^{Page 1 note 2} This paper (Luckey, P., “Die Ausziehung der n-ten Wurzel und der binomische Lehrsatz in der islamischen Mathematik”, Mathematische Annalen, 120 (1948): 217–74)CrossRefGoogle Scholar has been followed by a book (Luckey, P., Die Rechenkunst bei Gamšīd b. Mas';ūd al Kāšī mit Rückblicken auf die ältere Geschichte des Rechnens (Wiesbaden, 1951))Google Scholar where one can find biographical information about al-Kāshī. See also Rosenfeld, B. and Youschkevitch, A., “al-Kāshī”, Dictionary of Scientific Biography (1973), vol. VII, pp. 255–62Google Scholar, which refers to more recently available literature on this topic.

^{Page 1 note 3} In this paper, unless otherwise specified, “n ^th root” means “root of order higher than 3”. I do not discuss here the step from these roots to the general n ^th root.

^{Page 2 note 4} See p. 221. Later, on p. 248, he comes back to consider the possible origins of the Ruffini-Horner procedure, listing China among the possibilities without elaborating on this inclusion. In Luckey, P., “Zur islamischen Rechenkunst und Algebra des Mittelalters”, Forschungen und Fortschritte, 17/18 (1948): 199–204Google Scholar, he again mentions China, but without further remark.

^{Page 2 note 5} Compare Youschkevitch, A., Geschichte der Mathematik im Mittelalter (Leipzig, 1964), pp. 243 and 248.Google Scholar

^{Page 2 note 6} Rashed, R., “Résolution des équations numériques et algèbre: Šaraf al-Dīn al-Ṭūsī, Viète”, Archive for History of Exact Sciences, 12 (1974): 244–90.CrossRefGoogle Scholar Al-Ṭūsī's book On equations – he lived from the second half of the 12th century to the beginning of the 13th century – has since been edited and translated with detailed notes; see Rashed, R., Sharaf al-Dīn al-Ṭūsī: Œuvres mathématiques. Algèbre et géométrie au XII^e siècle, 2 vols. (Paris, 1984).Google Scholar

^{Page 2 note 7} Rashed, R., “L'extraction de la racine n ^ième et l'invention des fractions décimales (XI^e-XII^e siècles)”, Archive for History of Exact Sciences, 18 (1978): 191–243.Google Scholar

^{Page 2 note 8} Rashed, , “Résolution”, p. 250.Google Scholar

^{Page 2 note 9} I may mention Yan, Li, Zhongguo Shuxue Dagang (Beijing, 1958) (for example, p. 166);Google ScholarShiran, Du, “Shilun Song Yuan shiqi Zhongguo he Isilan Guojia jian de shuxue jiaoliu”, in Baocong, Qian (ed.), Song Yuan Shuxueshi Lunwenji (Beijing, 1966): 241–65;Google ScholarBaocong, Qian, Zhongguo Shuxue Shi, 2nd edn (Beijing, 1981);Google Scholar and Shiran, Du, “Zai lun Zhongguo he Alabo Guojia jian de shuxue jiaoliu”, Ziran Kexueshi Yanjiu, 3 (1984): 299–303.Google Scholar

^{Page 3 note 10} See Luckey, , “Die Ausziehung”, pp. 217–18Google Scholar; Youschkevitch, Geschichte, p. 242; and for a more recent account Rashed, R., “Al-Samaw'al, al-Bīrūnī et Brahmagupta: les méthodes d'interpolation”, Arabic Sciences and Philosophy, 1 (1991): 101–60, footnote 3 on p. 102.CrossRefGoogle Scholar

^{Page 3 note 11} Hence I will concentrate on their textual properties, steering clear of the algorithms' relations to geometry, of questions bearing on their genesis, etc.

¹² Let me make clear that I do not read Arabic, and so have had to rely on such translations as I have been able to find. The reader will judge whether this deficiency on my part affected the conclusions which I draw.

¹³ The earliest document transcribing the representation of numbers by counting rods, found in Dunhuang and written some time during the Tang dynasty, is reproduced in Yan, Li and Shiran, Du, Chinese Mathematics: A Concise History, trans. Crossley, John N. and Lun, Anthony W.C. (Oxford, 1987), p. 15Google Scholar, in a chapter (pp. 11–19) devoted to a detailed explanation of what in such a computing device may well have been arithmetic.

¹⁴ From the 10th century onwards at least, there were discussions of how to transfer computations on a counting board to paper. Al-Uqlīdisī would appear (see The Arithmetic of al-Uqlīdisī. Translation with notes by Saidan, A. S. (Amsterdam, 1978))Google Scholar to have been the first Arabic author to make the attempt. The question of the support of computations will prove to be quite important in our discussion. Before the 13th century, as far as we know, no written computation is to be found anywhere in a Chinese text.

