Page:Mind (New Series) Volume 8.djvu/354

340 BOBEKT LATTA: The reasoning is grounded on a more or less blind appeal to a system or systems that are presupposed without being thoroughly thought out. A considerable advance upon the ancient methods was made by Kepler, who introduced the notion of infinity in connexion with the solution of geometrical problems, and by Descartes, who invented the analytical geometry or geo- metry of co-ordinates. 1 The introduction of the idea that a finite figure or a finite area is reducible to an infinite j^mbex_of_elements was an explicit recognition of the inade- quacyoFthe^&sclidean postulates as principles of demonstra- tion, and it was the beginning of a train of thought which led inevitably to the Infinitesimal Calculus ; but, as Pascal pointed out in defending Cavalieri, the geometrical method which proceeds upon the principle that the infinitely little may be neglected differs only in manner of expression from the method of exhaustions used in the Greek Mathematics. 2 Both are ultimately based on reductio ad absurdum. On the other hand, the general effect of the changes intro- duced by Descartes was (1) to make the relation between the system of space and that of quantity in general more clear and definite, by finding (in the co-ordinates) units of space-relation, and (2) to substitute for the empirical reference to space that is implied in the use of a ruler and compasses a method by which figures and their properties may be shown by calculation (without drawing or construction) to follow from the nature of space as extension in three or in two dimensions. The Cartesian method in geometry is thus more positive, direct and explicit than the method of the Greeks. Eliminating the postulates of Euclid, or rather going beneath them to the grounds on which they rest and thinking out what they imply, it gives a more perfect demon- stration of the propositions of Euclid and solves more complex problems than the Greeks could have attempted. Neverthe- less, while the Cartesian geometry was much more positive and thorough in its method of demonstration than was the synthetic geometry, it still retained the doctrine or hypothesis of limits in a negative form. It was (considering plane geometry alone) on the right lines towards a positive solution 1 For a full history v. Gerhardt, Die Entdeckung der h'ohern Analysis, p. 6 sqq., and Cohen, Das Princip der Infinitesimal-Methode, 35 sqq. 2 So Leibniz says in a letter to Varignon that the infinitesimal calculus " donne directement et visiblement, et d'une maniere propre k marquer la source de 1'invention, ce que les anciens, comnie Archimede, donnoient par circuit dans leur reductions ad absurdum " (Gerhardt, Math. Schriften, iv., 92).