On the Measure of Solid Angles

One would think that such a nice formula should be old and well-known. But so far, I have not been able to find it in the literature. However, almost the same result was known to Euler [4, p. 215] and Lagrange [8, p. 340]. (The title above is a translation of the Latin title of [4].) We use the standard notations a, b, c for the sides of our spherical triangle T, and A, B, C for the angles. The area of T is its excess E = A + B + C rr. The old result can be written