Three stations, projected on the surface of the sea, give a spherical or spheroidal triangle according to the adoption of the sphere or the ellipsoid as the form of the surface. A spheroidal triangle differs from a spherical triangle, not only in that the curvatures of the sides are different one from another, but more especially in this that, while in the spherical triangle the normals to the surface at the angular points meet at the centre of the sphere, in the spheroidal triangle the normals at the angles A, B, C meet the axis of revolution of the spheroid in three different points, which we may designate [alpha], [beta], [gamma] respectively. Now the angle A of the triangle as measured by a theodolite is the inclination of the planes BA[alpha] and CA[alpha], and the angle at B is that contained by the planes AB[beta] and CB[beta]. But the planes AB[alpha] and AB[beta] containing the line AB in common cut the surface in two distinct plane curves. In order, therefore, that a spheroidal triangle may be exactly defined, it is necessary that the nature of the lines joining the three vertices be stated. In a mathematical point of view the most natural definition is that the sides be geodetic or shortest lines. C.C.G. Andrae, of Copenhagen, has also shown that other lines give a less convenient computation. K.F. Gauss, in his treatise, _Disquisitiones generales circa superficies curvas_, entered fully into the subject of geodetic (or geodesic) triangles, and investigated expressions for the angles of a geodetic triangle whose sides are given, not certainly finite expressions, but approximations inclusive of small quantities of the fourth order, the side of the triangle or its ratio to the radius of the nearly spherical surface being a small quantity of the first order. The terms of the fourth order, as given by Gauss for any surface in general, are very complicated even when the surface is a spheroid. If we retain small quantities of the second order only, and put [A], [B], [C] for the angles of the geodetic triangle, while A, B, C are those of a plane triangle having sides equal respectively to those of the geodetic triangle, then, [sigma] being the area of the plane triangle and [a], [b], [c] the measures of curvature at the angular points, [A] = A + [sigma](2[a] + [b] + [c])/12, [B] = B + [sigma]([a] + 2[b] + [c])/12, [C] = C + [sigma]([a] + [b] + 2[c])/12. For the sphere [a] = [b] = [r], and making this simplification, we obtain the theorem previously given by A.M. Legendre. With the terms of the fourth order, we have (after Andrae): [epsilon] [sigma] /m^2 - a^2 [a] - k \ [A] - A = --------- + -------k ( ---------k + ------- ), 3 3 \ 20 4k /