predicate to the subject, according to the principle of contradiction. That axiom, however, is a synthetical proposition a priori, and as such has the guarantee of pure, not empirical, perception, which is just as immediate and certain as the principle of contradiction itself, from which all demonstrations first derive their certainty. Ultimately this holds good of every geometrical theorem, and it is quite arbitrary where we draw the line between what is directly certain and what has first to be demonstrated. It surprises me that the eighth axiom is not rather attacked. "Figures which coincide with each other are equal to each other." For "coinciding with each other" is either a mere tautology or something purely empirical which does not belong to pure perception but to external sensuous experience. It presupposes that the figures may be moved; but only matter is movable in space. Therefore this appeal to coincidence leaves pure space – the one element of geometry – in order to pass over to what is material and empirical.
The reputed motto of the Platonic lecture-room, "(
Greek characters)," of which mathematicians are so proud, was no doubt inspired by the fact that Plato regarded the geometrical figures as intermediate existences between the eternal Ideas and particular things, as Aristotle frequently mentions in his "Metaphysics" (especially i. c. 6, p. 887, 998, et Scholia, p. 827, ed. Berol.) Moreover, the opposition between those self-existent eternal forms, or Ideas, and the transitory individual things, was most easily made comprehensible in geometrical figures, and thereby laid the foundation of the doctrine of Ideas, which is the central point of the philosophy of Plato, and indeed his only serious and decided theoretical dogma. In expounding it, therefore, he started from geometry. In the same sense we are told that he regarded geometry as a preliminary exercise through which the mind of the pupil accustomed itself to deal with incorporeal objects, having hitherto in practical life had only to
