An Arabic Finger-reckoning Rule Appropriated for Proofs in Algebra - Background

Numbers, magnitudes, and the place of algebra

Arabic arithmeticians (ḥussāb) calculated with any positive quantity that can be obtained from the unit through addition, subtraction, multiplication, division, and root extraction. Although they had no qualms about working with fractions and irrational roots, zero remained merely a placeholder in the base ten system of writing numbers, and we have no evidence that any medieval arithmetician considered the possibility of negative numbers.

Books on calculation (ḥisāb) taught methods of operating on known numbers, and many of them also explained techniques for finding unknown numbers. A typical sample problem for known numbers is this one from Ibn al-Yāsamīn's late twelfth-century Grafting of Opinions of the Work on Dust Figures (see Note 3, below):

If [someone] said to you, divide ten and a fourth by eight and a third. [Ibn al-Yāsamīn 1993, 172.5]

After working it out, the answer is found to be "one and two tenths and three tenths of a tenth". And a simple problem of finding an unknown number, from the same book, is:

If [someone] said to you, a quantity: you added its third and its fourth and got six. How much is the quantity? [Ibn al-Yāsamīn 1993, 195.9]

Here the answer is found to be "ten and two sevenths".

Today algebra is just about the only method taught for finding unknown numbers, but in medieval Islamicate countries several methods were practiced. Ibn al-Yāsamīn, in fact, solves the problem given above by four different methods: single false position, algebra, al-qiyās (see Note 4, below), and double false position. Like the other methods, algebra was a numerical technique, and was regarded as being part of arithmetic. So where I write about "Arabic arithmetic" keep in mind that it includes algebra. I will describe this algebra below, in the section "Arabic algebra".

And as in Greek mathematics, geometry and arithmetic were distinct domains. The objects of geometry were magnitudes: points, line segments, triangles, spheres, etc. Although magnitudes were regarded as being of a fundamentally different nature from numbers and independent of them, people fostered the connection between geometrical and arithmetical calculation by routinely assigning numerical measures to magnitudes.

Geometric vs. arithmetical proofs

People who have had only a little exposure to medieval Arabic algebra will often recall one or two names, usually al-Khwārizmī (ca. 825 CE) and al-Khayyām (Omar Khayyam, ca. 1075). People also tend to recall that these authors gave geometric proofs to the rules for solving simplified equations. Indeed, the proofs in the algebra books of many other authors are also based in geometric diagrams. Abū Kāmil (late 9th c.), Thābit ibn Qurra (late 9th c.), al-Karajī (early 11th c.), al-Samawʾal (12th c.), and Sharaf al-Dīn al-Ṭūsī (d. 1213) are the most well-known. But even the few people who have studied Arabic algebra in some depth can be excused for not knowing that it is just as common for the proofs of these rules to be based in arithmetic. The secondary literature has for the most part neglected the two sets of arithmetical proofs in two works of al-Karajī, as well as the arithmetical proofs in later authors like Ibn al-Yāsamīn (d. 1204), Ibn al-Bannāʾ (late 13th c.), al-Fārisī (late 13th c.), and Ibn al-Hāʾim (1387). (See Note 5, below.)

Al-Karajī had proven the rules for solving three-term quadratic equations via geometric diagrams with the appropriate references to Euclid's propositions II.5 and II.6 in his [Book of] al-Fakhrī on the Art of Algebra (henceforth al-Fakhrī). It was later, in his short treatise Causes of Calculation in Algebra and in his finger-reckoning book The Sufficient [Book] on Arithmetic, that he switched to arithmetical proofs. [See Note 6, below.] The other authors listed above were inspired directly or indirectly by him, though their proofs are by and large very different from each other.

Authors eschewed geometry in favor of arithmetic for different reasons. Al-Fārisī, for his part, held that because algebra belongs to arithmetic, proofs in algebra should properly be conducted arithmetically. This way Aristotle's prohibition of genus-crossing would be observed [Posterior Analytics Book I, part 7]. One should not prove results in the genus of arithmetic, like the rules for solving equations, by arguments in the genus of geometry. Al-Karajī and Ibn al-Hāʾim, on the other hand, both explained that proofs based in Euclid's Elements are simply too difficult for students learning algebra. Al-Karajī wrote in his short treatise Causes of Calculation his reason for moving away from the proofs based in geometry that he had previously given in al-Fakhrī:

… Then I saw that people seeking knowledge of calculation found it difficult to understand the correctness [of the rules] by means of those lines and figures, given their understanding by means of the tongue and hand. I discovered that many people found it very difficult when reading [the proofs] in books. And I decided because of this to make the proofs in this book easier. [Saidan 1986, 354.8]

Thus he switched to proofs based in arithmetic, which "puts an end to the incomprehensibility of the realm of lines and figures".

Note 3. The Arabic title of the book Grafting of Opinions of the Work on Dust Figures is Talqīḥ al-afkār fī l-ʿilm bi-rushūm al-ghubār. All translations in this article are by the author.

Note 4. Al-qiyās can be translated as "scaling". It consists of undoing the operations on the unknown, and works only for simple problems.

Note 5. I have recently published a paper outlining their arithmetical proofs, in [Oaks 2018].

Note 6. The Arabic titles of the three books [Book of] al-Fakhrī on the Art of Algebra, Causes of Calculation in Algebra, and The Sufficient [Book] on Arithmetic, are, respectively, al-Fakhrī fī ṣināʿat al-jabr wa-l-muqābala, ʿIlal ḥisāb al-jabr wa-l-muqābala, and al-Kāfī fī l-ḥisāb. These are published in [Saidan 1986, 95-351], [Saidan 1986, 353-369], and [al-Karajī 1986], respectively.