# Abū Kāmil Shujāʿ ibn Aslam : biography

'* (Latinized as Auoquamel'*, , also known as

**al-ḥāsib al-miṣrī**—lit. "the Egyptian calculator") (c. 850 – c. 930) was an Egyptian Muslim mathematician during the Islamic Golden Age. He is considered the first mathematician to systematically use and accept irrational numbers as solutions and coefficients to equations. His mathematical techniques were later adopted by Fibonacci, thus allowing Abu Kamil an important part in introducing algebra to Europe.

Abu Kamil made important contributions to algebra and geometry. He was the first Islamic mathematician to work easily with algebraic equations with powers higher than x^2 (up to x^8), and solved sets of non-linear simultaneous equations with three unknown variables. He wrote all problems rhetorically, and some of his books lacked any mathematical notation beside those of integers. For example, he uses the Arabic expression "māl māl shayʾ" ("square-square-thing") for x^5 (i.e., x^2\cdot x^2\cdot x).

## Life

Almost nothing is known about the life and career of Abu Kamil except that he was a successor of al-Khwarizmi, whom he never personally met.

## Notes

## On al-Khwarizmi

Abu Kamil was one of the earliest mathematicians to recognize al-Khwarizmi's contributions to algebra, defending him against Ibn Barza who attributed the authority and precedent in algebra to his grandfather, ʿAbd al-Hamīd ibn Turk. Abu Kamil wrote in the introduction of his *Algebra*:

I have studied with great attention the writings of the mathematicians, examined their assertions, and scrutinized what they explain in their works; I thus observed that the book by Muḥammad ibn Mūsā al-Khwārizmī known as *Algebra* is superior in the accuracy of its principle and the exactness of its argumentation. It thus behooves us, the community of mathematicians, to recognize his priority and to admit his knowledge and his superiority, as in writing his book on algebra he was an initiator and the discoverer of its principles, ...

## Works

### *Book of Algebra (Kitāb fī al-jabr wa al-muqābala)*

The *Algebra* is perhaps Abu Kamil's most influential work, which he intended to supersede and expand upon that of Al-Khwarizmi. Whereas the *Algebra* of al-Khwarizmi was geared towards the general public, Abu Kamil was addressing other mathematicians, or readers familiar with Euclid's *Elements*. In this book Abu Kamil solves systems of equations whose solutions are whole numbers and fractions, and accepted irrational numbers (in the form of a square root or fourth root) as solutions and coefficients to quadratic equations.Jacques Sesiano, "Islamic mathematics", p. 148, in

The first chapter teaches algebra by solving problems of application to geometry, often involving an unknown variable and square roots. The second chapter deals with the six types of problems found in Al-Khwarizmi's book, but some of which, especially those of x^2, were now worked out directly instead of first solving for x and accompanied with geometrical illustrations and proofs. The third chapter contains examples of quadratic irrationalities as solutions and coefficients. The fourth chapter shows how these irrationalities are used to solve problems involving polygons. The rest of the book contains solutions for sets of indeterminate equations, problems of application in realistic situations, and problems involving unrealistic situations intended for recreational mathematics.

A number of Islamic mathematicians wrote commentaries on this work, including al-Iṣṭakhrī al-Ḥāsib and ʿAli ibn Aḥmad al-ʿImrānī (d. 955-6), but both commentaries are now lost.

In Europe, similar material to this book is found in the writings of Fibonacci, and some sections were incorporated and improved upon in the Latin work of John of Seville, *Liber mahameleth*. A partial translation to Latin was done in the 14th-century by William of Luna, and in the 15th-century the whole work also appeared in a Hebrew translation by Mordekhai Finzi.