Islamic mathematics (c. 700—1600)

Muḥammad ibn Mūsā al-Ḵwārizmī

The Islamic Caliphate (Islamic Empire) established across the Middle East, North Africa, Iberia, and in parts of India in the 8th century made significant contributions towards mathematics.

Although most Islamic texts on mathematics were written in Arabic, they were not all written by Arabs, since much like the status of Greek in the Hellenistic world, Arabic was used as the written language of non-Arab scholars throughout the Islamic world at the time. Some of the most important Islamic mathematicians were Persian.

Muḥammad ibn Mūsā al-Ḵwārizmī, a 9th century Persian mathematician and astronomer to the Caliph of Baghdad, wrote several important books on the Hindu-Arabic numerals and on methods for solving equations. His book On the Calculation with Hindu Numerals, written about 825, along with the work of the Arab mathematician Al-Kindi, were instrumental in spreading Indian mathematics and Indian numerals to the West. The word algorithm is derived from the Latinization of his name, Algoritmi, and the word algebra from the title of one of his works, Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala (The Compendious Book on Calculation by Completion and Balancing). Al-Khwarizmi is often called the "father of algebra", for his preservation of ancient algebraic methods and for his original contributions to the field. [12] [13]

Further developments in algebra were made by Abu Bakr al-Karaji (953—1029) in his treatise al-Fakhri, where he extends the methodology to incorporate integer powers and integer roots of unknown quantities. In the 10th century, Abul Wafa translated the works of Diophantus into Arabic and developed the tangent function.

Omar Khayyam, the 12th century poet, was also a mathematician, and wrote Discussions of the Difficulties in Euclid, a book about flaws in Euclid's Elements. He gave a geometric solution to cubic equations, one of the most original developments in Islamic mathematics. He was also very influential in calendar reform. The Persian mathematician Nasir al-Din Tusi (Nasireddin) in the 13th century made advances in spherical trigonometry. He also wrote influential work on Euclid's parallel postulate.

In the 15th century, Ghiyath al-Kashi computed the value of π to the 16th decimal place. Kashi also had an algorithm for calculating nth roots, which was a special case of the methods given many centuries later by Ruffini and Horner. Other notable Islamic mathematicians are al-Samawal, Abu'l-Hasan al-Uqlidisi, Jamshid al-Kashi, Thabit ibn Qurra, Abu Kamil and Abu Sahl al-Kuhi.

During the time of the Ottoman Empire (from the 15th century) the development of Islamic mathematics became stagnant. This parallels the stagnation of mathematics when the Romans conquered the Hellenistic world.

Recent research paints a new picture of the debt that we owe to Islamic mathematics. Certainly many of the ideas which were previously thought to have been brilliant new conceptions due to European mathematicians of the sixteenth, seventeenth and eighteenth centuries are now known to have been developed by Arabic/Islamic mathematicians four centuries earlier. In many respects, the mathematics studied today is far closer in style to that of Islamic mathematics than to that of Hellenistic mathematics.