The arithmetic of Abu'l-Wafa

eter CHICAGO JOURNALS Science Soctety The Arithmetic of Abü'l-Wafa' Le Author(s): A. S. Saidan, Abu'l-Wafa', Muhammad ibn Muhammad ibn Yahya and al-Buzajani Source: Isis, Vol. 65, No. 3 (Sep., 1974), pp. 367-375 Published by: University of Chicago Press on behalf of History of Science Society Stable URL: http://www.jstor.org/stable/228959 Accessed: 10-11-2015 14:25 UTC Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor org/page/ JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship; For more information about JSTOR, please contact support@jstor org. History of Science Society and University of Chicago Press are collaborating with JSTOR to digitize, preserve and extend access lo Isis.

Author(s): A. S. Saidan, Abu'l-Wafa', Muhammad ibn Muhammad ibn Yahya and al-Buzajani Source: Isis, Vol. 65, No. 3 (Sep., 1974), pp. 367-375 Published by: University of Chicago Press on behalf of History of Science Society Stable URL: http://www.jstor.org/stable/228959 Accessed: 10-11-2015 14:25 UTC Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. History of Science Society and University of Chicago Press are collaborating with JSTOR to digitize, preserve and extend access to Isis. http://www.jstor.org

DOCUMENTS& TRANSLATIONS Arithmetic of Abu'l-Wafa' By A. S. Saidan* Abui'l-Wafa', Muhammad ibn Muhammad ibn Yahya, al-Buizajani (940-998) is known to the scholarly world for his contributions to astronomy and trigonometry.1 What is here called the arithmetic of Abui'l-Wafa'refers to a work of his preserved for us in MSS Leiden, Or. 103, and Cairo, Riyada 42 M. Medovoi is familiar with these two manuscripts and has written a study in Russian based mainly upon the former manuscript.2I have published the whole work in Arabic Arithmetic,using both manuscripts, with an introduction and commentaries based upon a comparison with other Arabic works of the same type.3 The purpose of this paper is to present an analysis of Abui'l-Wafa'swork in English. Two arithmetical systems co-existed in Islam: finger reckoning and Hindu arithmetic. The former, called in Arabic hisab al-yadd (hand arithmetic), !isib al-'uqiud (finger-jointarithmetic), or hisab al-RiumWa'l-'Arab(arithmetic of the Byzantines and the Arabs), seems to be a survival of a Greco-Babylonian manipulational practice. Hindu arithmetic, called al-iisib al-Hindi (Indian arithmetic), aisabal-takht (board arithmetic), or hisab al-ghubar (dust arithmetic), was imported from India with the numerals. Abui'l-Wafa's arithmetic is of the finger-reckoning type, which has the following important characteristics: It has no numerals. Numbers are written in words, and calculations are performed mentally. To remember the results of intermediary steps, calculators bent their finger joints in conventional ways which enabled them to indicate whole numbers from 1 to Society, 1956). E. S. Kennedy, "The Arabic Dec. Heritagein the Exact Sciences,"Al-Abbhath, 1970, 23:327-344. Heinrich Suter, Die Mathematiker und Astronomender Araber und ihre Werke(Leipzig:Teubner,1900). A. P. JuschkeGeorge Sarton, Introduction to the History of witsch,GeschichtederMathematikimMittelalter, Science, Vol. I (Baltimore:Publishedfor the trans. from the Russian by Viktor Ziegler Carnegie Institution of Washington by the (Leipzig:Teubner,1964). A. H. Sabra, "'Ilm alWilliams & Wilkins Company, 1927). Carl Hisab," Encyclopaedia of Islam (2nd ed.; Brockelmann, Geschichte der arabischen LitteraLotidon:Luzac & Co., 1971),pp. 1138-1141. tur, 2 vols. + 3 supplements(Weimar:E. Felber, 2 M. I. Medovoi, "Oh arifmetichestkom 1898-1902; Leipzig:C. F. Amelangs Verlag, Aby-l-Vafy" (The arithmetical treatise traktate 1901, 1909; Leiden:E. J. Brill, 1943-). Bernard by AbT'l-Wafa'), Istoriko-MathematicheskieCarra de Vaux, Les penseurs de l'Islam, Vol. II (Paris:P.Geuthner,1921).E. S. Kennedy,A Sur- Isseldovaniya,1960,13:253-324. 3 A. S. Saidan, Arabic Arithmetic.The Arithvey of Islamic Astronomical Tables (Transactions of the American Philosophical Society, N.S. Vol. meticof Abu'l-Waft' (in Arabic),PartI (Amman: 46, Pt. 2) (Philadelphia:AmericanPhilosophical Sam'iyyat'Ummal al Matabi', 1971). ReceivedNov. 1972:revised/acceptedMar. 1973. * Department of Mathematics, Jordanian University,Amman,Jordan. 1 The following are the major referencesfor the study of Abii'l-Wafa'and Arabicarithmetic:

