An Arabic Finger-reckoning Rule Appropriated for Proofs in Algebra - Ibn al-Bannāʾ and Arabic algebra

Ibn al-Bannāʾ

Abū l-ʿAbbās Aḥmad ibn Muḥammad ibn ʿUthmān al-Azdī al-Marrākushī is better known to us by the moniker Ibn al-Bannāʾ, which means "son of the builder". He lived his entire life in Morocco, from his birth in 1256 to his death in 1321. Ibn al-Bannāʾ was one of those brilliant medieval polymaths who wrote on a wide range of topics. He is known to have authored over a hundred works on mathematics, astronomy, logic, religion, law, language, philosophy, medicine, and the occult sciences. And like any good medieval polymath, Ibn al-Bannāʾ wrote textbooks in addition to more advanced works.

One of Ibn al-Bannāʾ's textbooks is titled Book on the Fundamentals and Preliminaries in Algebra (see Note 9, below). Like al-Karajī's al-Fakhrī, this book was written in the tradition of al-Khwārizmī and Abū Kāmil. But unlike those authors, Ibn al-Bannāʾ proves the rules for solving simplified equations in the setting of arithmetic rather than geometry. His approach was to complete the square in the context of the equation in each case, resting his arguments on the rule for squaring a binomial. For example, for the sample equation \(x^2+10x=39\) he adds 25 (the square of half of the 10) to both sides to get \(x^2+10x+25=64\), and he takes the square root of both sides to get \(x+5=8\), from which the solution falls out immediately. The steps in this process correspond to the steps in the rule, so the rule is proven. As befitting a textbook, these proofs are straightforward and easy to understand, which is why they are not the topic of this article.

Ibn al-Bannāʾ's Condensed Book from which I quoted above is another textbook, and in its time a very popular one. It is organized into two parts. In the first part he explains how to calculate with Indian (i.e. Arabic) numerals, and in the second part he covers the basic rules for solving problems by proportion, double false position, and algebra. He does not teach finger-reckoning in this book, but he does borrow some of the shortcuts from that system that might be useful for Indian calculation, including the "squaring" rule.

The Condensed Book is just too condensed to include proofs. It is so compact that even if Ibn al-Bannāʾ's main goal was to explain how to perform arithmetical operations, he did not include a single numerical example to illustrate the rules! To remedy this, he was the first among many to write a commentary on his little book that elaborates on the rules with examples, philosophical discussions, and proofs.

Ibn al-Bannāʾ completed his commentary in 1301, and gave it the title Lifting the Veil from the Face of the Operations of Arithmetic. (See Note 10, below.) But where the Condensed Book was a textbook on practical arithmetic for students learning to calculate with Indian numerals, many parts of Lifting the Veil were written for a more sophisticated reader. Ibn al-Bannāʾ engaged in philosophical investigations of the nature of number, the unit, and fractions, and many of his proofs were written at a level beyond the reach of arithmetic students.

It is in this book that Ibn al-Bannāʾ gave two more sets of proofs for the rules for solving simplified equations. These two sets are different from the proofs in his algebra book. The first set rests on the "squaring" rule from finger-reckoning, while the second set is based in arithmetical restatements of Euclid's propositions II.5 and II.6. And just as Euclid's propositions are removed from their geometrical context and redressed for a new purpose in algebra, the "squaring" rule is modified from its original form in finger-reckoning in preparation for the same purpose.

Arabic algebra

I cannot proceed any further without actually saying what these simplified equations are, so a brief overview of Arabic algebra is in order. Unlike algebra today, which encompasses any method of finding unknown numbers and even various kinds of formal, abstract reasoning, Arabic algebra (al-jabr wa-l-muqābala) was a specific technique of numerical problem-solving. It was characterized by its own technical vocabulary and procedures, and was practiced alongside other methods like single and double false position. Although algebra in Arabic was initially associated with the professional groups who practiced finger-reckoning, by Ibn al-Bannāʾ's time it had also become integrated into books on calculation with Indian numerals.

The powers of the unknown in Arabic algebra are given individual names. The first power is called a "root" (jidhr) or "thing" (shayʾ), and plays the role of our \(x\). Its square is called a māl, a word meaning "sum of money", "wealth", or "fund", and corresponds to our \(x^2\). Because the word māl was used in algebra in a sense unrelated to its quotidian meaning, and because there is no good English word corresponding to it, I leave it untranslated. I also write its plural with the English suffix: māls. Higher degree terms first appear in Abū Kāmil's Book on Algebra and in Qusṭā ibn Lūqā's translation of the Arithmetica of Diophantus, both from the latter ninth century. The cube of the "thing" is a "cube" (kaʿb), and higher powers were usually expressed as some combination of māl and kaʿb, like kaʿb kaʿb māl for the eighth power. Units were often counted in dirhams, a silver coin, but also frequently as "units" or "in number".

Equations were formed from these names. For example, Abū Kāmil sets up this equation to solve one of his problems:

[F]our ninths of a māl and four dirhams less two things and a third of a thing equal a thing and twenty-four dirhams. [Abū Kāmil 2012, 379.14]

Converting this into modern notation gives \[{4\over 9}x^2+4-2{1\over 3}x=x+24.\] Keep in mind that there is no negative quantity here. The "less" indicates that the positive \(2{1\over 3}x\) has been removed from the greater \({4\over 9}x^2+4\). Also, although books show the calculations in words, they were not worked out rhetorically. Calculations were performed mentally or in notation on some temporary surface like a dust-board. When needed, rhetorical versions were composed to communicate it to others in a book. Books were treated as transcriptions of lectures, and notation plays no role when listening to a teacher's recitation. Thus the lack of notation in books.

One way that equations differ from other forms of equating in Arabic mathematics is that they are stated using only these names. A simple example of an arithmetical equating that was not considered to be an algebraic equation is the enunciation to problem (I.17) in al-Karajī's al-Fakhrī:

Ten: you divided it into two parts, so the difference between them is equal (mithl) to the smaller part. [Saidan 1986, 173.15]

This question is framed in terms of the two unnamed parts of ten, and not with any algebraic name. We can restate it in algebraic terms by calling the parts \(x\) and \(10-x\) and setting up the equation \((10-x)-x=x\), but the enunciation as it stands belongs to arithmetic generally, and can be solved by methods other than algebra.

There is another way that equations are distinguished from other forms of equating. Several different words were used to mean "equal" in Arabic arithmetic, the most common being mithl, sawiya, and the prefix ka-, as well as the implied verb "to be". Algebraic equations, on the other hand, were always stated with the unusual verb ʿadala ("equal", "well-balanced"). This word was rarely used in mathematics outside the context of algebra. Noting which word Ibn al-Bannāʾ chooses to equate numbers and expressions will tell us how he understood his arguments.

Note 9. In Arabic, the title of Ibn al-Bannāʾ's Book on the Fundamentals and Preliminaries in Algebra is Kitāb al-uṣūl wa l-muqaddimāt fī l-jabr wa-l-muqābala. It is included in [Saidan 1986, 505-613].

Note 10. In Arabic, the title of Ibn al-Bannāʾ's Lifting the Veil from the Face of the Operations of Arithmetic is Rafʿ al-ḥijāb ʿan wujūh aʿmāl al-ḥisāb. It is available as [Ibn al-Bannāʾ 1994].