*The sixth and last in a **series. See the November 20 entry, “**What is algebra?**” for the first, the December 13 entry “**When did algebra begin?**” for the second, the December 19 entry “**The golden age of Arabic mathematics**” for the third, the December 25 entry “**al-Khwārizmī**”** for the fourth, and the December 30 entry “What is algebra good for?” for the fifth.*

History tends to focus on key individuals, when in fact most advances are the cumulative affect of the contributions of many. In the case of algebra, claims that al-Khwārizmī invented algebra are not sustainable. As I have explained in previous articles in this short series, the chain leading to algebra goes back at least to the ancient Babylonians, and to modern eyes Diophantus’s book *Arithmetica* was clearly a book on algebra. Nevertheless, al-Khwārizmī does deserve the credit for establishing algebra as a major collection of intellectual tools.

He can’t be credited with establishing it as a branch of mathematics, howver, since the mathematicians of the Arabic period did not view the methods they developed as anything other than a set of very valuable practical tools. (Likewise, Diophantus viewed his work as a sophisticated form of arithmetic, as the title of his famous work suggests.) Viewing algebra as a discipline in its own right came later.

Al-Khwārizmī’s greatness is in the same category as Euclid nine centures earlier, as Leonardo of Pisa four hundred years later, or as Steve Jobs in our own time: their impact on society and thence the course of history. None of these four were the original inventors or discoverers of the seminal developments we associate with their names. Their greatness was not one of original discovery – though both Euclid and al-Khwārizmī may well have contributed some of the methods they described in their seminal books and we know from his other works besides *Liber abbaci* that Leonardo was a first-rate, original mathematician. Rather, all four had the highly unusual ability to take a collection of powerful new ideas and package and present them to society in a manner that made them acceptable to – indeed eagerly sought-after by – a wide range of people. In our present-day society we tend to focus on the priority of discovery and invention, as epitomised by the status we award Nobel Laureates, but initial discovery would be of little value were others not able to take the new knowledge and use it to change society.

Each of Euclid, al-Khwārizmī, and Leonardo (and Steve Jobs) were followed by many others who carried the torch forwards, and they too deserve credit.

Among the hundreds of Arabic mathematicians who helped to develop and spread algebraic knowledge after al-Khwārizmī, several stand out as worthy of special mention. I’ll list a few.

**Abū Kāmil **The Egyptian born Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad ibn Shujā (c. 850 – c. 930) was the first major Arabic algebraist after al-Khwārizmī. By all accounts he was a prolific author. There are references to works with the titles *Book of fortune, Book of the key to fortune, Book of the adequate, Book on omens, Book of the kernel, Book of the two errors, *and* Book on augmentation and diminution*. None of these have survived. Works that did survive include the *Book on algebra*, the *Book of rare things in the art of calculation*, *Inheritance by means of algebra*, and the *Book on surveying and geometry*.

The *Book on algebra* (*Kitāb fi al-jabr wa al-muqābala*) is arguably Abū Kāmil’s most influential work. It expanded on al-Khwārizmī’s *Algebra*. Whereas the latter was aimed at the general public, Abū Kāmil wrote more for other mathematicians, assuming familiarity with Euclid’s *Elements*. He extended the range of polynomials studied beyond al-Khwārizmī to include 8^{th} powers.

**Al-Karajī **A century after Abū Kāmil did his work, around 1000 C.E., another major advance in algebra was made by the Persian mathematician and engineer *Abū Bakr ibn Mu**ḥ**ammad ibn al **Ḥ**usayn al-Karajī*, who lived from c. 953 to c. 1029. His three major works were *Al-Badi’ fi’l-hisab* (*Wonderful on calculation*), *Al-Fakhri fi’l-jabr wa’l-muqabala* (*Book of al-Fakhri on the Art of Algebra*), and *Al-Kafi fi’l-hisab* (*Sufficient on calculation*).

