Mathematical essay
The Cesàro Method for Spherical Trigonometry
My second-year undergraduate expository essay on using the Cesàro method to recover spherical trigonometric identities from Euclidean geometry.
15 pages Revised in 2025
Abstract
This essay develops the Cesàro method for spherical trigonometry. After introducing the basic geometry of the sphere, it establishes the circle-preserving and conformal properties of stereographic projection. The resulting planar triangle is then used to derive the spherical law of cosines, two of Delambre’s analogies, and one of Napier’s analogies. An appendix explains how analytic continuation relates the spherical and hyperbolic laws of cosines.