La thèse de Louis Gérard sur la géométrie non euclidienne (1892) : une nouvelle étape dans l’histoire de cette discipline

If the names of Nikolai Lobachevsky and János Bolyai are now inseparable from the challenge to Euclidean geometry, their works, which initiated this paradigm shift, went almost unnoticed when published in the 1820s and 1830s. It was not until the 1860s that their work was rediscovered and began to generate significant interest in the geometric theory based on the negation of the parallel postulate. The interpretation of non-Euclidean geometry within known theories (differential geometry and projective geometry) by Eugenio Beltrami [1868] and later Felix Klein [1871] contributed to its acceptance by the mathematical community. However, philosophical debates on the subject continued until the end of the 19th century. It was in this context that the French mathematician Louis Gérard defended, in 1892 at the Faculty of Sciences in Paris, a thesis titled Sur la géométrie non euclidienne [On Non-Euclidean Geometry]. Gérard’s thesis – the most significant work specifically on this subject published in French-speaking countries since the Memoirs of Joseph-Marie De Tilly [1870] and Camille Flye Sainte-Marie [1871] – has not yet been the subject of an in-depth study, and this gap is what the article under review seeks to address. First, a word about Louis Gérard. Born in 1859 in Grand, in the historical region of Lorraine, in eastern France, Gérard spent his entire career in secondary education. His thesis on non-Euclidean geometry thus represents his main contribution to mathematical research. The date of his death is unknown but is after 1939. As this article demonstrates, Gérard’s thesis serves as a transitional work between the contributions of the inventors of non-Euclidean geometry and the purely axiomatic research of the late 19th and early 20th centuries. For this reason, it deserves our attention.