Page:On the Fourfold Root, and On the Will in Nature.djvu/192

 certain physical theories, which present the phenomenon without being able to indicate its cause: for instance, Leidenfrost's experiment, inasmuch as it succeeds also in a platina crucible ; whereas the reason of being of a geometrical proposition which is discovered by intuition, like every knowledge we acquire, produces satisfaction. When once the reason of being is found, we base our conviction of the truth of the theorem upon that reason alone, and no longer upon the reason of knowing given us by the demonstration. Let us, for instance, take the sixth proposition of the first Book of Euclid:—

"If two angles of a triangle are equal, the sides also which subtend, or are opposite to, the equal angles shall be equal to one another." (See fig. 3.)

Which Euclid demonstrates as follows :— "Let a b c be a triangle having the angle a b c equal to the angle a c b, then the side a c must be equal to the side a b also.

"For, if side a b be not equal to side a c, one of them is greater than the other. Let a b be greater than a c ; and from b a cut off b d equal to c a, and draw d c. Then, in the triangles d b c, a b c, because d b is equal to a c, and b c is common to both triangles, the two sides d b and b c are equal to the two sides a c, a b, each to each ; and the angle d b c is equal to the angle a c b, therefore the base d c is equal to the base a b, and the triangle d b c is equal to the