New abū al-Majd ibn ‘Aṭiyya's multiplication and transmission to the Latin world

• Leila Hamouda (Université Tunis el-Manar)

Room: Amphithéâtre Gaston Darboux Abū al-Majd ibn ‘Aṭiyya ibn abī al-Majd al-Kātib wrote a mathematical text *Maqāla fī l-ḍarb wa l-qisma bi l-hindī min ghayr maḥw wa lā naql* before 1136, probably in Ifriqiya. To our knowledge, there is only one copy of this text, preserved at the British Library in the codex MS.7473 written in 1242. This text is divided into two parts. The first part presents a new algorithm for multiplying and dividing two integers, and the second gives a method for extracting the perfect fourth and fifth roots of an integer. In our presentation, we have chosen to focus on the part concerning multiplication. This new algorithm by abū al-Majd ibn ‘Aṭiyya is interesting not only because it is based on three lemmas that he proves through arithmetical-geometrical demonstrations, following a very precise hypothetic-deductive reasoning, but also because we have found that this new algorithm is taken up and explained in several texts written either in Latin in the beginning of the 13rd century or vernacular Italian from the 15th–16th centuries – beginning with Fibonacci in his *Liber Abbaci*, written in 1202, and ending with Tartaglia in his *La prima parte del general trattato di numeri, et misure* written between 1556 and 1560. However, we have not fond this algorithm in any Arabic mathematics books from the western or eastern regions of the Muslim-Arab world, written between the 9th and 15th centuries, among those that describe the operation of multiplication – noting that the copy of this text preserved at the British Library was written in an Oriental Arab calligraphy. Nevertheless, this algorithm is found, besides in the book of Fibonacci and of Tartaglia mentioned before, in the text *The Treviso Arithmetic* of 1478 written by an unknown author, and in Pacioli's *Summa de Arithmetica Geometria. Proportioni et Proportionalita* written in 1494. We have even observed the longevity of abū al-Majd ibn ‘Aṭyyia's algorithm up to the present day; indeed, we know that this algorithm still appears in Italian school textbooks. What does abū al-Majd ibn ‘Aṭyyia's algorithm of multiplication consist of, and what are the lemmas on which it is based? What are the strengths of the proofs of these lemmas? What is the name of abū al-Majd ibn ‘Aṭyyia's algorithm in these Latin and Italian texts? Why was this name chosen? Why did this algorithm spread and acquire an important place in the calculation traditions of the Latin world? What are the similarities and the differences between abū al-Majd ibn ‘Aṭyyia 's work and the works of these Latin mathematicians – not in a comparative spirit, but in a logic that will allow us to draw conclusions about the former's work through that of the other mathematicians? We will try to answer all these questions in our presentation.

Présentation du projet SNSF MediMath

• Eleonora Sammarchi (Université de Berne)

Room: Amphithéâtre Gaston Darboux

"Démontrer les équations" grâce à la géométrie : al-Khayyām et les Données d'Euclide

• Nicolas Grégoire-Veyrié (SNSF MediMath – Université de Berne)

Room: Amphithéâtre Gaston Darboux À la fin du IXe siècle, 'Umar al-Khayyām propose une résolution de toutes les formes d'équation cubique, à l'aide de sections coniques. Dans cet ouvrage transparaissent des éléments typiques mais mal connus de la géométrie ancienne : l'usage des Données d'Euclide et des échos des théories d'Analyse-Synthèse. Ceux-ci traduisent une réflexion certaine quant au fonctionnement d'une démonstration. Que peut-on apprendre de leurs relations à l'algèbre et de l'évolution conjointe de celle-ci avec la géométrie classique.

Présentation du Projet I3

• Graziana Ciola (Radbound University)

Room: Amphithéâtre Gaston Darboux

Torturing Cardano

• Mark Thakkar (ERC I3-Radbound University)

Room: Amphithéâtre Gaston Darboux Girolamo Cardano’s Ars magna (1545) famously introduced complex conjugates as apparent solutions to certain “impossible” problems. It is often said that he had to dismiss “the mental tortures” before operating with such numbers, but this is the result of a mistranslation that has largely gone undetected for decades. After establishing as much, I will show that this misunderstanding is the tip of a hitherto unnoticed iceberg.