J Indian Philos (2007) 35:543–575 DOI 10.1007/s10781-007-9026-4
Medieval Arabic Algebra as an Artificial Language Jeffrey A. Oaks
© Springer Science+Business Media B.V. 2007
Abstract Medieval Arabic algebra is a good example of an artificial language. Yet despite its abstract, formal structure, its utility was restricted to problem solving. Geometry was the branch of mathematics used for expressing theories. While algebra was an art concerned with finding specific unknown numbers, geometry dealt with general magnitudes. Algebra did possess the generosity needed to raise it to a more theoretical level—in the ninth century Abu¯ Ka¯mil reinterpreted the algebraic unknown ‘‘thing’’ to prove a general result. But mathematicians had no motive to rework their theories in algebraic form. Because it offered no advantage over geometry, algebra remained a practical art in both the Islamic world and in Europe until the scientific uphevals of the 17th–18th centuries. Keywords
Arabic algebra Æ artificial language Æ generosity Æ Abu¯ Ka¯mil
Introduction1 Before his return to England from India in 1811, Edward Strachey wrote a ¯ milı¯’s late sixteenth c. Arabic commentary and partial translation of al-ʿA 1 Notes on references. A semicolon (;) separates page numbers from line numbers, and a slash (/) separates the page of the Arabic text from the page of the translation. Example: (Al-Khwārizmī 1831, p. 3;9/5) refers to page 3 line 9 of the Arabic text and page 5 of the translation. The line number indicates the beginning of the referred passage, which may run on to several lines. All translations are mine (with some help from Haitham Alkhateeb) unless noted otherwise.
J. A. Oaks (&) Department of Mathematics & Computer Science, University of Indianapolis, Lilly Hall 213, 1400 East Hanna Avenue, Indianapolis, IN 46227, USA e-mail:
[email protected]
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textbook Essence of Arithmetic. He wrote ‘‘[it] is of considerable repute in India; it is thought to be the best treatise on Algebra, and it is almost the only book on ¯ milı¯’s book is a guide to basic arithmetic and the subject read here.’’2 Al-ʿA mensuration, with only a couple chapters on algebra at the end. Despite its brief treatment, the explanation of the rules of algebra and the sample problems in this book differ little from those in the earliest extant work on the topic, written by al-Khwa¯rizmı¯ near the start of the ninth century CE. At the level of elementary textbooks, Arabic algebra remained remarkably stable over the millennium from al-Khwa¯rizmı¯ to Strachey. An example from the Muslim west is the mathematics curriculum promulgated in 1875 at the al-Zaytu¯na mosque in Tunis. There the prescribed text for algebra was Ibn Gha¯zı¯’s Commentary, written about 1500 in the fashion of the arithmetic handbooks common in North Africa three centuries before that.3 Here, too, the basics of algebra are very close to what we see in the ninth century. This is not to say that algebra was stagnant over that vast expanse of time and place. While it had begun as a practical technique of problem solving, significant advances were made in algebra by various mathematicians after the publication of al-Khwa¯rizmı¯’s book. Prominent examples just from the ninth century include Qust: a¯ ibn Lu¯qa¯’s translation of Diophantus’ Arithmetica into contemporary Arabic algebra (ca. 860), Tha¯bit ibn Qurra’s proofs of the procedures for solving simplified quadratic equations using Euclidean geometry (latter ninth c.), and Abu¯ Ka¯mil’s work with irrational square roots (ca. 890). Later, al-Karajı¯ (ca. 1011/12) and al-Samawʾal (d. 1175) extended the range of powers of the unknown to include their reciprocals, and they explored the arithmetic of polynomials. ʿUmar al-Khayya¯mı¯ (1048–1131) and Sharaf al-Dı¯n al-Tu¯sı¯ (d. 1213) classified equations of degree three and less, ˙ ˙ solutions of irreducible cubic polynomials by the interand represented the sections of conic sections. All the while these developments were taking place, elementary textbooks on algebra continued to be written. It was the medieval concepts of polynomial and equation that determined the structure of algebraic solutions to problems. These concepts, which differ from ours, remained essentially unchanged throughout the span of Arabic algebra in both practical and scientific texts. In the section ‘‘Arabic Algebra: The Method’’ of this paper I explain Arabic polynomials and equations, and I outline the structure of algebraic solutions. The section ‘‘Notation’’ is a brief introduction to the algebraic notation which emerged from the textbook tradition in the Maghreb. In ‘‘Arabic Algebra as an Artificial Language’’ I characterize Arabic algebra as an artificial language, and generosity is addressed in ‘‘Generosity’’. Because Arabic algebra was treated as a method of solving specific problems, scientific applications were rare. The one instance of generosity I have found, in 2 (Strachey 1818, p. 167). For a brief account of Strachey’s duties and travels in India, see Sanders ¯ milı¯’s book in India, see De Young (1986). It circulated also in (1953, pp. 108ff). For more on al-ʿA Persian translation. 3 Ibn Gha¯zı¯’s book, Bughyat al-Tulla¯b fı¯ Sharh Munyat al-Hussa¯b (Aim of the Students in ˙ ˙ in Ibn Gha¯zı˙¯ (1983). Commentary on Desire of Reckoners) is published
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Abu¯ Ka¯mil, could have put algebra on par with geometry as the means of expressing theories, but it was ignored by later mathematicians. With geometry already serving the purpose, there was no motive for mathematicians to rework their ideas in algebraic form. It was only in the century from Descartes to Euler that algebra was transformed from a problem solving method using unknowns and equations to a tool for describing theories using variables and functions. With this transformation it finally surpassed geometry in its utility for describing nature and for reshaping our understanding of mathematics itself.
Sources I will be mentioning several Arabic books, but I list here only those I will be quoting. In chonological order they are:
4
Al-Khwa¯rizmı¯ (Abu¯ Jaʿfar Muhammad ibn Mu¯sa¯ al-Khwa¯rizmı¯). Kita¯b ˙ al-Mukhtasar fı¯ Hisa¯b al-Jabr wa’l-Muqa ¯ bala (Brief Book on Calculation ˙ ˙ by Algebra), written sometime 813-833 CE (Al-Khwa¯rizmı¯ 1831; al-Khwa¯rizmı¯ 1939). This work consists of three ‘‘books’’: (1) the algebra proper, which covers the rules of algebra with forty worked-out problems, (2) a section on the rule of three and mensuration problems, and (3) a long section comprised of worked-out inheritance problems. In this article I cite examples from the forty problems in part (1). These are referenced in Oaks and Alkhateeb (2005, Appendix A). Abu¯ Ka¯mil (Abu¯ Ka¯mil Shuja¯ʿ ibn Aslam ibn Muhammad ibn Shuja¯ʿ). ˙ 890–900 CE. The Kita¯b fı¯’l-Jabr wa’l-Muqa¯bala (Book of Algebra), ca. book is composed of three parts. In the first part Abu¯ Ka¯mil explains the rules of algebra, along with 74 worked-out problems. These problems are referenced in Oaks and Alkhateeb (2005, Appendix A). In the second part, On the Pentagon and Decagon, geometry problems are solved using algebra. The third part covers indeterminate analysis, again using algebra. A facsimile of the only known Arabic manuscript, copied in 1253, was published as Abu¯ Ka¯mil (1986), and Sami Chalhoub has recently edited and translated the book into German in Abu¯ Ka¯mil (2004). An English translation of Mordecai Finzi’s fifteenth-century Hebrew translation of the first part was published in Levey (1966). Abu¯ Ka¯mil lived in Egypt. Al-Karajı¯ (Fakhr al-Dı¯n Abu¯ Bakr Muhammad ibn al-Hasan al-Karajı¯ ˙ ˙ (or al-Karkhı¯)). Al-Fakhrı¯ fı¯ Sina¯ʿat al-Jabr wa’l-Muqa¯bala ([Book] of ˙ al-Fakhrı¯ on the Art of Algebra) (Saidan 1986, 95–308; Woepcke 1853). Al-Karajı¯, an Iranian mathematician who worked in Baghdad, wrote this treatise ca. 1011/12 CE.4 In it he follows the model of al-Khwa¯rizmı¯ and Abu¯ Ka¯mil, presenting algebra in two parts: first, an explanation of the rules of algebra, and second, a collection of 255 worked-out problems. Al-Kha¯zinı¯ (Abu¯ Mansu¯r ʿAbd al-Rahma¯n al-Kha¯zinı¯), Kita¯b Mı¯za¯n ˙ al-Hikma (Book of the˙ Balance of Wisdom) (Al-Kha¯zinı¯ 1940). This ˙ Al-Karajı¯ (1964, p. 13).
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treatise on the science of weights, written in 1121, is largely composed of extracts from previous works. Al-Samawʾal (Ibn Yahya¯ al-Maghribı¯ al-Samawʾal). Al-Ba¯hir fı¯ al-Jabr (The Dazzling [Book]˙ on Algebra), written in the mid-twelfth c. CE (Al-Samawʾal 1972). Al-Samawʾal worked in Baghdad and Iran. In this book he builds on the work of al-Karajı¯. Al-Hassa¯r (Abu¯ Bakr Muhammad ibn ʿAbdallah ibn ʿAyya¯sh al-Hassa¯r). ˙˙ ˙ ¯ r fı¯ Sanʿat ʿAmal al-Ghuba¯r (Book of ˙ Dem˙˙ Kita¯˙b al-Baya ¯ n wa’l-Tadhka ˙ onstration and Recollection in the Art of Dust-Board Reckoning). This arithmetic text was composed in the Maghreb, and contains some problems solved by al-jabr (algebra). The manuscript copied in Baghdad in 1194, soon after the author’s death, is available online (Al-Hassa¯r 1194). ˙ ˙ ˙Kita¯b fı¯hi Ibn Badr (Abu¯ ʿAbd Alla¯h Muhammad ibn ʿUmar ibn Badr). ˙ Ikhtisa¯r al-Jabr wa’l-Muqa¯bala (Brief Book on Algebra), written possibly ˙ in Muslim Spain, sometime after Abu¯ Ka¯mil, but before 1311 CE5 (Sa´nchez Pe´rez 1916; Saidan 1986, pp. 427–497). The first forty worked out problems are referenced in Oaks and Alkhateeb (2005, Appendix A). Ibn al-Ha¯ʾim (Abu¯’l-ʿAbba¯s Shiha¯b al-Dı¯n Ahmad ibn Muhammad ibn ˙ ¯nı¯ya (Commentary ˙ ʿIma¯d al-Dı¯n ibn ʿAlı¯). Sharh al-Urju¯za al-Ya¯smı on the Poem of al-Ya¯samı¯n). Edited and (partially) translated by Mahdi Abdeljaouad in Ibn al-Ha¯ʾim (2003). This 1387 treatise from the Maghreb takes the form of a commentary on Ibn al-Ya¯samı¯n’s famous poem on algebra. Al-Ka¯shı¯ (Ghiya¯th al-Dı¯n Jamshı¯d ibn Masʿu¯d al-Ka¯shı¯). Mifta¯h al-Hisa¯b ˙ (Key to Arithmetic) (Al-Ka¯shı¯ 1969). This general introduction˙ to arithmetic and mensuration, with a long chapter on algebra, was completed in 1427 in Samarqand. ¯ milı¯ (Baha¯ al-Dı¯n Muhammad ibn al-Husayn al-ʿA ¯ milı¯). Khula¯sat Al-ʿA ˙ ˙ ¯ ¯ al-Hisa¯b (Essence of Arithmetic) (Al-ʿAmilı¯ 1976). Al-ʿAmilı¯ worked ˙in Iran˙ and wrote this textbook on arithemtic and algebra ca. 1600 CE.
