Sciences, Philosophie, Histoire – UMR 7219, laboratoire SPHERE

Johannes Kepler published in 1624 a treatise on logarithms, the Chilias
Logarithmorum. In this treatise, he intended to give a legitimate proof of the
construction of logarithms, that of Neper being, in its opinion, tainted by the
use made of movement. Kepler based his construction on a general theory
of the measurement of ratios, of which Logarithms are a special case. We
propose here an analysis of Kepler's treatise, showing how his theory of the
measurement of ratios finds its roots in a medieval theory of ratios established
by Thomas Bradwardine and, after him, Nicole Oresme. We give in appendix
the French translation of the theoretical part of the Chilias Logarithmorum.

In 1544, in Nürnberg,
Michael Stifel published his Arithmetica
integra in three books. The third
book is a treatise of algebra which
brings into focus his «rule of Algebra
». This rule establishes, in four
steps, how to put into equation a given
problem and how to carry out its
resolution. Stifel pointed out at several
occasions the perfection of his
rule, which he demonstrated in the
numerous problems that are disseminated
throughout the third book. In
this study, we focus on the chapter 13
and on the appendix, which concludes
the book. They contain several
examples which Stifel drew from
Cardanos’s Practica arithmetice. Our
aim is to understand how Stifel took
up these problems, how he reappropriated
the methods of the Italian algebraist
and how he adapted these
examples to his rule of Algebra

Les méthodes de résolution des équations du troisième degré et du
quatrième degré par les algébristes italiens du XVIe siècle sont saluées par les
historiens des mathématiques comme un apport majeur à la théorie des équations.
Nous montrerons quels types de critiques ont suscités les travaux de Tartaglia
et Cardano sur les équations du troisième degré auprès de leurs contemporains.
Et nous verrons que Simon Stevin propose dans l'Arithmetique (1585)
un exposé des algorithmes de résolution de ces équations qui, sur plusieurs aspects,
présente des avancées significatives par rapport au traitement qu'en fait
Cardano dans l'Ars magna (1545). En particulier, un des mérites de Stevin est de
proposer des règles unifiées pour les équations sans terme du premier degré et
pour les équations complètes. Stevin fait aussi un pas décisif vers une meilleure
compréhension des méthodes de résolution en expliquant les origines des différents algorithmes.

Nous examinons dans cet article les interprétations numériques de la classification des lignes irrationnelles présentée par Euclide au livre X des Éléments, que proposent les deux algébristes de la Renaissance, Michael Stifel et Simon Stevin. Nous comparons ces interprétations en en soulignant les ressemblances et les différences et nous les replaçons dans le projet d'une arithmétique générale porté par les deux auteurs.

Si l’on prend en compte le nombre de manuscrits du De
magnete de Pierre de Maricourt (1269), il semble que ce traité fut
très populaire durant le Moyen Âge, mais on n’a que très peu de
témoignages de son utilisation durant cette période. Il est alors intéressant
de relever la référence à Pierre de Maricourt dans le De sex
inconvenientibus, un traité anonyme datant du début du XIVe siècle et
écrit à Oxford par quelqu’un qui connaissait très bien les théories du
mouvement de Thomas Bradwardine et de ses contemporains. Nous
verrons à cette occasion comment l’auteur discute la question de
l’attraction magnétique.

The famous French physician Jean Fernel published in 1528 in Paris the De proportionibus libri duo. This treatise belongs to the tradition of texts on proportion that follow Bradwardine's Tractatus de proportionibus seu de proportionibus velocitatum in motibus (1328). In the first book, Fernel presented a theory of ratios that is traditional but contains some distinctive features, on denominating ratios, on fractions, on irrational ratios. The second book is devoted to a theory of ratio of ratios of which I give an account in this paper. Résumé Le célèbre médecin français Jean Fernel publie à Paris, en 1528, le De proportionibus libri duo. Ce traité appartient à la tradition des textes sur la proportion issus du Tractatus de proportionibus seu de proportionibus velocitatum in motibus de Thomas Brad-wardine (1328). Dans le premier livre, Fernel présente une théorie des rapports qui est traditionnelle mais qui contient quelques traits distinctifs, sur la dénomination des rapports, sur les fractions, sur les rapports irrationnaux. Le deuxième livre est consacré à la théorie des rapports de rapports dont je donne un aperçu dans cet article.

