Abu Kamil
Egyptian mathematician of Abbasid era (c. 850 – 930)
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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 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.
The Muslim encyclopedist Ibn Khaldūn 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.
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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 ...