131 
 
REMARKS. 
 
 It is more than a century since Locke con- 
 ceived and maintained, after Hobbes and 
 perhaps others, that demonstrative reasoning 
 was apphcable to other subjects besides the 
 mathematics, and particularly to morahty. 
 Doctors Law and Hartley, the disciples and 
 successors of Locke, entered fully into his 
 views ; and Dr. Hartley especially was fond of 
 exhibiting his reasoning in a mathematical 
 form, and in some instances has very happily 
 applied algebraic formulae to illustrate, I do 
 not say to confirm, his trains of moral spe- 
 culation. 
 
 Mr. Whewell's Thoughts on Mathematics, 
 in which he affirms that mathematics afford 
 the best example of practical logic, and the 
 elaborate article in the Edinburgh Review, 
 
"A INTRODUCTORY REMARKS. 
 
 No. 126, on Mathematical Studies, which 
 treats generally of the influence of mathe- 
 matics upon the intellectual character and 
 powers, have in some degree recalled atten- 
 tion to the subject, and induced me to offer 
 some thoughts upon it, which I trust will not 
 appear altogether unworthy of perusal. 
 
 I do this with earnestness, and even anxiety; 
 not because I conceive that anything original 
 or remarkable will be found in the following 
 observations ; for I bear in mind an aphorism 
 of Dr. Johnson, " He who tries to say that 
 which has never been said before him, wdD 
 probably say that which will never be re- 
 peated after him ;" but because clear and 
 just views on this subject have a close and 
 important bearing upon the pursuit of science 
 of all kinds, whether physical or metaphy- 
 sical ; upon the attainment and diffusion of 
 truth ; upon the mental and moral improve- 
 ment, and consequently the harmony and 
 happiness, of man. These clear and just 
 views appear to be absent from the minds 
 and writings of many w^hose names are of 
 no small account in the Uterarv and scientific 
 
INTRODUCTORY REMARKS. d 
 
 world, although within easy reach of tiie in- 
 quiring, if they will use the glass supplied 
 by the plain and manly writers of the true 
 English school of philosophy. 
 
 I venture upon it further, because Mr. 
 Whewell, in the second edition of his pam- 
 phlet, declines going more at length into the 
 matters touched upon by the reviewer. He 
 has therefore left the field open to any one 
 who may dare to enter the lists against that 
 formidable and heavily-armed knight. 
 
 The interest wdiich may have been felt in 
 the papers alluded to has perhaps already 
 subsided, but the subject to which they re- 
 late is of permanent importance. Mr. 
 Whewell discussed the relative value of 
 different modes of pursuing mathematical 
 studies, assuming their usefulness and im- 
 portance. The reviewer, however, went 
 into a much wider field, namely, the 
 influence of mathematical studies upon the 
 mental powers and character in general. 
 And it was his strain of remark, so far as it 
 was of a metaphysical character, his obser- 
 vations about " two logics," " dissimilar de- 
 B 2 
 
4 INTRODUCTORY REMARKS. 
 
 velopments of thought," and "higher and 
 lower faculties," which induced me to review 
 the reviewer, and put together a few thoughts 
 upon the nature of reasoning and evidence in 
 general. With many of the positions of the 
 reviewer and of those Vv^hom he quotes, about 
 mathematical evidence and mathematical pro- 
 cesses, the matter of this Essay will be found 
 substantially to coincide ; but if there be 
 any truth in his view of the disqualification 
 of mere mathematicians for inquiries into 
 mental and moral philosoplw, I have en- 
 deavoured to approach more closely to the 
 sources of that disqualification, or rather so to 
 point out the distinction between the nature 
 of our thoughts and lang-uage on mathe- 
 matical and other subjects, as to furnish 
 some useful guidance to reasoners and in- 
 quirers in morals and rehgion. 
 
 Moreover, while it seemed to me that the 
 reviewer had merely taken occasion from 
 Mr. Whewell's pamphlet to propound cer- 
 tain semi-German notions, and heap together 
 certain Kantian phrases, than which none 
 are more utterly distasteful to a healthful 
 
DEFINITIONS. 5 
 
 English palate, it is perceptible, if I mis- 
 take not, that Mr. Whewell himself is not 
 altogether so clear and satisfactory as lie 
 might be, his own mind being apparently 
 tainted in some slight degree with German 
 phraseology and metaphysics. But I shall 
 not trespass further upon ground which lies 
 beyond the limits which it is my present in- 
 tention to occupy, nor detain the reader from 
 the real subject before us. 
 
 Now in order to perceive the possible ap- 
 plication of mathematical or demonstrative 
 reasoning to metaphysical subjects, the first 
 requisite is to understand exactly the nature 
 of that reasoning. In order to do this I ob- 
 serve, then, — ■ 
 
 First, That it is the important character- 
 istic of mathematical reasoning to proceed 
 from definitions. It sets out from these as 
 precise data, to which appeal is made in 
 every step of the demonstrative process. If 
 these definitions be not clearly understood, 
 if they be not fully granted and well laid up 
 in the mind, in vain does the student or 
 pupil attempt to proceed. The foundations 
 
DEFINITIONS'. 
 
 of the science v/ill then be broken up, the- 
 field of mathematical reasoning will then be 
 closed, for these are the gate of entrance. 
 
 It affects not the truth of the above 
 position to inquire and settle whether the 
 definitions are hypotheses or facts ; whether 
 they be explanations of terms, abstractions 
 of the mind, inductions from observation, or 
 assumptions which have no foundation in 
 the nature of things ; nor to inquire whether 
 the definitions of any particular treatise or 
 mathematician, from Euclid and his nume- 
 rous editors downwards to Newton and his 
 successors, are in every respect the best 
 possible, such as suit best the subsequent 
 course of reasoning, are most easily ad- 
 mitted by the student, or bring most clearly 
 before the mind the principle necessary for 
 future guidance ; nor to inquire how far the 
 postulates and axioms partake of the nature 
 of definitions, may be resolved into them, or 
 mRj be dispensed with altogether, without 
 injury to the study. These may be proper 
 subjects for the metaphysician or logician. 
 They may, to a certain degree, call for the 
 
DEFINITIONS. 7 
 
 early attention of the mathematical student ; 
 but the tutor who should begin with telling 
 his pupil all that has been said or might be 
 said about the definitions, postulates, and 
 axioms, would probably never get him over 
 the asses' bridge. Suffice it, as a matter of 
 fact, that when you open any elementary 
 treatise on mathematics, Euclid, the conic 
 sections, plane and spherical trigonometry, 
 and even books of arithmetic and algebra, 
 the first objects of careful attention to the 
 student are definitions. These are the foun 
 dations of his science, the elements of his 
 reasoning. To these he must adhere ; and 
 if there be anything inconsistent with them, 
 confusion ensues, demonstration ceases. 
 
 "It is in this last circumstance (I mean 
 the peculiarity of reasoning from definitions) 
 that the true theory of mathematical reason- 
 ing is to be found," says Dugald Stewart in 
 his chapter on Mathematical Reasoning, 
 chap. ii. sect. 3, of his second volume of the 
 Philosophy of the Human Mind, one of the 
 best portions of his writings, yet tinctured 
 deeply with his peculiar faults. This is the 
 
8 DEFINITIONS. 
 
 point upon which he rests ; and the writers 
 to whom he refers, and with quotations from 
 whom he is so fond of nibbhng, will be found 
 substantially to agree with him. 
 
 The following passage from Hobbes' Le- 
 viathan is also apt to my puqjose. " To 
 the priviledge of absurdity, no li^dng crea- 
 ture is subject, but man onely. And of 
 men, those are of all most subject to it, 
 that professe philosophy. For it is most 
 true that Cicero saith of them somewhere ; 
 that there can be nothing so absurd, but may 
 be found in the books of philosophers. And 
 the reason is manifest. For there is not one 
 of them that begins his ratiocination from 
 the definitions, or explications of the names 
 they are to use ; which is a method they are 
 to use onely in geometry ; whose conclusions 
 have THEREBY been made indisputable." — 
 Part i. chap. v. 
 
 The importance of the detinitions admits 
 of the following familiar illustrations. It 
 happened to me to commence the study of 
 Suclid with a youth who stumbled at the 
 first definition, — " A point is that which hat li 
 
DEFINITIONS. il 
 
 no parts, or wliich hath no magnitude." 
 " Then," said he, " it is a nonentity, — it is 
 nothing." He could not or he would not 
 admit such an abstraction ; and he began to 
 puzzle himself about the infinite divisibility of 
 matter, the nature and extension of ultimate 
 atoms, the impossibility of finding a given 
 place for that which had no parts, and so on. 
 He was not content to take this or any other 
 definition as a matter of course, and wait to 
 see how far the mathematician would be 
 consistent with himself in his subsequent 
 reasoning. He was determined to weigh and 
 settle the justness of every definition in his 
 own mind before he would proceed further ; 
 in short, he would concede nothing and dis- 
 pute everything. Consequently he never 
 took kindly to mathematical studies ; and 
 perhaps to this hour he looks upon mathe- 
 matics as a multitude of words about non- 
 entities, or things Avhich have no real ex- 
 istence, and consequently no practical value. 
 I by no means imply that in mathematics 
 the student is to begin with submitting to 
 authority, and not to think about the 
 B 5 
 
iO DEFINITIONS DECIDE 
 
 meaning of the language he uses ; nor 
 that the tutor should not be prepared to 
 defend his own preliminary statements. Of 
 this, perhaps, more hereafter. It is sufficient 
 to add at present, that Mr. De Morgan, in 
 his work on mathematical studies, published 
 by the Society for the Diffusion of Useful 
 Knowledge, ranks definition first among the 
 characteristics, and, I may say, as at the 
 foundation, of mathematical reasoning. 
 
 Secondly, It may appear to many super- 
 fluous, but it is important to observe that 
 the definitions on which mathematical rea- 
 soning depends are definitions properly so 
 called ; that is, they are explanations of terms 
 — determinations of that sense in which the 
 words employed as signs and instruments of 
 thought are to be taken, used, and under- 
 stood. 
 
 Every one who reads over the definitions 
 of Euclid must, I should think, immediately 
 assent to this. One definition may be 
 better than another of a line, or a straight 
 line, of a circle, or of parallel lines ; but 
 its superiority can only consist in fixing 
 
THE MEANING OF TERMS. 11 
 
 more clearly that sense of the word about 
 to be used, or that quality in the mind's con- 
 ception of the thing signified, (more simply, 
 the signification,) which alone is to be pre- 
 sent to the mind in its subsequent appli- 
 cation of the term. 
 
 Tlie definitions of geometry concern, it is 
 obvious, the meaning of the terms point, 
 line, straight fine, superficies, angle, triangle, 
 circle, and so on, so far as they are de- 
 finable. And the student would have 
 little occasion to pore over these definitions 
 if all the terms were previously familiar 
 to him, and all had that fixed and clear 
 meaning in his mind ; that is, were the signs 
 of those certain ideas of figure for which 
 they stand in the mind of the geometrical 
 reasoner. 
 
 If Mr. Dugald Stewart had kept Euclid 
 open before him when speaking of the 
 definitions, I can scarcely imagine he would 
 ever have called them hypotheses ; and if he 
 had not called them hypotheses, he w^ould 
 not have maintained that it is the peculiarity 
 of mathematical reasoning to employ hypo- 
 
12 DEFINITIONS DECIDE 
 
 theses instead of facts as the data on which 
 we proceed. See Stewart's Philosophy, 
 A^ol. ii. pp. 158, 160. 
 
 Take for example the ninth and eleventh 
 definitions. " A plane rectilineal angle is 
 the inclination of tw^o straight lines to one 
 another, which meet together but are not in 
 the same straight line." "■ An obtuse angle 
 is that which is greater than a right angle." 
 With what propriety can it be said that 
 these are h}^otheses? An hypothesis is 
 that mode of accounting for certain appear- 
 ances which, although probable, remains to 
 be verified by future experiments or obser- 
 vations ; or it is the supposed cause of certain 
 effects, whose adequacy or invariable an- 
 tecedency is assumed until disproved by 
 further investigation. This is the sense in 
 which the term hypothesis is generally em- 
 ployed. In this sense the Ptolemaic and 
 Tycho-Brahic systems of astronomy were 
 hypotheses ; in this sense the theory of 
 gravitation, as it first occurred to the mind 
 of Sir Isaac Newton, as accounting for the 
 phases and motions of the heavenly bodies. 
 
THE MEANING OF TERMS. 13 
 
 was an hypothesis ; in this sense the midu- 
 latory theory of hght is an hjq^othesis. 
 
 Now there is no analogy between this 
 meaning of the term hypothesis and the 
 definitions of geometry, or of any branch of 
 mathematical science. These definitions are 
 not imaginary explanations of given pheno- 
 mena, nor supposed causes of given effects ; 
 they are, as above said, simple explanations 
 of terms, or attempts, by the substitution of 
 other words in place of one general term, to 
 place before the mind, often assisted by 
 diagram or sensible representation to the 
 eye, that object of thought to which the 
 said term is invariably and solely to be ap- 
 plied. " Every general term," says Aristotle, 
 " is the abridgement of a definition." 
 . I observe that Mr. De Morgan, on the 
 Study of Mathematics, p. 70, places reason- 
 ing from hypothesis second among the 
 characteristics of geometrical reasoning. "•' In 
 the statement of every proposition," he says, 
 " certain connexions are supposed to exist, 
 from which it is asserted that certain con- 
 sequences will follow. Thus, in an isosceles 
 
14 DEFINITIONS DECIDE 
 
 triangle, the angles at the base are equal, or, 
 if a triangle be isosceles, the angles at the 
 base will be equal. Here the hypothesis or 
 supposition is, that the triangle has two 
 equal sides ; the consequence asserted is, that 
 the angles at the base, or third side, v.ill be 
 equal." 
 
 Let us remark, however, that still the 
 hypothesis implies a clear understanding of 
 the words employed, as in the above in- 
 stance, isosceles and triangle, both of which 
 have been clearly defined and are well under- 
 stood. Hypothesis here is strictly and merely 
 supposition ; a certain figure or relation of 
 lines is supposed or granted to exist, from 
 which certain consequences are deduced, 
 Tlie reasoning would not be vahd, or there 
 would be no reasoning at all, if the terms 
 employed did not in the first instance ex- 
 actly express the thing intended, — the object 
 of thought. Much of common reasoning is 
 reasoning from hypothesis in this sense ; 
 that is, it consists in supposing certain re- 
 lations to exist, and in showing that certain 
 consequences follow. 
 
THE MEANING OF TERMS. 15 
 
 It was not in this sense that Mr. Dugald 
 Stewart maintained that " in mathematics 
 we employ hypotheses instead of facts" as a 
 general proposition ; but rather with intent 
 to show that the whole of mathematics 
 rested upon assumptions, and therefore dif- 
 fered from reasonings which turn upon ob- 
 servation, and what he calls facts. He often 
 implies that mathematical reasoning remains 
 good, though there be no such things in 
 reality as points, lines, triangles, circles, and 
 squares in the mathematical sense. But if 
 it be so, even admitting all this, still let us 
 remember that the hypotheses or assump- 
 tions, so far as the definitions are included 
 in them, are of a certain kind ; namely, that 
 certain words shall invariably be associe^ted 
 with certain meanings or ideas, and no 
 other : for example, that you shall not 
 reason about a triangle as if it could possibly 
 mean a circle, nor about parallel lines as if 
 they were not equidistant at all points. 
 
 " It is not on the definition but the con- 
 ception," Mr. Whewell asserts, " that the 
 properties and demonstrations are built." 
 
16 DEFINITIONS DECIDE 
 
 But why separate definition and concep- 
 tion ? Are they not virtually the same thing ? 
 unless by definition we are to understand 
 mere words without signification, little black 
 marks upon a piece of white paper. The 
 definition is of value solely in fixing, and, as 
 it were, embodying the conception. Human 
 beings reason with words, which are the signs 
 or channels of ideas. You can only convey 
 your conception of a straight line or triangle 
 to another mind by a definition or descrip- 
 tion. It is the object of the definition to single 
 out that quality or property in the mind's 
 conception of the thing which distinguishes 
 it most completely from every other object 
 or thing whatsoever ; and which, by being so 
 distinguished, and having such settled pro- 
 perty, becomes the subject of reasoning. If 
 it belonged, for instance, to anything else 
 besides an angle to be composed of two 
 straight lines meeting together, but not in the 
 same straight line ; in these words we should 
 have no sufficient definition of an angle. 
 
 In the Appendix to his work on the Con- 
 nexion of Number and Magnitude, Mr. 
 
THE MEANING OF TERMS. 17 
 
 De Morgan makes some admirable and use- 
 ful observations on the definitions, postulates, 
 and axioms of Euclid ; and thus expresses 
 -himself: — "Some of the definitions contain 
 assumptions of certain conceptions existing, 
 to which names are to be given ; namely, 
 those of a point, a line, the extremities of a 
 line, a straight line, a surface, the extremi- 
 ties of a surface, a plane surface, a plane 
 angle, a plane rectihneal angle ; others 
 assume the possibility of certain relations 
 existing, as will appear from the form in. 
 which they are put." 
 
 He afterwards speaks of these as " in- 
 definable notions," and places the common 
 definitions of them in the light of postulates ; 
 thus, " Let it be granted that a point has no 
 parts or magnitude, and that we are con- 
 cerned with no other property of it, if there 
 be any." Again, he speaks of some of the 
 definitions, those from the eleventh to the 
 fourteenth, and from the nineteenth to the 
 twenty-third, as purely nominal, and there- 
 fore needing no remark. From the tenor of 
 his language, which the reader who is not 
 
18 DEFINITIONS DECIDE 
 
 acquainted with it will do well to consult, it 
 would seem that he considered the geometry 
 of Euclid as resting very much on common 
 notions (xoivrj swoia) which scarcely admit of 
 definition. Nevertheless I do not think that 
 his language countenances any separation 
 between the conception and what is usually 
 considered the definition, but the contrary. 
 The object of all the definitions clearly is 
 to associate a certain term with a particular 
 notion or conception, and thus to fix and 
 limit the meaning of the term. In his paper 
 on the study of mathematics, Mr. De Mor- 
 gan says, (p. 69.) " This {i.e. definition) is 
 merely substituting, instead of a description, 
 the name which it has been agreed to give to 
 whatever bears that description." 
 