¹⁵ I have already devoted several papers to the theme of square-root and cube-root extractions in Chinese texts. I present here only a summary of my conclusions, as necessary for my argument. See Chemla, K., “L'aspect algorithmique récurrent dans les mathématiques chinoises: Paysages d'algorithmes, algorithmes de paysages”, in Dhombres, Jean (ed.), Cahiers d'Histoire et de Philosophie des Sciences, 20 (1987): 86–104Google Scholar and “Should they read FORTRAN as if it were English?”, Bulletin of Chinese Studies, 1 (1987): 301–16,Google Scholar where I have given translations and interpretations of texts drawn from the Nine Chapters and the Zhang Qiujian's Mathematical Treatise, based on the edition by Baocong, Qian, Suanjing shishu, 2 vols. (Beijing, 1963).Google Scholar See also “Qu'apporte la prise en compte du parallélisme dans l'étude de textes mathématiques chinois? Du travail de l'historie du travail”, Extrême-Orient, Extrême-Occident, 11 (1989): 53–80.CrossRefGoogle Scholar Here I shall merely describe the algorithms which will be found (in English translation) in Appendices A, B, C below.

¹⁶ No known text contemporary with the Nine Chapters describes such an algorithm, but I proposed in Chemla, “Qu'apporte”, how we might reconstruct it.

¹⁷ So giving the approximate value (a. 10ⁿ)+(A-a ².10²ⁿ)/2a.10²ⁿ in the case of square root and (a. 10ⁿ)+(A-a ³.10³ⁿ)/3a ².10³ⁿ, in that of cube root. On the other possible ways of stating the end-result proposed either by the Nine Chapters or by an important third-century commentator, Liu Hui, see Chemla, K., “Des nombres irrationnels en Chine entre le premier et le troisième siècles”, Revue d'histoire des sciences, XLV (1992): 135–40.CrossRefGoogle Scholar

¹⁸ This leaves us with two numbers, one above the other on the counting board; they are the two significant coefficients of an equation a root of which has, as its integral part, b, namely:

This is exactly the way quadratic equations appear in the Nine Chapters. On this, see Chemla, K., “Elaboration of coherence between procedures in three separate worlds” (Beijing, 1984).Google ScholarProceedings of the Third International Conference on the History of Chinese Science (to appear). It is important to note that, at this period, equations were neither conceived nor written in a positional way. All this holds only partially for the cube root. The final configuration can be taken as an approximate value or as the starting point for the next digit. But it is far from representing a cubic equation.

¹⁹ In the cube root case, in contrast with the upper part of the table, which behaves like the configuration evolving during a division algorithm, two lines of auxiliary numbers are necessary, the middle and the lower. When starting the second phase of the computation, we are asked to put, in the middle line, 3a.10ⁿ, and in the lower 1. Some shifts take 3a.10ⁿ to become 3a.10ⁿ.10^2(n-1) and 1 to become 10^3(n-1). These shifts are performed in parallel for the two lines, and they replicate the corresponding shift in the auxiliary line of the square root extraction. In the same way, most of the changes that will affect these lines will be described as parallel.

²⁰ See Qian, , Suanjing, II, 325–7.Google Scholar I will call them either after the book's name or after Liu Xiaosun's name.

²¹ Note that, contrary to the former description, the successive digits of the quotient seem to be put in different lines above the dividend. Although, as I will detail more precisely below, the algorithm presented by Liu is found without significant change in most of the texts we know up to the 11th century, this feature of placing the various digits of the quotient on different lines is not copied but is peculiar to Liu's description.

²² The Sunzi suanjing is edited in Qian, , Suanjing, II, 275–322.Google Scholar

²³ Two problems of the Sunzi's Mathematical Treatise composed before the Zhang Qiujian's Mathematical Treatise involve the extraction of a square root, namely Problems 19 and 20 of its middle chapter (See Qian, Suanjing, II, 301–2). Neither number is a perfect square, and the result is given as the sum of an integer and a fraction. In the first problem, the given approximate value, a+(A-a ²)/2a, differs from the one given by Zhang Qiujian. In the latter one, there is a textual problem: three statements of the numerical value of the denominator are made, one in the answer to the problem, the two others at the end of the procedure, and they differ in the same way according to all sources. In one case, the mention of the denominator implies the approximate value as a+(A-a ²)/2a, in the two others, it implies it as a+ (A-a ²)/(2a+1). Dai Zhen, and following him Qian Baocong, both “corrected” the two last occurences to get the same approximate value as in the first problem. But it seems to me that two different approximate values may be meant by the two problems.

²⁴ A brief description of its content and an account of the biography of the author (who lived in the second half of the 6th century) are found in Qian, , Zhongguo, pp. 91–4.Google Scholar An edition of the text is given in Qian, , Suanjing, II, 437–84Google Scholar; the algorithms for the square root are on pp. 448–9.

²⁵ On the complicated history of this book see Qian, , Zhongguo, p. 79Google Scholar, who dates it from the 8th century. The root extraction is on p. 569 of Qian's edition. Note that in this algorithm only the integral part of the root is given, without any consideration of its fractional part. On pp. 558–9, root-extractions are listed among different kinds of “division”.

