Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/118

therefore their sum is

$$180^{\circ }+LL'_{1}L'L.$$

The form of the proof will require some modification and explanation, if the point ${\textstyle (3)}$ falls within the triangle. But, in general, we conclude

"The sum of the three angles of a triangle, which is formed of shortest lines upon an arbitrary curved surface, is equal to the sum of ${\textstyle 180^{\circ }}$ and the area of the triangle upon the auxiliary sphere, the boundary of which is formed by the points ${\textstyle L,}$ corresponding to the points in the boundary of the original triangle, and in such a manner that the area of the triangle may be regarded as positive or negative according as it is inclosed by its boundary in the same sense as the original figure or the contrary."

Wherefore we easily conclude also that the sum of all the angles of a polygon of ${\textstyle n}$ sides, which are shortest lines upon the curved surface, is [equal to] the sum of ${\textstyle (n-2)180^{\circ }+}$ the area of the polygon upon the sphere etc.

16.

If one curved surface can be completely developed upon another surface, then all lines upon the first surface will evidently retain their magnitudes after the development upon the other surface; likewise the angles which are formed by the intersection of two lines. Evidently, therefore, such lines also as are shortest lines upon one surface remain shortest lines after the development. Whence, if to any arbitrary polygon formed of shortest lines, while it is upon the first surface, there corresponds the figure of the zeniths upon the auxiliary sphere, the area of which is ${\textstyle A,}$ and if, on the other hand, there corresponds to the same polygon, after its development upon another surface, a figure of the zeniths upon the auxiliary sphere, the area of which is ${\textstyle A',}$ it follows at once that in every case

$$A=A'.$$

Although this proof originally presupposes the boundaries of the figures to be shortest lines, still it is easily seen that it holds generally, whatever the boundary may be. For, in fact, if the theorem is independent of the number of sides, nothing will prevent us from imagining for every polygon, of which some or all of its sides are not shortest lines, another of infinitely many sides all of which are shortest lines.

Further, it is clear that every figure retains also its area after the transformation by development.