 In regard to nominal definition, it is to my 
 purpose to quote here wdiat Dr. Whateley 
 says in his Elements of Logic, p. 155, fifth 
 edition ; and although I begin the quotation 
 in the middle of a sentence, no alteration is 
 made in its force or meaning : — " all that is 
 requisite for the purposes of reasoning 
 (which is the proper province of logic) is. 
 
THE MEANING OF TERMS. 19 
 
 that a term shall not be used in diiferent 
 senses ; a real definition of anything belongs 
 to the science or system which is employed 
 about that thing. It is to be noted, that in 
 mathematics (and indeed in all strict sciences) 
 the nominal and the real definition exactly 
 coincide ; the meaning of the word and the 
 nature of the thing being exactly the same. 
 This holds good also with respect to logical 
 terms, most legal, and many ethical terms." 
 
 Upon the whole we conclude that the 
 definitions of geometry settle the meaning of 
 terms.* 
 
 Thirdly, These terms are the signs of our 
 ideas of figure and quantity, including in the 
 latter term number and magnitude (both the 
 
 * Pascal, in his Reflexions sur la Geometrie en General, 
 justly observes, however, that many notions are assumed, 
 and terms are used in mathematics which are not defined. 
 " Cette judicieuse science est bien eloignee de definir ces 
 mots primitifs, espace, temps, mouvement, 4galit^, ma- 
 jorite, dhninution, tout, et les autres que le monde entend 
 de soi-meme." And this must be the case in all reason- 
 ing ; for, as definition is merely explaining one term by 
 many, it is obvious we might go on defining without 
 end, and not advance a step towards any valuable con^ 
 elusion. 
 
20 MATHEMATICAL CONCEPTIONS 
 
 how many and how great, quantus) ; which 
 ideas or notions come to us, so to speak, 
 originally from without ; i. e. they originate 
 in sensible impressions. They are not signifi- 
 cant merely of what passes within, or of 
 mental states, like the terms memory, the 
 will, judgement, attention, and desire, unless 
 indeed every sensation, such as of whiteness 
 or blackness, be considered a mental state, 
 and every idea an affection of the mind. 
 
 Here, perhaps, I am treading upon the 
 most doubtful, because metaphysical, ground. 
 Right or wrong, however, in what may be said 
 under this head, it will not invaUdate what 
 has been said about definition and its object. 
 It appears to me that mathematical reasoning 
 consists in tracing the relations of our ideas 
 of figure and quantity by means of exactly 
 defined symbols, whether words, diagrams, 
 or other symbols, one with another, in re- 
 spect of agreement or disagreement, equa- 
 lity, or inequality ; and these terms and ideas 
 receive clearness and strength by constant 
 application and reference to external things, 
 or sensible impressions ; and also by their 
 
ORIGINATE IN SENSIBLE IMPRESSIONS. 21 
 
 observed, clear, uniform, and well-defined 
 relation to each other. 
 
 The subject matter of mathematical reason- 
 ing may tlierefore be considered to be real 
 existencies, with as much justice as the sub- 
 ject matter of any other reasoning. For in 
 all reasoning, what has the mind before it 
 but its own abstractions or notions, and 
 terms affixed to those notions ? And who 
 can say that circles, angles, squares, lines, 
 have not as much foundation in, and refer- 
 ence to, things as they exist, as white, blue, 
 black, soft, hard, or other quahties of body, 
 solid, Hquid, brittle, or elastic ; or the ab- 
 stract ideas of space, time, beauty, honour, 
 virtue, and so on? Our ideas of number 
 and figure are ideas constantly forced upon 
 us by sensible objects, and all that fills this 
 visible diurnal sphere ; the terms significant 
 of these ideas are in constant use and appli- 
 cation in ordinary life. They are employed 
 by the humblest in station and education 
 with uniformity of meaning, with clearness 
 and accuracy for their purposes. It is true 
 thev mav not know anvthins: of the rela- 
 
22 DIAGRAMS. 
 
 tioiis and properties of triangles, squares, 
 circles, parallelograms, as traced by the ma- 
 thematician ; but the mathematician's skill 
 and wisdom consist only in having traced 
 and studied these relations by means of his 
 exact definitions, and by his deeper or more 
 frequent meditation on their several con- 
 nexions and consequences. The ideas or 
 notions of number and figure are common to 
 all minds. Attention and instruction only 
 are necessary to furnish them with the exact 
 definitions and new combinations. In num- 
 ber, it is obvious that the terms or figures are 
 themselves definitions, or their equivalents. 
 
 It is because the subject matter of mathe- 
 m^atical reasoning consists, in our ideas of 
 figures and magnitudes or quantities, that 
 the reasoning may be carried on by other 
 signs than words, viz. sensible diagrams. 
 The Arabic numerals, and the notations of 
 algebra, are artificial contrivances or abbre- 
 viated symbols for tracing the relations of 
 quantity as they are wanted, or as those re- 
 lations follow from the nature of the con- 
 trivances themselves. These diagrams, these 
 
DIAGRAMS. 23 
 
 figures and notations, are the signs and in- 
 struments of the mathematician's or alge- 
 braist's thoughts ; and it is because they are 
 alwaj^s of a clear and certain nature, and 
 bear a uniform, fixed, and definite relation 
 one to another, that the geometrical reason- 
 ing, and the arithmetical and algebraic pro- 
 cesses are the same to every mind. 
 
 Upon this circumstance, namely, the power 
 of fixing the attention and carrying on the 
 reasoning by means or help of sensible dia- 
 grams, Locke fastens, as, of the first import- 
 ance, and the great peculiarity in mathema- 
 tical studies. 
 
 " That which has given the advantage to 
 the ideas of quantity, and made them thought 
 more capable of certainty and demonstration, 
 is, first, that they can be set down and 
 represented by visible marks, which have a 
 greater and nearer correspondence with them 
 than any words or sounds whatsoever. Dia- 
 grams drawn on paper are copies of the ideas 
 in the mind, and not liable to the uncer- 
 tainty that words carry in their signification. 
 An angle, circle, or square, drawn in lines, 
 
24 DIAGRAMS. 
 
 lies open to the view, and cannot be mistaken : 
 it remains unchangeable, and may at leisure 
 be considered and exa.mined, and the demon- 
 stration be revised, and all the parts of it 
 may be gone over more than once without 
 any danger of the least change in the ideas. 
 This cannot be thus done in moral ideas ; we 
 have no sensible marks that resemble them, 
 whereby we can set them down." — (Book IV., 
 chap, iii., § 19.) 
 
 It matters not that he is the best mathe- 
 matician, or arithmetician, who needs least 
 the sensible diagram, or the figure on the 
 paper ; nor to say, with Mr. Stewart, that 
 the figure on paper cannot pretend to that 
 precise exactness which is the object of our 
 reasoning ; that the line we draw will have 
 some breadth, and the circle, however steady 
 the instrument and the hand, may deviate in 
 some point from equidistance. The most 
 skilful reasoners can only have a certain 
 idea of visible figure, and of the relation 
 of its several parts present to their minds, 
 which the less skilful require for facility and 
 permanency of reference on the paper. The 
 
DIAGRAMS. 25 
 
 diagram approaches sufficiently to sensible 
 exactness to keep before the mind that qua- 
 hty of the figure which is the sole object of the 
 reasoning ; and it is sufficient that the more 
 nearly the specific figure before us approaches 
 to exactness, the more applicable will the 
 reasoning be to that figure, — or, more cor- 
 rectly, it is only in so far as the figure fairly 
 represents the mind's view of its qualities 
 that the reasoning applies to it at all. There 
 is no such mystery in the most obscure of 
 the definitions as to niake us deny their re- 
 ference to a certain specific quality of ob- 
 jects, that is, to real existencies, in the 
 only practical sense of the words. The 
 constant application of mathematical rea- 
 sonings to the various branches of natural 
 philosophy, and the common use of mathe- 
 matical terms in the mathematical sense, 
 prove the contrary. We speak, for in- 
 stance, of the line between one shade of 
 colour and another, and length without 
 breadth is the only object of the mind's 
 contemplation in so speaking. Points and 
 angles are words oi perpetual occurrence : the 
 c 
 
26 MATHEMATICAL NOTIONS 
 
 former in the sense of the commencement or 
 termination of Unes, without being any de- 
 cided parts or given portions of the hne ; 
 and the latter in the sense of the meeting of 
 two or more hnes together, converging or 
 diverging v^dth more or less of rapidity or 
 extension. 
 
 It is perhaps of little consequence to de- 
 termine whence we get the notions or con- 
 ceptions upon which mathematical reasoning 
 turns, whilst it is certain we have the notions 
 and defined terms appropriate to them, ex- 
 cept in so far as it appears that in numerical 
 calculations, and in the geometry of Euclid, 
 there is a certain verification of the reasoning 
 by an appeal to the evidence of the senses. 
 In fact, it is hard to divine Vv^hence we get 
 notions of figure or quantity if it be not 
 from the sight and the touch, or from expe- 
 rience, — a word of extensive signification, 
 comprehending all the results of observation 
 and reflection. Those who say we do not 
 get these notions or conceptions from expe- 
 rience, would do well to tell us whence we do 
 get them, or produce the mathematician 
 
DERIVED FROM THE SENSES. 27 
 
 upon whom God has not bestowed the five 
 senses with which lie has happily blessed the 
 rest of mankind. 
 
 I will here venture a remark upon Mr. 
 Whewell's language, in his pamphlet on Ma- 
 thematical Studies (second edition, p. 32) : 
 
 " I mentioned it," says Mr. Whewell, " as 
 likely to make the study of mathematics 
 less beneficial as a mental discipline than it 
 might otherwise be if the first principles of 
 our knowledge be represented as borrowed 
 from experience, in such a manner that the 
 whole science becomes empirical only. 
 
 " I will not suppose that any person who 
 has paid any attention to mathematics does 
 not see clearly the difference between neces- 
 sary truths and empirical facts, — between 
 the evidence of the properties of a triangle 
 and that of the general laws of the structure 
 of plants. The peculiar character of mathe- 
 matical truth is, that it is necessarily and 
 inevitably true ; and one of the most im- 
 portant lessons which we learn from our ma- 
 thematical studies is a knowledge that there 
 c2 
 
28 MATHEMATICAL NOTIONS 
 
 are such truths, and a famiharity with their 
 form and character. 
 
 "' This lesson is not only lost, but read 
 backwards, if the student is taught that 
 there is no such difference, and that mathe- 
 matical truths themselves are learnt by ex- 
 perience. I can hardly suppose that any 
 mathematician would hold such an opinion 
 with regard to geometrical truths, although 
 it has been entertained by metaphysicians of 
 no inconsiderable acuteness, as Hume. We 
 might ask such persons how experience can 
 show, not only that a thing is, but that it 
 must be ; by what authority he, the mere 
 recorder of the actual occurrences of the 
 past, pronounces upon all possible cases, 
 though as yet to be tried hereafter only, or 
 probably never. Or, descending to par- 
 ticulars, when it is maintained that it is 
 from experience alone that we know that 
 two straight hues cannot enclose spa.ce, we 
 ask, who ever made the trial, and how ? 
 And w^e request to be informed in what way 
 he ascertained that the lines with which he 
 
DERIVED FROM THE SENSES. 29 
 
 made his experiment were accurately straight. 
 The fallacy is in this case, I conceive, too 
 palpable to require to be dwelt upon," 
 
 A meaning of the word empirical has 
 crept into our language lately, in conse- 
 quence of the freedom with which some phi- 
 losophers treat the king's English, and I fear 
 also from the bad translation of some Ger- 
 man writings, which it was not wont to 
 have, as if it v^ere simply equivalent to ex- 
 perimental ; whereby we are threatened with 
 the loss of a good word for a very important 
 idea, namely, that of quackery, or the ob- 
 servance of rules drawn from a narrow ex- 
 perience, in neglect or defiance of a large 
 and true experience. In eight instances of 
 the use of the word by our best old English 
 authors, which Johnson gives, it is inva- 
 riably associated with this latter meaning. 
 No fact, which is a fact, can deserve the 
 epithet empirical. 
 
 Mr. Whewell would have done well, if 
 we do not get our knowledge of the first 
 principles, or, as he better expresses it, 
 the fundamental conceptions of mathema- 
 
30 MATHEMATICAL NOTIONS 
 
 tical science from experience, to inform us 
 whence or how we do get them. As he has 
 not supphed that information, his reader 
 may be apt to pause ; and if he be a friend 
 of that wise and cautious old gentleman 
 hight Experience, he will not easily allow 
 the laugh to be turned against him. Just 
 so Mr. Dugald Stewart, in his remarks upon 
 demonstrative reasoning, says, "It is by no 
 means sufficient to account for the essential 
 distinction which every person must perceive 
 between the irresistible cogency of a mathe- 
 matical demonstration and that of any other 
 process of reasoning:" "that, in mathema- 
 tics, there is no such thing as an ambiguous 
 Vv^ord." But Mr. Stewart does not help his 
 reader to account for it in any other way. 
 Thus he first plunders him of an all-sufficient 
 principle, and then leaves him in the dark ; 
 nay, he lays it down as his own principle, 
 that it is the peculiarity of mathematics to 
 reason from definition, as if keeping to him- 
 self what he would not allow to another.* 
 
 * See on this the passage of Du Hamel's, quoted by 
 the Edinburgh reviewer, p. 427. 
 
DERIVED FROM THE SENSES. 31 
 
 It appears to me very reasonable to ask, 
 " What but experience can show, not only 
 that a thing is, but that it must he ?" — a 
 very general and perhaps useless proposition. 
 — Experientia clocet ; and let every man be 
 careful how he limits the extent and value of 
 her lessons. As we did not make our own 
 senses, nor the external world, we are 
 supplied by the constitution of our frame 
 with certain conceptions which are natural 
 to, and inseparable from, that frame. We 
 make words or signs for our conceptions, 
 and by use the words become indissolubly 
 associated with those conceptions ; and so 
 long as we make those signs or words stand 
 for those certain conceptions, so long, in 
 fact, as being signs they have signification, 
 we act the part of rational and consistent 
 beings. If a plain man be asked how he 
 knows that two straight lines cannot enclose 
 space, he may, in his turn, ask the questioner 
 whether he ever knew it otherwise ; and so 
 may force him to own that constant experi- 
 ence taught him that truth ; that nature had 
 furnished him with the notions of a line and 
 
32 EXPERIENCE. 
 
 of straightness, and the words belonging to 
 those notions ; that his mathematical studies 
 had built upon that experience ; and that, in 
 regard to ascertaining that any given lines be- 
 fore him were accurately straight, it was clear 
 that the straighter they were, in any conceiv- 
 able meaning of the term straight, the less 
 likely they were to enclose a space. If I 
 were asked how I know that in any right- 
 angled triangle the square which is described 
 upon the side subtending the right angle is 
 equal to the squares described upon the sides 
 which contain the right angle, my first an- 
 swer might be, that I knew it by studying 
 the 47th of the first book of Euchd. But the 
 study of Euclid forms a small part of my ex- 
 perience, which includes all my observations 
 and reflections upon the contents of Euchd, 
 and all the conceptions gathered from the 
 study of the relations traced and traceable be- 
 tween the various figures therein the subject 
 of meditation. In short, experience, like na- 
 ture, is a word of such very comprehensive im- 
 port, as containing within itself so completely 
 the sources of knowledge and instruction, that 
 
BEDDOES ON DEMONSTRATION. 33 
 
 whatever does not fall within the boundaries 
 of that wide domain, can be nothing short 
 of immediate inspiration.* If a friend asks 
 me to shoiu him that the thing must be 
 so, or in other words to furnish him with 
 Euclid's proof, or a mathematical proof of 
 the truth of that 47th proposition, that is 
 another question ; and then I should recal 
 the steps of the demonstration ; and what 
 demonstration is, is the matter into which we 
 are now inquiring. 
 
 There is a well-known work by Dr. Beddoes, 
 on the nature of Demonstrative Evidence, 
 which contains many uspful observations on 
 the connexion between language and thought, 
 in which he endeavours to show that Euclid's 
 reasoning begins from experiment and pro- 
 ceeds by experiments. There is an awkward- 
 
 * Of course it would be absurd to contend that the 
 truths and demonstrations of geometry are lessons of 
 mere experience in a sense strictly analogous to that in 
 which we apply the term to the observations and details 
 of our ordinary daily existence and sensation. We are 
 discussing solely the origin of the fundamental concep- 
 tions on which mathematical reasoning rests- — -the data 
 from which it starts. 
 
 c 5 
 
34 PLAYFAIR ON BEDD0E9. 
 
 ness in the phrase mental experiments, which 
 the Doctor uses, and which might have been 
 avoided by a different mode of stating his 
 argument or view ; and which seems to be 
 this, — that the fundamental notions or con- 
 ceptions from which mathematical reasoning 
 starts, and to which it appeals, are as much 
 the result of experience, and rest as much 
 upon the evidence of the senses, and the 
 natural meaning of our own words, in con- 
 nexion with that evidence, as the funda- 
 mentals of any physical science whatsoever ; 
 and he instances particularly the axioms, as 
 they are called, that " two straight lines 
 cannot enclose a space," and " the vvhole is 
 greater than its part," 
 
 In a review of a treatise of Leslie's, on 
 mathematics, attributed to Professor Play- 
 fair, in the twentieth volume of the Edin- 
 burgh Review, there are some remarks 
 upon this work of Dr. Beddoes, which, 
 coming from Professor Playfair, are entitled 
 to particular consideration. Playfair sug- 
 gests that Beddoes was no great mathe- 
 matician. But with submission, this is no 
 
HARTLEY ON PROPOSITIONS. 35 
 
 answer to Beddoes'" argument, and rather 
 too near an approach to the common tac- 
 tics of controversial writing, in which the 
 reader's attention is diverted from the ques- 
 tion, and the pursuit of truth, by some in- 
 sinuation against the character or abiUties of 
 an adversary. Playfair tells us that " geome- 
 trical reasoning is a process purely intellec- 
 tual, and resting ultimately on truths which 
 the mind intuitively perceives.'" Are we, then, 
 to rest here without going further, — without 
 venturing to ask what are " truths intuitively 
 perceived" ? In what sense this is true, the 
 present observations are meant to illustrate, 
 and, if I am not very much deceived, will 
 sufficiently, or in a great measure, help the 
 reader to understand. Meantime I beg to 
 call his attention to a remarkable and just 
 sentence of Hartley's, in his invaluable and 
 profound chapters on " Words, and the Ideas 
 associated with them, and on Propositions 
 and the Nature of Assent." "Rational 
 assent to any proposition may be defined 
 a readiness to affirm it to be true, proceeding 
 from a close association of the ideas sug- 
 
36 FORCE OF HABIT 
 
 gested by the proposition, with the idea or 
 internal feehng belonging to the word truth, or 
 of the terms of the proposition with the word 
 truth ;" and then follow some observations 
 on geometrical and mathematical reasoning, 
 which are as clear, beantiful, and unanswer- 
 able, as any observations upon abstract truths 
 Avithin the circle of human science and phi- 
 losophy. 
 
