Aspects Analytiques Dans la Mathematique De Sharaf Al-D\^in Al-T\^US\^I

TL;DR

This paper revisits Sharaf al-Din al-Tusi's mathematical work, highlighting its analytical aspects and foundational significance, challenging the view that it was confined to Euclidean geometry, and clarifying ambiguous passages.

Contribution

It demonstrates the analytical methods and reasoning in al-Tusi's work, establishing its importance in the history of mathematical analysis and questioning prior conjectures about its scope.

Findings

Al-Tusi introduces notions and procedures relevant to analysis.

His work extends beyond Euclidean geometry into advanced mathematics.

Clarifies previously ambiguous passages in the text.

Abstract

The analytical aspects of the "Trait\'e des \'equations" of Sharaf al-D\^in al-T\^us\^i (2nd half of the XIIth century) have been underlined by R. Rashed (1974, 1986). In the present paper, we consider again some of those aspects, when studying the "second part" of the "Trait\'e". We find out that al-T\^us\^i introduces certain notions, develops reasonings and uses calculating procedures that may allow his work to be an important reference in the history of the mathematical analysis. As such, one is led to examine the foundations of a J.P. Hogendijk conjecture (1989), which situates the advanced mathematics of this "2nd Part" in the only domain of Euclidian geometry. Simultaneously, we shall try to shed some light on a passage of the "Trait\'e", made unclear by al-T\^us\^i style and expression. Our study is essentially based on the texts to confirm the originality and the importance of these XIIth century mathematics. I would like to express my thanks to Prof. C. Houzel (Univ. Paris VII) and B. El-Mabsout (Univ. Paris VI), for reading the manuscript and providing their precious remarks.