Arabic Algebra: The Method The Background to Arabic Algebra By the time the first books on algebra appeared in Arabic early in the ninth century, the art was already part of the basic education of students in different trades. It was on the request of the caliph al-Ma¯ʾmu¯n (reigned 811–833 CE) that al-Khwa¯rizmı¯ composed his handbook on algebra. He relates in his introduction ‘‘Thus I wrote from the work on algebra a brief book which encompasses the fine and important parts of its calculations that people constantly require in cases of their inheritance, their legacies, their partition, their law-suits, and their trade, and in all their dealings with one another, such as the surveying of land, the digging of canals, geometry, and other 5
Saidan (1986, p. 409); Djebbar (2005, p. 133).
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various aspects and kinds are concerned.’’6 This was a time when ʿAbba¯sid patronage supported not only the translation and study of the foreign (mainly Greek) sciences, but also the cultivation and development of native traditions which had hitherto been transmitted orally, such as algebra and folk astronomy.7 Algebra was called al-jabr wa’l-muqa¯bala, literally ‘‘restoration and confrontation’’, after two steps in the simplification of equations.8 The name was frequently shortened to just al-jabr. Throughout the medieval period algebra coexisted with other problem solving techniques, such as proportion (al-arbaʿa al-mutana¯siba), single false position (al-ʿamal bi al-nisba), double false position (al-khata¯yn), working backwards (al-ʿamal bi-ʿaks), arithmetical reasoning (al-tahlı¯l), and geometry (al-handasati). Al-Hassar, for instance, poses prob˙ ˙ ˙ ˙ single false position, algebra, lems which he solves by three different methods: and double false position. Abu¯ Ka¯mil solves some problems by both geometry ¯ milı¯ uses proportion, algebra, double false position, and and algebra, and al-ʿA arithmetical reasoning.9 Textbooks typically present algebra in two sections. First, the basic rules are covered, including the solutions to the six simplified quadratic equations and the rules for operating on the powers of the unknown. Second comes a collection of worked out problems. The purpose of the first part is to prepare students to solve the problems in the second part. It is in these solved problems that we see algebra at work, so it is there that I focus my attention. Mathematics books in medieval Arabic were written in rhetorical form. In particular, the enunciations and solutions to worked-out problems by whatever method were expressed all in words, with the partial exception of HinduArabic numerals in some later texts. A notation for algebra did develop in western Africa in the later middle ages, but it served primarily as an aid toward producing a rhetorical solution. Polynomials and Equations What sets Arabic algebra apart from other problem solving techniques is its technical vocabulary, syntax, and transformational rules. While other arithmetical methods manipulate the given numbers of a problem to produce the answer, an algebraic solution entails naming an unknown quantity or magnitude, then setting up and solving an equation posed in terms of this name. The 6 Al-Khwa¯rizmı¯ (1939, p. 2;10/3); Al-Khwa¯rizmı¯ (1831, p. 16;1). Translation adapted from Gutas (1998, p. 113), which in turn was adapted from Rosen’s translation (al-Khwa¯rizmı¯ 1831, p. 3). 7
See Oaks and Alkhateeb (2005, Sect. 5.5) on the origins of Arabic algebra.
8
See Sect. 4.2 below. For a thorough explanation of the use of the words al-jabr and al-muqa¯bala in Arabic algebra, see Oaks and Alkhateeb (2007). 9 See Oaks and Alkhateeb (2005, Sect. 5.4) for an explanation of arithmetical reasoning. The ¯ milı¯. The latter text also provides the method is named al-tahlı¯l in al-Fa¯risı¯, al-Ka¯shı¯, and al-ʿA name for proportion. ˙ The name for single false position was provided to me by Mahdi Abdeljaouad. The other names are common.
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equation is a kind of artificial environment where the problem is resolved. This is true of modern elementary algebra, too. We name an unknown x, then we create and solve an equation expressed in terms of x and its powers. In Arabic algebra the first power of the unknown is named shayʾ (‘‘thing’’), and for historical reasons it is sometimes called jidhr (‘‘root’’). The square of ‘‘thing’’, our x2 , is named ma¯l (literally ‘‘sum of money’’, ‘‘treasure’’, ‘‘property’’).10 The cube of ‘‘thing’’ ðx3 Þ is kaʿb (‘‘cube’’), and the higher powers are written as combinations of ma¯l and kaʿb: ma¯l kaʿb ðx5 Þ; kaʿb kaʿb ðx6 Þ; ma¯l ma¯l kaʿb ðx7 Þ; ma¯l ma¯l ma¯l ma¯l ðx8 Þ; etc. Variations exist for x5 and x8 .11 Just as in Arabic arithmetic, simple numbers (constants) are counted in dirhams, a denomination of silver coin. Some algebraists substitute dirhams with a¯ha¯d (‘‘units’’), while others say min al-ʿadad (‘‘in number’’) instead, and ˙ in all texts the denomination is often dropped. So ‘‘ten dirhams’’, ‘‘ten units’’, ‘‘ten in number’’, and simply ‘‘ten’’ all take the meaning of our number 10. Al-Khwa¯rizmı¯ calls simple numbers (dirhams), roots (things), and ma¯ls the ‘‘numbers’’ required in calculating with algebra.12 Together with the other powers I call these the ‘‘algebraic numbers’’. A sample equation lifted from alKarajı¯’s problem (II-44) is ‘‘. . .ten things less a ma¯l, and that equals four things and five dirhams.’’13 In modern notation this is 10x x2 ¼ 4x þ 5. Medieval algebraists conceived of polynomials differently than we do today.14 For us, a polynomial is constructed from the powers of x with the operations of scalar multiplication and addition/subtraction. In other words, it is a linear combination of the powers. By contrast, Arabic polynomials contain no operations at all. In the expression ‘‘four things’’, the ‘‘four’’ is not multiplied by ‘‘things’’. Instead, it merely indicates how many things are present. Think of ‘‘four things’’ like ‘‘four bottles’’. Further, the phrase ‘‘four things and five dirhams’’ entails no addition, but is a collection of nine items of two different kinds. Think of it like ‘‘four bottles and five cans’’. The wa (‘‘and’’) connecting the things and the dirhams is the common conjunction. It does not take the meaning of the modern word ‘‘plus’’. The ‘‘ten things less a ma¯l’’ on the other side of the equation describes an amount which is a ma¯l short of ‘‘ten things’’. Think of it as a collection of ten items which have been diminished by a ma¯l. Similarly, if I take a bite out of an apple, I can describe the result as ‘‘an apple less a bite’’. ‘‘Ten things less a ma¯l’’, like the bitten apple, is a static object: it is only our description of it 10
Because there is no nice single-word English translation of ma¯l, and because the word was used in algebra with a meaning unrelated to its quotidian definition, I leave it untranslated. Also, I write the plural with the English suffix: ma¯ls. For a discussion of the origin of the terms shayʾ, jidhr, and ma¯l in Arabic algebra, see Oaks and Alkhateeb (2005).
11
Al-Khwa¯rizmı¯ only works with first and second degree polynomials. Powers from cube up first appear in Qusta¯ ibn Lu¯qa¯’s translation of Diophantus (Sesiano 1982, p. 45). ˙ 12 Al-Khwa¯rizmı¯ (1939, p. 16;13); Al-Khwa¯rizmı¯ (1831, p. 3;9/5). 13
Saidan (1986, p. 202;8).
14
See Oaks (forthcoming, Sect. 5) for a more thorough investigation of Arabic polynomials.
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which seems to imply a subtraction. Illa¯ (‘‘less’’) is the negative counterpart to ‘‘and’’. It does not mean ‘‘minus’’.15 I should say a little more about the difference between the medieval ‘‘and’’/ ‘‘less’’ and the modern ‘‘plus’’/‘‘minus’’. Our words ‘‘plus’’ and ‘‘minus’’ were borrowed from Latin as verbalizations of the modern symbols ‘‘þ’’ and ‘‘’’. They are used in a quasi-propositional sense to express mathematical operations. The Arabic words wa (‘‘and’’) and illa¯ (‘‘less’’), on the other hand, are conjunctions used in accordance with everyday use.16 While the two sides of al-Karajı¯’s equation may be the results of addition and subtraction, they do not express any mathematical operations themselves. The polynomials on both sides of the equation are inventories which list how many of each type of algebraic number are present, and how many of each type are lacking. Said another way, they are static aggregations of the various algebraic numbers. Al-Samawʾal works with this example of an eighth degree polynomial: ‘‘six ma¯ls cube cube and 28 ma¯ls ma¯l cube and six cubes cube and 38 ma¯ls ma¯l and 92 cubes and 30 things less 10 ma¯ls cube and two hundred ma¯ls’’17 ð6x8 þ 28x7 þ 6x6 þ 38x4 þ 92x3 þ 20x 10x5 200x2 Þ. This polynomial describes 190 objects of six types (powers) which have been diminished by 210 objects of two other types. Because of the difference between medieval and modern polynomials, I do not use modern notation. Instead, I render Arabic expressions and equations with a translation of the algebraic notation practiced in the late medieval Maghreb. Using abbreviations n ¼ dirhams/numbers/units, t ¼ things/roots, m ¼ ma¯ls, and ‘ for ‘‘less’’, Al-Karajı¯’s equation is 10t ‘ m1 ¼ 4t n5. I will describe the Arabic symbolism below in the section ‘‘Notation’’. The notation inserted in translations in all other sections are my additions to make the reading easier. One thing to keep in mind is the difference between operations and equations. Operations are performed in time, with a specified outcome, while equations assert the equivalence of two aggregations or collections of the algebraic numbers. Operations in Arabic are expressed in two ‘‘sentences’’. For example, Ibn Badr restates the enunciation to his problem (26) in terms of ‘‘things’’ in the beginning of his solution: ‘‘So since we divided (qasamna¯) a thing by a thing and ten, it resulted in (kharaja) three-fourths’’.18 The verb in the first sentence is the operation ‘‘divided’’, while the verb in the second sentence, ‘‘resulted’’, announces the outcome. After multiplying the 34 by the ‘‘a thing and ten’’, Ibn Badr arrives at the equation ‘‘three-fourths of a thing and seven dirhams and a half equals a thing’’ ð3t 7n1 ¼ 1t Þ. The verb expressing 4
2
equality in an equation is always a conjugation of ʿadala (‘‘to equal’’). This word is never used to announce the result of an operation.
