a straight line thrown into a form somewhat different; and a similar remark may be extended to the corresponding distribution of propositions into theorems and problems. Notwithstanding the

many conveniences with which this distribution is attended, it was evidently a matter of choice rather than that of necessity; all the truths of geometry easily admitting of being moulded into either shape, according to the fancy of the mathematician. As to the axioms, there cannot be a doubt, whatever opinion may be entertained of their utility or of their insignificance, that they stand precisely in the same relation to both classes of propositions.*

II.-Continua'ion of the Subject.-How far it is true that all Mathematical Evidence is resolvable into Identical Propositions.

I HAD occasion to take notice, in the first section of the preceding chapter, of a theory with respect to the nature of mathematical evidence, very different from that which I have been now attempting to explain. According to this theory (originally, I believe, proposed by Leibnitz) we are taught, that all mathematical evidence ultimately resolves into the perception of identity; the innumerable variety of propositions which have been discovered, or which remain to be discovered in the science, being only diversified expressions of the simple formula, a = a. A writer of great eminence, both as a mathematician and a philosopher, has lately given his sanction, in the strongest terms, to this doctrine; asserting, that all the prodigies performed by the geometrician are accomplished by the constant repetition of these words,-the same is the same. "Le géomêtre avance de supposition en supposition. Et rétournant sa pensée sous mille formes, c'est en répétant sans cesse, le même est le même, qu'il opère tous ses prodiges."

* In farther illustration of what is said above, on the subject of postulates and of problems, I transcribe, with pleasure, a short passage from a learned and interesting memoir, just published, by an author intimately and critically conversant with the classical remains of Greek geometry.

"The description of any geometrical line from the data by which it is defined, must always be assumed as possible, and is admitted as the legitimate means of a geometrical construction: it is therefore properly regarded as a postulate. Thus, the description of a straight line and of a circle are the postulates of plane geometry assumed by Euclid. The description of the three conic sections, according to the definitions of them, must also be regarded as postulates; and though not formally stated like those of Euclid, are in truth admitted as such by Apollonius, and all other writers on this branch of geometry. The same principle must be extended to all superior lines. "It is true, however, that the properties of such superior lines may be treated of, and the description of them may be assumed in the solution of problems, without an actual delineation of them. For it must be observed, that no lines whatever, not even the straight line or circle, can be truly represented to the senses according to the strict mathematical definitions; but this by no means affects the theoretical conclusions which are logically deduced from such definitions. It is only when geometry is applied to practice, either in mensuration, or in the arts connected with geometrical principles, that accuracy of delineation becomes important.”—See an Account of the Life and Writings of Robert Simson, M.D. By the Rev. William Trail, LL.D. Published by G. and W. Nicol,

London, 1812.

As this account of mathematical evidence is quite irreconcilable with the scope of the foregoing observations, it is necessary, before proceeding farther, to examine its real import and amount; and what the circumstances are from which it derives that plausibility which it has been so generally supposed to possess.

That all mathematical evidence resolves ultimately into the perception of identity, has been considered by some as a consequence of the commonly received doctrine, which represents the axioms of Euclid as the first principles of all our subsequent reasonings in geometry. Upon this view of the subject I have nothing to offer in addition to what I have already stated. The argument which I mean to combat at present, is of a more subtile and refined nature; and, at the same time, involves an admixture of important truth, which contributes not a little to the specious verisimilitude of the conclusion. It is founded on this simple consideration, that the geometrical notions of equality and of coincidence are the same; and that, even in comparing together spaces of different figures, all our conclusions ultimately lean with their whole weight on the imaginary application of one triangle to another;-the object of which imaginary application is merely to identify the two triangles together, in every circumstance connected both with magnitude and figure.*

Of the justness of the assumption on which this argument proceeds, I do not entertain the slightest doubt. Whoever has the curiosity to examine any one theorem in the elements of plane geometry, in which different spaces are compared together, will easily perceive, that the demonstration, when traced back to its first principles, terminates in the fourth proposition of Euclid's first book: a proposition of which the proof rests entirely on a supposed application of the one triangle to the other. In the case of equal triangles which differ in figure, this expedient of ideal superposition cannot be directly and immediately employed to evince their equality; but the demonstration will nevertheless be found to rest at bottom on the same species of evidence. In illustration of this doctrine, I shall only appeal to the thirty-seventh proposition of the first book, in which it is proved that triangles on the same base, and between the same parallels, are equal; a theorem which appears, from a very simple construction, to be only a few steps

* It was probably with a view to the establishment of this doctrine, that some foreign elementary writers have lately given the name of identical triangles to such as agree with each other, both in sides, in angles, and in area. The differences which may exist between them in respect of place, and of relative position (differences which do not at all enter into the reasonings of the geometer) seem to have been considered as of so little account in discriminating them as separate objects of thought, that it has been concluded they only form one and the same triangle, in the contemplation of the logician.

