*By Elena Ausejo*

This session—co-organized by Elena Ausejo (University of Zaragoza, Spain) and Hélène Gispert (University Paris-Sud, France)—took place on Monday and Tuesday, 9 and 10 July, 2007 and was divided into three subsessions. The following talks were presented:

*José M. Cobos, Area de Historia de la Ciencia, Universidad de Extremadura, Facultad de Biblioteconomí? y Documentacíon, Plazuela Ibn Marwan, 06071 Badajoz, Spainjcobos@unex.es*

Francisco Vera was born in Alconchel (Badajoz) in 1888 and died in Buenos Aires (Argentina) in 1967. He was mathematician, journalist, civil employee (at the National Audit Office), philosopher and fundamentally historian of the scientific ideas. Francisco Vera can be considered one of the most important historians of science in Spain. He was, like many others of that time, forced to go into exile—in his case in America—because of his political ideas after the Spanish Civil War (1936-39). The Dominican Republic, under dictator Trujillo, showed a wilful and interested predisposition to receive republican refugees. Colombia had neither a Spanish migratory tradition nor one from other countries. In addition, its economic structure in 1939 did not allow it to absorb a massive influx of Spaniards; the political factor determined whether Spanish refugees were accepted. Argentina, which had been receiving millions of immigrants since 1880, closed its doors in 1930. This talk followed Francisco Vera's life and work in these various republics.

in World War I

*David Aubin, Institut Mathématique de Jussieu, Université Paris 6, 175 rue de Chevaleret, 75013 Paris, Francedaubin@math.jussieu.fr*

During World War I, several mathematicians and scientists were employed in ballistics by most allied nations. They developed new computing procedures, computed new tables, and worked on various schemes for detecting enemy batteries. After the war, despite the many treatises and articles that were published about their crucial contributions, this production was soon all but forgotten. To understand the impact of mathematicians' involvement with ballistics, Aubin argued that we need to broaden the scope. Besides civilian scientists, military technicians (ballisticians) and artillerymen on the field were also involved. Artillery doctrines significantly evolved during the war creating a need for new tables. With a sense of urgency, ballisticians, who were the inheritors of a long scientific tradition, organized an adequate response to those needs. In these circumstances, ballisticians co-opted a few mathematicians, and they were rewarded by the military hierarchy. But the mathematicians involved often saw this work as a mere parenthesis in their scientific activity.

*J. J. Escribano, I.E.S. Valle del Cidacos, Dirección General de Educacíon del Gobierno de La Rioja, c/ Basconia s/n, 26500 Calahorra (La Rioja), Spainjesbriba@boj.pntic.mec.es
L. Españnol, and M. A. Martínez García, Departamento de Matemáticas y Computación, Universidad de La Rioja, c/ Luis de Ulloa s/n (Edif. Vives), 26004, Logroñnoluis.espanol@unirioja.es ; angeles.martinez@unirioja.es*

The century started in an atmosphere of "national regeneration," in which the advance of the sciences—hence also of mathematics—was a principal aim. This was true only until the Civil War of 1936-39. No theses were written during the Civil War, hence the real period finished in 1936. This talk explored the Spanish doctorate in mathematics between 1900 and 1936, the first year of the Spanish Civil War 1936-39. In particular, it looked at the legislative framework for the degree. Of importance here was the unsuccessful attempt, around 1920, to give autonomy to all the universities and to allow them to award the doctorate (until then the University of Madrid had the sole right to grant doctorates). The principal aim of the talk, however, was to give a catalogue of doctoral theses in mathematics what were defended in this period. Following the Plan Garca Álix (1900), after earning a four-year degree, a doctoral student took courses on Higher Analysis, Higher Geometry, and Astronomy for one more year, before writing a doctoral thesis on one of these subjects. In the second half of the period, doctoral theses on the History of Mathematics and Celestial Mechanics were also defended.

Ph.D. Degrees in Mathematics

*Juliette Leloup, Institut de Mathématiques de Jussieu, Université Paris 6, 175 rue de Chevaleret, 75013 Paris, Franceleloup@math.jussieu.fr*

The actual image of French mathematics in the interwar period is mainly based on testimonies of mathematicians who lived during this period; little else has been written on this subject. By analysing the corpus of Ph.D. degrees in mathematics defended in France between 1914 and 1945 (242) and the reports on them, this talk highlighted some of the dynamics behind mathematical research and its evolution. It explored the answer to questions such as: Which subjects were at the centre of interest? How much of the research was traditional mathematics and how much was on new themes? How did new subjects appear, in particular, probability? Which mathematicians were the most influential? What sort of consequences did the First World War have? How did the academic mathematical scene change? How was French mathematics influenced by foreign research? Which international links were maintained or encouraged? In its effort to provide answers to some of these questions, this talk sketched elements for a revised image of mathematical research in France of that time.

