Processing math: 100%

You are here

Mathematical Treasures - Al-Khwarizmi's Algebra

Author(s): 
Frank J. Swetz and Victor J. Katz

This is a page from al-Khwārizmī's algebra text, Kitāb al-jabr wa'l-muqābala, written in about 825, the first extant algebra text, by Muḥammad ibn Mūsā al-Khwārizmī. 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-Khwārizmī'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 b4. 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 b4. Thus, the area of the large square is, in modern notation, x2+bx+b24=(x+b2)2, and this is in turn equal to c+b24. The solution to the equation is then evident.

Index to Mathematical Treasures

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