On the measure of solid angles
Euler notes that plane angles may be measured by the circular arcs that they subtend. He also notes that a very clever geometer, Albert Girard (1595-1632), suggested measuring solid angles in the same way, by the part of a sphere that they subtend. He also defines the "area of an angle" of a sphere to be the area of the wedge-shaped region bounded by semicircles intersecting at the given angle: if the angle in radians measures a, then the area is 2ar2. If the radius is 1, as we will assume from now on, then the area is 2a. Euler then states and proves (with attribution) Girard's theorem: the area of a spherical triangle is always equal to the angle by which the sum of all three angles of the triangle exceeds two right angles. This gives the area of the triangle in terms of the angles. He then states his general problem: to give the area of the triangle in terms of the lengths of the sides. Euler manages to find a formula, which he follows with lots of examples. Then he derives a couple of rules for finding the measure of solid angles and wraps it up by measuring the solid angles of the regular solids.
Original Source Citation
Acta Academiae Scientiarum Imperialis Petropolitanae, Volume 1778: II, pp. 31-54.
Opera Omnia Citation
Series 1, Volume 26, pp.204-223.