Euler's work on spherical geometry: An overview with comments

TL;DR

This paper reviews Euler's extensive work on spherical geometry, highlighting his methods, formulae, and applications in areas like astronomy and cartography, emphasizing his contributions to the mathematical understanding of spheres.

Contribution

The paper provides a comprehensive overview of Euler's original methods and results in spherical geometry, connecting them to his broader scientific work.

Findings

Euler established key trigonometric formulae for spherical triangles.

He derived spherical Heron formulae for triangle areas.

Euler applied spherical geometry to polyhedra and mapping problems.

Abstract

We review Euler's work on spherical geometry. After an introduction concerning the general place that trigonometric formulae occupy in geometry, we start by the two memoirs of Euler on spherical trigonometry, in which he establishes the trigonometric formulae using different methods, namely, the calculus of variations in the first memoir, and classical methods of solid geometry in the other. In another memoir, Euler gives several formulae for the area of a spherical triangle in terms of its side lengths (these are ``spherical Heron formulae''). He uses this in the computation of numerical values of the solid angles of the five regular polyhedra, which is his goal in his memoir. We then review memoirs in which Euler systematically starts by establishing a theorem or a construction in Euclidean geometry and then proves an analogue in spherical geometry. We point out relations between Euler's memoirs on spherical trigonometry and works he did in astronomy, on the problem of drawing geographical maps, and in geomagnetism. We also review some other works of Euler involving spheres, including a memoir on the three-dimensional Apollonius problem and others concerning algebraic curves on the sphere. Even though these works are not properly on spherical geometry, they show Euler's interests in various questions related to spheres and we think that they are worth highlighting in such an overview. Beyond spherical geometry, the reader is invited to discover in this article an important facet of the work of the great Leonhard Euler. This article will appear as a chapter in the book ``Spherical geometry in the eighteenth century, I: Euler, Lagrange and Lambert'', Springer, 2026.