This idea is very explicitly stated, more than once, by Aristotle: oa v To TоGOV ¿v. "Those things are equal whose quantity is the same;" (Met. iv. c. 16;) and still more precisely in these remarkable words, εν τούτοις ή ισότης ένωσης; In mathematical quantities, equality is identity." (Met. x. c. 3.)

For some remarks on this last passage, see Note DD.

removed from the fourth of the same book, in which the supposed application of the one triangle to the other, is the only medium of comparison from which their equality is inferred.

In general, it seems to be almost self-evident, that the equality of two spaces can be demonstrated only by showing, either that the one might be applied to the other, so that their boundaries should exactly coincide; or that it is possible, by a geometrical construction, to divide them into compartments, in such a manner, that the sum of parts in the one may be proved to be equal to the sum of parts in the other, upon the principle of superposition. To devise the easiest and simplest constructions for attaining this end, is the object to which the skill and invention of the geometer is chiefly directed.

Nor is it the geometer alone who reasons upon this principle. If you wish to convince a person of plain understanding, who is quite unacquainted with mathematics, of the truth of one of Euclid's theorems, it can only be done by exhibiting to his eye, operations exactly analogous to those which the geometer presents to the understanding. A good example of this occurs in the sensible or experimental illustration which is sometimes given of the forty-seventh proposition of Euclid's first book. For this purpose, a card is cut into the form of a right angled triangle, and square pieces of card are adapted to the different sides; after which, by a simple and ingenious contrivance, the different squares are so dissected, that those of the two sides are made to cover the same space with the square of the hypotenuse. truth, this mode of comparison by a superposition, actual or ideal, is the only test of equality which it is possible to appeal to; and it is from this, as seems from a passage in Proclus to have been the opinion of Apollonius, that, in point of logical rigour, the definition of geometrical equality should have been taken. The subject is discussed at great length and with much

In

*I do not think, however, that it would be fair, on this account, to censure Euclid for the arrangement which he has adopted, as he has thereby most ingeniously and dexterously contrived to keep out of the view of the student some very puzzling questions, to which it is not possible to give a satisfactory answer till a considerable progress has been made in the elements. When it is stated in the form of a self-evident truth, that magnitudes which coincide, or which exactly fill the same space, are equal to one another; the beginner readily yields his assent to the proposition; and this assent, without going any farther, is all that is required in any of the demonstrations of the first six books: whereas, if the proposition were converted into a definition, by saying, " Equal magnitudes are those which coincide, or which exactly fill the same space;" the question would immediately occur, Are no magnitudes equal, but those to which this test of equality can be applied? Can the relation of equality not subsist between magnitudes which differ from each other in figure? In reply to this question, it would be necessary to explain the definition, by adding, That those magnitudes likewise are said to be equal, which are capable of being divided or dissected in such a manner that the parts of the one may severally coincide with the parts of the other;-a conception much too refined and complicated for the generality of students at their first outset ; and which, if it were fully and clearly apprehended, would plunge them at once into the profound speculation concerning the comparison of rectilinear with curvilinear figures.

acuteness, as well as learning, in one of the mathematical lectures of Dr. Barrow; to which I must refer those readers who may wish to see it more fully illustrated.

I am strongly inclined to suspect, that most of the writers who have maintained that all mathematical evidence resolves ultimately into the perception of identity, have had a secret reference in their own minds to the doctrine just stated; and that they have imposed on themselves, by using the words identity and equality as literally synonymous and convertible terms. This does not seem to be at all consistent, either in point of expression or of fact, with sound logic. When it is affirmed, for instance, that "if two straight lines in a circle intersect each other, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other;" can it with any propriety be said, that the relation between these rectangles may be expressed by the formula aa? Or, to take a case yet stronger, when it is affirmed, that "the area of a circle is equal to that of a triangle having the circumference for its base, and the radius for its altitude;" would it not be an obvious paralogism to infer from this proposition, that the triangle and the circle are one and the same thing? In this last instance, Dr. Barrow himself has thought it necessary, in order to reconcile the language of Archimedes with that of Euclid, to have recourse to a scholastic distinction between actual and potential coincidence; and, therefore, if we are to avail ourselves of the principle of superposition, in defence of the fashionable theory concerning mathematical evidence, we must, I apprehend, introduce a correspondent distinction between actual and potential identity.*

That I may not be accused, however, of misrepresenting the opinion which I am anxious to refute, I shall state it in the words of an author, who has made it the subject of a particular dissertation; and who appears to me to have done as much justice to his argument as any of its other defenders.

