World Library  
Flag as Inappropriate
Email this Article

Abū Kāmil Shujāʿ ibn Aslam

Abū Kāmil
Other names al-ḥāsib al-miṣrī
Born c. 850
Died c. 930
Era Islamic Golden Age
Region Egypt
Main interests
Algebra, Geometry
Notable ideas
  • Use of irrational numbers as solutions and coefficients to equations
Major works
The Book of Algebra

Abū Kāmil, Shujāʿ ibn Aslam ibn Muḥammad Ibn Shujāʿ (Latinized as Auoquamel,[1] Arabic: ابو كامل‎, also known as al-ḥāsib al-miṣrī—lit. "the Egyptian reckoner") (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.[2] His mathematical techniques were later adopted by Fibonacci, thus allowing Abu Kamil an important part in introducing algebra to Europe.[3]

Abu Kamil made important contributions to algebra and geometry.[4] He was the first Islamic mathematician to work easily with algebraic equations with powers higher than x^2 (up to x^8),[3][5] and solved sets of non-linear simultaneous equations with three unknown variables.[6] 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).[3][7]


  • Life 1
  • Works 2
    • Book of Algebra (Kitāb fī al-jabr wa al-muqābala) 2.1
    • Book of Rare Things in the Art of Calculation (Kitāb al-ṭarā’if fi’l-ḥisāb) 2.2
    • On the Pentagon and Decagon (Kitāb al-mukhammas wa’al-mu‘ashshar) 2.3
    • Book of Birds (Kitāb al-ṭair) 2.4
    • On Measurement and Geometry (Kitāb al-misāḥa wa al-handasa) 2.5
    • Lost works 2.6
  • Legacy 3
  • On al-Khwarizmi 4
  • Notes 5
  • References 6
  • Further reading 7


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]


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.[2][8] 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.[8] 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.[2]

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,[9] 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.[5][9] The third chapter contains examples of quadratic irrationalities as solutions and coefficients.[9] 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.[9]

A number of Islamic mathematicians wrote commentaries on this work, including al-Iṣṭakhrī al-Ḥāsib and ʿAli ibn Aḥmad al-ʿImrānī (d. 955-6),[10] 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.[11]

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 x^4 + 3125 = 125x^2 to calculate a numerical approximation for the side of a regular pentagon in a circle of radius 10.[12] He also uses the Golden Ratio in some of his calculations.[11] 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.[8] 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.[8]

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 geometry for non-mathematicians, like land surveyors and other government officials, which presents a set of rules for calculating the volume and surface area of solids (mainly rectangular parallelepipeds, right circular prisms, square pyramids, and circular cones). The first few chapters contain rules for determining the area, diagonal, perimeter, and other parameters for different types of triangles, rectangles and squares.[3]

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).[13]
  • 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 Islamic inheritance and discusses the opinions of known jurists.[9]

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), and Book of the Kernel (Kitāb al-ʿasīr).[5]


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][14] Many of his examples and algebraic techniques were later copied by Fibonacci in his Practica geometriae and other works.[5][11] Unmistakable borrowings, but without Abu Kamil being explicitly mentioned and perhaps mediated by lost treatises, are also found in Fibonacci's Liber Abaci.[15]

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.[3] 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, ...[8]


  1. ^ Rāshid, Rushdī; Régis Morelon (1996). Encyclopedia of the history of Arabic science 2. Routledge. p. 240.  
  2. ^ a b c Jacques Sesiano, "Islamic mathematics", p. 148, in  
  3. ^ a b c d e f g  .
  4. ^ a b c d e f Hartner, W. (1960). "ABŪ KĀMIL SHUDJĀʿ".  
  5. ^ a b c d e Levey, Martin (1970). "Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad ibn Shujāʿ".  
  6. ^ Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. pp. 518, 550.  
  7. ^  
  8. ^ a b c d e Sesiano, Jacques (2009-07-09). An introduction to the history of algebra: solving equations from Mesopotamian times to the Renaissance. AMS Bookstore.  
  9. ^ a b c d e f g h Sesiano, Jacques (1997-07-31). "Abū Kāmil". Encyclopaedia of the history of science, technology, and medicine in non-western cultures. Springer. pp. 4–5. 
  10. ^ Louis Charles Karpinski (1915). Robert of Chester's Latin Translation of the Algebra of Al-Khowarizmi, with an Introduction, Critical Notes and an English Version. Macmillan Co. 
  11. ^ a b c Livio, Mario (2003). The Golden Ratio. New York: Broadway. pp. 89–90, 92, 96.  
  12. ^ Ragep, F. J.; Sally P. Ragep; Steven John Livesey (1996). Tradition, transmission, transformation: proceedings of two conferences on pre-modern science held at the University of Oklahoma. BRILL. p. 48.  
  13. ^ Schwartz, R. K (2004). Issues in the Origin and Development of Hisab al-Khata’ayn (Calculation by Double False Position). Eighth North African Meeting on the History of Arab Mathematics. Radès, Tunisia.  Available online at: and
  14. ^ Karpinski, L. C. (1914-02-01). "The Algebra of Abu Kamil". The American Mathematical Monthly 21 (2): 37–48.  
  15. ^ Høyrup, J. (2009). Hesitating progress-the slow development toward algebraic symbolization in abbacus-and related manuscripts, c. 1300 to c. 1550: Contribution to the conference" Philosophical Aspects of Symbolic Reasoning in Early Modern Science and Mathematics", Ghent, 27–29 August 2009. Preprints 390. Berlin: Max Planck Institute for the History of Science. 


  • Sesiano, Jacques (2009-07-09). An introduction to the history of algebra: solving equations from Mesopotamian times to the Renaissance. AMS Bookstore.  
  • Levey, Martin (1970). "Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad ibn Shujāʿ".  
  • .  

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.  
  • Karpinski, L. C. (1914-02-01). "The Algebra of Abu Kamil". The American Mathematical Monthly 21 (2): 37–48.  
  • Herz-Fischler, Roger (June 1987). A Mathematical History of Division in Extreme and Mean Ratio. Wilfrid Laurier Univ Pr.  
  • Djebbar, Ahmed. Une histoire de la science arabe: Entretiens avec Jean Rosmorduc. Seuil (2001)
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from World eBook Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.