Euclid,

ElementsI 16©

translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 16: Upon one of the sides of any triangle being extended, the external angle is larger than each of the interior and opposite angles.

<(diagram 1) Let there be a triangle, ABG, and let one side of it, BG, be extended to D. I say that the exterior angle, that by AGD, is larger than each of the interior and opposite angles, the angles by GBA, BAG. (diagram 2) Let AG be bisected at E, (diagram 3) and let BE, being joined, be extended on a straight-line to Z, and let an equal, EZ, to BE be positioned, (diagram 4) and let ZG be joined, and let AG be drawn through to H. (diagram 5) And so since AE is equal to EG and BE to EZ, two, in fact, AE, EB, are respectively equal to two, GE, EZ. And an angle, that by AEB, is equal to an angle, that by ZEG. For it is at a vertex. (diagram 6) Therefore, a base, AB, is equal to a base, ZG, and triangle ABE is equal to triangle ZEG, and the remaining angles are respectively equal to the remaining angles, which the equal sides subtend. (diagram 7) Therefore, the angle by BAE equal to that by EGZ. But the angle by EGD is larger than that by EGZ. Therefore, the angle by AGD is larger than that by BAE. (diagram 8) Similarly, in fact, upon BG having been bisected, the angle by BGH, that is the angle by AGD, will be shown also larger than that by ABG. Therefore, upon one of the sides of any triangle being extended, the external angle is larger than each of the interior and opposite angles, just what it was required to show.

Note: apart from the Parallel Postulat (Post. 5), this theorem presupposes a different parallel postulate, namely that ZG is distinct from BG. Thus, spherical surfaces are implicitly ruled out. This is obvious, however, since the theorem would be false in a spherical space in any case. It is important to realize that the Parallel Postulate only rules out one form of non-Euclidean geometry, hyperbolic surfaces. It goes without saying that the geometry of the surfaces of spheres is an important part of Greek mathematics (cf. Theodosius). So it is striking that nothing in the *Elements* explicitly rules out a spherical geometry.