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From Abacus to Algorism: Theory and Practice in Medieval Arithmetic

Published online by Cambridge University Press:  05 January 2009

Extract

Even at the level of the most elementary arithmetical operations, procedures and practices change. During the twelfth and thirteenth centuries an unusually well documented development took place: at the beginning of the period the authors of elementary manuals of computation taught the use of the abacus, whereas at the end they described the method of calculation which came to be known as the algorism. Their ideas about number, however, were still largely drawn from Boethius's rendering of Nicomachus of Gerasa's Introduction to arithmetic in the Arithmetica and there had been little progress in attempting to reconcile Boethius's teaching on the theory of number with the rather different assumptions that underlie the methods of practical calculation. Boethius and Nicomachus, for example, emphasize that one is not a number but the source of number, and they are aware of the special problems posed by ‘two’. Nicomachus questions whether ‘two’ is anything more than an embodiment of the principle of ‘otherness’; for him, it is open to dispute whether it can be rated a number in its own right. For the teacher of the skills of calculation, ‘one’ and ‘two’ are merely digits like any other. By the fourteenth century, collections of textbook material on elementary arithmetic provided the student with instruction in both theory and practice. The abacus manuals are missing from many such collections because by then the abacus has apparently been relegated to the status of a simple practical aid, but the other elements in medieval arithmetical studies are variously covered.

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References

NOTES

1 Nicomachus, of Gerasa, Introduction to arithmetic, trans. d'Ooge, M. L. (London, 1926).Google Scholar

2 Boethius, , De institutione arithmetica, ed. Friedlein, G. (Leipzig, 1867).Google Scholar

3 On this problem and its classical sources, see d'Ooge, , op. cit. (1), p. 100.Google Scholar

4 Benedict, S. R., A comparative study of the early treatises introducing into Europe the Hindu art of reckoning (Concord, New Hampshire, 1914), p. 119Google Scholar. The author confines herself strictly to algorisms in this useful study of works which were available in print in the early years of this century.

5 A possible exception is the work mentioned in Benedict, , op. cit. (4), p. 13Google Scholar. This adds a series of mystical interpretations to its account of the fundamental operations with integers.

6 Some of the earliest treatises on the abacus are published in Bubnov, N. (ed.), Gerberti opera mathematica (Berlin, 1899)Google Scholar, including those of Gerbert of Aurillac, Abbo of Fleury, and Heriger of Louvain. Those treatises which were written from the mid-eleventh century onwards build upon this early work, which had been in circulation for-nearly a century. The later treatises include that of Garlandus Compotista, edited by Boncompagni, B. in ‘Scritti inediti relativi al calcolo dell'abaco’, Bullettino di bibliografia e di storia delle scienze matematiche e fisiche, x (1877), 595656Google Scholar; Adelard of Bath's Regule abaci, also edited by Boncompagni, ibid., xiv (1881), 1–134; and those of Turchillus and others, edited by Narducci, E. in ‘Intorno a due trattati inediti d'abaco’Google Scholar, ibid., xv (1882), 111–63; ‘Ralph of Laon: Liber de abaco’, ed. Nagl, A., in Abhandlungen zur Geschichte der Mathematik, v (1890), 96133.Google Scholar

7 Not all medieval writers accepted this etymology. The author of the preface to Sacrobosco's Algorismus in Bodleian Library MS. Laud. Misc. 644, f. 127v, gives several alternatives.

8 This was written in 1202, revised in 1228. See Benedict, , op. cit. (4), pp. 119–20.Google Scholar

9 Leonard of Pisa: Liber abaci, ed. Boncompagni, B. (Rome, 1857), p. 1Google Scholar. ‘But I have cleared up the mistakes in the algorism and the Pythagorean column by the Hindu method’.

10 Benedict, , op. cit. (4), p. 126.Google Scholar

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12 de Sacrobosco, Johannes, Algorismus vulgaris, ed. Curze, M. (Copenhagen, 1897)Google Scholar. Petrus de Dacia's commentary on the work is included in the edition.