In his Tractatus de proportionibus, Thomas Bradwardine describes some devices consisting of a magnet and pieces of iron, in order to put his rule of motion to the test. These devices, or similar ones, are also found in the Questiones super libros Physicorum of Nicole Oresme, and in three works by Blasius of Parma, namely the Questiones de ponderibus, the Questiones super libros Physicorum and the Questiones circa Tractatum proportionum. In this paper, I describe these devices, while examining the theory of magnetism that is in use and I analyse how the arguments on magnets demarcate the field of application of the rule of motion.

Professeur renommé au collège jésuite de Rome, Clavius était en relation avec la plupart des savants de son époque. Au tournant des XVIe et XVIIe siècles, son œuvre se situe à une période charnière entre tradition et modernité. Si on lui doit des résultats originaux, ses ouvrages sont surtout à visée pédagogique. C’est le cas pour son édition des Éléments d’Euclide, œuvre majeure et jalon significatif dans l’histoire de ce texte.
Cette étude montre comment Clavius s’approprie les Éléments et les nourrit d’énoncés et de concepts tirés d’autres ouvrages, principalement médiévaux, de commentaires pédagogiques et de réflexions polémiques. Elle est suivie d’une traduction du livre V, qui contient la théorie des proportions et illustre l’analyse des enjeux et de la structure du commentaire de Clavius.

Campanus’ 13th-century version of Euclid’s Elements is not
a translation but a comment written on the basis of Arabic and Latin 12th-century
sources and original works such as Jordanus’ Arithmetic. The study of numerical proportionality
in Book VII shows the nature and scope of Campanus’ work on Euclid’s
treatise as it was transmitted to him. We shall see that Campanus’ reflexions are
inscribed in the Euclidean project itself, which, mostly concerned with the logical
structure of the treatise, he sought to explain and strengthen. In this perspective,
he distinguished fundamental notions in this theory, such as “part” and “parts”, and
introduced the medieval notion of “denomination of a numerical ratio”.

In this paper, we study the reception of Euclid's Elements by a 13th century mathematician, Campanus. We are interested in the nature of his work on Book X, and its theory of irrationality. In Robert of Chester's version, which Campanus used for his edition, two non euclidean notions appear, the notion of straight lines ‘rational in length’ and ‘rational in power’. We asd the question if the introduction of these notions lead to a theory which is different from Euclid's. To answer that question, we examine the first definitions of Book X and some theorems of Bokk XIII, to which Campanus gives some additions called for by the introduction of the above notions.

The Hartree-Fock method, one of the first applications of the new quantum mechanics in the frame of the many-body problem, had been elaborated by Rayner Douglas Hartree in 1928 and Vladimir Fock in 1930. Promptly, the challenge of tedious computations was being discussed and it is well known that the application of the method benefited greatly from the development of computers from the mid-to-late 1950s. However, the years from 1930 to 1950 were in nothing years of stagnation as the method was the object of several considerations related to its mathematical formulation, possible extensions, and its conceptual understanding. Thus, with a focus on the respective attitudes of Hartree and Fock, in particular with respect to the concept of quantum exchange, the present work puts forward some mathematical and conceptual clarifications which played an important role for a better understanding of the many-body problem in quantum mechanics.

Scientific research — in particular, military and nuclear — had proven during the Second World War to have the potential to demonstrate the superiority of a country. Then, its internationalization in the post-war period led to its being considered a key element of cultural diplomacy. In 1968, during the organization of a conference on general relativity and gravitation in Tbilisi, geopolitics and science encountered abruptly. As a consequence of the Third Arab-Israeli War between Israel and the neighboring states of Egypt, Jordan, and Syria in June 1967, diplomatic relations between the USSR and Israel were interrupted. Then, Israeli physicists initially were not invited to join their colleagues in Georgia. Such a political decision provoked a tense crisis that afterwards would affect the institutional organization of the scientific community dealing with the theory of relativity. The present article proposes to examine this episode of the history of science as part of the history of Soviet cultural diplomacy. Indeed, focusing on Soviet actors of the Israeli crisis in 1968, it is possible to better understand the
political influences on scientists in the USSR. In this context, of particular interest is the international trajectory of the Head of the local organizing committee of the Tbilisi conference, Vladimir A. Fock. From his example, it is shown in the article that Soviet scientists primarily acted as representatives at the international level and were restricted in their movement. Their actions were under control of their scientific and political authorities.