Era V — Maragha and the Thirteenth Century (1200–1300)¶
After the Mongol conquest of Baghdad (1258), the scientific center of gravity moved to the great observatory at Maragha in northwestern Persia, founded 1259 under Hulagu Khan at the urging of Nasir al-Din al-Tusi. Maragha functioned as a research institute with a library, instrument workshops, international staff (including at least one Chinese astronomer), and a research program: repairing the physical inconsistencies of Ptolemaic astronomy with new mathematical devices. In the west, the Maghrebi school perfected the arithmetic textbook.
Nasir al-Din al-Tusi¶
Muhammad ibn Muhammad ibn al-Hasan al-Tusi (1201–1274) · Tus, Alamut, Maragha · trigonometry, astronomy, geometry, philosophy
The dominant scientific personality of the century: philosopher, theologian, astronomer, mathematician, and scientific administrator who persuaded a Mongol conqueror to fund an observatory.
Trigonometry as an independent science. His Kitab al-Shakl al-qatta ("Treatise on the Complete Quadrilateral," c. 1260) is the first comprehensive treatise in which trigonometry is developed for its own sake, detached from astronomy: systematic plane and spherical trigonometry, the six functions, the plane and spherical laws of sines with proofs, and the complete solution of triangles from given data, including the polar-triangle method for the case of three given angles on the sphere. Nothing comparable exists in Europe until Regiomontanus's De triangulis (written 1464), which covers much the same ground; whether Regiomontanus had indirect access to Tusi-school material remains debated among historians.
The Tusi couple. In his Tadhkira fi ilm al-hay'a ("Memoir on Astronomy," 1261), Tusi introduced the theorem that if one circle rolls inside another of twice the diameter, any point on the small circle traces a straight line, the "Tusi couple." It converts two uniform circular motions into linear oscillation, letting astronomers eliminate Ptolemy's equant and build planetary models from uniform circular motions only. Mathematically it is a special case of hypocycloid motion; mechanically it reappears in engineering as a straight-line linkage principle. The same geometric device, with the same diagram lettering in a key manuscript comparison, appears in Copernicus's De revolutionibus (1543), III.4; whether Copernicus reinvented it or received it via Byzantine Greek intermediaries (a Greek manuscript of Tusi-couple material reached Italy in the 15th century) is one of the liveliest debates in the history of astronomy. See Renaissance Connections.
Geometry and the parallels. His recension of Euclid and his discussion of the parallel postulate continued the Khayyam line; a related work published in Rome in 1594 under his name (now usually attributed to his son or school, "pseudo-Tusi") was read by John Wallis and cited by Saccheri in Euclides ab omni naevo vindicatus (1733), placing the Islamic parallels tradition documentably in the direct prehistory of non-Euclidean geometry.
Editions. His tahrir (recensions) of Euclid, Ptolemy, and the "middle books" (Archimedes, Apollonius, Menelaus, Theodosius, Autolycus, Thabit et al.) became the standard curriculum of Islamic mathematics for centuries, a one-man textbook publishing program.
Sources: DSB (S. H. Nasr, "al-Tusi"); F. J. Ragep, Nasir al-Din al-Tusi's Memoir on Astronomy (1993, edition and translation of the Tadhkira); Van Brummelen; Rosenfeld; Saliba, Islamic Science and the Making of the European Renaissance.
al-Urdi¶
Mu'ayyad al-Din al-Urdi (d. 1266) · Damascus, then Maragha · astronomy, geometry, engineering
Chief instrument builder of the Maragha observatory (his treatise describing its instruments survives) and author of Kitab al-Hay'a, which introduced Urdi's lemma: an equant-eliminating geometric transformation (an epicyclet device converting non-uniform motion about one center into uniform motion about another) developed before the observatory's founding. Urdi's lemma plus the Tusi couple form the standard toolkit of the Maragha planetary program, and the lemma reappears, again exactly, in Copernicus's models for the superior planets; Copernicus does not prove it, which historians (Saliba, Swerdlow) read as suggestive of inheritance rather than reinvention. That inference remains debated.
Sources: G. Saliba's edition and studies of al-Urdi; Swerdlow & Neugebauer, Mathematical Astronomy in Copernicus's De Revolutionibus; DSB.
Muhyi al-Din al-Maghribi¶
Muhyi al-Din al-Maghribi (c. 1220–1283) · al-Andalus/Maghreb, Damascus, Maragha · trigonometry, astronomy
A westerner who joined the Maragha staff after the fall of Damascus. He conducted a long observational program at Maragha (his records, analyzed by George Saliba, are among the best evidence of the observatory's actual working methods) and computed new planetary parameters. Mathematically, he worked on the classical problems (angle trisection, the regular heptagon), on Euclid's parallel postulate, and on precision computation of chords and sines, including an iterative determination of the chord of 1°, a line al-Kashi would perfect.
Sources: Saliba's studies of al-Maghribi's observations; DSB; Rosenfeld & Ihsanoglu's bio-bibliography.
Qutb al-Din al-Shirazi¶
Qutb al-Din Mahmud al-Shirazi (1236–1311) · Shiraz, Maragha, Tabriz · astronomy, optics, medicine
Student of Nasir al-Din al-Tusi at Maragha; polymath (he also produced a famous commentary on Avicenna's medical Canon). In astronomy his Nihayat al-Idrak and al-Tuhfa al-Shahiyya refined non-Ptolemaic planetary modeling, work continued by his student al-Farisi in optics and paralleled later by Ibn al-Shatir. In optics, he encouraged al-Farisi's re-examination of Ibn al-Haytham, which produced the first correct qualitative explanation of the rainbow (see al-Farisi, Era VI); Shirazi himself sketched the double-refraction-plus-reflection idea. The Shirazi–Farisi rainbow work is contemporary with, and independent of, Theodoric of Freiberg's similar European account (c. 1304–1310), a striking case of simultaneous discovery from the shared Alhazen inheritance.
Sources: DSB; Boyer, The Rainbow: From Myth to Mathematics; Saliba.
Ibn al-Banna¶
Ahmad ibn Muhammad ibn al-Banna al-Marrakushi (1256 – c. 1321) · Marrakesh, Fez · arithmetic, algebra, combinatorics
The great systematizer of the Maghrebi school. His Talkhis amal al-hisab ("Summary of Arithmetical Operations") became the arithmetic textbook of North Africa for two centuries, generating dozens of commentaries (including al-Qalasadi's, the vehicle for symbolic notation). It covers decimal arithmetic, fractions with the bar notation, proportion, root approximation (including the refinement \(\sqrt{a^2+r} \approx a + \frac{r}{2a+1}\)-type rules), false position, and algebra.
Combinatorics. In Raf al-Hijab, Ibn al-Banna derives rules equivalent to the formulas for combinations and permutations, \(\binom{n}{k}\) via the multiplicative rule \(\binom{n}{k}=\frac{n(n-1)\cdots(n-k+1)}{k!}\), motivated partly by a genuinely combinatorial tradition in Maghrebi scholarship (counting words, astrological conjunctions, and the like) that runs back through Ibn Munim's Fiqh al-Hisab (early 13th c., Marrakesh), where Pascal's-triangle tables are built for counting problems, one of the earliest places anywhere that the binomial table is read combinatorially rather than algebraically.
Sources: Djebbar's editions and studies; A. Djebbar & M. Aballagh on Ibn al-Banna; Katz (which discusses Ibn Munim's combinatorics); Lamrabet.