- THE MUSLIMS 425
explains the solution as follows (we put the legal principle that provides the equation
in capital letters):
Call the amount taken out of the debt thing. Add this to the property;
the sum is 10 dirhems plus thing. Subtract one-fifth of this, since he
has bequeathed one-fifth of his property to the friend. The remainder
is 8 dirhems plus | of thing. Then subtract the 1 dirhem extra that
is bequeathed to the friend. There remain 7 dirhems and | of thing.
Divide this between the two sons. The portion of each of them is
3£ dirhems plus § of thing. THIS MUST BE EQUAL TO THING.
Reduce it by subtracting | of thing from thing. Then you have |
of thing equal to 3| dirhems. Form a complete thing by adding to
this quantity | of itself. Now | of 3^ dirhems is 2| dirhems, so that
thing is 5| dirhems.Rosen (1831, p. 133) suggested that the many arbitrary principles used in these
problems were introduced by lawyers to protect the interests of next-of-kin against
those of other legatees.
5.2. Abu Kamil. A commentary on al-Khwarizmi's Algebra was written by the
mathematician Abu Kamil (ca. 850-930). His exposition of the subject contained
none of the legacy problems found in al-Khwarizmi's treatise, but after giving the
basic rules of algebra, it listed 69 problems of considerable intricacy to be solved.
For example, a paraphrase of Problem 10 is as follows:
The number 50 is divided by a certain number. If the divisor is
increased by 3, the quotient decreases by 3|. What is the divisor?Abu Kami! is also noteworthy because many of his problems were copied by
Leonardo of Pisa, one of the first to introduce the mathematics of the Muslims into
Europe.
5.3. Omar Khayyam. Although al-Khwarizmi did not consider any equations
of degree higher than 2, such equations were soon to be considered by Muslim
mathematicians. As we saw in Section 1 of Chapter 10, a link between geometry
and algebra appeared in the use of the rectangular hyperbola by Pappus to carry
out the neusis construction for trisecting an angle (Fig. 9 of Chapter 10). The
mathematician Omar Khayyam, of the late eleventh and early twelfth centuries
(see Amir-Moez, 1963), realized that a large class of geometric problems of this
type led to cubic equations that could be solved using conic sections. His treatise
on algebra^15 was largely occupied with the classification and solution of cubic
equations by this method.
Omar Khayyam did not have modern algebraic symbolism. He lived within the
confines of the universe constructed by the Greeks. His classification of equations,
like al-Khwarizmi's, is conditioned by the use of only positive numbers as data.
For that reason his classification is even more complicated than al-Khwarizmi's,
since he is considering cubic equations as well as quadratics. He lists 25 types of
equations (Kasir, 1931, pp. 51-52), six of which do not involve any cubic terms. The
particular cubic we shall consider is cubes plus squares plus sides equal number, or, as
(^15) This treatise was little noticed in Europe until a French translation by Franz Woepcke (1827-
1864) appeared in 1851 (Kasir, 1931, p. 7).