Abstract
This article presents an English translation and analysis of François-Joseph Servois’ (1767–1847) 1826 work Sur la théorie des parallèles (On the Theory of Parallels). This was Servois’ final mathematical publication. It considers the Parallel Postulate and explores ideas that would later contribute to the development of non-Euclidean geometry. Through a detailed examination of the sums of the angles of a triangle under geometric transformations, Servois investigated the possibility of proving Euclid’s Parallel Postulate. His approach, influenced by Adrien-Marie Legendre (1752–1833), anticipated methods that challenged classical Euclidean assumptions. This article ends with some possible activities in the classroom.
To complement the main text, we provide two supplements. Supplement 1 offers an annotated English translation of Servois’ Sur la théorie des parallèles, providing insight into his reasoning and historical context. Supplement 2 presents a note from Legendre’s Éléments de Géométrie, in which he reflected on the philosophical and mathematical implications of his attempted proof that: In any triangle, the sum of the three angles is equal to two right angles.