*Francisco A. González Redondo, Dpto. Álgebra, Facultad de Educacíon, Universidad Complutense de Madrid, C/ Rector Royo Villanova s/n, 28040 Madrid, Spainfglezr@edu.ucm.es*

Several authors have remarked upon the absence of a complete history of dimensional analysis. Some even consider that writing that history is an impossible task, due to the disparity of the existing conceptions of it and the number of scientific controversies which remain open. This talk presented some of the keys for breaking the vicious circle that has prevented the writing of that history. The new perspective is based upon fixing some special 'events' in order to delimit historical periods. The starting point of this history was the publication of P. W. Bridgman's book, *Dimensional Analysis* (1922) (although none of his biographers has realized the full potential of his approach). The second event is the publication of Fourier's "General Remarks" in his *Analytical Theory of Heat* (1822), where the concept of dimension is first applied to physical quantities as an evolution from geometrical dimensions. Fourier's contribution marks an historical break: before Fourier, dimensional analysis can be said to have been in its prehistory, while the period between Fourier and Bridgman could be considered the protohistory of the discipline. The solution of the problem of the history of dimensional analysis provides a complete historiographical model for all sciences. Through its development, both the history of every science and its teaching can be organized through a new perspective.

The Case of "Modern Mathematics" in France

*Hélène Gispert, Université Paris-Sud, 11 Bâtiment 407, Centre Universitaire, 91405 Orsay cedex, Francehelene.gispert@u-psud.fr*

The history of mathematical teaching has become, in France, a field actively studied by historians of mathematics. This owes to the fact that analysing this specific part of mathematical activity reveals, in a privileged way, the epistemological and social stakes of mathematics in any given period. This talk focused on contemporary times and the so-called "réforme des mathématiques modernes" in France. The talk first considered the very specific epistemological context of the reform in these years of "Bourbakisme" and of structuralism in all disciplines. It next emphasised the economical stakes of mathematics and mathematicians in these post-war decades, as illustrated by the serious interest and support of international organisations such as EOCE and OCDE in "modern math reform." The institutional educational context in 1960s France—when the first years of secondary level teaching just began to open to all children and not only to a narrow, socially privileged segment of the student population—and the difficulties and contradictions it provoked in the reform process were also explored. It was argued, finally, that this reform was thus promoted not just by mathematicians; its development in such a new educational context sharpened dissensions among mathematicians and others on both epistemological and social mathematical stakes.

*Jesús Hernádez, Dpto. Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, Campus de Cantoblanco, C/ Fco. Tomás y Valiente 7, 28049 Madrid, Spainjesus.hernandez@uam.es*

*This talk addressed different forms of presentation of basic notions in mathematics, the so-called "elements." It considered some classical texts, such as Euclid's Elements and the Grundlagen der Geometrie by Hilbert and compared them with the Éléments de Mathématique by Bourbaki. Thus, "classical" axiomatics was shown to have confronted "modern" axiomatics, the father of which, according to Dieudonné, was supposed to have been Hilbert. Similarities and differences were considered in some detail.*

*Christine Proust, RHESEIS, École Normale Supérieure, 1 rue Maurice Arnoux, 92120 Montrouge, FranceChristine.Proust@ens.fr*

*Les archives cunéiformes de nombreux musées archéologiques contiennent en abondance des "brouillons" d'écoliers, ces tablettes d'argile écrites dans les écoles de scribes de Mésopotamie à l'époque paléo-babylonienne (début du deuxième millénaire avant notre ère). Parmi elles, les tablettes mathématiques occupent une place importante, mais ont peu été étudiées jusqu'à une date récente. Quelles sont les raisons de l'éclatement et la dispersion des collections de tablettes scolaires à travers le monde, et du peu d'intérêt qu'elles ont suscité en histoire des sciences? En quoi leur étude peut-elle renouveler le regard porté aujourd'hui sur les mathématiques mésopotamiennes? Ces questions s'inscrivent dans le cadre plus général d'un programme de recherche sur les phénomènes de sélection des sources. Ici, quelques résultats récents ont été évoquées rapidement. Cette discussion a été suivie par une présentation de façon plus détaillée du cas d'une catégorie de tablettes scolaires destinée à l'apprentissage de l'écriture des nombres et des mesures, et à la mâtrise du système métrologique cohérent et standardisé en vigueur dans l'ensemble de la Mésopotamie paléo-babylonienne. Ces textes constituent la moitié de la documentation scolaire mathématique connue, et, dès les débuts de l'archéologie mésopotamienne, ils étaient accessibles aux chercheurs. Pourtant, ils sont pratiquement absents de l'historiographie. Dans sa présentation, Proust a montré comment leur réinsertion dans leurs archives d'origine éclaire d'un jour nouveau les pratiques de calcul, la conception des nombres, des surfaces et des volumes dans les mathématiques mésopotamiennes.*

