Sciences, Philosophie, Histoire – UMR 7219, laboratoire SPHERE

9:30 – 10:40
Maarten Bullynck (Univ. Paris 8, SPHERE)

Transfers and transformations of numerals. The brothers Humboldt at the crossroads of disciplines (1819-1835).

In 1819 Alexander von Humboldt presented his "Considérations générales sur les signes numériques des peuples’’ before the Paris Academy. Based upon a collection of data he had received from the various ethnographers and linguists in his scientific network, Alexander developed in this fragment a hypothetical history of the genesis of the hindu-arabic numeration system. The fragment would be reworked and updated in 1829 but it would remain but a fragment of a bigger, unfinished project. Nevertheless, the sketch would prove to be an important source of inspiration for the linguistic studies of his brother Wilhem (1830-1835) on the one hand, and for the prudent beginnings of a history of mathematics (Rosen, Woepcke, Nesselmann) on the other hand. Especially the reflections on the transfer and the transformation of numerals proved to be stimulating and led to the more general question of transfer between cultures and of contact between languages, problems that are still highly topical today.

10:40 – 11:50
Ivahn Smadja (Univ. Paris Diderot, SPHERE)

More on Brahmagupta in Germany : Which Quadrilaterals ? Which Proofs ?

As a result of Colebrooke’s 1817 English translations of Brahmagupta’s and Bhaskara’s mathematical works spreading across Europe among scholarly circles, the French geometer Michel Chasles pointed out the importance of a group of sutras within the mathematical chapters of Brahmagupta’s Brahma-sphuta-siddhanta, viz. BSS XII 21-38. He argued that in spite of the sutras being mere statements unsupported by individual proofs, the whole system of them made sense as providing a unified geometrical theory which, in his view, would solve completely, with precision and in all its generality, one single question, viz. how to construct a cyclic quadrilateral whose sides, diagonals, perpendiculars, segments and area, as well as the diameter of the circumscribed circle, may be expressed in rational numbers. A few years later, that interpretive puzzle and Chasles’ purported solution to it aroused rewewed interest in Germany. At about the same time he developed his theory of ideal numbers, E. E. Kummer also fell under the spell of Brahmagupta’s quadrilaterals. In connecting the problem of rational quadrilaterals, duly transposed into an algebraic framework, to Eulerian methods in Diophantine analysis, Kummer established, through reconstructing Brahmagupta’s methods, that Chasles wrongly imputed generality to Brahmagupta’s theory. Eventually, more than two decades later, by virtue of his conjoining philological precision and mathematical expertise, Hermann Hankel outlined an alternative interpretation of BSS XII 21-38, which would do justice to Chasles’ main insight, though in a minor key, owing to Kummer’s rectification. The present paper unfolds this reading against the backdrop of Chasles’ and Kummer’s previous attempts, thereby showing how it also qualifies as the cornerstone of Hankel’s characterization of so-called Indian intuitive ‘proofs’, by contrast to Greek deductive ones.

11:50 – 12:05 pause

12:05 – 13:15
Pierre Chaigneau (Univ. Paris Diderot, SPHERE)

Some links between Otto Neugebauer’s work and the history of ancient sciences as made before him in Germany.

Otto Neugebauer (1899-1990) entered the field of history of mathematics in 1926 with his doctoral dissertation, which focuses on the use of fractions in Old Egypt. For the next decades, he will be known as one of the greatest specialist of the sciences in Antiquity, bringing new methods and theses. This presentation look back at his influences. It is usual to recall his prestigious mathematical entourage at Göttingen, but the focus here will rather be on his references, in his first writings, to previous German historians of sciences, especially Hermann Hankel (1839-1873).

2 juin, 9:30 – 13:00

Autour d’Heinrich Weber

Séance organisée par Emmylou Haffner (Univ. Paris Diderot, SPHERE).

19 mai, 9:30 – 17:00

Matinée : Le Zīj al-Sindhind d’al-Khwarizmi

Séance organisée par Matthieu Husson (SPHERE), dans le cadre du projet ERC SAW "Mathematical Sciences in the Ancient World".

José Chabás (Universitat Pompeu Fabra)

Equations and velocities of the Sun and the Moon in the Zij al-Sindhind by al-Khwarizmi.

The astronomical tables ascribed to al-Khwārizmī (early 9th century) reached al-Andalus quite soon after their compilation in the East, and had a considerable success in the Iberian Peninsula, where they were used for several centuries. This so-called Indian tradition coexisted in the Peninsula with the astronomical practice stemming from Ptolemy’s works, also referred to as the Greek tradition, which was predominant both in the West and the East during the Middle Ages.

The solar and lunar equations in the Zij al-Sindhind present some basic features that make them differ from those compiled by Ptolemy and his followers. In turn, al-Khwarizmi’s tables for the solar and lunar velocities depend on the equations of the corresponding luminaries, and thus gave different results than those in the Greek tradition.

In this presentation we will explain the different models underlying the tables for the equations and the velocities of both luminaries, highlight the different parameters involved in them, and present the methods used in their compilation.

Michio Yano (Université de Kyoto)

Indian elements of al-Khwarizmi’s astronomy.

al-Khwārizmī’s astronomical work Zīj al-Sindhind (early ninth century) is very important because there survived only a few Indian texts on mathematical astronomy before him. Only ͞Aryabhata, Varāhamihira, Brahmagupta, and Bhāskara I predate him. So-called modern S͞uryasiddhānta was barely formed in the time of al-Khwārizmī. We know no original Sanskrit text(s) on which al-Khwārizmī ’s Arabic work is based, nor his Arabic translation is known to us. What we have is only the Latin translation of Adelad of Bath (early twelfth century).

The Latin text was published by Suter in 1914. Neugebauer published English translation with commentary in 1962. This is an indispensable work for further study. I am very much indebted to Neugebauer’s work. Although I have no knowledge of Latin and I am poorly informed of Islamic-Arabic astronomy, I would like to give some comments from the viewpoint of the history of Indian astronomy. Especially I want to compare al-Khwārizmī’s planetary theory with that of India and raise some questions.

Après-midi :

– Présentation des travaux de Sabine Rommevaux

Les nouvelles théories des rapports mathématiques du XIVe au XVIe siècle.