Nasir al-Din al-Tusi

How to use this course

Nasir al-Din al-Tusi was the leading scientist of the thirteenth-century Islamic world. He wrote about 150 works, in Arabic and Persian, on astronomy, mathematics, logic, ethics, theology and medicine. For historians of science he matters for three achievements. He found a way to make straight-line motion out of uniform circular motions, now called the Tusi couple, and used it to build planetary models without Ptolemy’s equant. He wrote the first treatise to present trigonometry as a subject in its own right, apart from astronomy. And he persuaded the Mongol ruler Hülegü to build at Maragha, in north-west Iran, an observatory with a large staff, a library and a lasting endowment.

His life is better documented than that of most medieval scholars, but parts of it are disputed, above all his conduct when the Ismaili fortresses and then Baghdad fell to the Mongols. Part I keeps the documented facts apart from the accusations and legends. Part II is technical. Its exercises use modern notation for arguments he made in the language of spheres and arcs.

This course follows the course on Claudius Ptolemy, whose Almagest al-Tusi edited, and the course on Ibn al-Haytham, whose Doubts concerning Ptolemy set the problem he solved. Lesson 07 of the course on Nicolaus Copernicus shows the couple at work in De revolutionibus.

Timeline

Most dates in al-Tusi’s life are given in his sources in the hijri calendar. Some events, including his studies, are dated only roughly. One modern biography gives his birth and death a day earlier, as 17 February 1201 and 25 June 1274.

Tus, Nishapur and the Mongol storm, 1201–c. 1230

Muḥammad ibn Muḥammad ibn al-Ḥasan al-Ṭūsī was born in Tus, in Khorasan in north-eastern Iran, on 18 February 1201. The honorific Naṣīr al-Dīn, “helper of the faith”, was added later, and Persian writers often call him simply Khwāja, “the master”. His family belonged to the Twelver Shiʿi community. His father, a jurist, taught him the religious sciences and encouraged him to study the rational ones too. A maternal uncle is said to have taught him logic and natural philosophy, and other teachers taught him algebra and geometry.

As a young man he went to Nishapur, the great city of Khorasan, where he studied philosophy in the tradition of Ibn Sīnā (Avicenna) with Farīd al-Dīn Dāmād and medicine with Quṭb al-Dīn al-Miṣrī. He studied advanced mathematics with Kamāl al-Dīn ibn Yūnus, one of the best mathematicians of the age. The Dictionary of Scientific Biography places this study at Nishapur; more recent accounts place it at Mosul in northern Iraq, where Ibn Yūnus taught, in the 1220s.

The invasion

In 1219–21 the armies of Chinggis Khan overran Khwarazm and Khorasan. Nishapur was destroyed in 1221, and Tus suffered too. The world of patronage in which a young scholar might have found a post at a court or a college was broken. Al-Tusi was then about twenty. For the next three decades the Ismaili fortresses of Quhistan and the Alborz mountains were among the few places in Iran that the Mongols did not control.

Among the Ismailis, c. 1230–1256

The Nizari Ismailis, a Shiʿi community distinct from al-Tusi’s own, held a chain of mountain fortresses in Iran, with their centre at Alamut. In the early 1230s al-Tusi entered the service of Nāṣir al-Dīn Muḥtasham, the Ismaili governor of Quhistan, and later lived at Alamut and other strongholds. Whether he went willingly, and how free he was to leave, has been debated since the Middle Ages. The Dictionary of Scientific Biography judged that he was probably not free to go. Some of the works of these years are written in an Ismaili idiom; after 1256 he wrote as a Twelver Shiʿi.

The works of the fortress years

• Akhlāq-i Nāṣirī (the Nasirean Ethics), written for his patron in the 1230s, became the most widely read work on ethics in Persian.
• Risāla-yi Muʿīniyya (1235), a Persian summary of astronomy. In it he promised an “elegant” solution to a problem in Ptolemy’s lunar model without giving it.
• Ḥall-i mushkilāt-i Muʿīniyya (Solution of the difficulties of the Muʿīniyya), completed in October 1245, an appendix that first presents the device now called the Tusi couple, applied to the Moon.
• Taḥrīr al-Majisṭī, completed on 13 February 1247, a recension of the Almagest with his own comments, including a second version of the couple.
• Recensions of the “intermediate books” of Greek mathematics, the works students read between Euclid and Ptolemy, by Autolycus, Aristarchus, Theodosius, Hypsicles, Menelaus and others. Together with his recensions of Euclid and Ptolemy they became standard teaching texts for centuries.

