Historical Context¶
The Setting¶
The mathematical tradition covered in this reference arose within the Islamic world in the broad geographic sense: the territories from al-Andalus (Islamic Spain) and the Maghreb in the west, through Egypt, Syria, and Iraq, to Persia, Central Asia, and northern India in the east. The scholars who worked in this tradition were predominantly Muslim, but the scientific community also included Christian, Jewish, and Sabian participants (Thabit ibn Qurra, one of the greatest geometers of the ninth century, was a Sabian from Harran; Hunayn ibn Ishaq, the master translator, was a Nestorian Christian). The unifying medium was the Arabic language, which served the role Latin later served in Europe and English serves in science today. Historians therefore sometimes call this tradition "Arabic mathematics" (after the language) and sometimes "Islamic mathematics" (after the civilization); both labels describe the same body of work.
The Translation Movement into Arabic (c. 750–950)¶
The Abbasid caliphate, established in 750 with its capital at Baghdad from 762, sponsored a deliberate, generously funded campaign to translate the scientific heritage of earlier civilizations into Arabic. Three streams mattered most for mathematics:
The Greek stream. Euclid's Elements, Ptolemy's Almagest, the works of Archimedes, Apollonius's Conics, Diophantus's Arithmetica, and Menelaus's Spherics were translated, often multiple times with increasing precision. Translators such as al-Hajjaj ibn Yusuf ibn Matar, Hunayn ibn Ishaq, his son Ishaq ibn Hunayn, and Thabit ibn Qurra (who revised many translations) made Baghdad the custodian of Greek mathematics at a time when much of it was inaccessible in Western Europe.
The Indian stream. Sanskrit astronomical works (the siddhanta tradition, rendered in Arabic as Sindhind) arrived at the Abbasid court in the 770s, carrying with them the decimal positional numeral system, the zero, and Indian trigonometry based on the sine rather than the Greek chord.
The Persian stream. Sasanian astronomical tables (zij) and administrative computational practice contributed to the applied side of the tradition.
The House of Wisdom¶
The Bayt al-Hikma (House of Wisdom) in Baghdad, associated especially with the caliph al-Ma'mun (r. 813–833), functioned as a library, translation center, and research institution. Its exact institutional nature is debated by historians, and popular accounts sometimes overstate its resemblance to a modern university. What is not debated is the roster of mathematicians connected with al-Ma'mun's patronage: al-Khwarizmi, the three Banu Musa brothers, and al-Kindi all worked in this Baghdad milieu, and the observational astronomy program al-Ma'mun sponsored (including a measurement of the Earth's circumference) set the pattern of state-supported exact science that later courts imitated.
Why Mathematics Flourished¶
Several forces sustained mathematical work across nine centuries:
Religious practice generated computational problems. Determining prayer times requires spherical astronomy; finding the qibla (direction of Mecca) from any city is a genuinely hard problem in spherical trigonometry; the lunar calendar demands visibility calculations; and Islamic inheritance law ('ilm al-fara'id) requires exact fractional arithmetic, which al-Khwarizmi treated at length in his algebra. A specialized professional, the muwaqqit (timekeeper attached to a mosque), emerged; al-Khalili in fourteenth-century Damascus, who computed universal auxiliary tables for spherical astronomy, held such a post.
Commerce and administration demanded arithmetic. The vast trade networks of the Islamic world ran on computation, and treatises on hisab (practical reckoning) were bestsellers of the manuscript era.
Courts competed for prestige. From Baghdad to Cairo (the Fatimid observatory), Rayy, Isfahan (where Omar Khayyam reformed the calendar), Maragha (the great observatory founded in 1259), and Samarkand (Ulugh Beg's observatory and madrasa, c. 1420s), rulers funded observatories and salaried scholars in a pattern that resembles modern national laboratories more than anything in contemporary Europe.
Astronomy and astrology paid the bills. As in Kepler's Europe centuries later, the demand for astronomical tables and astrological services financed a great deal of pure mathematics.
Periodization Used in This Reference¶
| Era | Span | Centers | Signature achievements |
|---|---|---|---|
| I. Foundations | 750–900 | Baghdad | Translation movement; al-Khwarizmi's algebra; Thabit's geometry and number theory |
| II. Tenth century | 900–1000 | Baghdad, Rayy, Cairo, Córdoba | Decimal fractions (al-Uqlidisi); trigonometric functions completed (Abu al-Wafa); geometric constructions |
| III. Golden age | c. 1000–1100 | Ghazna, Cairo, Isfahan, al-Andalus | Ibn al-Haytham's optics; al-Biruni's geodesy; al-Karaji's algebra of polynomials; Khayyam's cubics |
| IV. Twelfth century | 1100–1200 | Persia, al-Andalus, Maghreb | al-Samawal's symbolic methods; Sharaf al-Din al-Tusi's analysis of cubics |
| V. Maragha | 1200–1300 | Maragha, Maghreb | Tusi's trigonometry as independent science; planetary theory (Tusi couple) |
| VI. Fourteenth century | 1300–1400 | Damascus, Persia, Maghreb | Ibn al-Shatir's planetary models; al-Farisi's number theory and optics; al-Khalili's tables |
| VII. Samarkand | 1400–1500 | Samarkand, Maghreb, Cairo | al-Kashi's computational mathematics; Ulugh Beg's star catalog; al-Qalasadi's algebraic notation |
| VIII. Later scholars | 1500 onward | Istanbul, Persia, Maghreb | Taqi al-Din's observatory; continued textbook tradition |
What Came Before and After¶
This tradition did not arise from nothing and did not vanish into nothing. It consciously absorbed Greek, Indian, and Persian mathematics, and it was in turn absorbed, through the Latin translation movement of the twelfth century and the commercial mathematics of the Italian abacus schools, into the European tradition that leads to the Renaissance algebraists, to Viète and Descartes, and onward to the modern discipline. The Transmission chapters follow that story in detail.
Primary secondary sources for this chapter: J. L. Berggren, Episodes in the Mathematics of Medieval Islam; Victor J. Katz, A History of Mathematics; George Saliba, Islamic Science and the Making of the European Renaissance; Dimitri Gutas, Greek Thought, Arabic Culture. Full details in the Bibliography.