II.

Continuation 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 géomètre 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."

As this account of mathematical evidence appears to me 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.†

It is more than probable, that this theory was suggested to Leibnitz by some very curious observations in Aristotle's Metaphysics, Book IV. chap. iii. and iv.

I must here observe, in justice to my friend M. Prevost, that the two doctrines which I have represented in the above paragraph as quite irreconcilable, seem to be regarded by him as not only consistent with each other, but as little more than different modes of stating the same proposition. The remarks with which he has favored me on this point will be found in the Appendix annexed to this volume. At present, it may suffice to mention, that none of the following reasonings apply to that particular view of the question which he has taken. Indeed, I consider the difference of opinion between us, as to the subject now under consideration, as chiefly verbal. On the subject of the preceding article, our opinions are exactly the same. See Appendix.

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

* 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 : Ira ŵv rò woròn "Those things are equal whose quantity is the same;' (Met. iv. c. 16.) and still more precisely in these remarkable words, iv roúrois à idórns ivórns; " In mathematical quantities, equality is identity." (Met. x. c. 3.)

For some remarks on this last passage, See Note (F.)

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 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 hypothenuse. In truth, this mode of comparison by a superposition, actual or ideal, is the only test of equality which it is possi

ble 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 rigor, the definition of geometrical equality should have been taken.* The subject is discussed at great length, and with much 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 a =a? 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 propo

*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.

sition, 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 formulâ, a= a. Sunt veritates identicæ, sub variâ formâ 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 quædam longâ ratiociniorum serie, alia autem breviore viâ, ad primum omnium principium reducantur, et in illud resolvantur. Sic v. g. propositio 2+24 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 existentiâ non agunt geometræ, sed ea hypothetice tantum subintelligitur. Inde summa oritur certitudo ratiocinia

"Cum demonstravit Archimedes circulum æquari rectangulo triangulo cujus basis radio circuli, cathetus peripheriæ exæquetur, nil ille, siquis propius attendat, aliud quicquam quam aream circuli ceu polygoni regularis indefinite multa latera habentis, in tot dividi posse minutissima triangula, quæ totidem exilissimis dicti trianguli trigonis æquentur; eorum verò triangulorum æqualitas e solà congruentiâ 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 Mathematica, Lect. V.