Page:Spherical Trigonometry (1914).djvu/30

Two triangles, $ABC,A''BC$, which have a side $BC$ common, and whose other sides belong to the same great circles, are called colunar triangles, as they together make up a lune. $A''$ is the point diametrically opposite to $A$ on the sphere.

If $A'',B'',C''$ be diametrically opposite to $A$, $B$, $C$ respectively, the triangle $ABC$ has three colunar triangles, namely, $A''BC$, $B''CA$, and $C''AB$.

Antipodal triangles are triangles whose respective vertices are diametrically opposite to one another in pairs; such, for example, are the triangles $ABC$, $A''B''C''$.

25. Polar triangle.^[1] Let $ABC$ be any spherical triangle, and let the points $A'$, $B'$, $C'$ be those poles of the arcs $BC$, $CA$, $AB$ respectively which lie on the same sides of them as the opposite angles $A$, $B$, $C$; then the triangle $A'B'C'$ is said to be the polar triangle of the triangle $ABC$.

Since there are two poles for each side of a spherical triangle, eight triangles can be formed having for their angular points poles of the sides of the given triangle; but there is only one triangle in which these poles $A'$, $B'$, $C'$ lie towards the same parts with the corresponding angles $A$, $B$, $C$; and this is the triangle which is known under the name of the polar triangle.

1. The discovery of the polar triangle is due to Snellius. Its use is explained in his Trigonometria, (Lib. III, Prop. VIII), published at Leyden in 1627.