# 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" ($$x^2 + bx = c$$). 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 $$\frac{b}{4}.$$ 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 $$\frac{b}{4}.$$ Thus, the area of the large square is, in modern notation, $x^2 + bx + {\frac{b^2}{4}} = {\left(x + {\frac{b}{2}}\right)}^2,$ and this is in turn equal to $$c + {\frac{b^2}{4}}.$$ The solution to the equation is then evident.

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