Notes on angles and solid angles, in relation with Euler's memoir De mensura angulorum solidorum

TL;DR

This paper explores the historical development of angles and solid angles, focusing on Euler's work and its philosophical implications, linking ancient Greek, Arab, and Renaissance mathematicians to modern mathematics.

Contribution

It extends the study of solid angles to the general concept of angles, highlighting historical and philosophical perspectives from ancient to modern times.

Findings

Euler's questions relate to ancient mathematical and philosophical issues.

The study reveals the continuity of mathematical ideas over 2500 years.

Mathematics and philosophy are deeply interconnected in the development of angle concepts.

Abstract

We provide some historical context to the study of solid angles carried out by Euler in his memoir \emph{De mensura angulorum solidorum} (On the measure of solid angles). We extend our study to the general notion of angle (not only solid). While doing so, we explore some works by Ancient Greek mathematicians and others by Arabs mathematicians of the Middle-Ages as well as some later Western authors from the Renaissance. In particular, we review the Pythagorean anthyphairetical perspective on angles which establishes the basis of the important relation between the mathematical notion of angle and the philosophical concept of finitization of the Infinite. In doing so, we shall show that questions addressed by Euler lead us to questions raised about 2500 years ago. At the same time, we highlight the fact that mathematics in those times is also today's mathematics. The reader can also see in this study the intermingling between mathematics and philosophy. This paper will appear in the book \emph{Spherical geometry in the Eighteenth Century, I: Euler, Lagrange and Lambert}, ed. R. Caddeo and A. Papadopoulos, Springer, 2026.