Abū Kāmil Shujāʿ ibn Aslam
Abū Kāmil, Shujāʿ ibn Aslam ibn Muḥammad Ibn Shujāʿ (Latinized as Auoquamel,[1] Arabic: أبو كامل شجاع بن أسلم بن محمد بن شجاع, also known as alḥāsib almiṣrī—lit. "the Egyptian reckoner") (c. 850 – c. 930) was an Egyptian 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]
Abū Kāmil  

Born  c. 850 
Died  c. 930 
Residence  Egypt 
Other names  alḥāsib almiṣrī 
Academic background  
Influences  AlKhwarizmi 
Academic work  
Era  Islamic Golden Age 
Main interests  Algebra, geometry 
Notable works  The Book of Algebra 
Notable ideas 

Influenced  AlKaraji, Fibonacci 
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 (up to ),[3][5] and solved sets of nonlinear simultaneous equations with three unknown variables.[6] He illustrated the rules of signs for expanding the multiplication .[7] He also enumerated all the possible solutions to some of his problems. 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" ("squaresquarething") for (as ).[3][8]
The muslim encyclopedist Ibn Khaldūn classified Abū Kāmil as the second greatest algebraist chronologically after alKhwarizmi.[9]
Life
Almost nothing is known about the life and career of Abu Kamil except that he was a successor of alKhwarizmi, whom he never personally met.[3]
Works
Book of Algebra (Kitāb fī aljabr wa almuqābala)
The Algebra is perhaps Abu Kamil's most influential work, which he intended to supersede and expand upon that of AlKhwarizmi.[2][10] Whereas the Algebra of alKhwarizmi was geared towards the general public, Abu Kamil was addressing other mathematicians, or readers familiar with Euclid's Elements.[10] 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 AlKhwarizmi's book,[11] but some of which, especially those of , were now worked out directly instead of first solving for and accompanied with geometrical illustrations and proofs.[5][11] The third chapter contains examples of quadratic irrationalities as solutions and coefficients.[11] 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.[11]
A number of Islamic mathematicians wrote commentaries on this work, including alIṣṭakhrī alḤāsib and ʿAli ibn Aḥmad alʿImrānī (d. 9556),[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.[11] 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.[11]
Book of Rare Things in the Art of Calculation (Kitāb alṭarā’if fi’lḥisāb)
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 almukhammas wa’almu‘ashshar)
In this treatise algebraic methods are used to solve geometrical problems.[4] Abu Kamil uses the equation 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 alṭair)
A small treatise teaching how to solve indeterminate linear systems with positive integral solutions.[10] 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 seventysix 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.[10]
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.[11]
On Measurement and Geometry (Kitāb almisāḥa wa alhandasa)
A manual of geometry for nonmathematicians, 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 alkhaṭaʾayn).[15]
 Book on Augmentation and Diminution (Kitāb aljamʿ wa altafrīq), 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 alwaṣāyā bi aljabr wa almuqābala), which contains algebraic solutions for problems of Islamic inheritance and discusses the opinions of known jurists.[11]
Ibn alNadim in his Fihrist listed the following additional titles: Book of Fortune (Kitāb alfalāḥ), Book of the Key to Fortune (Kitāb miftāḥ alfalāḥ), Book of the Adequate (Kitāb alkifāya), and Book of the Kernel (Kitāb alʿasīr).[5]
Legacy
The works of Abu Kamil influenced other mathematicians, like alKaraji 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 alKhwarizmi
Abu Kamil was one of the earliest mathematicians to recognize alKhwarizmi's contributions to algebra, defending him against Ibn Barza who attributed the authority and precedent in algebra to his grandfather, 'Abd alHamīd 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 alKhwārizmī 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, ...[10]
Notes
 Rāshid, Rushdī; Régis Morelon (1996). Encyclopedia of the history of Arabic science. 2. Routledge. p. 240. ISBN 9780415124119.
 Sesiano, Jacques (2000). "Islamic mathematics". In Selin, Helaine; D'Ambrosio, Ubiratàn (eds.). Mathematics Across Cultures: The History of NonWestern Mathematics. Springer. p. 148. ISBN 1402002602.
 O'Connor, John J.; Robertson, Edmund F., "Abū Kāmil Shujāʿ ibn Aslam", MacTutor History of Mathematics archive, University of St Andrews.
