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This thesis presents a number of linked views of court-sponsored mathematical practice in sixteenth- and early seventeenth-century Medici Tuscany, drawn from various sources of different types. Egnatio Danti's Le scienze matematiche n'dotte in tavole (1577), a series of 'tables' or trees outlining the range and scope of the mathematical sciences, as presented by a Medici court cosmographer, provides the framework for investigation into the mathematical practice of a range of mathematicians and 'practical mathematicians' working at or for the Medici court between 1540 and 1609 (under the first three grand dukes of Florence and Tuscany). Further views of mathematical practice are then sought through closer examination of the careers of a number of Florentine 'practical mathematicians', using official court records, instruments and maps, printed and manuscript writings, and private letters. Examination of these differing views of mathematics, both public and formulated (Danti's tables), and more derivative (views taken from the official court records and private letters), aims to contribute towards an understanding of Florentine courtly mathematics in its working setting. My identification of Medici mathematicians is taken from the activities of practitioners and categories used by their patrons and employers. Using the term 'practical mathematician' as defined explicitly in Danti's tables, this thesis will test the usefulness of this actor's categorisation. Comparison of the careers and activities of a range of 'practical mathematicians', more or less concerned with the mathematical basis of their activities, from cosmographers and teachers at the Medici court, to ducal military and hydraulic engineers and a ducal scenographer, reveals certain similarities of mathematical practice between these men, and a range of mathematical practice in Medici Tuscany much in keeping with the view of the mathematical sciences presented in Danti's tables.

This thesis aims to introduce ideas and methods from the emerging field of digital humanities into the study of history of mathematics, through case studies relating to the role of correspondence, commu- nication, and collaboration in Leonhard Euler's mathematical practice. Euler's known correspondence numbers almost three thousand letters, exchanged with hundreds of correspondents from across Europe. The correspondence is a vital source for understanding Euler's mathem- atics, but it has not yet been examined in great detail; this thesis is a contribution towards such a study. The thesis is motivated by a case study which highlights the cent- ral role of correspondence and personal contact in Euler's work on continued fractions. A desire for better understanding of the corres- pondence leads to the use of methods from the digital humanities, a relatively young field which has been evolving rapidly since the begin- ning of the 21st century. The thesis considers the particular challenges encountered when using such methods in the study of eighteenth- century mathematical texts. A database is used to facilitate the explor- ation and comprehension of Euler's correspondence. This enables the identification of a corpus of letters, all connected with the same math- ematical topic, which would be suitable for further study. A prototype digital edition of one of these letters is presented, featuring a tran- scription, editorial annotations, and digital facsimiles of the original manuscript. Finally, it is shown how existing digital tools that were designed for use in other fields, such as mathematics and cartography, may be appropriated to aid understanding of primary sources in the history of mathematics.

Gaston Darboux (1842-1917) est un mathématicien français né à Nîmes, dont la seconde partie de carrière est connue pour ses travaux et son enseignement de Géométrie. Avec une méthodologie inspirée des approches biographiques modernes, nous reconstruisons le parcours de la première partie de sa vie pour comprendre le mathématicien et le professeur qu'il devient en 1878. Nous commençons par présenter son parcours atypique d'étudiant qui se termine à l'Ecole Normale. Puis, pour cerner son identité mathématique, nous étudions la théorie des focales, méconnue, ainsi que la théorie des surfaces orthogonales et celle des cyclides où Darboux joue un rôle important. Nous nous penchons ensuite sur la réception immédiate de ces travaux, mais également dans un second temps sur la naissance d'un journal scientifique intitulé « Bulletin des Sciences Mathématiques et Astronomiques », dont Darboux devient le rédacteur en chef dès sa création en 1869. Une analyse particulière est alors consacrée à ce journal, qui montre la force des influences mutuelles entre le rédacteur et son périodique. En suivant les dynamiques de son cheminement intellectuel, notre enquête bascule dès lors sur les mathématiques du théorème des bornes, en Analyse, et de la théorie des solutions singulières des équations différentielles. Pour finir, nous examinons la réception de ces nouveaux travaux, ce qui nous offre les clefs d'interprétation de son parcours et des facettes de ses multiples identités lorsque sa trajectoire institutionnelle le fait se tourner à nouveau vers la Géométrie en 1878. / Gaston Darboux (1842-1917) is a french mathematician born in Nîmes, whose second part of the career is well known for his works and his teachings in Geometry. Using a methodology close to the modern biographical approaches, we rebuild the first part of his life trajectory in order to understand the mathematician and the teacher he will become in 1878. We start with a presentation of his uncommon journey as a student, that ends in the Ecole Normale. To portray his mathematical identity, we study the focal theory, mainly unknown, but also the theory of orthogonal and cyclide surfaces, where Darboux plays a major role. Then we focus on the immediate reception of these works, but also on the other hand on the birth of a scientific journal entitled “Bulletin des Sciences Mathématiques et Astronomiques” of which Darboux becomes chief editor right at its inception. A very specific investigation is devoted to this periodical, underlining the strength of the mutual influences occurring between the editor and its journal. Following the dynamic of his line of thought, our inquiry tackles next the mathematics of the extreme value theorem, in analysis, and these of the theory of the singular solutions of differential equations. Lastly, we consider the reception of these new works, which allow us to understand properly his path and the aspects of his multiple identities in 1878, at the time when his institutional trajectory lets him turn to Geometry again.