A taḥrīr was more than an edition. Al-Tusi restated each work, filled gaps in the proofs, added alternative proofs and noted where he disagreed. For Euclid’s fifth postulate this led him to the work described in Lesson 08.

Hülegü, Baghdad and Maragha, 1256–1259

In 1256 the Mongol prince Hülegü, brother of the Great Khan Möngke, led a campaign against the Ismailis. The fortresses surrendered one by one, and Alamut fell in December. Al-Tusi was among those who came out. Later writers accused him of having urged the surrender to save himself; others say he negotiated it to save lives. The sources do not settle the question. Hülegü, who took astrology seriously, took him into his service as adviser and astrologer, and later gave him charge of the religious endowments (awqāf) across his realm.

The fall of Baghdad

In February 1258 Hülegü took Baghdad. The city was sacked and the last Abbasid caliph, al-Mustaʿṣim, was put to death. Al-Tusi accompanied the army. Some later Sunni authors blamed him, as a Shiʿi, for encouraging the attack; the charge rests on polemic written long afterwards. He is credited, by sympathetic sources, with protecting scholars and books, and he later visited Shiʿi centres in Iraq such as Hilla.

The observatory

Hülegü made Maragha, in Azerbaijan, his capital. In 1259 construction began on an observatory on a hill west of the town, under al-Tusi’s direction. Hülegü funded it through an endowment, a waqf, administered by al-Tusi, which allowed the institution to outlive its founder. Lesson 09 describes its buildings, instruments and staff.

The Tusi couple

For the general reader. Ancient astronomy allowed only uniform circular motions, because the heavens were made of spheres that turn evenly. Some of Ptolemy’s models needed a body to move closer and farther along a straight line. Al-Tusi showed how two spheres, each turning uniformly, can produce exactly that.

Geometric proof

Let \(O\) be the centre of the large circle, \(K\) the centre of the small one, and let the small circle pass through \(O\), as it always does, since its diameter equals the large radius. Start with the marked point \(P\) at the point of tangency \(T_0\) on the large circle. When \(OK\) has turned through angle \(\theta\), the point of tangency \(T\) is at angle \(\theta\) on the large circle, and the arc \(T_0T\) of the large circle has length \(2r\theta\). Rolling without slipping means the arc \(TP\) on the small circle has the same length, so it subtends the angle \(2\theta\) at \(K\). By the inscribed angle theorem, this arc subtends \(\theta\) at \(O\), which lies on the small circle. So \(\angle TOP=\theta=\angle TOT_0\), and \(P\) lies on the line \(OT_0\).

Vector proof

Put \(O\) at 0 and \(T_0\) at \(2r\) on the real axis. The centre is \(K=re^{i\theta}\). Relative to fixed directions, the small circle turns at angle \(-\theta\), so

The point moves with simple harmonic motion along the diameter \([-2r,2r]\). Kennedy, who named the device in 1966, described it this way: two linked vectors of equal length, the second turning at twice the rate of the first and in the opposite sense relative to it.

The first statement is in the Ḥall of 1245; the version in the Taḥrīr al-Majisṭī of 1247 is the one most often cited, and the Tadhkira of 1261 gives the fullest treatment, with a proof.

Proclus, in his commentary on Euclid, had discussed the reverse question, circular motion composed from straight motions. Nicole Oresme described a related rolling construction in the fourteenth century. In the nineteenth century the same mechanism appeared in engines as a straight-line linkage, and Gerolamo Cardano’s name is often attached to it.

Planets without the equant

For the general reader. In Ptolemy’s planetary models the carrier of each planet turns evenly about a point, the equant, that is not its centre. Ibn al-Haytham had objected that no real sphere can do this. Al-Tusi kept the even turning about the equant and used a couple to slide the planet’s epicycle in and out, so that it follows Ptolemy’s circle closely while every sphere turns evenly about its own centre.

