ject, the problems of geometry are not less hypothetical and speculative (or, to adopt the phraseology of some late writers, not less objects of pure reason) than the theorems; the possibility of drawing a mathematical straight line, and of describing a mathematical circle, being assumed in the construction of every problem, in a way quite analogous to that in which the enunciation of a theorem assumes the existence of straight lines and of circles corresponding to their mathematical definitions. The reasoning, therefore, on which the solution of a problem rests, is not less demonstrative than that which is employed in proof of a theorem. Grant the possibility of the three operations described in the postulates, and the correctness of the solution is as mathematically certain, as the truth of any property of the triangle or of the circle. The three postulates of Euclid are, indeed, nothing more than the definitions of a circle and 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 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.*

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.

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)

"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 Simpson, M. D. By the Rev. William Trail, LL. D. Published by G. and W. Nicol, London, 1812.

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."

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 immedi

* 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.

a wr

This idea is very explicitly stated, more than once, by Aristotle: To modov iv. "Those things are equal whose quantity is the same;" (Met. iv. c. 15.) and still more precisely in these remarkable words, v T8T015 À towns ivorus; “ In mathematical quantities, equality is identity.” (Met. x.

c. 3.

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

ately 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