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Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad Ibn Shujāʿ ( Latinized as Auoquamel, , also known as ''Al-ḥāsib al-miṣrī''—lit. "The Egyptian Calculator") (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 year ...
mathematician during the
Islamic Golden Age The Islamic Golden Age was a period of scientific, economic, and cultural flourishing in the history of Islam, traditionally dated from the 8th century to the 13th century. This period is traditionally understood to have begun during the reign o ...
. He is considered the first mathematician to systematically use and accept
irrational number In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, ...
s as solutions and
coefficients In mathematics, a coefficient is a multiplicative factor involved in some term of a polynomial, a series, or any other type of expression. It may be a number without units, in which case it is known as a numerical factor. It may also be a ...
to equations. His mathematical techniques were later adopted by
Fibonacci Leonardo Bonacci ( – ), commonly known as Fibonacci, was an Italians, Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". The name he is commonly called, ''Fibonacci ...
, thus allowing Abu Kamil an important part in introducing algebra to
Europe Europe is a continent located entirely in the Northern Hemisphere and mostly in the Eastern Hemisphere. It is bordered by the Arctic Ocean to the north, the Atlantic Ocean to the west, the Mediterranean Sea to the south, and Asia to the east ...
. Abu Kamil made important contributions to
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
and
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
. 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 In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single e ...
with three unknown
variables Variable may refer to: Computer science * Variable (computer science), a symbolic name associated with a value and whose associated value may be changed Mathematics * Variable (mathematics), a symbol that represents a quantity in a mathemat ...
. 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 Mathematical notation consists of using glossary of mathematical symbols, symbols for representing operation (mathematics), operations, unspecified numbers, relation (mathematics), relations, and any other mathematical objects and assembling ...
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 is a reference work or compendium providing summaries of knowledge, either general or special, in a particular field or discipline. Encyclopedias are divided into articles or entries that are arranged alphabetically by artic ...
Ibn Khaldūn Ibn Khaldun (27 May 1332 – 17 March 1406, 732–808 Hijri year, AH) was an Arabs, Arab Islamic scholar, historian, philosopher and sociologist. He is widely acknowledged to be one of the greatest social scientists of the Middle Ages, and cons ...
classified Abū Kāmil as the second greatest algebraist chronologically after
al-Khwarizmi Muhammad ibn Musa al-Khwarizmi , or simply al-Khwarizmi, was a mathematician active during the Islamic Golden Age, who produced Arabic-language works in mathematics, astronomy, and geography. Around 820, he worked at the House of Wisdom in B ...
.


