Available online at www.sciencedirect.com Historia Mathematica 38 (2011) 338–367 www.elsevier.com/locate/yhmat Cube root extraction in medieval mathematics Bo Göran Johansson Department of Computer Communication and Software Engineering, Gotland University, Visby SE-621 57, Sweden Available online 28 October 2010 Abstract The algorithms used in Arabic and medieval European mathematics for extracting cube roots are studied with respect to algebraic structure and use of external memory (dust board, table, paper). They can be separated into two distinct groups. One contains methods used in the eastern regions from the 11th century, closely con- nected to Chinese techniques, and very uniform in structure. The other group, showing much wider variation, contains early Indian methods and techniques developed in central and western parts of the Arabic areas and in Europe. This study supports hypotheses previously formulated by Luckey and Chemla on an early scientific connection between China and Persia. Ó 2010 Elsevier Inc. All rights reserved. Résumé Les algorithmes employé en les mathématiques arabes et européennes médiévales pour l’extraction des racines cubiques sont étudiées par rapport à la structure algébrique et l’emploi de la mémoire externe (table de poussière, papier). On peut séparer deux groupes distincts. Un groupe contiens les méthodes utilisés dans les régions orientaux depuis le XIe siècle, étroitement liés avec des techniques chinoises, avec une structure très uniforme. L’autre groupe, qui montre une variété plus grande, contiens des techniques développés dans les régions arabes centraux et occidentaux et dans l’Europe. Ce traité support la thèse présentée par Luckey et Chemla sur une connexion scientifique entre la Chine et la Perse. Ó 2010 Elsevier Inc. All rights reserved. MSC: 01A30; 01A35 Keywords: Medieval mathematics; Algorithms; Root extraction E-mail address: [email protected] 0315-0860/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.hm.2010.08.001 Cube root extraction in medieval mathematics 339 1. Introduction In this paper, I will compare algorithms used for the extraction of cube roots in Chinese, Arabic,1 and medieval European mathematics from the 11th to the 14th century. Earlier studies by Luckey [1948] and Chemla [1994] have shown that one can distinguish two dis- tinct main streams, one in China and the eastern parts of the Arabic regions, and one in the central and western Arabic regions, and later also in Europe. This study aims at giving a detailed analysis of these two traditions. The former turns out to be homogenous, with very small variations in the technique. In contrast to this, the latter shows a large variation in the structure of the algorithms, within which the European algorismus tradition turns out to show some specific features. When analyzing the algorithms, I will distinguish between what I call the deep structure and the surface structure of the algorithm (these terms have been introduced in a rather different connection by Chomsky). The deep structure comprises a sequence of instructions, corresponding to performing algebraic operations, and storing in and reading from the memory. One important aspect of deep structure is the degree of nonredundancy, i.e., how the algorithm carries information from one step to the next, avoiding repetition of operations. The algebraic parts of the deep structure may be interpreted in terms of the mathematical language of today. It should be noted, however, that the transcription into modern algebraic language sometimes may tell too much and sometimes too little (e.g., different sequences of manipulations on the counting surface might be interpreted by the same algebraic formula). The memory could be various kinds of counting surfaces on which numbers were laid out (with rods or stones) or written down. But we also find “read-only” memories in the form of tables (e.g., sexagesimal multiplication tables). One might say that the deep structure is part of the mathematical grammar. It is univer- sal in character, and we often find the same structure in techniques from different times and cultures. In the history of mathematics, focus has often been laid on the deep structure, above all on the algebraic structure; the use of memory is more rarely analyzed. The surface structure includes the kind of medium used as a memory and the way the numbers that occur in the computations are written (or “laid out”) in, or erased from, the memory. The way the algorithm is described in the primary sources should also be included in the surface structure. Chemla [1994, 209] puts heavy stress on these features: “there are similarities and differences in the way of recounting an algorithm, and in the manner in which it sets the numbers on a surface. These similarities can often indicate historical connections more precisely than a modern translation.” It is worth pointing out that the deep structure might sometimes be dependent on the surface structure; if the memory is limited (e.g., if you use mental memory only), this will affect the possible methods of calculation considerably. 