²⁶ See Yan, Li, Zhongguo Shuxue, pp. 84–6;Google Scholar see also the calendrical part of the Jiu Tang shu in the Lidai tianwen lüli deng zhi huipian, 7, p. 2106. The algorithm here slightly diverges from the others in that both divisors are added before being eliminated. This has no substantial effect on the lay-out of the array, the computation, or the final approximation taken.

²⁷ The text of this lost book is quoted by Yang Hui in the Xugu zhaiqi suanfa, and has recently been translated in Lay-Yong, Lam, A Critical Study of the Yang Hui Suan Fa, A Thirteenth-century Chinese Mathematical Treatise (Singapore, 1977), pp. 177–8.Google Scholar The text for the square root is found in Yongle Dadian, Chapter 16344, p. 12.

²⁸ See Shuchun, Guo, “Jia Xian Huangdi jiuzhang suanjing xicao chu tan. Xiangjie jiuzhang suanfa jiegou shixi”, Ziran Kexueshi Yanjiu, 7 (1988): 328–34Google Scholar for an attempt to reconstruct parts of his commentary. Shuchun, Guo, “Jia Xian de shuxue chengjiu”, Ziran Bianzhengfa Tongxun, 11 (1989): 53–61Google Scholar, describes his contributions to mathematics.

²⁹ Qian, , Zhongguo, pp. 144–54Google Scholar argues that his writings were composed during 1023–1050, while Baocong, Qian, “zengcheng kaifangfa de lishi fazhan”, in Qian, (ed.), Song Yuan: 36–59Google Scholar, describes the sources on whose basis we know something of his works. See also Yan, Li, Zhongguo Shuxue, pp. 165–85.Google Scholar

³⁰ I have set these together in Appendix C. The first method is instanced in the extraction of a cube root, the second one by finding a cube root and also a fourth root. We shall first describe those methods before dealing with the question of whether they can be attributed to Jia Xian. For the sake of convenience, I name them after him, since, as we shall see, he is the first known author whose name is associated with them.

³¹ The result is not kept in a specific line but directly added to or subtracted from the line immediately above the one which is multiplied. Note that this reproduces the basic step of multiplication or division: the line underneath is multiplied by the digit of the multiplier or of the quotient, so as to be respectively added to or subtracted from a middle line, the line of the result, which is the one just above it in this case. This remark has important consequences, which I will develop later.

³² For a mathematical description of the algorithm, see Libbrecht, Ulrich, Chinese Mathematics in the Thirteenth Century, the Shu-shu chiu-chang of Ch'in Chiu-shao (Boston, 1973), pp. 177–211Google Scholar or Rongzhao, Mei, “Jia Xian de zengchengkaifangfa”, Ziran Kexueshi Yanjiu, 8 (1989): 1–8.Google Scholar

³³ See Qian, , “zengcheng”, p. 38.Google Scholar As for the root that is found, I am not aware of any text ascribed to Jia Xian where the result is other than an integer. Yet in all subsequent texts that use this procedure to deal with any kind of equations, the algorithm finishes with a three-line array in which the lowest row, viz, the denominator of the fractional part of the root, comes from adding “all divisors”. Again, if we compare the procedures described either in the Nine Chapters, Zhang Qiujian's Mathematical Treatise, or in the works of Jia Xian's followers, the terminal approximation is obtained by an identical use of the final three-line configuration, adding on all the divisors remaining on the table. Thus the algorithms themselves are intimately connected with the setting chosen and with the divisors extant on the surface when we need either to get the next digit or to obtain a last approximation to the root. Note that the approximation obtained in this way in Jia Xian's algorithm produces, uniformly for all root extractions, the following value:

that is a+(A-a ⁿ)/[(a+1)ⁿ-a ⁿ], when the integral portion of the root is a.

³⁴ It is not wholly clear from Yang Hui's text whether Jia Xian himself described this algorithm for the fourth-root extraction. From the analysis of the Xiangjie Jiuzhang suanfa proposed in Guo, “Jia Xian Huangdi”, we may believe so. There is an additional argument for associating Jia Xian's name with this extension, as I will show below.

³⁵ The top row is multiplied by a lower row to be added (multiplication) or subtracted (division) to a middle row. No other computation, no coefficient, are necessary.

³⁶ This conclusion departs from the one given in Ling, Wang and Needham, Joseph, “Horner's method in Chinese mathematics: its origin in the root-extraction procedures of the Han dynasty”, T'oung Pao, 43 (1955): 345–401Google Scholar, since these authors described the procedure found in the Nine Chapters as a Ruffini-Horner procedure.

³⁷ Yang Hui quotes two algorithms to perform root-extractions: this is, as far as I know, the first extant Chinese document in which this is performed in two different ways, both named by qualifying the general term for “root extraction” (kaifangfa). The name of this one, licheng shisuo pingfangfa or lifangfa, is understood by Qian, “zengcheng”, pp. 37–8, as “square- (resp. cube-) root extraction with the help of a table of numbers”. Qian Baocong supposes that this refers to the triangle of binomial coefficients. I will not discuss this point here. Whatever the case, this algorithm uses these coefficients. The name of the other one, analysed above, zengcheng pingfangfa or lifangfa (“square- (resp. cube-) root extraction by addition and multiplication”) probably indicates its basic components.

