Abu Kamil

Abū Kāmil, Shujā ibn Aslam
Born	c. 850
Died	c. 930
Other names	al-ḥāsib al-miṣrī
Academic background
Influences	al-Khawarizmi
Academic work
Era	Islamic Golden Age
Main interests	Algebra, Geometry
Notable works	The Book of Algebra
Influenced	al-Karaji, Fibonacci

Abū Kāmil, Shujāʿ ibn Aslam ibn Muḥammad Ibn Shujāʿ (Arabic: ابو كامل, also known as al-Ḥasib al-Miṣrī—literally, "the Egyptian calculator") (c. 850 – c. 930) was an Egyptian Muslim mathematician during the Islamic Golden Age. Abu Kamil made important contributions to the field of algebra, and considered as the first mathematician to systematically use and accept irrational numbers as algebraic objects. His mathematical techniques were later adopted by Fibonacci, thus allowing Abu Kamil an important part in introducing algebra to Europe.

Another achievement of Abu Kamil is solving non-linear simultaneous equations with three unknown variables. He was also the first Islamic mathematician to work easily with algebraic equations with powers higher than $x^{2}$ (up to $x^{8}$). He wrote all problems rhetorically, and some of his books lacked any mathematical notation beside those of integers. For example, he uses the expression "square square root" for $x^{5}$ (i.e., $x^{2}\cdot x^{2}\cdot x$).

Works

The Book of Algebra

Kitāb fi al-jabr wa al-muqābala, in which Abu Kamil solved 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. The Algebra is perhaps Abu Kamil's most influential work, which he intended to supersede and expand upon that of Al-Khawarizmi. Whereas the Algebra of al-Khawarizmi was intended towards a general public, Abu Kamil was addressing other mathematicians, or readers familiar with Euclid's Elements.

The first section of this book teaches algebra by solving problems of application to geometry, often involving an unknown variable and square roots. The second section deals with the six types of problems found in Al-Khawarizmi's book, but some of these problem, especially those of $x^{2}$, was worked out directly instead of first solving for $x$, and now also accompanied with geometrical illustrations and proofs. The third section contains examples of quadratic irrationalities as solutions and coefficients. In the fourth section, these irrationalities are used to solve problems involving polygons. The rest of this book contains sets of indeterminate equations and systems, problems of application encountered in realistic situations, and another set of problems describing 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. 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.

The Book of Rare Things in the Art of Calculation

Kitāb al-ṭarā’if fi’l-ḥisāb, describes a number of systematic procedures for finding integral solutions for indeterminate equations. Such procedures were not known to many of his peers, and some of which are not found in Diophantus's Arithmetica. In this book he describes one problem for which he found 2,678 solutions.

On the Pentagon and Decagon

Kitāb … al-mukhammas wa’al-mu‘ashshar. Some of the calculations in this book used the Golden Ratio.

The Book of Birds

Kitāb al-ṭair, a small treatise teaching how to solve indeterminate linear systems with positive integral solutions. The title is derived from a type of problems known in the middle ages which involve the purchase of different species of birds. Abu Kamil wrote in the introduction:

I found myself before a problem that I solved and for which I discovered a great many solutions; looking deeper for its solutions, I obtained two thousand six hundred and seventy-six correct ones. My astonishment about that was great, but I found out that, when I recounted this discovery, those who did not know me were arrogant, shocked, and suspicious of me. I thus decided to write a book on this kind of calculations, with the purpose of facilitating its treatment and making it more accessible.

According to Jacques Sesiano, Abu Kamil remained seemingly unparalleled throughout the middle ages in trying to find all the possible solutions to some of his problems.

On Measurement and Geometry

Kitāb al-misāḥa wa al-handasa, a manual of geometry for non-mathematicians, like land surveyors and other government officials.

The Book of Estate Sharing using Algebra

Kitāb al-waṣāyā bi al-jabr wa al-muqābala, contains algebraic solutions for problems of Islamic inheritance and discusses the opinions of known jurists.

Book of the Two Errors

Kitāb al-khaṭaʾayn, a now lost treatise on the use of double false position.

Other Works

Ibn al-Nadīm in his Fihrist listed the following additional titles: Book of Fortune (Kitāb al-falāḥ ), Book of the Key to Fortune (Kitāb miftāḥ al-falāḥ), Book of the Adequate (Kitāb al-kifāya), Book of the Kernel (Kitāb al-ʿasīr), Book on Augmentation and Diminution (Kitāb al-jamʿ wa’l-tafrīq).

Influence

Abu Kamil's works influenced other mathematicians, like al-Karaji and Fibonacci, and as such had a lasting impact on the development of algebra. Many of his examples and algebraic techniques were later copied by Fibonacci in his Practica geometriae and other works.

Example of problems solved by Abu Kamil

Non-linear equations

Abu Kamil solved the non-linear simultaneous equations with three unknown variables:

(1) x + y + z = 10, (2) x^2 + y^2 = z^2, (3) xy = z^2.

He does so by first making an arbitrary non-zero guess, x_0, for x (he choses x=1), and then solves (2, 3) for corresponding y_0 and z_0. Since 2 and 3 are homogeneous, any solution of (2, 3) will also be a solution if x, y and z are multiplied by any constant. In particular, if they are multiplied by 10 / (x_0 + y_0 + z_0) then they will still solve (2, 3), and will also solve (1) by construction.

On al-Khawarizmi

Almost nothing is known about the life and career of Abu Kamil except that he was an immediate successor of al-Khawarizmi, whom he never personally met. He was also one of the earliest mathematicians to recognize Al-Khwarizmi's contributions to algebra, defending him against Ibn Barza who attributed some of the achievements of al-Khawarizmi 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-Khawārizmī 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, ...

Further reading

• Djebbar, Ahmed. Une histoire de la science arabe: Entretiens avec Jean Rosmorduc. Seuil (2001)
• Levey, Martin (1970). "Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad ibn Shujā". Dictionary of Scientific Biography. Vol. 1. New York: Charles Scribner's Sons. pp. 30–32. ISBN 0684101149.

External links

• Levey, Martin (2008) [1970–80]. "Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad ibn Shujā". Complete Dictionary of Scientific Biography. Encyclopedia.com. {{cite encyclopedia}}: External link in |title= (help)
• O'Connor, John J.; Robertson, Edmund F., "Abu Kamil", MacTutor History of Mathematics Archive, University of St Andrews