15 16 17 18
See Oaks and Alkhateeb (2007, Sect. 3.5) for a discussion of illa¯. Both wa and illa¯ are used in telling time: 6:25 is often spoken as ‘‘six and a half less five’’. Al-Samawʾal (1972, p. 48;1). Al-Samawʾal uses Hindu-Arabic numerals for only some numbers. Saidan (1986, p. 464;9); Sa´nchez Pe´rez (1916, p. 65/44;8).
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The Structure of Algebraic Solutions to Problems There are three stages in the algebraic solution to a problem in medieval Arabic: Stage 1: the establishment of an equation in terms of the algebraic numbers, Stage 2: the simplification of the equation to one of six canonical types, Stage 3: the application of the proper procedure to arrive at the answer. Medieval algebraists did not enumerate these stages in their solutions, though some, like al-Karajı¯ and Ibn al-Ha¯ʾim, explained the basic steps.19 Consider the following example, Abu¯ Ka¯mil’s problem (3):20 [Enunciation]
[Stage 1]
So if he said ten: you divided it into two parts. So you divided each one of them by the other. Then you subtracted one of them from the other, so there remained five-sixths of a dirham.21 Its rule is that you make one of the two parts a thing, and the other ten less a thing. So you multiply a thing by its same, so it yields a ma¯l. Then you multiply ten less a thing by its same, so it yields a ma¯l and a hundred dirhams less twenty things. So subtract one of them from the other, so there remains a hundred dirhams less twenty things. So remember it. Then multiply one of the two parts by the other, which is a thing by ten less a thing, so it yields ten things less a ma¯l. So multiply it by what remained from the two divisions, after you subtracted one of them from the other, which is five-sixths of a dirham. So it yields eight things and a third thing less fivesixths of a ma¯l equals a hundred dirhams less twenty roots h i t m n t . ‘ 1 ‘ 5 ¼ 83
[Stage 2]
100
6
20
So restore the hundred dirhams by the roots [i.e. the twenty things] and add it to the eight things and a third of a thing less five-sixths of a ma¯l. So it yields twenty-eight things and a third of h a thing lessifive-sixths of a ma¯l equals a hundred dirhams t 2813
‘
m 5 6
n ¼ 100 . So restore it by the five-sixths of a ma¯l and
add it to the hundred dirhams, so it yields a hundred dirhams and five-sixths h of a ma¯li equals twenty-eight things and a third of a thing
n m 100 56
¼ 28t 1 . 3
So complete the five-sixths of a ma¯l so that it becomes one ma¯l. And to complete it you add to it the same as its fifth.22 So 19
Al-Karajı¯ (1986, p. 169;6); Ibn al-Ha¯ʾim (2003, pp. 219/81–224/89).
20
Abu¯ Ka¯mil (1986, p. 56;1).
21
If we name the parts P1 and P2 (so P1 þ P2 ¼ 10), the problem poses
In modern notation, þ Alkhateeb (2007, Sect. 3.3). 22
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5 2 6x
1 5 2 5 ð6 x Þ
P1 P2
PP21 ! 56.
! x . This technique is called regula infusa. See Oaks and 2
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add to everything you have the same as its fifth, so it yields a ma¯l and a hundred dirhams and twenty dirhams equals thirtyh i four things m n ¼ t . 1 120
34
23
So halve the roots. So multiply it by its same, and subtract from what is gathered a hundred and twenty dirhams. So there remains a hundred sixty-nine dirhams. So its root is thirteen. So subtract it from half the things, which is seventeen. So there remains four, which is one of the two parts, and the other is six.
[Stage 3]
A student today would likely begin the solution with the equation x 10 x 5 ¼ ; 10 x x 6
and then simplify it to the form ax2 þ bx þ c ¼ 0. But the two sides of an Arabic equation are ideally polynomials, so Abu¯ Ka¯mil first manipulates and works through the operations outlined in the enunciation before arriving at n x2 the equation 8t1 ‘ m5 ¼ 100 ‘ 20t . In a modern equation, like 5 þ 2x3 ¼ pffiffiffiffiffiffiffiffi , the 22x 3 6 two sides are real numbers whose construction from the single unknown x is made transparent by the operations indicated in the notation. An Arabic equation, instead, asserts the equivalence of two static aggregations of the different algebraic numbers. In stage 2 the equation is simplified. Because only positive numbers were acknowledged, there were six types of simplified equation of degree two or less: Type Type Type Type Type Type
1: 2: 3: 4: 5: 6:
ma¯ls equals roots (in modern terms, ax2 ¼ bx), ma¯ls equals numbers ðax2 ¼ cÞ; roots equals numbers ðbx ¼ cÞ; ma¯ls and roots equals numbers ðax2 þ bx ¼ cÞ; ma¯ls and numbers equals roots ðax2 þ c ¼ bxÞ; ma¯ls equals roots and numbers ðax2 ¼ bx þ cÞ.
To simplify 8t1 ‘ 3
m 5 6
n ¼ 100 ‘
t 20,
n Abu¯ Ka¯mil needs to restore the deficient 100 to
n , and also to restore the deficient 8t1. After that the number of ma¯ls is a whole 100 3
n ¼ 34t is of type 5. set to one.24 The resulting simplified equation m1 120 The solution to the simplified equation is worked out in stage 3. For each type of equation a special procedure is followed. For the simple types 1–3, one only needs to divide a number by another. For each of the compound equations of types 4–6 the procedure is more complex. Although al-Khwa¯rizmı¯ and
23
i.e. take half the number of roots (things): half of 34 is 17.
Traditionally an equation was considered simplified even with a 6¼ 1, but in practice algebraists set a to one as part of stage 2. 24
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some later algebraists give geometric proofs of these procedures, the solutions were applied in problems without reference to geometry. Developments The two oldest texts on algebra which have come down to us are al-Khwa¯rizmı¯’s Brief Book on Calculation by Algebra and a fragment of the Algebra of Ibn Turk.25 Both were composed early in the ninth century during the reign of al-Ma¯ʾmu¯n, and deal only with polynomials of degree one and two. Neither book can be assumed to represent the extent of algebraic practice at the time, since one is a self-proclaimed abridgement and the other is incomplete. So Qust:a¯ ibn Lu¯qa¯’s translation of Diophantus (ca. 860), which is the earliest text we have with higher powers, may or may not have introduced cube ðx3 Þ, ma¯l ma¯l ðx4 Þ, ma¯l cube (x5 ), etc. In either case, higher powers up to a ma¯l ma¯l ma¯l ma¯l ðx8 Þ were part of the mainstream of algebra by the end of the century. Al-Karajı¯, working in the beginning of the eleventh century, is the algebraist who introduced what we call the negative powers of the unknown. He and later mathematicians prefixed an algebraic number with the word juzuʿ (‘‘part’’) to indicate its reciprocal. So ‘‘five parts of a ma¯l’’ corresponds to our 5x2 . While this extension of the range of powers was presented in many textbooks, ‘‘parts’’ were rarely used in worked-out problems.26 The ways algebraists handled roots and divisions had a bigger impact on problem solving than the extension of powers. The five basic operations encountered in Arabic algebra are addition, subtraction, multiplication, division, and square root. The set of polynomials is closed only under the first three operations, so any divisions or roots specified in the enunciation compel the algebraist to take one of three alternatives:
Manipulate the operations to eliminate roots or divisions before setting up an equation. We saw above that Abu¯ Ka¯mil does this with divisions in problem (3). Give names to the results of divisions. Abu¯ Ka¯mil and other algebraists sometimes name the results of dividing ten less a thing by a thing, and of dividing a thing by ten less a thing, as ‘‘dinar’’ and ‘‘fals’’. These were denominations of gold and copper coins. Their product is one (silver) dirham. The names ‘‘dinar’’ and ‘‘fals’’ are then operated on just like the other algebraic numbers. Admit roots or divisions to equations. Abu¯ Ka¯mil’s book is the earliest we have which takes the third alternative with respect to square roots. He and later algebraists handled irrational roots of
25
Ibn Turk’s fragment is edited and translated in Sayili (1962).
26
I have found only one example of ‘‘parts’’ in a problem. It occurs in a solution to al-Fa¯risı¯’s problem (28) (Al-Fa¯risı¯ 1994, p. 555;11).