 But, fourthly, whencesoever w^e get the 
 notions or conceptions with which we are 
 concerned m mathematical reasoning, I 
 think it must be admitted, that habit, i. e. 
 the constant recurrence of the same simple 
 ideas of number and figure, and the constant 
 association of the same terms with the same 
 ideas, has much to do with that feehng of 
 certainty and satisfaction, that readiness and 
 confidence of assent, wiiich we recognise in 
 connexion with the processes of arithmetic, 
 algebra, and geometry. 
 
 How much there is in habit may be easily 
 and irresistibly showm. Thus we say that 
 2 and 3 make 5, and the three angles of 
 a triangle are equal to two right angles ; 
 
IN MATHEMATICAL REASONING. 37 
 
 and we feel the truth as we pronounce 
 the words. But if we take higher numbers, 
 and more advanced propositions, — if we say 
 that nine thousand six hundred and seventy- 
 three (9673) times seventy-three thousand 
 six hundred and nine (73,609) make 
 712,019,857, or upwards of seven hundred 
 and twelve millions ; or if we take some of 
 the propositions relating to proportion in the 
 fifth book, or go on to the more abstruse calcu- 
 lations in algebra, trigonometry, and fluxions, 
 will our assent be so ready? Who will 
 assert it ? And why ? — because we are not in 
 the habit of attending to high numbers and 
 advanced propositions. Doubt, ignorance, 
 and difficulty attach themselves to our terms. 
 He who has just risen from calculations, or the 
 study of mathematics, will feel a confidence in 
 terms and propositions which others do not. 
 A ready accountant casts up with a glance or 
 two a long column of accounts ; he perceives 
 the relation of each item to the whole 
 amount in a space of time that appears incre- 
 dibly short to one wholly unaccustomed to 
 such work. Those who are in the habit of 
 
38 FEWNESS OF TERMS AND PREMISES 
 
 estimating the number of persons in a 
 crowded room or assembly, can tell by 
 looking at the mass, wdth reference to the 
 space occupied, how many may be present 
 with much more correctness than another 
 who should try for the first time to count the 
 heads. So the bare statement of a pro- 
 position, and a glance at the diagram, mil 
 enable the quick mathem.atician to under- 
 stand the whole demonstration, and to re- 
 peat the various steps of the process faith- 
 fully to another ; while he who is slow at 
 combining the ideas of figure, notwithstand- 
 ing ever so careful reading of the proof, will 
 be still at a loss to perceive its cogency ; and 
 will pass from the words to the figure, and 
 the figure to the words, without being a 
 whit the wiser, or having any distinct idea of 
 what he is about, or where he is, present to 
 the mind. The elaborate paper of Sir W. 
 Hamilton, of Dublin, to the Royal Society*, 
 appears a chaos of warring elements, a mere 
 jumble of letters and figures to the tyro 
 
 * " On a General Method in Dynamics." — Phil. Trans. 
 1834, Pt.ii.,p. 247. 
 
IN MATHEMATICAL REASONING. 39 
 
 in algebraic studies ; " monstrum horrendum 
 inform' ingens cui lumen ademtum;" but to 
 that of the learned reader, and to his own 
 eye, it appears as the harmonious and beau- 
 tiful arrangement of simple* elements, each 
 having its due place and force, combining to 
 one noble, important, and useful result. 
 
 Further, in geometrical and mathematical 
 reasoning the premises are few ; the terms 
 employed are few ; and the mind is only en- 
 gaged in tracing the relations of a few dis- 
 tinct simple ideas, which are fixed by sen- 
 sible impressions. The whole vocabulary of 
 Euclid may be comprised in a couple of 
 pages. Each book turns upon a few defi- 
 nitions. The whole volume is filled with 
 repetitions of the same terms, with appeals 
 to the same brief premises ; attention is 
 more or less frequently recalled to each pro- 
 position as it passes in review, and which 
 ranks, when proved, among the foregone 
 premises. The notations of algebra are 
 comparatively few ; the letters which stand 
 for unknown quantities derive their meaning 
 solely from connexion with, and relation to, 
 the known quantities, at least in their first 
 
40 SAMENESS OF TERMS 
 
 use ; and at last from their relation to each 
 other, in consequence of an extended mean- 
 ing in the symbols, with which meaning, by 
 habitual contemplation, the mind becomes 
 famihar. Among the fig-ures of arithmetic 
 there are but nine units ; after ten you begin 
 with new relations of the first nine ; hun- 
 dreds are combinations of tens, thousands of 
 hundreds, and so on. And with regard to 
 the higher numbers, we can always make 
 clear their value to the senses ; for though 
 we could form not the least notion how many 
 men there might be in a field of battle, or 
 how many grains of corn in a sack, by 
 looking at them in the mass, yet divide 
 them into companies of thousands, of hun- 
 dreds, and tens, and by this arrangement the 
 mind gains a clear and practical sense of the 
 number. It is doubtless by understanding the 
 number and character, and the due arrange- 
 ment of his forces, that a commander-in-chief 
 is enabled to dispose of them to the best ad- 
 vantage, and form the order of battle. 
 
 Our ideas of number and figure are what 
 Locke calls " distinct simple modes ;" and 
 however varied in combination or relations, 
 
AND SIMPLICITY OF IDEAS. 41 
 
 the same signs or terms are invariably con- 
 nected with the sam.e conformations of figure, 
 and the same relations of number. Put 
 down a three-sided figure in lines, or any 
 four or more of the Arabic numerals in a 
 line, as 4565, and every human being using 
 the English language would express the 
 relation in the same terms, — w^ould pro- 
 nounce the one a triangle, and read the other 
 four thousand five hundred and sixty-five. 
 
 " The idea of two is as distinct from that 
 of one," says Locke, B. II., chap, xiii., "■ as 
 blueness from heat, or either of them from 
 any number ; and yet it is made up only of 
 that simple idea of an unit repeated ; and 
 repetitions of this kind joined together, 
 make those distinct simple modes of a dozen, 
 a gross, a million." 
 
 Thus also he speaks concerning figure, 
 § 6 : " The mind having a power to repeat 
 the idea of any length directly stretched out, 
 and join it to another in the same direction, 
 which is to double the length of that straight 
 line, or else join another with what inclina- 
 tion it thinks fit, and so make what sort of 
 
42 FEWNESS OF TERMS 
 
 angle it pleases ; and being able also to 
 shorten any line it imagines by taking from 
 it one-half, or one-fourth, or what part it 
 pleases, without being able to come to an 
 end of any such division, it can make an 
 angle of any bigness ; so also the lines that 
 are its sides of any length it pleases, which 
 joining again to other lines of different 
 lengths, and at different angles, till it has 
 wholly enclosed any space ; it is evident that 
 it can multiply figures, both in their shape 
 and capacity, in infinitum ; all which are but 
 so many different simple modes of space." 
 
 There does not appear any advantage, but 
 the contrary, in the use of the term " mode,'" 
 and alternating it with " idea,'" as Locke 
 does in this and in other parts of his Essay; 
 but whether ideas or modes, it is e^ddent 
 they are simple, because they do not admit 
 of being resolved into other ideas or notions 
 still simpler, but result at once from uni- 
 formity in the structure and impressions of 
 the senses, which uniformity laj's the foun- 
 dation for language and reasoning. 
 
 The simplicity and uniformity of the sen- 
 
IN MATHEMATICAL REASONING. 43 
 
 sible impressions of space, or figure and 
 number, and the comparative fewness of the 
 terms or symbols in use in mathematical 
 reasoning, constantly associated with the 
 same impressions, — terms or symbols w4iich 
 are in fact human contrivances for conveying 
 those impressions from one mind to another, 
 — these things are to be borne in mind, and 
 duly weighed, in estimating the nature of 
 demonstrative evidence. Nor let any man 
 despise mathematical studies, or think them 
 a mere ringing of changes upon the same set 
 of bells, because the terms employed are 
 few, and the original simple ideas few ; 
 otherwise, let him despise the EngHsh lan- 
 guage, or language in general, because there 
 are only twenty-six letters in the alphabet. 
 For what endless varieties of thought, — what 
 worlds of wisdom, — what vast structures of 
 science, are these twenty-six letters, all-suffi- 
 cient ! And what would human life be with- 
 out them ? 
 
 But we have not yet analysed the nature 
 of mathematical reasoning. We have said 
 that mathematical reasoning sets out from 
 
44 FINAL CHARACTERISTIC 
 
 definitions ; that these definitions settle the 
 meaning of terms ; that these terms are the 
 signs of onr ideas of figure and quantity, of 
 numbers and magnitudes ; that these ideas 
 are among the simplest, clearest, with which 
 our minds and senses are conversant ; that 
 the terms in use, and the simple ideas 
 to which they are uniformly appropriated, 
 are comparatively few ; that the constancy 
 of connexion between the terms and ideas, 
 that is, habit, has much to do with that 
 feeling of assent and conviction to which the 
 reasoning gives rise, by which the processes 
 are accompanied, as any one must perceive 
 who begins to instruct children in arithmetic 
 or in geometry. 
 
 But further, fifthly, and lastly, the demon- 
 strative quality of mathematical reasoning 
 consists essentially in this, — the perception 
 of the agreement or disagreement of certain 
 ideas and certain terms with other inter- 
 mediate ideas and terms, which are used as 
 a measure or test of truth, such ideas and 
 terms having been previously selected by the 
 mind for a measure or test ; in other words, 
 
OF DEMONSTRATION. 45 
 
 a means or standard of comparison. To 
 demonstrate is to show that a certain propo- 
 sition not granted to be true is true by virtue 
 of some premise previously admitted or as- 
 sumed as a criterion of truth, or by virtue of 
 some other truth previously demonstrated."* In 
 other words, to demonstrate is to discover 
 and trace farther and new relations amongst 
 our ideas by comparing them one with an- 
 other ; which new relations will be of deter- 
 minate and constant character, in proportion 
 as the intermediate ideas which are used as a 
 means of comparison, as a measure, are 
 themselves of distinct and constant character 
 and value. Or again, it is to show that 
 ideas not clearly perceived to harmonize or 
 agree, do harmonize, by comparison, with 
 other ideas whose agreement is clearly per- 
 ceived. Thus in the first proposition of 
 Euclid, the sides of a given triangle are 
 proved to be equal when they are ail shown 
 to belong to the class of lines which radiate 
 
 * The reader may consult the papers on mathematics, 
 by Mr. De Morgan, who gives this just account of de- 
 monstration. 
 
46 FINAL CHARACTERISTIC 
 
 from the centre to the circumference of the 
 same circle, or of equal circles ; of which 
 class of lines equality is previously premised 
 in the fifteenth definition. 
 
 This essential quality of demonstrative 
 reasoning is thus distinctly laid down by the 
 great master Locke, book iv., c. ii. : — 
 
 " The next degree of knowledge is where 
 the mind perceives the agreement or dis- 
 agreement of any ideas, but not immediately. 
 Though wherever the mind perceives the 
 agreement or disagreement of any of its ideas 
 there be certain knowledge, yet it does not 
 always happen that the mind sees that 
 agreement or disagreement which there is 
 between them, even vvhere it is discoverable ; 
 and in that case remains in ignorance, at 
 most gets no further than a probable con- 
 jecture. The reason why the mind cannot 
 always perceive presently the agreement or 
 disagreement between two ideas, is because 
 those ideas, concerning whose agreement or 
 disagreement the inquiry is made, cannot by 
 the mind be so put together as to shew it. 
 In this case, then, when the mind cannot so 
 
OF DEMONSTRATION. 47 
 
 bring its ideas together, as by their imme- 
 diate comparison, and, as it were, juxta- 
 position and apphcation one to another, to 
 perceive their agreement or disagreement, it 
 is fain, by the intervention of other ideas, 
 (one or more as it happens,) to discover the 
 a2;reement or disa2;reement Vvhich it searches ; 
 and this is what we call reasoning.''' 
 
 Again, § 3 : — 
 
 " Those intervening ideas which serve to 
 show the agreement of any two others are 
 called proofs ; and where the agreement and 
 disagreement is by this means plainly and 
 clearly perceived, it is called demonstration." 
 
 Again, § 7 : — 
 
 " In every step reason makes in demon- 
 strative knowledge, there is an intuitive 
 knowledge of that agreement or disagreement 
 it seeks with the next intermediate idea, 
 which it uses as a proof ; for if it were not so, 
 that yet would need a proof; since without 
 the perception of such agreement or dis- 
 agreement there is no knowledge. If it be 
 perceived by itself, it is intuitive knowledge ; 
 if it cannot be perceived by itself, there is 
 
48 FINAL CHARACTERISTIC 
 
 need of some intervening idea, as a common 
 measure to show their agreement or disagree- 
 ment. * * * =^ So that to make a thing a 
 demonstration, it is necessary to perceive 
 the immediate agreement of the intervening 
 ideas, whereby the agreement or disagree- 
 ment of the two ideas under examination 
 (whereof the one is alw^ays the first and the 
 other the last in the account) is found." 
 
 It is not without reason that Locke dwells 
 upon this ; and he repeats himself in ch, xv. 
 of the fourth book on Probabihty, which 
 the reader may consult. 
 
 In this analysis of demonstrative or mathe- 
 matical reasoning, it is finally to be obseiwed 
 that the definitions are used as the primary 
 common measures or tests ; they are the 
 original ideas or settled notions by means of 
 which the relations of other ideas one with 
 another are traced, and the agreement or 
 disagreement ascertained and settled, and 
 by which the new relations, so ascertained, 
 become themselves of determinate and con- 
 stant character. Each book begins with its 
 necessary definitions ; and each proposition, 
 
OF DEMONSTRATIVE REASONING. 49 
 
 when settled, becomes itself a premise or test, 
 by help of which further relations are traced, 
 and new agreements or disagreements as- 
 certained and fixed. The mind is con- 
 tinually reverting to its original simple no- 
 tions, builds carefully upon them, and not only 
 has a power to retrace, but is very frequently 
 employed in carefully retracing every step 
 of its progress. Thus we return to the point 
 from which we set out, that definition is the 
 basis of mathematical reasoning, and gives 
 it its peculiarly fixed, clear, and cer- 
 tain character. 
 
 The reader who may doubt whether 
 this be a correct or perfect analysis of 
 mathematical or demonstrative reasoning, 
 is requested, by a careful examination of 
 mathematical works, to supply the deficiency. 
 Let him apply it to the most simple or the 
 most abstruse propositions and demonstra- 
 tions, and say what essential quality of such 
 reasoning has been omitted. 
 
50 STEWART CONTRASTED 
 
 SECTION 11. 
 
 Having now analysed with all the com- 
 pleteness in our power, the nature of de- 
 monstrative reasoning, we are prepared for 
 the inquiry, wiiether it differs from other 
 reasoning, or reasoning in general, in any 
 respects or particulars whatsoever. And, 
 if it do not so differ, we are then prepared 
 for the important inquiry, how the cogency 
 and certainty of mathematical science can be 
 applied to and obtained in moral, political, 
 metaphysical, and religious subjects. 
 
 Now the tendency and almost the object 
 of Mr. Dugald Stewart's chapters on mathe- 
 matical demonstration, and on the Aristo- 
 telian logic, is to draw a broad hne of dis- 
 tinction between mathematical reasoning, 
 mathematical evidence, and other kinds of 
 reasoning, other kinds of evidence. " JNIa- 
 thematical definitions," he says, (vol. ii., 
 p. 156, 4to,) " are of a nature essentially dif- 
 ferent from the definitions employed in any 
 
WITH WHATELY. 51 
 
 of the other sciences." Agam, p. 157, he 
 speaks of " the essential distinction which 
 everyperson must perceive between the irre- 
 sistible cogency of a mathematical demon- 
 stration and that of any other process of 
 reasoning." He repeats this idea in various 
 places. I need only refer to p. 203, 
 where he says, " If the account which has 
 been given of the nature of demonstrative 
 evidence be admitted, the province over 
 which it extends must be limited almost en- 
 tirely to the objects of pure mathematics." 
 
 But what says Dr. Whately in his Ele- 
 ments of Logic ? 
 
 " One of the chief impediments to the attain- 
 ment of a just view of the nature and objects of 
 logic^ is the not fully understanding, or not suffi- 
 ciently keeping in mind, the sameness of the 
 reasoning 2}rocess in all cases. If, as the ordinary 
 mode of speaking would seem to indicate, mathe- 
 matical reasoning, and theological, and meta- 
 physical, and political, &c., were essentially differ- 
 ent from each other, i. e. different kinds of reason- 
 ing, it would follow that, supposing there could 
 be at all any such science as we have described 
 logic, there must be so many different species, 
 or at least different branches of logic. And such 
 D 2 
 
52 EDINBURGH REVIEWER 
 
 is perhaps the most prevaihng notion/^ — 3rd. ed. 
 p. 23. 
 
