Era VI — The Fourteenth Century (1300–1400)¶
Damascus becomes a center of astronomical computation of staggering ambition; number theory and optics reach new peaks in Persia; the Maghrebi textbook tradition rolls on.
Kamal al-Din al-Farisi¶
Kamal al-Din al-Farisi (c. 1267–1319) · Tabriz · optics, number theory
Student of Qutb al-Din al-Shirazi and the deepest mathematical mind of his generation in two separate fields.
Optics. His Tanqih al-Manazir ("Revision of the Optics") is a critical commentary on Ibn al-Haytham that in places surpasses the master: using a water-filled glass sphere as a scaled model of a raindrop (an explicit act of experimental modeling), al-Farisi explained the primary rainbow as two refractions with one internal reflection and the secondary bow as two refractions with two internal reflections, with the color order reversal accounted for. This is the correct geometry of the rainbow, contemporaneous with and independent of Theodoric of Freiberg; the quantitative completion had to wait for Descartes (1637) and Newton (dispersion).
Number theory. His treatise on amicable numbers gives a new proof of Thabit ibn Qurra's rule, and to build it he developed the theory of divisors of a number from its prime factorization, including a statement and use of the existence side of what is now called the fundamental theorem of arithmetic, together with combinatorial rules for counting divisors (linking the divisor count to combinations). Rashed has argued this fused number theory with the new combinatorics; the essential-uniqueness side of unique factorization is treated as understood rather than proved, so historians phrase the credit carefully.
Sources: Rashed, "Materials for the study of the history of amicable numbers and combinatorial analysis" and The Development of Arabic Mathematics; Boyer, The Rainbow; DSB (R. Rashed, "Kamal al-Din al-Farisi").
Ibn al-Shatir¶
Ala al-Din Ali ibn Ibrahim ibn al-Shatir (1304 – c. 1375) · Damascus · astronomy
Muwaqqit (mosque timekeeper) at the Umayyad Mosque, and the culmination of the Maragha-style planetary program. In his Nihayat al-Sul ("The Final Quest Concerning the Rectification of Principles"), he built planetary, lunar, and solar models that are entirely equant-free and eccentric-free, using only combinations of uniform circular motions (epicycles on epicycles, with the Urdi lemma and Tusi couple among the tools). Two results stand out for the European story:
- His lunar model eliminates Ptolemy's absurd near-doubling of the Moon's apparent size, and is mathematically identical to Copernicus's lunar model in De revolutionibus (as shown by Roberts and Kennedy in the 1950s).
- His Mercury model and his solar model likewise reappear, parameter conventions and devices matching, in Copernicus's Commentariolus and De revolutionibus. Copernicus adds the heliocentric transformation; the geometric machinery is Damascene. Direct textual transmission is undocumented; possible routes (Byzantine manuscripts, Padua-trained orientalists, Jewish intermediaries such as the circle around Moses Galeano, who knew both Damascus and Veneto) are actively researched.
He also built a famous sundial for the Umayyad Mosque (fragments survive; a replica stands in place) and fine astronomical instruments.
Sources: V. Roberts, "The Solar and Lunar Theory of Ibn ash-Shatir," Isis 48 (1957); E. S. Kennedy & Roberts's follow-ups; Kennedy & Ghanem (eds.), The Life and Work of Ibn al-Shatir; Saliba; Swerdlow & Neugebauer.
al-Khalili¶
Shams al-Din Muhammad al-Khalili (fl. c. 1365) · Damascus · spherical astronomy, computation
Ibn al-Shatir's colleague as muwaqqit in Damascus, and the author of what David King called the most impressive computational achievement of medieval astronomy: universal auxiliary tables for spherical astronomy, some 13,000+ entries, tabulating functions of two and three arguments that solve the standard problems (prayer times, qibla) for any latitude, together with a qibla table giving the direction of Mecca for a grid of coordinates covering the Islamic world, computed to minutes of arc with remarkable accuracy. In modern terms, he tabulated general solutions to the spherical trigonometric equations rather than case-by-case answers, an industrial-scale exercise in function tabulation, error control, and interpolation, i.e., numerical analysis as a discipline of practice.
Sources: D. A. King, "Al-Khalili's Auxiliary Tables for Solving Problems of Spherical Astronomy," Journal for the History of Astronomy 4 (1973), and King's In Synchrony with the Heavens; DSB.
Ibn Muhammad al-Umawi¶
Yaish ibn Ibrahim al-Umawi (fl. 14th century; dating uncertain, possibly 15th) · Andalusi origin, active in Damascus · arithmetic
Author of Marasim al-Intisab fi Ilm al-Hisab, a compact arithmetic notable to historians because it is a western (Andalusi/Maghrebi) arithmetic written for an eastern audience, documenting the flow of techniques (including divisibility tests and checking rules, e.g., casting out nines, sevens, elevens) between the two halves of the Islamic world. Saidan edited and studied the text.
Sources: A. S. Saidan's edition; Rosenfeld & Ihsanoglu.
Ibn Qunfudh and the commentary tradition¶
Ibn Qunfudh al-Qasantini (1339–1407) · Constantine, Fez · arithmetic, algebra (commentaries)
Included as a representative of the mathematical commentary culture that kept the Maghrebi school alive: his commentaries on Ibn al-Banna's Talkhis transmitted and lightly extended the material, preserving worked examples of the Maghrebi symbolic notation. The commentary genre is easy to undervalue; it was the tradition's mechanism of quality control, pedagogy, and incremental improvement, and manuscripts of these commentaries are how much of our knowledge of Maghrebi notation survives.
Sources: Djebbar; Abdeljaouad, "The Eight Hundredth Anniversary of Maghribian algebraic symbolism"-line studies; Lamrabet.