2010/09/08 by Leonhard Euler, Euler, Leonhard, Jordan Bell +1
Mathematics · #01A50 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #History and Overview (math.HO) #History and Theory of Mathematics #Mathematics and Applications #math.CA #math.HO #msc:01A50
paper · pdf · doi:10.48550/arxiv.1009.1439
9 pages, E561
arxiv created 2010/09/08 · openalex publication_date 2010/09/08 · arxiv updated 2010/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This is a translation of Euler's 1773 "Variae observationes circa angulos in progressione geometrica progredientes", E561 in the Eneström index. I translated this paper as a result of my study of Euler's work on the infinite product ∏k=1^∞ (1-zk). If one instead considers the finite product ∏k=1n (1-zk), one can study its behavior on the unit circle. The absolute value of ∏k=1n (1-eikθ) is 2n |∏k=1n sin kθ/2|. My interest in the product ∏k=1n sin kθ/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 ∏k=1n sin kθ/2, but it has identities which I had not seen before. The identities have a form similar to Viète's infinite product ∏k=1^∞ cos θ/2k=\fracsinθθ.