Page:Spherical Trigonometry (1914).djvu/27

§19] Let $$AB$$, $$BC$$, $$CA$$ be the arcs of great circles in which the planes cut the sphere; then $$ABC$$ is a spherical triangle, and the arcs $$AB$$, $$BC$$, $$CA$$ are its sides. Suppose $$Ab$$ the tangent at $$A$$ to the arc $$AB$$, and $$Ac$$ the tangent at $$A$$ to the arc $$AC$$, the tangents



being drawn from $$A$$ towards $$B$$ and $$C$$ respectively; then the angle $$bAc$$ is one of the angles of the spherical triangle. Similarly angles formed in like manner at $$B$$ and $$C$$ are the other angles of the spherical triangle.

19. The principal part of a treatise on Spherical Trigonometry consists of theorems relating to spherical triangles; it is therefore necessary to obtain an accurate conception of a spherical triangle and its parts.

It will be seen that what are called sides of a spherical triangle are really arcs of great circles, and these arcs are proportional to the three plane angles which form the solid angle corresponding to the spherical triangle. Thus, in the figure of the preceding Article, the arc $$AB$$ forms one side of the spherical triangle $$ABC$$, and the plane angle $$AOB$$ is measured by the fraction $$\dfrac{\mathrm{arc\ } AB} {\mathrm{radius\ } OA}$$; and thus the arc $$AB$$ is proportional to the angle $$AOB$$ so long as we keep to the same sphere.

And from the proposition proved in Article 9 it follows that the angles of a spherical triangle are the inclinations of the plane faces which form the solid angle.