KANT'S PROOF OF THE PROPOSITION, ETC. 509 and in his private notes (Lose Blatter), which are full of mathematical calculations upon knotty points, that occupied his attention from moment to moment. And yet there is not a single instance in all Kant's writings of his using the Cartesian method in geometry. He repeatedly praises it as being " rich in discovery " (erfindungsreich), he quotes its formulae with a facility which betokens long acquaintance with them, yet in his own investigations he altogether dis- dains its aid. And Kant was not content to limit his mathematical speculations to the pure geometrical method without giving good reasons for his preference. In Dynamics Newton had often gone considerably out of his way to preserve the purity of his procedure ; his Principia explain the motion of a body, moving, under certain forces, by actually following out its path on paper from point to point before the eyes of the reader. Thus by strictly geometrical methods the orbits of the moving masses are depicted in concrete, and when the process is finished the nature of these orbits is intuitively deducible from the nature of their construction. In direct contrast with this somewhat slow and laborious process the Leibnitian method simply writes down a symbolical expres- sion for the forces acting on a mass, and by a single swift stroke (Integration) another is deduced, which is the perfect algebraic expression for the curve described. So far, the analysis appears to have been concerned with mere signs in the abstract, but a serious difficulty now presents itself, when we try to interpret our results. We work with symbols but these have no signification, except indeed as interpreted, and for this interpretation the analytical method must ulti- mately call in the aid of pure geometry, for every equation used depends absolutely for its meaning upon the propositions of the Euclideans. If we use an example quoted by Kant himself we might suppose the formula, ax = y' 2 , to be the result of an algebraic analysis of the data in some problem or other : now we know that this represents a certain curve described, but it is only by a knowledge of the pure geo- metrical properties of Conic Sections that we are enabled to identify it with a parabola. It is a distinct merit of Kant's, though, I believe, he has never received the full credit of it, that he clearly perceived and as boldly proclaimed the incompleteness of the Cartesian method at the very time when it was generally considered to be an instrument of mathematical research far more powerful than the pure geometry, which it really involves, and from which it was originally derived. And it was this