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monomials in a way which might seem peculiar to the modern reader. For them, the number (coefficient) of a term was required to be rational. pffiffimffi The square root of m8 , for instance, was as ‘‘root of eight ma¯ls’’ expressed 8 rather than ‘‘froot of eightg things’’ ptffiffi8 : This applied even when the ‘‘number’’ of a term had both rational and irrational parts. In problem (44) Abu¯ Ka¯mil writes ‘‘twenty pffiffiffiffiffi ffi m Þ; while we would express it things and the root of two ma¯ls’’27 ð20t 200 phundred ffiffiffiffiffiffiffiffi as the single term ð20 þ 200Þx. Perhaps this restriction to rational quantities was due to the fact that in medieval Arabic all numbers are cardinal numbers. One can have ten apples, ten dirhams, ten things, or ten cubes, but ‘‘ten’’ cannot stand on its own. It must be ten of something. Fractions also make sense, like five-sixths of a pffiffiffi dirham. But having 8 things is as inconceivable as having e fingers. Irrational roots naturally arise in geometry, which likely accounts for their presence in algebra problems. Also beginning with Abu¯ Ka¯mil we occasionally find roots of polynomials. For instance, he sets up and solves this equation in problem (61): maffiffiffiffiffiffi ¯ l and ‘‘a p m c two things equals the root of twenty ma¯ls and four cubes’’.28 m1 2t ¼ 20 4 . The incorporation of roots led to some potentially ambiguous readings. To pffiffiffiffiffi m express mm Abu ¯ Ka¯mil cannot write ‘‘the root of two ma¯ls ma¯l less a ‘ 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi m ma¯l’’, since this could be misinterpreted as mm 2 ‘ 1 . To make it clear he writes ‘‘The root of two ma¯ls ma¯l, subtracting from it a ma¯l’’.29 And he exffiffiffiffiffiffiffiffiffiffiffiffiffiffi qp ffiffiffiffiffi ffi mm m presses ¯ ls ma¯l and two ma¯ls, whose root is 20 2 as ‘‘the root of twenty ma taken’’,30 since the simpler phrase ‘‘root of the root of twenty ma¯ls ma¯l and qffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi mm m two ma¯ls’’ could be taken as meaning 20 2 . The phrases ‘‘subtracting from it’’ and ‘‘whose root is taken’’ are inefficient rhetorical patches concocted to handle aggregation. Al-Karajı¯ was the first to freely incorporate divisions in equations.31 For instance, in problem (III-10) he sets up and solves the equation ‘‘ten less a thing divided by a thing and a thing divided by ten less a thing, and that equals two dirhams and a sixth.’’32 In both notations, this is n 10
‘
t 1
t 1
t 1 n 10
‘
t 1
¼
n 2 16
or
10 x x 1 þ ¼2 : x 10 x 6
27
Abu¯ Ka¯mil (1986, p. 93;16).
28
Abu¯ Ka¯mil (1986, p. 121;18).
29
Abu¯ Ka¯mil (1986, p. 93;7).
30
Abu¯ Ka¯mil, (1986, p. 118;7).
31
With one exception each, Al-Khwa¯rizmı¯ and Abu¯ Ka¯mil avoided divisions in equations.
32
Saidan (1986, p. 213;2).
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The operation of division, like other operations, was always distinguished from its result. For example, al-Karajı¯ writes ‘‘Then divide ten less a thing by a thing. It yields ten less a thing divided by a thing.’’33 The phrase ‘‘ten less a thing divided by a thing’’ is the long, descriptive name given to the result of the division.
Notation Calculation and Oral Texts Education in medieval Islam was centered around memorization and recitation.34 Arithmetic and algebra books preserve traces of this approach to teaching and learning. Problems begin ‘‘If [a questioner] says to you’’, and often when an intermediate result is to be set aside, the student is instructed ‘‘So remember it’’. Further testimony are the memorization-friendly poems explaining the rules of arithmetic and algebra. Both Ibn al-Ha¯ʾim’s and Ibn Gha¯zı¯’s books are commentaries on such poems. Because of the emphasis on oral instruction in Islamic education, written texts read like transcripts of lectures. Notation serves no purpose in recitation, so the early books are devoid of symbols—even numbers were written out in words. J. L. Berggren describes the Hindu-Arabic numerals in Ku¯shya¯r ibn Labba¯n’s book Principles of Hindu Reckoning (early eleventh c.): ‘‘In the text of his book Ku¯shya¯r writes out, in words, all the names of the numbers, and it is only when he is actually exhibiting what is written down on the dustboard that he uses the Hindu-Arabic ciphers. A reason for this may be that explanations were considered as text and therefore written out in words, like any other text. The examples of what was written on the dust board, however, may have been viewed as illustrations, much like a diagram in a geometrical argument, and they were there to show what the calculator would actually see on the dust board.’’35 A Notation for Algebra in the Maghreb Eventually a notation for algebra was devised. The evidence we have so far suggests that it originated in the twelfth century, and was restricted to the
33
Saidan (1986 p. 215;3).
34
On Islamic education see Makdisi (1981); Pedersen (1984); Berkey (1992); Chamberlain (1994).
35
Berggren (1986, p. 32). For Ku¯shyar’s book see Ku¯shya¯r ibn Labba¯n (1965).
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Medieval Arabic Algebra as an Artificial Language
555
western portion of North Africa. In particular, the notation was most likely unknown to the more famous scientific algebraists in the East, such as al-Karajı¯ and al-Khayya¯mı¯. The examples I show here are taken from Mahdi Abdeljaouad’s article ‘‘Le manuscrit de Jerba’’.36 There he traces the development of the symbols, and he gives a description of a particular manuscript of Ibn al-Ha¯ʾim’s Sharh al-Urju¯za, copied in 1747 by a Tunisian named Muhammad al-Ba¯z. Ibn al-Ha¯ʾim wrote his book without notation, and al-Ba¯z ˙ in the margins of his copy solutions to all the problems in symbols. added Here is a sample equation: m t n = 1 1 30
Reading the Arabic from right to left, the first term has the letter mı¯m ( ) above a 1, indicating one ma¯l. This is separated from the other two terms by an elongated la¯m , the last letter in the word taʿdil (‘‘equals’’). It looks like a backwards ‘‘L’’, and functions like our ‘‘=’’ sign. On the left side of the equation the letter shı¯n ( ), the first letter of the word shayʾ (‘‘thing’’),37 is atop the number 1, indicating one ‘‘thing’’. The last term consists of the letter ʿayn ( ), the first letter in ʿadad (‘‘number’’), above the number ‘‘30’’. Note that no symbol is used for ‘‘and’’. Each side of the equation is an inventory of the algebraic numbers.38 In this next equation the symbol for ‘‘less’’ is the Arabic word illa¯, but without the initial aleph. It looks like an upside-down ‘:39 m 10
Square root is indicated by the letter jı¯m ( terms. Jı¯m is the first letter in jidhr (‘‘root’’):40
n n = 20 20
) stretched above the
m t 1 3
36
Abdeljaouad (2002).
37
The shı¯n should have three dots above it, but for the notation they are dropped. Al-Ba¯z dropped the dots on other letters as well.
38
Rarely the wa (‘‘and’’) was included in the notation.
39
Abdeljaouad (2002, p. 26).
40
Abdeljaouad (2002, p. 30). The dot is omitted in the letter.
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556
J. A. Oaks
There was a notation for ‘‘divided by’’, also. The first two letters of the word maqsu¯m (‘‘divided’’) extend for the length of the terms: . This is a fancy version of the bar already used by Arabic mathematicians for writing simple fractions. To mimick the Arabic, I translate it now as ‘‘d—v’’ for ‘‘divided’’.41
d
n 10 m 1
t 1
v
The notations for roots and division is superior to the rhetorical presentation of the same operations. While algebraists had to resort to cumbersome phrases to verbalize the roots or divisions of polynomials, the notation makes it simple. One just extends the symbols across all the affected terms. Because Arabic equations do not use addition, subtraction, or multiplication, this is all that was needed. The notation was not restricted to writing equations. In some books every stage of the solution, from naming an unknown ‘‘thing’’ all the way to the answer, is conducted with symbols. The earliest known text with an entire problem solved in notation is Ibn Gha¯zı¯’s Commentary (ca. 1500).42 The manuscript of Ibn al-Ha¯ʾim copied by al-Ba¯z in 1747 is remarkable in that the margins of almost every page are filled with notational solutions to the rhetorical problems in the text. Here is a sample problem as Ibn al-Ha¯ʾim wrote it: And if he said a quantity: its third and a dirham is multiplied by its fourth and a dirham, so it becomes twenty dirhams. How much is it? So make the quantity a thing, and multiply its third and a dirham by its fourth and a dirham: it comes to a dirham and a third and a fourth of a thing and half a sixth of a ma¯l, and that equals twenty, which is the fourth type [of equation]. So work it out, so it yields the thing is twelve, which is the desired quantity.43
41
Abdeljaouad (2002, p. 27). Again, the dots are omitted in the notation.
42
Ibn Gha¯zı¯ (1983, p. 302).
43
Ibn al-Ha¯ʾim (2003, p. 227;10).
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Medieval Arabic Algebra as an Artificial Language
557
In the left margin al-Ba¯z wrote: Ababic notation44
My translation
Note that the verbal solution instructs the reader to ‘‘work it out’’ once the equation is set up, while the symbolic solution in the margin carries the calculations through to the end. Like Hindu-Arabic numeration, the algebraic notation was used to work through a problem off to the side. It was a calculation aid, designed as preparation for the solution in words. It would not have been used in a formal text. The notation which we find in manuscripts is either included to instruct the student on its use (as we find in Ibn Gha¯zı¯’s tract), or it appears in the margins (as in al-Ba¯z’s manuscript). 44
I have rewritten the Arabic notation myself because I have no clear photo of it.
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Syncopated or Symbolic? In 1842 G.H.F. Nesselmann identified three stages in the history of algebra, based on the evolution of notation.45 His divisions provide a semiotic distinction between abbreviations and symbols: 1. Rhetorical algebra, in which the algebra is expressed entirely in words. 2. Syncopated algebra, where abbreviations stand in place of words. 3. Symbolic algebra, in which a language of signs is employed, independent of oral expression. Medieval Arabic texts fall in the first category. Not only are whole problems solved in words, but often there is not even any paragraphy, or visual signs such as indentation or blank space, to indicate where one problem ends and another begins. Examples of syncopated algebra are the abbreviations used by Diophantus, and the algebra found in some Renaissance works. In his ˜ .4.ce.’’ for ‘‘3 cose 1494 Summa de Arithmetica Luca Pacioli writes ‘‘3.co.m meno 4 censi’’ (‘‘3 things less 4 ma¯ls’’).46 The abbreviations are to be expanded into words and pronounced, just like the ubiquitous shortcuts he and others take in ordinary text. For example, he writes ‘‘q¯sta co¯clusio¯e’’ for ‘‘questa conclusione’’ conclusion’’).47 Our modern algebra is symbolic. p ffiffiffiffiffiffiffiffiffiffiffiffi(‘‘this ffi 3 When we see x 1 x3 ¼ 3x 1 on a page, we process the symbols directly, without recourse to real or imagined speech. Because modern mathematics is (generally) not pronounced, we can make good use of visual arrangements, 1 X 1 p2 like ¼ . 2 k 6 k¼1 Although the individual symbols in the Arabic notation are mostly abbreviations, their visual layout makes it inconvenient to expand them into words and read them out loud—especially for roots of polynomials and division. This is a two-dimensional notation, the vertical dimension linking the algebraic power with its number, and the horizontal linking the different powers to form an equation. It falls into Nesselmann’s third category, ‘‘symbolic algebra’’.48 Medieval versus Modern Notation C. H. Edwards wrote that Leibnitz’ ‘‘infinitessimal calculus is the supreme example, in all of science and mathematics, of a system of notation and terminology so perfectly mated with its subject as to faithfully mirror the basic logical operations and processes of that subject’’.49 While choosing an appropriate notation for calculus is a bit more tricky than it is for medieval 45
Nesselmann (1842, p. 302). See also Rotman (2000, pp. 44ff).
