In the absence of the Arabic text of al-Khwārizmī's Arithmetic (ca. 825), which has not yet been found, the oldest Latin adaptations from the twelfth century are the only evidence documenting the genesis and the first spreading of a decimal arithmetic that uses nine figures and zero, i.e. the Indian reckoning known in the Middle Ages as algorismus. This paper studies these texts, their content, their sources, and identifies their authors and the milieus in which they were written.
En l'absence du texte arabe de l'Arithmétique d'al-Khwārizmī (ca. 825), jusqu'ici introuvable, les plus anciennes versions latines remaniées du XIe siècle constituent notre seule source de connaissance de la genèse et de la première diffusion d'une arithmétique de position utilisant neuf chiffres et le zéro, le ‘calcul indien’ connu du Moyen Age sous le nom d'algorismus. Le présent article analyse ces textes, leur contenu, leurs sources, et en détermine les auteurs ou les milieux dans lesquels ils furent élaborés.
1 See further on the erroneous identification of this author with the translator John of Seville (Iohannes Hispalensis).
2 Boncompagni, B., Iohannis Hispalensis liber algorismi de pratica arismetrice, Trattati d'aritmetica 2 (Roma, 1857), p. 118.
3 Boncompagni, B., Il Liber Abbaci, Scritti di Leonardo Pisano 1 (Roma, 1857), p. 177.
4 We don't believe however that it is by means of counters of abacus that “Hindu-Arabic” numerals spread in the West but rather by means of the manuscripts about the Indian reckoning. See on that subject Allard, A., “L'époque d'Adélard de Bath et les chiffres arabes dans les manuscrits latins d'arithmétique,” in Burnett, C. (ed.) Adelard of Bath, an English scientist and Arabist of the early twelfth century, The Warburg Institute, Surveys and Texts 14 (London, 1987), pp. 37–43, and Beaujouan, G., “Etude palèographique sur la ‘rotation’ des chiffres et l'emploi des apices du Xe au XIIe siècle,” Revue d'Histoire des Sciences, 1 (1948): 301–13. It should be noted that Gerbert mentions twice an opuscule now lost, De multiplicatione et diuisione, composed by Joseph the Wise (Josephus Hispanus). The work probably contented itself with describing the two most difficult operations performed with an abacus.
5 About this author's true name and his role in the beginnings of algebra, see Rashed, R., Entre arithmétique et algèbre. Recherches sur l'histoire des mathématiques arabes (Paris, 1984), pp. 17–29.
6 Some elements in Algebra follow after the Indian reckoning in John of Toledo's work. Cf. Boncompagni, Iohannis Hispalensis liber algorismi, pp. 93–136. See also recent editions: Hugues, B.B., “Gerard of Cremona's translation of al-Khwārizmī's Al-Jabr: a critical edition,” Mediaeval Studies, 48 (1986): 211–63;Hugues, B.B., Robert of Chester's Latin translation of al-Khwārizmī's Al-Jabr, Boethius 14 (Stuttgart, 1989);Kaunzner, W., “Die lateinische Algebra in Ms Lyell 52 der Bodleian Library Oxford, früher Ms Admont 612,” Sitzungsber. d. Öster. Akad. d. Wis., Phil.-Hist. Kl., 375 (1986), pp. 47–89.
7 This interpolation, already underlined by Suter, is now unanimously accepted. See, for example, Juschkewitsch, A.P., “Ueber ein Werk des Abū 'Abdallah Muhammad ibn Mūsā al-Huwārizmī al Maĝusī zur Arithmetik der Inder,” in Schriftenreihe f. Gesch. d. Naturwis. Technik u. Medizin, Beiheft z.60 Geburtstag v.G. Harig (Leipzig, 1964), pp. 21–63, esp. p. 24, n. 14.
8 Ruska, J., “Zur ältesten arabischen Algebra und Rechenkunst,” Sitzungsber. d. Heidelb. Akad. d. Wis., Phil.-Hist. Kl., 2 (1917), pp. 18–19.
9 Cf. Rashed, Entre arithmétique et algèbre, p. 19, n. 6.
10 See further on the passage of the introduction of Dixit Algorizmi… about al-Khwārizmī's Algebra.
11 Among others, Ibn Labbān's and al-Uqlīdisī's now well-known works are to be mentioned. About the foundation of Arabic arithmetic, see Rashed, Entre arithmétique et algèbre, pp. 63–67.