³⁸ Even though this algorithm is identical to Liu's (see, Mei, “Jia Xian”, p. 4), Yang Hui explicitly attributes it to Jia Xian in the commentary to Chapter 5 and again in the Compilation where its title is preceded by Jia Xian's name (see Appendix C). Does Yang Hui refer to Jia Xian's general formulation – a characteristic feature of Jia Xian's procedures, see Guo, , “Jia Xian de shuxue chengjiu”, pp. 56–8-,Google Scholar or does this mean that Jia Xian extended it to extract roots of higher order? It is difficult to conclude on this point. In the Compilation…, the method by addition and multiplication comes next, as if both procedures were belonging to the same text, since they share the same first two steps (see Appendix C).

³⁹ See the text in Appendix E. The chapter dealing with root extractions from which it comes is lost, but parts of it are preserved in the encyclopedia Yongle Dadian (Chapter 16344, pp. 5–6). For a presentation of both methods and a description of the triangle of binomial coefficients one may consult Lay-Yong, Lam, “The Chinese connection between the Pascal triangle and the solution of numerical equations of any degree”, Historia Mathematica, 7 (1980): 407–24.CrossRefGoogle Scholar

⁴⁰ Or perhaps, what amounts to the same: “Origin of the procedure of root extractions”, “Origin of how root extraction produces the divisors”.

⁴¹ This term is the same one as that given to the method of root extraction ascribed to Jia Xian (see above), namely the method using binomial coefficients.

⁴² Between the two “1”, from left to right, we find the numbers which appear from bottom to top, between the “corner” and the “number-product”, in the usual presentation of the root-extraction associated with the line.

⁴³ Or perhaps “method for finding the sides (lian) through extracting by addition and multiplication”.

⁴⁴ The successive rows of the table are in fact obtained in the set-up of the root extraction, viz, in a set of superimposed lines. The line n is generated in a column with n positions. I owe to A. Honuchi the remark that this differs from the way in which the numbers are actually arranged in the table. For a detailed description of the lay-outs mentioned in the previous quotation, see Qian, “zengcheng”.

⁴⁵ If we observe not only the final array, but the successive states of the columns described explicitly (see appendix E), the whole triangle appears to hav been generated, even though in another position (See Mei, “Jia Xian”, p. 6). The n ^th root extraction thus involves the coefficients put into play by all root-extractions of lesser degree. The table can be seen as either containing all the transient numbers that appear in the root extraction corresponding to its last row, or containing the coefficients of all root-extractions till the n ^th one. It is again what remains constant through the changes.

⁴⁶ The structure of this sentence mirrors Yang Hui's commentary quoted above, which yields the same alternative understandings.

⁴⁷ Or, in terms of a vertical lay-out of the computations, as the sum of the numbers contained in the column under it. This way of generating the triangle of binomial coefficients is different from the one in al-Karajī's text, as quoted by al-Samaw'al (see Rashed, R., “L'induction mathématique: al-Karajī, al-Samaw'al”, Archive for History of Exact Sciences, 9 (1972): 1–21).CrossRefGoogle Scholar There it is built out of the classical relationship

Yet, as Qian, , Zhongguo, p. 154 and again pp. 221–4Google Scholar, stresses, both ways of generating the table are met with in al-Kāshī's Key to Arithmetic (compare Luckey, “Die Ausziehung”, pp. 267–73).

⁴⁸ That al-Karajī gives another mode of generation of this triangle also has to do with his own aim in constructing the triangle, namely to compute (a+b)ⁿ, which he does by induction, deducing the expansion of (a+b)ⁿ from that of (a+b)ⁿ⁻¹.

⁴⁹ Allard, André, “The Arabic origins and development of Latin algorisms in the twelfth century”, Arabic Sciences and Philosophy, 1 (1991): 233–83CrossRefGoogle Scholar, uses the same method to analyse the diffusion of al-Khwārizmī's book on arithmetic in medieval Latin texts.

⁵⁰ This slight transformation was inspired by structuralist texts by R. Jakobson or C. Lévi-Strauss.

⁵¹ The idea presented in this paragraph has been developed in Chemla, “Elaboration of coherence”. I do not mean to assert here that there are only two such traditions.

⁵² See Rashed, , “Résolution”, pp. 250–1Google Scholar and Saidan, , The Arithmetic, pp. 437–67.Google Scholar R. Rashed, “L'extraction des racines carrées et cubiques”, in idem (ed.), La science arabe. Développements et prolongements latin et hébraïque (Paris, forthcoming), presents the set of Arabic texts concerning square and cube root extractions currently available today: approximations, algorithms and their proofs.

⁵³ On this 10th century mathematician see Saidan, The Arithmetic. A translation of the algorithm which his Kitāb al-fuṣūl fi al-ḥisāb al-hindī gives for a square-root extraction is given on pp. 76–9.