2. Preliminaries The extraction of higher roots in Arabic mathematics was first studied by Luckey [1948]. His paper focuses on the work of Jamshıd al-Kashı from 1427, The Key to 1 I use the term “Arabic mathematics” instead of the alternative “Islamic mathematics,” since the main language of the mathematical works was Arabic (even though some of the mathematicians mentioned here were non-Arabs). 340 B.G. Johansson Arithmetic (Miftah: al-h: isab ). Here al-Kashı extracts the fifth root of a given decimal number and the sixth root of a sexagesimal number and also gives a few more exam- ples, leaving the details to the reader. In his analysis of the algorithm, Luckey compares it with considerably later works by Ruffini [1804] and Horner [1819] and finds that “The comparison shows that K. [al-Kashı ] calculates after the Ruffini–Horner method. All numbers in his tables are also found in another order in the modern scheme, and in both patterns all numbers are created in the same order and by the same com- putations” [transl. from Luckey, 1948, 244]. In his paper, Luckey also considers the extraction of the cube root by al-Nasawı (11th century), and draws the conclu- sion that the Ruffini–Horner method is present in this case as well; “K. [al-Kashı ] is, as a whole, close to the 400 years older N. [al-Nasawı], in the use of mathe- matical terms and even in the words used in his text” [transl. from Luckey, 1948, 252]. As a contrast, when Luckey describes the techniques used by Indian and medieval Euro- pean mathematicians to extract the cube root (roots of higher order are not found with them), he finds another method used exclusively, which he characterizes as following “the complete binomial theorem,” i.e., using all the terms of the formula (a + b)3 = a3 + 3a2b +3ab2 + b3 separately in the calculations. Luckey discusses, in particular, the works Liber abbaci by Leonardo Pisano and other 13th- and 14th-century works by, among oth- ers, Sacrobosco and Levi ben Gerson (Gersonides). Since Luckey, more Arabic works have been published that use methods related to those of al-Nasawı and al-Kashı . The work of Kushy ar ibn Labban, Principles of Hindu Reckoning (Kitab fıUs: ul h: isab al-hind), was known already to Luckey and was published in English translation by Levey and Petruck [1965]. Other works by Ibn Ibrahı m al- Uqlıdisı [Saidan, 1978], Ibn T: ahir al-Baghdadı [Saidan, 1985], and Nas:ır al-Dın al-T: usı [Saidan, 1967], all treating extraction of the cube root, were edited and/or translated after Luckey’s time. The generalization of the algorithm to higher roots has been shown to have been known already in the 12th century by al-Samaw’al al-Maghribı[Rashed, 1978] and also by Nas:ır al-Dın al-T: usı in the 13th century [Saidan, 1967]. The possible Chinese influence, assumed already by Luckey, was brought forward further by Chemla [1994], who analyzed the techniques used in the Nine Chapters on the Mathematical Art (probably completed in the first century B.C.) and by Zhang Qiujian (fifth century A.D.) and Jia Xian (11th century), comparing these with the tech- niques used by Kushy ar ibn Labban, al-Samaw’al and al-Kashı . Chemla points out the great similarities between the algorithms used by these authors and emphasizes the essen- tial difference between these algorithms and the techniques used in Indian mathematics and by other Arabic authors, such as al-Uqlıdisı. In this article, I have added the works of Ibn T: ahir and al-T: usı to the former group and, as a contrast, I have analyzed a selection of representative works from the western Arabic regions and from medieval European mathematics. In all algorithms discussed here, numbers are written in a positional system, to base ten or sixty. In some cases, the calculations might be purely mental, but normally some kind of external memory is also used: numbers may be written or laid out in a number of rows on a board (allowing erasing techniques) or written on paper. The cube root is then deter- mined one digit at a time, starting with the highest digit, creating a sequence of approx- imations from below, which Luckey [1948, 226] called the “Methode der sukzessiven Stellenbestimmung.” Cube root extraction in medieval mathematics 341 First, the digits of the given number are counted (beginning from the right); the first, fourth, seventh places and so forth I here call cube places. The number of cube places equals the number of digits of the cube root. For example (from Leonardo Pisano), given the num- ber N = 9 876 543, the cube places are situated at the digits 3, 6 and 9, so the cube root is a three-digit number.