Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/63

The general investigations developed in this treatise will, in the conclusion, be applied to the theory of triangles of shortest lines, of which we shall introduce only a couple of important theorems. If ${\textstyle a,}$ ${\textstyle b,}$ ${\textstyle c}$ be the sides of such a triangle (they will be regarded as magnitudes of the first order); ${\textstyle A,}$ ${\textstyle B,}$ ${\textstyle C}$ the angles opposite; ${\textstyle \alpha ,}$ ${\textstyle \beta ,}$ ${\textstyle \gamma }$ the measures of curvature at the angular points; ${\textstyle \sigma }$ the area of the triangle, then, to magnitudes of the fourth order, ${\textstyle {\frac {1}{3}}(\alpha +\beta +\gamma )\sigma }$ is the excess of the sum ${\textstyle A+B+C}$ over two right angles. Further, with the same degree of exactness, the angles of a plane rectilinear triangle whose sides are ${\textstyle a,}$ ${\textstyle b,}$ ${\textstyle c,}$ are respectively

$${\begin{aligned}A&-{\tfrac {1}{12}}(2\alpha +\beta +\gamma )\sigma \\B&-{\tfrac {1}{12}}(\alpha +2\beta +\gamma )\sigma \\C&-{\tfrac {1}{12}}(\alpha +\beta +2\gamma )\sigma .\end{aligned}}$$

We see immediately that this last theorem is a generalization of the familiar theorem first established by Legendre. By means of this theorem we obtain the angles of a plane triangle, correct to magnitudes of the fourth order, if we diminish each angle of the corresponding spherical triangle by one-third of the spherical excess. In the case of non-spherical surfaces, we must apply unequal reductions to the angles, and this inequality, generally speaking, is a magnitude of the third order. However, even if the whole surface differs only a little from the spherical form, it will still involve also a factor denoting the degree of the deviation from the spherical form. It is unquestionably important for the higher geodesy that we be able to calculate the inequalities of those reductions and thereby obtain the thorough conviction that, for all measurable triangles on the surface of the earth, they are to be regarded as quite insensible. So it is, for example, in the case of the greatest triangle of the triangulation carried out by the author. The greatest side of this triangle is almost fifteen geographical^[1] miles, and the excess of the sum of its three angles over two right angles amounts almost to fifteen seconds. The three reductions of the angles of the plane triangle are ${\textstyle 4''.95113,}$ ${\textstyle 4''.95104,}$ ${\textstyle 4''.95131.}$ Besides, the author also developed the missing terms of the fourth order in the above expressions. Those for the sphere possess a very simple form. However, in the case of measurable triangles upon the earth’s surface, they are quite insensible. And in the example here introduced they would have diminished the first reduction by only two units in the fifth decimal place and increased the third by the same amount.

1. This German geographical mile is four minutes of arc at the equator, namely, ${\textstyle 7.42}$ kilometers, and is equal to about ${\textstyle 4.6}$ English statute miles. [Translators.]