Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/102

[and] also, if ${\textstyle L'}$ is any other point on the sphere,

$$\cos(1)L.\cos(1)L'+\cos(2)L.\cos(2)L'+\cos(3)L.\cos(3)L'=\cos LL'.$$

We shall add here another theorem, which has appeared nowhere else, as far as we know, and which can often be used with advantage.

Let ${\textstyle L,}$ ${\textstyle L',}$ ${\textstyle L'',}$ ${\textstyle L'''}$ be four points on the sphere, and ${\textstyle A}$ the angle which ${\textstyle LL'''}$ and ${\textstyle L'L''}$ make at their point of intersection. [Then we have]

$$\cos LL'.\cos L''L'''-\cos LL''.\cos L'L'''=\sin LL'''.\sin L'L''.\cos A.$$

The proof is easily obtained in the following way. Let

$$AL=t,\qquad AL'=t',\qquad AL''=t'',\qquad AL'''=t''';$$

we have then

$${\begin{alignedat}{6}&\cos LL'&&=\cos t&&\cos t'&&+\sin t&&\sin t'&&\cos A,\\&\cos L''L'''&&=\cos t''&&\cos t'''&&+\sin t''&&\sin t'''&&\cos A,\\&\cos LL''&&=\cos t&&\cos t''&&+\sin t&&\sin t''&&\cos A,\\&\cos L'L'''&&=\cos t'&&\cos t'''&&+\sin t'&&\sin t'''&&\cos A.\end{alignedat}}$$

Therefore

$${\begin{alignedat}{1}\cos LL'&\cos L''L'''-\cos LL''\cos L'L''\\&=\cos A\{\cos t\cos t'\sin t''\sin t'''+\cos t''\cos t'''\sin t\sin t'\\&\qquad \qquad -\cos t\cos t''\sin t'\sin t'''-\cos t'\cos t'''\sin t\sin t''\}\\&=\cos A(\cos t\sin t'''-\cos t'''\sin t)(\cos t'\sin t''-\cos t''\sin t')\\&=\cos A\sin(t'''-t)\sin(t''-t')\\&=\cos A\sin LL'''\sin L'L''.\end{alignedat}}$$

Since each of the two great circles goes out from ${\textstyle A}$ in two opposite directions, two supplementary angles are formed at this point. But it is seen from our analysis that those branches must be chosen, which go in the same sense from ${\textstyle L}$ toward ${\textstyle L'''}$ and from ${\textstyle L'}$ toward ${\textstyle L''.}$

Instead of the angle ${\textstyle A,}$ we can take also the distance of the pole of the great circle ${\textstyle LL'''}$ from the pole of the great circle ${\textstyle L'L''.}$ However, since every great circle has two poles, we see that we must join those about which the great circles run in the same sense from ${\textstyle L}$ toward ${\textstyle L'''}$ and from ${\textstyle L'}$ toward ${\textstyle L'',}$ respectively.

The development of the special case, where one or both of the arcs ${\textstyle LL'''}$ and ${\textstyle L'L''}$ are ${\textstyle 90^{\circ },}$ we leave to the reader.

6) Another useful theorem is obtained from the following analysis. Let ${\textstyle L,}$ ${\textstyle L',}$ ${\textstyle L''}$ be three points upon the sphere and put