English Title

A universal spherical trigonometry, derived briefly and from first principles


Leonhard Euler

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Original Language


Archive Notes

A scan of this work, along with its figures, can also be found at Archive.org. An English translation by Ian Bruce is available at 17centurymaths.com.

Content Summary

This is an introduction to 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, and cos c = cos a cos b + sin a sin b cos C. Then, he gives a duality theorem: given a spherical triangle with angles A, B, C and sides a, b, 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.

Published as

Journal article

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Written Date


Original Source Citation

Acta Academiae Scientiarum Imperialis Petropolitanae, Volume 1779: I, pp. 72-86.

Opera Omnia Citation

Series 1, Volume 26, pp.224-236.

Record Created