Al-Khwarizmi's Algebra

This is a page from al-Khwarimi'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.