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):-- [Illustration: _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 _c 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 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 c_." My demonstration of the same proposition would be as follows (see fig. 5):-- [Illustration: _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_.