13 A list of the abacus treatises available in print has already been given. Benedict gives details of the editions of algorism treatises known to her. Note, in particular, those edited by Boncompagni, B. in his Trattati di arithmetica (2 vols., Rome, 1857)Google Scholar, and Halliwell, J. O. (ed.), Rara mathematica (London, 1839)Google Scholar, which includes Alexander of Villa Dei's Carmen de algorismo. To these might be added two algorisms edited by Karpinski, L. C. in his ‘Two twelfth century algorisms’, Isis, iii (19201921), 396413.Google Scholar

14 Turchillus, , op. cit. (6), p. 135Google Scholar. ‘All the multitude of number proceeds from unity and, in a sense, it comes to an end in ten’.

15 Ibid. ‘They have bound number, which is infinite, by some kind of limit as if it were finite’.

16 Boethius, , Arithmetica, op. cit. (2), p. 12, lines 14–15.Google Scholar

17 MS. Laud. Misc. 644, f. 127v. ‘A number is a collection of unities proceeding from unities’.

18 It may be worth re-emphasizing here that the question whether unity is itself a number, which preoccupies arithmetical theorists of the day, is irrelevant to the concerns of practical computation.

19 MS. Laud. Misc. 644, f.i27v.

20 Cf. Boethius's commentary on the Categories, in Migne, J. P. (ed.), Patrologia latina, lxiv (Paris, 1891)Google Scholar, cols. 200–2, and Petrus Abaelardus Dialectica, ed. deRijk, L. M. (Assen, 1956), p. 56.Google Scholar

21 Sacrobosco, , op. cit. (12), p. 6.Google Scholar

22 Bubnov, , op. cit. (6), pp. 257–61, 275–9Google Scholar, and Ralph, of Laon, , op. cit. (6), p. 107.Google Scholar

23 See Benedict, , op. cit. (4), p. 32Google Scholar. On ciphers, see, too, Mendelsohn, C. J., ‘A bibliographical note on the De cifris of Leone Battista Alberti’, Isis, lxxxv (1940), 4851.CrossRefGoogle Scholar

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25 Isidore, , Etymologiae, ed. Lindsay, W. M. (2 vols., Oxford, 1911), iiGoogle Scholar. Book xviii, Chapter 9, 3.

26 d'Alverny, M. T., M élanges offerts à Andr é Piganiol (Paris, 1966), p. 1526.Google Scholar

27 Compare the usages of the three manuscripts considered in Karpinski, , op. cit. (13), p. 403.Google Scholar

28 Differentia is, however, found in some of the earliest abacus treatises. See Bubnov, , op. cit. (6), pp. 13, 14, 15, 17.Google Scholar

29 The treatise is edited by Boncompagni, , in Bullettino, x (1877), op. cit. (6), 608Google Scholar. ‘But a digit is a smaller number, that is, one, two, up to nine. An articulas is a larger number, that is, ten, twenty, thirty, and it is so-called because it “joins”, for it joins and unites the smaller number with itself’.

30 Ibid., p. 626.

31 Turchillus, , op. cit. (6), p. 138.Google Scholar

32 Sacrobosco, , op. cit. (12), p. 1.Google Scholar

33 Benedict, , op. cit. (4), p. 15.Google Scholar

34 Sacrobosco, , op. cit. (12), p. 8.Google Scholar

35 Garlandus, , op. cit. (6), p. 595.Google Scholar

36 Turchillus, , op. cit. (6), p. 137Google Scholar. ‘This science consists in multiplying and dividing. For we must first learn to multiply, that is, to bring together or make one sum from different or similar sums, and then we must learn to divide, that is, to divide one sum equally among many’.

37 Sacrobosco, , op. cit. (12), p. 8Google Scholar. ‘Multiplication is the discovery, for two given numbers, of a third which contains one of the others as many times as there are units in the remaining number’.

38 Ibid., p. 11. ‘Division is the distribution of the greater of two numbers by the lesser into as many parts as there are units in the lesser’.