46
Pacioli (1494, fol. 112a; 3rd line from bottom).
47
Pacioli (1494, fol. 116a;3).
48
Abdeljaouad also makes a case for the Arabic notation as fully symbolic (Abdeljaouad 2002).
49
Edwards (1979, p. 232).
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Medieval Arabic Algebra as an Artificial Language
559
algebra, it is true that the Maghrebean notation is also ‘‘perfectly mated with its subject.’’ The medieval concept of polynomial is faithfully represented in the notation as an inventory of the different powers, and the two operations of square root and division are represented accurately and efficiently as well. The question then arises: Is the Arabic notation compatible with our modern algebra? If we render the formula F ¼ MA in the old symbols, it would be something like F1 by M1 ¼ A1 . If this seems awkward, try to write m 0 c2 Einstein’s formula E ¼ qffiffiffiffiffiffiffiffiffiffiffiffi with the notation! The dissonance between 2 1 cv2 the medieval notation and modern formulas stems from the different concepts of polynomial and equation. From a semiotic perspective, the t in the term 8t is of a different nature than the x in 8x. In the Arabic 8t , the t tells us the power, and the 8 tells us how many we have. Their relationship is asymmetric: 8t is nonsense. It is the 8t as a unit which constitutes a term, so t alone cannot be operated on to produce t2 or 1 1t . In the modern 8x, both the 8 and the x are numbers in a symmetric relationship, and x can stand alone: x2 and 1 1x are valid expressions. Both the medieval and modern algebraic notations were designed to suit contemporary concepts, and they are not interchangeable.
Arabic Algebra as an Artificial Language Abstraction Al-Khwa¯rizmı¯ opens his algebra book with an appeal to the primacy of arithmetic in calculation: ‘‘When I considered what people generally want in calculating, I found that it is always a number.’’50 What he is telling us is that whether one wishes to find an unknown line segment, weight, span of time, or amount of money, calculations are always numerical. In fact, we find problems about all kinds of quantifiable objects solved by algebra. Below are examples from different books. I give the enunciation, the assignment of the unknown to the abstract ‘‘thing’’, and the first equation. Arithmetic. ‘‘Ten: you divided it into two parts. Then you divided one of them by the other. So the result of the division is four.’’ The solution begins ‘‘So the rule for this is that you make one of the two parts a thing . . .’’ The equation is set up as ‘‘four things equals . . . ten less a thing.’’ (al-Khwa¯rizmı¯’s problem (T3).51 ) Geometry. ‘‘So we begin by finding the value of a chord of a fifth of a known circle from its diameter, which is that you make the known circle circle ABDEG, and its diameter ten in number, which is line HE. And an equilateral equiangular pentagon is inscribed in the circle, which is pentagon ABDGE. So if we wanted to learn the value of each side of this pentagon . . .’’ In the solution he m mm writes ‘‘We make line ED a thing . . .’’ and the equation m9 cc1 ‘ mm 6 ¼ 4 ‘ 1 is 625
25
50
Al-Khwa¯rizmı¯ (1831, p. 3;1/5); al-Khwa¯rizmı¯ (1939, p. 16;7). Rosen’s translation.
51
Al-Khwa¯rizmı¯ (1939, p. 36;4).
25
123
560
J. A. Oaks
set up (in words) and solved. (From the second part of Abu¯ Ka¯mil’s Algebra, On the Pentagon and Decagon.52 ) Metrology. ‘‘A piece of jewelry is made of gold and pearl, and its weight is three mithqals, and its value is twenty-four dinars. And the value of a mithqal of gold is five dinars, and of pearl is fifteen dinars. We want to know the weight of each of them.’’ In the solution the assignment ‘‘We suppose that the weight of the gold is a thing’’ is made, and the equation ‘‘. . . forty-five dinars less ten things, which equals the twenty-four dinars’’ is set up and solved. (From al-Ka¯shı¯’s Mifta¯h al-Hisa¯b.53 ) ˙ suppose ˙ Finance. ‘‘So if he said: you hired [someone] by the month for forty dirhams and a ring. He worked five days and took the ring. How much is the ring worth?’’ In the solution the assignment is ‘‘So make its value a thing.’’ The equation is set up as ‘‘The five things equals forty dirhams.’’ (al-Karajı¯’s al-Fakhrı¯, problem (I-21).54 ) Horology. ‘‘[What] remains of the night is a quarter of what has elapsed and half of what remains. How much is the remainder, and how much has elapsed?’’ ‘‘So make the elapsed [portion of the night] a thing.’’ He h i then t t figures that the remainder is a half thing, so ‘‘the sum of that 1 and 1 equals 2 12 hours.’’ (al-Karajı¯’s al-Fakhrı¯, problem (I-48).55 ) People. ‘‘Fifty dirhams were divided among [a number of] men, so each one of them got something. Then you added to them three men, and you divided fifty among them. So what each one of the latter [men] got is less than what the former [men] got by three and three-quarters of a dirham.’’ The question is to find the number of former men. Ibn Badr writes in the solution ‘‘you make the [number of] original men a thing’’, and the equation is set up as ‘‘. . . a ma¯l and a quarter of a ma¯l and three things and three-quarters of a thing. So this equals fifty.’’ (Ibn Badr’s problem (28).56 ) In each of these problems an unknown magnitude or amount is named ‘‘thing’’, and an equation is then framed in terms of the algebraic numbers. So ‘‘thing’’, as an abstract number, can represent any quantifiable object. Stage 1 of the solution is the transformation of the context-specific enunciation to the context-free realm of abstract equation. Vocabulary and Syntax The building blocks, or ‘‘words’’, of Arabic algebra are the powers ‘‘dirham’’, ‘‘thing’’, ma¯l, ‘‘cube’’, etc. These are combined in stage 1 by arithmetical operations into polynomials (and occasionally roots and divisions), which are the ‘‘phrases’’ of the language. The two operations 52
Abu¯ Ka¯mil (1986, p. 134;18).
53
Al-Ka¯shı¯ (1969, p. 229;26). The whole problem is translated in the Appendix.
54
Saidan (1986, p. 174). The whole problem is translated in the Appendix.
55
Saidan (1986, p. 185). Sa´nchez Pe´rez (1916, p. 45;13); Saidan (1986, p. 465;5). Saidan mistakenly writes ‘‘threefourths’’ instead of ‘‘three and three-fourths’’. 56
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Medieval Arabic Algebra as an Artificial Language
561
performed in the beginning of Abu¯ Ka¯mil’s problem (12) are a good illustration of the construction of such phrases. The enunciation and stage 1 are: So if he said to you a quantity: you subtract from it its third and two dirhams. Then you multiply what remains by itself. So it brings back the quantity with twenty-four dirhams added to it. So its rule is that you make the quantity a thing. So you subtract from it its third and two dirhams, so there remains two-thirds of a thing less two dirhams. So you multiply it by itself, so it yields four-ninths of a ma¯l and four dirhams less two things and two-thirds of a thing equals a thing and twenty-four dirhams.57 In notation, the two operations are: t n t n results in 2 ‘ : subtracted from 1 2 2 3 m n t t n multiplied by itself results in 4 ‘ 2 2 ‘ 2 2 4 3 9 3: t
1 3
t m n t n : ‘ 2¼ 4 2 3 1 24 9 4 Equations are the ‘‘sentences’’ of the language. Here are two more examples, one with a root and one with a division: ‘‘Twenty-two ma¯ls less the root of three hundred eighty-four ma¯ls ma¯l equals two cubes.’’58
This last result is then run right into the equation, which is
rffiffiffiffiffiffiffiffi mm c ¼ 384 2
m ‘ 22
pffiffiffiffiffiffiffiffiffiffiffiffi 22x2 384x4 ¼ 2x3
‘‘Twenty less two things divided by a thing equals six things less four.’’59 ‘
n 20
t 1
t 2
¼
t n ‘ 6 4
20 2x ¼ 6x 4 x
The two operations found in equations, square root and division, both admit recursion. We find roots of polynomials which themselves contain irrational roots, and division can be compounded. In one problem Abu¯ Ka¯mil works with the term ‘‘the root of a ma¯l ma ¯ l and a quarter ma¯l ma¯l less half a ma¯l, whose r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi qffiffiffiffiffi mm m 60 root is taken’’ ¯ ʾim 11 ‘ 1 , and in one sample calculation Ibn al-Ha 4
2
divides ‘‘twenty dirhams divided by a thing’’ by ‘‘a thing and two dirhams’’ to get ‘‘twenty dirhams divided by a thing, dividing all that by a thing and two dirhams.’’61 Compound divisions were generally avoided, since they could 57
Abu¯ Ka¯mil (1986, p. 69;1).
58
From Abu¯ Ka¯mil’s problem (54) (Abu¯ Ka¯mil 1986, p. 101;7).
59
From Ibn al-Ha¯ʾim’s problem (I-7) (Ibn al-Ha¯ʾim 2003, p. 238;19).
60
Problem (60) (Abu¯ Ka¯mil 1986, p. 116;12).
61
Ibn al-Ha¯ʾim (2003, p. 162;4).
