Legendre's theorem on spherical triangles

In geometry, Legendre's theorem on spherical triangles, named after Adrien-Marie Legendre, is stated as follows: The theorem was very important in simplifying the heavy numerical work in calculating the results of traditional (pre-GPS and pre-computer) geodetic surveys from about 1800 until the middle of the twentieth century.

The theorem was stated by Legendre (1787) who provided a proof[1] in a supplement to the report of the measurement of the French meridional arc used in the definition of the metre.

Tropfke (1903) maintains that the method was in common use by surveyors at the time and may have been used as early as 1740 by La Condamine for the calculation of the Peruvian meridional arc.

[3] Girard's theorem states that the spherical excess of a triangle, E, is equal to its area, Δ, and therefore Legendre's theorem may be written as The excess, or area, of small triangles is very small.

For example, consider an equilateral spherical triangle with sides of 60 km on a spherical Earth of radius 6371 km; the side corresponds to an angular distance of 60/6371=.0094, or approximately 10−2 radians (subtending an angle of 0.57° at the centre).

are calculated by dividing the true lengths by the square root of the product of the principal radii of curvature[5] at the median latitude of the vertices (in place of a spherical radius).