A. S. SAIDAN -A-%. The initial page of the MS Leiden Or. 103. 9,999.Thissamedevicewasrepeatedto indicatenumbersfromten thousandonward. It contains three sets of fractions.The first, here called the Arabic traditional system,supposedlyarosefromthe factthatin Arabicthereare only nine singlenames for fractions.Thesearethe namesof 1/2, 1/3, ...., 1/10.Thesefractionsarecalledthe ru'Fs,literally"heads,"that is, principalfractions.In addition,thereare the following subsets: (1) Compositefractions,whichconsistof two categories: (i) 2/3, 3/4, ... . 9/10 (ii) Combinations of ru'us,like 1/2 + 1/7, . (2) Fractionsof fractions,like 1/2 x 1/6. (3) Asamm,that is, "deaf,"fractions.These are fractionslike 1/11, 2/11, which cannotbe expressedexactlyin termsof the othertypes. Rules are stated, and always traditionallyobserved,to express every common fractiona/b in terms of ru'iis,combinationsthereof, and/or fractionsof fractions. Thus1/12is expressedas 1/2 x 1/6,whichis saidto be betterthan 1/3 x 1/4. Similarly, 5/12 is expressedas 1/3 + 1/2 x 1/6, whereas7/12 is convertedto 1/3 + 1/4. Eventhe asammfractionshaveto be approximated. This practicewas graduallyovershadowedby the Indianconceptof the common fractiona/b, but it survivedtill late in the MiddleAges. Arabicauthorsseemto have thought that it was of Arabic origin, but we have reasonsto believethat it was a traditionalpracticecomingdown from the ancientEgyptianconceptof the common fraction;Proclusin the fifth centuryexpressed23/25 in the same way that Abu'lWafa'wouldhavedone. The secondset of fractionsis that of the scaleof sixty.This was mainlythe concern of astronomersbut foundits wayto problemsof everydaylife and servedthe purposes that percentageservetoday.