*Al-Fakhri* is regarded as one of the key works on the path that led to the final separation of algebra from geometry as a discipline in its own right. Al-Karajī gave a systematic treatment of reducible higher-degree equations. He studied the algebra of exponents, and was the first to state explicitly that the sequence *x*, *x*^{2}, *x*^{3}, … could be extended indefinitely, and likewise the reciprocals 1/*x*, 1/*x*^{2}, 1/*x*^{3}, …

**Omar Khayyám **Shortly after al-Karajī died, another famous Arab scholar came onto the scene: Omar Khayyám. Although in the West he is better known today as a poet, he was a first rate mathematician.

Al-Khayyám, more fully *Ghiyath al-Din Abu’l-Fath Umar ibn Ibrahim al-Nisaburi al-Khayyámi*, was born on 18 May, 1048 in Nishapur, Persia (now Iran), and died there on 4 December, 1131. As a young man he studied philosophy, and went on to be an outstanding mathematician and philosopher. By the time he was 25, he had written several books, covering arithmetic, geometry, algebra, and music. His major work in algebra was an analysis of polynomial equations titled *Treatise on the Proofs of Algebra Problems*.

Al-Khayyám approached mathematics primarily as a geometer, firmly rooted in the Greek tradition. Whereas Abū Kāmil and al-Karajī presented algebra as a method for numerical problem-solving, al-Khayyám viewed it as a tool for theoretical geometers.

**al-Samaw****ʾ****al **Several further advances in algebra were made around the mid-twelfth century by a teenager (yes, that’s right, a teenager) called Ibn Yaḥyā al-Maghribī al-Samawʾal, who was born around 1130 in Baghdad. His parents were Jewish, his father a literature scholar and Rabbi from Morocco, his mother from Basra, in Iraq.

Although his initial interest as a child was to become a doctor, al-Samawʾal proved to be a child prodigy in mathematics, and the study of medicine was soon relegated to second place (but not abandoned). He began to study the Hindu methods of calculation when he was thirteen or so. Rapidly finding himself ahead of his teachers, he continued on his own, reading the works of Abū Kāmil, al-Karajī, and others. By the time he was eighteen years old he had read almost all the available mathematical literature. He wrote his most famous treatise, *al-Bahir fi’l-jabr* (*The brilliant in algebra),* when he was just nineteen years old.

Mathematicians before al-Samaw’al had begun to develop what contemporary historians have called the “arithmetization of algebra”. Al-Samaw’al was perhaps the first to give this development a precise description, writing that it involved “operating on unknowns using all the arithmetical tools, in the same way as the arithmetician operates on the known.” This can be regarded as a significant step toward the development of modern algebra.

In all, al-Samaw’al is reported to have written 85 books or articles, though most have not survived. He died in Maragha, Iran, around 1180.

Further advances in algebra were made in the Maghreb in the twelfth to fifteenth century, by a highly organized teacher-student network linked to mosque and madrasah teaching. The Maghrebs used abbreviations for both unknowns and their powers and for operations; another innovation in the chain that culminated in the development of modern symbolic algebra in Europe in the 16^{th} century.

* * *

For the next episode in the development and growth of algebra, when the ideas found their way to Europe, see my recent book The Man of Numbers: Fibonacci’s Arithmetic Revolution. (And for a comparison between Fibonacci’s role and that of Steve Jobs, see the companion e-book Leonardo and Steve.)

**Acknowledgement
**I am greatful to Professor Jeffrey Oaks of the University of Indianapolis for his assistance in the prepartion of the essays in this series. In particular, he supplied me with preprints of his forhtcoming articles for Springer Verlag’s upcoming

*Encyclopedia of Sciences and Religions*(2012): “Mathematics and Islam”, “Arithmetic and Islam”, “Algebra and Islam”, and “Geometry and Islam”, which I drew on heavily. He also commented in detail on a more substantial work from which these essays were abridged.