12 Allard, A., “The influence of Arabic mathematics in medieval West,” in Arabic Science (London, 1991) (to press).
13 This comparison was first made by Reinaud, M., “Mémoire géographique, historique et scientifique sur l'Inde, antérieurement au milieu du XIe siècle de l'ère chrétienne, d'après les écrivains arabes, persans et chinois,” Mémoires Ac. Inscr. et Belles Lettres, 18 (1849).
14 The two hundred-odd known manuscripts and several editions printed between 1488 and 1582 of John of Sacrobosco's Algorismus Vulgaris (beginning of the 13th century) testify, for example, this spreading. See also Vogel, K., Die Practica des Algorismus Ratisbonensis (München, 1954);Tropfke, J., Geschichte der Elementarmathematik. Arithmetik und Algebra, rev. Vogel, K., Reich, K., Gericke, H. (Berlin and New York, 1980);Allard, A., Muhammad ibn Mūsā al-Khwārizmī Le Calcul Indien. Edition critique, traduction et commentaire des versions latines remaniées du XIIe siècle (Tunis and Namur, 1991) (to press).
15 For the convenience of this account, the works are referred to by their incipit or their shortened title and more often by an abbreviation. The references of the editions are given later on.
16 Boncompagni, B., Algoritmi de Numero Indorum, Trattati d'aritmetica 1 (Roma, 1857);Vogel, K., Mohammed ibn Musa Alchwarizmi's Algorismus. Das früheste Lehrbuch zum rechnen mit indische Ziffern (Aalen, 1963); Juschkewitsch, “Uber ein Werk des al-Ḫuwārizmī”; Allard, Le Calcul Indien, pp. 1–22.
17 Rashed, Entre arithmétique et algèbre, p. 20, after Juschkewitsch, “Ueber ein Werk des al-Huwārizmī,” p. 25. Also Toomer, G.J., “Al-Khwārizmī, Abū Ja'far Muhammad ibn Mūsā,” Dict. Sc. Biography (New York, 1973), vol. VII, pp 358–65, esp. p. 360; the author notes however that the translation “is probably much altered from the original.”
18 About the formation of Latin vocabulary, see Allard, A., “La formation du vocabulaire latin de l'arithmétique médiévale” in Weijers, O. (ed.) Méthodes et Instruments du travail intellectuel au Moyen Age (Turnhout, 1990), vol. III, pp. 137–81. The Arabic origin revealed by some expressions in DA was already underlined by Woepcke, F., Sur l'introduction de l'arithmétique indienne en Occident et sur deux documents importants publiés par le Prince Don Balthasar Boncompagni et relatifs à ce point de l'Histoire des Sciences (Roma, 1859), pp. 18–19.
19 Allard, Le Calcul Indien, pp. 1, 1 and 12; 2, 11.
20 Ibid., p. 1, 1–5.
21 Ibid., p. 1, 12; 2, 23.
22 See further on our chapter about the fundamental operations of Latin algorisms.
23 Allard, Le Calcul Indien, p. 8, 30–31. We'll see however that the authenticity of this example is questionable.
24 Allard, Le Calcul Indien, p. 1, 28–36.
25 Hugues, “Gerard of Cremona's translation,” p. 233, 6–7.
26 We correct quite considerably Hugues, Robert of Chester's Latin translation, p. 29, 8–10, which maintained, for the translation of Algebra, an interpolation about the definition of the unit due to Johann Scheubel and obviously inspired by a Latin translation of Euclid's Elements.
27 The pages mentioned about the examples are those of our edition: Allard, Le Calcul Indien.
28 Allard, Le Calcul Indien, pp. 15–16.
29 Ibid., pp. 150–63.
30 Ibid., p. 163.
31 The word uices indicate however, in Fibonacci's Liber abaci, the multiplication of integers. Thus, 7 uices 7 fiunt 49 (“7 successions of 7 are 49” or “7 multiplied by 7 is 49”). LA and LP use the word about the succession of seasons: Allard, Le Calcul Indien, p. 62, 15–16.
32 Ibid., p. 6, 20–33.
33 This order of operations is not common to all the Latin texts studied. The same variety can be observed in the Arabic works.
