Era VIII — Later Scholars (1500 onward)¶
The tradition did not stop in 1500, though its relative position changed as printed European mathematics accelerated. The Ottoman, Safavid, and Mughal worlds sustained observatories, madrasa curricula, and occasional genuine novelty into the seventeenth century and beyond.
Ali Qushji¶
Ala al-Din Ali ibn Muhammad al-Qushji (c. 1403–1474) · Samarkand, Tabriz, Istanbul · astronomy, mathematics
(Chronologically fifteenth-century, placed here as the bridge figure.) A member of Ulugh Beg's Samarkand team who completed work on the Zij-i Sultani after the murders and dispersals, then migrated westward, ending as professor at Hagia Sophia's madrasa in Mehmed II's Istanbul, thereby physically carrying the Samarkand tradition to the Ottoman capital, decades before and after the fall of Constantinople funneled Greek manuscripts (and possibly Arabic astronomical ideas via Byzantine scholars) to Italy. His theoretical writings argued that astronomy should free itself from Aristotelian physics and entertained the discussability of the Earth's rotation, arguing observation could not decide it, positions with obvious resonance in the Copernican debates, and studied by historians (Ragep) as evidence of the conceptual ferment available on the eve of Copernicus.
Sources: F. J. Ragep, "Ali Qushji and Regiomontanus" (Journal for the History of Astronomy, 2005) and related papers; Fazlıoğlu; DSB supplement.
Taqi al-Din¶
Taqi al-Din Muhammad ibn Ma'ruf (1526–1585) · Damascus, Cairo, Istanbul · astronomy, mathematics, engineering
Founder and director of the short-lived Istanbul observatory (built c. 1575–1577, demolished 1580 after political-religious intrigue), exactly contemporary with Tycho Brahe's Uraniborg and comparably equipped; the two programs are the twin peaks of pre-telescopic observation. Taqi al-Din's Sidrat muntaha al-afkar records observations including of the great comet of 1577. Mathematically notable: he used decimal fractions systematically in trigonometric tables (extending al-Kashi's program), employed a form of prosthaphaeresis-style computation, and described mechanical clocks with alarm and striking trains (al-Kawakib al-durriyya, 1559), advocating clocks as observational instruments, and steam-driven and pump devices in his engineering treatises.
Sources: Sevim Tekeli's studies of Taqi al-Din; Aydin Sayili, The Observatory in Islam; İhsanoğlu's Ottoman science surveys.
Baha al-Din al-Amili¶
Baha al-Din Muhammad al-Amili (1547–1621) · Lebanon, Safavid Persia (Isfahan) · mathematics, architecture, theology
Sheikh al-Islam of Isfahan under Shah Abbas and a celebrated polymath of Safavid culture. His Khulasat al-Hisab ("Essentials of Arithmetic") is a compact masterpiece of pedagogy, arithmetic, algebra, mensuration, that became the standard mathematics textbook from the Ottoman lands through Persia to Mughal India for over two centuries, translated into Persian repeatedly and into German by G. H. F. Nesselmann (1843), one of the first Arabic mathematical texts studied by modern European historians. He famously listed seven problems unsolved "since the ancients" (including perfect-number questions and the impossibility-flavored equation \(x^3+y^3=z^3\)). His era also engaged decimal fractions and the classical heritage, evidence of a living, if no longer frontier-setting, mathematical culture.
Sources: Nesselmann's translation; DSB; Rosenfeld & Ihsanoglu.
Ibn Hamza al-Maghribi and the logarithm question¶
Ibn Hamza al-Maghribi (fl. late 16th century) · Ottoman lands · arithmetic
A Turkish-Algerian mathematician whose Tuhfat al-A'dad (c. 1591, written in Turkish) discussed the correspondence between geometric progressions and the arithmetic progression of their exponents, the conceptual kernel from which Napier's logarithms (1614) grew (a kernel also present in al-Samawal's exponent rule and in European writers like Stifel, 1544). Occasional popular claims that Ibn Hamza "invented logarithms" overstate the case: recognizing the exponent correspondence is not constructing logarithm tables. He is included here precisely to model how this reference treats inflated claims, by stating exactly what the source supports.
Sources: Salih Zeki's early-20th-century Ottoman history of science (Asar-i Bakiye), the primary modern notice; discussions in Turkish history-of-science literature. Direct manuscript scholarship in Western languages is thin, and Claude flags this entry as resting on a thinner evidentiary base than others in this reference.
The long tail¶
Mathematical activity in Arabic and Persian continued: commentaries and super-commentaries on al-Amili and Ibn al-Banna, timekeeping tables maintained in Cairo and Damascus into the 19th century (David King has documented this "in synchrony with the heavens" culture in detail), and the Mughal court's Sanskrit-Persian mathematical exchanges. When the modern research university arrived in the Middle East in the 19th–20th centuries, it grafted onto a culture that had never entirely stopped calculating.
Sources: King, In Synchrony with the Heavens; İhsanoğlu (ed.), History of the Ottoman State, Society and Civilisation (science chapters).