 Again, p. 25, lie says, — 
 
 " Supposing it to have been perceived that the 
 operation of reasoning is in all cases the same, the 
 analysis of that operation could not fail to strike 
 the mind as an interesting matter of inquiry." 
 
 And thus, p. 50 : — 
 
 " Whatever the subject matter of an argument 
 may be, the reasoning itself, considered by itself, 
 is in every case the same process; and if the 
 writers against logic had kept this in mind, they 
 would have been cautious of expressing their con- 
 tempt of what they call syllogistic reasoning, 
 which is in truth all reasoning." 
 
 Let us contrast with this, — for there is 
 nothing more instructive than bringing into 
 juxtaposition the different aspects in which 
 these recondite matters are presented to our 
 attention, — let us contrast with this the 
 barely intelligible language of the Edinburgh 
 reviewer, p. 413. 
 
 " Now as all matter is either necessary or con- 
 tingent, (a distinction Avhich may be here roughly 
 assumed to coincide with mathematical or non- 
 
CONTRASTED WITH WHATELY. 53 
 
 mathematical;,) we have thus^ besides a theoretic or 
 general logic, two practical or special logics in 
 their highest universality or contrast. 
 
 "theoretical LOGIC. 
 
 Practical logic, as spe- Practical logic, as spe- 
 cially applied to neces- cially applied to con- 
 sary matters = mathe- tingent matter ::= philo- 
 matical reasoning. sophy and general rea- 
 
 soning." 
 
 He says, p. 422,— 
 
 " How opposite are the habitudes of mind which 
 the study of the mathematical and. the study of 
 the philosophical sciences require and cultivate, 
 has attracted the attention of observers from the 
 most ancient times. Tlie principle of this con- 
 trast lies in their different objects, in their differ- 
 ent ends, and in the different modes of considering 
 their objects," 
 
 He speaks also of matliematics as " de- 
 termining dissimilar developments of thought 
 from other sciences," as " not developing the 
 higher faculties," as " dependant on the 
 lower imagination." 
 
 Again, p. 422 : — 
 
 "Mathematics, departing from certain original 
 hypotheses, and these hypotheses exclusively de- 
 termining every movement of their procedvire, and 
 
54 GENERAL INFLUENCE 
 
 the images or vicarious syraljols^ about which 
 they are conversant, being clear and simple, the 
 deductions of these sciences are apodictic or 
 demonstrative; that is, the possibility of the con- 
 trary is at every step seen to be excluded in the 
 very comprehension of the terms. On the other 
 hand, in philosophy, (with the exception of the 
 science of logic,) and in our reasonings in general, 
 such demonstrative certainty is rarely to be at- 
 tained ; probable certainty, i. e. where we are never 
 conscious of the impossibility of the contrary, is all 
 that can be compassed ; and this also not being 
 internally evolved from any fundamental data, 
 must be sought for, collected, and applied from 
 without." 
 
 " From this general contrast it will be seen how 
 an excessive study of the mathematical sciences, 
 not only does not prepare, but absolutely incapaci- 
 tates the mind for those intellectual energies which 
 philosophy and life require. ^Ye are thus dis- 
 qualified for observation either internal or external, 
 for abstraction and generalization, and for common 
 reasoning." 
 
 Now common reasoning we conceive to be 
 very bad reasoning ; such reasoning as fails 
 to satisfy the man who is seeking for clear 
 and exact views, who fears to be misled by 
 words, and who remembers that fine phrase- 
 ology teaches nothing. It may be observed 
 
OF MATHEMATICAL STUDIES. 55 
 
 here, that whatever force or justness there is 
 in the reviewer's general course of observation, 
 it all lies in the word " excessive" — " an ex- 
 cessive study of the mathematical sciences." 
 And it is perfectly obvious that he who is con- 
 versant only with mathematical notions and 
 mathematical processes, may be ignorant of 
 many other objects of human attention, 
 which come nearer home to the business and 
 bosoms, the pleasures and pains, of mankind 
 at large. He who is always dwelling in 
 circles and squares, ellipses and parabolas, 
 differentials and integrals, may have a pro- 
 portionally confined range of thought. He 
 will not understand the feehngs and thoughts 
 of other men ; and he may fancy, from the 
 habitual association of his ideas, or from his 
 determining everything in the same way, that 
 he can ascertain the precise quantity of en- 
 joyment which a company of aldermen de- 
 rive from eating and drinking, by means of 
 the differential or integral calculus, and de- 
 termine the relative merits of Homer and 
 Virgil by the rule and compasses. But what 
 then ? Shall matliematical studies not be 
 
56 GENERAL INFLUENCE 
 
 valued as an essential part of the training of the 
 youthful mind ? Is Mr. Whewell's sentiment 
 invalidated, that they are the best practical ex- 
 emplification and exercise of logic ? If there 
 be one mode of stud3dng mathematics better 
 than another, shall not a mathematical pro- 
 fessor discuss this question, and endeavour 
 to settle which is best ? How many sciences 
 are there which require for their pursuit, 
 comprehension, and enjoyment, a thorough 
 knowledge of the higher branches of mathe- 
 matics, such as astronomy, optics, dynamics, 
 and all those which go under the name of the 
 mixed sciences. Who would undervalue 
 the hisrhest mathematical attainment when 
 
 o 
 
 applied to these branches of science ; and 
 not rather regret, when he sees the mathe- 
 matician soaring in the clouds and lost in 
 the dim distance of algebraic formulae, his 
 inability to follow ? ' ' Non omnes possumus 
 omnia." But we can all enjoy and apply 
 those practical and simple conclusions, for 
 the establishment of which the most pro- 
 found mathematical investigations are oft- 
 times necessary. 
 
OF MATHEMATICAL STUDIES. 57 
 
 If the question be, What degree of time 
 and attention should be given up to mathe- 
 matical studies in a thoroughly comprehen- 
 sive course of academic education ? or, How 
 far exclusive encouragement should be given 
 to high mathematical attainment in an uni- 
 versity? (which the reviewer has in part 
 raised and discussed,) this may be settled 
 without depreciating the importance and 
 value of mathematics for the discipline of 
 the youthful mind. You have then to take 
 into account the great and general purposes 
 of education, the whole constitution of the 
 human mind, the condition, and wants of 
 society at large, the fitness of an individual 
 for the particular station which he is de- 
 signed to occupy, and the kind of knowledge 
 which his meditated profession may re- 
 quire. 
 
 It is curious to contrast the reviewer's 
 statement of the injurious influence of 
 mathematical science in disqualifying for ob- 
 servation, either internal or external, for 
 abstraction and generalization, with the in- 
 tellectual character of Sir Isaac Newton 
 D 5 
 
58 herschel's character 
 
 drawn by Sir John Herschel in his Treatise 
 on the Study of Natural Philosophy, p. 271. 
 
 " His wonderful combination of mathematical 
 skill with physical research enabled him to invent, 
 at pleasure, new and unheard-of methods of in- 
 vestigating the effects of those causes which his 
 clear and penetrating mind detected in operation. 
 Whatever dej)artment of science he touched, he 
 may be said to have formed afresh. Ascending by 
 a series of close-compacted inductive arguments 
 to the highest axioms of dynamical science, he 
 succeeded in applying them to the complete ex- 
 planation of all the great astronomical phenomena, 
 and many of the minuter and more enigmatical 
 ones. In doing this, he had every thing to create ; 
 the mathematics of his age proved totally inade- 
 quate to grapple with the numerous difficulties 
 which were to be overcome. * * * Of the optical 
 discoveries of Newton we have already spoken ; 
 and if the magnitude of the objects of his astrono- 
 mical discoveries excite our admiration of the 
 mental powers which could so familiarly grasp 
 them, the minuteness of the researches into which 
 he there set the first example of entering, is no less 
 calculated to produce a corresponding impression. 
 Whichever way Ave turn our view, we find our- 
 selves compelled to l)ow before his genius, and to 
 assign to the name of Newton a place in our 
 veneration which belongs to no other in the an- 
 nals of science. His era marks the accomplished 
 
OF SIR ISAAC NEWTON. 59 
 
 maturity of the human reason as ai^plied to such 
 objects. Every thing which went before might 
 be more properly compared to the first imperfect 
 attempts of childhood, or the essays of inexpert 
 though promising adolescence. Whatever has 
 been since performed, hoAvever great in itself, and 
 worthy of so splendid and auspicious a beginning, 
 has never, in point of intellectual effort, surpassed 
 that astonishing one which produced the Prin- 
 cipia." 
 
 I refer to this treatise with a strong feeling 
 of interest, because it is evident from the 
 observations on nomenclature, and on science 
 generally, that Herschel's clear English 
 mind duly estimates the importance of 
 settled terms with settled meanings ; and 
 while he dwells on the necessity of having 
 exact and uniform standards of measure- 
 ment and value, his reader is set upon the 
 inquiry into the nature and purposes of 
 measures or tests. He who can perceive the 
 importance of a proper use of words in 
 physical science, must feel that importance 
 also in metaphysical. Without it, in fact, 
 we can have nothing worthy of the name of 
 science. Sir John Herschel would probably 
 
60 
 
 LOGIC DEFINED. 
 
 smile at the idea of mathematical science 
 disqualifying for generalization and abstrac- 
 tion or any useful exercise of mind. 
 
 I have indulged in these references to 
 Dugald Stewart, Dr. Whately, the Edin- 
 burgh reviewer, and Sir John Herschel, 
 with a view to place before the reader in an 
 easy manner the different lights in which the 
 same objects, or objects closely alUed in 
 nature and character, are presented to our 
 attention, and the necessity of close and 
 cautious investigation. 
 
 Now bearing in mind the foregone analysis 
 of geometrical or demonstrative reasoning, 
 in order to perceive its connexion with logic, 
 it is necessary to understand what logic is. 
 Is Dr. Whately right or wrong when he says 
 the reasoning process is the same in all cases ? 
 If he is right, of course it follows that 
 mathematical or geometrical reasoning is 
 but one illustration or practical application 
 of logic. 
 
 I am unable to attach any other consistent 
 meaning to the term logic, than that it is an- 
 other word — the Greek word — for reasoning. 
 
LOGIC DEFINED. 61 
 
 As a science it investigates the principles of 
 reasoning, or analyses and determines the 
 process of the mind in reasoning ; as an art 
 it is the practical application or exemplifica- 
 tion of the rules so deduced. On this 
 point nothing can be clearer and more satis- 
 factory than Dr. Whately's observations in 
 his preface and throughout his treatise. 
 
 Yet notwithstanding this clearness, and 
 notwithstanding Dr. Whately's correction of 
 the error of Watts in considering logic as "the 
 right use of reason," "a method of invigo- 
 rating and properly directing all the powers 
 of the mind," a writer on logic in the edition 
 of the Encyclopaedia Britannica now in the 
 course of publication, says, " Logic may be 
 defined as the science of the laws of thought 
 considered as thought. This is the central 
 notion towards which the various views of 
 the science, from Aristotle downwards, gravi- 
 tate ; it is the one definition in which others, 
 apparently the most opposite, find their com- 
 plement and reconciliation." Then, by way 
 of elucidating this definition, the writer 
 (whom from his use of the term laws, and the 
 
62 LOGIC AS TREATED 
 
 epithets, contingent, necessary, universal, di- 
 rigible, and so on, I could suspect to be the 
 Edinburgh reviewer already alluded to) pro- 
 ceeds to tell us, first, that logic is conversant 
 about thought ; in the second place, about 
 thought considered as thought ; and in the 
 third place, it is the science of the laws of 
 thought, because it is conversant about the 
 universal and necessary in thought. 
 
 These are the remarks of a writer who 
 comments on the erroneous definition of 
 logic, in an article which the editor of the 
 Encyclopaedia has reprinted ; an article 
 which tells us that " Logic is the art of 
 properly conducting reason in the knowledge 
 of things, whether for instructing ourselves 
 or others ; or it may be defined the science 
 of human thought, inasmuch as it traces the 
 progress of knowledge ; and that its business 
 is to evolve the laws of human thought, and 
 the proper manner of conducting the reason, 
 in order to the attainment of truth and 
 knowledge." And while the writer comments 
 on this article, he further tells us that from 
 Aristotle downwards the purity of the science 
 
BY THE ENCYCLOPAEDIA BRITANNICA. 63 
 
 has been contaminated by foreign infusions. 
 He speaks of Dr. Wliately's Elements 
 as vague and vacillating in its views, its 
 doctrines neither being developed from the 
 primary laws of thought, nor combined to- 
 gether as the essential parts of one necessary 
 whole. In short, being desirous to make 
 something more of a subject than has ever 
 yet been made of it, and to see further 
 into things than any one else has seen, he 
 plunges into darkness and a wood of words, 
 
 " hunc tegit omnis 
 
 Lucus, et obscuris claudunt convallibus umbrae," 
 
 or, like many of his brethren, he is so 
 blinded by the mists of his own land that he 
 cannot enjoy the cheerful^ sun and daylight 
 loved by the children of the south ; and 
 when he is pleased to consider thought as 
 thought, he forgets that no one in his senses 
 was likely to mistake it for " cakes and ale." 
 Indeed, but for the eminence to which the 
 Encyclopaedia Britannica aspires, and is in 
 part deservedly raised as an authority in the 
 sciences, he might be benevolently left to the 
 condition and neglect in which the New 
 
64 
 
 LOGIC AS TREATED 
 
 Poor-Law leaves those who will not help 
 themselves when they can. 
 
 Seriously, however, when we talk of the 
 science of the laws of thought, do we know 
 what we are talking about ? With confidence, 
 I answer No. The vdiole subject of meta- 
 physics, the whole state of our knowledge 
 and language on the nature, qualities, powers, 
 and affections of the human understanding, 
 as may be inferred from the article meta- 
 physics in this very Encyclopsedia, is such, 
 that to talk of the laws of thought, to speak 
 of primary laws, which implies secondary, 
 and universal and necessary, which implies 
 particular and contingent, is to talk of no- 
 body knows what. What has logic to do 
 with the laws by which thoughts come and 
 go in the mind of a child or of a maniac ? for 
 I suppose a maniac has thoughts, and, if so, 
 is subject to laws of thought. True enough, 
 all logic supposes a thinking mind ; but so 
 does every other science ; so do carpentry and 
 masonry ; and wherever we have thinking 
 minds, there we have minds subject no 
 doubt to what we are pleased to call laws. 
 
BY THE ENCYCLOPAEDIA BRITANNICA. 65 
 
 But to set the mind hunting after the gene- 
 ral laws of thought, under pretence of study- 
 ing logic, is to entrap the student into un- 
 looked-for difficulties, to leave Aristotle 
 utterly in the lurch, to give us our labour 
 for our pains, and to bring us, after a 
 fatiguing hunt, like Spenser's good knight, 
 only to the cave of despair. If logic be the 
 science of the laws of thought, what is the 
 province of mental philosophy ? I do not 
 question that the one touches closely the 
 province of the other, but science used for- 
 merly to consist in nicely distinguishing 
 rather than confounding the things that dif- 
 fer, howsoever minute that difference. At the 
 risk of appearing merely to reprint what my 
 reader may find elsewhere, but what cannot 
 be too strongly impressed upon the mind, 
 I must use the words of Dr. Whately, and 
 say that "the attempt to comprehend so 
 wide a field is no extension of science, but a 
 mere verbal generalization, which leads only 
 to vague and barren declamation. In every 
 pursuit, the more precise and definite oux 
 
66 NATURE OF REASONING 
 
 object, the more likely we are to attain some 
 valuable result." 
 
 Without further discussion, I must assume 
 that logic is but another word for reasoning; 
 and the object of it as a science is to ascer- 
 tain the process of the mind, to which we 
 specially give that name. Now we have 
 analysed the nature of mathematical reason- 
 ing, or, in other words, we have examined 
 the process of the mind in that reasoning. 
 Can we, then, abstract what is peculiar to 
 the mathematics, and talk of reasoning in 
 general without regard to any particulars ? 
 May we not ask what is meant by reasoning 
 as a term standing alone ? Is there one 
 determinate process of mind to which the 
 term reasoning is peculiarly and alone appro- 
 priate ? 
 
 Nature and the senses give me the idea of 
 a man and of a horse. I suppose the body 
 and legs of a horse joined to the breast and 
 head of a man, and call that supposition or 
 conception by the name of a centaur. Doing 
 this, would you say I reason? No, I only 
 
EXAMINED. 67 
 
 imagine, and give a name to the object of 
 my imagination, which are indeed important 
 elements of the reasoning powers. But when 
 I say all animals have feeling, no vegetable 
 has feeling, therefore no vegetable is an 
 animal ; you would say I reason, although 
 from the very obviousness of the words, and 
 from their arrangement, and the smallness 
 of the effort of which we are conscious in 
 following that arrangement, the portion of 
 reason concerned, if we could divide reason 
 into measureable portions, is almost too in- 
 significant to be worthy of the name. But 
 if this be reasoning, what have we ? 
 
 First of all, words, or audible sounds asso- 
 ciated with many sensible impressions or 
 objects — animals. 
 
 Secondly, These objects classified, and 
 viewed in a common relation or under the 
 affirmation — having feelings. 
 
 And thirdly. Other objects, viewed under 
 a different relation, having no feeling, there- 
 fore excluded from this class, no vegetables 
 animals, or vegetables no animals. 
 
 In this who can detect any thing but the 
 
68 
 
 WHATELY S LOGIC 
 
 results or lessons of human experience or 
 registered observation, classified, and clothed 
 in appropriate language, — that which is 
 affirmed of one being denied of the other 
 class, — language being to us the means 
 and very element of thought, at least of 
 thought conveyed from one mind to an- 
 other ? and hence the beauty of the Greek 
 word T^oyog, which is at once verbum and 
 ratio — the audible sound and mental ap- 
 prehension. 
 