34 It is an imaginary title, “Algoritmi de numero Indorum,” which is a free translation from Arabic and which is sometimes accepted so by some authors. As a precaution we prefer to refer to it by its incipit.
35 According to R. Thomson's current research on the Cambridge manuscripts.
36 Juschkewitsch, “Ueber ein Werk des al-Ḫuwārizmī,” pp. 22–23.
37 We could also mention successively, but without proof, all the known translators of Arabic into Latin in the 12th century.
38 Chasles, M., “Aperçu historique sur l'origine et le développement des méthodes en géométrie,” Mém. Acad. Roy. Sc. et Belles Lettres de Bruxelles, 11 (1857), pp.510–11.
39 It is the Latin manuscript 16208 of the Bibliothèque Nationale of Paris. Cf. Libri, G., Histoire des sciences mathématiques en Italie (Paris, 1838), vol. I, pp. 47 and 298.
40 Nagl, A., “Ueber eine Algorismus-Schrift des XII Jahrhunderts und über die Verbreitung der indisch-arabischen Rechenkunst und Zahlzeichen im christl. Abendlande,” Zeitschr. f. Math. u. Phys., Hist.-Liter. Abt., 34 (1889), pp. 129–46 and 161–70.
41 Curtze, M., “Ueber eine Algorismus-Schrift des XII Jahrhunderts,” Abhandl. z. Gesch. d. Math., 8 (1898), pp. 3–27.
42 Tannery, P., “Sur l'auteur d'un texte algorithmique du douzième siècle publié par Curtze,” Bibliotheca Mathematica, 3. Folge, 5 (1904), repr. Mémoires Scientifiques, 5, pp. 343–45.
43 Haskins, C.H., Studies in the History of Mediaeval Science (Cambridge, 1924), p. 24. On the same basis as Haskins, the attribution to Pedro Alfonso was later suggested by Vallicrosa, J.M. Millás y, “La aportación astronómica de Pedro Alfonso,” Sefarad, 3 (1943): 65–105, esp. p. 83, then positively recognized by Lemay, R., “The hispanic origin of our present numeral forms,” Viator, 8 (1977): 435–62, esp. p. 446, n. 6.
44 Allard, Le Calcul Indien, pp. 23–61.
45 Some additions are peculiar to only one of the two manuscripts; they are always independent of the Indian reckoning and of the general content of LY.
46 Although the fourth book of Liber Ysagogarum refers to music (“De musicis ac geometricis rationibus”), it virtually contains geometry only. The purpose of our temporary edition of 1975, like Curtze's edition, was to study the arithmetical part of the work only.Dickey, B.G., Adelard of Bath. An examination based on heretofore unexamined manuscripts. Ph.D. Dissert., Univ. of Toronto (1982) (unpublished), realized an edition, so far unpublished, of the whole of LY and De opere astrolapsus. But both the arithmetical part and the author's conclusions in his comments are so different from ours that we can't refer to them. The parts of our colleague's work that are independent of the Indian reckoning are a major contribution to the knowledge of mediaeval science and deserved to be edited. We refer to them on several occasions.
47 Dickey, Adelard of Bath, pp. 83–84, rightly points out the fact that Walcher of Malvern (d. 1135), Pedro Alfonso's disciple, frequently used in his De drecone Indian and Roman numerals without mentioning on that point his master's heritage, contrary to what he said about the sexagesimal fractions. It is difficult to imagine that Pedro Alfonso, if he knew Indian numerals, wouldn't have used them but would have passed them instead on to his disciple. Indian numerals are probably the product of the copyist of De dracone or it is likely that Walcher of Malvern knew them thanks to someone other than Pedro Alfonso.
48 The reproduced tables, except for some corrections, from Dickey, Adelard of Bath, p. 318, can easily be compared with the transcription of Neugebauer, O., The astronomical tables of al-Khwārizmī, Hist. Fibs. Skr. Dan. Vid. Selks., IV, 2 (1962), pp. 137 and 143–45.