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J. A. Oaks
always be simplified to a single division. The rhetorical expression of such compound terms is awkward, but the notation handles aggregation and recursion efficiently. So at least in notation Arabic algebra was able ‘‘to make infinite use of finite means’’.62 There are specific rules for transforming one ‘‘sentence’’ into another. This is the simplification of equations, performed in stage 2 of the solution. I include the Arabic words most commonly associated with the step.63 al-jabr (restoration). Restore a diminished term.64 n t n n t n ¼ becomes ¼ ‘ 20 80 100 20 80 100 n n n The diminished 100 is restored, turning 100 ‘ 20t into a whole 100 . To balance t the equation, 20 is added to the other side of the equation. al-muqa¯bala (confrontation). Confront (take the difference of) like terms on opposite sides of the equation.65 n m m m n ¼ becomes ¼ 11 44 12 1 44 Here the
m 12
is confronted with the m1 , leaving
m 11
on one side.
ikma¯l/radda (completion/returning). Set the number of the highest power term to 1. ‘‘Completion’’ is used to raise the number to one, and ‘‘returning’’ to lower the number to one.66 n m t n m t ¼ becomes 1 2¼ 300 1 20 100 6 3 3 rffiffiffiffiffiffiffi rffiffiffiffi rffiffiffiffiffiffiffiffi t mm n m m n ¼ becomes ¼ 2 2 30 1 2 450 This can be accomplished by multiplying by the reciprocal of the number of ma¯ls, or by regula infusa.67
Square both sides. Used when the square root of a polynomial is present.68 t n ‘ ¼ 1 1 14
rffiffiffiffiffiffiffi mn 11
becomes
m 1
n t m ‘ 1¼ 1 11 2 1 82 2
62
Staal (2006, p. 131).
63
For a full explanation of these rules, see Oaks and Alkhateeb (2007).
64
From Abu¯ Ka¯mil’s problem (1) (Abu¯ Ka¯mil 1986, p. 49;16).
n 1
65
Al-Khwa¯rizmı¯’s problem (17) (Al-Khwa¯rizmı¯ 1939, p. 46;10; Al-Khwa¯rizmı¯ 1831, p. 39;9/55). First example: Ibn Badr’s problem (27), (Sa´nchez Pe´rez 1916, p. 45;2/66; Saidan 1986, p. 464;19). Saidan forgot to write thulth (‘‘third’’) before the first ma¯l. Second example: Abu¯ Ka¯mil’s problem (40), (Abu¯ Ka¯mil 1986, p. 87;21).
66
67
Abu¯ Ka¯mil uses regula infusa in problem (3). See footnote 22 above.
68
Al-Karajı¯’s problem (IV-34) (Saidan 1986, p. 261;5). I have rendered al-Karajı¯’s ‘‘a dirham and 9 a half and half an eighth of a dirham’’ in the medieval way, as 1 11 82. This is our 1 16.
123
Medieval Arabic Algebra as an Artificial Language
563
Multiply both sides by a denominator to eliminate division.69 n 100
m 10 n 10
‘
t 1
¼
t 52 12
becomes
m t t m n ‘ ‘ ¼ 525 52 12 10 1000 100
Stage 3 of the solution follows a predetermined algorithm using the quantities of the algebraic numbers. There was a numerical recipe for each of the six types of equation. Setting up an equation in stage 1 can be challenging, while the simplification and solution of equations in stages 2 and 3 follow straightforward rules. So in many solved problems stage 3 is omitted, and often stage 2 is not shown as well. The Mode of Representing an Artificial Language Rhetorical Arabic algebra exhibits an informal vocabulary. Some of the concepts are represented by more than one word, such as dirham/unit/number for constants and thing/root for the first degree unknown. I have found three different words used to mean ‘‘less’’: illa¯, ghayr, and siwan. Even the words which describe the steps of simplification were not fixed. More than one verb was used for each of restoration (jabr, ikma¯l), completion (ikma¯l, jabr, tama¯m) and returning (radda, hatta). And muqa¯bala, the word used to ‘‘confront’’ like ˙ ˙two ˙ terms, was also used in other ways in the worked-out problems.70 Still more variation is found for the operations of addition, subtraction, etc. This should not be surprising, since the operational verbs are borrowed from arithmetic. For subtraction I have found at least eight different words, whose meanings differ like the English ‘‘take away’’, ‘‘drop’’, ‘‘cast away’’, ‘‘throw down’’, ‘‘diminish’’, etc. With the exception of thing/root, all these concepts were expressed with words according to their ordinary meanings. The only truly technical terms in Arabic algebra are the names of the powers (which are not always unique) and ʿadala, the word used to mean ‘‘equal’’ in equations. In notation this variation is erased. Al-Ba¯z uses a single symbol for each concept, no matter how many different words Ibn al-Ha¯ʾim used to express it. As I noted above, there was some variation in the notation from one author to another, but each mathematician stuck to a specific scheme. Arabic algebra began as a spoken language, and it was only after several centuries that a notation was devised. But because the notation was created with an eye toward producing a version in words, there remained a natural isomorphism between them. Rhetorical and symbolic Arabic algebra are different ways of expressing the same artificial language. So the actual symbols or words used are not an essential part of a language. What is essential
69
Al-Karajı¯’s problem (III-18) (Saidan 1986, p. 218;20).
70
See Oaks and Alkhateeb (2005) for an account of thing/root, and Oaks and Alkhateeb (2007) for the words used in simplifying equations.
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is the structure of the meanings of words, expressions, sentences, and transformations.
Generosity Arabic Algebra to European Algebra As mathematicians in the eighteenth century were well aware, modern European algebra is a very generous artificial language. Frits Staal has written about the almost miraculous utility of algebra for describing the physical sciences, which not only made them accessible to people who could not otherwise penetrate the Latin of Galileo or Newton, but which also opened the door for rapid advancement.71 Because modern algebra traces its origin back to its Arabic counterpart, I should say some words about the historical link between the two before I address generosity. Europeans first learned of algebra in the twelfth century from Arabic sources through two different channels: the translations of Arabic works made in Spain, and the contact between Italian and Muslim merchants in the Mediterranean. In Spain the algebra texts of al-Khwa¯rizmı¯ and Abu¯ Ka¯mil were rendered into Latin, along with other practical treatises utilizing algebra.72 The most well-known Italian merchant of the time is Leonardo of Pisa (Fibonacci), who included a long chapter on algebra in his Liber Abaci, written in Latin in 1202. By the latter thirteenth century arithmetic textbooks were being composed in Italian for use in the abacus schools of central and northern Italy. The peak period of activity for these schools was the 14–16th centuries, and abacus textbooks cover the same material as their Arabic counterparts. The chapters on algebra which frequently conclude these books read as if they were translations from the Arabic. This tradition, in turn, was the basis for the work of Cardano, Vie`te, Fermat, and Descartes. Modern European algebra has exhibited both internal and external generosity. Internal generosity involves new insights into algebra/arithmetic as a result of a rethinking of current concepts. An example is the introduction and later acceptance of complex numbers which originated with the solution to cubic equations in the sixteenth century. External generosity relates to an application of algebra to another branch of mathematics, or to the physical world. Here the parameters are reinterpreted or generalized, leading to a new understanding. An example is the rise of quantum mechanics stemming (at least in part) from a reinterpretation of Planck’s formula.73 To ascertain the nature and extent of generosity in Arabic algebra, I first consider applications
71
Staal (2006).
72
Among the latter are the Liber Augmenti et Diminutionis and the Liber Mensurationum, neither of which survives in Arabic.
73
See Staal (2006, p. 128). Of course the two types were often intertwined.
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outside arithmetic for evidence of external generosity. I then look to work in arithmetic itself. Scientific Applications of Arabic Algebra I review here the few scientific applications of Arabic algebra known to me. I do not include textbook exercises which only appear to be applications, such as most of the examples given above in the section ‘‘Abstraction’’. Those problems may derive from real applications, but their primary purpose is to teach the method of algebra. I also exclude applications to practical mensuration and finance, such as al-Khwa¯rizmı¯’s problems in geometry and inheritance. Algebra was already used to solve such problems in the earliest texts, so these are not applications of an existing method to a new area. The earliest application of algebra to geometry we possess is Abu¯ Ka¯mil’s On the Pentagon and Decagon, the second part of his Algebra. He solves several problems concerning regular polygons, mostly 5- and 10-gons. We saw the enunciation of the first problem above in the section ‘‘Abstraction’’: to find the side of a pentagon inscribed in a circle of diameter 10. Jan Hogendijk wrote ‘‘. . .Abu¯ Ka¯mil shows that his algebraic methods can be used to find easy solutions to geometric problems that were difficult or even insoluble for his predecessors.’’74 Al-Ma¯ha¯nı¯, also working in the late ninth century, attempted to solve a problem of Archimedes using algebra. The problem is to cut a sphere by a plane so that the two volumes are in a give ratio.75 Al-Ma¯ha¯nı¯ transformed the problem into an algebraic equation with ‘‘cubes, ma¯ls, and numbers’’.76 Unfortunately, he was unable to solve the cubic equation. In an interesting twist, Abu¯ Jaʿfar al-Kha¯zinı¯, working in the tenth century, solved the equation by representing its solution in terms of the intersection of conic sections—so what began as an application of algebra to geometry turned into an application of geometry to algebra.77 This approach of solving cubic equations by intersecting conic sections became the hallmark of al-Khayya¯mı¯. He, and later Sharaf al-Dı¯n al-Tu¯sı¯, classified and solved all 25 simplified equations of ˙ many of them with the aid of conic sections. But the degree three or less, solutions to cubic equations as a function of their coefficients eluded medieval algebraists. It was only in the sixteenth century, with the work of del Ferro and Tartaglia, that a numerical solution to Archimedes’ problem became possible. An application of algebra to metrology is preserved in ʿAbd al-Rahma¯n ˙ to al-Kha¯zinı¯’s Book of the Balance of Wisdom. In the chapter attributed al-Khayya¯mı¯ we find a problem on determining the amount of gold in a mixed body of gold and silver. The solution uses the ratio of the weight of the body in 74
Abu¯ Ka¯mil (1986, Introduction p. 1).
75
On the Sphere and Cylinder, Proposition II.4. See Netz (2004) for a discussion of the history of this problem.
76
Woepcke (1851, p. 2;7).
77
Woepcke (1851, p. 2); Kasir (1931, p. 43).