The Tadhkira

Al-Tadhkira fī ʿilm al-hayʾa (Memoir on Astronomy), first issued at Maragha in 1261, describes the cosmos as a system of solid spheres. Its second book catalogues the difficulties in Ptolemy’s models, perhaps the most thorough medieval account of them, and presents his alternatives. F. Jamil Ragep’s edition and translation (1993) is the standard modern study.

The equant

Take the deferent radius as 1. Put the Earth \(O\) at 0, the deferent centre \(C\) at \(e\), and the equant \(Q\) at \(2e\) on the real axis, toward the apogee. Ptolemy’s epicycle centre \(Z\) lies on the circle \(|Z-e|=1\), and its direction from \(Q\) is the mean anomaly \(M\), which grows uniformly. So \(Z=2e+\rho(M)e^{iM}\), where \(\rho\) solves \(|e+\rho e^{iM}|=1\):

Al-Tusi’s model, in modern terms

The following is a modern reconstruction of the effect of his arrangement for the upper planets. Let a sphere centred at \(Q\) turn uniformly, carrying the line \(QZ\) at angle \(M\). On this line, a couple whose large circle has diameter \(2e\) moves the epicycle centre in and out, completing one oscillation per revolution, nearest \(Q\) at apogee. Then

The direction is exact because both points lie on the ray from \(Q\) at angle \(M\); the distance follows from the expansion of \(\rho\). In this sense al-Tusi achieved what Ibn al-Haytham had demanded, a physical arrangement of evenly turning spheres that reproduces Ptolemy’s predictions.

The Moon, latitude and Mercury

The first application of the couple, in 1245, was to the Moon: Ptolemy’s lunar model made the epicycle’s centre move non-uniformly, and al-Tusi’s replacement used a couple to produce the required variation. For the latitudes of the planets, which required the planes of the epicycles to tilt back and forth, the Tadhkira gives a second, “curvilinear” version in which the oscillation takes place along an arc of a great circle on a sphere. The equality there is approximate, and al-Tusi said so. For Mercury, whose Ptolemaic model has a moving deferent centre, the Tadhkira concedes that he had found no satisfactory solution.

The Maragha school and Copernicus

For the general reader. Al-Tusi was not alone. His colleagues at Maragha and their successors produced several sets of equant-free models over the next century, culminating in those of Ibn al-Shāṭir in Damascus. Two centuries later Copernicus used the same devices. Whether he learned of them, and how, is one of the main open questions in the history of astronomy.

Al-ʿUrḍī’s lemma

Muʾayyad al-Dīn al-ʿUrḍī, who came to Maragha from Damascus and built its instruments, had worked out his own equant-free models before he arrived. They rest on a simple lemma.

Proof. In complex terms, \(D=A+\ell e^{i\alpha}\) and \(C=B+\ell e^{i\alpha}\), so \(C-D=B-A\).

The ʿUrḍī model

Put the deferent centre at \(\tfrac32e\), halfway between Ptolemy’s centre and equant. Let the epicycle centre ride on a small epicycle of radius \(\tfrac e2\), carried by the deferent and turning so that

A direct computation gives \(Z_U-2e=e^{iM}-\tfrac e2\big(1+e^{2iM}\big)=e^{iM}(1-e\cos M)\). This is exactly al-Tusi’s \(Z_T\) of Lesson 05. The two models put the epicycle centre at the same point. The ʿUrḍī lemma is the geometric form of this identity: the uniform motion about the deferent centre, shifted by the small epicycle, is seen as uniform from the equant.

From Maragha to Damascus

Quṭb al-Dīn al-Shīrāzī, al-Tusi’s most distinguished student, produced further models. Ibn al-Shāṭir (d. 1375), timekeeper of the Umayyad Mosque in Damascus, built models without equant or eccentric for the Sun, Moon and planets, using epicyclets and the couple. His lunar model and his model for Mercury are the same as those Copernicus later used.

Transmission

In De revolutionibus III.4 Copernicus proves the couple theorem and uses it for the slow oscillation he attributed to the equinoxes. Historians have pointed to the close match between his diagram and al-Tusi’s, including the lettering of corresponding points. A Byzantine route is the most discussed: Gregory Chioniades (d. c. 1320) brought astronomy from Iran into Greek, including a lunar model with the couple, and Greek manuscripts containing the couple were in Italy by the fifteenth century, when Copernicus studied there. No document shows that Copernicus saw them, and some historians still argue for independent discovery.