49 Allard, Le Calcul Indien, pp. 27, 18–23; 37, 1–3; 38, 5–7.
50 But unlike the Latin text, it matters little that the Arab author has to consider negative numbers since he operates in the same way for the multiplication of 3 by 5. Cf. Juschkewitsch, , “Abū'I-Wafā' al-Būzjānī,” Dict. Sc. Biography (New York, 1970), vol. I, p. 41,and Tropfke, Geschichte der Elementarmathematik, p. 219. The method of LY also appears in an anonymous Latin algorism of Salem cloister that was certainly composed at the beginning of the 13th century, but the method is applicable here for the numbers between 5 and 10.Cf. Cantor, M., “Ueber ein Codex des Klosters Salem,” Zeitschr. f. Math. u. Phys., 10 (1985): 3–16, esp. p. 5. This method is also found in the oldest algorism known in French (13th c.) (ed. Waters, E.G., “A thirteenth century Algorism in French verse,” Isis, 11 (1928): 45–84).
51 It corresponds perhaps to the unclear formula in an addition of a manuscript of LA. It also appears in Helcep Sarracenicum, in the algorism of Salem and in John of Sacrobosco's Algorismus Vulgaris. Cf. Burnett, C., “Algorismi decentior est diligentia: the Arithmetic of Adelard of Bath and his Circle,” in Folkerts, M. (ed.) Mathematische Problem im Mittelalter. Der arabische und lateinische Sprachbereich (1991), p. 18;Cantor, “Ueber ein Codex des Klosters Salem,” p. 5;Curtze, M., Petri Philomeni de Dacia in Algorismum Vulgarem Johannis de Sacrobosco Commentarius una cum Algorismo ipso (Copenhague, 1897), p. 8.
52 Cf. Busard, H.L.L., The first Latin translation of Euclid's Elements commonly ascribed to Adelard of Bath, Pont. Inst. of Mediaeval Studies, Studies and Texts 64 (Toronto, 1983), p. 196.
53 Dickey, Adelard of Bath, p. 310.
54 We choose here an example that is quite convincing. Actually, Elements III, 20, 25, 36 (35 in Adelard's version) and VI,4 are almost perfectly alike. About these comparisons, see Folkerts, , “Adelard's versions of Euclid's Elements,” in Burnett, C. (ed.) Adelard of Bath, and English scientist and Analist of the early twelfth century, The Warburg Institute, Surveys and Texts 14 (London, 1987), pp. 55–68, esp. pp. 62–63.
55 Elements I, axiom 5.
56 Dickey, Adelard of Bath, p. 94.
57 Cf. Reuter, J.H.L., Petrus Alfonsi. An examination of his works, their scientific content and their background, Phil.D. Dissert., Oxford, St. Hilda's College (1975) (unpublished).
58 Successively elwazat, tadil, albuht, buht, elaug, emulkaam, obvious transcriptions of al-wasat, ta'dīl, al-buht, al-auj, al-muqawwam. The author of LY I also knew a Zīj which he attributed erroneously to Ptolemy.
59 Suter, H., Die astronomischen Tafeln des Muḥammed ibn Mūsā al-Khwārizmī in der Bearbeitung des Maslama al-Majrītī und der lateinischen Uebersetzung des Athelard von Bath, Danske Vid. Selks. Skr., 7 Raekke, Hist. og Filos. Afd., III, 1 (Copenhague, 1914), p. 9. The value given for the circumference of the Earth and the value π = 22/7 have the result 7636 testified by LY I.
60 Thus, the ecentricus or the epiciclus of LY I are only referred to by their function in Adelard. Cf. Dickey, Adelard of Bath, p. 159.
61 8° ahead of schedule of the Sun in comparison with the zodiac in 900 years and equal delay of the 900 years that follow.
62 Three circles are situated above Saturn in Pedro Alfonso against two in Adelard of Bath and LY I. This question was carefully studied by Dickey, Adelrad of Bath, pp. 99–101.
