Page:On the Fourfold Root, and On the Will in Nature.djvu/194

 of such a reason of being for each relation of Space, as we are of the necessity of a cause for each change. In complicated theorems it will, of course, be very difficult to show that reason of being ; and this is not the place for difficult geometrical researches. Therefore, to make my meaning some what clearer, I will now try to bring back to its reason of being a moderately complicated proposition, in which nevertheless that reason is not immediately evident. Passing over the intermediate theorems, I take the 16th :

"In every triangle in which one side has been produced, the exterior angle is greater than either of the interior opposite angles."



This Euclid demonstrates in the following manner (see fig. 4):—

"Let a b c be a triangle ; and let the side b c be produced to d; then the exterior angle a c d shall be greater than either of the interior opposite angles b a c or c b a. Bisect the side a c at e, and join b e ; produce b e to f, making e f equal to e b, and join f c. Produce a c to g. Because a e is equal to e c, and b e to e f ; the two sides a e, e b, are equal to the two sides c e, e f, each to each ; and the angle a e b is equal to the angle c e f, because they are opposite vertical angles ; therefore the base a b is equal to the base e f, and the triangle a e b is equal to the triangle c e f, and the remaining angles of one triangle to the remaining angles