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air to its weight in water, which is a way of measuring specific gravity. The general derivation is given in terms of geometry, in which the weights are represented by line segments. An example for a mixed body with weights in the ratio 10:1034 follows (he implicitly assumes in the solution that the weights are in fact 10 and 1034). Al-Khayya¯mı¯ works the problem using ratios in the geometric diagram to find that the weights of the gold and silver are each 5. He follows this with a solution to the same problem by algebra. The weight of the portion of gold in air is named ‘‘a thing’’, and he sets up and solves the equation ‘‘ten and three-quarters [less] a thing and a tenth of a thing equals ten and a half less a thing and half a tenth of a thing’’78 ð10n 3 ‘ 1t1 ¼ 10n 1 ‘ 1t1 Þ. 4
10
2
20
Al-Khayya¯mı¯ solves a similar problem by algebra later in the treatise.79 It should be no surprise that a mathematician who did such important work toward the solution of cubic equations would utilize algebra when the possibility arose. It is doubtful that algebra was used in practice for similar problems, since the rule for finding the weight using ratios is arguably simpler. The one application of algebra to trigonometry I know is found in al-Ka¯shı¯’s ‘‘Treatise on the calculation of the sine of one degree’’. Trigonometric identities make it possible to compute the sines of angles only of the form 2n 3 (n is any integer). Thus one can find sinð3 Þ, sinð1 12 Þ, sinð34 Þ, which bypass sinð1 Þ. But sinð1 Þ is necessary to fill out the sine table. This problem had usually been overcome through interpolation of nearby values, but alKa¯shı¯ takes a different approach. He calls sinð1 Þ ‘‘a thing’’, and he sets up the equation ‘‘four ma¯ls less one three-thousand six-hundredth of a ma¯l ma¯l are equal to a ma¯l and 6 16 49 7 59 8 56 29 40 [eighths of] things.’’80 m 4
‘
mm
1 3600
¼ m1
t
1 8A
; where A is the long hexadecimal number . This is simplified
to a cubic equation, whose root al-Ka¯shı¯ extracts according to a process equivalent to the Ruffini-Horner algorithm. I have come across no other application of Arabic algebra after scanning over 100 different texts from various branches of mathematics and science. Surely there are others, but those I have found indicate that scientists used algebra as a method of finding the solutions to specific problems. It was not a vehicle for expressing theories—that job fell to geometry. Internal Generosity The extension of the range of powers and the introduction of roots and divisions to equations, which I described above in the section ‘‘Developments’’, did not lead to any fundamental rethinking of the notions of polynomial or 78
The section begins at Al-Kha¯zinı¯ (1940, p. 87;21). The algebraic solution begins at p. 90;19. Al-Khayya¯mı¯ mentions in his derivation that the weight of an object in air is greater than its weight in water, but in the example the numbers are reversed.
79
Al-Kha¯zinı¯ (1940, p. 125;19).
80
Rosenfeld and Hogendijk (2002–2003, p. 41/57;1). Their translation, except I put back ma¯l for their square.
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equation. Roots and divisions certainly made equations more abstract, but they were only appended to the existing method. Had a new conception taken hold we would expect to see some change in the solutions to quadratic problems which do not call for roots or divisions. But these continued to be solved the old way, by working all the operations first. Algebra was applied within arithmetic to solve problems on indeterminate analysis. We not only have Qust: a¯’s translation of Diophantus, but there was also a native Arabic tradition of indeterminate analysis. The third part of Abu¯ Ka¯mil’s Algebra is devoted to the topic, as well as much of al-Karajı¯’s book al-Badı¯ʿ fı¯ al-Hisa¯b.81 Many indeterminate problems also appear in later ˙ algebra and arithmetic books mixed with ordinary problems. I find no evidence of generosity in these applications. Algebra was also applied to various problems on summing integers. One of many examples, from al-Hassar’s arithmetic book, is to find n such that ˙ ˙ ˙ second solution is by algebra: ‘‘So make the 1 þ 2 þ 3 þ þ n ¼ 55.82 His number a thing, and you add to it one. It yields one and a thing. So multiply it by half of the thing. . .’’ But again, the algebra here is nothing different from what is found in other kinds of problems. I have found one application of algebra to arithmetic which truly represents a new way of thinking. It is found in Abu¯ Ka¯mil’s Algebra, in the third solution to problem (61). There he would like to invoke a certain arithmetical pffiffiffi identity: pffiffiffi given two unequal numbers, which we can call x and y, if x þ n x ¼ y n y, pffiffiffi pffiffiffi then y ¼ x þ n (n ¼ 1, 2, 3, etc.). He expresses the identity as follows: [Suppose that] for two different numbers, we subtract from the larger its root, and one adds to the smaller its root, so they are equal. Then the root of the larger number is larger than the root of the smaller number by one, since he said a root and a root. And if he said we subtract from the larger two of its roots and one adds to the smaller two of its roots, so the two are equivalent. Then the root of the larger number is larger than the root of the smaller by two. And if he said three roots and three roots, so then the root of the larger is larger than the root of the smaller by three. . .83 Two proofs of the identity are given, the first by geometry and the second by algebra. The geometry proof is nothing unusual. The two numbers are represented by squares, so their roots are line segments. Then areas and segments are compared to show that the identity holds. The algebraic proof begins: Its reason (ʿilla) by means of algebra (al-jabr) is true and clear. And that is consider two quantities, where the root of one of them is greater than the root of the other by a dirham. So if you subtracted from the larger its root, and you added to the smaller its root, they 81
The Marvelous [Book] on Arithmetic (Al-Karajı¯ 1964).
82
Al-Hassar (1194, fol. 76r;11). ˙ ˙˙ Abu¯ Ka¯mil (1986, p. 122;15).
83
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became equal, as I mentioned to you. This is that you make the root of one of the two quantities a thing, and the other a thing and a dirham. So you multiply each one of them by itself, so it yields the smaller is a ma¯l and the larger is a ma¯l and two things and a dirham. So if you added to the smaller its root, which is a thing, and you subtracted from the larger its root, which is a thing and a dirham, there remained from the larger a ma¯l and a thing, and the smaller ended up as a ma¯l and a thing, so they are equal.84 He then shows the case n ¼ 2 in the same way. This is the only proof using algebra I have seen in medieval Arabic. It is also the only instance I know in Arabic algebra where ‘‘a thing’’ represents an arbitrary quantity, and not a specific unknown number which will be found by setting up and solving an equation. Unknown, Arbitrary, and Variable Numbers This is a good time to pause and consider the different types of unidentified numbers which have been the object of study by mathematicians from ancient to modern times. There are three basic kinds: unknown numbers, arbitrary numbers, and variable numbers. Examples of unknown numbers are those we encounter in Arabic algebra. ‘‘Thing’’ in the equation ‘‘Four things equals ten less a thing’’ is an unknown but determined number which will be revealed by solving the equation. Unknown numbers can be found by many other methods, too. When solving a problem by geometry, for instance, an unknown number is often represented by a line segment. Areas and segments are manipulated and compared to arrive at the solution. Arbitrary numbers are fixed yet undetermined. They have been fundamental to the writing of proofs since the time of Euclid. Heath’s translation of Proposition VII.29 of the Elements begins: Any prime number is prime to any number which it does not measure. Let A be a prime number, and let it not measure B; I say that B; A are prime to one another. ______________A __________________B85
84
Abu¯ Ka¯mil (1986, p. 124;7). Abu¯ Ka¯mil does not seem bothered by the fact that he proves the converse of his claim.
85
Euclid (1956, p. 330).
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The number represented by the segment labeled A86 is not an unknown like the algebraic ‘‘thing’’, since its value will not be found. But at the same time it is fixed as soon as Euclid draws it. Arbitrary numbers are still common in mathematics proofs today. For example, to show that x > 1 implies x2 > x, we can begin the proof with ‘‘Let x 2 R such that x > 1. Then. . .’’. As soon as the first sentence is completed, x is locked into a particular value. Like the Unknown Soldier, or like a card drawn face down from a deck, x is both fixed and anonymous. Because it is chosen arbitrarily, can function as any number greater than 1. The generality provided by the anonymity of the arbitrary number or magnitude is what made Greek geometry ideal for expressing theories. In metrology, al-Khayya¯mı¯ is able to represent the weights of gold and silver in air and water with segments, which in turn represent arbitrary numbers. With these segments he derives a general method for finding the weights. He only uses algebra to solve a specific example. Abu¯ Ka¯mil’s algebraic proof of the arithmetical identity is significant because ‘‘thing’’ is an arbitrary number, not an unknown number. Medieval algebraists did have the capacity to lift algebra from its place in textbook problem-solving to a tool for investigating the relationships of general quantities on par with geometry. The fact that it follows a geometric proof speaks for this—what geometry did, so algebra can do too. Abu¯ Ka¯mil’s declaration that the proof is ‘‘true and clear’’ is testimony to his awareness of the leap he was making. But no one picked up on his idea. Geometry was already the established setting for proof in mathematics, so there was no good motive for reworking theories in algebraic form. What Abu¯ Ka¯mil offers is only an alternative to geometry, not an improvement over it.87 Variable numbers took some time to develop. Graeco-Arabic and medieval European mathematicians generally obeyed Aristotle’s dictum that ‘‘. . .mathematical objects, with the exception of those of astronomy, are without motion.’’88 Even Archimedes, who often applied mechanics to geometry, excluded motion from his rigorous proofs. It was the blossoming of dynamics and computational mathematics in the seventeenth century which led to a breakdown of the Aristotelian division of
86
Reviel Netz has shown that the letters in Greek diagrams are not symbols for points or segments or regions, but rather they point to particular geometrical objects in the diagram. He writes ‘‘. . . the letter alpha signifies the point [in our case a segment] next to which it stands, not by virtue of its being a symbol for it, but simply because it stands next to it. The letters in the diagram are useful signposts. They do not stand for objects, they stand on them.’’ Netz (1999, p. 47) The A and B in the ekthesis to Euclid VII.29 are not symbols for arbitrary numbers. Instead, they label line segments which represent numbers. For Euclid it is the segment in the diagram which stands for the number. The letter functions like a tag. 87
Another feature of geometry which made it ideal for theories is that several independent quantities or magnitudes could be represented. But Abu¯ Ka¯mil and later algebraists solved problems by algebra naming up to four independent unknowns (Sesiano 2000, 149; Djebbar 2005, 57, 148; Suter 1910/11).
88
Metaphysics 989b 32, translated in Aristotle (1966).
