En 1849, Arthur Cayley et George Salmon démontrent que toute surface cubique contient exactement vingt-sept droites. Résultat célèbre de la deuxième moitié du 19ème siècle, ce théorème a notamment donné lieu à des recherches sur une équation algébrique particulière appelée "équation aux vingt-sept droites". Dans notre thèse, nous étudions les rapprochements entre groupes, équations et géométrie opérés dans ces recherches. Après un travail préparatoire mettant en place certains points mathématiques et chronologiques associés aux vingt-sept droites, nous nous intéressons au Traité des substitutions et des équations algébriques de Camille Jordan, publié en 1870. Cet ouvrage contient une section consacrée à l'équation aux vingt-sept droites dont nous analysons en détail les mathématiques. Pour mettre en contexte certains points, un corpus plus large est ensuite construit autour des "équations de la géométrie", famille d'équations associées à des configurations géométriques dont les vingt-sept droites ne sont qu'un exemple. Ce corpus s'étend de 1847 à 1896, et ses principaux auteurs sont Jordan, Alfred Clebsch et Felix Klein. Dans le but de rendre compte de l'organisation particulière du savoir partagé dans le corpus, nous discutons et utilisons alors la notion de "culture". Enfin, nous étudions précisément deux textes du corpus proposant de géométriser certaines parties de l'algèbre et nous montrons en quoi les équations de la géométrie ont participé à une compréhension géométrique de la théorie des substitutions ainsi qu'à l'élaboration des idées du Programme d'Erlangen de Klein (1872). / In 1849, Arthur Cayley and George Salmon proved that every cubic surface contains exactly twenty-seven lines. A famous result in the second half of the 19th century, this theorem gave rise to research about a particular algebraic equation called the "twenty-seven lines equation." In our thesis, we study how groups, equations, and geometry interact throughout this research. After a preparatory work presenting some mathematical and chronological points about the twenty-seven lines, we look into Camille Jordan's Traité des substitutions et des équations algébriques, published in 1870. This book contained a section devoted to the twenty-seven lines equation, the mathematics of which we thoroughly study. In order to contextualize some elements, a larger corpus is then built around "geometrical equations," a family of equations linked to geometrical configurations among which the twenty-seven lines are just one example. The corpus extends from 1847 to 1896 and its main authors are Jordan, Alfred Clebsch, and Felix Klein. Aiming at describing the particular organization of the knowledge shared in the corpus, we then discuss and use the notion of "culture." Finally, we closely study two texts of the corpus, each of them presenting a geometrization of a part of algebra, and we ascertain that geometrical equations participated to a geometrical understanding of substitution theory as well as the elaboration of the ideas of Klein's Erlanger Programm (1872).

La parution en 1750 de l'Introduction à l'analyse des lignes courbes algébriques, unique ouvrage publié de Gabriel Cramer, professeur de mathématiques à l'Académie de Genève, est un jalon important dans l'histoire de la géométrie des courbes et de l'algèbre. Il s'est passé près de dix années entre le moment où le Genevois a écrit les premières lignes de son traité des courbes, à l'automne 1740, et sa publication effective : il s'agit de l'œuvre d'une vie.Ce livre a une histoire, à la fois intellectuelle et matérielle, qui s'inscrit dans les contextes scientifiques, professionnels, académiques et sociaux dans lesquels évoluent son auteur puis ses lecteurs. De la genèse d'un texte manuscrit en devenir dont nous suivrons les évolutionsau cours du processus d'écriture et au gré des événements de la vie de son auteur, aux différentes lectures mathématiciennes et historiennes du texte publié qui en sont faites dans les quelque deux siècles qui ont suivi sa publication, je souhaite ici écrire, pour autant que cette expression ait un sens - ce que je m'attacherai à démontrer - une « biographie » de l'Introduction de Gabriel Cramer / The publication in 1750 of the Introduction à l'analyse des lignes courbes algébriques, the only published work by Gabriel Cramer, professor of mathematics at the Geneva Academy, is an important milestone in the history of geometry of curves and algebra. Almost ten years passed between the time when the Genevan wrote the first lines of his treatise on curves in the autumn of 1740 and its actual publication, making it a lifetime achievement.This book has a history, both intellectual and material, which takes place in the scientific, professional, academic and social contexts in which evolve its author and its readers. From the genesis of a handwritten text as a work in progress of which we will follow the evolutions during the process of writing and according to the events of its author's life, to the various mathematicians and historians' readings of the published text which are made in the two centuries following its publication, I would like to write here, insofar as this expression makes sense - which I shall endeavour to demonstrate - a « biography » of Gabriel Cramer's Introduction

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-09-29T19:20:16Z No. of bitstreams: 1 alinesouzareis.pdf: 273728 bytes, checksum: 48ebd086f8af423a6fe43cf974ce8847 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-10-09T19:38:40Z (GMT) No. of bitstreams: 1 alinesouzareis.pdf: 273728 bytes, checksum: 48ebd086f8af423a6fe43cf974ce8847 (MD5) / Made available in DSpace on 2017-10-09T19:38:40Z (GMT). No. of bitstreams: 1 alinesouzareis.pdf: 273728 bytes, checksum: 48ebd086f8af423a6fe43cf974ce8847 (MD5) Previous issue date: 2017-08-05 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Este trabalho surgiu a partir de uma preocupação do desenvolvimento do raciocínio algébrico dos estudantes, visto que, muitos deles não se sentem confortáveis quando começam a estudar as incógnitas dentro da Matemática. Propomos a utilização da metodologia da Resolução de Problemas aliada a História da Álgebra, no desenvolvimento da linguagem algébrica, como método facilitador da compreensão dos conceitos relacionados à equação polinomial do primeiro grau. Buscamos, a partir de estudos bibliográficos referentes à História da Álgebra e Resolução de Problemas Matemáticos, propor atividades pedagógicas que abordam o desenvolvimento da linguagem algébrica. As atividades aqui descritas são proposta a serem aplicadas com turmas de 7o ano do Ensino Fundamental, ao longo de oito semanas, com encontros semanais de 1 hora e 40 minutos. Esperamos ampliar a formação do estudante contribuindo para a construção de conceitos relativos à abstração e generalização matemática, pois acreditamos ser primordial que o aluno compreenda a transição da linguagem verbal para a linguagem algébrica . / This work arose from a concern for the development of students’ algebraic reasoning, since many of them do not feel comfortable when they begin to study the variables within Mathematics. We propose the use of the Problem Solving methodology and the History of Algebra, in the development of algebraic language, as a facilitating method for understanding the concepts related to the first-degree polynomial equation. From bibliographic studies concerning the History of Algebra and Resolution of Mathematical Problems, we have proposed pedagogical activities that deal with the development of algebraic language. The activities are proposed to be applied with 7th grade classes of elementary school, over eight weeks, with weekly meetings of 1 hour and 40 minutes. We hope to broaden student training by contributing to the construction of concepts related to abstraction and mathematical generalization, as we believe it is paramount that the student understands the transition from verbal to algebraic language.