*Betsabé Caunedo, Dpto. Historia Medieval, Facultad Filosofía y Letras, Universidad Autónoma de Madrid, Campus de Cantoblanco, C/ Fco. Tomás y Valiente 7, 28049 Madrid, Spainbetsabe.caunedo@uam.es*

*This talk concentrated on three Castilian manuscripts concerning commercial arithmetic, all of which were written in the 14th century. The first manuscript, considered to be the most important, is the Libro de Arismética, which is kept among the manuscripts at the Royal College of San Isidoro in León (manuscript 46). As recently as the year 2000, a study of it has allowed us to revise backward by over 100 years, the date which, up until then, had been used when speaking about treatises on mercantile arithmetic in the Peninsula. Knowledge of the existence of this manuscript and its study has also obliged us seriously to question if there are not, in fact, more studies of this type buried in the different Castilian archives. In just three years, we have located two additional manuscripts on this discipline, which we believe also to date from the 14th century. This means that the Libro de Arismética is not only a magnificent copy, but is also rare, unique, and therefore exceptional. These three manuscripts are proof of an authentic technical literature and of the existence of didactic activity in Castile at the service of the active, flourishing commerce, typical of late mediaeval Castile.*

*Maryvonne Spiesser, Labo Émile Picard, UFR MIG Université Paul Sabatier, F-31062 Toulouse Cedex 9, Francespiesser@picard.ups-tlse.fr*

*Le XVe siècle est traversé, en France et particulièrement en Occitanie, par un mouvement dynamique de production de traités de mathématique pratique en langue vulgaire, qui visent en premier lieu les milieux du commerce. En arithmétique, cohabitent les ouvrages de calcul avec les jetons et les "algorismes," fondés sur les techniques d'écriture et de calcul indo-arabes. Cette présentation s'est intéressée à ces derniers. C'est Guy Beaujouan qui a lancé leur étude dans les années 50. Le corpus actuellement connu et étudié est limité à une quinzaine de titres, en français ou en occitan, la plupart issus des régions méridionales. Ici on s'est posé la question de leur origine, de leur circulation et de leur diffusion. Malgré une structure assez homogène, les disparités entre ces ouvrages sont notables et nous interpellent sur le milieu des auteurs, les publics visés, les cadres d'enseignement. Quelques exemples, dont celui du Triparty en la science des nombres de Nicolas Chuquet (1484), ont servi à montrer la complexité de ce courant mathématique qui va se transformer au siècle suivant au contact de l'humanisme, mais dont l'esprit persistera longtemps dans l'enseignement élémentaire français.*

*Elena Ausejo, Facultad de Ciencias, Universidad de Zaragoza, Ciudad Universitaria, E-50009 Zaragoza, Spainichs@unizar.es*

*Juan de Yciar (1522-90), the most important caligrapher of the Spanish Renaissance, is also the author of works such as Libro Subtilissimo, por el qual se enseña a escreuir y contar perfectamente el qual lleua el mesmo orden que lleua vn maestro con su discipulo Hecho y experimentado por Iuan de Yciar Vizcayn, and Arte Breue y Prouechoso de cuenta Castellana y Arithmetica, donde se muestran las cinco reglas de guarismo por la cuenta castellana, y reglas de memoria. Y agora nueuame[n]te en esta postrera impression se han añadido vnas cuentas muy graciosas y prouechosas, sacadas del libro de Fray Iuan de Ortega : y mas al cabo va añadida vna cuenta abreuiada de marauedis (both printed in Zaragoza by mid 16th-century), which prove his interest in teaching counting together with writing. An even rarer book (one single copy in the British Library) is purely mathematical, Libro intitulado aritmética práctica muy provechoso para toda persona que quisiere ejercitarse en aprender a contar (1549). This talk discussed the second copy of the latter work, which has recently been found in Spain.*

*Juan Riera Palmero, Área de Historia de la Ciencia, Facultad de Medicina, Universidad de Valladolid, Avda. Ramón y Cajal s/n, 47005 Valladolid, Spainjriera@med.uva.es*

*This talk explored the activities of the Spanish Inquisition at the end of the 18th century, after the French Revolution of 1789, when the Spanish Bourbons tried to stop the arrival of revolutionary ideas into Spain. The Inquisition was charged with this task, which also affected scientific institutions such as the Academy of Mathematics of Barcelona, founded in 1739.*

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