⁵⁴ As we know it, this is the book whose diffusion Allard, “The Arabic origins”, analyses with the help of the method just described.

⁵⁵ M. Levey and M. Petruck, in Kūshyār ibn Labbān. Principles of Hindu Reckoning. A Translation with introduction and notes (Madison, Wisconsin, 1965), give a translation of his Principles of Hindu Reckoning. In Appendix D I reproduce the square and cube root algorithms there presented. On the life and achievements of this scientist, who lived around 1000, see Saidan, A. S., “Kushyār ibn Labbān ibn Bāshahrī, Abū 'l-Ḥasan, al-Jīlī”, Dictionary of Scientific Biography (1973), vol. VII, pp. 531–3.Google Scholar

⁵⁶ In my argument, Kūshyār is less important for being the first to give an account of this algorithm than for recording an algorithm whose textual description can be contrasted with al-Uqlīdisī's: one whose description of the lay-out of the array was followed in substance by some subsequent Arabic mathematicians. I here use names as tags to identify the types of the algorithms and without primary intention to specify the original contrivers of these. The kind of result which I am pursuing should not be dependent on identifying the author of the first Arabic text to use this or that subset of algorithms.

⁵⁷ See Saidan, , The Arithmetic, Book II, chapter 18.Google Scholar

⁵⁸ On this kind of number in Chinese, Indian and Arabic texts see Chemla, Karine, Djebbar, Ahmed, and Mazars, Guy, “Quelques points communs dans des textes arabes, chinois, indiens”, in Benoit, P., Chemla, K., and Ritter, J. (ed.), Histoire de fractions, fractions d'histoire (Basel, 1992), pp. 262–76.Google Scholar

⁵⁹ See Datta, B. and Singh, A. N., History of Hindu Mathematics, 2nd edn (Bombay, 1962), pp. 169–80;Google ScholarShukla, K. S., The Patiganita of Sridharacarya with an Ancient Sanskrit Commentary, ed. with an Introduction, English Translation with Notes (Lucknow, 1959), pp. 9–14;Google ScholarPrakash, Satya, A Critical Study of Brahmagupta and his Works (New Delhi, 1986), pp. 166–72;Google ScholarBag, A. K., Mathematics in Ancient and Medieval India (Delhi, 1979), pp. 78–81;Google ScholarHayashi, Takao, The Bakhshali Manuscript Medieval Ph. D. Thesis (Brown University, 1985), pp. 179–80.Google Scholar The only variation such texts present is that the bottom line sometimes carries just the root and sometimes its double. Such fractional approximations to square roots are found in Indian texts, but they are not directly appended to computing roots in a place-value system of numbers.

⁶⁰ Allard, “The Arabic origins” presents a detailed and up-to-date analysis of the possible relationships of al-Khwārizmī's Arithmetic to the 12th century Latin treatises on the same topic. He discusses the other sources which also were drawn upon in the latter and then assesses with what confidence we may use them to reconstruct al-Khwārizmī's original book. The reader will find all the relevant bibliography in his paper.

⁶¹ It is in this tradition that we may put authors like al-Ḥaṣṣār; compare his algorithm for square-root extraction in Suter, H., “Das Rechenbuch des Abû Zakarîjâ el-Hassâr”, Bibliotheca Mathematica, Third Series, 2 (1901): 12–40, pp. 21–3.Google Scholar

⁶² I should make it clear that if the result in the square-root extraction is the final configuration, in the cube root extraction it is the three upper rows of the final configuration, once the row below has been used to build up the denominator. Kūshyār does not use the final configuration stricto sensu. The fact that the setting of the algorithm is changing in such a way that the result appears as a part of it, is the point which we take to be a possible hint of a link with Chinese texts. Still the way the approximate value is obtained from the table of numbers is different. If we look at the Chinese algorithms which terminate in an approximation to the value, for most of them, the denominator of the fraction is obtained as the sum of all divisors left on the table when the computation is over. This leads to differing values according to the setting. In Kūshyār's algorithms the denominator of the approximations is produced by the same procedure as the one which prepares the line below the number whose root is to be extracted. But, as it is clear in the case of the cube root, not all the lines below the number enter the denominator. The ensuing approximation to the root is a+(A-a ³)/(3a ²+1), the same as in such algorithms as Liu's. The additional 1 in the denominator, which is effected by the borrowed row in Chinese texts, is introduced by Kūshyār as an alien element. Saidan, A. S., “al-Nasawī, Abū 'l-Ḥasan, ‘Alī ibn Aḥmad”, Dictionary of Scientific Biography (1974), vol. IX, pp. 614–5Google Scholar, notices that better approximations, such as a+(A-a ³)/(3a ²+3a+1), that is a+(A-a ³)/((a+1)³-a ³), were available in contemporary Arabic mathematical texts. This formula yields mutatis mutandis the same approximation in the case of square-root extraction, but they diverge from cube root onwards. This led Suter, Heinrich, “Über das Rechenbuch des Alî ben Ahmed el-Nasawî”, Bibliotheca Mathematica, Third Series, 7 (1906–1907): 113–19Google Scholar, to modify al-Nasawī's text (see below) in order to get this approximation. Luckey, , “Die Ausziehung”, p. 264Google Scholar, criticizes this on the basis of all extant manuscripts, If Kūshyār had added all lines below the number, it would indeed produce this approximation. Nevertheless his account does have rather the property which I stress: the same procedure produces both the denominator in the approximation and a line for further computation of a digit. Approximations of the type a+(A-a ⁿ)/((a+1)ⁿ-a ⁿ) do in fact appear together with “Ruffini-Horner” algorithms in our sense, but let me come back to this point below.