Trigonometry as a science

For the general reader. For the Greeks, trigonometry was a tool inside astronomy, built on one theorem of Menelaus about a figure of four intersecting arcs. Astronomers of the ninth to eleventh centuries introduced the sine, tangent and other functions and found simpler theorems. Al-Tusi gathered this work into a treatise that, for the first time, presented plane and spherical trigonometry as a subject in its own right.

The treatise

Kashf al-qināʿ ʿan asrār al-shakl al-qaṭṭāʿ (Uncovering the secrets of the secant figure), known in Europe as the Treatise on the Quadrilateral, has five books. Book I treats compound ratios, and argues that every ratio can be treated as a number. Book II treats the plane secant figure, Books III and IV the spherical one, and Book V gives methods that dispense with the figure altogether, using the sine theorem and related rules. It builds on Abū al-Wafāʾ, Abū Naṣr Manṣūr ibn ʿIrāq and al-Bīrūnī.

Proof of (b). Drop the great-circle arc \(h\) from \(C\) perpendicular to \(AB\). In each of the two right triangles, the rule for a right spherical triangle, \(\sin(\text{side opposite an angle})=\sin(\text{hypotenuse})\sin(\text{angle})\), gives \(\sin h=\sin b\sin A\) and \(\sin h=\sin a\sin B\). Hence \(\sin a/\sin A=\sin b/\sin B\). The right-triangle rule itself follows from a short computation with vectors, or, as in the treatise, from the secant figure. Part (a) is the limit of (b) for small triangles, or follows the same way with \(h=b\sin A=a\sin B\).

Right triangles and the polar triangle

For a right spherical triangle (\(C=90^\circ\)) the treatise derives the relations that solve all its cases, including

For a triangle known only by its three angles, al-Tusi used the polar triangle. The poles of the three sides form a second triangle whose sides are the supplements of the first triangle’s angles, and whose angles are the supplements of its sides. A problem about angles becomes a problem about sides.

The parallel postulate

For the general reader. Euclid’s fifth postulate, about when two lines meet, seemed to many readers to need a proof. Ibn al-Haytham and Omar Khayyam had tried. Al-Tusi criticized them and tried again, by studying a quadrilateral with two equal sides perpendicular to a base. His arguments failed, as all such arguments must, but on the way he proved theorems that hold in non-Euclidean geometry too.

The texts

Al-Tusi treated the question in his recension of Euclid and in a separate work, al-Risāla al-shāfiya ʿan al-shakk fī al-khuṭūṭ al-mutawāziya (The treatise that heals doubt about parallel lines). A different Arabic recension of Euclid, printed at Rome in 1594 under his name, is now generally attributed to a later writer of his school. It was this printed text, with its argument on parallels, that John Wallis discussed in 1663 and that Girolamo Saccheri knew when he wrote Euclides ab omni naevo vindicatus (1733).

Proof. Triangles \(DAB\) and \(CBA\) are congruent (SAS: \(AD=BC\), right angles at \(A\) and \(B\), common side \(AB\)), so \(DB=CA\). Then triangles \(ADC\) and \(BCD\) are congruent (SSS), and \(\angle ADC=\angle BCD\).

There are three possibilities: both summit angles right, both obtuse, or both acute. Al-Tusi tried to exclude the obtuse and acute cases. In the acute case, he noted, the angle sum of a triangle would be less than two right angles, which is a property of the hyperbolic geometry found by Nikolai Lobachevsky and János Bolyai in the nineteenth century. To exclude it, he relied on a claim about how the distance between two lines behaves, which is equivalent to the postulate he wanted to prove. Saccheri’s later treatment followed the same three-hypothesis plan.

The observatory and the Ilkhani tables

For the general reader. Maragha is often called the first observatory in the modern sense: a permanent institution with buildings, large instruments, a library, a school and a salaried staff, funded by an endowment. Its main product was a set of astronomical tables for the Mongol rulers, the Zīj-i Īlkhānī.