39 Benedict, , op. cit. (4)Google Scholar, gives a number of variations passim.

40 Sacrobosco, , op. cit. (12), p. 4.Google Scholar

41 Ganz, S., ‘The invention of the decimal fractions and the application of the exponential calculus by Immanuel Bonfils of Tarascon c. 1350’, Isis, lxix (1936), 1645.Google Scholar

42 MS. Laud. Misc. 644, f.121r, col. 2.

43 ‘The Algorismus Demonstratus of Jordanus Nemorarius’, edited by H. L. L. Busard, in Centaurus, xv (1970), 193227.Google Scholar

44 Weisheipl, J. A., ‘The curriculum of the faculty of arts at Oxford in the early fourteenth century’, Medieval studies, xxvi (1964), 161.Google Scholar

45 Ibid., p. 170.

46 Ker, N. R., Mediaeval manuscripts in British libraries (London, 1969), pp. 194–5Google Scholar, gives a description and further references.

47 Thorndike, and Kibre, , op. cit. (11), col. 1448.Google Scholar

48 Ibid., col. 1016.

49 Ibid., col. 1238.

50 Ibid., col. 1272.

51 Sacrobosco, , op. cit. (12)Google Scholar. Curze used only one manuscript, from Munich, Codex Latinus Monacensis 11067. Thorndike and Kibre give thirteen manuscripts known to them, but it is very probable that a search will yield more, if Benedict is right in believing that this was a very popular and influential textbook; see Benedict, , op. cit. (4), p. 126Google Scholar. The altered incipit and explicit have helped the treatise to avoid identification in more recent times. Whatever explanation is to be advanced for the altered preface we have here, to all intents and purposes, a hitherto unnoted copy of Sacrobosco's Algorismus vulgaris. Here is the preface:

MS. Laud Misc. 644, f. 127v. Quoniam dicetur a boetio in principio arismetice omnia quaecumque a primeva rerum origine constituta sunt ratione numerorum videntur esse formata.a Et sicut sint sic cognosci habent. Unde in universa rerum cognitione est numerandi ars necessaria. Cum igitur hec scientia de qua ad presens intendimus circa numerum consistat, primo videndum est quid sit proprium nomen ipsius et unde dicatur. Deinde quid sit numerus et quot species numeri. Est autem nomen huius artis argorismus, et dicetur ab ‘argos’ quod est ‘ars’ et rismus quod est numerus, inde argorismus est ars numerandi. Vel dicetur ab ‘on’ quod est ‘in’ et ‘gogos’ ‘ductio’ et ‘rismus’ numerus [et] est introductio in numero. Tercio modo dicetur ab algo inventore. Secundum quosdam vocatur argorismus a quodam graeco quod est alba arena, quare haec ars solet doceri in tali arena. Alio modo dicetur ab algis quare greci primo talem artem invenerunt. Numerus dupliciter notificatur, materialiter, ut numerus est unitas collective, formaliter, ut numerus est unitatum collectio ex unitatibus profusa.b De unitate vero dicit Boetius potentialiter est omnis numerus.c Unitas enim est qua unaquaque res dicitur una.

Sacrobosco, , op. cit. (4), p. 1Google Scholar. Omnia quae a primeva rerum origine processerunt, ratione numerorum formata sunt,a et quemadmodum sunt, sic cognosci habent: unde in universa rerum cognitione est ars numerandi cooperativa. Hanc igitur scientiam numerandi compendiosam quidam philosophus edidit nomine Aigus, unde et Algorismus nuncupatur, [quae] vel ars numerandi, vel ars introductoria in numerum interpretatur. Numerus quidem dupliciter notificatur, materialiter et formaliter. Materialiter enim numerus est unitates collectae, formaliter est multitudo ex unitatibus profusa.b Unitas vero est, qua unaquaque res dicitur una.

a. Cf. Boethius, , Arithmetica, op. cit. (2), p. 12Google Scholar, lines 14–15.

b. Ibid., p. 13, lines 11–12.

c. Ibid., p. 16, lines 23–5.