"Omnes mathematicorum propositiones sunt identicæ, et repræsentantur hac formula, a = a. Sunt veritates identicæ, sub varia forma expressæ, imo ipsum, quod dicitur contradictionis principium, vario modo enunciatum et involutum ; siquidem omnes hujus generis propositiones reverâ in eo continentur. Secundum nostram autem intelligendi facultatem ea est propositionum differentia, quod quæ

"Cum demonstravit Archimedes circulum æquari rectangulo triangulo cujus basis radio circuli, cathetus peripheriæ ex æquetur, nil ille, siquis propius attendat, aliud quicquam quam aream circuli seu polygoni regularis indefinite multa latera habentis, in tot dividi posse minutissima triangula, quæ totidem exilissimis dicti trianguli trigonis ææquentur; eorum verò triangulorum æqualitas è sola congruentia demonstratur in elementis. Unde consequenter Archimedes circuli cum triangulo (sibi quantumvis dissimili) congruentiam demonstravit. Ita congruentiæ nihil obstat figurarum dissimilitudo; verùm seu similes sive dissimiles sint, modò æquales, semper poterunt, semper posse debebunt congruere. Igitur octavum axioma vel nullo modo conversum valet, aut universaliter converti potest; nullo modo, si quæ isthic habetur congruentia designet actualem congruentiam; universim, si de potentiali tantùm accipiatur."-Lectiones Mathematicæ, Sect. V.

dam longa ratiociniorum serie, alia autem breviore via, ad primum omnium principium reducantur, et in illud resolvantur. Sic v. g., propositio 2 + 2 = 4 statim huc cedit 1 + 1 + 1 + 1 = 1 + 1 + 1 + 1; i. e. idem est idem; et proprie loquendo, hoc modo enunciari debet. Si contingat, adesse vel existere quatuor entia, tum existunt quatuor entia; nam de existentia non agunt geometræ, sed ea hypothetice tantum subintelligitur. Inde summa oritur certitudo ratiocinia perspicienti; observat nempe idearum identitatem; et hæc est evidentia assensum immediate cogens, quam mathematicam aut geometricam vocamus. Mathesi tamen sua natura priva non est et propria; oritur etenim ex identitatis perceptione, quæ locum habere potest, etiamsi ideæ non repræsentent extensum."*

=

a,

With respect to this passage I have only to remark, (and the same thing is observable of every other attempt which has been made to support the opinion in question,) that the author confounds two things essentially different;-the nature of the truths which are the objects of a science, and the nature of the evidence by which these truths are established. Granting, for the sake of argument, that all mathematical propositions may be represented by the formula a it would not therefore follow, that every step of the reasoning leading to these conclusions, was a proposition of the same nature; and that, to feel the full force of a mathematical demonstration, it is sufficient to be convinced of this maxim, that every thing may be truly predicated of itself; or, in plain English, that the same is the same A paper written in cipher, and the interpretation. of that paper by a skilful decipherer, may, in like manner, be considered as, to all intents and purposes, one and the same thing. They are so, in fact, just as much as one side of an algebraical equation is the same thing with the other. But does it therefore follow, that the whole evidence upon which the art of deciphering proceeds, resolves into the perception of identity?

It may be fairly questioned, too, whether it can, with strict correctness, be said even of the simple arithmetical equation 2 + 2 = 4, that it may be represented by the formula a = a. The one is

a proposition asserting the equivalence of two different expressions ;-to ascertain which equivalence may, in numberless cases, be an object of the highest importance. The other is altogether unmeaning and nugatory, and cannot, by any possible supposition, admit of the slightest application of a practical nature. What opinion then shall we form of the proposition a a, when considered as the representative of such a formula as the binomial theorem of Sir Isaac Newton? When applied to the equation 2 + 2 = 4,

*The above extract (from a dissertation printed at Berlin in 1764) has long had a very extensive circulation in this country, in consequence of its being quoted by Dr. Beattie, in his Essay on Truth, (see p. 221, 2nd edit.) As the learned author of the essay has not given the slightest intimation of his own opinion on the subject, the doctrine in question has, I suspect, been considered as in some measure sanctioned by his authority. It is only in this way that I can account for the facility with which it has been admitted by so many of our northern logicians,