⁶³ On the life and career of this Persian student of Kūshyār see Saidan, “Al-Nasawī”. Suter, “Über das Rechenbuch”, is mainly devoted to his algorithms for square- and cube-root extractions and also translates numerical examples given by al-Nasawī. Luckey, , “Die Ausziehung”, pp. 245–55Google Scholar provides a more complete translation of the text for the cube-root extraction, since he includes al-Nasawī's general description of the algorithms, which appears to be very close to Kūshyā's, even to the extent that the places where some tables of numbers are included in the text are the same as those in Kūshyār. I also note in Appendix D that a missing part of Kūshyār's text can be recovered from the corresponding portion of al-Nasawī's. Moreover the latter makes it clear in his account of the general procedure what the final approximation shall be, whereas Kūshyār does not include this point to the general statement of the algorithm. It appears then that the result has actually to be taken from the final configuration. We should, however, treat with caution the text of al-Nasawī's statement of his general algorithm as given by Luckey, , “Die Ausziehung”, p. 249Google Scholar, since some steps are missing. The omission might be Luckey's, since he makes no comment that his account is incomplete. The corresponding part is not missing in the text of the ensuing, examples, as translated either by him (p. 250) or by Suter, “Über das Rechenbuch”. Moreover, the missing part can be found in Kūshyār's text. It seems that Luckey, p. 251, further omits from his translation the addition of 1 to the middle line in preparation for the final approximation, since this last step is described in the general procedure and is given in Suter's translation. Although al-Nasawī's text is indeed very close to Kūshyār's, there are some systematic differences that I discuss below.

⁶⁴ This indeed fits in with what we know of the historical contacts between India and the Arabic-speaking world in the medieval period.

⁶⁵ R. Rashed pointed out during the seminar I gave that this might have come from China by way of an Indian text. This is certainly possible, and it would be very interesting to find hints of such a second arithmetical tradition in Indian mathematical texts. Indeed, if there is a historical connection between the Chinese tradition and that in which Kūshyār should be placed, there are problems One is with the title of Kūshyār's book Principks of Indian Reckoning. What does the term “Indian” refer to? Does it mean that he found the method in an Indian text? Or does it signify rather that computations are made in the “Indian reckoning” in the decimal place- value system?

⁶⁶ In Chemla, “Elaboration of Coherence”, I analysed the system of roles played by the borrowed rod in Liu's algorithms, contrasting this with their equivalent in setting 10²ⁿ or 10³ⁿ and deriving the final approximation in Kūshyār's algorithms. Notice here that Kūshyār's settings are yet closer to a positional notation than al-Uqlīdisī's, as the cube-root extraction goes through such arrays as

⁶⁷ To be precise, the general procedure is not formulated in the optimal terms for this first part to be re-usable for each phase. Let me explain what I mean by considering only the numerical values of the entries in the rows, and not their positions. In the line below we might find either a (in the first phase) or 3a+b (in the next one). This has to become a ² in the first case and (3a+b)b in the second one in order for it to be added to the middle line. The most general formulation of this, “multiply the uppermost by the lowest”, is the one actually used in the worked example. But the prescription of the general procedure “multiply it by itself”, which gets a ² at the initial step, cannot yield (3a+b)b at the second one. We can see here the algebraic relevance of choice of formulation. A sentence may refer to one or more differing computations depending on its formulation. A close watch on this at a time when no algebraic symbolism was used needs to be kept. In the case outlined above, the importance of making use of the general pattern is clear: This is what brings us close to the Ruffini-Horner algorithm here; and it depends on how the uppermost row is described to be used.

Note, too, that the step in the general procedure which ordains “double the number in its lowest position”, in the first phase, means that we are to double a to be 2a, but in the second one it means that we should pass from 3a+b to 3a+2b. Although this way of expressing the first and the following steps is being given as the same, there is a departure here from the “Rufilni-Horner” procedure in that the top row is not implied in the description.

⁶⁸ I return to al-Kāshī's Key to Arithmetic below; in it, by my definition a Ruffini-Horner procedure is indeed used.

⁶⁹ Indeed al-Nasawī's way of describing the algorithm undoes the homogeneity displayed in Kūshyā's formulation. The main reason for this is that he does not use the general formulation bringing into play the top row, but only particular computations involving the numbers put in the lower rows. His description of the first phase in the general procedure makes use of the same formulation as Kūshyār's (compare above): “multiply it by itself” here means from a get a ². But in his example, he goes on using the same formulation, which compels him to give a different description of what needs to be done in getting from (3a+b) to (3a+b)b: “multiply this (b) with the three below (3a) and with itself”, so yielding 3ab+b ².

