algebra,” namely, “restoring” (al-jabr) and (al-muq¯abala) “compensating,” came to play their crucial role. Let us consider an example to see how exactly they worked:2 You divide ten into two parts; you multiply each part by itself; you add <the products>, and then you add to them the difference between the two parts before they are multiplied, so that you get fifty-four dirhams.
In this problem, we are asked to divide the number 10 into two parts that satisfy a certain condition. This is of course quite similar to the kinds of problems solved by Diophantus. Diophantus was translated into Arabic somewhat later (see below) and al-Khw¯arizm¯ı was not aware of his texts. But these problems were of a rather general kind, which in several mathematical cultures were known and solved in various ways. A main innovation involved in al-Khw¯arizm¯ı’s treatise was the systematic way in which he approached, analyzed and classified all possible cases of those problems involving squares of the unknown. It is also interesting to notice the use of the term “dirham” here to mean just “numbers.” The issue of the terminology used in Arabic mathematical texts is very interesting in itself and revealing about underlying attitudes towards numbers, but it goes beyond the scope of this book, and I will not go into the details.3 Let us follow the steps of al-Khw¯arizm¯ı’s solution to this problem. This is where the focus of our interest lies now. I describe the steps rhetorically, as he did, and I add to them symbolic formulations that help us follow his way of reasoning. After we have gone through this, however, I will stress the places where the symbolic diverts from what al-Khw¯arizm¯ı did rhetorically: 1 You multiply ten minus a thing by itself; the result is one hundred plus a square, minus twenty things. [(10 – x)2 = 100 + x2 – 20x] 2 You multiply the thing that remains from ten by itself; the result is a square. [x2 ] 3 Then, you add all of that, and the result is one hundred plus two squares minus twenty things. [100 + 2x2 – 20x] 4 But he said: you add to them the difference between the two parts before they are multiplied. You say: the difference between them is ten minus two things. [(10 – x) – x] 5 The sum of this is therefore one hundred and ten plus two squares minus twenty-two things, equal to fifty-four dirhams. [110 + 2x2 – 22x = 54] Al-Khw¯arizm¯ı has arrived at this point at an “equation,” which is a rhetorically formulated condition. He obviously displays a remarkable ability to conduct this kind of complex reasoning in a purely rhetorical manner. But now the question is how do we proceed from this expression to one that we can recognize as embodying one of the six canonical cases. He does it, using “restoring” and “compensating,” as follows: 6 One hundred and ten dirhams plus two squares are equal to fifty-four dirhams plus twenty-two things. [110 + 2x2 = 54 + 22x] 2 3
(Rashed 2009, pp. 158 ff.). See (Oaks and Alkhateeb 2007).
98 | NUMBERS IN MEDIEVAL ISLAM