The third set arose from the local units of money, weights, and measures. Where 1 dirham = 6 danigs = 48 habbas,the danig is used to denote 1/6, and the habba 1/48. However intricate it may look to us, finger reckoning was the system from which Arabic algebra, trigonometry, and higher mathematical notions were developed. It was already well established in Islam before the Hindu system made its first appearance in the Arab world. The Hindu system brought the numeral forms, the concept of the common fraction, and well-defined algorithms on the dust board. The amalgamation of the two systems resulted in the arithmetic handed over to the Latin arithmeticians of the Middle Ages. Arabic works on finger reckoning deal invariably with ratio, multiplication, and division in application to everyday problems. Finding the unknown quantity leads to algebra, and mensuration leads to root extraction and to trigonometry. Arabic works of Hindu arithmetic, on the other hand, start by introducing the nine numeral forms, the concept of place value, and the zero form, and then deal with addition, subtraction, multiplication, division, and roots. Business arithmetic, and therefore algebra and trigonometry, hardly found a place in early works of this type. They appear in later works where the two types are unified. TEXT AND MANUSCRIPTSOF THE ARITHMETIC Abul-Wafa's treatise is the earliest work on finger reckoning that has survived, and the most important. The author gave it the name Fi ma Ya1itajuilai-h! al-Kuttab wa'l 'Ummal wa-Ghairuhummin 'Ilm al-Hisiab(On what scribes, officials, and others need of the science of arithmetic). As the name implies, it was written for state officials, and therefore besides its value for historians of science it gives insight into tenth-century life in Islam from the administrativepoint of view. The treatise is divided into seven parts called manizil, literally "places." That is why the Cairo copy gives it the shorter name of Kitab al-Maniizil al-Sab', book of the seven places. Each part is in seven chapters. The Leiden copy (112 folios), consists of the first three parts. The Cairo copy (230 folios), starts with part [I, chapter 2, and goes to the end. It bears two sets of pagination: a recent one that extends from 1 to 230, and an old one that extends from 50 to 265 (the remainingfifteen folios which do not have this old pagination are in a different hand.) There are a few folios missing: part II, chapters 3 and 4; part IV, chapter 7; and part VII, chapters 1 and 2. The first folio is rather worn out. The book ends on folio 225 with the statement that copying was finished on Friday, 3 Hijja, 487 (Dec. 15, 1094). Folios 226-230 give problems of types discussed in the text and contain nothing of interest to us except that some numbers are denoted by Hindu numerals. CONTENTS OF THE ARITHMETIC The book starts with a long table of contents (pp. 65-69 in my printed edition, which will hereafterbe noted in parentheses). The following summary will probably be more meaningful to the modern reader: Part I, On Ratio (pp. 71-122). Chapter 1 (pp. 71-74): definition of ratio, the four types of Arabic traditional fractions, rules to be observed in expressingratios in the traditional way.

A. S. SAIDAN Chapter2 (pp. 75-78): principal, composite, and fractions of fractions expressed as parts of 60. Chapter 3 (pp. 79-80): expressing n/60 in the traditional way, n being a whole number. 1/4 + 1/6 Example: 25 - 15 + 10 -* 25/60 Chapter4 (pp. 81-89): expressing n/60 in the traditional way, n being a whole number plus a principal or composite fraction. 10 + 1 1/3 - (11 1/3)/60 = 1/6 + 1/5 x 1/9 Example: 11 1/3 Chapter 5 (pp. 90-112): expressing n/60 in the traditional way, n being a whole number plus a fraction of a fraction. Chapter6 (pp. 113-119): Section 1: expressing a/b first as x/60, secondly in the traditional way. 1/5 + 1/6 + 1/10 Example: 7/15 -28/60 = (12 + 10 + 6)/60 fractions. Section 2: approximatinga.amm 11/60 approximately Example: 3/17 = (10 10/17)/60 (10 + 1)/60 + 1/6 + 1/6 x 1/10 For a better approximation, note that 10/17 (35 3/5)160 35/60 approximately 1/3 + 1/4 (10 1/3 + 1/4)/60 3/17 (6 + 3 1/3 + 1 - 1/4)/60 1/10 + 1/2 x 1/9 + 1/6 x 1/8 For a still better approximation the 3/5 which was neglected above can be treated still further. Chapter7(pp. 120-122): further examples. Part II, On Multiplicationand Division (pp. 123-201). Chapter 1 (pp. 124-131): definitions and preliminary remarks. Multiplication and division are defined according to Euclid and Nicomachus. The operations are split into the following types, where H stands for whole numbers and Ffor fractions: (i) H x H, F x F, H x F, (H + F) x H, (HF+ F) x F, and (H + F) x (4 + F) (ii) H/H, F/F, H/F, F/FL, (H + F)/H, H/(H + F), (HA+ F)/F, F/(H + F), and (H + F)/(H + F) Chapter2 (pp. 132-159): multiplication and division of whole numbers. 1. Every number is composed of maritib, places, and 'uqiid, literally, knots or finger knots. Thus three hundred and two is composed of three in the hundreds place and two in the units place; the three and two are the 'uqiid. Places are units, tens, hundreds, repeated for thousands, repeated again for thousands of thousands and so on. 'Uqiudmaybe one, two, . . . up to nine. 2. Multiplication of places by places. The rules here given lead to the formula lom X Ion = lom+n 3. Division of places. Rules are here given leading to l0I . 10'10`