 I prefer, however, taking a work of au- 
 thority like Dr. Whately's as a guide for the 
 course of thought which it appears most 
 useful and important to pursue. In ana- 
 lysing the operation of the mind in reason- 
 ing, Dr. Whately says, " It will be found 
 that every conclusion is deduced in reality 
 from two other propositions, thence called 
 premises." He contends there must be two 
 propositions, and says, (section third,) of a 
 valid argument, "It is impossible for any 
 one who admits both premises to avoid ad- 
 mitting the conclusion." Then, after giving 
 an example of the true syllogism, he says 
 
CONSIDERED. , 69 
 
 there is this maxim resulting from it, " that 
 whatever is predicated universally of any 
 class of things, may be predicated in like 
 manner of any thing comprehended in that 
 class," — the celebrated principle called the 
 dictum de omni et nullo of Aristotle. After 
 some observations on the substitution of 
 letters and symbols for the terms of the 
 syllogism, on apparent arguments, on the 
 importance of finding a proper middle 
 term, on generalization and abstraction, 
 he winds up the analysis with the remark, 
 " that it consists in referring the term we 
 are speaking of to some class, viz. a mid- 
 die term, which term again is referred 
 to or excluded from (as the case may be) 
 another class, viz. the term which we wish 
 to affirm or deny of the subject of the 
 conclusion." 
 
 With a very strong sense of the value 
 of Dr. Whately's Elements, — of the cor- 
 rectness, and usefulness of the principles and 
 views therein detailed, — it may be permit- 
 ted me to observe, that even that work is, 
 in some degree, deficient in the rigid 
 
70 whately's logic 
 
 propriety of language which the subject 
 demanded, and which might easily have been 
 given to it. The analytical outline of logic 
 can scarcely be regarded as a successful and 
 complete elucidation of the science. Dr. 
 Whately himself calls it an imperfect and 
 irregular sketch. 
 
 For as the analytical outline, and the syn- 
 thetical compendium, appear in juxtaposi- 
 tion, the reader naturally expects that they 
 should answer exactly the one to the other, 
 the analysis being the resolution of the whole 
 into the parts, or, if the reader like it 
 better, the tracing of given effects to the 
 causes from which they spring ; — the syn- 
 thesis, — the enumeration of the several 
 parts which combine to make the whole, or 
 the advance from the cause to the varied 
 effects or consequences. But this corre- 
 spondence is by DO means so clear as it 
 might have been, — as it ought to be. For in- 
 stance : having in the analysis stated that 
 the operation of reasoning is in all the cases 
 the same, (p. 25, fifth edition,) and that in 
 every instance in which we reason a certain 
 
CONSIDERED. 71 
 
 process takes place in the mind, which is 
 one and the same in all cases, Dr. Whately 
 opens the compendium by saying, " There 
 are three operations or states of the mind 
 which are immediately concerned in argu- 
 ment." Again, after having in the analysis 
 described the process in reasoning as the 
 deduction of a conclusion from two other 
 propositions, thence called premises, in the 
 compendium he says, " Reasoning is the 
 act of proceeding from one judgement to an- 
 other, founded upon that one, or the result 
 of it." These discrepancies may be more 
 apparent than real ; they may be of slight 
 consequence : but the careful reader is, to a 
 certain degree, distracted. And as the great 
 object of the study of logic is to clear and 
 to brace the mind, — as it is but the athletics 
 and gymnastics of the reasoning faculties, — 
 as clearness and strength are entirely depend- 
 ant on perfect precision in the use of terms, 
 — so the teacher of logic should avoid a ver- 
 bal discrepancy as fatal to his science, as the 
 man under training should avoid diluents 
 and laxatives of every kind. 
 
72 ANALYSIS AND SYNTHESIS. 
 
 I am aware that some may think I have 
 drawn too strictly the parallehsm between 
 the analytical and synthetical modes ; but, 
 after a careful perusal of Mr. Dugald 
 Stewart's remarks upon the use of these 
 terms in ancient and modern philosophy, 
 showing that authority may be pleaded for 
 using them in an exactly opposite and mu- 
 tually convertible sense, and that he himself 
 is at a loss to give them precise meaning, — - 
 sometimes confuting in the notes what he lays 
 down in the text ; — after reading also what 
 Maclaurin says about these modes, in his ac- 
 count of Sir I. Newton's discoveries, — I cannot 
 help considering them in a very simple and 
 obvious light, as different or opposite modes 
 of going over the same or a precisely similar 
 path ; according to the simile of Condillac, 
 one being up and the other down the hill : 
 only instead of saying,' with Condillac, that 
 as the two methods are contrary to one an- 
 other, if the one be good the other must be 
 bad, I rather say both may be good, accord- 
 ing to the position and view which we as- 
 sume, and the walk which, for the time, we 
 
ANALYSIS OF ARGUMENTS. / <J 
 
 please to take. Dr. Whately, at any rate, 
 cannot object to a rigid parallelism, since, in 
 his Introduction, he uses the words as they 
 are used in chemistry ; and by S5Titliesis he 
 appears to understand the enumeration of ele- 
 mentary substances, thence proceeding on- 
 wards to simple combinations or more com- 
 plex substances ; by analysis only resolving 
 these last, namely, the complex substances, 
 step by step, into their simple elements. 
 He should therefore obviously have begun 
 his synthetical compendium with a clear and 
 explicit enumeration of those very elements 
 into which, in his analysis, he had resolved 
 the science or art under review. 
 
 But the first and second sections of the 
 third chapter and second book, on Argu- 
 ments, contain a sufficient analysis of argu- 
 ments, which are "Reasonings expressed in 
 words ;" and with such reasonings chiefly, if 
 not entirely, is logic concerned. " An 
 argument," says Dr. Whately, "is an ex- 
 pression in which, from something laid 
 down and granted as true, {i. e. the premises,) 
 
 E 
 
74 SYLLOGISM. 
 
 something else {i. e. the conclusion) beyond 
 this must be admitted to be true as following 
 necessarily from the other." Again, a syl- 
 logism, — (and let us remember that every just 
 argument may be reduced into the form of a 
 perfect or pure categorical syllogism, which 
 is sufficiently obvious to the student of logic,) 
 — " a sjdlogism is an argument so expressed 
 that the conclusiveness of it is manifest 
 from the mere force of the expression." 
 
 I conceive this to be a just and sufficient 
 account of syllogistic reasoning, taken in 
 connexion with what was before said respect- 
 ing terms. By logic, the force of all reasoning, 
 or the correctness of syllogism, is shown to 
 depend entirely upon the degree of exactness 
 and comprehensiveness of meaning in the 
 terms employed, or, what is virtually the 
 same thing, of distinctness in the things 
 signified ; and the chief purpose or use of 
 the study is to call the attention of the mind 
 to those forms of expression, and to fix those 
 forms upon the memory, which are always 
 essential to strict, legitimate, and convincing 
 
SYLLOGISM. /O 
 
 inference. Now the value or correctness of 
 a syllogism depends , mainly upon one prin- 
 cipal term, called the middle term. 
 
 " Every syllogism," says Dr. Whately, 
 "has three, and only three, terms ; viz. the 
 middle term, and the two terms (or ex- 
 tremes, as they are commonly called,) of 
 the conclusion or question. The middle 
 term (called, by the older logicians, argu- 
 mentum.) is that with which each of them is 
 separately compared, in order to judge of 
 their agreement or disagreement." Again 
 he says, " Every argument consists of two 
 parts : that which is proved, and that by 
 means of which it is proved." And again, 
 " The axioms or canons by which the vali- 
 dity of pure syllogisms is to be explained 
 are these ; viz. first, if two terms agree with 
 one and the same third, they agree with 
 each other ; secondly, if one term agrees 
 and another disagrees with one and the same 
 third, these two disagree with each other." 
 
 It is obvious that here we have but dif- 
 ferent views or statements of substantial!}^ 
 the same thing, namely, the use of one 
 E 2 
 
76 SYLLOGISM. 
 
 term, called a middle or third, as a means of 
 comparison between two other terms; in 
 other words, a reference of two terms to one 
 and the same object of comparison, used as 
 a measure or test of their agreement or dis- 
 agreement,^ — in short, the celebrated dictum 
 of Aristotle, that whatever may be predi- 
 cated universally of a class of objects may be 
 predicated of every individual comprehended 
 in it ; which is analogous to the axiom, or 
 common notion of equality, that things 
 which are equal to the same are equal to one 
 another, or that the whole is made up of all 
 the parts, 
 
 A syllogism, to make a homety simile, is 
 a kind of two -pronged fork ; the middle 
 term is the handle which unites the prongs, 
 and enables us to seize conveniently the ob- 
 ject of our thought, and feed our mental 
 appetite v/ith food convenient for it. 
 
 E. g. All horned animals are ruminant ; 
 the ox is a horned animal ; therefore the ox 
 is ruminant. Supposing the habits of the ox, 
 in particular, unknown or doubtful, but the 
 circumstance of its possessing horns evident, 
 
SYLLOGISM. *7l 
 
 you infer that that may be predicated of the 
 ox which you have previously predicated of 
 the class to which it evidently belongs ; and 
 although the syllogism is evidently of little 
 use in cases of experimental knowledge, 
 where the validity both of the premises and 
 the conclusion depends upon one and the 
 same process of observation, yet is it evi- 
 dent that the first premise of the above ex- 
 ample can only be true by involving the 
 truth of the conclusion. 
 
 Horned animals 
 
 \ 
 
 I i 
 
 ruminate the ox 
 
 Wliatever exhibits marks of design must 
 have an intelligent creator ; the universe ex- 
 hibits marks of design ; therefore the uni- 
 verse must have an intelligent creator. 
 
 Design 
 
 \ ': 
 
 I I 
 
 Creator the universe 
 
 All reasoning is included in the term Logic ; 
 mathematics is reasoning upon figure and 
 quantity ; therefore mathematics is a branch 
 of logic. 
 
78 
 
 SYLLOGISM. 
 
 Logic 
 
 I 
 
 reasoning mathematics 
 
 The whole mystery of syllogism consists, 
 therefore, simply in referring the two things 
 whose relation you wish to ascertain, to some 
 common class signified by what is called the 
 middle term ; or, regarding the conclusion 
 as a truth you w^ish to demonstrate, the 
 demonstration consists in referring it to 
 some test, i. e. to some more general pro- 
 position, whose truth, whose use as a test, 
 is previously admitted, agreed upon, or 
 assumed.* 
 
 * My attention was arrested by the ^^gnette attached 
 to the original edition of Hobbes' Le\dathan after writing 
 the above, as exhibiting a similar idea of logic to the 
 eye. 
 
 The connexion between logic and just classification 
 
SYLLOGISM. 79 
 
 In examining the nature of syllogistic rea- 
 soning, it is now evident that you have, — 
 
 In the first place, something laid down, 
 granted as true, or assumed as necessary, for 
 the subsequent proceeding of the mind. It 
 does not affect the truth of this position to 
 determine whether you must always have two 
 propositions, as in a regular syllogism, before 
 a conclusion can be drawn ; or whether the 
 mind, having admitted one judgement or pro- 
 position, be led on from that one to an- 
 other inevitably following from it. Suffice it, 
 that in every argument you must begin with 
 something laid down, granted, or assumed ; 
 in other words, you must have some datum 
 or data, as points from which to start, or 
 ground on which to rest. 
 
 Secondly. Having something laid down 
 or granted, it is the characteristic of all cor- 
 
 proves that Hobbes is about right, though he meets with 
 Mr. Stewart's particular reprobation, in saying that 
 " when a man reasoneth, he does nothing else but con- 
 ceive a sum total from addition of parcels, or conceive a 
 remainder from subtraction of one sum from an- 
 other," &c. 
 
80 COxMPARISON OF LOGICAL 
 
 rect or logical reasoning, that the conclusion 
 necessarily follows from the premises. It is 
 involved in the premises ; it is a consequence 
 inevitably connected with them. An absolute 
 necessity is the quality of all sound reasoning. 
 
 How then does logical or common reason- 
 ing differ from w^iat we call mathematical or 
 demonstrative reasoning ? In both we have 
 data from which to start, and conclusions 
 inevitably resulting, involved in the meaning 
 of the terms, i. e. in our conception of the 
 things signified. 
 
 Will any man assert that logical reason- 
 ing is not demonstrative in the mathematical 
 sense of the term? that is, does not con- 
 vey a feeling of certainty to the mind in the 
 justness and necessity of the conclusion. For 
 my own part, I feel no difference in respect 
 of absolute certainty. Having admitted that 
 " Yvhatever exhibits marks of design must 
 have an intelligent author," and that " the 
 universe exhibits marks of design," it appears 
 to me as inevitable to admit that "there- 
 fore the universe must have an intelligent 
 author," as to admit that the three angles of 
 
AND MATHEMATICAL REASONING. 81 
 
 a triangle are equal to two right angles, hav- 
 ing admitted that the sum of the angles 
 on the same side of a straight line, at the 
 same point, is equal to two right angles. The 
 conclusion is involved in the meaning of the 
 terms : in the latter case it rests upon the 
 notion of equality ; in the former, the notion 
 of design. It must be confessed, however, 
 that the notion of design is not so imme- 
 diately an idea of sensation, to use Locke's 
 phraseology, as that of equahty. It is an 
 idea drawn rather from reflection, or what 
 we are conscious of when the mind turns in- 
 ward upon itself. In common language, two 
 things are said to be equal vv^hen there are no 
 sensible marks of difference between them. 
 But an object of nature or of art will indicate 
 design to an observer very much in propor- 
 tion to the observer's power of appreciating 
 the end aimed at, and the means employed ; 
 and surely it is impossible to separate the 
 idea of design from the perception of means 
 and ends. 
 
 It is not, then, in the circumstance of start- 
 ing from certain data or given principles ; it 
 E 5 
 
82 COMPARISON OF LOGICAL 
 
 is not in the necessity of the conclusion, as 
 resulting inevitably from the data, that logical 
 or common reasoning, and mathematical 
 reasoning, differ from one another. We must 
 seek for that difference elsewhere. When, 
 therefore, the Edinburgh reviewer, after 
 Kant, talks of necessary and contingent mat- 
 ter, as distinguishing two sorts of logic, one 
 of which he equals to mathematical reason- 
 ing, the other to general reasoning, it is a 
 talk without meaning, or at least without 
 clear and sufficient meaning. For in all logic, 
 as in all mathematics, the conclusion is 
 equally necessary, equally contingent; equally 
 necessary in the sense of inevitably following 
 from, or being involved in, the data ; equally 
 contingent upon the comprehension or force 
 of the terms ; that is, the degree of clearness 
 and exactness in the things signified. 
 
 But, again, if in all reasonings you start 
 from some data, and in all reasonings 5^ou 
 have necessary conclusions, where lies the 
 difference between what we call mathema- 
 tical and common reasoning? Obviously 
 we must seek it in the nature or difference of 
 
AND MATHEMATICAL REASONING. 83 
 
 the data : and here, perhaps, we shall come 
 to see in what sense it is true, if in any sense 
 it be true, that in mathematics you have ne- 
 cessary, in logic contingent, matter. In ma- 
 thematics we have, as I have shown, defini- 
 tions, i. e. exact terms, significant of certain 
 clear ideas of figure and quantity ; and we 
 are employed in tracing the relations of these 
 ideas one with another. In all other reason- 
 ing we have also terms, and these terms are, 
 or ought to be, signs of ideas. But while, 
 in mathematical reasoning, we are concerned 
 with ideas of figure and quantity, (to say no- 
 thing of forces and motion, with whatever else 
 can be viewed and treated mathematically,) in 
 logic we have every variety of term and of 
 idea. There is no proposition of any kind, no 
 number of words which can be put together 
 so as to form a proposition, to whatever sub- 
 ject it may relate, which may not form part 
 of a syllogism. It is owing to this compre- 
 hensiveness or vastness of logic, in its prac- 
 tical application, that its true nature is so 
 little understood. In all reasoning, in all 
 thought, as communicated from one mind to 
 
84 COMPARISON OF LOGICAL 
 
 another, we have, firstly, the terms ; se- 
 condly, the things signified : but the sole 
 question is, how far the terms are exact, and 
 the things signified clear. 
 
 To put this in another light. If, as we 
 have endeavoured to show, the essence of 
 logic, of all arguments or syllogisms, consist 
 in the reference of tw^o terms to one and the 
 same common term or object of comparison, 
 used as a measure or test of agreement or dis- 
 agreement ; and if, as we have also shown, the 
 essence of mathematical or geometrical rea- 
 soning consist in the perception of the agree- 
 ment or disagreement of certain ideas and 
 terms, by means of other intermediate ideas 
 and terms, previously admitted as a test ; it 
 remains only to inquire what are the peculiar 
 measures, or what is the excellence of the 
 tests, in mathematical reasoning; for by 
 these it obtains its character of force and 
 clearness, that is, of demonstration. 
 
 Have you in mathematics better data, bet- 
 ter measures, than in other subjects ? The 
 only answer is, that in mathematics you are 
 conversant with ideas of figure and quantity, 
 
AND MATHEMATICAL REASONING. 85 
 
 and that you have certain well-defined terms 
 always associated with the same simple ideas 
 or impressions ; the miiformity in the struc- 
 ture of the senses, and in the sensible impres- 
 sions made upon them, laying the foundation 
 for that peculiar constancy of language, and 
 consequent clearness of reasoning, which we 
 recognize in connexion with figures and 
 numbers. 
 