Site and staff

The buildings stood on a flat-topped hill west of Maragha, on a site about 400 m by 150 m: a central tower, circular platforms, a foundry and workshop, living quarters and a library. Excavations in 1972 uncovered the foundations. The staff included Muʾayyad al-Dīn al-ʿUrḍī, who designed and built the instruments; Quṭb al-Dīn al-Shīrāzī; Muḥyī al-Dīn al-Maghribī; Najm al-Dīn Dabīrān al-Kātibī al-Qazwīnī, the logician; al-Tusi’s sons Ṣadr al-Dīn and Aṣīl al-Dīn; and a Chinese scholar whose name the sources give as Fao Mun-ji. Ṣadr al-Dīn succeeded his father as director.

Instruments

Al-ʿUrḍī’s instruments, built in 1261–62, included a large mural quadrant fixed in the meridian, an armillary sphere of five rings with a sighting alidade, a solstitial armill, an azimuth ring with two quadrants, and a parallactic ruler. Size was the point: a larger graduated arc allows finer divisions.

The tables

The Zīj-i Īlkhānī, in Persian, was completed in 1272 under Hülegü’s son Abaqa, after about twelve years of work. It contains tables for the Sun, Moon and planets, a star list, and chronological material that includes the Chinese calendar. It takes the precession of the equinoxes as 51″ per year. Later astronomers found that many of its planetary parameters were taken from earlier tables rather than from new observations, and criticized them. Muḥyī al-Dīn al-Maghribī wrote an independent work based on Maragha observations. John Greaves published part of the tables in Latin in 1650.

Roots and binomial coefficients

For the general reader. Al-Tusi also wrote on arithmetic. In a treatise of 1265 he explained how to extract roots of any degree from a whole number, digit by digit, using the coefficients of \((x+y)^n\). The method probably continues the work of the school of al-Karajī and al-Samawʾal in the eleventh and twelfth centuries.

The method

To find the \(n\)-th root of \(N\), guess the leading part \(x\) of the root (the tens digit, say, times 10), and seek the next digit \(y\) from

The coefficients come from the triangle in which each entry is the sum of the two above it, \(\binom nk=\binom{n-1}{k-1}+\binom{n-1}{k}\). In Jawāmiʿ al-ḥisāb al-Tusi took as his example the sixth root of \(244{,}140{,}626\). The root is 25, with remainder 1; the computation is laid out in a table resembling the scheme later named after William Horner.

Scholar of the Ilkhanate, 1259–1274

In his last fifteen years al-Tusi was the most powerful scholar in the Mongol state. He ran the observatory, administered religious endowments throughout the Ilkhanate, and continued to write. Hülegü died in 1265 and Abaqa succeeded him; al-Tusi kept his position. The Tadhkira (1261), the Jawāmiʿ al-ḥisāb (1265) and the Ilkhani tables (1272) all belong to these years, as do much of his philosophy and theology.

• Philosophy. His commentary on Ibn Sīnā’s al-Ishārāt wa-l-tanbīhāt defended Avicenna against the criticisms of Fakhr al-Dīn al-Rāzī and helped revive Peripatetic philosophy in Iran.
• Theology. Tajrīd al-iʿtiqād, a compact statement of Twelver Shiʿi doctrine in the form of philosophical theology, attracted hundreds of commentaries and glosses.
• Logic. Asās al-iqtibās, in Persian, and other works in the tradition of Ibn Sīnā.
• Other subjects. A Persian translation of al-Ṣūfī’s Book of the Fixed Stars, a work on precious stones, the Tanksūkh-nāma, based largely on al-Bīrūnī, and works on medicine and astrology.

In 1274 he left Maragha for Baghdad with a group of students. He fell ill there and died on 26 June 1274. He was buried near the shrine of the seventh Shiʿi imam, Mūsā al-Kāẓim, at Kadhimiya on the edge of Baghdad. His students included Quṭb al-Dīn al-Shīrāzī, astronomer and physician, and the theologian al-ʿAllāma al-Ḥillī.

The long reach of Maragha

The Maragha observatory declined after Abaqa’s death in 1282, and by the early fourteenth century it had fallen out of use, through lack of funds and earthquake damage. Its model lived on. Ulugh Beg’s observatory in Samarkand, built in the 1420s, followed it in scale and organization, and through Samarkand the idea of a large state-funded observatory reached the Ottoman world and, by the time of Tycho Brahe, Europe.

