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An Arabic Finger-reckoning Rule Appropriated for Proofs in Algebra - Reading the language more closely

Jeffrey A. Oaks (University of Indianapolis)

Reading the language more closely

I have already explained two ways that equations in algebra can be differentiated from the equating of numbers in arithmetic. Equations are stated in terms of the names of the powers ("thing"/"root", "māl", "cube", etc.), and the two sides are always equated with the verb ʿadala.

There is another important way that arithmetical problems and rules, like the "squaring" rule, will differ from algebraic equations. I will illustrate this by comparing the finger-reckoning rule with Abū Kāmil's equation. Recall the rule:

You take half of the sum of the two numbers and you square it. You subtract from the result a square of half of the difference between them. The remainder is the result of the multiplication.

The equation is:

twenty dirhams and three things and a third of a thing equal four ninths of a māl.

Recall, also, that I rewrote both with modern algebraic notation, the rule as \(\left({1\over 2}(p+q)\right)^2-\left({1\over 2}|p-q|\right)^2=pq\) and the equation as \(20+3{1\over 3}x={4\over 9}x^2\).

We have already seen that the numbers in the rule are not expressed with algebraic names from which we could build composite expressions through operations. Here the numbers are only referred to as "the two numbers" and "them". I converted the rule into an algebraic identity by naming the numbers \(p\) and \(q\) and forming expressions from them. For instructors of mathematics this act of naming is second nature. To pick a simpler example, we can translate "the sum of two consecutive squares" into \(n^2+(n+1)^2\) with hardly a thought, but to many of our students this is a formidable task.

There is still another, perhaps more critical, difference between the rule and the equation, one that lies at the heart of what differentiates medieval from modern algebra. The rule, you will notice, is presented as a sequence of operations to be performed, with a specified outcome. One is instructed to take half of a sum, to square it, and to subtract another amount also obtained through a halving and a squaring, and the result of these operations is the product of the two numbers. The equation, by contrast, asserts the equivalence of two fixed amounts.

Let's examine the equation more closely. The expression "twenty dirhams and three things and a third of a thing" exhibits no operations at all. It is simply a collection of two kinds of object, \(20\) of one and \(3{\frac{1}{3}}\) of the other. It is like saying "twenty pounds and three and a third ounces". The expression does not call for the operation of addition, but instead indicates the aggregation of the two kinds with the common conjunction "and" (wa). Similarly, the "less" (illā) in expressions like "ten less a thing" (our \(10-x\)) is not the operation of subtraction, but is the negative counterpart to "and". It indicates that the ten is deficient, lacking a "thing".

There is no scalar multiplication in the equation, either. It would be just as silly to say that there is scalar multiplication in "three things and a third of a thing" as it would be to say that there is scalar multiplication in the expression "three and a third ounces". Likewise, the "four ninths of a māl" is spoken much like "four ninths of a kilogram" (to stick with units of weight/mass).

The two sides of an Arabic equation ideally contain no operations at all. (Exceptions were sometimes made for division, but there is no space to go into that here!) One can, of course, perform operations on algebraic expressions. For instance, we can multiply a thing by ten less a thing to get ten things less a māl. It is the individual expressions themselves that lack operations.

It is for this reason that Arabic algebraists insisted on working out all operations prior to setting up their equations. The two sides of the equation should be specified amounts, without any unresolved operations. Where our \[{4\over 9}x^2+4-2{1\over 3}x=x+24\] is built from the operations of addition, subtraction, scalar multiplication, and exponentiation, both sides of its medieval counterpart are aggregations of different kinds of number. Thus modern notation is not only inadequate for expressing medieval arithmetical rules, it also does not reliably express medieval algebraic equations!

I mention all this because it helps us better understand the unfolding of Ibn al-Bannāʾ's proofs. For each proof he has two pieces to work with: the rule and the equation. All the work takes place in the context of the rule, and he uses the equation only to substitute values. For his type 4 proof he cannot start off by stating some rhetorical equivalent of \[39x^2+\left({\frac{1}{2}}(39-x^2)\right)^2 = \left({\frac{1}{2}}(39-x^2)\right)^2\] and then work through the operations. Such a monstrosity could have been formulated, but it violates the idea of a "composite number" (polynomial) in Arabic. Instead, he must perform the operations first, and only when all additions, subtractions, and multiplications have been completed can he confront the two parts to form new, and simple, equations.

Note. For detailed studies on the ideas briefly outlined in this section, see [Oaks 2009] for the "aggregations interpretation" and [Oaks 2010] on the verb ʿadala.

Jeffrey A. Oaks (University of Indianapolis), "An Arabic Finger-reckoning Rule Appropriated for Proofs in Algebra - Reading the language more closely," Convergence (December 2018)