 It follows, therefore, that it is not in any 
 theories about generalization and abstraction, 
 it is not in any difference between the higher 
 faculties and the common faculties, that you 
 are to seek for an explanation of the cogency 
 of mathematical and the weakness of other 
 reasoning. It is because your abstraction is 
 so comparatively easy, and your generaliza- 
 tion so complete ; and because you have cer- 
 tain exact terms and symbols, used as known 
 and admitted measures, or criteria of proof, 
 about the application of which there neither 
 is nor can be a possibility of doubt, that 
 mathematical reasoning is so satisfactory.* 
 
 * Perhaps this comparative easiness of the abstraction 
 and generahzation in the mathematical sciences is the 
 
86 COMPARISON OF LOGICAL 
 
 If in that reasoning there were the shghtest 
 ambiguity in terms ; if an angle were mis- 
 taken for a chord ; if A B were mistaken for 
 any other hne than that intended in the 
 geometrical proof ; the reasoning would be 
 vitiated, the demonstration lost, the link of 
 concatenation broken ; just as in arithmetic, 
 if 3^on mistook a 3 for a 5, your answer 
 would be wrong ; or, in more dignified phrase, 
 your postulatum would exhibit an erratum. 
 So, in logic, if the sense of your terms be 
 changed in the premises and conclusion, the 
 force of the syllogism is destroyed ; the bolt 
 of the stable is drawn, and the horse gone. 
 
 Think not there is no classification (and 
 wherever classification is, there is of course 
 
 main thing intended by the revievv^er when he speaks of 
 the lower faculties only being employed and developed in 
 ■the study ; but the word " faculties" makes sad havoc in 
 the writings of some philosophers, who would lead' you 
 to suppose man had as many distinct faculties as a cen- 
 tipede has legs. A faculty is simply a power; every 
 action and every thought may be attributed to a distinct 
 faculty for that action and that thought ; and we may, if 
 we please, talk of a faculty for sitting and a faculty for 
 walking ; but we gain confusion and lose distinctness by 
 such phraseology. 
 
AND MATHEMATICAL REASONING. 0/ 
 
 abstraction and generalization,) in mathema- 
 tics ; for in the very first proposition you 
 prove your triangle to be equilateral, when 
 the sides are shown to belong to the class of 
 lines which radiate from the centre to the 
 circumference of the same circle, or of equal 
 circles, of which class of lines you have pre- 
 mised equality as the characteristic ; and he 
 who should dispute this premise must look 
 out for another proof of the first proposition 
 than that with which his Euchd has furnished 
 him. 
 
 It is to be observed particularly, that the 
 definitions in geometry, and, I think, in all 
 the branches of mathematics, are classifica- 
 tions or abstractions of a very simple kind : 
 in geometry, of certain sensible impressions, 
 derived partly from sight and partly from 
 touch. Thus, a line is defined, " Length 
 without breadth." It has been suggested 
 that the words " without breadth" are unne- 
 cessary ; but their use is perhaps expedient, 
 in order to confine attention to that which is 
 exclusively the object of the thought or rea- 
 soning. If any "given fine before us have 
 
88 COMPARISON OF LOGICAL 
 
 sensible breadth, that breadth forms no part 
 of or consideration in the reasoning. Now 
 more or less of what we call length is found 
 in all visible and tangible objects. There is 
 not an object which we see or handle that 
 has not some outline, straight or curved. To 
 this quality of objects, common to all, but 
 peculiar to none, we give the name hne or 
 length. It is an instance of abstraction, sim- 
 ple and complete ; and when w^e have lines 
 of a certain kind, we give them a correspond- 
 ing denomination, as straight, curved, waving. 
 It is not however with lines, in this very ge- 
 neral or abstract character, that geometrical 
 reasoning has much to do ; but it is concerned 
 with lines in more precise and limited cha- 
 racter ; as they are related to each other in 
 a certain fixed, perceptible, or conceivable 
 manner : for instance, as perpendicular or 
 inclined, as • meeting in a point and forming 
 angles, as parallel or equidistant, as forming 
 squares, circles, elhpses, parabolas, or other 
 curves, with fixed properties and relations. 
 Hence figures are classed as triangles and 
 parallelograms, as equilateral and equiangu- 
 
AND MATHEMATICAL REASONING. 89 
 
 lar, as pentagons, hexagons, polygons, and 
 so on. And with regard to all these classes 
 of figures, or definite arrangements of lines, 
 the mathematical reasoning is strictly syllo- 
 gistic ; as in the fifth proposition of the first 
 book, the proof of the equality of the angles 
 at the base of an isosceles triangle turns upon 
 bringing the angles in question within a cer- 
 tain class, viz. the class of angles subtended 
 by equal bases, in triangles which have two 
 sides of the one equal to two sides of the 
 other, of which equality is demonstrated in 
 the fourth proposition : and let us remember 
 that every proposition in Euclid is demon- 
 strated as true, not merely of the individual 
 diagram before the student, but of its class, 
 of which class the said diagram is, in respect 
 of the reasoning, a perfect and sufficient ex- 
 ample. Thus the angles of all triangles are 
 equal to two right angles, whatever be the 
 length of the sides, whether they be right- 
 angled or obtuse ; whether the lines be black 
 or blue ; whether it be the triangle on the 
 paper, or a supposed triangle, formed by 
 lines conceived to meet in the centres of the 
 
90 COMPARISON OF LOGICAL 
 
 earth, the sun, and the moon. It is the 
 simplicity and perfection of the classes ; the 
 accuracy with which every term marks and 
 defines the class ; and the never-faihng con- 
 nexion between the terms and the sensi- 
 ble impressions, and the ease and certainty 
 with which the sensible impressions lead to 
 and support the mathematical conceptions 
 and definitions ; these things help to make, 
 if they do not, as I conceive, themselves 
 make, the proof so cogent and the assent 
 so firm in geometry. 
 
 The dependence of the reasoning upon a 
 clear apprehension of the definition, starting 
 from it and adhering to it, becomes still 
 more clear, if we look at the subject of pro- 
 portion upon which Mr. Whewell has made 
 some but not very distinct remarks. The 
 fifth book of Euclid, which treats of pro- 
 portion, Mr. De Morgan calls, in con- 
 junction with Aristotle's logic, the most in- 
 disputable treatise that ever was written. On 
 the other hand, Leslie, in the fourth Preli- 
 minary Dissertation to the Encyclopaecha 
 Britannica, says that it cannot possibly be 
 
AND MATHEMATICAL REASONING. 91 
 
 taught. The whole difficulty seems clearly 
 to lie in the necessity of enlarging the mind's 
 view of proportion, previously and strongly 
 associated with numbers, i. e. with arith- 
 metical proportion, to magnitudes whose re- 
 lation to each other cannot be expressed in 
 numbers ; and this difficulty can only be 
 overcome by the assiduous study of such 
 magnitudes, and of a book, or books, in 
 which such magnitudes and their relations 
 are brought before the mind. The mind, by 
 the study of the subject, grows to the ap- 
 prehension of the definition, which is a ge- 
 neral principle or view of a certain mutual 
 relation of magnitudes, involving the truth of 
 the propositions to which it is afterwards ap-^ 
 plied. 
 
92 LANGUAGE AS CONNECTED 
 
 SECTION III. 
 
 Having now shown that the object of the 
 science of logic is to call the attention to 
 those forms of expression which are essential 
 to valid arguments, in which the conclusion 
 is necessarily involved in the premises, and 
 the mind is led to perceive a connexion or 
 relation which it did not before perceive 
 between its ideas and terms, — that it resolves 
 itself very much into just classification ; 
 having also shown that mathematical reason- 
 ing owes its clearness and cogency to the 
 simplicity and clearness of its subject mat- 
 ter, — its abstractions and classifications, or 
 relations of figures and quantities, being 
 marked by defined terms, which are the 
 media of mathematical proof, — we have to in- 
 quire how far other notions, besides those of 
 figure and quantities, are susceptible of exact 
 definition and exact language, and thereby 
 of exact comparison one with another. This 
 is the sum and substance of the question, — 
 
WITH GENERAL REASONING. 93 
 
 the susceptibility of exact comparison be- 
 tween our notions of other subjects than 
 figure and quantity ; subjects less connected 
 with sensible impressions, and in which our 
 reasoning cannot be assisted or verified by 
 an immediate appeal to the evidence of the 
 senses. 
 
 I may be as certain, and doubtless I am, 
 that there Hved a celebrated orator named 
 Cicero, at Rome, as that the angles at the base 
 of an isosceles triangle are equal to one an- 
 other ; but it is evident that the ideas asso- 
 ciated with the words Cicero, celebrity, ora- 
 tory, Rome, are of a far more varied and com- 
 plex character than the ideas associated with 
 the terms of the above or any mathematical 
 proposition ; and of a hundred persons who 
 w^ill equally readily assent to the historical or 
 moral proposition, the ideas associated with its 
 terms will differ by a thousand modifications 
 and varieties. Here, then, lies the difficulty. 
 Forms and magnitudes visible to the eye, 
 and weighed by the hand, can be compared, 
 and their exact difference can be estimated 
 and described. But who shall compare and 
 
94 LANGUAGE AS CONNECTED 
 
 estimate tlie exact difference between two 
 tastes and two sounds ? and how indefinite 
 must those terms remain which are not only- 
 associated with a number of such sensible 
 impressions incapable of comparison, but 
 with trains of sensation and emotion, the 
 traces of which pass away with the moment 
 in wdiich they have birth, and which are, 
 perhaps, wholly similar in no two existing 
 minds. 
 
 It is often said, by way of distinguishing 
 mathematical science from all other kinds 
 of reasoning, that the mathematics are hu- 
 man contrivances for attaining human ends; 
 and that in the reasoning v/e are only evolving 
 the nature and properties of these contri- 
 vances. Thus in numbers, we have fractions 
 and decimals ; in algebra, equations and roots ; 
 so in the affairs of life w^e have invented 
 common measures with which to compare 
 magnitudes and quantities : thus we have 
 the foot, the yard, the pound weight, the 
 ton, the pint, the bushel ; by means of which 
 quantities, distances, and magnitudes, are re- 
 lativelv ascertained and settled. So we have 
 
WITH GENERAL REASONING. 95 
 
 thermometers, barometers, and chronometers, 
 for measuring atmospheric weight, heat, and 
 time. These are some of the received mea- 
 sures or tests for ascertaining and determin- 
 ing the degrees of difference by which one 
 quantity varies from another, of whatsoever 
 can be measured or tested. 
 
 Let us, hov/ever, remember that all lan- 
 guage is a human contrivance for expressing 
 human thoughts in all their wide relations 
 and variety ; and although the language which 
 we daily use may have, and most general 
 terms have, very different ideas associated 
 therewith in different minds, yet reasoners 
 have a certain power over its use. We can 
 examine, control, and the exact reasoner 
 always seeks to control by examination and 
 reflection, the associated ideas. It should 
 seem, indeed, that where we have no com- 
 mon measures or exact standards of compa- 
 rison, we cannot come to any exact conclu- 
 sions. And this must be the case in all ques- 
 tions of degree, in things that cannot be re- 
 duced to measure ; where we must be con- 
 tent to use our comparatives and superlatives 
 
96 LANGUAGE AS CONNECTED 
 
 indefinitely, and satisfy ourselves with the 
 old maxim, ' De gustibus non disputandmn.' 
 But it is in our power to approach to exact- 
 ness, (in mathematical phrase, as nearly as lue 
 please,) by using well-defined terms associated 
 with uniform sensible impressions, and by dis- 
 tinguishing things that differ in kind, if not 
 in degree. Y\^e may scrupulously avoid un- 
 necessary changes of terms, when the subject 
 matter of our reasoning is the same ; leave 
 nothing to be understood which it is possible 
 to express ; and beware of using relatives 
 and pronouns, to which there is no clear or 
 certain antecedent. 
 
 It is also often said, by way of further dis- 
 tinguishing the metaphysical sciences from 
 the mathematical, that the latter turns upon 
 human abstractions or hypotheses, and you 
 begin with definitions ; whilst the former 
 turns upon facts, and you end with definitions. 
 But is this an adequate distinction ? Every 
 one, doubtless, must be aware of a certain 
 difference between what are called inquiries 
 into facts, and the pursuit of a train of 
 mathematical reasoning. But what is that 
 
WITH GENERAL REASONING. 97 
 
 difference? What is the process of mind 
 in the two cases ? In mathematical pro- 
 positions you have all the facts before you, 
 and are deducing logically the consequences 
 which flow from the acknowledged data. 
 In what are called inquiries into fact, you 
 are generally testing the correctness of some 
 verbal statement, which indicates the ex- 
 istence of some but perhaps by no means 
 clear impressions upon the human mind ; or 
 you are seeking to supply the want of impres- 
 sions on your awn senses, by considering 
 the nature and evidence of the impressions 
 on the senses of others. These impressions 
 you gather from what is called testimony ; 
 and testimony introduces all the ambiguities 
 and difficulties of language. Here you must 
 have various methods of examination or 
 tests of truth dra^vm from experience and 
 suited to the particular case under exami- 
 nation, into which it is unnecessary to enter 
 at length. 
 
 It is often difficult, we say, to ascertain 
 what is the fact. In general all the difficulty 
 arises from, or is increased by, the indefinite- 
 
 F 
 
98 LANGUAGE AS CONNECTED 
 
 ness of the terms in which the statement of 
 fact is enwrapped by the witnesses, and the 
 want of care, caution, and skill in observing 
 and registering observations. The extreme 
 rapidity with which the human mind min- 
 gles its own inferences with its observations 
 or sensible impressions ; the confidence 
 with which it attaches imaginary or expe- 
 rienced causes to the perceived effects, and 
 the difficulty of dissociating the one from 
 the other, give rise to various embarrass- 
 ments not merely with regard to reliance 
 upon the statements, but in ascertaining 
 even the meaning of others, — embarrass- 
 ments which most writers of reputation in 
 the abstruse subjects of morals and religion 
 rather increase than help to overcome. 
 However, " it is possible," says the sweet- 
 minded Hartley, " for two persons of intelh- 
 gent and candid minds to understand one 
 other." 
 
 Then, as to ending with definitions, the 
 metaphysician who talks of that as his end, 
 while it is the mathematician's beginning, 
 should remember that this is only another 
 
WITH GENERAL REASONING. 99 
 
 mode of stating that his metaphysical dis- 
 cussions are disputes about words. As such 
 no doubt they have their importance, the 
 question being, What are the ideas associated 
 with, or the mental phenomena to be classed 
 under, certain terms which exercise exten- 
 sive sway over the thoughts, feelings, and 
 conduct of men? Let the metaphysician, 
 then, take care of the road by which he 
 proposes to pursue this end. Let him take 
 heed whilst he is in pursuit of a proper defi- 
 nition of a certain term, of such a definition 
 as will be acceded to by the student, that he 
 does not embarrass himself and his reader by 
 a multitude of other still less defined or 
 definable expressions, which thicken the 
 darkness and the difficulty, and conduct to 
 dismay and to despair. 
 
 These simple observations will, I conceive, 
 go some way towards illustrating the ques- 
 tion, in what manner, if in any manner, 
 demonstrative reasoning may be connected 
 with and obtained in metaphysical and 
 moral science. Logic calls our attention 
 to the nature and force of terms ; to those 
 f2 
 
100 DEMONSTRATION 
 
 forms of expression which are esseritialty 
 connected with vaUd reasonings — reason- 
 ings to w^iich the mind assents through 
 the very nature of the terms. Mathe- 
 matical science exhibits the perfection of 
 such reasoning and such terms. It works 
 with symbols significant of certain clear and 
 uniform notions of figures and magnitudes. 
 In its higher branches it is indeed the very 
 science of symbols ; its results arising out of 
 the conceived and admitted nature of the 
 symbols themselves, and their uniform rela- 
 tions one with another. 
 
 Before saying a few^ words on the applica- 
 tion of demonstrative reasoning to physical 
 and metaphysical science, I cannot forbear 
 remarking how groundless is the common no- 
 tion that demonstrative reasoning is stronger 
 than all other kinds of evidence, — than the 
 evidence of the senses or of testimony ;* a 
 
 * There is some logical impropriety in connecting the 
 words demonstration and evidence, and in talking of 
 demonstrative evidence. In demonstrative reasoning we 
 trace the harmony subsisting among om- agreed princi- 
 ples or admitted notions and conceptions, and the con- 
 sequences or connected notions. The senses (and testi- 
 
NOT ALWAYS NECESSARY. 101 
 
 notion which makes some men call for 
 demonstration where it is evidently absmxl 
 to make that call, unless it be remembered 
 that the nature of every proposition must 
 determine the nature of the demonstration, 
 or the kind and criteria of proof. But who 
 shall undertake to demonstrate by any ad- 
 mitted criteria of proof that milk is wdiite 
 and the sky sometimes appears blue ? These 
 are simple instances of names attaching to 
 the objects of sense, and all communion of 
 mind depends on agreeing to give similar 
 names to similar impressions. And what 
 folly would it be when my servant testifies 
 that a crowd is in the streets, or Mr. Smith 
 is in the study, to ask him to demonstrate 
 these affirmations and propositions. The 
 conduct of life depends upon notions and 
 habitual statements which require no demon- 
 stration, no laboured process of inquiry and 
 proof; and with the general language of 
 human intercourse, the associated ideas are 
 
 mony, which must be ultimately referred to impressions 
 on the senses) evidence the existence of certain causes of 
 sensation external to ourselves, independent of our ovv^n 
 minds, and supply the fundamental conceptions from 
 which we reason. 
 
102 PHYSICAL SCIENCE. 
 
 sufficiently clear and uniform for the harmony 
 and happiness and the ordinary wants of 
 society. But when men aspire to what is 
 called science, then must they take heed of 
 language, as the ladder on w^hose rails the 
 foot must rest in every step of the ascent. 
 
 Now with regard to physical science, it is 
 obvious man is simply the observer and 
 registrar of external nature. " Homo est 
 nature minister," is the short and obvious 
 maxim of the natural philosopher, and it 
 is evident that the sciences which w^e call 
 inductive merely give us the combined re- 
 sults of human observation in various spe- 
 cified departments, arranged, classified, and 
 marked, and mathematical science is essen- 
 tial to perfect our observations. 
 
