#### English Title

A universal spherical trigonometry, derived briefly and from first principles

#### Enestrom Number

524

#### Fuss Index

333

#### Original Language

Latin

#### 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

#### Published Date

1782

#### Written Date

1781

#### 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

2018-09-25