⁷⁰ Luckey, , “Die Ausziehung”, p. 235Google Scholar, did reveal that the procedures for choosing the successive digits given by al-Nasawī (and Kūshyār, in fact), on the one hand, and by al-Kāshī on the other differed. Since the Chinese texts as we now know them do not make this precise, I have not commented here on this point. Luckey's wise remarks about tracking wrong methods as they were adduced is another way in which we may distinguish traditions. We see that again they oppose Kūshyār and al-Nasawī, on the one hand, and al-Kāshī on the other, as the final appended approximation does (see below). All this supports my contention that these methods are not identical. We do know that the question of how to choose the successive digits was carried further by Arabic authors (see Rashed, , “Résolution”, pp. 262–4Google Scholar and “L'extraction”, pp. 208–13).

⁷¹ An account of al-Samaw'al's life is given in Anbouba, A., “Al-Samaw'al, Ibn Yaḥyā al-Maghribī”, Dictionary of Scientific Biography (1975), vol. XII, pp. 91–5.Google Scholar On his mathematical treatise al-Bāhir, see Ahmad, S. and Rashed, R., al-Bāhir en algèbre d'As-Samaw'al (Damas 1972).Google Scholar The present algorithm for root extraction is set down in a treatise on arithmetic he wrote two years before his death in 1174, a part of which has been identified by Rashed who (see Rashed, “L'extraction”, pp. 198–207) describes n-th root algorithm in painting in the background to the first systematic treatment of decimal numbers which al-Samaw'al also gives. I follow this account in my presentation of al-Samaw'al's procedure for extracting a fifth root.

⁷² See Rashed, , “L'extraction”, p. 199.Google Scholar

⁷³ For a mathematical description of these equations see Rashed, , “L'extraction”, pp. 202 ff.Google Scholar I take their standard form to be

⁷⁴ There is however one difference: these digits are repeated in the same positions in the bottom row, where they are used to compose the coefficients.

⁷⁵ See for instance Libbrecht, Chinese Mathematics. Al-Samaw'al in his al-Bāhir does in fact consider polynomials written in a positional way (see Ahmad and Rashed, al-Bāhir, Introduction, pp. 12–36), but they are written horizontally (see Rashed, “L'extraction”, p. 223). Hence, this positional notation for equations as it appeared on the array to which the algorithm is applied is divorced from the positional writing of polynomials as described by al-Samaw'al. In China the two positional notations had been closely connected both in the case of equations (by extant 12th century texts) and polynomials (in texts of the 13th century, though these mention previous ones now lost). On p. 228 for instance, Rashed mentions the link of this representation with al-Samaw'al's specific treatment of decimal numbers.

⁷⁶ See Rashed, , “L'extraction”, p. 200.Google Scholar Note that the first digit is 6.

⁷⁷ In order not to get too tangled in details that are of no importance here, I replace al-Samaw'al's particular numbers by their algebraic equivalent in a and its powers. Moreover I only quote the first part of the procedure, the following ones sharing a similar structure. The full original text is described and translated in Rashed, “L'extraction”, p. 202. To fill up the other lines of the table, al-Samaw'al uses respectively the same procedure.

⁷⁸ This table supposes that we work on paper and not on a dust-board, ensuring that all stages in the computation are kept. This leads to an organisation in a triangular array, and not in a column.

⁷⁹ See Rashed, , “L'induction”, p. 7.Google Scholar Here I give only an excerpt from it. Its mode of generation, as I have earlier remarked, follows the rule

differing from the mode of formation employed in al-Samaw'al's root extraction. We have seen that Jia Xian's algorithm is also correlated with his description of the generation of the triangle. I have mentioned that al-Kāshī, who for both decimal and sexagesimal place-value systems gives the same algorithm as al-Samaw'al for performing the computation only in a column on paper (see Luckey, “Die Ausziehung”, pp. 239–45), alludes to both modes of generation.

⁸⁰ It seems to me, however, not impossible that he in fact did this. This would lead to a slightly different reconstruction of the setting for the part of the procedure which he does not tell us explicitly. An argument in favour of this is that when al-Samaw'al refers to the fourteen computations needing to be performed to fill up the rows of the next underlying equation, he seems to include the computations of the powers of a. But if we look at the second table, these numbers are arranged in a row, notwithstanding that he specifies them to be in a column in the part he describes explicitly. So it may be that the computations were arranged in such a way that the transformations were performed column by column rather than line by line. This would involve turning the second type of array 90^° anticlockwise. The table would thus be presented so as to have the successive transformations of the column in successive columns, as follows:

Moreover such a lay-out would make the computational scheme akin to those described by Kūshyār and al-Nasawī. Thus not only would the way of positioning the digits in the first line and the first disposition of the computations be common, but also the whole setting. The difference between the two kinds of algorithm would then lie only in the way in which the computation is described, al-Samaw'al's procedure would be a true “Ruffini-Horner” in my definition of it. Finally, this lay-out would make it closer to al-Kāshī's.