52 f. 131r, ‘In dandis accipiendisque’ (Boethius, , op. cit. [2], p. 3Google Scholar, line 2) to ‘propriis laboris’ (ibid., p. 4, line 12) f. 131v, ‘orbi infigendo’ to ‘hoc studio’ (ibid., p. 4, line 20). The scribe now moves in mid-sentence to ‘nobilitas quodam’ (ibid., p. 7, line 25). In column 2, after ‘per se est’ (ibid., p. 9, line 1), he returns to take up up the prologue where he left off (ibid., p. 4, line 20) with ‘longis tractus otiis labor adiecit’. On f.132r, at ‘quam probater’ (ibid., p. 5, line 24), the text jumps again, to ‘inter omnes priscae autoritatis’ (ibid., p. 7, line 1), runs to ‘prudentiae’ (ibid., p. 7, line 24), moves to ‘speculator integritas’ (ibid., p. 9, line 2); from this point the text runs on correctly to the end of chapter three of the Arithmetica.

53 Thorndike, and Kibre, , op. cit. (11), col. 1567.Google Scholar

54 Summary of and commentary on the Arithmetica appear several times in Thorndike and Kibre's columns; see op. cit. (11), cols. 767, 153, 783, 641, 1175.

55 MS. Laud. Misc. 644, f. 133r.

56 Ibid. ‘Arithmetic is the science of absolute multitude which considers the power of numbers… It differs from geometry in what I have called “multitude”, and also from astronomy, for these deal not with multitude but with magnitude. In that it is called “absolute” it differs from music, which although it deals with multitude, is not concerned with absolute multitude but with multitude in relation to sounds’.

57 Boedthius, , op. cit. (2), pp. 89.Google Scholar

58 f. 133r. ‘The difference between this study and that of the algorism is that in the latter, number is considered according to its uses.’

60 Ibid., ff. 134r–135v. Cf. Boethius, , op. cit. (2), p. 13Google Scholar, line 13 to p. 15, line 24.

61 Cf. The commentaries on Boethius of Gilbert of Poitiers, ed. Häring, N. M. (Toronto, 1966)Google Scholar, and The commentaries on Boethius of Thierry of Chartres and his school, ed. Häring, N. M. (Toronto, 1971).Google Scholar

62 Southern, R. W., ‘Lanfranc of Bec and Berangar of Tours’, Studies presented to F. M. Powicke (Oxford, 1948), pp. 41–2.Google Scholar

63 Pseudo-Apuleius, , Apulei de philosophia libri, ed. Thomas, P. (Leipzig, 1921), pp. 176–94Google Scholar, for the text of the Periermenias, and pp. 181, 187 for references to equipollent propositions.

64 Abbo of Fleury, Syllogismorum categoricorum et hypotheticorum enodatio, ed. van de Vyver, A. (Bruges, 1966), p. 85.Google Scholar

65 Ibid., p. 41, line 30 to p. 42, line 1. ‘Then we must know that some propositions are called equipollent, which some call equimodi, because they signify the same thing in different ways, as in: “No man is a stone” and: “Every man is not a stone”’.

66 Ibid, following.

67 Lanfranc, , De corpore et sanguine DominiGoogle Scholar, in Migne, , op. cit. (20), cl. col. 417.Google Scholar

68 Ibid. ‘I hide my art by using equipollent propositions’.

69 Compotista, Garlandus, Dialectica, ed. de Rijk, L. M. (Assen, 1959), p. 131Google Scholar, lines 32–3. ‘Some categorical propositions are equipollent to certain hypothetical propositions, both “connected” and “disjoined”.’

70 Abelard, , op. cit. (20), pp. 191, 199, 228, 305.Google Scholar

71 Garlandus, , op. cit. (6), p. 606Google Scholar. ‘Take an emiscla and ten more, and divide them into two fractions which are equivalent to them, that is a semiuncia and a duella. Do the same with the other two fractions, to one of which a tremissis and a siliqua are equipollent, and to the other of which, the uncia, a sextans and a bisse are equipollent. If you ask why they are rendered into their equivalents…’

72 On Turchillus, see Haskins, C., Studies in the history of mediaeval science (London, 1927), pp. 327–35.Google Scholar

73 Turchillus, , op. cit. (6), p. 138Google Scholar. ‘And if in any of the columns the total of the similar or dissimilar characters exceeds that of the column, cast them out and take their equivalent and place it in the next column along, according to its proper place’.

74 ‘Every expression containing denominations in the nominative and the genitive is the equivalent of any other where the nominative is changed to the genitive and die genitive to the nominative, as in: “a third of a fourth and a fourth of a third”.’