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

Athanase Papadopoulos, Vladimir Turaev

TL;DR

The article surveys Euler's pivotal contributions to spherical geometry, highlighting how his work on spherical trigonometry—via both variational methods and classical Euclidean demonstrations—shaped the field and its applications. It emphasizes Girard's formula for spherical area, spherical Heron-type area relations, and Lexell's locus problem, and traces the Euclidean-to-spherical methodological bridge through Pappus-type constructions and other memoirs. The discussion situates Euler's results within geography, astronomy, and geomagnetism, and extends to higher-dimensional analogues such as the three-dimensional Apollonius problem, illustrating Euler's broad influence on later non-Euclidean developments. Overall, the piece presents Euler as a central figure whose geometric, analytic, and applied insights bridged classical geometry and later transformative ideas in geometry and its applications, as part of a forthcoming Springer volume (2026).

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.

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

TL;DR

The article surveys Euler's pivotal contributions to spherical geometry, highlighting how his work on spherical trigonometry—via both variational methods and classical Euclidean demonstrations—shaped the field and its applications. It emphasizes Girard's formula for spherical area, spherical Heron-type area relations, and Lexell's locus problem, and traces the Euclidean-to-spherical methodological bridge through Pappus-type constructions and other memoirs. The discussion situates Euler's results within geography, astronomy, and geomagnetism, and extends to higher-dimensional analogues such as the three-dimensional Apollonius problem, illustrating Euler's broad influence on later non-Euclidean developments. Overall, the piece presents Euler as a central figure whose geometric, analytic, and applied insights bridged classical geometry and later transformative ideas in geometry and its applications, as part of a forthcoming Springer volume (2026).

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.

Paper Structure

This paper contains 9 sections, 8 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Why spherical trigonometry?
  3. On Euler's work on spherical trigonometry
  4. On the measure of solid angles
  5. The locus of spherical triangles with a given area
  6. Other memoirs: from Euclidean to spherical geometry
  7. Geography, astronomy and geomagnetism
  8. Curves on the sphere, the spheroid, etc.
  9. Another problem on spheres

Figures (3)

  • Figure 1: From Wallis' proof of Girard's Theorem (Wallis2, Vol. 2, p. 875), the construction for the proof of Girard's formula. The region FAHDF is a lune.
  • Figure 2: From Euler's memoir Euler-Mensura-T, the construction for the proof of Girard's formula. The region ACBDA (left) is a lune. In the middle figure, we have three lunes.
  • Figure 3: A plate from Euler's memoir Solutio facilis problematis, quo quaeritur sphaera, quae datas quatuor sphaeras utcunque dispositas contingat