63 Allard, Le Calcul Indien, pp. 23–24.
64 For example, Physics, V, 1.
65 Burnett, C., Hermann of Carinthia. De Essentiis (Leyde, 1982), p. 248.
66 Allard, Le Calcul Indien, pp. 23–24.
67 Nagy, A., Die philosophischen Abhandlungen des Ja'qub ben Ishāq al-Kindī (Münster, 1897), pp. 35–37.
68 We thank our colleague Ahmed Hasnaoui, who kindly helped us to make the mentioned comparisons.
69 Boncompagni, Iohannis Hispalensis liber algorismi, pp. 25–93.
70 It seems that such is the true name of the author, as Lemay, R., “De la scolastique à l'histoire par le truchement de la philologie: itinéraire d'un médiéviste entre Europe et Islam,” in La diffusione delle scienze islamiche nel medio evo europeo, Acc. Naz. d. Lincei (Roma, 1987), pp. 400–535, esp. pp. 410–26, pointed out. But we are not convinced that lohannes Hispalensis is a parent, or even a son, of the count Sisnando Davidiz and that he must be identified with “Avendauth philosophus” mentioned in some Latin manuscripts of Avicenna.
71 Vallicrosa, J.M. Millás y, “Una obra astronomica desconocida de Johannes Avendaut Hispanus,” Osiris, 1 (1936): 451–75, esp. p. 460.
72 Several authors accepted these attributions, like Wappler, E., “Beitrag zur Geschichte der Mathematik,” Abhandl. z. Gesch. d. Math., 5 (1890), p. 158;Steinschneider, M., “Ueber die mathematischen Handschriften der Amplonianischen Sammlung zu Erfurt,” Bibliotheca Mathematica (1891), p. 47;Vogel, Mohammed ibn Musa Alchwarizmi's Algorismus, p. 43. Some authors, like Carmody, F.J., Arabic Astronomical and Astrological Sciences in Latin Translation. A Critical Bibliography (Berkeley, 1956), p. 47, mix up the text of LA with Gerard of Cremona's translation of al-Khwārizmī's Algebra.
73 This crucial observation is the result of research by d'Alverny, M.-T., “Translations and translators,” in Benson, R. and Constable, G. (eds.), Renaissance and Renewal in the Twelfth Century (Oxford, 1982), pp. 421–62, esp. pp. 458–59, on Gerard of Cremona's translations.
74 See, for instance, the preface of De regimine sanitatis. The Latin on the thirty or forty of John of Seville's translations is very bad and is modelled on Arabic style; it is not the fact for LA.
75 LA and LP are edited together in Allard, Le Calcul Indien, pp. 62–224.
76 Such was still, after Eneström, G., “Ueber den Bearbeiter oder Uebersetzer des von Boncompagni (1857) herausgegeben ‘Liber algorismi de pratica arismetrice’,” Bibliotheca Mathematica, 6, 3 (1905), p. 114, our position at the time of our temporary edition of 1975.
77 This continuation of LA has the title “De multiplicatione digitorum inter se.” Cf. Boncompagni, Iohannis Hispalensis liber algorismi, p. 97.
78 Ibid., pp. 99–100.
79 Cf. Martinori, E., La moneta. Vocabulario generale (Roma, 1915), p. 266.
81 About numerals in Latin manuscripts, see Allard, “L'époque d'Adélard de Bath.”
82 This hypothesis comes from an erroneous identification of an assumed “Iohannes Avendauth” with Iohannes Hispalensis. The doubts on this question expressed by Haskins were totally confirmed by d'Alverny, “Translations and translators,” pp. 444–45.
83 It should be reminded again that, except for a mere statement of intention in one of John of Seville's works, none of the mentioned authors have shown any interest in the Indian reckoning or Hebraic culture; Pedro Alfonso's Dialogi cum Judaeo are even a deliberate refutation of Judaism.
84 Cf. d'Alverny, “Translations and translators,” pp. 439–58.
85 Cf. d'Alverny, , “Avendauth?,” Homenaje a Millás-Vallicrosa (Barcelona, 1954), vol. I, pp. 19–43.
86 Twelve lunar months in common year and thirteen lunar months in embolismic years. Cf. Tannery, P., “Sur la division du temps en instants au Moyen Age,” Bibliotheca Mathematica, 3. Folge, 6 (1905), repr. Mémoires Scientifiques, 5, pp. 346–47. But we don't think that we should see in these elements a Latin survival of al-Khwārizmī's work about the Jewish calendar.Cf. Kennedy, E.S., “Al-Khwārizmī on the Jewish calendar,” Scripta Mathematica, 27 (1964), pp. 55–59.
87 The manuscript is mentioned by d'Alverny, “Avendauth?,” p. 40, and by Sánchez-Albornoz, C., “Observaciones a unas paginas de Lemay sobre los traductores Toledanos,” Cuadernos de Hist. de España, 41–42 (1965): 313–324, esp. p. 323, n. 49.
