Annotations to a certain passage of Descartes for finding the quadrature of the circle

TL;DR

This paper explores Descartes' approach to squaring the circle using the quadratrix, reconstructs his argument, and derives related infinite series and product formulas involving trigonometric functions.

Contribution

Euler reconstructs Descartes' argument for circle quadrature, deriving new infinite series and product formulas related to the quadratrix and circle rectification.

Findings

Derived an infinite series involving tangent functions.

Established a product formula involving secant functions.

Connected Descartes' geometric approach to modern analysis.

Abstract

Translation from the Latin of "Annotationes in locum quendam Cartesii ad circuli quadraturam spectantem" (1763). The passage Euler is referring to is the "Excerpta" in part 6, p. 6 of Descartes' 1701 "Opuscula posthuma". Before reading this paper I had not heard of the "quadratrix" before, and I recommend learning a bit about it before reading this. I found Thomas Heath, "A history of Greek mathematics", vol. I, chapter VII to be helpful, in particular pp. 226-230. The quadratrix is a "mechanical curve" that can be used to rectify the circle. The usual problem of squaring the circle is to construct a square with the same area (or perimeter) as a given circle, in a finite number of steps using compass and straightedge. Descartes worked in the reverse direction: from a given square he constructed the radius of a circle with the same perimeter, but in an infinite number of steps. In this paper Euler reconstructs Descartes' argument and develops some consequences of it. Euler finds that \[ \sum_{n=0}^\infty \frac{1}{2^n} \tan \frac{1}{2^n}\phi = \frac{1}{\phi} - 2\cot 2\phi. \] Integrating this yields \[ \prod_{n=1}^\infty \sec \frac{1}{2^n} \phi = \frac{2\phi}{\sin 2\phi}. \] I'd like to thank Davide Crippa from the University of Paris 7 for some helpful back and forth about this paper. One of the only citations to this paper that I have found is in Pietro Ferroni, De calculo integralium exercitatio mathematica, Allegrini, Florence, 1792, pp. xxi--xxiii. The full text of it is available on Google Books.