Page:Spherical Trigonometry (1914).djvu/36

37. If one angle of a spherical triangle be greater than another, the side opposite the greater angle is greater than the side opposite the less angle.

Let $ABC$ be a spherical triangle, and let the angle $ABC$ be greater than the angle $BAC$: then the side $AC$ will be greater than the side $BC$. At $B$ make the angle $ABD$ equal to the angle $BAD$; then $BD$ is equal to $AD$ (Art. 34), and $BD+DC$ is greater than $BC$ (Art. 29); therefore $AD+DC$ is greater than $BC$; that is, $AC$ is greater than $BC$.

38. If one side of a spherical triangle be greater than another, the angle opposite the greater side is greater than the angle opposite the less side.

This follows from the preceding Article by means of the polar triangle.

Or thus; suppose the side $AC$ greater than the side $BC$, then the angle $ABC$ will be greater than the angle $BAC$. For the angle $ABC$ cannot be less than the angle $BAC$ by Art. 37, and the angle $ABC$ cannot be equal to the angle $BAC$ by Art. 36; therefore the angle $ABC$ must be greater than the angle $BAC$.

This Chapter might be extended; but it is unnecessary to do so because the Trigonometrical formulae of the next Chapter supply an easy method of investigating the theorems of Spherical Geometry. See, for example, Arts. 67, and 68.

39. Note.―The foundation of the science of Spherical Trigonometry is attributed to the astronomer Hipparchus (150 B.C.). Fundamental theorems of the subject are found in the Sphaerica of Menelaus and in