Life

Almost nothing is known about the life and career of Abu Kamil except that he was a successor of
al-Khwarizmi Muhammad ibn Musa al-Khwarizmi , or simply al-Khwarizmi, was a mathematician active during the Islamic Golden Age, who produced Arabic-language works in mathematics, astronomy, and geography. Around 820, he worked at the House of Wisdom in B ...
, 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 Muhammad ibn Musa al-Khwarizmi , or simply al-Khwarizmi, was a mathematician active during the Islamic Golden Age, who produced Arabic-language works in mathematics, astronomy, and geography. Around 820, he worked at the House of Wisdom in B ...
. Whereas the ''Algebra'' of al-Khwarizmi was geared towards the general public, Abu Kamil was addressing other mathematicians, or readers familiar with
Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...
's ''Elements''. In this book Abu Kamil solves systems of
equation In mathematics, an equation is a mathematical 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 ...
s whose solutions are whole numbers and
fractions A fraction (from , "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, thre ...
, and accepted
irrational numbers In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, ...
(in the form of a
square root In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
or fourth root) as solutions and
coefficient In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...
s to
quadratic equation In mathematics, a quadratic equation () is an equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where the variable (mathematics), variable represents an unknown number, and , , and represent known numbers, where . (If and ...
s. 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 In mathematics, a quadratic irrational number (also known as a quadratic irrational or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the rational number ...
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 made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon' ...
. The rest of the book contains solutions for sets of
indeterminate equation In mathematics, particularly in number theory, an indeterminate system has fewer equations than unknowns but an additional a set of constraints on the unknowns, such as restrictions that the values be integers. In modern times indeterminate equati ...
s, problems of application in realistic situations, and problems involving unrealistic situations intended for
recreational mathematics Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research-and-application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...
. 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 Leonardo Bonacci ( – ), commonly known as Fibonacci, was an Italians, Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". The name he is commonly called, ''Fibonacci ...
, and some sections were incorporated and improved upon in the Latin work of
John of Seville John of Seville (Latin: ''Johannes Hispalensis'' or ''Johannes Hispaniensis'') (fl. 1133-53) was one of the main translators from Arabic into Castilian in partnership with Dominicus Gundissalinus during the early days of the Toledo School of Tr ...
, ''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 equation In mathematics, particularly in number theory, an indeterminate system has fewer equations than unknowns but an additional a set of constraints on the unknowns, such as restrictions that the values be integers. In modern times indeterminate equati ...
s. It is also the earliest known Arabic work where solutions are sought to the type of indeterminate equations found in
Diophantus Diophantus of Alexandria () (; ) was a Greek mathematician who was the author of the '' Arithmetica'' in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations. Although Jose ...
's ''
Arithmetica Diophantus of Alexandria () (; ) was a Greek mathematics, Greek mathematician who was the author of the ''Arithmetica'' in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations ...
''. 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 () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...
in a circle of diameter 10. He also uses the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...
in some of his calculations.
Fibonacci Leonardo Bonacci ( – ), commonly known as Fibonacci, was an Italians, Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". The name he is commonly called, ''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 system In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstractio ...
s 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 Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
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. Three equiva ...
s, right circular
prism PRISM is a code name for a program under which the United States National Security Agency (NSA) collects internet communications from various U.S. internet companies. The program is also known by the SIGAD . PRISM collects stored internet ...
s,
square pyramid In geometry, a square pyramid is a Pyramid (geometry), pyramid with a square base and four triangles, having a total of five faces. If the Apex (geometry), apex of the pyramid is directly above the center of the square, it is a ''right square p ...
s, and circular
cones In geometry, a cone is a three-dimensional figure that tapers smoothly from a flat base (typically a circle) to a point not contained in the base, called the ''apex'' or '' vertex''. A cone is formed by a set of line segments, half-lines, ...
). The first few chapters contain rules for determining the
area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
,
diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek � ...
,
perimeter A perimeter is the length of a closed boundary that encompasses, surrounds, or outlines either a two-dimensional shape or a one-dimensional line. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimet ...
, 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, 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 German historian, Orientalist and mathematician. He is remembered for publishing editions and translations of medieval Arabic mathematical manuscripts and for his research on the propagation o ...
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 Islamic Inheritance jurisprudence is a field of fiqh, Islamic jurisprudence () that deals with inheritance, a topic that is prominently dealt with in the Qur'an. It is often called ''Mīrāth'' (, literally "inheritance"), and its branch of Shar ...
and discusses the opinions of known
jurists A jurist is a person with expert knowledge of law; someone who analyzes and comments on law. This person is usually a specialist legal scholar, mostly (but not always) with a formal education in law (a law degree) and often a legal practition ...
.
Ibn al-Nadim Abū al-Faraj Muḥammad ibn Isḥāq an-Nadīm (), also Ibn Abī Yaʿqūb Isḥāq ibn Muḥammad ibn Isḥāq al-Warrāq, and commonly known by the '' nasab'' (patronymic) Ibn an-Nadīm (; died 17 September 995 or 998), was an important Muslim ...
in his ''
Fihrist The () (''The Book Catalogue'') is a compendium of the knowledge and literature of tenth-century Islam compiled by Ibn al-Nadim (d. 998). It references approx. 10,000 books and 2,000 authors.''The Biographical Dictionary of the Society for the ...
'' 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 (; c. 953 – c. 1029) was a 10th-century Persian mathematician and engineer who flourished at Baghdad. He was born in Karaj, a city near Tehran. His three principal surviving works are mathematical: ''Al-Badi' fi'l-hisab'' (''Wonderful on ...
and
Fibonacci Leonardo Bonacci ( – ), commonly known as Fibonacci, was an Italians, Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". The name he is commonly called, ''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 The or (Latin for "The Book of Calculation") was a 1202 Latin work on arithmetic by Leonardo of Pisa, posthumously known as Fibonacci. It is primarily famous for introducing both base-10 positional notation and the symbols known as Arabic n ...
''.


On al-Khwarizmi

Abu Kamil was one of the earliest mathematicians to recognize
al-Khwarizmi Muhammad ibn Musa al-Khwarizmi , or simply al-Khwarizmi, was a mathematician active during the Islamic Golden Age, who produced Arabic-language works in mathematics, astronomy, and geography. Around 820, he worked at the House of Wisdom in B ...
's contributions to
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, 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 () was a ninth-century mathematician. Not much is known about his life. The two records of him, one by Ibn Nadim and the other by al-Qifti are not identical. Al-Qifi mentions his name as ʿAbd al-Hamīd ibn Wase ibn Tu ...
. 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

* * *


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