- THE MUSLIMS 423
Colebrooke also notes (p. Ixxi) that Mohammed Abu'l-Wafa al-Buzjani (940-
998) wrote a translation or commentary on the Arithmetica of Diophantus. This
work, however, is now lost. Apart from these possible influences of Greek and Hindu
algebra, whose effect is difficult to measure, it appears that the progress of algebra
in the Islamic world was an indigenous growth. We shall trace that growth through
several of its most prominent representatives, starting with the man recognized as
its originator, Muhammed ibn Musa al-Khwarizmi.
5.1. Al-Khwarizmi. Besides the words algebra and algorithm, there is a common
English word whose use is traceable to Arabic influence (although it is not an Arabic
word), namely root in the sense of a square or cube root or a root of an equation.
The Greek picture of the square root was the side of a square, and the word side
(pleura) was used accordingly. The Muslim mathematicians apparently thought of
the root as the part from which the equation was generated and used the word
jadhr accordingly. According to al-Daffa (1973, p. 80), translations into Latin from
Greek use the word latus while those from Arabic use radix. In English the word
side lost out completely in the competition.
Al-Khwarizmi's numbers correspond to what we call positive real numbers.
Theoretically, such a number could be defined by any convergent sequence of ra-
tional numbers, but in practice some rule is needed to generate the terms of the
sequence. For that reason, it is more accurate to describe al-Khwarizmi's numbers
as positive algebraic numbers, since all of his numbers are generated by equations
with rational coefficients. The absence of negative numbers prevented al-Khwarizmi
from writing all quadratic equations in the single form "squares plus roots plus num-
bers equal zero" (ox^2 + bx + c = 0). Instead, he had to consider three basic cases
and three others, in which either the square or linear term is missing. He described
the solution of "squares plus roots equal numbers" by the example of "a square
plus 10 roots equal 39 dirhems." (A dirhem is a unit of money.) Al-Khwarizmi's
solution of this problem is to draw a square of unspecified size (the side of the
square is the desired unknown) to represent the square (Fig. 3). To add 10 roots,
he then attaches to each side a rectangle of length equal to the side of the square
and width 2^ (since 4 · 2^ = 10). The resulting cross-shaped figure has, by the
condition of the problem, area equal to 39. He then fills in the four corners of the
figure (literally "completing the square"). The total area of these four squares is
4 • (25)^2 = 25. Since 39 + 25 = 64, the completed square has side 8. Since this
square was obtained by adding rectangles of side 2\ to each side of the original
square, it follows that the original square had side 3.
This case is the one al-Khwarizmi considers first and is the simplest to un-
derstand. His figures for the other two cases of quadratic equations are more
complicated, but all are based on Euclid's geometric illustration of the identity
((a + b)/2)^2 + ((a - b)/2)^2 = ab (Fig. 17 of Chapter 10).
Al-Khwarizmi did not consider any cubic equations. Roughly the first third of
the book is devoted to various examples of pure mathematical problems leading to
quadratic equations, causing the reader to be somewhat skeptical of his claim to be
presenting the material needed in commerce and law. In fact, there are no genuine
applications of quadratic equations in the book. But if quadratic equations have no
practical applications (outside of technology, of course), there are occasions when
a practical problem requires solving linear equations. Al-Khwarizmi found many