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

Stelios Negrepontis, Athanase Papadopoulos

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.

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

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.

Paper Structure

This paper contains 17 sections, 17 equations.

Table of Contents

  1. Preamble: On the mathematical notion of angle
  2. Introduction
  3. The Pythagoreans, Plato and Aristotle
  4. On plane angles in Euclid's Elements
  5. The Pythagorean principle of the Infinite, interpreted as the infinite anthyphairesis of the diameter to the side of the square, is the cause of the acute and obtuse angles
  6. Two more useful arithmetical propositions
  7. The Pythagorean principle of the Finite, interpreted as the Finitization of the infinite anthyphairesis of the diameter to the side of a square, is the cause of the right angle
  8. Our interpretation of Postulate 4 of the Elements: All right angles are equal
  9. Angles, the parallel postulate, Euler and Lagrange
  10. Solid angles
  11. Angles and solid angles as magnitudes
  12. Angles and solid angles in the Arabic mathematical literature
  13. Some notes on solid angles from the Renaissance
  14. On the notion of angle in spherical geometry
  15. In guise of a conclusion: Some modern works on the notion of angle
  16. ...and 2 more sections