Felix Klein's Erlanger Programm (1872) has been extensively studied by historians. If the early geometrical works in Klein's career are now well-known, his links to the theory of algebraic equations before 1872 remain only evoked in the historiography. The aim of this paper is precisely to study this algebraic background, centered around particular equations arising from geometry, and participating on the elaboration of the Erlanger Programm. Another result of the investigation is to complete the historiography of algebraic equations, in which those "geometrical equations" do not appear.
RésuméLe Programme d'Erlangen de Felix Klein (1872) a été abondamment étudié par les historiens. Si les premiers travaux géo-métriques de la carrière de Klein sont maintenant bien connus, ses rapports à la théorie des équations algébriques antérieurs à 1872 ne sont qu'évoqués dans l'historiographie. Le but de cet article est justement d'étudier ce contexte algébrique, centré autour d'équations algébriques particulières provenant de la géométrie, qui participe à l'élaboration du Programme d'Erlangen. Un autre résultat de ce travail est la complétion de l'historiographie des équations algébriques, dans laquelle ces "équations de la géométrie" n'apparaissent pas.
This article examines the research of Louis J. Mordell on the Diophantine equation y 2 − k = x 3 as it appeared in one of his first papers, published in 1914. After presenting a number of elements relating to Mordell's mathematical youth and his (problematic) writing, we analyze the 1914 paper by following the three approaches he developed therein, respectively based on the quadratic reciprocity law, on ideal numbers, and on binary cubic forms. This analysis allows us to describe many of the difficulties in reading and understanding Mordell's proofs, difficulties which we make explicit and comment on in depth.
Argument This paper challenges the use of the notion of "culture" to describe a particular organization of mathematical knowledge, shared by a few mathematicians over a short period of time in the second half of the nineteenth century. This knowledge relates to "geometrical equations," objects that proved crucial for the mechanisms of encounters between equation theory, substitution theory, and geometry at that time, although they were not well-defined mathematical objects. The description of the mathematical collective activities linked to "geometrical equations," and especially the technical aspects of these activities, is made on the basis of a sociological definition of "culture." More precisely, after an examination of the social organization of the group of mathematicians, I argue that these activities form an intricate system of patterns, symbols, and values, for which I suggest a characterization as a "cultural system."
This article is aimed at throwing new light on the history of the notion of genus, whose paternity is usually attributed to Bernhard Riemann while its original name Geschlecht is often credited to Alfred Clebsch. By comparing the approaches of the two mathematicians, we show that Clebsch's act of naming was rooted in a projective geometric reinterpretation of Riemann's research, and that his Geschlecht was actually a different notion than that of Riemann. We also prove that until the beginning of the 1880s, mathematicians clearly distinguished between the notions of Clebsch and Riemann, the former being mainly associated with algebraic curves, and the latter with surfaces and Riemann surfaces. In the concluding remarks, we discuss the historiographic issues raised by the use of phrases like "the genus of a Riemann surface"-which began to appear in some works of Felix Klein at the very end of the 1870s-to describe Riemann's original research.
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