Various observations on angles proceeding in geometric progression

TL;DR

This paper explores properties of angles in geometric progression, focusing on products of sine functions and their relation to Euler's identities, with historical insights and mathematical observations.

Contribution

It provides a translation and analysis of Euler's 1773 observations on angles in geometric progression, highlighting identities related to sine products and their connection to Euler's work.

Findings

Identifies identities involving products of sine functions in geometric progression

Connects Euler's identities to Viète's infinite product

Provides historical context for trigonometric product identities

Abstract

This is a translation of Euler's 1773 "Variae observationes circa angulos in progressione geometrica progredientes", E561 in the Enestr{\"o}m index. I translated this paper as a result of my study of Euler's work on the infinite product $\prod_{k=1}^\infty (1-z^k)$. If one instead considers the finite product $\prod_{k=1}^n (1-z^k)$, one can study its behavior on the unit circle. The absolute value of $\prod_{k=1}^n (1-e^{ik\theta})$ is $2^n |\prod_{k=1}^n \sin k\theta/2|$. My interest in the product $\prod_{k=1}^n \sin k\theta/2$ has inspired me to become acquainted with Euler's papers on trigonometric identities, in particular E447, E561, and E562. E561 says nothing about the product $\prod_{k=1}^n \sin k\theta/2$, but it has identities which I had not seen before. The identities have a form similar to Vi\`ete's infinite product $\prod_{k=1}^\infty \cos \theta/2^k=\frac{\sin\theta}{\theta}$.