 As we range the walks of time and space, 
 every new object or relation which presents 
 itself to attention, a new term, significant 
 of that object or relation, must be given. 
 That same term will suffice for the same 
 object or its counterpart, when the mind 
 meets with it again. And the uniformity 
 and variety in nature gives rise to the uni- 
 formity and variety in the structure and use 
 
PHYSICAL SCIENCE. 103 
 
 of human language, which, as a whole, 
 may be considered a mystical radiation from 
 nature, imprinting its pictures upon the 
 subtle ground-work of the mind. A term 
 becomes general by being apphed to many 
 similar objects or impressions. The uni- 
 formity in the objects and processes leads 
 us analogically to apply the term laws to 
 the phenomena and processes of nature, as 
 science in its various departments reveals 
 and registers the order, succession, and 
 character of these phenomena. Each sci- 
 ence having its own pecuHar phenomena to 
 register, requires its own pecuhar terms ; 
 nor can we, where hew objects of perception, 
 new relations of thought, are to be expressed 
 and held forth to the contemplation of 
 the mind, object to many new and there- 
 fore hard terms. But in these days the 
 student of nature must be warned against 
 supposing that a mere knowledge of terms 
 is scientific knowledge, although it must be 
 admitted as an indispensable and an un- 
 avoidable part thereof. The learner of a 
 new science cannot but be as the child, whose 
 
]04 PHYSICAL SCIENCE. 
 
 understanding grows to the meaning and 
 right use of the language of the world around 
 him, nor must he complain of difficulties and 
 impediments. Happy the learner, however, 
 w^ho is in the hands of judicious guides, wiio 
 consider terms subsidiary to instruction in 
 things, w4th whom books are, as they 
 assuredly ought to be in physical science, 
 not the substitutes for the companionship of 
 nature, but the aids to interpret her lessons, 
 and to observe and to arrange her instruc- 
 tions. The botanist must keep in the fields 
 and the garden; the chemist in the laboratory; 
 the geologist in the quarry, by the hill side 
 or under the cliff; the astronomer must sweep 
 the heavens with his glass, and report to 
 others 
 
 " Of fields of radiance, whose unfading light 
 Has travell'd the profound six thousand years. 
 Nor yet arrived in sight of mortal things." 
 
 And it is the beauty of physical science, 
 wdien legitimately and lovingly pursued, that 
 it calls us into communion w^ith the Creator 
 as he reveals himself in his w^orks, and away 
 from the perverse disputes and vain jang- 
 
PHYSICAL SCIENCE. 105 
 
 lings of men. In physical science there is 
 comparatively little of tiresome useless argu- 
 mentation ; the facts on which the classifica- 
 tions and conclusions rest are evident to those 
 senses of whose use not even the Fall has 
 deprived the children of Adam. And though 
 the senses may sometimes deceive us from 
 a kind of natural difference in keenness or 
 constitutional imperfection, or through hasty 
 inferences and casual associations, yet the 
 philosopher who builds his system of science, 
 physical or metaphysical, upon other ground, 
 who thinks the root of the tree of life and 
 knowledge is not in that plain but all-sup- 
 porting soil, had better return at once to the 
 speculations of the schoolmen, and puzzle 
 himself with inquiries into the necessary 
 attributes of spirits that have never inhabited 
 a body, or dilate upon that pleasant and 
 edifying subject, " cliim.?era bombinans in 
 vacuo." 
 
 Vanity, partiality for their own habits of 
 study, and their own modes of classifica- 
 tion, may even in physical science lead men 
 to differ widely and warmly ; but the differ- 
 F 5 
 
106 PHYSICAL SCIENCE. 
 
 ence will hardly be very sore, if unconnected 
 with any worldly interests. Questions of fact, 
 and questions of the meaning of terms in 
 most matters of pure science, would be settled 
 without length of debate, if men did not wish 
 to appear wiser than they are ; if they were 
 content patiently to learn and mildly to in- 
 struct ; if they sought the knowledge and 
 love of nature rather than the estimation of 
 men ; if they looked upon themselves as 
 mutual interpreters, and mutual servants of 
 the will of Him, who has made of one blood 
 all nations of the earth to dwell together 
 upon its face, and bound the vast family of 
 man together by the strongest ties of mutual 
 interest, making it their chief and noblest 
 happiness to benefit and assist each other. 
 
 In the pursuit of physical science it would 
 be very easy to repeat some of the principal 
 rules to be observed ; but they are already in 
 various forms and in abundance before the 
 studious and inquiring world. To observe and 
 register carefully, not to generalize too fast, to 
 take special care of your premises before you 
 arrive at your conclusions, to build the super- 
 
WORKS OF REFERENCE. 107 
 
 structure of system upon the solid ground- 
 work of clear and well-ascertained facts, and 
 with the strong masonry of well-defined and * 
 chastised language — the cement of human 
 knowledge — these are the short and simple, 
 but universal rules for all philosophy. The 
 Novum Organum of Bacon ; the first 
 of the preliminary dissertations prefixed to 
 BufFon's Theorie du Monde ; the admirable 
 Reflexions sur la Geometric, in the Pensees 
 de Pascal ; Hartley's invaluable Observations 
 on Propositions and the Nature of Assent ; 
 the last book of Locke's Essay on the Human 
 Understanding ; Herschel's Treatise on the 
 Study of Natural Philosophy ; Dr. Whately's 
 Logic ; some of the remarks of Laplace, 
 in his Essay on Probabilities, to which I 
 shall hereafter advert, and of Cuvier in the 
 Preface and Introduction to his Animal 
 Kingdom ; and Mr. De Morgan's treatises 
 published by the Society for the Diffusion of 
 Useful Knowledge ; — are among the works of 
 most importance in connexion with the subject 
 of this Essay. The following passage from the 
 preface to Cuvier's first edition of his Regne 
 
108 EXTRACT FROM CUVIER. 
 
 Animal will be acceptable to the reader who 
 is not previously acquainted with it, and 
 bears closely upon our present course of 
 thought, " The habit necessarily acquired 
 in the study of natural history, of the mental 
 classification of a great number of ideas, is one 
 of the advantages of that science which is sel- 
 dom observed, and which, when it shall have 
 been generally introduced into the s^^stem of 
 common education, will become perhaps the 
 principal one. By it the student is exercised 
 in that part of logic which is termed method, 
 just as he is by geometry in that of syllo- 
 gism ; because natural history is the science 
 which requires the most precise methods, as 
 geometry is that which demands the most 
 vigorous reasoning. Now this art of method, 
 once well acquired, may be applied with in- 
 finite advantage to studies the most foreign 
 to natural history. Every discussion mi- 
 plying a classification of facts, every inquiry 
 w^hich demands a distribution of materials, is 
 performed according to the same laws ; and 
 the young man who had cultivated this 
 science merely for amusement, is surprised, 
 
METAPHYSICAL SCIENCE. 109 
 
 when he makes the experiment, at the facih- 
 ties it affords him in disentanghng all kinds 
 of affairs. It is not less useful in solitude. 
 Sufficiently comprehensive to satisfy the 
 most powerful mind, sufficiently various and 
 interesting to calm the most agitated soul, 
 it consoles the unhappy, it soothes animosi- 
 ties. Once elevated to the contemplation of 
 that harmony of nature irresistibly regulated 
 by Providence, how weak and insignificant 
 appear those causes which it has been pleased 
 to leave dependant on the arbitrary will of 
 man. How astonishing to behold so many 
 examples of fine genius consuming them- 
 selves so vainly for their own happiness, or 
 that of others, in the pursuit of empty specu- 
 lations, whose very traces a few years suffice 
 to sweep away." 
 
 But is the motto "homo est naturse mi- 
 nister" applicable solely to physical and not 
 to metaphysical, moral, and political sci- 
 ence ? Can we study the human mind and 
 human interests without a careful observa- 
 tion of the phenomena or facts ? Is man to 
 interpret his mental and moral constitution 
 
110 METAPHYSICAL SCIENCE. 
 
 by means of some a ??norf principles, which 
 are to be received, without question, upon 
 the ipse dixit of this or that philosopher? 
 There is indeed a growing perception of the 
 importance of the Baconian method, in moral 
 and political science as well as in other de- 
 partments of human inquiry, notwithstand- 
 ing the tendency of many productions of 
 eminence to draw away men's attention from 
 the palpable instruction of nature to the 
 cloudy, obscure, inconsistent, and unmean- 
 ing language of the schools. But a slight 
 acquaintance with the most popular works 
 on ethical and metaphysical philosophy, will 
 lead to the conclusion that they are sadly 
 deficient in that exact language which is 
 the essential characteristic of all true science, 
 and must be its foundation. 
 
 " A plain and unadorned style," says 
 Lord Bacon, "is the proper style for philo- 
 sophy." Yet how far from being plain, — 
 how studied in variety of phrase, — how re- 
 dundant in poetic and metaphoric ornament, 
 are the pages of a Stewart and Mackintosh ! 
 Their pages seem to have been written under 
 
STEWART AND MACKINTOSH. Ill 
 
 the impression that no effort of art should be 
 spared to beguile the reader in the study of 
 matters so dry and so laborious as ethics and 
 metaphysics. But the result of this is to 
 give the real student, who is carefully looking 
 out for facts and principles, more trouble 
 than can be estimated in finding the object of 
 his search, — in separating the important 
 from the trivial, — in testing, by the applica- 
 tion of precise logic, the presence and 
 quantity of meaning in a cloudy solution of 
 verbiage. Their dissertations leave hardly 
 any definite impression on the mind. Nor 
 is this owing to any peculiar difficulty in 
 coming at the facts which lay the foundation 
 for principles, so much as to an erroneous 
 view of the true method, or from a bad 
 habit of philosophising, or perhaps to an in- 
 disposition for that patient and cautious exa- 
 mination of details, with that simplicity 
 and exactness of phrase, which have few 
 charms for any but the obscure devotees of 
 truth. 
 
 Metaphysical discussions are pre-eminently 
 discussions about words. Ideas, feelings, 
 
112 STEWART AND MACKINTOSH. 
 
 principles, faculties, powers, affections, ma- 
 terialism, spiritualism, necessit}^, free-will, 
 cu7n multis aliis, occur so constantly in these 
 discussions, with so Httle exactness of mean- 
 ing, and such variety of indistinct associated 
 notions, that the student is still, like Milton's 
 angels, " in wandering mazes lost." 
 
 The dissertations of Stewart and Mackin- 
 tosh, prefixed to the Encyclopaedia Britannica, 
 enjoy a high degree of reputation. They are 
 spoken of by critics of eminence as inimitable 
 and invaluable. Reviews of great authority 
 delight in recording the highest estimation of 
 their merit. They are unquestionably use- 
 ful and pleasing ; but of what are they the 
 history ? If we compare them with the ac- 
 companying dissertation by Playfair, on the 
 History of Natural Philosophy, they must 
 sink in the comparison, inasmuch as they 
 fail to give that systematic view of real 
 though gradual additions to human know- 
 ledge, — of discoveries, with the authors, and 
 times of discovery, which can alone properly 
 constitute a history of any science or phi- 
 losophy. They do not fasten upon im- 
 
Stewart's estimate of locke. 113 
 
 portant epochs of improvement in the mode 
 of studying and treating the philosophy oi 
 mind. They do not give the details of writers 
 and their systems, in the clear and exact 
 manner most essential to the reader, nor sum 
 up their respective merits and defects by the 
 application of any well-stated, established, and 
 acknowledged principles or criteria of excel- 
 lence. 
 
 Mr. Stewart makes the chief merit of 
 Locke to consist in the inquiring spirit 
 with which he imbues his reader. Speaking' 
 of the general effect of Locke's discussions, 
 in preparing the thinking part of his readers 
 to a degree till then unknown, for the un- 
 shackled use of their own reason, he says, 
 "This has always appeared to me the most 
 characteristical feature of Locke's Essay; and 
 that to which it is chiefly indebted for its 
 immense influence on the philosophy of the 
 eighteenth century." The Essay of Locke 
 is certainly the production of a profoundly 
 thoughtful mind ; and the diligent reader, 
 called into sympathy with such a mind, must 
 partake, and imbibe in some degree, the spirit 
 
114 
 
 of the great and good philosopher. But 
 surely the merits of Locke, as the founder of 
 a school of metaphysicians, are of a much 
 higher order than such an estimate of his 
 chief characteristics implies. Locke not only 
 prepares the reader to reason, but prepares 
 him to reason well ; to know himself, and 
 man, and nature, better. It is difficult for 
 us in these times to conceive the clouds and 
 mists which the Essay on the Human Under- 
 standing dissipated, and how it illuminated 
 the whole horizon of human thought. Cud- 
 worth's Intellectual System is a bock which 
 sets the reader upon using his own reason ; 
 and it requires a vastly superior stock of eru- 
 dition to weigh and consider its contents. But 
 Cudworth's learning oppresses the mind with 
 its weight, and lies upon the intellect like a 
 London fog upon the chest ; whereas Locke 
 clears and braces the understanding. He 
 has laid the broad and deep basis for a true 
 intellectual system in his grand division of 
 Sensation and Reflection ; the world without 
 us, and the world within. His second chap- 
 ter, on the Origin of our Ideas, and his last 
 
115 
 
 chapter, on Knowledge, Reason, Faith, and 
 Judgement, are full of rich, just, and invalua- 
 ble matter. In the 28th chapter of the 2nd 
 book he has given, incidentally, an analysis of 
 moral laws, which must be in part followed by 
 every other inquirer into ethics and jurispru- 
 dence ; and the few words of Mr. Austin (in 
 a note, p. 174 of his work on Jurisprudence) 
 are far more worthy of the subject than all 
 Mr. Stewart's elaborate though elegant 
 phraseology. " Allowing for defects, which 
 were nearly inevitable, his analysis is stri- 
 kingly accurate. It evinces that matchless 
 power of precise and just thinking, with that 
 rehgious regard for general utility and truth, 
 which marked the incomparable man who 
 emancipated human reason from the yoke of 
 mystery and jargon. And from this, his in- 
 cidental excursions into the field of law and 
 morality, and from other passages of his 
 essay wherein he touches upon them, we 
 may infer the important services which he 
 would have rendered to the science of ethics 
 if, complying with the instances of Molyneux, 
 he had examined the subject exactly." 
 
 Mr. Stewart's estimate of the defects 
 
116 LOCKE. 
 
 and excellencies of Locke can never sa- 
 tisfy the deep and cautious investigator of 
 the constitution and powers of the human 
 understanding ; and his everlasting, untiring 
 intimations that the principles of Locke's 
 Essay, if carried out, or not guarded by 
 Locke's admissions, lead to the scepticism of 
 Hume, are beyond measure unsatisfactory 
 and wearisome to those who neither see nor 
 understand the consequence, and who find it 
 difficult to make out, at least from Mr. 
 Stewart's statements, in what the scepticism 
 of Hume precisely consisted, or how far it 
 extended, save in the celebrated Essay on 
 Miracles, where the uniformity of nature, 
 or our own experience of it, is contrast- 
 ed, in a loose manner, with the variable- 
 ness of human testimon}^ and perhaps in a 
 general want of principles and convictions in 
 all the objects of metaphysical inquiry. 
 There are two very different states of mind 
 often confounded under the term scepti- 
 cism ; one is that of an inquirer, who is de- 
 sirous of placing his own meaning, and the 
 meaning of others, before him in the clear- 
 est and most satisfactory light, and pro- 
 
MACKINTOSH. 117 
 
 portioning the degree of his assent to the 
 strength of the evidence ; the other, that of 
 a man who concludes positively against cer- 
 tain opinions and propositions, as having no 
 foundation in truth, merely because he can- 
 not keep steadily before him the chain of 
 arg-uments or evidence by which they are 
 supported. The former is allied to what 
 Hartley calls a " rehgious scepticism,'"' which 
 receives nothing rashly, and is always pre- 
 pared and seeking for evidence, with a view 
 of concluding wisely and rationally. The lat- 
 ter is less allied to scepticism than to dog- 
 matism, and becomes dogmatism when a 
 man, losing sight of the degrees of evidence 
 by which the opinions he controverts are sup- 
 ported, thinks to bear them down by the 
 strength, rather than the justness, of his 
 assertions. It is that of a man who is not 
 seeking for truth, but doubts its existence ; 
 who is in the habit of questioning every 
 thing, and believing nothing. 
 
 But if Mr. Stewart's work be open to cri- 
 ticism, what shall we say of Sir James Mack- 
 intosh, wdiose Dissertation excites the prin- 
 
118 MILL ON MACKINTOSH. 
 
 cipal reviews to rapture, and which Mr. 
 Whewell undertakes to preface and to edit ? 
 That it is well — very well — worth perusal, may 
 be asserted, partly because so little is written 
 and so little is read with becoming care in 
 such departments of human inquiry. But that 
 the more it is read and examined, the less 
 satisfactory it will appear, may also be assert- 
 ed ; and he who does not choose to rest his 
 justification of this assertion upon any ela- 
 borate showing of his own, may shelter him- 
 self behind Mill's Fragment on Mackintosh, 
 and ask. Where is the answer to that book ? 
 He may ask why the laudatory reviewers never 
 allude to its existence, and why, since it con- 
 tains so vigorous, hearty, and relentless an at- 
 tack upon the merits of Sir James, there is no 
 attempt to protect his fame, to meet and rebut 
 its arguments. Surely Mr. Mill was no puny 
 adversary — no mere fly, that stings but im- 
 pedes not the noble racer in his dazzling 
 course. The peaceful student of nature and 
 of truth may not admire the somewhat bear- 
 ish style in which Mr. Mill shakes and tears 
 his prey ; and with his Fragment strews the 
 
MILL ON MACKINTOSH. 119 
 
 ground with fragments of Sir James's mangled 
 Dissertation. He may feel that the arrow from 
 Mr. Mill's quiver, when he touches upon Sir 
 James's " dandy philosophy" and lack-a- 
 daisical style, has been dipped too cruelly in 
 poison, tinged too darkly with the " gall of 
 bitterness." Yet while his feelings of pity 
 and tenderness are roused, and he cries out 
 for mercy ! mercy ! on Sir James, it is a cry 
 for mercy rather than for justice. The soft 
 heart wishes to spare, and not to trample on 
 the fallen. But Mr. Mill, roused by the ap- 
 parent, though as he calls it affected candour, 
 with real injustice to Mr. Bentham, descends 
 like a second Achilles to avenge his friend, 
 and drags his victim thrice round the field 
 of his defeat. 
 
