history, and of other sciences which relate to facts, cannot, in one single instance, apply to the definitions of geometry; inasmuch as these definitions are neither the result of observations nor of comparisons, but the hypotheses, or first principles, on which the whole sci

ence rests.

If the foregoing account of demonstrative evidence be just, it follows, that no chain of reasoning whatever can deserve the name of a demonstration (at least in the mathematical sense of that word) which is not ultimately resolvable into hypotheses or definitions.* It has been already shown, that this is the case with geometry; and it is also manifestly the case with arithmetic, another science to which, in common with geometry, we apply the word mathematical. The simple arithmetical equations 2+2= 4; 2+3= 5, and other elementary propositions of the same sort, are (as was formerly observed) mere definitions; † perfectly analogous, in this respect, to those at the beginning of Euclid; and it is from a few fundamental principles of this sort, or at least from principles which are essentially of the same description, that all the more complicated results. * in the science are derived.

To this general conclusion, with respect to the nature of mathematical demonstration, an exception may perhaps be, at first sight, apprehended to occur, in our reasonings concerning geometrical problems: all of these reasonings (as is well known) resting ultimately upon a particular class of principles called postulates, which are commonly understood to be so very nearly akin to axioms, that both might, without impropriety, be

Although the account given by Locke of what constitutes a demonstration be different from that which I have here proposed, he admits the converse of this doctrine as manifest; viz. That if we reason accurately from our own definitions, our conclusions will possess demonstrative evidence; and "hence," he observes with great truth, "it comes to pass, that one may often meet with very clear and coherent discourses, that amount yet to nothing." He afterwards remarks, that "one may make demonstrations and undoubted propositions in words, and yet thereby advance not one jot in the knowledge of the truth of things." "Of this sort," he adds, "a man may find an infinite number of propositions, reasonings, and conclusions, in books of metaphysics, school-divinity, and some sort of natural philosophy; and, after all, know as little of God, spirits, or bodies, as he did before he set out."Essay on Human Understanding, Book IV. chap. viii.

† See page 22.

comprehended under the same name.

"The defini

tion of a postulate," says the learned and ingenious Dr. Hutton, "will nearly agree also to an axiom, which is a self-evident theorem, as a postulate is a self-evident problem." * The same author, in another part of his work, quotes a remark from Dr. Barrow, that "there is the same affinity between postulates and problems, as between axioms and theorems." + Dr. Wallis, too, appears, from the following passage, to have had a decided leaning to this opinion: "According to some, the difference between axioms and postulates is analogous to that between theorems and problems; the former expressing truths which are self-evident, and from which other propositions may be deduced; the latter, operations which may be easily performed, and by the help of which more difficult constructions may be effected." He afterwards adds, "This account of the distinction between postulates and axioms seems not ill-adapted to the division of mathematical propositions into problems and theorems. And, indeed, if both postulates and axioms were to be comprehended under either of these names, the innovation would not, in my opinion, afford much ground for censure."

In opposition to these very high authorities, I have no hesitation to assert, that it is with the definitions of Euclid, and not with the axioms, that the postulates ought to be compared, in respect of their logical character and importance ;-inasmuch as all the demonstrations in plane geometry are ultimately founded on the former, and all the constructions which it recognises as legitimate, may be resolved ultimately into the latter. To this remark it may be added, that, according to Euclid's view of the subject, the problems of geometry are not less hypothetical and speculative 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

* Mathematical Dictionary, Art. Postulate. † Ibid. Art. Hypothesis.

Wallisii Opera. Vol. II. pp. 667, 668.

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.

"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.

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: tra av rò moròv "Those things are equal whose quantity is the same;" (Met. iv. c. 16.) and still more precisely in these remarkable words, v roúrois ǹ ¡œórns ivórns; "In mathematical quantities, equality is identity." (Met. x. c. 3.)

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