In January 1882 Beyel received his doctorate from the University of Zürich for his thesis Centrische Kollineation nter Ordnung in der Ebene vermittelt durch Ähnlichkeitspunkte von Kreisen ; it was re-conferred to him on the occasion of its 50th anniversary. In 1883 he habilitated at the Polytechnic, as Privatdozent. He held his post until 1834, lecturing mainly on geometry, in particular projective geometry. Beyel was a member of the German Mathematical Society.
In 1889 he married Lydia Magdalena Schalch (1860-1946). Their son Franz did a PhD in German literature and worked as a teacher in Basel.
Beyel was a prolific writer. His most important book, written in 1922, is Der mathematische Gedanke in der Welt. Plaudereien und Betrachtungen eines alten Mathematikers , in which he writes about mathematical topics and mathematics in everyday life for lay people. Reprints of this book are still sold today; a review  describes it as 'a declaration of love to mathematics'. Beyel also wrote a number of geometry books, such as Geometrische Studien (1886) and Darstellende Geometrie: Mit einer Sammlung von 1800 Dispositionen zu Aufgaben aus der darstellenden Geometrie (1901). The latter is a concise collection of the basic principles of orthogonal parallel projection intended for students; it is based on Beyel's lectures . Moreover, Beyel published numerous papers on geometry: Two examples are LVII Sätze über das orthogonale Dreieck (1889) and Der kubische Kreis mit Doppelpunkt (1897).
In addition to his mathematical publications he wrote several articles on the state of the cinema, on political issues, and against immoral literature. He was one of the founders of the Schweizerische Monatshefte für Politik und Kultur founded in 1921. Its current name is "Swiss Month" (Schweizer Monat). He regularly reviewed books for this publication.
Beyel joined the organising committee of the first International Congress of Mathematicians in December 1896, but he did not hold a particular position.
Article by: Stefanie Eminger, University of St Andrews