⁸¹ If the lay-out is on the two kinds of tables alternately, the positional setting of the underlying equations will of course be correspondingly alternately horizontal and vertical.

⁸² In all these ways, al-Samaw'al's procedure recalls al-Kāshī's; compare Rashed, , “L'extraction”, pp. 207–9.Google Scholar The main difference between them would seem to be that al-Samaw'al apparently refers to a computation to be performed on an erasable surface, whereas al-Kāshī describes a computation wholly preserved on paper. Luckey's description of the transformation of procedures from one medium to the other is illuminating, but my argument here leads to conclude somewhat differently that the procedure on an erasable surface corresponding to the one al-Kāshī describes for paper is not al-Nasawī's but al-Samaw'al's. Furthermore, the truncated positional notation, the positioning of the digits in the top row, the final approximations – on which see below –, all connect al-Samaw'al's and al-Kāshī's procedures. Luckey insists that the results given by al-Kāshī's procedure are all of form

as they appear in the final array, with a denominator produced as the sum of all lines below to what 1 is added. I agree with Du, “Shilun”, pp. 255–9, when in his analysis of al-Kāshī's method, he gives it to be identical to Jia Xian's. The same conclusion is also to be found in Qian, Zhongguo, p. 221, and restated in Du, “Zai lun”.

⁸³ A strong argument in support of this is that all these texts share a number of common features in their setting.

⁸⁴ See Rashed, , “L'extraction”, pp. 213–16.Google Scholar

⁸⁵ Note that 1 is added as in Kūshyā's procedure. In Chinese texts this was done by adding on the borrowed rod. I will discuss the occurrence of this approximation in Chinese texts in the second part of this paper. In the example given, however, (see Rashed, “L'extraction”, p. 214), the denominator ensues not as the total sum of the rows but from a computation mixing together powers of a and their binomial coefficients. I have not seen enough of his text to determine whether such approximations are appended to al-Samaw'al's algorithms.

⁸⁶ Not to mention such a systematic display of the result in decimal numbers as al-Samaw'al proposes. We find it in Chinese texts, but this is not done systematically.

⁸⁷ Hence I agree with conclusion of Rashed, , “L'extraction”, pp. 198–213Google Scholar, on al-Samaw'al's text.

^{Page 265 note 1} I transcribe the text in bold letters, Yang Hui's commentary in plain ones.

^{Page 265 note 2} Whereas it is more frequent in the Song Yuan period to find fa (method) or cao (detailed commentary) when introducing an algorithm or details about its operation working, Yang Hui here uses the term shu (procedure) that we find in older texts like the Nine Chapters or the Zhang Qiujian's Mathematical Treatise.

^{Page 265 note 3} This term Ji refers to a number which has been obtained through a product and hence to the measure of an area, a volume, etc. It is often used to designate the number whose root is sought (see procedures above).

^{Page 265 note 4} Yu refers to the bottom divisor on the counting surface.

^{Page 265 note 5} We have already seen this term, used passim to designate divisors in higher-root extractions; compare the third method in Appendix C for extracting a fourth root.

^{Page 265 note 6} fang can designate square or cube, but in any case it refers to the first divisor put in the line immediately below the dividend. It seems that lian refers to any line below, fang (be it square, cube or whatsoever) to a line above. The entire algorithm is here alluded to in a single sentence which indicates that a line below has to be multiplied by the quotient to make a line above: the key step of the method “by addition and multiplication”. This way of designating the whole by its part is common in Chinese texts.

^{Page 265 note 7} We follow the edition in Qian, Zhongguo, pp. 150–1, which is closer to the original text. Qian has there changed his mind from what he earlier wrote in Qian, “zengcheng”, regarding the completeness of the text, a point I will not discuss here.

^{Page 266 note 8} Here “position”, wei, refers to the various lines of the array on which the algorithm is worked. The counting rod of the corner gives the coefficient of highest degree in the underlying equation. Yang Hui specifies what has to be done in the instance of a sixth-root extraction: the rod is the last line of the table, and we have to leave five rows empty between the number and the rod. These are related to the number of positions in the corresponding row of the table of binomial coefficients.

^{Page 266 note 9} The term used to express addition carries with it the connotation that this be to an empty line. The “previous positions” are above, in agreement with Chinese way of writing from top of bottom.

^{Page 266 note 10} We get in this way the second oblique line of the table, if we start from the right. All numbers in this triangular table will be generated as sums starting from the far right oblique, which figurates the counting rod of the corner below. The table is bringing together all the coefficients needed for any root-extraction, and its generation shows their close interdependency.

^{Page 266 note 11} 10 is the sum of the coefficients below. The commentary makes precise for each number which number is to be added to it by the procedure of addition and multiplication in order to get the new column of coefficients described in the following phase. In the case of this 10, this is the last change to which the number in this position is submitted, since later the changes only affect lower rows.