Abu Kamil

Egyptian mathematician of Abbasid era (c. 850 – 930)

Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad Ibn Shujāʿ (Latinized as Auoquamel, أبو كامل شجاع بن أسلم بن محمد بن شجاع, also known as Al ḥāsib al miṣrī—lit. "The Egyptian Calculator") (c. 850 – c. 930) was a prominent 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. He illustrated the rules of signs for expanding the multiplication 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 ). One notable feature of his works was enumerating all the possible solutions to a given equation.

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.

The Algebra is perhaps Abu Kamil's most influential work, which he intended to supersede and expand upon that of Al Khwarizmi. 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 both commentaries are now lost. 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 uses the golden ratio in some of his calculations. Fibonacci knew about this treatise and made extensive use of it in his Practica geometriae.</blockquote>

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.

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. 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. Many of his examples and algebraic techniques were later copied by Fibonacci in his Practica geometriae and other works.

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,