Page:Mind (New Series) Volume 8.djvu/536

 522 BRUCE MCEWEN analytical proposition. And the illustration given is the use of an angle, which being " common to two triangles," often helps us to prove them equal in every respect. Now in Euclid the first case of such procedure occurs in the proof of the fifth proposition, where we have the following argu- ment. (We presuppose Euclid's construction.) In the triangles FAC, GAB, FA = GA, and AC = AB, and the angle A is common to the two triangles, therefore they are equal in every respect. Here the conclusion follows directly from the fourth proposition which proves the equality of two triangles by superposition. But it must be noticed that, when we super- pose the triangle FAC upon the triangle GAB, we must place the angle BAG in the entirety new position CAB. In fact we must invert the angle or, putting it with empirical crudeness, we must turn the paper over, and then we see that the angle is not changed. I should distinctly deny it to be an analytic proposition, following from the law of Contradiction, that the angle A is equal to itself inverted in this way. 1 That A = A inverted is a proposition eminently synthetic but in Geometry undoubtedly true. While no geometrician would ever think of denying the Law of Identity, yet I would assure any of Kant's critics that an instance of an identical proposi- tion openly formulated in Mathematics cannot be found. In his Doctrine of Method Kant lays down that "the ground-evidence of Mathematics depends upon Definitions, Axioms and Demonstrations " ; and we have traced the gradual recognition by Kant of the synthetic and intuitive nature of all three of these ' moments ' of Mathematics. It is true that Kant appears rather to neglect the Arithmetical part of Mathematics, preferring to discuss the nature of 1 A clear apprehension of this fact would go far towards removing the objection commonly felt by mathematical purists to the simple proof of the fifth Proposition which is obtained by inverting the triangle BAG and superposing it upon itself without any further construction.