Naṣīr al-Dīn al-Ṭūsī

Naṣīr al-Dīn al-Ṭūsī was born on 18 February 1201 in Ṭūs and died on 26 June 1274 in Kāẓimayn, a shrine district within greater Baghdad (Britannica records the death place simply as Baghdad; the older MacTutor entry gives Kāẓimayn). His father was a jurist in the Twelfth Imam tradition; his early education was in Ṭūs and Nishapur, and one of his teachers in Nishapur was Kamāl al-Dīn ibn Yūnus, the student, in turn, of the older Sharaf al-Dīn al-Ṭūsī. The two Ṭūsīs are connected through this intermediate teacher.

Alamūt, then the Mongols

Around 1227, with Genghis Khan's forces already through Khorāsān, al-Ṭūsī accepted sanctuary from an Ismāʻīlī governor and eventually moved to the Ismāʻīlī capital at Alamūt. The years at Alamūt are when he wrote much of his philosophy and logic, an Ismāʻīlī theology (Taṣawwurāt, “Notions”), and his commentary on the Almagest (Taḥrīr al-Majisṭī, 1247). In 1256 Hülegü Khan's forces took Alamūt; al-Ṭūsī was among those taken. Whether he was present at, or complicit in, the fall of Baghdad in 1258 is contested by modern historians; Britannica describes the question as “controversial” and lists both sides. The later record is clear: he became scientific adviser to the Mongol court, and in 1259 obtained Hülegü's support for an observatory.

The Marāgha observatory

Construction began in 1259 near Hülegü's capital at Marāgha (now in Azerbaijan, northwestern Iran). The observatory was operational by 1262, supported by an endowment dedicated to research, and staffed with astronomers and instrument makers from across the Mongol domains, including Chinese scholars. Britannica records that research continued at Marāgha for at least twenty five years after al-Ṭūsī's death; some of its instrument designs would later influence the Samarkand observatory of Ulugh Beg, where al-Kāshī would work (see the next entry).

The observatory's main product was the Zīj-i Īlkhānī (“Īlkhānic Tables,” c. 1271), a set of planetary tables based on twelve years of observation.

The Tadhkira and the Ṭūsī couple

His most consequential astronomical work is al-Tadhkira fī ʻilm al-hayʻa (“Memoir on Astronomy”). Its most quoted device is a geometric construction now called the Ṭūsī couple: given a small circle rolling inside a larger circle of exactly twice its radius, a point on the small circle traces a linear diameter of the larger one. Two uniform circular motions, combined, produce oscillation along a line.

He used the couple to reform Ptolemy's planetary models, specifically, to replace variable circular motions (the equant) with combinations that are uniform. The resulting “Ṭūsī couple” and the associated revised constructions later appear in the astronomy of the Marāgha school, and the same geometric device turns up two and a half centuries later in Copernicus's De Revolutionibus (Book III, chapter 4).

The route of the device from Marāgha to Renaissance Europe remains contested. Britannica reports that “most historians of Islamic astronomy believe that the planetary models developed at Marāgha found their way to Europe (perhaps via Byzantium) and provided Nicolaus Copernicus with inspiration for his astronomical models”, which presents a transmission hypothesis with limited documentary evidence. F. Jamil Ragep and others have argued the case in detail; other scholars have proposed that Copernicus took a similar construction from Proclus's commentary on the first book of Euclid, which was in Latin circulation. We present the transmission question as open.

Trigonometry as an independent subject

Before al-Ṭūsī, trigonometry appeared as a service discipline inside astronomy: it existed to make tables of sines usable by astronomers. His Treatise on the Quadrilateral is the first surviving exposition of plane and spherical trigonometry as an autonomous mathematical subject with its own definitions, propositions, and problems.

The treatise gives a clean statement of the law of sines for plane triangles,

\(\dfrac{a}{\sin A} \;=\; \dfrac{b}{\sin B} \;=\; \dfrac{c}{\sin C},\)

along with its spherical counterpart. His earlier commentary on the Almagest (1247) had already produced sine tables accurate to three sexagesimal places at every half degree.

The nth root manuscript

A 1265 manuscript in al-Ṭūsī's hand contains a worked treatment of nth root extraction, including the binomial coefficients arranged in the pattern we now call Pascal's triangle. MacTutor's biographical entry is careful about attribution here: the method belongs to al-Karajī's school (see the archive entry for al-Karajī); al-Ṭūsī's manuscript preserves a version of that tradition. This page follows the same attribution.

Related curriculum links in the curriculum

Naṣīr al-Dīn reaches MOTA's geometry and trigonometry strands: the sine rule for plane and spherical triangles is still the tool every olympiad student needs to move between angles and lengths, and the Ṭūsī couple is the cleanest classroom example of decomposing a rectilinear motion into circular components, a habit that reappears every time a student expresses harmonic motion as the sum of two rotations.

Sources

1. MacTutor History of Mathematics, University of St Andrews, Naṣīr al-Dīn al-Ṭūsī biography (birth 18 February 1201 Ṭūs; death 26 June 1274 Kāẓimayn; education under Kamāl al-Dīn ibn Yūnus, whose own teacher was Sharaf al-Dīn al-Ṭūsī; Alamūt years to 1256; Hülegü's support for the observatory from 1259 and its operation from 1262; the eight sphere lunar model in the Tadhkira; the sine tables to three sexagesimal places per half degree in the Taḥrīr al-Majisṭī of 1247; the Treatise on the Quadrilateral as the first independent exposition of trigonometry; the 1265 nth root manuscript as a version of al-Karajī's school's methods). https://mathshistory.st-andrews.ac.uk/Biographies/Al-Tusi_Nasir/
2. Encyclopædia Britannica, “Naṣīr al-Dīn al-Ṭūsī”, secondary verification of birth and death dates. Britannica lists Baghdad as the death place, while MacTutor lists Kāẓimayn, a shrine district within greater Baghdad, so the two accounts are compatible. The entry also supports the contested framing of his role in the Mongol capture of Baghdad, the observatory chronology and endowment, the 1271 date of the Zīj-i Īlkhānī, the Ṭūsī couple, and the transmission claim about Copernicus. https://www.britannica.com/biography/Nasir-al-Din-al-Tusi
3. F. Jamil Ragep, Naṣīr al-Dīn al-Ṭūsī's Memoir on Astronomy (al-Tadhkira fī ʻilm al-hayʻa), Sources in the History of Mathematics and Physical Sciences 12, 2 vols., New York: Springer-Verlag, 1993, the standard critical edition, English translation, and commentary on the Tadhkira; the source for detailed discussions of the Ṭūsī couple construction, the reform of Ptolemaic planetary models, and the transmission to Copernicus question.