Assessment

• Astronomy: the couple and the equant-free models. With his colleagues and successors, from al-ʿUrḍī and al-Shīrāzī to Ibn al-Shāṭir, he created a tradition of non-Ptolemaic planetary theory whose devices reappear in Copernicus. The Tadhkira was studied, and commented on, for centuries.
• Mathematics: trigonometry as an independent discipline; the sine theorem in both plane and spherical form given a central place; the polar triangle; theorems on the quadrilateral that Saccheri later developed; root extraction by binomial coefficients.
• Teaching texts: his recensions of Euclid, Ptolemy and the intermediate books were the standard forms in which the Islamic world read Greek mathematics for centuries.
• Philosophy and religion: the Nasirean Ethics, the commentary on Ibn Sīnā and the Tajrīd remain studied texts.
• Honours: the lunar crater Nasireddin and asteroid 10269 Tusi; the K. N. Toosi University of Technology in Tehran; the Shamakhy Astrophysical Observatory in Azerbaijan bears his name.

Connections

People linked to Nasir al-Din al-Tusi across this collection. A name without a link has no page here yet. See Nasir al-Din al-Tusi in the network of all 337 lives →

Teachers and advisors

• Kamal al-Din ibn Yunus 1220s

Students

• al-Allama al-Hilli
• Qutb al-Din al-Shirazi

Influenced by

• Claudius Ptolemy
• Euclid
• Ibn al-Haytham
• Ibn Sina
• Omar Khayyam

Influenced

• Giovanni Girolamo Saccheri
• John Wallis
• Nicolaus Copernicus
• Regiomontanus
• Ibn al-Shatir

Colleagues and collaborators

• Mu'ayyad al-Din al-Urdi 1259-1266
• Muhyi al-Din al-Maghribi

Family

• Asil al-Din al-Tusi
• Sadr al-Din al-Tusi

Also mentioned in this life

• al-Biruni
• Aristarchus of Samos
• János Bolyai
• Tycho Brahe
• Gerolamo Cardano
• Joseph Fourier
• Nikolai Lobachevsky
• Proclus

Fields

• astronomy 50 lives
• trigonometry 9 lives
• geometry 32 lives
• philosophy 12 lives
• theology 8 lives

On CoursesHub

• Astrophysics for Beginners

Sources and further reading

Biography and context

1. S. H. Nasr, al-Ṭūsī, Muḥammad ibn Muḥammad ibn al-Ḥasan, Dictionary of Scientific Biography, vol. 13, Scribner, 1976.
2. J. J. O’Connor and E. F. Robertson, Nasir al-Din al-Tusi, MacTutor History of Mathematics.
3. F. J. Ragep, Ṭūsī: Abū Jaʿfar Muḥammad ibn Muḥammad ibn al-Ḥasan Naṣīr al-Dīn al-Ṭūsī, Biographical Encyclopedia of Astronomers.
4. Nasir al-Din al-Tusi and Maragheh observatory, Wikipedia.
5. A. Sayılı, The Observatory in Islam, Turkish Historical Society, 1960.

Works and mathematics

1. F. J. Ragep, Naṣīr al-Dīn al-Ṭūsī’s Memoir on Astronomy (al-Tadhkira fī ʿilm al-hayʾa), 2 vols., Springer, 1993.
2. F. J. Ragep, From Tūn to Toruń: the twists and turns of the Ṭūsī-couple, in R. Feldhay and F. J. Ragep (eds.), Before Copernicus, McGill-Queen’s University Press, 2017.
3. Tusi couple, Wikipedia.
4. E. S. Kennedy, Late medieval planetary theory, Isis 57 (1966).
5. G. Saliba, A History of Arabic Astronomy: Planetary Theories during the Golden Age of Islam, New York University Press, 1994; Islamic Science and the Making of the European Renaissance, MIT Press, 2007.
6. N. Swerdlow and O. Neugebauer, Mathematical Astronomy in Copernicus’s De Revolutionibus, Springer, 1984.
7. A. Carathéodory Pacha, Traité du quadrilatère attribué à Nassiruddin-el-Toussy, Constantinople, 1891.
8. Spherical trigonometry before the modern era: the treatise of Nasir al-Din al-Tusi, arXiv:2605.03516.
9. G. Van Brummelen, The Mathematics of the Heavens and the Earth: The Early History of Trigonometry, Princeton University Press, 2009.
10. B. A. Rosenfeld, A History of Non-Euclidean Geometry, Springer, 1988.