Born : c. 850
Died : c. 930
Residence : Egypt
Main : Algebra, geometry.
Abū Kāmil,
Shujāʿ ibn Aslam ibn Muḥammad Ibn Shujāʿ (Latinized as Auoquamel, Arabic: 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 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. 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
(up to), and solved sets of non-linear simultaneous equations with three
unknown variables. He illustrated the rules of signs for expanding the
multiplication. 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" ("square-square-thing") for (as).
The muslim
encyclopedist Ibn Khaldūn classified Abū Kāmil as the second greatest
algebraist chronologically after al-Khwarizmi.
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 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.
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 , were now
worked out directly instead of first solving for 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.
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's Arithmetica. 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 to calculate
a numerical approximation for the side of a regular pentagon 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 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.
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 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 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. 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.
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