88 This identification is largely detailed and argued by Lemay, R., “Dans l'Espagne du XIIe siècle: les traductions de l'arabe au latin,” Annales: Economies, Sociétés, Civilisations, 18 (1963): 639–65, esp. pp. 647–54. It is refuted by Sánchez-Albornoz, “Observaciones.”
89 Lemay, “De la scolastique à l'histoire,” p. 418, maintains positively that John of Seville, John David and Avendauth are one and only one person. So far, it is the argumentation of d'Alverny, “Avendauth?,” pp. 19–43, that has convinced us better. Lemay, “The hispanic origin of our present numeral forms,” p. 446, n. 6, even says that the author of LY I was certainly Pedro Alfonso: we have shown that this attribution must be rejected.
90 Cf. Baur, L., Dominicus Gundissalinus. De diuisione philosophiae, Beiträge z. Gesch. d. Philos. d. Mittelalters, IV, 2–3 (1903), p. 91.
91 “Sed destructio rei non est aliud quam separatio formae a materia” (P.L. 63, col. 1075).
92 De immortalitate animae often uses Avendauth's De anima and De processione mundi Hermann of Carinthia's De essentiis; De ortu scientiarum uses al-Farabi's (and Hugues of St-Victor's) works and De unitate et uno Ibn Gabirol's.
93 One won't forget that DA reveals an influence of traditional Latin arithmetic coming from Boethius.
94 Allard, Le Calcul Indien, p. 23.
95 For example, an account on the names of the one week's days linked to the planets. Cf. Allard, Le Calcul Indien, pp. 30, I, 13–31, I, 6.
96 The meaning of the abbreviations is given pp. 236 and 243. It should be noted that many identical passages in LA and LP lead to gather these two texts together in a special way. The chapter about common fractions being interrupted ex abrupto in DA at the beginning of the multiplication of by , we have put the numbers that have to do with this text between brackets.
97 Besides the way of arranging, in the absence of tierces, 12°30' 45″ 50IV. We haven't posted this example common to all the texts, considering that it didn't reflect, strictly speaking, an operation of Indian reckoning.
98 We have here the only coincidence between LY III and LA. One finds accessorily once more a proof that LP cannot be a reshaping of LA.
99 The most part of Latin translations from Arabic in the 12th century relate to the astronomy. The John of Seville's translations are on this point exemplary.
100 This insertion is the fact of LY I and of DA. It is expressed once more in the third person plural of the Latin verbs. Cf. Allard, Le Calcul Indien, p.42, 18–19.
101 The proposed diagram shows some relations but does not aim to establish a chronological hierarchy. By studying the manuscripts themselves, we can observe privileged relations between LY I and LY III.
102 We have seen earlier that Boethius' influence is felt at various stages in all the treatises and that even DA is not an exception to the rule.
103 One finds a detailed study about these numerals in Allard, “L'époque d'Adélard de Bath.”
104 About these uses before the 12th century, see Beaujouan, , “Etude paléographique sur la ‘rotation’ des chiffres.”
105 This distinction has been made traditional since old studies, especially Smith, D.E. and Karpinski, L.C., The Hindu-Arabic Numerals (Boston and London, 1911), and Gandz, S., “The origin of the Gubar numerals,” Isis, 16 (1931): 393–424.
106 Cf. Pidal, G. Menéndez, “Los llamados numerales árabes en Occidente,” Boletin d.l. Real Ac. de la Historia, 145 (1959), p. 188. A recent publication about numerals in Arabic documents of Spain speaks of the ghubār numerals, similar to those of the Latin manuscripts of the 12th and 13th centuries, only in late documents of the 15th and 16th centuries, in the provinces of Aragon and of Valence. It is certain, however, that hindī numerals were known in the 12th century, at least thanks to the translations of works which were inspired by al-Khwārizmī's arithmetic.Cf. Labarta, A. and Barceló, C., Numeros y cifras en los documentos arábigohispanos (Cordoba, 1988).