OF ABO'L-WAFA' 4. Multiplication and division of the 'uquid.The multiplication table from 1 x 1 to 9 x 9 is assumed known. 5. The basic algorithm of multiplication appears from the following example, expressing in numerals what the author states in words: 46 x 28: 2 places in each; 2 x 2 -4, .. 4 multiplications: 40 x 20 = 800; 40 x 8 = 320; 6 x 20 = 120; 6 x 8 48 total 1288 Other devices are suggested, mainly for keeping order and avoiding oversight. The following is worth noting. The author suggests it to avoid Indian schemes, but it is rather Indian in disguise: To multiply 489 by 67, go on as follows: six seven nine four eight twenty-four forty-eight fifty-four twenty-eight fifty-six sixty-three Add up, starting from the left, inserting the units and carrying the tens as units in the next place to the right: thus you start by putting down three and carrying six to the second place. Finally you obtain three seven two three six which means that the answer is 32763. Further examples follow. For the zero the curved form is suggested. 6. Division is taken up gradually: division by numbers less than ten, division by tens, division by a number of two places, division by a number of many places. The schemes correspond generally to the schemes of multiplication. Chapter3 (pp. 160-173): the common denominator, addition, and subtraction of fractions. When we think of a fraction we associate it with the picture of two numbers one above the other, the lower being the denominator. Finger reckoning did not have such a picture. Instead, the computer thought of two thirds and associated it with the (least) whole number whose two thirds made a whole number. What we call the denominator was called by him the makhraj(outcome), or the number from which the fraction comes out (as a whole number). In this chapter Euclid's algorithm of the common factor is given and used to yield the common denominator. This is utilized in addition and subtraction of fractions. Another method of addition and subtraction is given under the name of the method of scribes. This requires converting fractions to the scale of sixty and reconverting the sum or differenceto the traditional Arabic form. Chapter4 (pp. 174-185): multiplication and division of fractions. Here two kinds of fractions are discussed. Those we have so far encountered are absolute fractions. Two schemes for multiplying and dividing these are given: one scheme is much like what we now use, and the other resorts to the scale of sixty. The other kind of fractions is a set of what are called al-kuszural-muntasibs(related fractions). These arise from the units of currency in use. The arithmetic of these is much like that of duodecimals in the former British monetary system. What makes it more complicated is that the units of currency were never standardized in the Islamic world; they changed from time to time and from place to place.

A. S. SAIDAN Chapters5, 6 (pp. 186-192): further multiplication and division of fractions. Taking again H and F to denote whole numbers and fractions respectively, the types here discussed are: H x F, H/F, F/H, (H + F) x H, (H + F(/H, H/)H + F), (H + F) x F, (H + F)/F, F/(H + F), (H + F) x (H + F), and (H + F)/(H + F) Chapter7 (pp. 193-201): short methods of multiplication and division. Here we meet one of the characteristic topics of finger reckoning that we hardly find in the Hindu schemes-shortcuts in multiplication and division. Finger reckoning faded away gradually, but these shortcuts continued to grow. The following are the rules given by Abui'l-Wafa': 1. Rules of multiplying and dividing by 5, 2 1/2, 33 1/3, . .. and generally, 10/m, n being any integer, and m a small integer like 2, 3, 4,4... 2. The above rules are utilized in multiplying by 15, 15 + 1, ... Thus 15 x n = n + 1/2n tens; 14n (n + 1/2) tens - n. 3. (lOim+ a) (lOIm+ b) = (lOm + a + b) x lOm + ba. Thus 59 x 54 = (59 + 4) fifties + 9 x 4. (lOmn + a + bn) x lOm + ab. Thus 64 x 28 = 4. (lOmn + a) (iOn + b) 4 3 twenties 8. x x + 8) (64 + 5. (lOm + a) (lOm + b) = (lOIm+ a + b - 10) x (m + 1) tens + (10 - a) (10 - b). Thus 59 x 54 (59 + 4 - 10) sixties + (10 - 9) (10 - 4). This rule is - 2, which is called applied to the simple case of 5 x 3. Here lOm + a + b - 10 a debt of 2. This is probably the earliest Arabic referenceto negative quantities. 6. Multiplication and division of powers of 10. Thus 1200/16 (12/16 = 3/4) of a hundred. PartIII, Mensuration(pp. 202-276). Chapters1, 2 (pp. 205-221): units of length, area, and volume; conversion. Chapter3 (pp. 222-233): mensuration of the circle. The following are the rules here given. (d diameter,r radius, A = area, c circumference,I = arc length, h = height of segment): 3. c (22/7)d 2. A -d2 x 11/14 1. A = r x c/2 x 7/8 6. A-c2/1 5. A-(c/2)2 x 7/22 4. c =V a4A x 22/7 9. d =h + (/2)2/h 7. h = r - \/r2 - (1/2)2 8.1/2 = Vh(d - n) 10. A table of sines: If a semicircle is drawn with center M and diameter AMB, and points C1, C2, . . . are taken so that the arcs AC1 = C1C2= C2C3= ... - one unit, then, with v = 22/7, the semicircle is divided into 22 equal arcs. zQAMC1 - 907/ 11 = ml, LAMC2= 2ml- M2, L/AMC3= 3m, iM3, and so on. The author gives the lengths of the arcs AC1, AC2, . . . and the lengths of their chords. Since chord ACr= 2r sin 1/2 mr, the lengths of the chords gives sines of angles from 00 to 900 at intervals of 900/22. Abui'l-Wafa'does not show how he calculates these lengths, but he says that he uses Ptolemy's theorem with approximations. His figures are correct to three decimal places. This table is henceforth used to find sines of given angles or arcs of given sines. The famous geometrical theorem attributed to Ptolemy is the one used in building up the sine tables in the Almagest. Al-Biruni, however, and other Muslim