Le thème principal de ce travail de thèse est de montrer l’interaction existant entre les jeux et les mathématiques au travers d’une catégorie de jeux bien particuliers : les jeux combinatoires. Ces jeux se font sans hasard, sans information cachée et pour chacun des deux joueurs il existe une façon optimale de jouer. Les premiers exemples rencontrés se trouvent dans des écrits de la Renaissance. Les jeux se diffusent aux 17ème et 18ème siècles dans le cadre des récréations mathématiques, genre littéraire et éditorial nouveau qui propose une pratique ludique des sciences fondée sur le défi à l’entendement. L’analyse des jeux combinatoires intéresse ensuite les mathématiciens du début du 20ème siècle, notamment pour les jeux de type Nim. La thèse s’attache à retracer le développement de la théorie mathématique qui se construit autour des jeux combinatoires et aboutit au corps des nombres surréels de John Conway en 1976. En parallèle, elle montre qu’un autre résultat fondamental, attribué à Zermelo (1912), sur la détermination du jeu d’Échecs permet aux jeux combinatoires de s’implanter sur un plan technologique et culturel. Nous voyons les premières machines électromécaniques destinées à jouer au Nim apparaître vers 1940 et se confronter au public lors d’expositions et de salons scientifiques. La naissance des ordinateurs dans les années 1950 ouvre de nouvelles voies pour la programmation du jeu d’Échecs, jeu combinatoire par excellence. La thèse fait revivre les moments forts, faits d’espoirs et de déceptions, qu’a traversés la recherche en programmation d’Échecs, depuis ses débuts jusqu’à la victoire du programme Deep Blue sur le champion du monde Garry Kasparov en 1997. / The main theme of this thesis is to point out the interaction between games and mathematics by means of a category of very specific games, the combinatorial games. These games are no chance games of perfect information and either player (Arthur or Bertha) can force a win, or both players can force at least a draw. The first examples of combinatorial games can be found in Renaissance works. Throughout the seventeenth and eighteenth centuries, games spread as part of recreational mathematics, a new literary and editorial genre that offered an entertaining practice of science based on a challenge to understanding. Then, the analysis of combinatorial games, especially Nim games, aroused the interest of the early-twentieth-century mathematicians. This thesis is devoted to trace the development of the mathematical theory that was formulated around combinatorial games and that led to John Conway’s Field of Surreal Numbers in 1976. In parallel, it shows that another fundamental result on Chess determination, attributed to Zermelo (1912), enabled combinatorial games to become established on a cultural and technological level. Around 1940 appeared the first electromechanical machines, designed to play Nim and to meet the challenges of the audience during scientific exhibitions. The emergence of computers during the 1950s opened new paths for programming Chess, the ultimate combinatorial game. This work brings the highlights, made of hopes and disappointments, which the Chess programming research went through, since its very beginning up to the victory for Deep Blue program over the world champion Garry Kasparov in 1997.

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