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Mathematical Treasures - Al-Khwarizmi's Algebra

Frank J. Swetz and Victor J. Katz


This is a page from al-Khwarizmi's algebra text, Kitab al-jabr wa l-muqabala, written in about 825, the first extant algebra text, by Muhammad ibn Musa al-Khwarizmi. This copy itself is undated, however. It corresponds to page 15 in the translation by Frederic Rosen: The Algebra of Muhammed ben Musa (London: Oriental Translation Fund, 1831), which is also available in a reprinting in the series on Islamic Mathematics and Astronomy, from the Institute for the History of Arabic-Islamic Science at the Johann Wolfgang Goethe University, Frankfurt am Main. On this page is al-Khwarizmi's proof of the rule for solving a quadratic equation of the form "squares plus roots equal numbers" (x2 + bx = c). The central square in the diagram represents the square on the unknown. The four rectangles on the four sides of the square each have width b/4. Thus the area of the central square plus the four rectangles is c. The square is then completed by adding the four corner squares, each of side b/4. Thus, the area of the large square is, in modern notation, x2 + bx + b2/4 = (x + b/2)2, and this is in turn equal to c + b2/4. The solution to the equation is then evident.

Frank J. Swetz and Victor J. Katz, "Mathematical Treasures - Al-Khwarizmi's Algebra," Convergence (January 2011)


Mathematical Treasures from the Smith and Plimpton Collections at Columbia University