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mathematics from physics. The first appearance of variable magnitudes is found in the algebraic geometry of Descartes and Fermat. Descartes opens Book II of his Geometry by criticizing the ancients for separating ‘‘geometrical’’ from ‘‘mechanical’’ curves.89 He then proceeds to represent curves algebraically, using the variables x and y to denote perpendicular line segments. In an equation like yy ¼ 2y xy þ 5x xx,90 x and y vary to trace out the graph of the curve. For the first time an equation was set up which was not to be solved. Fermat hit upon the same idea of using algebra to describe curves at about the same time. This was no application of algebra to geometry. In the late sixteenth century Franc¸ois Vie`te had discarded the arithmetical foundation of medieval algebra, to reconstruct it in the framework of Euclidean geometry. Both Descartes and Fermat built on Vie`ta’s ideas, so their work is pure geometry expressed in the new (geometric) algebra. Variable numbers came about a century later, when algebra returned to its place in arithmetic, free from its dependence on geometry and mechanics. I cannot recount the causes of this shift, which occurred in a particularly volatile period in the history of mathematics. I will say only that by the mid-eighteenth century the notion of an algebraic function had emerged, largely through the work of Johann Bernoulli and Euler. It was the function, whose value is determined by a variable number, which caused a revolution in science.91 Numerical functions and variables are more versatile and accessible than even the dynamical geometry of Fermat and Descartes. Scientists in the mideighteenth century finally had a motive for casting away geometry in favor of algebra for describing nature. D’Alembert, for example, found the general 2 2 solution to the vibrarting string problem @@t2u ¼ a2 @@xu2 as /ðat þ xÞ þ wðat xÞ, where / and w are twice-differentiable functions. To solve this, or even to express it, in terms of geometry would be a daunting job for the best mathematician. Algebra had come a long way from its medieval role as a problem solving technique with unknowns and equations to the modern algebra of functions and variables. Conclusion: Generosity Requires a Motive Frits Staal wrote ‘‘. . . the so-called ‘European’ revolution was not a product of Europe or of world history, but was triggered by the introduction of formulas and equations that were easy to understand, independent of the varieties and vagarities of natural languages, and led to a language that is universal.’’92 But do we not find that these are all characteristics of Arabic algebra as well? We have seen that Arabic algebra is universal in that it can be used to solve problems in any quantifiable branch of inquiry, and in notation it is expressed 89
Descartes (1954, p. 40).
90
Descartes (1954, p. 77).
91
See Youschkevitch (1976) for an accout of the development of functions.
92
Staal (2006, p. 129).
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with easy-to-understand equations which are free of the problems of natural language. Yet no revolution was initiated by Arabic algebra. This is not because it lacked generosity—Abu¯ Ka¯mil’s proof shows that algebra had the potential to become something much more than a method of problem solving. What stood in the way was Greek geometry, an edifice which already satisfied the theoretical needs of mathematicians and scientists. There was no motive to oust geometry at this time in history. It was only with the rapid changes in 17-18th century European mathematics that algebra was transformed, and the supremacy of geometry was finally challenged. Acknowledgements I thank Mahdi Abdeljaouad for allowing me to reproduce photographs of the Jerba MS from his article (Abdeljaouad 2002), and Haitham Alkhateeb for his help with some of the translations.
Appendix I include here translations of five problems of various types. Al-Khwa¯rizmı¯’s problem (1).93 So if a questioner asked, ten: you divided it into two parts. Then you multiplied one of them by the other, so it yielded twenty-one dirhams. So you knew that one of the two parts of the ten is a thing, and the other is ten less a thing. So multiply a thing by ten less a thing. So it yields ten things less a ma¯l equals twenty-one. So restore the ten things by the ma¯l and add it to the twenty-one. So it yields ten things equals twenty-one dirhams and a ma¯l. So cast away half the roots, so there remains five. So multiply it by itself: it yields twenty-five. So cast away from it the twenty-one which is with the ma¯l, so there remains four. So take its root, which is two. So subtract it from half the roots, which is five. There remains three, and that is one of the two parts. And if you wished, you added the root of the four to half the roots, so it yields seven, which is one of the two parts. And this is a problem which you work out by adding and subtracting.94 Al-Karajı¯’s al-Fakhrı¯, problem (I-21).95 So if he said: suppose you hired [someone] by the month for forty dirhams and a ring. He worked five days and took the ring. How much is the ring worth? So make its value a thing. So his due for the remaining twenty-five days, after the five [days] which he was paid for, is five things. Since his due for every five days is a thing, the five things equals forty dirhams. So the one thing equals eight dirhams, [which is] the value of the ring.
93
Al-Khwa¯rizmı¯ (1939, p. 38;14); al-Khwa¯rizmı¯ (1831, p. 30;2/41).
94
i.e. the problem has two solutions, which are found by adding and subtracting the 2. Only type 5 equations yield two (positive) solutions.
95
Saidan (1986, p. 174).
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Ibn Badr’s problem (28).96 Fifty dirhams were divided among [a number of] men, so each one of them got something. Then you added to them three men, and you divided fifty among them. So what each one of the latter [men] got is less than what the former [men] got by three and three-quarters of a dirham. Its rule is that you multiply the former men by the difference between what one of the former got, and one of the latter. So what you gathered, you divided it by the difference between the former men and the latter. So what results, you multiplied it by the latter men. So what you gathered, you confronted it with the divided number.97 And that is that you make the former men a thing. So you multiply that by the difference [between] what one of the latter men got, and one of the former, and that is three and three-quarters. That gives you three things and three-quarters of a thing. So divide that by the difference between the former men and the latter, and that is three. It results for you in a thing and a quarter of a thing. So multiply it by the latter men, which is a thing and three. It results for you in a ma¯l and a quarter of a ma¯l and three things and three-quarters of a thing. So it equals fifty. So you take four-fifths of everything you have, so you get a ma¯l and three things equals forty. So you work it out [according] to what you did in the fourth [problem]. Your result is that the former men is five. So if you divided the fifty dirhams by it, your result is that [the share] one of them is ten. So if you added to them three, and you divided fifty by what is gathered, it results in [the share of] every one of the latter is six and a quarter. And [the share of] each one of the former is ten. So the difference between one of the former and one of the latter is three and three-quarters. Al-Ka¯shı¯.98 A piece of jewelry is made of gold and pearl, and its weight is three mithqals, and its value is twenty-four dinars. And the value of a mithqal of gold is five dinars, and of pearl is fifteen dinars. We want to know the weight of each of them. So by algebra (al-jabr wa’l-muqa¯bala), we suppose that the weight of the gold is a thing. Then its cost is five things, and that leaves the weight of the pearl as three mithqals less a thing. We multiplied it by the value of a mithqal of it, which is fifteen. It resulted in forty-five dinars less fifteen things, which is the cost of the pearl. We summed the two values. It amounted to forty-five dinars less ten things, which equals the twenty-four dinars, the value of the jewelry.
96 Sa´nchez Pe´rez (1916, p. 45;13); Saidan (1986, p. 465;5). I altered the word order in some phrases to make the reading easier. For instance, instead of saying ‘‘So if you divided by it the fifty dirhams’’ I wrote ‘‘So if you divided the fifty dirhams by it’’. 97
This paragraph is a general plan for the solution. To ‘‘confront’’ two polynomials means to equate them and solve the equation.
98
Al-Ka¯shı¯ (1969, p. 229;4).
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After that restore the diminished term and confront them [i.e. the 45 and the 24]. It yields twenty-one dinars equals the ten things, which is the first of the simple [equations]. We divided the number [21] by the number of things [10]. The result of the division is two and a tenth, which is the unknown thing, which is the weight of the gold. So that leaves the weight of the pearl as ninetenths of a mithqal. After the algebraic solution, al-Ka¯shı¯ gives an algorithm for finding the solution. This algorithm is proven by geometry. ¯ milı¯’s problem (3).99 Al-ʿA A quantity: we added to it its fifth and five dirhams, and we subtracted from the outcome its third and five dirhams: nothing remains. So by algebra (al-jabr), suppose the quantity is a thing, and add to it its fifth and five dirhams. It becomes a thing and a fifth of a thing and five dirhams. Then subtract from a thing and a fifth of a thing and five dirhams its third, so there remains four-fifths of a thing and three dirhams and a third. And if you subtracted from it five, nothing remains, so it equals the five. And then subtracting like [terms], four-fifths of a thing equals a dirham and two-thirds. So 1 divide one and two-thirds by four-fifths. It results in two and half a sixth ½2 12 , which is the sought-after [quantity]. Two other solutions follow, by double false position and arithmetical rea¯ milı¯ explains the rules for solving quadratic equations, soning. Although al-ʿA his sample problems are all linear.
References Note: IM&A stands for the series Islamic Mathematics and Astronomy, published in Frankfurt am Main by the Institute for the History of Arabic-Islamic Science and the Johann Wolfgang Goethe University. Abdeljaouad, M. (2002). Le manuscrit mathe´matique de Jerba: Une pratique des symboles alge´briques maghre´bins en pleine maturite´. Quaderni de Ricerca in Didattica del G.R.I.M., 11, 110–173. Also published in Vol. 2, pp. 9–98 In: Abdallah El Idrissi and Ezzaim Laabid (Eds.), Actes du 7e`me Colloque Maghre´bin sur l’Histoire des Mathe´matiques Arabes. 2 vols. Marrakech: E´cole Normale Supe´rieure, 2005. Also available online at: http://math.unipa.it/~grim/ MahdiAbdjQuad11.pdf. I refer to the pagination of the online pdf file. Abu¯ Ka¯mil (1986). Kita¯b fı¯ al-Jabr wa’l-Muqa¯bala. (A facsimile edition of MS Istanbul, Kara Mustafa Pas¸a 379, copied in 1253 C.E. Edited by Jan P. Hogendijk.) Frankfurt am Main: Institute for the History of Arabic-Islamic Science at the Johann Wolfgang Goethe University. Abu¯ Ka¯mil (2004). Die algebra: Kitab al-Gabr wal-muqabala des Abu Kamil Soga ibn Aslam. Edited and translated by Sami Chalhoub. Aleppo: Institute for the History of Arabic Science. ¯ milı¯ (Mathematical Works of Baha¯ʾ al-Dı¯n ¯ milı¯ (1976). Riya¯d¯ıya¯t Baha¯ʾ al-Dı¯n al-ʿA Al-ʿA ˙ ¯. Aleppo: Maʿhad al-Tura¯th al-ʿIlm al-ʿArabı¯, Ja¯miʿat ˙ Halab. ¯ milı¯), edited by ˙Jala¯l Shawqı al-ʿA ˙ Aristotle (1966). Aristotle’s Metaphysics, translated by Hippocrates G. Apostle. Bloomington: Indiana University. Berggren J. L. (1986). Episodes in the mathematics of Medieval Islam. New York: Springer. Berkey, J. (1992). The Transmission of Knowledge in Medieval Cairo. Princeton: Princeton University.
99
¯ milı¯ (1976, p. 133). Al-ʿA
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