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KANT'S PROOF OF THE PROPOSITION, ETC. 517 naturally enough, though perhaps rashly, that such axioms really were general principles applicable not only in Geometry and Arithmetic, but also, as we have already remarked, in all human reasoning, and therefore we might conclude that their proof was not completely attained if we attempted it mathematically. We might decide, and, I firmly believe, Kant's opinion in 1781 would support our decision, that Mathematics is here borrowing general principles proved analytically by Philosophy and applying them for her own particular purposes, not as axioms, for, as Kant states in his Logic ( 36), " analytical propositions never are axioms (axiomata) but depend upon the identity of conceptions (ac- roamata) ". But, as a matter of fact, if any one should try upon Kantian principles to defend Kant's statement, that we have no " axioms " de quantitate, we could at once deny this to be true within the sphere of Mathematics, by pro- ducing a positive instance of such an axiom, an instance which, so far as I see, admits of no doubt, namely the eighth axiom of Euclid : " Magnitudes which can be made to coin- cide are equal ". Its very phraseology indicates that we are to superpose two magnitudes in pure intuition in order to see their equality ; it certainly is an axiom de quantitate (I should say it is the fundamental axiom of quantity in Geo- metry), and no analysis of conceptions could ever produce it. Now I do not know what the test for Equality in general may be, I have serious doubt whether any such exist, but in Geometry I have a truly geometrical and spatial test of geometrical equality in the coincidence of superposed figures. And if a pure spatial intuition of the coincidence of two figures be the only means by which we can know them to be equal, then all axioms as to geometrical equality directly involve an intuition of this nature and therefore are like the eighth axiom synthetic. Indeed, whenever we compare any two geometrical figures, whether with regard to equality or inequality, we must in the end resort to superposition and an intuitive perception of their relation to each other, and for this reason I consider all axioms de quantitate to be per- fectly synthetic. But Kant's subsequent silence on this point makes it for ever impossible to discover whether such considerations had any real influence upon him or not. True, the passage we have opposed was neither obliterated nor altered in the second edition of the Critique, and yet I think there is in that work quite sufficient indication that Kant was well aware of the deficiencies of his earlier position. In the fifth paragraph of the Introduction, which appeared for the first