Abū Kāmil
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Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad Ibn Shujāʿ ( Latinized as Auoquamel, ar, أبو كامل شجاع بن أسلم بن محمد بن شجاع, also known as ''Al-ḥāsib al-miṣrī''—lit. "the Egyptian reckoner") (c. 850 – c. 930) was a prominent
Egyptian Egyptian describes something of, from, or related to Egypt. Egyptian or Egyptians may refer to: Nations and ethnic groups * Egyptians, a national group in North Africa ** Egyptian culture, a complex and stable culture with thousands of years of ...
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 illustrated the rules of signs for expanding the multiplication (a \pm b)(c \pm d). 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 (as x^5 = x^2\cdot x^2\cdot x). One notable feature of his works was enumerating all the possible solutions to a given equation. The Muslim
encyclopedist An encyclopedia (American English) or encyclopædia (British English) is a reference work or compendium providing summaries of knowledge either general or special to a particular field or discipline. Encyclopedias are divided into article ( ...
Ibn Khaldūn Ibn Khaldun (; ar, أبو زيد عبد الرحمن بن محمد بن خلدون الحضرمي, ; 27 May 1332 – 17 March 1406, 732-808 AH) was an Arab The Historical Muhammad', Irving M. Zeitlin, (Polity Press, 2007), p. 21; "It is, of ...
classified Abū Kāmil as the second greatest algebraist chronologically after al-Khwarizmi.


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.


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
equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
s whose solutions are whole numbers and fractions, and accepted irrational numbers (in the form of a square root or fourth root) as solutions and
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
s to quadratic equations. 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 In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
. 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.


''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. It is also the earliest known Arabic work where solutions are sought to the type of indeterminate equations found in
Diophantus Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...
's ''
Arithmetica ''Arithmetica'' ( grc-gre, Ἀριθμητικά) is an Ancient Greek text on mathematics written by the mathematician Diophantus () in the 3rd century AD. It is a collection of 130 algebraic problems giving numerical solutions of determinate e ...
''. However, Abu Kamil explains certain methods not found in any extant copy of the ''Arithmetica''. He also describes one problem for which he found 2,678 solutions.


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

In this treatise algebraic methods are used to solve geometrical problems. Abu Kamil uses the equation x^4 + 3125 = 125x^2 to calculate a numerical approximation for the side of a regular
pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...
in a circle of diameter 10. He also uses the golden ratio in some of his calculations. Fibonacci knew about this treatise and made extensive use of it in his ''Practica geometriae''.


''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 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.
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, which presents a set of rules for calculating the volume and surface area of solids (mainly rectangular
parallelepiped In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidea ...
s, right circular prisms,
square pyramid In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it is a right square pyramid, and has symmetry. If all edge lengths are equal, it is an equilateral square pyramid, ...
s, 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.


Lost works

Some of Abu Kamil's lost works include: * A treatise on the use of double
false position In mathematics, the ''regula falsi'', method of false position, or false position method is a very old method for solving an equation with one unknown; this method, in modified form, is still in use. In simple terms, the method is the trial and er ...
, known as the ''Book of the Two Errors'' (''Kitāb al-khaṭaʾayn''). * ''Book on Augmentation and Diminution'' (''Kitāb al-jamʿ wa al-tafrīq''), which gained more attention after historian
Franz Woepcke Franz Woepcke (6 May 1826 – 25 March 1864) was a historian, Orientalist and mathematician. He is remembered for publishing editions and translations of medieval Arabic mathematical manuscripts and for his research on the propagation of the H ...
linked it with an anonymous Latin work, ''Liber augmenti et diminutionis''. * ''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. Ibn al-Nadim 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'').


Legacy

The works of Abu Kamil influenced other mathematicians, like
al-Karaji ( fa, ابو بکر محمد بن الحسن الکرجی; c. 953 – c. 1029) was a 10th-century Persian people, Persian mathematician and engineer who flourished at Baghdad. He was born in Karaj, a city near Tehran. His three principal sur ...
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. Unmistakable borrowings, but without Abu Kamil being explicitly mentioned and perhaps mediated by lost treatises, are also found in Fibonacci's '' Liber Abaci''.


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,
'Abd al-Hamīd ibn Turk ( fl. 830), known also as ( ar, ابومحمد عبدالحمید بن واسع بن ترک الجیلی) was a ninth-century Muslim mathematician. Not much is known about his life. The two records of him, one by Ibn Nadim and the other by al-Q ...
. 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, ...


Notes


References

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Further reading

* * * * Djebbar, Ahmed. ''Une histoire de la science arabe'': Entretiens avec Jean Rosmorduc. Seuil (2001) {{DEFAULTSORT:Abu Kamil 9th-century mathematicians 10th-century mathematicians 9th-century people from the Abbasid Caliphate 10th-century people from the Abbasid Caliphate Mathematicians from the Abbasid Caliphate Algebraists 850s births 930 deaths Year of birth uncertain Year of death uncertain Medieval Egyptian mathematicians Mathematicians who worked on Islamic inheritance