astronomers pointed out that this method of building up the table is laborious. Al-Birtini wrote a treatise called Istikhraj al-awtarfi al-daira (extracting [lengths of] circular chords), in which he suggests easier methods. Abud'l-Wafa'prepared and used for astronomical purposes a more elaborate sine table in which the sine is given for each minute of the argument, to three sexagesimal places.4 But it is unlikely that he improved upon Ptolemy's method of calculating these chords. For intermediaryangles he uses linear interpolation. 11. He gives the formulas for the areas of circular sector and segment. Chapters4, 5 (pp. 234-261): mensuration of triangles, quadrilaterals,polygons, and composite figures. About the triangle, practically everything we usually find in books of numerical geometry is given here, including the area in terms of the perimeterand the cosine rule. The same applies to quadrilaterals, except that the author makes no mention of the cyclic quadrilateral. If a regular polygon has n sides of length 1 each, and the inscribed circle is of radius r, then the area of the polygon is 1/2nlr. To find this r he uses his table of sines but adds a rule which he attributes to the Indians and criticizes as inaccurate and not built on deductive reasoning. The rule is 4r2 = 2/9 12 {n/2(`1) + 3} The composite figureshe treats are: 1. rectilinearfigures surmounted by circular segments, and 2. areas included between two intLrsectingcirculararcs. Chapter6 (pp. 262-268): volumes. The solids here considered are the cube, the cuboid, the prism, the cylinder, the cone, the frustum of the cylinder, the frustum of the cone, the sphere, and the spherical sector. Chapter7 (pp. 269-276): on finding distances. A crude instrument is suggested for finding distances, heights, and depths where these cannot be found by direct measurement.The instrument consists of a rectangular board half an armlength wide and two or more armlengths long. All sides are graduated. To one corner a pointer is attached carryingtwo small hollow blocks. Let us say we want to know the width of a river we cannot cross. The board is fixed on a stand of a known height at one bank. The pointer is directed so that if you look through the blocks you see the other bank. By finding at which points along the perimeter the pointer meets the board, we can easily find the width of the river. The principle is obviously that of similar triangles. Part IV, On Taxes (pp. 277-301). Chapters1, 2 (pp. 278-282): kinds of taxes and principles that govern their evaluation. The following are the main ideas: 1. Landowners pay to the state in money or in kind, at rates that differ, but payment involves three kinds of dues: 1. payment per unit of land, called the tasq. I Kennedy,Surveyof IslamicAstronomicalTables,p. 145.

