Page:Mind (New Series) Volume 8.djvu/524

 510 BRUCE MCEWEN : clear insight into what is fundamental in Mathematics that enabled Kant, as every student knows, to confine the question of the nature of all geometry with the most systematic con- sistency within the limits of the Euclidean method. As early as 1763, in his treatise on the Principles of Natural Theology, Kant takes his stand definitely within these limits and announces the doctrine that "the definitions of Mathe- matics are entirely synthetic, while those of Philosophy are analytic ". " The conceptions, which I expound in Mathe- matics, are not given before the definition, but have their origin always from it. A ' cone ' may mean what it will, but, within the realm of Mathematics, it is formed by the arbitrary representation of the revolution of a right-angled triangle about one of its sides. Here and in all other cases the exposition finds its origin through Synthesis." The definitions of Philosophy on the other hand, Kant at present considers to be analytic in general, yet instances may be produced, he says, " from which we might draw the con- clusion that the expositions of philosophers may sometimes be synthetic and those of the mathematician analytic ". " Leibniz," he explains, "formed the conception of a simple substance, which had only confused representations, and called it a slumbering Monad. Here he has not expounded (erklart) this monad, but merely formed a conception of it (erdacht), for the conception was not given to him, but was first created by him." Kant then is quite willing to believe that philosophic definitions may be of two kinds but he sturdily denies that the same is true of Mathematics. " The expositions of the mathematicians," he says, " have often been analytic in their nature, I admit it, but yet there has always been an error involved. For example, Wolf con- sidered geometrical similarity from a philosophic point of view, so as to bring this particular case under the general conception of similarity." But Euclid's definition of simi- larity runs : ' Rectilinear figures are said to be similar, when they are equi-angular and have the sides about the equal angles proportional ' with a corresponding definition, if need be, for curvilinear figures, so that " for Geometry nothing depends on the general definition of similarity ". Here then we have Kant's first detailed account of the nature of mathematical definitions, and this account he never saw fit to alter, save perhaps by multiplying and improving his illustrations of its truth. Throughout the remainder of his writings it is always laid down that the mathematician in defining proceeds " by constructing his conceptions, and not by analysing them ". When this