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."
![](http://upload.wikimedia.org/wikipedia/commons/thumb/f/f9/4Fold_Root_Figure_4m.gif/300px-4Fold_Root_Figure_4m.gif)
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