107 Addere, duplatio, impar, multiplicatio…
108 Cf. Allard, La formation du vocabulaire latin de l'arithmétique médiévale.
109 Besides, in the Jewish authors mentioned, the division comes immediately after the multiplication.
110 There are texts of the “A type” according to the designation of the author of the edition of al-Uqlīdisīs Arithmetic. Cf. Saidan, A.S., The Arithmetic of al-Uqlīdisī (Dordrecht and Boston, 1978), pp. 19–21. It should be noted, however, that the author's considerations about Latin algorisms, pp. 22–23, are totally erroneous. He believes, for instance, that DA and Boncompagni's Algoritmi de numero Indorum are two different texts.
111 The authors are of “H/HA types” in A.S. Saidan, The Arithmetic of al-Uqlīdisī, pp. 21–29.
112 Cf. Allard, Le Calcul Indien, pp. 58–59 and 206–17.
113 Cf. Rashed, Entre arithmétique et algèbre, p. 121.
114 Allard, Le Calcul Indien, pp. 218, 1–219, 15.
115 Ibid., pp. 219–22.
116 We apply the indications given by the texts that, in this part, do not give any example.
117 About this question of the Arab author's invention of decimal fractions, see Rashed, Entre arithmétique et algèbre, pp. 120–32.
118 Cf. J. Tropfke, Geschichte der Elementarmathematik, pp. 248–63 and Juschkewitsch, A.P., Geschichte der Mathematik in Mittelalter (Leipzig, 1964), pp. 362–65.
119 This fact convinces us once more that the author of LA can't be the translator John of Seville whose many known works never reach this level of knowledge.
120 For example, Vogel, Mohammed ibn Musa Alchwarizmi's Algorismus, p. 48. Cf. Allard, Le Calcul Indien, pp. 46–49.
121 Miss G. L'Huillier nevertheless pointed out the important part played by LA in the elaboration of John of Murs' Quadripartitum numerorum (via the manuscript 15461 of Paris, that the author once handled). We kindly thank her for her help.
122 Burnett, “Algorismi decentior est diligentia.”
123 Allard, Le Calcul Indien, p. 27, 22–23; Burnett, “Algorismi decentior est diligentia,” p. 18, § 43–44.
124 Allard, Le Calcul Indien, p. 94, app. crit. and p. XVI.
125 Ibid., pp. 67, 27–68, 15.
126 Burnett, “Algorismi decentior est diligentia,” p. 17, § 5–8.
127 Ibid., p. 10 and n. 78.
128 Crossley, J.N. and Henry, A.S., “Thus Spake al-Khwārizmī: A Translation of the text of Cambridge University Library MS Ii.VI.5.,” Historia Mathematica, 17 (1990):103–31, have recently published an English translation of Dixit Algorizmi… (DA). Proposing some corrections that are not pertinent to Vogel, Mohammed ibn Musa Alchwarizmi's Algorismus, the authors attempted to translate the text literally, giving an irrelevant value to the punctuation of the Cambridge manuscript and without taking into account the features of mediaeval Latin. The introduction of the translation takes up again the genesis of Latin algorisms but without examining them critically: many of them are nowadays considered as completely unfounded. This translation and its comments require to be used with caution.
129 On the whole, this method matches the one christened by Benedict, S.R., Comparative study of early treatises introducing into Europe the Hindu art of reckoning, Ph.D. Dissert., Univ. Michigan (1924). But the numerous mistakes contained in this work make its use quite unreliable.
130 Allard, Le Calcul Indien, pp. 8, 3–6 and 9, 1.
131 Ibid., p. 11, 20–27.
132 Ed. Allard, A., “A propos d'un algorisme latin de Frankenthal: une méthode de recherche,” Janus, 65 (1978): 119–41.
133 Ed. Karpinski, L.C., “Two Twelth Century Algorisms,” Isis, 3 (1921): 396–413.
135 Ed. Halliwell, J.O., Rara Mathematica (London, 1841), pp. 73–83.
136 Ibid., pp. 1–26; Curtze, Petri Philomeni de Dacia, pp. 1–19.
137 Ed. Boncompagni, Il Liber Abbaci, pp. 22–23.
138 Levey, M. and Petruck, M., Kūshyār ibn Labbān. Principles of Hindu Reckoning (Madison, 1966), p. 12.
139 Saidan, The Arithmetic of al-Uqlīdisī, p. 375.
* I am grateful to F. Decolle and J.-P. Nyssen for all their help in improving the English of this article.
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