Abu Kamil

Abu Kamil أبو كامل
Born	c. 850
Died	c. 930
Other names	Al-ḥāsib al-miṣrī
Academic background
Influences	Al-Khwarizmi
Academic work
Era	Islamic Golden Age (Middle Abbasid era)
Main interests	Algebra, geometry
Notable works	The Book of Algebra
Notable ideas	Use of irrational numbers as solutions and coefficients to equations
Influenced	Al-Karaji, Fibonacci

Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad Ibn Shujāʿ (

Abu Kamil made important contributions to

Islamic mathematician

to work easily with algebraic equations with powers higher than ${\displaystyle x^{2}}$ (up to ${\displaystyle x^{8}}$), He illustrated the rules of signs for expanding the multiplication ${\displaystyle (a\pm b)(c\pm d)}$.[7] 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 ${\displaystyle x^{5}}$ (as ${\displaystyle x^{5}=x^{2}\cdot x^{2}\cdot x}$).^[3][8] One notable feature of his works was enumerating all the possible solutions to a given equation.^[9]

The Muslim

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.^[3]

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

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

but some of which, especially those of ${\displaystyle x^{2}}$, were now worked out directly instead of first solving for ${\displaystyle x}$ and accompanied with geometrical illustrations and proofs.

A number of Islamic mathematicians wrote commentaries on this work, including al-Iṣṭakhrī al-Ḥāsib and ʿAli ibn Aḥmad al-ʿImrānī (d. 955-6),[12] but both commentaries are now lost.^[4]

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.^[9] 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.^[9]

Book of Rare Things in the Art of Calculation (Kitāb al-ṭarā’if fi’l-ḥisāb)

Abu Kamil describes a number of systematic procedures for finding integral solutions for indeterminate equations.^[4] It is also the earliest known Arabic work where solutions are sought to the type of indeterminate equations found in Diophantus's Arithmetica. However, Abu Kamil explains certain methods not found in any extant copy of the Arithmetica.^[3] He also describes one problem for which he found 2,678 solutions.^[13]

On the Pentagon and Decagon (Kitāb al-mukhammas wa’al-mu‘ashshar)

In this treatise algebraic methods are used to solve geometrical problems.^[4] Abu Kamil uses the equation ${\displaystyle x^{4}+3125=125x^{2}}$ to calculate a numerical approximation for the side of a regular pentagon in a circle of diameter 10.^[14] He also uses the golden ratio in some of his calculations.^[13] Fibonacci knew about this treatise and made extensive use of it in his Practica geometriae.^[4]

Book of Birds (Kitāb al-ṭair)

A small treatise teaching how to solve indeterminate linear systems with positive integral solutions.^[11] The title is derived from a type of problems known in the east 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.^[11]

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.^[9]

On Measurement and Geometry (Kitāb al-misāḥa wa al-handasa)

A manual of

Lost works

Some of Abu Kamil's lost works include:

• A treatise on the use of double
  false position, known as the Book of the Two Errors (Kitāb al-khaṭaʾayn).^[15]
• Book on Augmentation and Diminution (Kitāb al-jamʿ wa al-tafrīq), which gained more attention after historian Franz Woepcke linked it with an anonymous Latin work, Liber augmenti et diminutionis.^[4]
• Book of Estate Sharing using Algebra (Kitāb al-waṣāyā bi al-jabr wa al-muqābala), which contains algebraic solutions for problems of
  jurists.^[9]

Legacy

The works of Abu Kamil influenced other mathematicians, like al-Karaji and Fibonacci, and as such had a lasting impact on the development of algebra.^[5][16] Many of his examples and algebraic techniques were later copied by Fibonacci in his Practica geometriae and other works.^[5][13] Unmistakable borrowings, but without Abu Kamil being explicitly mentioned and perhaps mediated by lost treatises, are also found in Fibonacci's Liber Abaci.^[17]

On al-Khwarizmi

Abu Kamil was one of the earliest mathematicians to recognize

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, ...[11]

Notes

References

• Sesiano, Jacques (2009-07-09). An introduction to the history of algebra: solving equations from Mesopotamian times to the Renaissance. AMS Bookstore.
  ISBN 978-0-8218-4473-1
  .
• Levey, Martin (1970). "Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad ibn Shujāʿ".
  ISBN 0-684-10114-9
  .
• O'Connor, John J.; Robertson, Edmund F., "Abu Kamil", MacTutor History of Mathematics Archive, University of St Andrews

Further reading

• Yadegari, Mohammad (1978-06-01). "The Use of Mathematical Induction by Abū Kāmil Shujā' Ibn Aslam (850-930)". Isis. 69 (2): 259–262.
  S2CID 144112534
  .
• Karpinski, L. C. (1914-02-01). "The Algebra of Abu Kamil". The American Mathematical Monthly. 21 (2): 37–48.
  JSTOR 2972073
  .
• Herz-Fischler, Roger (June 1987). A Mathematical History of Division in Extreme and Mean Ratio. Wilfrid Laurier Univ Pr.
  ISBN 0-88920-152-8
  .
• Djebbar, Ahmed. Une histoire de la science arabe: Entretiens avec Jean Rosmorduc. Seuil (2001)