## E524 -- Trigonometria sphaerica universa, ex primis principiis breviter et dilucide derivata

(A universal spherical trigonometry, derived briefly and from first prinicples)

Summary:

This is an introduction to the basics of spherical trigonometry. Euler derives the formulas for a triangle ABC on a unit sphere, where the sides of the triangle are a, b, c: sin C/sin c = sin A/sin a, cos A sin c = cos a sin b - sin a cos b cos C, cos c = cos a cos b + sin a sin b cos C. Then he gives a duality theorem: Given whatever spherical triangle whose angles are A, B, and C and sides a, b, and c, it is always possible to exhibit an analogous triangle whose angles are complementary to the sides of the first triangle, and the sides are the complements of the angles.

According to the records, it was presented to the St. Petersburg Academy on March 12, 1781.

Publication:
• Originally published in Acta Academiae Scientarum Imperialis Petropolitinae 3, 1782, pp. 72-86
• Opera Omnia: Series 1, Volume 26, pp. 224 - 236
Documents Available:
• Original Publication: E524