In which theorem are the angles A, B, and C of a spherical triangle equal to the corresponding angles of a plane triangle whose sides are a, b, and c, each increased by one-third the spherical excess of the triangles?

• A. Helmert's
• B. Crandall's
• C. Legendre's
• D. Laplace's

Answer: (C)

Legendre's theorem shows that for a small spherical triangle, you can replace its spherical angles with the angles of a plane triangle whose sides are the original spherical sides increased by one-third of the spherical excess. The spherical excess E is A + B + C − π. If you take the plane triangle with sides a + E/3, b + E/3, c + E/3, its angles are exactly A, B, and C. This works because the curvature of the sphere causes the sum of the angles to exceed π by E, and spreading that excess equally among the three sides aligns the plane-triangle angles with the spherical ones. The other names don't describe this precise relation between spherical angles and a nearby plane triangle.