*The third in a series. See the November 20 entry, “What is algebra?” for the first and the December 13 entry “When did algebra begin?” for the second.*

On 14 September 786,** **Harun al-Rashid became the fifth Caliph of the Abbasid dynasty. From his court in the capital city of Baghdad, Harun ruled over the vast Islamic empire, stretching from the Mediterranean to India. He brought culture into his court and encouraged the widespread pursuit of learning.

Al-Rashid had two sons, the elder al-Amin, the younger al-Mamun. Harun died in 809 and there was an armed conflict between the brothers.** **Al-Mamun won the armed struggle and al-Amin was defeated and killed in 813. Following this, al-Mamun became Caliph and ruled the empire from Baghdad.

Al-Mamun continued the patronage of learning started by his father. With his encouragement, scholars of the time set about collecting and writing down in books all available practical knowledge, much of which had hitherto been transmitted only orally, including mathematics and folk astronomy. They translated into Arabic works of Greek and Indian science.

Many of the works collected and created may have been housed in a library called the House of Wisdom, though there is no evidence to support the commonly repeated claims that (1) it was massive, (2) it was founded by al-Mamun, or (3) translations were carried out there.

The tradition of learning, writing, and translation begun by al-Rashid and al-Mamun continued for the next quarter century, making the Islamic civilization the center of world knowledge. The aristocracy and other wealthy groups within Muslim society supported the appropriation of all practical and scientific knowledge they could acquire. They employed scholars to translate into Arabic works by Indian, Sasanian, and especially Greek authors, and mathematicians recorded on paper all that was known of arithmetic, algebra, and mensuration, which had hitherto been communicated orally by traders. In addition to the mathematical sciences (arithmetic, geometry, optics, mathematical astronomy, etc.), they also translated texts on geography, astrology, philosophy, medicine, agriculture, alchemy, and even falconry.

Greek works formed the bulk of the material translated. In addition, the more scientifically oriented mathematicians adopted the Greek tradition of definitions, axioms, and propositions with rigorous proof, and astronomers embraced the Greek idea of geometric models of planetary motion. Within this framework, Indian techniques were incorporated into this new Arabic/Islamic mathematics.

In addition to the translations, scholars wrote commentaries and criticisms of the ancient mathematics and made their own original contributions. For example, in the 9^{th} century, Thābit ibn Qurra (d. 901) translated several works of Archimedes, wrote commentaries on Euclid’s *Elements* and Ptolemy’s *Almagest*, critiqued Euclid’s definition for the composition of ratios of numbers, and derived and proved new formulas for volumes of solids of revolution.

When the sources of Greek and other foreign texts was finally exhausted, scholars continued to produce new results in all branches of mathematics. For instance, in the 11^{th} century, Ibn al-Haytham made major contributions to optics and geometry, and at the start of the 12^{th} century, al-Khāyyamī wrote his book on algebra.

Over a thousand mathematical manuscripts from the period have survived, about half of them dating before the 15^{th} century.

Al-Khwārizmī, who may have studied and worked in the House of Wisdom, was one of the earliest contibutors to this vast undertaking, and arguably had the most impact of all the mathematicians involved. But his books – he wrote one on Hindu arithmetic in addition to the one on algebra – should be viewed as part of this larger movement.

At the time, algebra was viewed primarily as a practical, numerical problem solving technique, not the autonomous branch of mathematics it became later. Indeed, the greatest contribution of Arabic mathematical work to society was its development as a set of practical tools.

Three systems of practical calculation were taught and practiced in the medieval Islamic world: finger reckoning, Hindu arithmetic, and the base 60 system of the astronomers. Merchants preferred finger-reckoning, which worked for numbers up to 10,000. Finger reckoning was used to solve problems by various methods, such as double false position and algebra. Al-Khwārizmī is known to have written a work, now lost, called *Book of Adding and Subtracting*, in the early 9^{th} century, which was probably devoted to the use of finger reckoning. (If so, it was probably the earliest written text on the subject.)

The Arabic mathematicians referred to the numerals 1, 2, 3, etc., as Hindī numerals, because they acquired the system from India. These numerals were already in use in the Middle East by the 7^{th} century CE. The earliest known Arabic text describing the system is al-Khwārizmī’s *Book on Hind**ī** Reckoning*, written in the early 9^{th} century, which survives only in Latin translation. The original algorithms for calculating in this system were devised for use on a dust board, where erasing is easy. In the middle of the 10^{th} century, al-Uqlīdisī introduced new algorithms for use with pen and paper. The Arabic mathematicians introduced the concept of decimal fractions, wihich al-Uqlīdisī described for the first time.

Unlike Diophantus, most of the Arabic authors, including al-Khwārizmī, wrote their algebra almost entirely in words. For example, where we would write down the symbolic equation *x* + 1 = 2, they might write “The thing plus one equals two” (and very occasionally “The thing plus 1 equals 2”). This is generally known as the *rhetorical *form, and remained in common use right up to the 16^{th} century. This is, however, a notational distinction, not one of content. Commentators who refer to “rhetorical algebra” as being a form of algebra distinct from “literal algebra” are in error. For, although the Arabic authors wrote their books rhetorically, with no notation even for numbers, they did not solve problems rhetorically. Throughout most of Arabic algebra, problems were worked out on some ephemeral surface, by writing the coefficients and numbers in Hindu form. For example, they would write

1 2 1

to mean *x*^{2} + 2*x* + 1. Later, Arabic scholars in the Maghreb developed a truly algebraic notation, with symbols for the words representing the powers of the unknown, but even they they would resort to rhetorical text to communicate the result of a calculation.

*Symbolic algebra*, where full symbolism is used, is generally credited in the first instance to the French mathematician François Viète (1540 –1603), followed by René Descartes (1596 – 1650), though traces can be discerned in the writings of some Arabic mathematicians as early as the 13^{th} century.

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In my next two articles in this short series, I’ll say a bit about al-Khwārizmī and take a look at the contents of his seminal book on algebra. In particular, I’ll give his answer to that perennial student question, “What is algebra good for?”

*I use terms like “Arabic mathematics” in the standard historical fashion to refer to the mathematics done where and when the primary language for scholastic texts was Arabic. Mathematics, like all of science, belongs to the world.*