Page:On the Fourfold Root, and On the Will in Nature.djvu/195

 of the other, each to each, to which the equal sides are opposite ; therefore the angle b a e is equal to the angle e c f. But the angle e c d is greater than the angle e c f Therefore the angle a c d is greater than the angle a b c."

"In the same manner, if the side b c be bisected, and the side a c be produced to g, it may be demonstrated that the angle b c g, that is, the opposite vertical angle a c d is greater than the angle a b e."

My demonstration of the same proposition would be as follows (see fig. 5) :



For the angle b a c to be even equal to, let alone greater than, the angle a c d, the line b a toward c a would have to lie in the same direction as b d (for this is precisely what is meant by equality of the angles), i.e., it must be parallel with b d ; that is to say, b a and b d must never meet ; but in order to form a triangle they must meet (reason of being), and must thus do the contrary of that which would be required for the angle b a c to be of the same size as the angle a c d.

For the angle a b c to be even equal to, let alone greater than, the angle a c d, line b a must lie in the same direction towards b d as a c (for this is what is meant by equality of the angles), i.e., it must be parallel with a c, that is to say, b a and a c must never meet ; but in order to form a triangle b a and a c must meet and must thus do the contrary of that which would be required for the angle a b c to be of the same size as a c d.

By all this I do not mean to suggest the introduction of