 Hartner, W. (1960). "ABŪ KĀMIL SHUDJĀʿ". Encyclopaedia of Islam. 1 (2nd ed.). Brill Academic Publishers. pp. 132–3. ISBN 9004081143.
 Levey, Martin (1970). "Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad ibn Shujāʿ". Dictionary of Scientific Biography. 1. New York: Charles Scribner's Sons. pp. 30–32. ISBN 0684101149.
 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. ISBN 9780691114859.
 Mat Rofa Bin Ismail (2008), Helaine Selin (ed.), "Algebra in Islamic Mathematics", Encyclopaedia of the History of Science, Technology, and Medicine in NonWestern Cultures (2nd ed.), Springer, 1, p. 114, ISBN 9781402045592
 Bashmakova, Izabella Grigorʹevna; Galina S. Smirnova (20000115). The beginnings and evolution of algebra. Cambridge University Press. p. 52. ISBN 9780883853290.
 Sesiano, Jacques (2008). "Abū Kāmil". Encyclopaedia of the History of Science, Technology, and Medicine in NonWestern Cultures. Springer Netherlands: 7–8. doi:10.1007/9781402044250_9198. ISBN 9781402045592.
 Sesiano, Jacques (20090709). An introduction to the history of algebra: solving equations from Mesopotamian times to the Renaissance. AMS Bookstore. ISBN 9780821844731.
 Sesiano, Jacques (19970731). "Abū Kāmil". Encyclopaedia of the history of science, technology, and medicine in nonwestern cultures. Springer. pp. 4–5.
 Louis Charles Karpinski (1915). Robert of Chester's Latin Translation of the Algebra of AlKhowarizmi, with an Introduction, Critical Notes and an English Version. Macmillan Co.
 Livio, Mario (2003). The Golden Ratio. New York: Broadway. pp. 89–90, 92, 96. ISBN 0767908163.
 Ragep, F. J.; Sally P. Ragep; Steven John Livesey (1996). Tradition, transmission, transformation: proceedings of two conferences on premodern science held at the University of Oklahoma. BRILL. p. 48. ISBN 9789004101197.
 Schwartz, R. K (2004). Issues in the Origin and Development of Hisab alKhata'ayn (Calculation by Double False Position). Eighth North African Meeting on the History of Arab Mathematics. Radès, Tunisia. Available online at: http://facstaff.uindy.edu/~oaks/Biblio/COMHISMA8paper.doc Archived 20110915 at the Wayback Machine and "Archived copy" (PDF). Archived from the original (PDF) on 20140516. Retrieved 20120608.CS1 maint: archived copy as title (link)
 Karpinski, L. C. (19140201). "The Algebra of Abu Kamil". The American Mathematical Monthly. 21 (2): 37–48. doi:10.2307/2972073. ISSN 00029890. JSTOR 2972073.
 Høyrup, J. (2009). Hesitating progressthe slow development toward algebraic symbolization in abbacusand 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.
References
 Sesiano, Jacques (20090709). An introduction to the history of algebra: solving equations from Mesopotamian times to the Renaissance. AMS Bookstore. ISBN 9780821844731.
 Levey, Martin (1970). "Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad ibn Shujāʿ". Dictionary of Scientific Biography. 1. New York: Charles Scribner's Sons. pp. 30–32. ISBN 0684101149.
 O'Connor, John J.; Robertson, Edmund F., "Abū Kāmil Shujāʿ ibn Aslam", MacTutor History of Mathematics archive, University of St Andrews.
Further reading
 Yadegari, Mohammad (19780601). "The Use of Mathematical Induction by Abū Kāmil Shujā' Ibn Aslam (850930)". Isis. 69 (2): 259–262. doi:10.1086/352009. ISSN 00211753. JSTOR 230435.
 Karpinski, L. C. (19140201). "The Algebra of Abu Kamil". The American Mathematical Monthly. 21 (2): 37–48. doi:10.2307/2972073. ISSN 00029890. JSTOR 2972073.
 HerzFischler, Roger (June 1987). A Mathematical History of Division in Extreme and Mean Ratio. Wilfrid Laurier Univ Pr. ISBN 0889201528.
 Djebbar, Ahmed. Une histoire de la science arabe: Entretiens avec Jean Rosmorduc. Seuil (2001)