A. S. SAIDAN ii. payment to the surveyor, called the ayin. iii. payment to the accountant, called the rawaj. 2. The Euclidean notion of proportion applies to all. Chapters3-7 (pp. 283-301): problems of tax calculations. The following six units are involved: 1 dirham = 6 danigs = 60 'ushairs of currency I100'ushairsof capacity 1jarib =10 gafizes Manipulations involving these units require special attention. We should remember that each of these has a twofold significance. Thus a ddnig is a unit of money and may also denote one sixth. Similarly the gafiz is a unit of capacity and may also denote one tenth. Thus a ddnigs x b gafizes = a/6 x b gafizes if the required result is expected to give some capacity. It is b/10 x a danigs if the result is that of money. The nature of the problem decides which result should be taken. Part V, On Exchange and Shares (pp. 302-329). Chapter 1 (pp. 303-306): kinds of kurrs and conversion from one kind to another. The kurr, a unit of capacity for measuring crops, varied in value from place to place. Four such values are given with tables of conversion. Chapters2, 3, 4 (pp. 307-315): kinds of crops, their standard values, exchange. Crops are divided according to price into four major kinds. Under barley, for example, are placed all crops whose price is more or less equal to that of barley. The relative values are given and the problem of exchange arises. The problem becomes worth skillful consideration when the crops to be exchanged are given in different kurrs.A conversion table is supplied for this purpose. Chapters5, 6, 7 (pp. 316-329): problems related to harvest. 1. Distribution of shares; the share of the state and that of the tax collector. 2. Assessment of total price. 3. Harvest valued by differentunits of capacity. Part VI, MiscellaneousTopics (pp. 320-346). Chapter1 (pp. 331-336): conversion of payment in kind to cash. Chapter2 (pp. 337-340): exchange of money units. Chapter3 (p. 341): payment in kind or cash with respect to weight. Chapter4 (pp. 342, 343): pay and annuities of soldiers. Chapter5 (pp. 344, 345): fodder for animals as entitlement of owners in civil service. Chapter6 (p. 346): impost dues; (not complete). Chapter 7: problems related to mail and official journeys (missing from the manuscript). Part VII, FurtherBusiness Topics ChapterI (missing): units of weight. Chapter2 (missing, except for a few lines, pp. 346, 347): conversion of weights and related topics. Chapter3 (pp. 348-353): assessment of wages. Chapter4 (pp. 354-357): wages paid in kind.

ChapterS (p. 358): wages for mortaring and similar labor. Chapter6 (pp. 359-360): wages for bricklaying. Chapter7 (pp. 361-367): five problems, on fractions and streams filling a well. TOPICS NOT TREATEDBY ABU'L-WAFA' Abui'l-Waf&'sarithmetic is lengthy compared to the other Arabic books on finger reckoning that have come down to us. Yet we miss in it some important topics that we find in the others. Algebra: In connection with proportion all other works consider the unknown and usually present in a seemingly traditional way the types of equations that may be encountered. Abui'l-Wafa'ignores these. In one case at least he obtains a quadratic equation and gives its positive root without telling how he does it. He states, however (p. 132), that he has written a commentary to the algebra of al-Khwarizml. This is probably why he overlooks algebra in this work: he has dealt with it elsewhere. Extraction of roots: In all Arabic works of arithmetic this is included as a basic operation. Abui'l-Wafa'gives the roots encountered in connection with mensuration but does not tell how he obtains them. Is this because he had discussed this in some other work? I am inclined to give a negative answer. Finger reckoners must have had mental ways for obtaining roots, exactly or approximately, but the practice of presenting these as a branch of arithmetic came probably with Hindu arithmetic, which Abua'l-Wafa'did not master. Casting out nines: All other works, whether of finger reckoning or of Hindu arithmetic, use casting out nines for checking results. Later ones use casting out eights or elevens as well. Some medieval Arabic commentators state that casting out nines is an Indian practice. The odds are, therefore, that this is a practice which Abfu'l-Wafa'did not know of. There are places in this work where ignorance of the Indian practice is shown and others where it is deprecated.