 Seriously, he convicts Sir James over and 
 over again of those superficial statements, of 
 that vagueness and confusedness of lan- 
 guage, which have hitherto been, and which 
 continue to be, the great impediments to 
 sound progress in the philosophy of mind. 
 Yet, to do Sir James some justice, he is free 
 from that aversion to acknowledge the full 
 
120 HARTLEY. 
 
 extent and importance of the ''law of asso- 
 ciation," which is characteristic of many, I 
 believe most, of the Scotch metaphysical 
 writers. He shares not in Dugald Stewart's 
 antipathy to Hartley. He is the first meta- 
 physical writer of considerable repute of late 
 years, as far as 1 know, wdio seems to be 
 somewhat duly aware of the merit and im- 
 portance of the Hartleian theoiy and prin- 
 ciples. He is wrong when he affects to see 
 deeper into the moral constitution of man 
 than others who have gone before him. He 
 is convicted of error and absurdity in sup- 
 posing that those of whom he was writing 
 had overlooked a distinction which he had 
 sagacity to discover between thoughts and 
 feehngs, perceptions and emotions. And even 
 while he admits that Hartley vras a great 
 philosopher and a good man, there is hardly 
 a sentence upon the nature of Hartley's sys- 
 tem which conveys to the reader a distinct 
 idea of it, — of its apprehended merits or 
 defects. "Nothing," says Sir James, "more 
 evidently points out the distinction of the 
 Hartleian svstem from all svstems called self- 
 
HARTLEY. 121 
 
 ish, not to say its superiority in respect to dis- 
 interestedness over all moral systems before 
 Butler and Hutcheson, than that excellent 
 part of it which relates to the Rule of Life." 
 Now the Rule of Life contains the whole 
 moral system of Hartley, and is a deduction 
 from the inquiry which Hartley instituted 
 into the frame and constitution of man. 
 And in the whole compass of human philo- 
 sophy does any thing surpass that rule of 
 life ? But for Butler and Hutcheson, it is 
 evident that Hartley would have been the 
 favourite philosopher of Mackintosh. Now 
 since, according to his own showing, Butler 
 and Hutcheson have really no system, but 
 merely assert that human nature has certain 
 characteristics, which no one ever denied, 
 and which the philosophers of the Hartleian 
 school have resolved, or tried to resolve, 
 into the true elementary principles, we may 
 perhaps, notwithstanding Mr. Mill's caustic 
 animadversions, admit that the Dissertation 
 of Sir James Mackintosh will, after all, help 
 the student of mental philosophy to the 
 quarter where hght is to be found. But he 
 
 G 
 
122 PRESENT STATE 
 
 must coatirme to be a student, and verify 
 statements and weigh opinions for himself. 
 
 It ma^T- appear unwarrantable to some 
 readers that I should have lapsed from an 
 inquiry into the nature of mathematical rea- 
 soning and its application to science, sud- 
 denly to a criticism upon the Dissertations 
 of Stewart and Mackintosh. But the reason 
 is obvious : these Dissertations appear to 
 be the most popular, and so far the most 
 important, treatises published of late upon 
 the mental and moral constitution of man, 
 which is doubtless the noblest of all human 
 objects of inquiry and contemplation. I 
 see or fancy that these Dissertations, both 
 admitting and lamenting the ambiguity of 
 language as the greatest obstacle in the way 
 of the moral philosopher, are utterly defi-. 
 cient in earnest and successful efforts to 
 grapple with the difficulty and overcome it ; 
 nay, increase rather than diminish the evil. 
 True it is, that while we confine ourselves to 
 that which we can rigidly and logically prove, 
 in the present state of ethical science, 
 we shall go but a little wa}", and we shall 
 
OF ETHICAL SCIENCE. 123 
 
 soon come to all the conclusions which can 
 be safely concluded, whether gathered from 
 our most comprehensive inductions or flow- 
 ing from the nature of our conceptions and 
 terms. 
 
 How little can be known ! 
 
 This is tlie wise man's sigh : how far we err ! 
 This is the good man's not unfrequent pang." 
 
 The Excursion. 
 
 But it is better to make sure of a little than 
 to darken counsel by words without know- 
 ledge ; it is better to build a small house fit 
 and furnished to dwell in, than ruin our 
 fortunes by undertaking a palace which we 
 can neither finish nor inhabit ; and let us 
 remember that philosophy is, after all, only 
 strong sense, or the combined result of 
 larger observation and deeper reflection in a 
 little better dress than that of ordinary 
 life. When a writer or reader sits down to 
 philosophize, or to logical composition, he 
 should consider first what it is he proposes 
 to prove ; and secondly, by what means 
 he proposes to prove it ; and he will soon 
 find that he must make the nature and use 
 g2 
 
124 LAPLACE ON PROBABILITIES. 
 
 of language his principal study, as it is the 
 main instrument of thought and communi- 
 cation ; and in the proper use of it, consists 
 the chief difference between the strong and 
 the weak mind. 
 
 " Scribencli recte sapere est et principium et fons." 
 
 In considering the application of mathe- 
 matical reasoning to moral and metaphysical 
 subjects, it would be unpardonable to over- 
 look the Philosophical Essay of Laplace, on 
 Probabilities, reviewed by Playfair in the 
 tY\"enty-third volume of the Edinburgh Review. 
 Laplace has touched upon the applicability 
 of the calculus to moral sciences in the fol- 
 lowing order of topics : — First, the proba- 
 bilities of testimony ; second, the votes and 
 decisions of assemblies ; third, the judgements 
 of legal tribunals ; fourth, tables of mortality, 
 the mean duration of life, of marriages, and 
 any social relations ; fifth, illusions in the 
 estimate of probabilities. 
 
 Although many just and some curious and 
 novel observations are made by Laplace 
 u.nder these heads, vet their general efiect is 
 
LAPLACE ON PROBABILITIES. 125 
 
 to show the wide difference which exists be- 
 tween those events which, by a strict and 
 clear resemblance, can be classed together 
 and subjected to numerical calculation, and 
 those which are of more complicated charac- 
 ter, have a less perfect -analogy, differ from 
 each other by many circumstances peculiar 
 to each case, — diversities growing out of the 
 general mental, and moral constitution of 
 man, and which cannot, therefore, be so 
 legitimately classed, nor so successfully 
 brought within the category of number. 
 It is indeed somewhat singular, that La- 
 place should have placed tables of mor- 
 tality and the duration of life, and the re- 
 lative proportion of births, marriages, and 
 deaths, to population, — events of such uni- 
 formity of character, and so easily and na- 
 turally classed, — in connexion with subjects 
 so wide and general as the nature of testi- 
 mony, the decisions of popular assemblies 
 and legal tribunals, and the illusions of ima- 
 gination, — subjects which he has touched 
 but slightly. Statistical calculations ought 
 not indeed to be overlooked by the moralist, 
 
126 PLAYFAIR ON LAPLACE. 
 
 since they furnish that extended view of 
 the phenomena of Hfe and of the moral 
 world, which is essential to the deduction of 
 sound moral rules and principles, and af- 
 fords great help to the interpretation of the 
 laws and ends of Divine Providence in the 
 administration of the world. Any collection 
 of analogous phenomena in the physical 
 world is of importance in the moral, since 
 such collections lay the foundation of general 
 rules, and supply the exact knowledge which 
 is of the greatest use in the conduct of life. 
 But in collecting analogies we must not 
 overlook differences, and be duly cautious 
 in the apphcation of general maxims, which 
 are apt to mislead as well as to instruct. 
 Playfair has made some just remarks on the 
 vagueness of Laplace's language and views 
 on the subject of testimony ; and it is easy 
 to conceive that Laplace's devotedness to 
 the exact sciences may have disposed him to 
 an unreasonable mistrust and disregard of all 
 conclusions of the pure intellect, drawn from 
 the phenomena of history and the moral 
 world. But when Playfair intimates that no 
 
LAPLACE ON ASSOCIATION. 127 
 
 conclusion founded on the application of the 
 calculus to moral probabilities should be al- 
 lowed to interfere luith the truths of religion, 
 he seems to imply that such an interference 
 may be anticipated, and rationally feared, and 
 to forget that, to the philosophic mind, truth 
 in all departments must be ever consistent 
 with itself; that though the atoms of the 
 universe may be weighed and measured, and 
 every seeming accident shall be reduced to 
 order and to rule, the reasons for adoring the 
 Creator, and trusting in him and obeying 
 him, and for loving our neighbour as our- 
 selves, will only be proportionably multi- 
 plied. 
 
 It is to me truly interesting to observe La- 
 place perceiving and recording the impor- 
 tance of that principle of association into 
 which Hartley, foUov/ed by Mill, resolves all 
 reasoning. " The most fertile of all the 
 principles of psychology," (Laplace, p. 224, 
 of the 8vo. edition of his Essay,) " is that of 
 the association of all things which have had 
 in the sensorium a simultaneous or regularly 
 successive existence ; an association whereby 
 
128 LAPLACE ON ASSOCIATION. 
 
 the return of one calls up all the others con- 
 nected with it ; the objects which we have 
 formerly seen awaken the traces of things 
 which in the first view were associated with 
 them. These traces call up in the same 
 manner those of other objects, and so on in 
 succession, so that by means of one thing 
 presented to the mind, we can recal an infi- 
 nity of others, and rest our attention upon 
 whatever we wish to consider. To this 
 principle the employment of signs and lan- 
 guage for recalling sensations and ideas be- 
 longs ; it accounts for the formation and 
 analysis of complex, abstract, and general 
 ideas, and for reasoning. Many philosophers 
 have well developed this principle, which up 
 to the present time constitutes the real part 
 of metaphysics." " I have endeavoured to 
 show in these papers," says Hartley, at the 
 conclusion of his 99th proposition, " that 
 all reasoning, as well as affection, is the mere 
 result of association." 
 
 Some further extracts from Laplace might 
 perhaps be given with interest to the reader, 
 but it does not appear to me that Laplace 
 
Austin's jurisprudence. 129 
 
 adds any thing to the ingenious observations 
 and speculations of Hartley, in his chapter 
 on the Deduction of Rules for the ascertain- 
 ment of Truth, and advancement of Know- 
 ledge, from the mathematical Methods of con- 
 sidering Quantity. Mr. Dugald Stewart, in 
 the Preliminary Dissertation to his Philoso- 
 phical Essays, was pleased to sneer at some 
 of the most refined and natural conclusions 
 of that great and good philosopher, Hartley, 
 in the chapter referred to, and to consider 
 them as evidence of the unsoundness of his 
 understanding. But Mr. Stewart is very 
 often unfortunate in his selection of the sen- 
 timents of the greatest English metaphysi- 
 cians, which he marks for peculiar reproba- 
 tion and contempt. 
 
 I have now brought together, and laid 
 before the reader, some thoughts which have 
 occurred to me when reading the more po- 
 pular scientific publications of the day. I am 
 not without hope of awakening the attention 
 of a few, into whose hands these pages may 
 fall, to the contents and merits of authors 
 whose works they may not already suffi- 
 
130 Austin's jurisprudence. 
 
 ciently appreciate, but which it is good " to 
 feed upon, as insects on a leaf, till the whole 
 heart be coloured vv^itli the fibre." The rea- 
 der w^ho wishes to see one of the best examples 
 which our times have afforded of strictly lo- 
 gical or demonstrative reasoning applied to 
 moral and metaphysical subjects, will do vvell 
 to study the w^ork of Mr. John Austin, if he is 
 not already acquainted with it, entitled, " The 
 Province of Jurisprudence Determined." 
 being the substance of lectures delivered 
 at the University of London as Professor of 
 Jurisprudence. Contrary to the practice of 
 those numerous writers who confound that 
 of which they are treating with every thing 
 else with which it can, by possibility, be con- 
 , founded, Mr. Austin carefully distinguishes 
 his subject from every thing with which it is, 
 howsoever remotely, related. He is one of 
 the few writers who does not fear to repeat 
 himself, so much as to be misunderstood by 
 his reader. He w^ould rather give you a whole 
 sentence tAvo or three times over, than intro- 
 duce a pronoun whose antecedent might be 
 doubtful or nonexistent. He seeks not, bv 
 
Austin's jurisprudence. 131 
 
 showy phrase, to impose on the understanding 
 of his reader, or to smooth over with received 
 verbiage the difficulties of his subject and the 
 absence of thought ; and if he sometimes 
 wearies with the repetition of a lengthened 
 phrase, for which a convenient abbreviation 
 might have been advantageously adopted, we 
 could forgive much more than this to a writer 
 who feels so acutely the audacity of the para- 
 dox, " that men really should think distinctly 
 and speak with a meaning." As it bears very 
 closely, indeed, upon the topic discussed in 
 these pages, and as the matter is itself of the 
 highest importance in connexion with ethical 
 inquiries, I shall not scruple to close this 
 essay with the following extract from Mr. 
 Austin's work, feeling, as I do, its fullness 
 of truth : — 
 
 " If there were a reading public nume- 
 rous, discerning, and impartial, the science 
 of ethics, and all the various sciences which 
 are nearly related to ethics, would advance 
 with unexampled rapidity. 
 
 " By the hope of obtaining the approba- 
 tion which it would bestow upon genuine 
 
132 Austin's jurisprudence. 
 
 merit, writers would be incited to the patient 
 research and reflection which are not less re- 
 quisite to the improvement of ethical than 
 to the advancement of mathematical science. 
 
 " Slight and incoherent thinking v/ould be 
 received with general contempt, though it 
 were cased in polished periods, studded Vvith 
 brilliant metaphors. Ethics would be con- 
 sidered by readers, and therefore treated by 
 writers, as the matter or subject of a sci- 
 ence ; as a subject for persevering and ac- 
 curate investigation, and not as a theme for 
 childish and babbling rhetoric. 
 
 " This general demand for truth, (though 
 it were clothed in homely guise,) and this 
 general contempt of falsehood and nonsense, 
 (though they were decked with rhetorical 
 graces,) would improve the method and the 
 style of inquiries into ethics, and into the 
 various sciences which are nearly related to 
 ethics. The writers would attend to the 
 suggestions of Hobbes and of Locke, and 
 would imitate the method so successfully 
 pursued by geometers ; though such is the 
 variety of the premises which some of their 
 
Austin's jurisprudence. - 133 
 
 inquiries involve, and such are the com- 
 plexity and ambiguity of some of the terms, 
 that they would often fall short of the per- 
 fect exactness and coherency which the few- 
 ness of his premises, and the simphcity and 
 definiteness of his expressions, enable the 
 geometer to reach. But though they would 
 often fall short of geometrical exactness and 
 coherency, they might always approach, and 
 would often attain, to them : they would 
 acquire the art and the habit of defining 
 their leading terms ; of steadily adhering to 
 the meanings announced by the definitions ; 
 of carefally examining, and distinctly stating, 
 their premises ; and of deducing the con- 
 sequences of their premise's with logical 
 rigour. Without rejecting embellishments 
 which might happen to fall in their way, the 
 only excellencies of style for which they 
 would seek are precision, clearness, and con- 
 ciseness ; the first being absolutely requisite 
 to the successful prosecution of inquirj^ 
 whilst the others enable the reader to seize 
 the m.eaning with certainty, and spare him 
 unnecessary fatigue. 
 
134 Austin's jurisprudence. 
 
 " And what is equally important, the pro- 
 tection afforded by this public to diligent and 
 honest writers, would inspire into writers 
 upon ethics, and upon the nearly-related 
 sciences, the spirit of dispassionate inquiry, 
 the indifferency or impartiality in the pursuit 
 of truth, which is just as requisite to the 
 detection of truth as continued and close 
 attention, or sincerity and simplicity of pur- 
 pose. Relying on the discernment and the 
 justice of a numerous and powerful public, 
 shielded by its countenance from the shafts 
 of the hypocrite and the bigot, — ^indifferent 
 to the idle whistling of that harmless storm, 
 they would scrutinize established institutions 
 and current or received opinions fearlessly, 
 but coolly, with the freedom which is de- 
 manded by general utility, but without the 
 antipathy which is begotten by the dread of 
 persecution, and wliich is scarcely less ad- 
 verse than ' the love of things ancient' to 
 the rapid advancement of science. 
 
 " This patience in investigation, this dis- 
 tinctness and accuracy of method, this free- 
 dom and ' indifferency ' in the pursuit of the 
 
Austin's jurisprudence. 135 
 
 useful and the true, would thoroughly dispel 
 the obscurity by which the science is clouded, 
 and would clear it from most of its uncer- 
 tainties. The wish, the hope, the prediction 
 of Mr. Locke, would, in time, be accom- 
 plished, and ' ethics v\ould rank with the 
 sciences which are capable of demonstration.' 
 The adepts in ethical as well as in mathema- 
 tical science would commonly agree in their 
 results ; and as the jar of their conclusions 
 gradually subsided, a body of doctrine and 
 authority to which the multitude might trust 
 would emerge from the existing chaos." 
 
 FINIS. 
 
 R. Kinder, Printer, 21, Great New Street. 
 

 
 
 
 Deacidified using the Bookkeeper process. 
 Neutralizing agent: Magnesium Oxide 
 Treatment Date: Sept. 2004 
 
 PreservationTechnologies 
 
 A WORLD LEADER IN PAPER PRESERVATION 
 
 1 1 1 Thomson Park Drive 
 Cranberry Township. PA 16066 
 (724)779-2111 
 

 5 3ai) ^^^ 3&:i3B> : 
 
 
 ^:^~s?iii^